16-Civ-B3 Geotechnical Design · December 2015

Question 6 of 9: Cantilever Retaining Wall — Factors of Safety against Overturning and Sliding

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. Professional Engineers Ontario / Engineers Canada National Examinations, December 2015 — 98-Civ-B3 Geotechnical Design. Three hours, OPEN BOOK, any non-communicating calculator. Section A carries five discussion questions of 7 marks each (answer any four); Section B carries four design questions of 24 marks each (answer any three); the examinable total is 4 × 7 + 3 × 24 = 100 marks. All nine questions are worked below, because the set is a study resource rather than a timed attempt.

Reference texts (98-Civ-B3 / 16-Civ-B3 Geotechnical Design).

B. M. Das, Principles of Foundation Engineering, 9th ed. — subsurface exploration (Ch. 2), bearing capacity (Ch. 3), settlement of shallow foundations (Ch. 5), stress increase in a soil mass (Ch. 6), retaining walls (Ch. 8), pile foundations (Ch. 11).
B. M. Das, Principles of Geotechnical Engineering, 9th ed. — consolidation (Ch. 11), shear strength (Ch. 12), lateral earth pressure (Ch. 13), slope stability (Ch. 15).
Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM), 4th ed. — the governing Canadian practice document for site investigation, in-situ testing, bearing resistance, deep foundations, earth-retaining structures and expansive soils.
R. F. Craig, Craig's Soil Mechanics, 9th ed. — effective stress, earth pressure theory and slope stability.
D. P. Coduto, Foundation Design: Principles and Practices, 3rd ed. — in-situ testing correlations and settlement of shallow foundations.
M. J. Tomlinson & J. Woodward, Pile Design and Construction Practice, 6th ed. — shaft adhesion in clay, bored and augered piles, load testing.

Sources of design charts and assumed values (page-1 Note 6). Note 6 of this paper requires the candidate to identify the source of every design chart used and of every value assumed in the absence of data. They are named where used and collected here:

Q6 — Rankine active coefficient for a sloping backfill, Das, Principles of Foundation Engineering, Eq. (8.5); base friction and adhesion mobilisation factors $k_1 = k_2 = \tfrac{2}{3}$ after Das §8.5; Rankine passive coefficient $K_p = \tan^2(45^{\circ} + \phi'_2/2)$. Assumed: stem height $H = 10.0$ m (the exam omits it — see the callout in Q6); reinforced concrete $\gamma_c = 24$ kN/m$^3$ (CFEM §4; CSA A23.3 normal-density concrete); the backfill is fully drained so no water force acts.
Q7 — overburden correction $C_N$ after Liao & Whitman (1986); $\phi'$ from $(N_1)_{60}$ after Peck, Hanson & Thornburn (1974) as fitted by Wolff (1989), cross-checked against Hatanaka & Uchida (1996); bearing capacity factors from Das Table 3.3 (Prandtl–Reissner $N_q$, Vesic $N_{\gamma} = 2(N_q+1)\tan\phi'$), shape factors after De Beer (1970) and depth factors after Hansen (1970), Das Table 3.4; settlement from Meyerhof's (1965) SPT expression, Das Eq. (5.42), cross-checked by Schmertmann's strain-influence method with $E_s = 500(N_{60}+15)$ kPa. Assumed: founding depth $D_f = 2.0$ m; the sand is uniform to at least $2B$ below the base; tolerable settlement 25 mm.
Q8 — compression index from Terzaghi & Peck (1967), $C_c = 0.009(LL-10)$; stress increase by the 2:1 method (Das §6.2) with a Boussinesq rectangular-area cross-check (Das Table 6.6); Simpson weighting of $\Delta\sigma'$ prescribed on the exam paper itself. Assumed: $\gamma_w = 9.81$ kN/m$^3$; the clay is saturated so $e_0 = wG_s$; the sand layers are incompressible relative to the clay.
Q9 — undrained ($\phi_u = 0$) mass procedure, Das, Principles of Geotechnical Engineering, §15.5; drained comparison by the ordinary method of slices and Bishop's simplified method, Das §15.11–15.12, and by the infinite-slope criterion, Das Eq. (15.10). Assumed: no external water force and no seismic loading; the sliding mass is homogeneous.

