Earth pressure

Behind the wall, the backfill pushes with Rankine’s active pressure on the virtual back, and the water and the surcharge add to it; in front, if the wall is embedded, the soil resists with Rankine’s passive pressure.

Active thrust

The active thrust is Rankine’s, on the virtual back: the vertical plane through the end of the heel, from the base to the backfill surface. Its height is

$$H' = H + b\tan\beta$$

with $b$ the horizontal distance from the back of the crest to the virtual back. Each backfill stratum has its own coefficient, defined for $\beta \le \phi$ (a steeper backfill is refused: it would slide on its own) and equal to $\tan^2(45^\circ - \phi/2)$ on level ground:

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

At a depth $z$ below the top of the virtual back, the pressure is

$$\sigma_a = (\sigma'_v + q)\,K_a - 2c\sqrt{K_a}$$

with $\sigma'_v$ the vertical effective stress ($\gamma$ above the water table, $\gamma'$ below), so a surcharge adds $K_a q$ all the way down. The diagram breaks at each stratum boundary and at the water table. The thrust $P_a$ is parallel to the backfill surface: $P_a\cos\beta$ pushes the wall, and $P_a\sin\beta$ bears down on the virtual back, at $x = B$.

Tension crack

Soil can’t pull on the wall, so the design diagram is zero wherever $\sigma_a$ is negative. In a cohesive backfill a crack opens from the top of the virtual back down to $z_c$, where the pressure turns compressive; if it ends in a dry top stratum,

$$z_c = \frac{1}{\gamma}\left(\frac{2c}{\sqrt{K_a}} - q\right)$$

The crack is taken full of rain water down to its bottom, or to the water table if higher; over that depth $d$ the water pushes $\tfrac{1}{2}\gamma_w d^2$, at $2d/3$ below the top of the virtual back.

The report also gives the textbook thrust, before the crack opens: the whole diagram, its tension included, and no water in the crack. It’s a reference; the checks use the design thrust.

Water

Below the water table behind the wall, the soil pushes with $\gamma'$ and the water on its own, with its hydrostatic thrust at a third of its height $h_w$ above the base.

Passive resistance

If the wall is embedded, the soil in front of the toe resists with Rankine’s passive pressure for level ground, down to the base:

$$K_p = \tan^2\!\left(45^\circ + \frac{\phi}{2}\right), \qquad \sigma_p = \sigma'_v K_p + 2c\sqrt{K_p}$$

The thrust of the water in front, $\tfrac{1}{2}\gamma_w h^2$ with $h$ its height above the base, adds to it.