Active thrust, 6 m wall, sand over clay

Assumptions: smooth vertical wall, horizontal backfill, Rankine theory; sand above the water table is taken at the given unit weight; the clay is saturated, so the effective stress uses $\gamma' = \gamma_{sat} - \gamma_w = 19.00 - 9.81 = 9.19\ \text{kN/m}^3$.

Coefficients

$$\begin{aligned} K_{a1} &= \tan^2(45^\circ-30^\circ/2) = 0.3333 \\ K_{a2} &= \tan^2(45^\circ-30^\circ/2) = 0.3333,\quad 2c\sqrt{K_{a2}} = 2\times10\times0.5774 = 11.55\ \text{kN/m}^2 \end{aligned}$$

Pressures (kN/m$^2$)

• Sand, top: $0$.
• Sand, bottom ($z=3$ m): $p = K_{a1}\gamma H_1 = 0.3333\times17.5\times3 = 17.50$.
• Clay, top: $\sigma_v' = 17.5\times3 = 52.50$, so $p = 0.3333\times52.50 - 11.55 = 5.95$.
• Clay, bottom ($z=6$ m): $\sigma_v' = 52.50 + 9.19\times3 = 80.07$, so $p = 0.3333\times80.07 - 11.55 = 15.14$.
• Water pressure at the base: $u = 9.81\times3 = 29.43$ (zero at the water table).

 0 --|
     |\
sand | \
     |  \
 WT -|---17.50
     |  eff 5.95
clay |   \
     |    \ eff 15.14
 base|-----> + water 29.43

Forces per metre run and lever arms about the base

$$\bar y = \frac{\sum M}{\sum F} = \frac{189.72}{102.04} = 1.86\ \text{m above the base}$$

Answer: total thrust $P = 102.0$ kN/m (water included), acting horizontally at 1.86 m above the base of the wall.