https://minewiki.engineering.queensu.ca/mediawiki/api.php?action=feedcontributions&user=CPeebles&feedformat=atomQueensMineDesignWiki - User contributions [en]2023-10-04T06:49:03ZUser contributionsMediaWiki 1.27.4https://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5463Stability Graph Method2015-02-06T20:10:29Z<p>CPeebles: /* Calculation of Max Allowable Span */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
*<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
*<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
*<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
*A : is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
*B : is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
*C : is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
*<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
*<math> N_w = cos(T_w)*cos(P_w) </math><br />
*<math> E_w = sin(T_w)*cos(P_w) </math><br />
*<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
*<math> N_j = cos(T_j)*cos(P_j) </math><br />
*<math> E_j = sin(T_j)*cos(P_j) </math><br />
*<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w \cdot j</math>, between the stope face and the joint set<br />
<br />
*<math>w \cdot j = N_w \cdot N_j + E_w \cdot E_j + D_w \cdot D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
*<math>\alpha = cos^-1(w \cdot j)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
*<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
*<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
*<math>S = (w*h) / (2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5174Stability Graph Method2015-02-06T01:07:33Z<p>CPeebles: /* Case Example */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
*<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
*<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
*<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
*A : is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
*B : is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
*C : is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
*<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
*<math> N_w = cos(T_w)*cos(P_w) </math><br />
*<math> E_w = sin(T_w)*cos(P_w) </math><br />
*<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
*<math> N_j = cos(T_j)*cos(P_j) </math><br />
*<math> E_j = sin(T_j)*cos(P_j) </math><br />
*<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w*j</math>, between the stope face and the joint set<br />
<br />
*<math>w*j = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
*<math>\alpha = cos^-1(w*j)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
*<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
*<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
*<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5173Stability Graph Method2015-02-06T01:07:14Z<p>CPeebles: /* Case Example */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
*<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
*<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
*<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
*A : is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
*B : is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
*C : is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
*<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
*<math> N_w = cos(T_w)*cos(P_w) </math><br />
*<math> E_w = sin(T_w)*cos(P_w) </math><br />
*<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
*<math> N_j = cos(T_j)*cos(P_j) </math><br />
*<math> E_j = sin(T_j)*cos(P_j) </math><br />
*<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w*j</math>, between the stope face and the joint set<br />
<br />
*<math>w*j = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
*<math>\alpha = cos^-1(w*j)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
*<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5172Stability Graph Method2015-02-06T01:06:52Z<p>CPeebles: /* Joint Orientation Factor, B */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
*<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
*<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
*<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
*A : is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
*B : is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
*C : is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
*<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
*<math> N_w = cos(T_w)*cos(P_w) </math><br />
*<math> E_w = sin(T_w)*cos(P_w) </math><br />
*<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
*<math> N_j = cos(T_j)*cos(P_j) </math><br />
*<math> E_j = sin(T_j)*cos(P_j) </math><br />
*<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w*j</math>, between the stope face and the joint set<br />
<br />
*<math>w*j = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
*<math>\alpha = cos^-1(w*j)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5171Stability Graph Method2015-02-06T01:06:28Z<p>CPeebles: /* Joint Orientation Factor, B */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
*<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
*<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
*<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
*A : is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
*B : is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
*C : is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
*<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
*<math> N_w = cos(T_w)*cos(P_w) </math><br />
*<math> E_w = sin(T_w)*cos(P_w) </math><br />
*<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
*<math> N_j = cos(T_j)*cos(P_j) </math><br />
*<math> E_j = sin(T_j)*cos(P_j) </math><br />
*<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
*<math>w*j = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
*<math>\alpha = cos^-1(w*j)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5170Stability Graph Method2015-02-06T01:05:54Z<p>CPeebles: /* Joint Orientation Factor, B */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
*<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
*<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
*<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
*A : is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
