Give the wall’s height and the backfill behind it. Rischio’s engine works out Rankine’s active thrust as the studio does on a retaining wall, with the tension crack of a cohesive soil and the water. The page adds the at-rest and passive thrusts and Coulomb’s coefficients to compare. It’s free and needs no account.

Calculator · Rischio's engine · Rankine's earth pressure

Wall and backfill

Wall height · H

Unit weight · γ

kN/m³

Cohesion · c'

kPa

Friction angle · φ'

Surface slope · β

Load, wall and water

Uniform surcharge · q

kPa

0 if there's none.

Wall friction · δ'

For Coulomb only; often about ⅔ φ'.

Water tableNo, drained

These data give no result: check the values.

Earth pressure coefficients

Quantity	Value
Active, Rankine · Kacos β (cos β − √(cos²β − cos²φ')) / (cos β + √(cos²β − cos²φ'))	0.350
Active, Coulomb · Kavertical back, δ' = 20°	0.340
At rest · K0(1 − sin φ') (1 + sin β)	0.587
Passive, Rankine · Kpcos β (cos β + √(cos²β − cos²φ')) / (cos β − √(cos²β − cos²φ'))	2.775
Passive, Coulomb · Kpδ' = 20°; overestimates Kp once δ' passes φ'/2	10.903

Rischio's engine calculated this. In the studio you keep the model, change it on the drawing and get the calculation report, every step shown.

Start free

This calculator is offered for information and technical support. It doesn't replace professional judgment or the reviews due, and its use is the user's responsibility.

How it calculates

Lateral earth pressure is the horizontal push of the soil on a wall, a fraction $K$ of the vertical effective stress: $\sigma_h = K\,\sigma'_v$ . The coefficient depends on how the wall moves.

Active, $K_a$ : the wall moves away from the soil just enough for the soil behind it to reach failure. It’s the smallest push, and the one retaining walls are designed for.
At rest, $K_0$ : the wall doesn’t move, as in a basement wall braced by its floors.
Passive, $K_p$ : the wall is pushed into the soil, which resists with the largest pressure, as in front of a wall’s toe.

Rankine’s active coefficient, with the backfill’s surface sloping at $\beta$ , is

K_a = \cos\beta\,\frac{\cos\beta - \sqrt{\cos^2\beta - \cos^2\phi'}}{\cos\beta + \sqrt{\cos^2\beta - \cos^2\phi'}}

which is $\tan^2(45° - \phi'/2)$ on level ground. The passive coefficient swaps the signs in the middle. The engine integrates $\sigma_a = (\sigma'_v + q)\,K_a - 2c'\sqrt{K_a}$ down the wall, parallel to the surface. A cohesive soil can’t pull on the wall, so where $\sigma_a$ would be negative a tension crack opens; the engine takes it full of water and says so in a warning. The passive pressure is $\sigma_p = (\sigma'_v + q)\,K_p + 2c'\sqrt{K_p}$ .

At rest, $K_0 = (1 - \sin\phi')(1 + \sin\beta)$ : Jaky’s coefficient for a normally consolidated soil, with the slope factor that US Army Corps of Engineers EM 1110-2-2502 gives. An overconsolidated soil pushes more.

Coulomb’s coefficients come from a sliding wedge with friction $\delta'$ between the wall and the soil. On a vertical back:

K_a = \frac{\cos^2\phi'}{\cos\delta' \left[1 + \sqrt{\dfrac{\sin(\phi' + \delta')\,\sin(\phi' - \beta)}{\cos\delta'\,\cos\beta}}\right]^2}

Its thrust leans at $\delta'$ to the horizontal. For the passive case, Coulomb’s flat wedge overestimates $K_p$ once $\delta'$ passes about $\phi'/2$ , so treat that value with care. The calculator gives Coulomb’s thrusts only for a cohesionless soil, the case the formula covers.

For every state, the page integrates the same diagram the engine does: $\sigma'_v$ with the submerged unit weight below the water table, and the water’s own thrust, $P_w = \tfrac{1}{2}\gamma_w h_w^2$ , apart. The engine’s method is in the docs, under earth pressure.

How to use it

Give the wall’s height and the backfill: its unit weight, cohesion, friction angle and the slope of its surface.
Add a uniform surcharge if there’s one, and the wall friction $\delta'$ for Coulomb’s coefficients.
Turn on the water table if the backfill isn’t drained, with its depth below the top of the wall.

The results show Rankine’s active thrust from the engine, with its height above the base and the pressure diagram on the wall, then a table of the active, at-rest and passive thrusts by both theories and one of the coefficients. Restore the example brings back a 3 m wall with its backfill at 10°: Rankine’s $K_a$ is 0.350 and Coulomb’s, with $\delta' = 20°$ , 0.340. To check the wall that holds this backfill, use the retaining wall calculator, or design it in Rischio’s retaining wall tool.

Frequently asked questions

What is lateral earth pressure?

The horizontal pressure that soil puts on a structure beside it, such as a retaining wall, a basement wall or a sheet pile. It grows with depth, because the vertical stress does, and it depends on the soil’s strength and on how much the structure moves.

What is the difference between active, at-rest and passive pressure?

They’re three states of the same soil. Active is the smallest push, once the wall has moved away enough. At rest is the push on a wall that can’t move. Passive is the largest resistance, once the wall has pushed into the soil. For $\phi' = 30°$ on level ground, $K_a = 0.33$ , $K_0 = 0.50$ and $K_p = 3.0$ .

What is the difference between Rankine and Coulomb?

Rankine assumes a smooth wall, so its thrust is parallel to the backfill’s surface. Coulomb takes the friction between the wall and the soil and a sliding wedge, so its thrust leans at $\delta'$ . In the active case, Coulomb’s coefficient comes out a little smaller. In the passive case, Coulomb’s can be much larger and is unconservative.

Where does the active thrust act?

On a dry, cohesionless backfill with nothing on top, the pressure grows linearly from zero at the top, so the thrust acts at a third of the height above the base. A surcharge adds a uniform pressure and raises it, and water changes it too. The calculator gives that height for each case.

Does the wall need to be designed for active or at-rest pressure?

For active pressure when it can move away from the soil a little, as a cantilever or gravity retaining wall does by tilting or sliding. A wall that can’t move, such as a basement wall tied to its floors or a wall on rock, is usually designed for at-rest pressure.

Lateral earth pressure calculator: Rankine and Coulomb