Mine Slope Stability and Excavation Design Exercises

These exercises practise first-pass calculations used in mine slope stability, quarry face review, highwall control, and excavation design. They connect geometry, groundwater, effective stress, shear resistance, sliding checks, monitoring trends, rockfall controls, haul-road access, support demand, and geotechnical risk ranking.

Assume simplified nominal values unless an exercise states otherwise. Real slope design requires a site-specific ground model, structural mapping, hydrogeology, laboratory and field testing, blast-damage review, staged excavation assessment, monitoring, trigger-action plans, operating controls, and competent geotechnical judgement.

How to Use These Exercises

For each problem:

1. define the excavation stage, slope scale, material domain, and failure mode;
2. separate geometry, groundwater, strength, surcharge, and monitoring assumptions;
3. keep units consistent between metres, kN, kPa, degrees, days, and millimetres;
4. state whether the result is a geometry check, stability screen, monitoring trigger, or operating decision;
5. identify the field evidence that would validate or challenge the calculation.

The most common mistake is treating a calculated factor of safety as a complete design result. A slope calculation is useful only when the failure mechanism, geological structure, water condition, construction quality, and inspection evidence match the model.

Use the exercises as operating gates: revise slope geometry, clean or extend drainage, restrict access, increase monitoring frequency, require support, reject a haul-road grade, or escalate geotechnical review when movement, pore pressure, rockfall capacity, or support utilization no longer matches the design basis.

Exercise 1: Overall Slope Angle

An open-pit wall has a vertical height H=120\ \text{m} between the toe and crest reference points. The horizontal projection between those points is L=210\ \text{m}.

Estimate the overall slope angle measured from the horizontal.

Solution

Overall slope angle:

Engineering Comment

The result is a geometry descriptor, not a stability proof. The same overall angle can be acceptable or unsafe depending on discontinuity orientation, groundwater, blasting damage, bench condition, loading, and consequence of failure.

Exercise 2: Repeated Bench Geometry

A trial inter-ramp geometry uses six repeated bench modules. Each module has bench height H_b=12\ \text{m}, bench face angle \theta_b=70^\circ, and catch berm width B=7.5\ \text{m}.

Estimate the approximate inter-ramp angle for the repeated geometry.

Solution

Horizontal projection of one bench face:

Horizontal projection per module:

Total height and horizontal projection:

Inter-ramp angle:

Engineering Comment

This estimate assumes the constructed benches match the design geometry. Overbreak, poor scaling, ramp interruptions, blocked berms, or damaged faces can reduce catch capacity and change the effective slope angle.

Exercise 3: Hydrostatic Force in a Water-Filled Tension Crack

A tension crack behind a slope can hold water to depth h_w=10\ \text{m} after heavy rainfall. Use \gamma_w=9.81\ \text{kN/m}^3.

Estimate the hydrostatic resultant per metre out-of-plane width.

Solution

For triangular hydrostatic pressure:

Engineering Comment

Water-filled cracks can add destabilizing force and increase pore pressure along discontinuities. Crack inspection, drainage, rainfall triggers, and temporary exclusion zones are slope controls, not administrative details.

Exercise 4: Effective Stress and Shear Strength

At a potential sliding surface, the total normal stress is \sigma_n=420\ \text{kPa} and pore pressure is u=176\ \text{kPa}. The effective cohesion is c'=20\ \text{kPa} and the effective friction angle is \phi'=32^\circ.

Estimate effective normal stress and Mohr-Coulomb shear strength.

Solution

Effective normal stress:

Shear strength:

Engineering Comment

The water pressure has removed a large part of the effective normal stress. A drainage or pore-pressure trend can therefore change slope capacity even when the excavation geometry has not changed.

Exercise 5: Planar Sliding Screen

A block can slide on a discontinuity dipping at \alpha=34^\circ. The block weight is W=920\ \text{kN/m}, the water-pressure resultant normal to the plane is U=180\ \text{kN/m}, effective cohesion is neglected, and the effective friction angle is \phi'=36^\circ.

Estimate the screening factor of safety. Then estimate the additional resisting force needed to reach FS=1.20, assuming support acts directly opposite sliding.

Solution

Driving force:

Effective normal force:

Frictional resistance:

Factor of safety:

Required resistance for FS=1.20:

Additional resisting force:

Engineering Comment

This is a simplified planar check. A real design must verify kinematic feasibility, persistence, end release, water pressure distribution, seismic or blast loading, support orientation, support reliability, and whether the assumed sliding surface is actually present.

