Active pressure on wall, cohesive soil

For a purely cohesive soil, $\phi=0$, $K_a = 1$, so Rankine's active pressure at depth $z$ is

$$p_a = \gamma_t z - 2c$$

Pressure diagram

• At the top ($z=0$): $p_a = -2c$ (tension).
• Zero at depth $z_0 = 2c/\gamma_t$.
• At the base ($z=H$): $p_a = \gamma_t H - 2c$.

 z=0  |<-2c            (tension, ignored)
      |
 z0   |----+ 0         z0 = 2c/gt
      |     \
      |      \
 H    |-------+ gt*H - 2c

Soil cannot carry tension, so a tension crack forms to depth $z_0$ and the tensile part is ignored. The pressure then acts only on a triangle of height $(H - z_0)$ with base value $\gamma_t H - 2c = \gamma_t(H - z_0)$.

Magnitude of active force

$$\begin{aligned} P_a &= \tfrac12(\gamma_t H - 2c)(H - z_0) = \tfrac12\gamma_t (H - z_0)^2 \\ &= \tfrac12\gamma_t H^2 - 2cH + \frac{2c^2}{\gamma_t} \end{aligned}$$

Line of action is at the centroid of the triangle, one-third of its height above the base:

$$\bar{y} = \frac{H - z_0}{3} = \frac{H - 2c/\gamma_t}{3} \ \text{(from the base)}$$

If the tension zone were not neglected (no crack), $P_a = \tfrac12\gamma_t H^2 - 2cH$, but this under-estimates the thrust on the wall, so the crack is allowed for in design.