*B : is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
*C : is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
*<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
*<math> N_w = cos(T_w)*cos(P_w) </math><br />
*<math> E_w = sin(T_w)*cos(P_w) </math><br />
*<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
*<math> N_j = cos(T_j)*cos(P_j) </math><br />
*<math> E_j = sin(T_j)*cos(P_j) </math><br />
*<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
*<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
*<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5169Stability Graph Method2015-02-06T01:05:29Z<p>CPeebles: /* Shape factor, S */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
*<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
*<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
*<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
*A : is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
*B : is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
*C : is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
*<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5168Stability Graph Method2015-02-06T01:05:18Z<p>CPeebles: /* Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
*<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
*<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
*<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
*A : is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
*B : is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
*C : is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5166Stability Graph Method2015-02-06T01:04:33Z<p>CPeebles: /* Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
A : is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B : is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C : is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5165Stability Graph Method2015-02-06T01:04:14Z<p>CPeebles: /* Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
<math>RQD/Jn</math> : is the measure of block size for a jointed rock mass<br />
<br />
<math>Jr/Ja</math> : is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5164Stability Graph Method2015-02-06T01:03:58Z<p>CPeebles: /* Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
<math>Q' = (RQD/Jn) x (Jr/Ja)</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
<math>RQD/Jn</math>: is the measure of block size for a jointed rock mass<br />
<br />
<math>Jr/Ja</math>: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5163Stability Graph Method2015-02-06T01:03:34Z<p>CPeebles: /* Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
where:<br />
<br />
<math>Q' = RQD/Jn x Jr/Ja</math>, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
<math>RQD/Jn</math>: is the measure of block size for a jointed rock mass<br />
<br />
<math>Jr/Ja</math>: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
<math>HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)</math><br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5162Stability Graph Method2015-02-06T01:02:34Z<p>CPeebles: /* Cablebolt-Supported Stopes */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5161Stability Graph Method2015-02-06T01:01:42Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5160Stability Graph Method2015-02-06T01:01:26Z<p>CPeebles: /* Rock Stress Factor A */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5159Stability Graph Method2015-02-06T01:01:06Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2 <br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5158Stability Graph Method2015-02-06T01:00:57Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2 <br />
[[File:phase_example.PNG]]<br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5157Stability Graph Method2015-02-06T01:00:25Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2 Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5156Stability Graph Method2015-02-06T01:00:08Z<p>CPeebles: /* Rock Stress Factor A */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2 Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5155Stability Graph Method2015-02-06T00:59:09Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]]<br />
[[File:phase_example.PNG]]<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 2: Determining max induced compressive stress using Phase 2<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5153Stability Graph Method2015-02-06T00:55:03Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]][[File:phase_example.PNG|thumb|right|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5152Stability Graph Method2015-02-06T00:54:47Z<p>CPeebles: /* Rock Stress Factor A */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|left|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]][[File:phase_example.PNG|thumb|right|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5150Stability Graph Method2015-02-06T00:53:47Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|right|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]][[File:phase_example.PNG|thumb|right|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5148Stability Graph Method2015-02-06T00:51:50Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|left|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]][[File:phase_example.PNG|thumb|left|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5147Stability Graph Method2015-02-06T00:51:29Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|left|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]][[File:phase_example.PNG|thumb|left|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5146Stability Graph Method2015-02-06T00:51:14Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|left|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]][[File:phase_example.PNG|thumb|left|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5145Stability Graph Method2015-02-06T00:50:54Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|left|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]] [[File:phase_example.PNG|thumb|left|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5144Stability Graph Method2015-02-06T00:49:31Z<p>CPeebles: /* Rock Stress Factor A */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|right|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]] <br />
<br />
