Why is the soil (and surcharge) on the heel included in vertical loads?

It adds to the weight and stabilising moment about the toe

What is FS against overturning and how found?

757.14/250.47 = 3.02 (restoring/overturning moment about the toe)

With e = 0.263 m, where is q_max = 141.3 kN/m^2?

At the toe, since the resultant is on the toe side of centre

2074 Asoj · 10 marks

Stability of cantilever wall, 6 m, q = 35

Short answer

Maximum base pressure is 141.3 kN/m², minimum 53.4 kN/m², computed from vertical resultant eccentricity 0.263 m. Factor of safety against overturning is 3.02 and against sliding 1.68.

Determine the maximum and minimum pressure under the base of the cantilever retaining wall as shown in the figure below and also the factor of safety against sliding and overturning. The approximate shear strength parameters for the soil are $c = 0$ , $\phi = 41^\circ$ . The unit weight of soil and concrete are $16\ \text{kN/m}^3$ and $24\ \text{kN/m}^3$ respectively. The water table is below the base of the wall. Take $\delta = 27^\circ$ on the base of the wall. [Figure: cantilever wall 6 m high, stem top width 0.4 m, heel length 2 m, base slab thickness 0.5 m, total base width 3.5 m, backfill with surcharge $q = 35\ \text{kN/m}^2$ .]

Assumptions (figure values as given): total height 6.0 m (stem 5.5 m + slab 0.5 m), stem 0.4 m thick throughout, heel 2.0 m, base width 3.5 m, so the toe is $3.5-2.0-0.4 = 1.1$ m. Rankine active pressure acts on a vertical plane through the heel; the surcharge on the heel counts as a vertical load; passive resistance and soil over the toe are neglected.

1. Active thrust

K_a = \tan^2(45^\circ-20.5^\circ) = 0.2077

Part	Force (kN/m)	Arm above base (m)	Moment
Surcharge $K_a q H = 0.2077\times35\times6$	43.61	3.00	130.84
Soil $\tfrac12 K_a\gamma H^2$	59.81	2.00	119.63
Total	103.43		250.47

2. Vertical loads (about the toe)

Part	Force (kN/m)	Arm (m)	Moment
Stem 0.4 x 5.5 x 24	52.80	1.300	68.64
Base slab 3.5 x 0.5 x 24	42.00	1.750	73.50
Soil on heel 2 x 5.5 x 16	176.00	2.500	440.00
Surcharge on heel 35 x 2	70.00	2.500	175.00
Total	340.80		757.14

3. Base pressure

\bar x = \frac{757.14-250.47}{340.80} = 1.487\ \text{m},\qquad e = 1.75-1.487 = 0.263\ \text{m} < B/6 = 0.583\ \text{m}

q_{max,min} = \frac{340.80}{3.5}\left(1\pm\frac{6\times0.263}{3.5}\right)

q_{max} = 141.3\ \text{kN/m}^2\ (\text{toe}),\qquad q_{min} = 53.4\ \text{kN/m}^2\ (\text{heel})

4. Factors of safety

FS_{ot} = \frac{757.14}{250.47} = 3.02\ (>2)

FS_s = \frac{340.80\times\tan27^\circ}{103.43} = 1.68\ (>1.5)

Answer: $q_{max} = 141.3$ kN/m $^2$ , $q_{min} = 53.4$ kN/m $^2$ ; $FS_{overturning} = 3.02$ ; $FS_{sliding} = 1.68$ . Both are adequate.

✓ Answer checked by Claude Sonnet 5.5

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Stability of cantilever wall, 6 m, q = 35

1. Active thrust

2. Vertical loads (about the toe)

3. Base pressure

4. Factors of safety

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