Ground engineering · Rock mechanics
Rock wedge stability (3-D, Hoek & Bray) with stereonet calculator
Tetrahedral wedge formed by two discontinuities, the slope face and the upper ground surface. Geometry by plane intersections, then the vector solution for the normal reactions on both planes (or sliding on one, or lift-off) with cohesion, friction, water, a horizontal seismic force and a rock bolt. Includes the Markland kinematic test and an equal-area stereonet.
Standard: Hoek & Bray Rock Slope Engineering (vector method) · Markland kinematic test · GEO5 Rock Stability / RocPlane / Swedge style
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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 |
|---|---|---|
| Plane A dip | 45 | ° |
| Plane A dip direction | 105 | ° |
| Plane A cohesion | 24 | kPa |
| Plane A friction angle | 20 | ° |
| Plane B dip | 70 | ° |
| Plane B dip direction | 235 | ° |
| Plane B cohesion | 48 | kPa |
| Plane B friction angle | 30 | ° |
| Slope face dip | 65 | ° |
| Slope face dip direction | 185 | ° |
| Upper surface dip | 12 | ° |
| Upper surface dip direction | 195 | ° |
| Height of the crest above the wedge toe | 40 | m |
| Rock unit weight | 25 | kN/m³ |
| Max water head on the planes (0 = dry) | 0 | m |
| Horizontal seismic coefficient | 0 | |
| Rock bolt force (total) | 0 | kN |
| Bolt trend | 0 | ° |
| Bolt plunge (below horizontal) | 0 | ° |
| Required FS | 1.5 |
Worked example — results
| Result | Value | Unit |
|---|---|---|
| Line of intersection plunge / trend | 31.2 / 157.7 | ° |
| Wedge volume | 13,151 | m³ |
| Wedge weight | 328,776 | kN |
| Plane A / B areas | 1,388 / 1,399 | m² |
| Normal reaction A / B | 262,316 / 161,046 | kN |
| Driving force along the intersection | 170,298 | kN |
| Failure mode | sliding on both planes (along the intersection) | |
| Factor of safety | 1.70 |
- ✓ Kinematically possible (plunge < face dip, daylights)
- ✓ FS ≥ 1.5 — 1.70
Method, formulas and assumptions
- Unit normals from dip / dip direction; intersection i = nA × nB (downward). Vertices from three-plane intersections (A, B, face, upper surface) with the face through the wedge toe and the upper surface at height H above it; volume = |V1·(V2 × V3)|/6.
- Resultant F = W + kh·W (horizontal, along the intersection trend) + bolt + water (γw·hw/3 × area on each plane). NA = −(F·nA − (F·nB)(nA·nB))/(1 − (nA·nB)²), NB likewise.
- Both positive: FS = (NA tanφA + NB tanφB + cA·AA + cB·AB)/(F·i). One positive: sliding on that plane. Both negative: lift-off (FS = 0).
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For preliminary design and education. Check every result against the current standard and your own engineering judgement before use.