[[File:phase_example.PNG|thumb|right|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5143Stability Graph Method2015-02-06T00:49:01Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|left|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]] <br />
<br />
[[File:phase_example.PNG|thumb|center|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5142Stability Graph Method2015-02-06T00:48:45Z<p>CPeebles: /* Calculation of Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|left|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]] <br />
<br />
[[File:phase_example.PNG|thumb|left|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5141Stability Graph Method2015-02-06T00:48:27Z<p>CPeebles: /* Rock Stress Factor A */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG|thumb|left|Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]] <br />
<br />
[[File:phase_example.PNG|thumb|right|Figure 2: Determining max induced compressive stress using Phase 2]]<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5138Stability Graph Method2015-02-06T00:46:52Z<p>CPeebles: /* Cablebolt-Supported Stopes */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|center|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5136Stability Graph Method2015-02-06T00:46:19Z<p>CPeebles: /* Cablebolt-Supported Stopes */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5134Stability Graph Method2015-02-06T00:45:46Z<p>CPeebles: /* Cablebolt-Supported Stopes */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
[[File:Supported_Stability_Graph.PNG|thumb|right|462x480px|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5133Stability Graph Method2015-02-06T00:45:17Z<p>CPeebles: </p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG|thumb|right|462x480px|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
<br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5132Stability Graph Method2015-02-06T00:44:22Z<p>CPeebles: /* Cablebolt-Supported Stopes */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG|thumb|right|462x480px|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5130Stability Graph Method2015-02-06T00:43:14Z<p>CPeebles: /* Cablebolt-Supported Stopes */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG|thumb|left|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5129Stability Graph Method2015-02-06T00:42:59Z<p>CPeebles: /* Cablebolt-Supported Stopes */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG|thumb|right|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5128Stability Graph Method2015-02-06T00:42:38Z<p>CPeebles: /* Cablebolt-Supported Stopes */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG|thumb|right|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=5127Stability Graph Method2015-02-06T00:42:10Z<p>CPeebles: /* Cablebolt-Supported Stopes */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG|thumb|Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=4926Stability Graph Method2015-02-05T17:25:42Z<p>CPeebles: /* References */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=4925Stability Graph Method2015-02-05T17:25:27Z<p>CPeebles: /* References */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
<br />
3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=4924Stability Graph Method2015-02-05T17:25:08Z<p>CPeebles: /* Input Parameters */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice. <ref name="McKinnon"> Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University. </ref><br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
<br />
2. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
<br />
3. Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University.<br />
<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=4922Stability Graph Method2015-02-05T17:23:29Z<p>CPeebles: </p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress effects twice.<br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
<br />
2. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
<br />
3. Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University.<br />
<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=4921Stability Graph Method2015-02-05T17:21:10Z<p>CPeebles: </p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress affects twice.<br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories of unsupported open stopes compiled by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories of open stopes supported by cablebolts compiled by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the joint sets into DIPS the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle, alpha, for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor is chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii are determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it can be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. When a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
<br />
2. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
<br />
3. Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University.<br />
<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=4906Stability Graph Method2015-02-05T17:06:25Z<p>CPeebles: </p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so Jw/SRF is removed to not account for the stress affects twice.<br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability.<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability.<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor, S, also known as the hydraulic radius (HR) is defined by the ratio of a stope's free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]].<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress, the measurement must be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor is used in calculating the stability number; this is done because the joint set with the lowest B factor will be the compromising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure caused by the presence of joint sets in the rock mass must to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
