Tension cracks in cohesive backfill

In a purely cohesive soil ($\phi=0$) the active pressure at depth $z$ is $p_a = \gamma z - 2c$, which is negative (tension) near the surface.

• The tension zone extends to a depth $z_0 = \dfrac{2c}{\gamma}$ (zero pressure point). Soil cannot carry tension for long, so tension cracks form to this depth and the soil there gives no pressure on the wall.
• The pressure diagram is therefore a triangle from $z_0$ to the base, with maximum value $\gamma H - 2c$ at the bottom. Thus the active thrust is

$$P_a = \tfrac12(\gamma H-2c)(H-z_0) = \tfrac12\gamma H^2 - 2cH + \frac{2c^2}{\gamma}$$

and it acts at $(H-z_0)/3$ above the base.

• Neglecting the cracks (taking tension) would under-estimate the thrust. If the cracks fill with water, an extra hydrostatic thrust $\tfrac12\gamma_w z_0^2$ acts on the wall and increases pressure.