Coulomb Earth Pressure Theory: Active & Passive Formula, Wall Friction & Worked Example

Coulomb earth pressure theory gives the lateral force that soil puts on a retaining wall by looking at a wedge of soil that slides behind the wall. Unlike Rankine’s theory, it can include friction between the wall and the soil, a sloping wall back and a sloping backfill. That makes it the standard method for gravity walls, abutments and any wall where the back face is rough or inclined.

This guide gives the Coulomb active and passive coefficients (K_a and K_p), explains every symbol, compares Coulomb with Rankine, and works through a full example you can check by hand.

Quick answer: Coulomb theory in one minute

• Charles-Augustin de Coulomb proposed it in 1776, 81 years before Rankine (1857).
• It assumes a plane failure surface and finds the soil wedge that gives the largest active thrust (or the smallest passive resistance).
• Total active thrust: P_a = ½ · K_a · γ · H², acting at H/3 above the base and inclined at angle δ (wall friction) to the normal of the wall back.
• For a vertical, smooth wall with level backfill (δ = 0), Coulomb gives exactly the same K_a as Rankine: (1 − sin φ) / (1 + sin φ).
• For active pressure it is accurate. For passive pressure with large wall friction it overestimates resistance, so engineers limit δ or use a log-spiral method.

Assumptions of Coulomb’s theory

1. The backfill is dry, homogeneous, isotropic and cohesionless (c = 0) in the classic form.
2. The failure surface is a plane passing through the heel of the wall.
3. The sliding wedge behaves as a rigid body.
4. Friction acts on the failure plane (angle φ) and on the wall back (angle δ).
5. The wall moves enough to develop the full active (or passive) state.
6. The problem is two-dimensional: results are per metre length of wall.

Coulomb active earth pressure coefficient (K_a)

Symbols used (angles in degrees):

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The active coefficient is:

K_a = sin²(α + φ) ÷ { sin²α · sin(α − δ) · [ 1 + √( sin(φ + δ) · sin(φ − β) ÷ ( sin(α − δ) · sin(α + β) ) ) ]² }

The total active thrust per metre run is P_a = ½ K_a γ H². Because the pressure grows linearly with depth (triangular diagram), P_a acts at H/3 above the base. Its line of action is tilted at δ from the normal to the wall back, so it has a horizontal and a vertical component.

Check: Coulomb reduces to Rankine

Put α = 90°, β = 0 and δ = 0. The formula becomes cos²φ ÷ (1 + sin φ)², which simplifies to (1 − sin φ) ÷ (1 + sin φ): exactly the Rankine active coefficient. So Rankine is a special case of Coulomb for a smooth vertical wall.

Coulomb passive earth pressure coefficient (K_p)

K_p = sin²(α − φ) ÷ { sin²α · sin(α + δ) · [ 1 − √( sin(φ + δ) · sin(φ + β) ÷ ( sin(α + δ) · sin(α + β) ) ) ]² }

Total passive resistance: P_p = ½ K_p γ H². Caution: a real passive failure surface is strongly curved. The plane-surface assumption makes Coulomb K_p unsafe (too high) when δ is large. Common practice is to use δ no more than about φ/3 for passive calculations, or to use log-spiral (Caquot–Kerisel) values.

What value of wall friction δ should you use?

These are common design assumptions. Always follow the code or geotechnical report that governs your project.

Coulomb K_a values table (vertical wall, level backfill)

Wall friction lowers K_a by roughly 10%. It also tilts the thrust downward, which helps a wall resist both sliding and overturning.

Worked example: active thrust on a 6 m wall

Given: vertical wall back (α = 90°), level backfill (β = 0), H = 6 m, sand with γ = 18 kN/m³ and φ = 30°, rough concrete wall with δ = 20° (⅔ φ).

