# Stability Graph Method

## Contents

## History

Empirical databases such as the 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. ^{[1]}

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. ^{[1]}

It is important to note that the database assembled for the modified stability graph reflects Canadian practice, and is bias towards Canadian ground conditions

## Input Parameters

### Modified Stability Number, N'

N' = Q' x A x B x C

where:

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. ^{[2]}

RQD/Jn: is the measure of block size for a jointed rock mass

Jr/Ja: is the measure of joint surface strength and stiffness

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.

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.

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.

### Shape factor, S

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:

HR = Area(m^2)/Perimeter(m) = w x h/2(w + h)

## Calculation of Input Parameters

### Q'

The input parameters for Q' can be measured and calculated as described in Site investigation and rock mass characterization.

### Rock Stress Factor A

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 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.

Figure 1: Rock Stress Factor A (Potvin, 1988) for Stability Graph analysis ^{[1]}

Figure 2: Determining max induced compressive stress using Phase 2

### Joint Orientation Factor, B

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 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.

Figure 3: Joint Orientation Factor B (Potvin, 1988) for Stability Graph analysis ^{[1]}

Figure 4: Measuring the angle between two planes using DIPS

*Direct Calculation of Interplane Angle*

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):

For the stope face:

For the joint plane:

Next calculate the dot product, **Failed to parse (unknown function "\cdotj"): {\displaystyle w\cdotj}**
, between the stope face and the joint set

**Failed to parse (unknown function "\cdotj"): {\displaystyle w\cdotj = N_w*N_j + E_w*E_j + D_w*D_j}**

The true interplane angle, , is given by:

**Failed to parse (unknown function "\cdotj"): {\displaystyle \alpha = cos^-1(w\cdotj)}**

### Gravity Adjustment Factor, C

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.

Figure 5: Gravity Adjustment Factor, C, for Stability Graph analysis ^{[1]}

## Determination of S

Given modifed stability number, the hydraulic radius can be determined using the Stability Graphs.

### No-Support Stopes

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. ^{[1]}

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.

Figure 6: Database of unsupported stopes ^{[1]}

### Cablebolt-Supported Stopes

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. ^{[1]}

Figure 7: Database of cablebolt supported stopes ^{[1]}

## Cable Support Guidelines

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)^{[1]}. 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.

Figure 8: Design Chart for cablebolt density requirements ^{[1]} Figure 9: Design chart for minimum cablebolt length required ^{[1]}

## Case Example

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.

The rockmass exhibits the following characteristics:

Rockmass Properties | |
---|---|

RQD | 86.29 |

Jn | 9 |

Jr | 3 |

Ja | 1 |

USC | 75 MPa |

E | 24 GPa |

Sigma h | 22.7 MPa |

Sigma v | 31.9 MPa |

The Rockmass also has 3 Joint Sets with the following dip/dip directions

Joint #1 | Joint #2 | Joint #3 | |||
---|---|---|---|---|---|

D | DD | D | DD | D | DD |

43 | 294 | 56 | 163 | 33 | 105 |

**Q'**

**Rock Stress Factor, A**
Given the stress conditions and rock conditions the max induced compressive stress in the hanging wall and the back are:

Stope face | Induced stress |
---|---|

Hanging wall | 5.36 MPa |

Back | 52.9 MPa |

Given the max induced stress and the USC of each stope face, the following ratio are calculated corresponding to the given A factor

Stope Face | Ratio | A Factor |
---|---|---|

Hanging wall | 14 | 1 |

Back | 1.4 | 0.1 |

**Joint Orientation Factor, B**

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.

Stope face | Joint #1 | Joint #2 | Joint #3 | B Factor |
---|---|---|---|---|

Hanging wall | 61 | 50 | 82 | 0.6 |

Back | 43 | 56 | 33 | 0.275 |

**Gravity Adjustment Factor, C**

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.

Stope face | Mode of Failure | C Factor |
---|---|---|

Hanging wall | Slabbing | 5.46 |

Back | Slabbing | 2 |

**N'**

Stope face | Q' | A | B | C | N' |
---|---|---|---|---|---|

Hanging wall | 21.6 | 1.0 | 0.6 | 5.46 | 94.2 |

Back | 21.6 | 0.1 | 0.275 | 2.0 | 1.19 |

Based on the un-supported stability graph the following hydraulic radii are determined

Hanging Wall | 12.8 |
---|---|

Back | 3 |

### Calculation of Max Allowable Span

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.

Max Allowable Span | |
---|---|

Hanging Wall | 42m |

Back | 8.6m |

## Summary

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. ^{[1]}

## References

- ↑
^{1.00}^{1.01}^{1.02}^{1.03}^{1.04}^{1.05}^{1.06}^{1.07}^{1.08}^{1.09}^{1.10}^{1.11}^{1.12}Hutchinson, D. J., & Diederichs, M. S. (1996). Cablebolting in Underground Mines. Richmond: BiTech Publishers. - ↑ Mckinnon, S. (2014). Empirical design of open stope dimensions. Kingston: Queen's University.

3. Mawdesley, C., Truman, R., & Whiten, W. (2003). Extending the Mathews stability graph for open-stope design. EBSCO Publishing.

4. Stewart, S., & Forsyth, W. (1995, August). The Mathew's method for open stope design. Retrieved from Centre for Excellence in Mining Innovation: [1]