A rock mechanics calculation example is most useful when it follows the sequence used in a real excavation review: establish whether movement is kinematically possible, define the rock block, resolve forces on the controlling discontinuity, and then test the sensitivity of the result. The arithmetic is usually straightforward. The engineering judgement lies in deciding whether the assumed geometry, shear strength and groundwater conditions represent the ground that will actually be exposed.
A planar sliding block beside an excavation
Consider a cut in rock where a persistent discontinuity daylights in the excavation face. The face is steep enough that the discontinuity can release a block, and the joint dips out of the slope at 30 degrees. A structural mapping programme suggests that the block can be represented in a two-dimensional section with a width of 1 m normal to the section.
The purpose of the calculation is not to claim that all discontinuities behave as simple planar slides. It is to establish a transparent first assessment. If mapping identifies two intersecting joints, a wedge analysis is more appropriate. If the rock mass is heavily fractured, circular or non-circular rock mass failure may need consideration instead.
For this example, use the following preliminary design values:
- Cross-sectional area of potential block: 12 m²
- Unit width of analysis: 1 m
- Rock unit weight, γ: 26 kN/m³
- Dip of sliding plane, α: 30 degrees
- Sliding-plane length: 8.5 m
- Effective friction angle, φ’: 32 degrees
- Effective cohesion, c’: 10 kPa
- Resultant water force acting normal to the plane, U: 70 kN
The cohesion value should be treated carefully. A clean, persistent joint may have negligible effective cohesion at design scale, particularly where weathering, infilling, blasting damage or repeated movement are possible. Here it is retained only to demonstrate the calculation. A sensitivity case with c’ = 0 is essential.
Step 1: Calculate the block weight
For a unit width, the block volume is the cross-sectional area multiplied by the width:
“`text V = 12 m² × 1 m = 12 m³ “`
The block weight is therefore:
“`text W = γV = 26 kN/m³ × 12 m³ = 312 kN “`
This value represents the gravitational load on the potential sliding block. In a full design model, additional loads may also be relevant. These can include crest surcharge, support loads, seismic action, anchor prestress, or temporary construction loads. They should not be added by habit. Each load needs a credible load path to the block being analysed.
Step 2: Resolve the driving and normal forces
The component of the block weight parallel to the discontinuity drives sliding:
“`text T = W sin α T = 312 × sin 30° = 156 kN “`
The component normal to the plane provides the initial contact force:
“`text N = W cos α N = 312 × cos 30° = 270 kN “`
At this stage, the dry calculation appears simple. The resisting force is the sum of frictional resistance and cohesion over the plane area. For a one-metre-wide block, the joint area is:
“`text A = 8.5 m × 1 m = 8.5 m² “`
The cohesive contribution is:
“`text c’A = 10 kN/m² × 8.5 m² = 85 kN “`
The dry frictional component is:
“`text N tan φ’ = 270 × tan 32° = 169 kN “`
The total dry resistance is 254 kN. The corresponding factor of safety is:
“`text FSdry = (c’A + N tan φ’) / T FSdry = (85 + 169) / 156 = 1.63 “`
A factor of safety of 1.63 may appear satisfactory for some temporary situations, subject to the project basis of design. It does not, however, answer the central question for a joint-controlled failure: what happens after water enters the discontinuity?
Step 3: Include water pressure
Water pressure reduces the effective normal force and, therefore, the available frictional resistance. In this simplified representation, the water force is taken as 70 kN acting normal to the sliding plane:
“`text N’ = N – U N’ = 270 – 70 = 200 kN “`
The factor of safety becomes:
“`text FSwater = [c’A + (N – U) tan φ’] / T FSwater = [85 + (200 × tan 32°)] / 156 FSwater = 1.35 “`
The reduction from 1.63 to 1.35 is material. It shows why drainage details, rainfall response and the continuity of open joints can govern an excavation design even where the intact rock is strong.
The water force must be based on a defensible pressure distribution. A triangular distribution may be suitable where pressure increases with depth from a drained upper end. A measured or modelled piezometric distribution is preferable where groundwater conditions are complex. Water may also generate uplift, flow along the discontinuity or pressure in tension cracks. Those effects are not automatically captured by applying a single average pressure.
The result that often changes the decision
Now remove cohesion from the calculation. This is not an academic exercise. It is a realistic lower-bound case for a rough, weathered or persistent discontinuity where apparent cohesion cannot be relied upon over the full plane.
“`text FSwater, c’=0 = [(270 – 70) tan 32°] / 156 FSwater, c’=0 = 0.80 “`
The same block changes from apparently acceptable to clearly unstable. That contrast is the value of a transparent rock mechanics calculation example: it identifies which assumptions deserve further investigation before support is selected.
It would be inappropriate to respond by simply choosing a higher cohesion value. The better response may be additional structural mapping, scanline measurements, borehole information, water observations after rainfall, or a review of joint infill and roughness. If the potential plane is confirmed, drainage, scaling, rock bolts, dowels, mesh, shotcrete or a flatter excavation geometry can then be evaluated against a defined failure mechanism.
Checks before relying on the calculation
A numerical factor of safety has meaning only if the assumed failure is possible. First, confirm kinematics. The discontinuity must dip out of the excavation and daylight in the face. Its dip direction must be suitably aligned with the face, and the release surfaces required to form a block must exist.
Secondly, confirm persistence and block geometry. A discontinuity visible over a short exposure may terminate harmlessly, or it may continue behind the face and define a large release surface. The assumed 12 m² block should be traceable to mapped structures, not merely selected because it produces a convenient result.
Thirdly, use shear-strength parameters that match the discontinuity condition. Peak strength may be relevant for a rough, fresh joint with limited displacement. Residual strength is more appropriate where the plane is slickensided, infilled, previously displaced or likely to undergo sustained movement. Scale effects and blast damage can reduce field performance relative to small laboratory specimens.
Finally, apply the project’s design framework consistently. Some projects use allowable-stress calculations with target factors of safety, while others use partial factors and limit-state verification. The required margin depends on consequences, uncertainty, permanence, monitoring and the reliability of drainage or support. There is no universal factor of safety that substitutes for a defined design basis.
Making the calculation practical in design work
For repeated assessments, calculation software should make the assumptions visible rather than hide them behind a single output value. The engineer needs to inspect geometry, loads, water pressure, effective normal force, shear resistance and the resulting factor of safety in detail. Unit handling matters as much as the equations: mixing MPa, kPa, metres and millimetres is an easy way to produce a credible-looking but incorrect result.
A practical workflow is to create a dry baseline, add groundwater, then vary friction angle, cohesion, block volume and discontinuity dip within plausible bounds. This quickly reveals whether the design is governed by a single uncertain parameter or remains acceptable across the expected range. For engineers working between site and office, purpose-built macOS and iOS tools such as those developed by Psicons AB can help keep this setup, calculation and result review consistent across devices.
The most useful next step is rarely a more elaborate equation. It is usually a targeted field observation that reduces the uncertainty exposed by the calculation.
