Lateral Earth Pressure Calculator (Rankine)

Understanding the lateral earth pressure calculator — concept and principle

Retaining walls and basement walls must resist the sideways push of soil. This calculator uses Rankine's theory for a dry, cohesionless backfill to give the active, passive and at-rest earth pressure coefficients and the active and passive thrust per metre run of wall.

How it is calculated

• Active coefficient: K_a = tan²(45° − φ/2).
• Passive coefficient: K_p = tan²(45° + φ/2) = 1/K_a.
• At-rest coefficient (Jaky): K₀ = 1 − sin φ.
• Lateral pressure at depth z: σ_h = K γ z, rising linearly from zero at the top.
• Thrust per metre: P = ½ K γ H², acting at H/3 above the base.

φ is the friction angle, γ the unit weight (kN/m³) and H the wall height (m).

The principle

Soil at rest pushes sideways with a fraction K₀ of the vertical stress. If the wall moves slightly away from the soil, the soil behind it shears along a plane inclined at 45° + φ/2 to the horizontal, and the pressure drops to the minimum active value. If the wall is pushed into the soil, the soil resists until it fails along a flatter plane, giving the much larger passive value. Rankine (1857) derived these limits from the Mohr circle at failure, assuming a vertical, smooth wall and a horizontal ground surface. Since pressure grows linearly with depth, the pressure diagram is a triangle, whose area is the thrust and whose centroid is at one-third of the height.

Worked example

1. Inputs: φ = 30°, γ = 18 kN/m³, H = 4 m.
2. K_a = tan²30° = 0.3333; K_p = 1/0.3333 = 3; K₀ = 1 − sin 30° = 0.5.
3. Active pressure at the base = 0.3333 × 18 × 4 = 24 kPa.
4. Active thrust = 0.5 × 0.3333 × 18 × 4² = 48 kN/m, acting 1.33 m above the base.
5. Passive thrust = 0.5 × 3 × 18 × 16 = 432 kN/m.

How to read the result

Use the active thrust for walls that can yield a little, such as free-standing cantilever walls, and the at-rest thrust for rigid walls braced at the top, such as basement walls. Active conditions need only a small outward movement, while full passive resistance needs much larger movement, so designers usually take only part of the passive value. Water behind the wall adds full hydrostatic pressure, so drainage matters.

Common mistakes

• Using K_p = 1/K_a when the wall is rough or the backfill slopes; Coulomb or log-spiral methods are needed then.
• Forgetting surcharge loads such as traffic, which add K × q over the full height.
• Ignoring groundwater or poor drainage behind the wall.
• Applying active pressure to rigid walls that cannot move.

Frequently asked questions

Why is the thrust located at H/3?

The pressure diagram is a triangle, and a triangle's centroid lies one-third of its height from the wide end.

Does cohesion reduce active pressure?

Yes, by 2c√K_a, which can create a tension crack zone near the top; this calculator assumes c = 0.

Is Jaky's formula valid for all soils?

It suits normally consolidated soils; overconsolidated soils have higher K₀.

Sources: Rankine, W. J. M. (1857), On the stability of loose earth, Philosophical Transactions of the Royal Society of London, 147 · Das, B. M., Principles of Foundation Engineering, Cengage