Ground engineering · Rock mechanics
Rock slope kinematic analysis (planar, wedge, toppling) calculator
Discontinuity sets mapped in a rock cut are tested against the slope face. Planar sliding needs a set dipping within ±20° of the face direction, flatter than the face and steeper than its friction angle. Every pair of sets forms a wedge whose line of intersection must daylight (plunge less than the face's apparent dip in that direction) and plunge steeper than φ. Flexural toppling needs a steep set dipping into the slope with (90° − dip) + φ ≤ face dip. Results are drawn on a lower-hemisphere stereonet.
Standard: Markland (1972) test · Hoek & Bray (1981) · Goodman flexural toppling criterion · lower-hemisphere equal-angle stereonet
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Inputs in the worked example
The live calculator starts with these values; change any of them and every result, check and diagram updates instantly.
| Input | Value | Unit |
|---|---|---|
| Joint sets: dip (°), dip direction (°), friction angle (°) | 38, 160, 30 72, 230, 32 65, 95, 30 80, 345, 28 | |
| Slope face dip | 60 | ° |
| Slope face dip direction | 170 | ° |
| Planar / toppling direction limit | 20 | ° |
Worked example — results
| Result | Value | Unit |
|---|---|---|
| J1 38/160 (φ 30°) | PLANAR SLIDING possible · no toppling | |
| J2 72/230 (φ 32°) | no planar · no toppling | |
| J3 65/95 (φ 30°) | no planar · no toppling | |
| J4 80/345 (φ 28°) | no planar · TOPPLING possible | |
| Wedge J1–J2: intersection 37.9° → 155° | apparent face dip 59.1° · WEDGE possible | |
| Wedge J1–J3: intersection 37.9° → 164° | apparent face dip 59.8° · WEDGE possible | |
| Wedge J1–J4: intersection 3.4° → 74° | apparent face dip -9.6° · stable kinematically | |
| Wedge J2–J3: intersection 44.0° → 158° | apparent face dip 59.5° · WEDGE possible | |
| Wedge J2–J4: intersection 64.6° → 277° | apparent face dip -26.6° · stable kinematically | |
| Wedge J3–J4: intersection 59.6° → 58° | apparent face dip -33.5° · stable kinematically |
- ✗ No planar sliding mechanism — 1 set(s): check stability
- ✗ No wedge mechanism — 3 wedge(s): check stability
- ✗ No flexural toppling mechanism — 1 set(s)
Method, formulas and assumptions
- Planar (Markland): |αj − αf| ≤ 20°, φ < ψj < ψf.
- Wedge: intersection line of each pair (cross product of poles) must plunge steeper than φ (lower of the pair) and flatter than the face's apparent dip in its trend, tan ψa = tan ψf·cos(trend − αf), and trend within 90° of the face.
- Flexural toppling (Goodman): set dips into the slope within ±20° of the opposite direction and (90° − ψj) + φ ≤ ψf.
- A kinematically possible mechanism then needs a stability analysis (Planar failure, 3-D wedge tools).
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For preliminary design and education. Check every result against the current standard and your own engineering judgement before use.