Exercise 6: Monitoring Velocity and Acceleration

A slope prism records cumulative displacement of 14.2\ \text{mm} on day 0, 18.0\ \text{mm} on day 2, and 24.4\ \text{mm} on day 4.

Estimate the two average velocities and the change in velocity per day. A trigger is defined as velocity above 3.0\ \text{mm/day} with increasing velocity.

Solution

First two-day velocity:

Second two-day velocity:

Average acceleration over the interval between velocity estimates:

Engineering Comment

The latest velocity exceeds the trigger and is increasing. The response should follow the trigger-action plan: verify instrumentation, inspect cracks and water conditions, control access, review blasting and rainfall history, and escalate geotechnical review.

Exercise 7: Catch Berm Capacity Screen

After a blast, a bench face is expected to shed 0.18\ \text{m}^3 of ravelled material per metre of face length. The exposed length is 120\ \text{m}. The cleaned catch berm has an estimated effective trap cross-section of 0.32\ \text{m}^2 per metre of length.

Estimate the ravel volume, trap volume, and capacity ratio.

Solution

Expected ravel volume:

Estimated trap volume:

Capacity ratio:

Engineering Comment

The berm appears to have capacity for this simplified ravel estimate, but the conclusion depends on berm cleanliness, block size, rebound, face height, impact energy, access, water or snow accumulation, and whether prior falls have already consumed the catch volume.

Exercise 8: Haul-Road Grade on a Pit Ramp

A pit ramp gains 48\ \text{m} of elevation over 620\ \text{m} of centerline distance. The operating limit for the haul fleet is 8.0\% grade.

Check the ramp grade and margin.

Solution

Ramp grade:

Margin to the operating limit:

Engineering Comment

The nominal grade is just inside the stated limit. The review should also check rolling resistance, curve radius, drainage, road width, safety berms, sight distance, braking distance, surface condition, and whether loaded and empty travel have different constraints.

Exercise 9: Excavation Support Load Utilization

A fractured brow requires a simplified support line load of 55\ \text{kN/m}. Rock bolts are installed at spacing s=1.4\ \text{m} along the supported line. Each bolt has allowable tensile capacity T_{allow}=95\ \text{kN}.

Estimate bolt load and utilization. Then check utilization if corrosion or installation uncertainty reduces allowable capacity to 80\ \text{kN}.

Solution

Load assigned to each bolt:

Utilization with nominal allowable capacity:

Utilization with reduced allowable capacity:

Engineering Comment

The support is highly sensitive to capacity reduction. Support review should address bolt orientation, bond length, plate seating, corrosion protection, installation quality, inspection access, and whether the assumed load path is compatible with the actual failure mechanism.

Exercise 10: Geotechnical Risk Ranking for Blocked Drains

A final wall relies on horizontal drains to control pore pressure. A failure-mode review assigns blocked drains severity S=9, occurrence O=5, and detection ranking D=6.

After cleaning access is improved and inspections are scheduled, occurrence is estimated at O=3 and detection at D=3. Compare the traditional risk priority numbers.

Solution

Initial risk priority number:

Revised risk priority number:

Reduction:

Engineering Comment

The revised ranking is lower, but high-consequence geotechnical hazards still require engineering controls. The drain program must be validated with piezometer trends, inspections, cleaning records, rainfall response, access checks, and clear escalation triggers.

Review Checklist

When reviewing a mine slope or excavation calculation, ask:

• Is the failure mode explicit: planar sliding, wedge failure, toppling, circular failure, ravelling, buckling, or support overload?
• Does the ground model include structural fabric, persistence, weathering, blast damage, excavation stage, groundwater, and material-domain boundaries?
• Are pore pressures, crack water, rainfall response, drainage condition, and piezometer evidence connected to the stability result?
• Are monitoring triggers linked to survey quality, instrument reliability, displacement rate, acceleration, visual inspection, and access control?
• Are catch berms, haul roads, support systems, and exclusion zones reviewed as operating controls, not only geometry features?
• Does the response plan define who acts, what evidence confirms the trigger, what area is restricted, and what condition closes the action?
• Is continued operation blocked when field evidence contradicts the assumed mechanism, water condition, support capacity, or constructed geometry?

Good slope engineering keeps the calculation tied to the mine face. Geometry, water, structure, monitoring, and operating discipline must all support the same ground-behaviour interpretation.