Step 1: terms of the formula

• sin²(α + φ) = sin²(120°) = 0.866² = 0.750
• sin²α = sin²(90°) = 1
• sin(α − δ) = sin(70°) = 0.9397
• sin(φ + δ) · sin(φ − β) = sin 50° · sin 30° = 0.7660 × 0.5 = 0.3830
• sin(α − δ) · sin(α + β) = 0.9397 × 1 = 0.9397
• √(0.3830 ÷ 0.9397) = √0.4076 = 0.6384

Step 2: K_a
K_a = 0.750 ÷ [ 0.9397 × (1 + 0.6384)² ] = 0.750 ÷ (0.9397 × 2.6844) = 0.750 ÷ 2.5225 = 0.297

Step 3: total thrust
P_a = ½ × 0.297 × 18 × 6² = 96.3 kN per metre of wall, acting 6/3 = 2.0 m above the base.

Step 4: components (thrust tilted δ = 20° from the horizontal normal)
Horizontal: 96.3 × cos 20° = 90.5 kN/m
Vertical (downward on the wall): 96.3 × sin 20° = 32.9 kN/m
Overturning moment about the base: 90.5 × 2.0 = 181 kN·m/m

Step 5: compare with Rankine
Rankine K_a = 0.333 → P_a = ½ × 0.333 × 18 × 36 = 108 kN/m, all horizontal, moment 216 kN·m/m. Coulomb’s horizontal thrust is about 16% lower, and its 32.9 kN/m vertical component adds to the wall’s resistance. Rankine is therefore the more conservative (safer, less economical) choice for this wall.

Passive side of the same soil

For φ = 30°, vertical wall, level ground: Rankine K_p = 3.0. Coulomb with δ = 10° (φ/3) gives K_p ≈ 4.14. With δ = 20° it gives about 6.1, which is too optimistic for design. This is why δ is limited on the passive side.

Coulomb vs Rankine: differences at a glance

Limitations of Coulomb’s theory

• The real failure surface is curved. The plane assumption is acceptable for active pressure but not for passive pressure with high δ.
• The classic formula is for dry cohesionless soil. Water pressure, surcharge and cohesion must be added separately. For example, add the full hydrostatic pressure below the water table and use the submerged unit weight for the soil.
• It gives the total thrust, not the true pressure distribution. A triangular distribution is assumed.
• The formula is not valid when β is greater than φ: the slope itself would be unstable.

Where Coulomb theory is used

• Gravity and semi-gravity retaining walls with rough or battered backs
• Bridge abutments and wing walls
• Basement walls and sheet pile walls (active side)
• The graphical Culmann method, which is Coulomb’s wedge analysis done by drawing, useful for irregular backfill and line loads
• The Mononobe–Okabe method for earthquake earth pressure, which is Coulomb’s wedge with added seismic inertia forces

Want to check a full cantilever wall for overturning, sliding and bearing? Try our free Retaining Wall Design Calculator, and read about the soil under the wall in Safe Bearing Capacity of Soil.

FAQ

What is Coulomb’s earth pressure theory?

It is a method that finds the lateral thrust of soil on a wall by analysing the equilibrium of a rigid soil wedge that slides on a plane surface. It includes friction on the wall back.

Is Coulomb or Rankine more conservative?

For active pressure, Rankine usually gives a larger thrust, so it is more conservative. For passive pressure, Coulomb with large wall friction gives values that are too high, so it is unconservative.

When does Coulomb give the same answer as Rankine?

When the wall back is vertical, the backfill is level and the wall is smooth (δ = 0). Both give K_a = (1 − sin φ)/(1 + sin φ).

Where does the Coulomb active thrust act?

At one third of the wall height above the base (for a triangular pressure diagram), inclined at the wall friction angle δ to the normal of the wall back.

What is a typical value of wall friction angle?

For rough concrete against sand, about two thirds of φ is common for the active case. Smoother walls use about half of φ, and passive calculations usually limit δ to about one third of φ.

Related guide: Retaining Wall Cost per Foot