The hydraulic radius for a given free stope face can be determined using the stability graph, given a modifed stability number.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories complied by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories complied by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to a given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the Joint sets into dip the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle alpha for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor was chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii can be determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it could be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. For when a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
<br />
2. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
<br />
3. Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University.<br />
<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=4903Stability Graph Method2015-02-05T17:00:09Z<p>CPeebles: /* History */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were developed as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N, which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S, which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method generates dimensions for each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N', to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so to not account for the same input twice Jw/SRF is removed.<br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor S also known as the hydraulic radius (HR) is defined by the ratio of a stope free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]]<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress the measurement needs to be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor will be used in calculating the stability number. This is done because the joint set with the lowest B factor will be the comprising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure due to the present joint sets in the rock mass needs to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
With the hydraulic radius for a given free stope face can be determined using the stability graph, given a modifed stability number.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories complied by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories complied by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to a given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the Joint sets into dip the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle alpha for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor was chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii can be determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it could be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. For when a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
<br />
2. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
<br />
3. Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University.<br />
<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=4901Stability Graph Method2015-02-05T16:46:21Z<p>CPeebles: /* References */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were develop as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations. Mathews (1981) developed an empirical method to dimension stopes based on a stability number, N which defines the rock mass's ability to stand up to given ground conditions, and shape factor, S which is the stope face hydraulic radius that accounts for the geometry of the stope surface. The method dimensions each active stope face based on, N and S. The initial stability graph developed by Mathews is based on 50 case histories. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Potvin (1988) further expanded the original stability graph with an additional 175 case histories and introduced the modified stability number, N' to replace Mathews stability number. The modified stability number is similar to the N value developed by Mathews, but has different factor weightings. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
The database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions<br />
<br />
== Input Parameters ==<br />
<br />
===Modified Stability Number, N'===<br />
<br />
N' = Q' x A x B x C<br />
<br />
where:<br />
<br />
Q' = RQD/Jn x Jr/Ja, Q' is used instead of Q because the , A, factor accounts for the strength of the rock so to not account for the same input twice Jw/SRF is removed.<br />
<br />
RQD/Jn: is the measure of block size for a jointed rock mass<br />
<br />
Jr/Ja: is the measure of joint surface strength and stiffness<br />
<br />
A: is the measure of the ratio of intact rock strength to induced stress. As the maximum compressive stress acting parallel to a free stope face approaches the unaxial compressive strength (UCS) of the rock, factor A degrades to reflect the related instability due to rock yield.<br />
<br />
B: is the measure of the relative orientation of dominant jointing with respect to a free stope face. Joints forming shallow angle (10-30 degrees) with the free face are likely to become unstable, where joints perpendicular to the free face have little influence on stability<br />
<br />
C: is the measure of the influence of gravity on the stability of the face being considered. The back of the stope or structural weaknesses of a stope oriented unfavorably with respect to gravity sliding have a maximum impact on stability<br />
<br />
===Shape factor, S===<br />
<br />
The shape factor S also known as the hydraulic radius (HR) is defined by the ratio of a stope free face's area to perimeter, calculated as:<br />
<br />
HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)<br />
<br />
== Calculation of Input Parameters ==<br />
<br />
===Q'===<br />
<br />
The input parameters for Q' can be measured and calculated as described in [[Site investigation and rock mass characterization]]<br />
<br />
===Rock Stress Factor A===<br />
<br />
Factor A, can be determined using Figure 1. The UCS of the rock mass can be determined by a unaxial compressive strength test, and the max induced compressive stress can be determined using numerical software such as RocScience [[Numerical modelling|Phase 2]]. When the Stability graph method was developed the max induced compressive stress was consistently measured at the center of the free stope face being measured, so when measuring the max induced stress the measurement needs to be taken at the center of the free stope face being measured, as seen in Figure 2.<br />