Section A — discussion questions (7 marks each; answer any four)

Section B — design questions (24 marks each; answer any three)

Question 6: Cantilever Retaining Wall — Factors of Safety against Overturning and Sliding (24 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Check — the stem height H is not given by the exam. The question lists $x_1$ to $x_5$, $D$ and $\alpha$, but Figure 1 labels the stem height only as $H$ and no value appears anywhere in the paper. Page-1 Note 7 instructs the candidate to exercise sound engineering judgment where information is missing, and Note 1 to state the assumption in the answer book, so this solution adopts $H = 10.0$ m and states why. With $x_5 = 1.2$ m the total wall height is $H' = 11.2$ m, and the base width $B = 0.75 + 1.5 + 4.0 = 6.25$ m then gives $B/H' = 0.56$, squarely inside the 0.5 to 0.7 band of the standard cantilever-wall proportioning rules (Das, Principles of Foundation Engineering, Fig. 8.2); the base thickness ratio $x_5/H' = 0.107$ is likewise in the usual $H'/12$ to $H'/10$ range. Any other height in that band is equally defensible, so a sensitivity study for $H = 8$ m and $H = 12$ m is reported after the main calculation and the conclusions are shown to be unchanged. Also assumed: reinforced concrete $\gamma_c = 24$ kN/m$^3$; the backfill is drained by weepholes and a granular blanket so no water force acts; base friction and adhesion mobilised at two-thirds of $\phi'_2$ and $c'_2$.

Given.

Quantity	Symbol	Value
Stem thickness, top / base	$x_1$ / $x_3$	0.5 m / 1.5 m
Toe projection	$x_2$	0.75 m
Heel projection	$x_4$	4.0 m
Base slab thickness	$x_5$	1.2 m
Depth to underside of base in front	$D$	2.0 m
Backfill surface slope	$\alpha$	$10^{\circ}$
Stem height (assumed, see callout)	$H$	10.0 m
Backfill unit weight and friction angle	$\gamma_1,\ \phi'_1$	16.8 kN/m$^3$, $34^{\circ}$ ($c_1 = 0$)
Foundation soil	$\gamma_2,\ \phi'_2,\ c'_2$	17.6 kN/m$^3$, $30^{\circ}$, 10 kN/m$^2$
Concrete unit weight (assumed)	$\gamma_c$	24 kN/m$^3$

Find. The factor of safety of the wall against overturning about the toe and against sliding along the base, together with a check that the resultant remains within the middle third of the base.

[Figure not reproduced: Figure Q6 — cantilever retaining wall, redrawn from the exam's Figure 1. The active thrust is taken on the vertical Rankine plane rising from the back of the heel, so the soil wedge between that plane and the stem is counted as part of the resisting weight. See the official exam paper.]

Approach. Take the active thrust by Rankine theory on the vertical plane through the back of the heel (valid because the stem's back face is vertical and the retained wedge is bounded by that plane), then take moments about the toe of all weights and of the thrust to obtain the overturning factor of safety, and finally compare the mobilised base friction plus adhesion with the horizontal component of the thrust for sliding.

Establish the height of the Rankine plane and the active earth-pressure coefficient. The virtual back rises from the rear of the heel to the sloping backfill surface, so its height exceeds the wall height by the rise of the backfill across the heel: $$H' = x_5 + H + x_4\tan\alpha = 1.2 + 10.0 + 4.0\tan 10^{\circ} = 11.905\ \text{m}$$ For a backfill sloping at $\alpha$ the Rankine coefficient is $$K_a=\cos\alpha\,\frac{\cos\alpha-\sqrt{\cos^{2}\alpha-\cos^{2}\phi'_1}}{\cos\alpha+\sqrt{\cos^{2}\alpha-\cos^{2}\phi'_1}}$$ With $\alpha = 10^{\circ}$ and $\phi'_1 = 34^{\circ}$, $\cos\alpha = 0.98481$ and $\cos\phi'_1 = 0.82904$, so $\sqrt{0.96985-0.68730}=0.53155$ and $$\boxed{K_a = 0.98481\times\frac{0.98481-0.53155}{0.98481+0.53155}=0.2944}$$
Compute the active thrust and resolve it. The thrust on the virtual back acts parallel to the ground surface, that is at $\alpha$ to the horizontal: $$P_a=\tfrac{1}{2}\gamma_1 K_a H'^{2}=\tfrac{1}{2}(16.8)(0.2944)(11.905)^{2}=350.5\ \text{kN/m}$$ Resolving, $P_h = P_a\cos\alpha = 345.2$ kN/m and $P_v = P_a\sin\alpha = 60.9$ kN/m. The horizontal component drives both overturning and sliding; the vertical component acts downward on the virtual back at the heel and is therefore stabilising for both.
Take the overturning moment about the toe. The thrust acts at one third of the height of the pressure diagram, and only its horizontal component has a lever arm about the toe (the vertical component is counted with the weights): $$M_o=P_h\frac{H'}{3}=345.2\times\frac{11.905}{3}=345.2\times3.968=\boxed{1369.7\ \text{kN}\cdot\text{m/m}}$$

