Trapezoidal masonry wall, overturning

Geometry and assumptions: top width 1 m, base 3 m, height 4 m, vertical back face against the soil and an inclined front face (the toe is the front bottom corner). The back is vertical and no wall friction is given, so Rankine's theory for a surcharge angle $\beta = 20^\circ$ is used. Passive resistance is ignored.

1. Active thrust (Rankine, sloping backfill)

$$K_a = \cos\beta\,\frac{\cos\beta-\sqrt{\cos^2\beta-\cos^2\phi}}{\cos\beta+\sqrt{\cos^2\beta-\cos^2\phi}} = 0.4142$$ $$P_a = \tfrac12 K_a\gamma H^2 = 0.5\times0.4142\times20\times4^2 = 66.27\ \text{kN/m}$$

It acts at $H/3 = 1.333$ m above the base, parallel to the backfill slope ($20^\circ$):

$$P_h = 62.28\ \text{kN/m},\qquad P_v = 22.67\ \text{kN/m}$$

2. Moments about the toe

$$M_o = 62.28\times1.333 = 83.03\ \text{kN m/m}$$

3. Base pressures

$$\bar x = \frac{436.00-83.03}{214.67} = 1.644\ \text{m},\qquad e = \frac{B}{2}-\bar x = 1.5-1.644 = -0.144\ \text{m},\qquad |e| = 0.144\ \text{m} < B/6 = 0.5\ \text{m}$$ $$q_{max} = 92.2\ \text{kN/m}^2\ (\text{heel}),\qquad q_{min} = 50.9\ \text{kN/m}^2\ (\text{toe})$$

4. Factors of safety

$$FS_{ot} = \frac{M_r}{M_o} = \frac{436.00}{83.03} = 5.25\ (>2,\ \text{safe})$$

Sliding (for information) with $\mu = 0.45$: $FS_s = 214.67\times0.45/62.28 = 1.55$.

Answer: $q_{max} = 92.2$ kN/m$^2$, $q_{min} = 50.9$ kN/m$^2$, $FS_{overturning} = 5.25$.