Earth Pressure Calculator — Coulomb's Theory
Charles-Augustin de Coulomb formulated in 1776 the general theory of earth pressures on rigid walls considering wall-soil friction (δ), the inclination of the wall face (α) and the slope of the backfill (β). It is more realistic than Rankine because it includes three key geometric variables and reduces active pressure by 5 % to 15 % when real friction exists at the contact. This calculator provides Ka, Kp, the total thrust Pa and its point of application for gravity walls, bridge abutments with compacted backfill, anchored walls and sheet piles. If your wall is vertical and wall-soil friction is negligible, Rankine is equivalent and faster; in all other cases use Coulomb.
What is it and when is it applied?
Coulomb assumes that the soil fails by a triangular wedge sliding along a straight failure plane from the base of the wall to the surface of the backfill. By establishing force equilibrium on the wedge (weight, thrust on the wall and reaction on the failure plane), Ka is obtained by maximisation and Kp by minimisation of the thrust function. Applicable to concrete or masonry gravity walls, bridge abutments, cantilever walls, anchored walls and retaining walls with compacted backfill. It should not be used when the backfill is heterogeneous in layers (a zonal method is required) or when seismic thrusts are predominant (use Mononobe-Okabe).
Applied formulas
Active coefficient (Coulomb):
Ka = cos²(φ − α) / [cos²α · cos(α + δ) · (1 + √(sin(φ + δ)·sin(φ − β) / (cos(α + δ)·cos(α − β))))²]
Passive coefficient:
Kp = cos²(φ + α) / [cos²α · cos(α − δ) · (1 − √(sin(φ − δ)·sin(φ + β) / (cos(α − δ)·cos(α − β))))²]
Notation: φ = internal friction angle of soil; δ = wall-soil friction (≈ 2/3 φ for smooth concrete, 1/2 φ for very smooth concrete, equal to φ for rough concrete); α = inclination of the wall face from vertical (positive when the wall leans towards the soil); β = backfill slope (0 if horizontal).
Total thrust due to self-weight of backfill:
Pa = 0.5 · γ · H² · Ka (applied at H/3 from the base, inclined at δ to the normal of the wall face)
Horizontal and vertical components: Pah = Pa · cos(α + δ); Pav = Pa · sin(α + δ)
Calculation example
| Parameter | Value |
|---|---|
| Wall height H | 5.0 m |
| Backfill unit weight γ | 19 kN/m³ |
| Friction angle φ | 34° |
| Wall-soil friction δ | 2/3 · 34 = 22.7° |
| Backface inclination α | 10° (towards the backfill) |
| Backfill slope β | 0° |
| Cohesion c | 0 |
| Water table | Not detected |
We calculate Ka with the angles in radians: cos²(φ − α) = cos²(34 − 10) = cos²24° = 0.835. cos²α = cos²10° = 0.970. cos(α + δ) = cos(10 + 22.7) = cos(32.7°) = 0.841. sin(φ + δ) = sin(56.7°) = 0.836. sin(φ − β) = sin(34 − 0) = 0.559. cos(α − β) = cos(10°) = 0.985. Numerator: 0.835. Denominator root: (0.836·0.559)/(0.841·0.985) = 0.467/0.828 = 0.564; √0.564 = 0.751. (1 + 0.751)² = (1.751)² = 3.066. Total denominator: 0.970 · 0.841 · 3.066 = 2.501. Ka = 0.835 / 2.501 = 0.334. Comparison with Rankine (same data without δ or α): Ka-Rankine = tan²(45 − 17) = tan²28° = 0.283. Difference +18 % because the backface is inclined. With the actual component, the thrust projected onto the horizontal is smaller: Pa = 0.5 · 19 · 5² · 0.334 = 0.5 · 19 · 25 · 0.334 = 79.3 kN/m. Components: Pah = 79.3 · cos(32.7°) = 79.3 · 0.841 = 66.7 kN/m; Pav = 79.3 · sin(32.7°) = 79.3 · 0.540 = 42.8 kN/m. The Pav is favourable to the wall because it increases the effective weight resisting sliding.
Result: Ka = 0.334 · Pa = 79.3 kN/m · Pah = 66.7 kN/m · Pav = 42.8 kN/m (favourable).
Interpretation of results
The vertical component of the thrust (Pav) only exists when δ ≠ 0 and is always favourable for the wall: it increases the vertical load on the foundation and therefore the frictional force resisting sliding. For this reason, a wall with a rough backface (high δ) is better than a smooth one with the same geometry. For conservative sliding design, it is advisable to reduce δ to 1/2·φ; for overturning, 2/3·φ is maintained. In temporary structures (shoring, temporary sheet piles), δ = 0 is used as a conservative assumption.
Reference standards
- BS EN 1997-1 (Eurocode 7) — Geotechnical design — Part 1: General rules
- Coulomb, C.A. (1776). Essai sur une application des règles de maximis et minimis
- BS EN 1997-1 (Eurocode 7) — Annex C, analytical methods
- BS EN 1997-1 (Eurocode 7) — Geotechnical design — Part 1: General rules
- BS 5930 — Code of practice for ground investigations
- BS 1377-2 — Methods of test for soils for civil engineering purposes — Part 2: Classification tests
Frequently asked questions
What value of δ do I use in practice?
For concrete cast against granular soil: δ ≈ 2/3·φ. For very smooth concrete or when a geomembrane is present: δ ≈ 1/2·φ. For rough stone masonry: δ ≈ φ. For steel (sheet piles) in sand: δ = 17-22° typical. BS 5930 provides comprehensive tables. A zero value is reserved for conservative temporary design.
Is Coulomb also valid for passive pressure?
Yes, but with a caveat: Coulomb overestimates Kp when δ is large. For δ > φ/3, it is advisable to use the logarithmic spiral method (Caquot-Kerisel 1948) which is more accurate. In the design of anchored walls and port sheet piles, Caquot-Kerisel is always used for passive pressure.
How do I handle multiple backfill layers?
Classical Coulomb assumes a single homogeneous material. For layered backfill with different φ and γ, calculate the thrust in sections (each stratum) summing the contributions. A more accurate alternative: the wedge method with a bilinear failure surface or, directly, finite element analysis with geotechnical software.
What if the backfill has cohesion?
Coulomb with cohesion becomes more complex because a tension zone appears near the crest and the critical plane must be found iteratively. In practice, cohesion in the backfill is neglected (it is usually clean compacted sand by requirement) and the design is conservatively based only on φ. If the backfill is cohesive due to site constraints, use Rankine with the tension crack expression or a numerical tool.