Assemble the resisting weights and their moments about the toe. The free body is the wall plus the soil column standing on the heel, bounded on the right by the virtual back. Taking the toe as the origin:

Component	Computation	Weight (kN/m)	Arm (m)	Moment (kN·m/m)
Stem, rectangular part	$0.5\times10.0\times24$	120.0	2.000	240.0
Stem, battered triangle	$\tfrac12(1.5-0.5)\times10.0\times24$	120.0	1.417	170.0
Base slab	$6.25\times1.2\times24$	180.0	3.125	562.5
Backfill over the heel	$4.0\times10.0\times16.8$	672.0	4.250	2856.0
Sloping backfill wedge	$\tfrac12(4.0)(4.0\tan10^{\circ})\times16.8$	23.7	4.917	116.5
Soil over the toe	$0.75\times(2.0-1.2)\times17.6$	10.6	0.375	4.0
$P_v$ on the virtual back	$P_a\sin10^{\circ}$	60.9	6.250	380.4
Totals		1187.1		4329.3

The soil standing on the toe contributes only 4 kN·m/m and could be neglected without changing the answer; it is retained here because it is genuinely present.

Factor of safety against overturning. Dividing the resisting moment by the overturning moment, $$FS_{(overturning)}=\frac{\sum M_R}{M_o}=\frac{4329.3}{1369.7}=\boxed{3.16}$$ This comfortably exceeds the usual requirement of 2.0 (CFEM and Das both cite 2 to 3 for overturning), so the wall is safe against rotation about the toe.
Assemble the resistance to sliding. Because the base is cast against the $\gamma_2, c'_2, \phi'_2$ soil rather than against a rock surface, the friction angle and adhesion mobilised at the interface are taken at two-thirds of the soil values, which is the conventional allowance for construction disturbance: $$\delta'=\tfrac{2}{3}\phi'_2=20^{\circ},\qquad c_a=\tfrac{2}{3}c'_2=6.67\ \text{kPa}$$ The frictional and adhesive components are then $$\begin{aligned} F_{friction}&=\left(\sum V\right)\tan\delta'=1187.1\times\tan 20^{\circ}=432.1\ \text{kN/m}\\ F_{adhesion}&=B\,c_a=6.25\times6.67=41.7\ \text{kN/m} \end{aligned}$$
Evaluate the passive resistance available in front of the toe. With $\phi'_2 = 30^{\circ}$ the Rankine passive coefficient is $K_p=\tan^{2}(45^{\circ}+\phi'_2/2)=\tan^{2}60^{\circ}=3.0$, and over the full embedded depth $D = 2.0$ m, $$P_p=\tfrac12 K_p\gamma_2 D^{2}+2c'_2\sqrt{K_p}\,D=\tfrac12(3.0)(17.6)(2.0)^{2}+2(10)(1.732)(2.0)=105.6+69.3=174.9\ \text{kN/m}$$
Factor of safety against sliding. Two answers must be quoted, because the passive wedge in front of a wall is frequently removed by services, scour or landscaping and many authorities require the check without it: $$\begin{aligned} FS_{(sliding)}\Big|_{\text{no }P_p}&=\frac{432.1+41.7}{345.2}=\boxed{1.37}\\ FS_{(sliding)}\Big|_{\text{with }P_p}&=\frac{432.1+41.7+174.9}{345.2}=\boxed{1.88} \end{aligned}$$ Neglecting the passive wedge the wall does not meet the usual requirement of 1.5; counting it, it does. The honest engineering statement is therefore that sliding governs this wall and that its adequacy depends on soil that is easily disturbed.
Check the position of the resultant on the base. The line of action of the resultant crosses the base at $\bar{x}=(\sum M_R-M_o)/\sum V=(4329.3-1369.7)/1187.1=2.493$ m from the toe, so the eccentricity is $$e=\frac{B}{2}-\bar{x}=3.125-2.493=0.632\ \text{m}\ <\ \frac{B}{6}=1.042\ \text{m}$$ The resultant lies inside the middle third, so the whole base remains in compression, and the contact pressures are $$q_{max,min}=\frac{\sum V}{B}\left(1\pm\frac{6e}{B}\right)=190.0\left(1\pm0.607\right) \Rightarrow q_{max}=305\ \text{kPa},\quad q_{min}=75\ \text{kPa}$$ A bearing-capacity check on the eccentrically loaded base would follow in a full design.

