Coulomb’s made the following assumptions in the development of his theory.

The assumptions are:

- The soil is isotropic and homogeneous.
- The surface of rupture is a plane.
- The failure wedge is a rigid body.
- There is friction between and the wall.
- Back of wall need not be vertical.
- Failure is two dimensional.
- The soil is cohesionless.
- Coulomb’s equation of shear strength is valid.

Coulomb made his derivation based on limit equilibrium approach.

Active Case

Figure 3.25 below shows the cross section of a retaining Wall. Equilibrium analysis of failure wedge ABC involves:

- Weight of wedge ABC (magnitude and direction known)
- P
_{a}(direction known, magnitude unknown) - R (direction known, magnitude unknown)

Hence, from the triangle of forces can be drawn and P_{a} can be determined.

Weight of wedge ABC

From Î” ABC,

Area of Î” ABC = ½ AD Ã— BC

BC/AB = {sin(Î± + Î²)/ sin(Ï´ – Î²)}

BC = AB {sin(Î± + Î²)/ sin(Ï´ – Î²)}

Again, AD = AB sin[180^{0} – (Î± + Ï´)] = AB sin(Î± + Ï´)

**Triangle of Forces for W, P _{a} and R**

From the sine rule,

Substituting the value of W from the equation into we get,

In order to get the maximum value of P_{a},

Where,

Note:

When, Î² = 0 (leveled backfilled), Î´ = 0 (no wall friction), then k_{a} = K_{a} = (1 – sinÉ¸)/(1 + sinÉ¸). The point of application of P_{a} is at a distance of H/3 above the base of the wall.