<br />
[[File:Rock_Stress_Factor_A.PNG]] <br />
<br />
Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:phase_example.PNG]]<br />
<br />
Figure 2: Determining max induced compressive stress using Phase 2<br />
<br />
===Joint Orientation Factor, B===<br />
<br />
Factor B, is determined using Figure 3. The true angle between a joint set and a free face of a stope can be determined by hand or using software such as RocScience's [[Numerical modelling|DIPS]]. The interface seen in DIPS to measure the angle between two planes can be seen in Figure 4. The first point is the pole of the stope face being analysed, and the second point is the pole of the joint set. There is a potential for a rock mass to have multiple joint sets, so the B factor has to be calculated for each joint set, the joint set with the lowest B factor will be used in calculating the stability number. This is done because the joint set with the lowest B factor will be the comprising joint set, and be the most likely mode of failure in the stope.<br />
<br />
[[File:Joint_Orientation_Factor_B.PNG]]<br />
<br />
Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:DIPS_example.PNG]]<br />
<br />
Figure 4: Measuring the angle between two planes using DIPS<br />
<br />
''Direct Calculation of Interplane Angle''<br />
<br />
Given the trend and plunge of the pole vector of the stope face, ''w'', and a joint set,''j'', the interplane angle alpha can be determined with respect to the global coordinate grid (North, East, Down):<br />
<br />
For the stope face:<br />
<br />
<math> N_w = cos(T_w)*cos(P_w) </math><br />
<math> E_w = sin(T_w)*cos(P_w) </math><br />
<math> N_w = sin(P_w) </math><br />
<br />
For the joint plane:<br />
<br />
<math> N_j = cos(T_j)*cos(P_j) </math><br />
<math> E_j = sin(T_j)*cos(P_j) </math><br />
<math> N_j = sin(P_j) </math><br />
<br />
Next calculate the dot product, <math>w\cdotj</math>, between the stope face and the joint set<br />
<br />
<math>w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j</math><br />
<br />
The true interplane angle, <math>\alpha</math>, is given by:<br />
<br />
<math>\alpha = cos^-1(w\cdotj)</math><br />
<br />
===Gravity Adjustment Factor, C===<br />
<br />
Factor C, is determined using Figure 5. Before determining, C, the most likely mode of failure due to the present joint sets in the rock mass needs to be determined.<br />
<br />
[[File:Gravity_Adjustment_Factor_C.png]]<br />
<br />
Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Determination of S==<br />
<br />
With the hydraulic radius for a given free stope face can be determined using the stability graph, given a modifed stability number.<br />
<br />
===No-Support Stopes===<br />
<br />
Based on 176 case histories complied by Potvin and 13 by Nickson of unsupported open stopes, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 6. Stable stopes exhibited little or no deterioration during the service life of the stope. Unstable stopes exhibited limited wall failure and/or fallout involving less than 30% of the face area. Caved stopes suffered unacceptable failure. The transition zone seen in Figure 6 is a buffer zone between the stable and caved zones. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
Using N' calculated from the input parameters, the corresponding stope face hydraulic radius can be determined using Figure 6. The intercept of the N' value calculated with the upper boundary of the transition zone is used to find the corresponding hydraulic radius no-support limit. <br />
<br />
[[File:No_Support_Stability_Graph.PNG]]<br />
<br />
Figure 6: Database of unsupported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
===Cablebolt-Supported Stopes===<br />
<br />
Based on 66 case histories complied by Potvin and 46 by Nickson of open stopes supported by cablebolts, the modified stability number, N', and the shape factor, S, for each case are plotted on the Stability Graph shown in Figure 7. The three zones of the stability graph are shifted right, due to the limited effectiveness of cablebolts. The upper bound of the transition zone indicates the maximum stable hydraulic radius for cablebolted stopes, below this bound there is limited confidence of cablebolt effectiveness until it is assumed no useful support is provided by cablebolts. Below the lower bound of the transition zone caving is inevitable. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
[[File:Supported_Stability_Graph.PNG]]<br />
<br />
Figure 7: Database of cablebolt supported stopes <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
== Cable Support Guidelines ==<br />
<br />
Based on Potvin database, rough guidelines were developed for cablebolt design, Figure 8 and 9 are the design charts for cablebolt length and density representing the case histories of the database. The cablebolt density design chart, Figure 8, relates suggested cablebolt density and the empirical parameter, (RQD/Jn)/HR. This parameter measures the rock mass block size relative to the excavation size. The design chart also has three zones representing different degrees of conservatism, for example when designing a non-entry stope versus a main haulage drift two different degrees conservatism will be used. In practice it is common to use the , B, design line. Potvin also determined the point at which cablebolts become ineffective, as shown by the shaded grey areas. It was found when the block size (RQD/Jn) became significantly smaller relative to the excavation span (HR) cablebolts become ineffective, as well the minmum practical density that cablebolts should be deployed at his 0.6 bolts/m^2 which works out a 3x3m square pattern. The cablebolt length design chart, Figure 9, follows the logical rule of thumb in ground support of relating cablebolt length to excavation span (HR)<ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref>. The representative line of the database follows the trend of, length = 1.5 x HR, up to the practical max cablebolt length of 15m at a hydraulic radius of 10m.<br />
<br />
<br />
<br />
[[File:Cablebolt_density_design_chart.PNG]]<br />
[[File:Cablebolt_length_design_chart.PNG]]<br />
<br />
