Tension crack and active pressure, layered clay

Data: $H = 6$ m; top 3 m: $\gamma_1 = 18$, $c_1 = 18$; bottom 3 m: $\gamma_2 = 20$, $c_2 = 25$. Saturated clay is taken as $\phi_u = 0$, so $K_a = 1$ and $p_a = \gamma z - 2c$.

Depth of tension crack (where $p_a = 0$ in the top layer):

$$z_0 = \frac{2c_1}{\gamma_1} = \frac{2\times18}{18} = 2.0\ \text{m}$$

Pressures (kN/m$^2$)

• $z=0$: $-36$ (tension, ignored); $z = 2$ m: $0$.
• $z = 3$ m (top layer): $18\times3 - 36 = 18.0$.
• $z = 3$ m (bottom layer): $54 - 2\times25 = 4.0$.
• $z = 6$ m: $54 + 20\times3 - 50 = 64.0$.

 0 |
   | crack zone (2 m)
 2 |\
   | \
 3 |--18 / 4
   |      \
   |       \
 6 |--------64

Forces (per metre) and arm above the base

The trapezoid has parallel sides 4 (top) and 64 (at the base) and height 3 m, so its centroid is measured from the base side:

$$\bar y = \frac{3\,(2\times4+64)}{3\,(4+64)} = 1.059\ \text{m}$$ $$\bar y = 138.0/111.0 = 1.24\ \text{m above the base}$$

Answer: tension crack depth $= 2.0$ m; active thrust $P_a = 111$ kN/m (horizontal), acting 1.24 m above the base of the wall.