Stability of cantilever wall, q = 50 kPa

Assumptions (dimensions read from the scan): stem 4.00 m high and 0.30 m thick, base slab 0.45 m thick (total height 4.45 m), base width 2.80 m, heel 1.90 m so toe $= 2.80-1.90-0.30 = 0.60$ m; embedment 1.0 m (soil over the toe is 0.55 m high). Soil $\gamma = 18$ kN/m$^3$, $\phi = 40^\circ$, concrete 25 kN/m$^3$, base friction $\delta = 25^\circ$, surcharge 50 kPa. Rankine active pressure on a vertical plane through the heel.

1. Active thrust

$$K_a = \tan^2(45^\circ-20^\circ) = 0.2174$$

2. Vertical loads (about the toe)

3. Checks

• Overturning: $FS_{ot} = 497.21/165.13 = 3.01$ (>2, safe).
• Sliding: $FS_s = 299.24\tan25^\circ/87.13 = 1.60$ (>1.5, safe even without passive resistance; with the passive force on the 1 m embedment, $P_p = \tfrac12K_p\gamma D^2 = 41.4$ kN/m, it rises to 2.08).
• Base pressure: $\bar x = (497.21-165.13)/299.24 = 1.110$ m, $e = 1.4-1.110 = 0.290$ m $< B/6 = 0.467$ m, so the whole base is in compression.

$$q_{max} = 173.3\ \text{kN/m}^2,\qquad q_{min} = 40.4\ \text{kN/m}^2$$

$q_{max} < q_a = 600$ kN/m$^2$. Safe.

Answer: the wall is stable. $FS_{ot} = 3.01$, $FS_{sliding} = 1.60$, $q_{max} = 173.3$ kN/m$^2$ $<600$ kN/m$^2$, $e < B/6$.