18-Env-A3 Geotechnical and Hydrogeological Engineering · December 2015
Question 6 of 6: Cantilever Retaining Wall — Active Force and Overturning Stability
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2015 — 04-Env-A3 / Geotechnical & Hydrogeological Engineering. 3 hours duration; open book exam, any non-communicating calculator permitted. The first five questions as they appear in the answer book are marked (20 marks each, 100 marks total); all six are solved below for completeness.
Reference texts. Braja M. Das, Principles of Geotechnical Engineering (9th ed.) — weight–volume relations, compaction, lateral earth pressure and retaining-wall stability chapters; Craig & Knappett, Craig's Soil Mechanics (8th ed.) — seepage/flow-net theory and Rankine earth-pressure cross-reference; Freeze & Cherry, Groundwater (1979) — Darcy's law, anisotropic layered media and the Dupuit–Thiem equation for radial flow to a well.
Question 6: Cantilever Retaining Wall — Active Force and Overturning Stability (20 marks)
Check: reading Figure 4, the black footing bar terminates flush with the stem's own back face — this is an L-shaped wall with ALL footing width on the toe (front) side and ZERO heel behind the stem. The retained backfill therefore bears directly on the back face of both the stem and the footing, over the full height $H=5.0+0.5=5.5$ m, and none of the backfill's own weight sits on the footing to help resist overturning.
Given.
Given data
Quantity
Symbol
Value
Stem height / thickness
$h_{\text{stem}},t_{\text{stem}}$
5.0 m, 0.5 m
Footing thickness
$t_{\text{foot}}$
0.5 m
Toe width (point B to stem face)
—
1.5 m
Unit weight, reinforced concrete
$\gamma_c$
22 kN/m³
Sand friction angle / cohesion
$\phi,c$
30°, 0
Saturated unit weight of backfill
$\gamma_{sat}$
22 kN/m³
Find. (a) active force on the wall; (b) factor of safety against overturning about the toe, point B.
L-shaped cantilever wall: 2.0 m total footing width, zero heel; overturning is checked about the toe, point B.
Approach. Compute the single-layer submerged Rankine active force (buoyant $\gamma'$) plus the separate hydrostatic water thrust over the full retained height $H=5.5$ m, both acting at $H/3$ above the base; compute the resisting moment from the concrete self-weight alone about toe B (there is no heel soil to add); take the ratio for FS.
Rankine coefficient and effective active force. With $\gamma'=\gamma_{sat}-\gamma_w=22-9.81=12.19\ \text{kN/m}^3$ and $K_a=\dfrac{1-\sin30^\circ}{1+\sin30^\circ}=\dfrac13$:
$$P_{a,\text{eff}}=\tfrac12K_a\gamma'H^2=\tfrac12\left(\tfrac13\right)(12.19)(5.5)^2=\boxed{61.5\ \text{kN/m}}.$$
Hydrostatic water thrust (water table at grade, full 5.5 m height):
$$P_w=\tfrac12\gamma_wH^2=\tfrac12(9.81)(5.5)^2=\boxed{148.4\ \text{kN/m}}.$$
(a) Total active force, acting at $H/3=1.833$ m above the base (both components are triangles of the same height):
$$P_a=P_{a,\text{eff}}+P_w=61.5+148.4=\boxed{209.8\ \text{kN/m}}.$$
Overturning moment about toe B.
$$M_o=P_a\times\frac{H}{3}=209.8\times1.833=\boxed{384.7\ \text{kN}\cdot\text{m/m}}.$$
Resisting moment from the concrete self-weight only (stem + footing; there is no heel soil):
$$W_{\text{stem}}=0.5(5.0)(22)=55.0\ \text{kN/m at }x=1.75\ \text{m from B},$$
$$W_{\text{foot}}=2.0(0.5)(22)=22.0\ \text{kN/m at }x=1.00\ \text{m from B},$$
$$M_r=55.0(1.75)+22.0(1.00)=\boxed{118.3\ \text{kN}\cdot\text{m/m}}.$$
(b) Factor of safety against overturning.
$$FS_{\text{OT}}=\frac{M_r}{M_o}=\frac{118.3}{384.7}=\boxed{0.31}.$$
Check: $FS_{\text{OT}}=0.31\ll1.5$ — the wall as detailed is NOT stable against overturning. With zero heel, none of the retained soil's weight contributes to the resisting moment, leaving only the concrete self-weight to resist a 5.5 m column of saturated backfill plus full hydrostatic thrust. A real design would need a heel extension under the backfill (so the soil's own weight adds a large stabilizing moment) or a substantially wider/deeper footing; this is reported as computed rather than adjusted, since recognizing the inadequacy is the point of the check.