Figure 8: Design Chart for cablebolt density requirements <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref> Figure 9: Design chart for minimum cablebolt length required <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==Case Example==<br />
<br />
A stope dipping at 65 degrees, and 20m wide is located 800m below the surface. The mine levels are spaced every 60m. Determine the max span that the stope can be. <br />
<br />
The rockmass exhibits the following characteristics:<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Rockmass Properties<br />
|-<br />
|RQD<br />
|86.29<br />
|-<br />
|Jn<br />
|9<br />
|-<br />
|Jr<br />
|3<br />
|-<br />
|Ja<br />
|1<br />
|-<br />
|USC<br />
|75 MPa<br />
|-<br />
|E<br />
|24 GPa<br />
|-<br />
|Sigma h<br />
|22.7 MPa<br />
|-<br />
|Sigma v<br />
|31.9 MPa<br />
|}<br />
<br />
The Rockmass also has 3 Joint Sets with the following dip/dip directions<br />
<br />
{| class="wikitable"<br />
!colspan="2"|Joint #1<br />
!colspan="2"|Joint #2<br />
!colspan="2"|Joint #3<br />
|-<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|D<br />
|DD<br />
|-<br />
|43<br />
|294<br />
|56<br />
|163<br />
|33<br />
|105<br />
|}<br />
<br />
'''Q''''<br />
<br />
<math>Q' = (RQD/Jn)*(Jr/Ja) = (86.3/9)*(3/1) = 21.6</math><br />
<br />
'''Rock Stress Factor, A'''<br />
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Induced stress<br />
|-<br />
|Hanging wall<br />
|5.36 MPa<br />
|-<br />
|Back<br />
|52.9 MPa<br />
|}<br />
<br />
Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to a given A factor<br />
<br />
{| class="wikitable"<br />
!|Stope Face<br />
!|Ratio<br />
!|A Factor<br />
|-<br />
|Hanging wall<br />
|14<br />
|1<br />
|-<br />
|Back<br />
|1.4<br />
|0.1<br />
|}<br />
<br />
'''Joint Orientation Factor, B'''<br />
<br />
Inputting the Joint sets into dip the true angle between the joint sets and the hanging wall, and the back were determined. In the table below are the true angle alpha for each joint set and corresponding B factor used for determining the stability number, N'. The lowest B factor was chosen, due to the corresponding joint set is the most likely to cause failure and compromise the stope face. <br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Joint #1<br />
!|Joint #2<br />
!|Joint #3<br />
!|B Factor<br />
|-<br />
|Hanging wall<br />
|61<br />
|50<br />
|82<br />
|0.6<br />
|-<br />
|Back<br />
|43<br />
|56<br />
|33<br />
|0.275<br />
|}<br />
<br />
'''Gravity Adjustment Factor, C'''<br />
<br />
The gravity adjustment factor is based on the joint set that will most likely be the cause for failure. In the table below are the corresponding C factor for each stope face.<br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Mode of Failure<br />
!|C Factor<br />
|-<br />
|Hanging wall<br />
|Slabbing<br />
|5.46<br />
|-<br />
|Back<br />
|Slabbing<br />
|2<br />
|}<br />
<br />
'''N''''<br />
<br />
<math>N' = Q' x A x B x C</math><br />
<br />
{| class="wikitable"<br />
!|Stope face<br />
!|Q'<br />
!|A<br />
!|B<br />
!|C<br />
!|N'<br />
|-<br />
|Hanging wall<br />
|21.6<br />
|1.0<br />
|0.6<br />
|5.46<br />
|94.2<br />
|-<br />
|Back<br />
|21.6<br />
|0.1<br />
|0.275<br />
|2.0<br />
|1.19<br />
|}<br />
<br />
Based on the un-supported stability graph the following hydraulic radii can be determined<br />
<br />
{| class="wikitable"<br />
!|Hanging Wall<br />
|12.8<br />
|-<br />
!|Back<br />
|3<br />
|}<br />
<br />
===Calculation of Max Allowable Span===<br />
<br />
Based on the equation to determine the hydraulic radius, the max allowable span can be back calculated. As seen in the table below the limiting stope face is the back which usually the case when the horizontal in-situ stress exceeds the vertical in-situ stress, if the vertical in-situ stress was greater it could be expected that the hanging wall's max allowable span be the limiting dimension. <br />
<br />
<math>S = (w*h)/(2(h+w))</math><br />
<br />
{| class="wikitable"<br />
!colspan="2"|Max Allowable Span<br />
|-<br />
|Hanging Wall<br />
|42m<br />
|-<br />
|Back<br />
|8.6m<br />
|}<br />
<br />
==Summary==<br />
<br />
The stability graph method provides a rough estimation for designing stope excavations, and should only be used as a guideline in determining stope size and required ground support. In the early stages of a project the rockmass is never well defined, and so a range of input values should be used in determining, N', for a more robust stope design. There are limitations to the stability graph method, the estimation of the max stable hydraulic radius is based on the assumption the span is fully bounded. For when a stope is surrounded by backfilled stopes, the backfill must be tight to the walls and back or else the the surrounding is not considered to be supporting elements. In addition the database compiled to developed the stability graph is largely based on Canadian mines, making the design charts bias towards Canadian conditions. For the stability graph to better reflect the ground conditions in which design work is being done, an up-to-date local database should be kept to calibrate the stability graph, based on the local rockmass parameters, stope dimensions and stability status. <ref name="Diederichs">Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. </ref><br />
<br />
==References==<br />
<br />
<references /><br />
<br />
2. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.<br />
<br />
3. Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University.<br />
<br />
4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: <br />
[https://www.minewiki.org/index.php/The_Mathew%27s_method_for_open_stope_design]</div>CPeebleshttps://minewiki.engineering.queensu.ca/mediawiki/index.php?title=Stability_Graph_Method&diff=4900Stability Graph Method2015-02-05T16:46:10Z<p>CPeebles: /* References */</p>
<hr />
<div><br />
== History ==<br />
<br />
Empirical databases such as the [[Site investigation and rock mass characterization|Q and RMR]] systems were develop as a tool to help guide engineers when designing excavations underground, these databases are primarily based on civil engineering tunnel cases at low to moderate depth. These tunnels were designed as permanent openings with high traffic. The Q and RMR systems are very important to tunnel design work, however can be over conservative when applied to temporary or non-entry excavations