The wall is therefore secure against overturning and against bearing eccentricity, and marginal against sliding.

Result	Symbol	Value
Height of the Rankine virtual back	$H'$	11.91 m
Rankine active coefficient (sloping backfill)	$K_a$	0.2944
Active thrust on the virtual back	$P_a$	350.5 kN/m
Horizontal / vertical components	$P_h$ / $P_v$	345.2 / 60.9 kN/m
Overturning moment about the toe	$M_o$	1369.7 kN·m/m
Total vertical force	$\sum V$	1187.1 kN/m
Total resisting moment about the toe	$\sum M_R$	4329.3 kN·m/m
Factor of safety, overturning	$FS_{(o)}$	3.16 (≥ 2 required — OK)
Factor of safety, sliding (passive wedge ignored)	$FS_{(s)}$	1.37 (< 1.5 — not acceptable)
Factor of safety, sliding (passive wedge included)	$FS_{(s)}$	1.88 (≥ 1.5 — OK)
Eccentricity of the resultant	$e$	0.632 m < B/6 = 1.042 m — OK
Base contact pressures	$q_{max}$ / $q_{min}$	305 / 75 kPa

Sensitivity to the assumed stem height, and the design recommendation. Repeating the whole calculation for the two other plausible heights gives:

$H$ (m)	$H'$ on the virtual back (m)	$P_a$ (kN/m)	$FS$ overturning	$FS$ sliding, no $P_p$	$FS$ sliding, with $P_p$
8.0	9.91	242.6	4.51	1.68	2.41
10.0 (adopted)	11.91	350.5	3.16	1.37	1.88
12.0	13.91	478.1	2.35	1.16	1.54

Across the whole plausible range the conclusion is the same: overturning is never critical, and sliding is always the governing check, falling below 1.5 without the passive wedge for every height at or above 10 m. The design response is standard — add a shear key beneath the base to force the failure surface into undisturbed soil and mobilise the full $\phi'_2$ and $c'_2$ rather than two-thirds of them, and provide a properly drained backfill so that the assumption of no water force remains true. Recomputing the adopted case with the full soil parameters, $\sum V\tan30^{\circ}+Bc'_2=685.4+62.5=747.9$ kN/m, raises the sliding factor of safety without any passive wedge from 1.37 to 2.17.

Topic. External stability of a cantilever retaining wall with an inclined granular backfill.

Key relations. $K_a=\cos\alpha\dfrac{\cos\alpha-\sqrt{\cos^{2}\alpha-\cos^{2}\phi'}}{\cos\alpha+\sqrt{\cos^{2}\alpha-\cos^{2}\phi'}}$; $P_a=\tfrac12\gamma K_a H'^{2}$ acting at $H'/3$ and inclined at $\alpha$; $FS_{(o)}=\sum M_R/\sum M_o$; $FS_{(s)}=\left[\sum V\tan(k_1\phi'_2)+Bk_2c'_2+P_p\right]/P_h$; $e=B/2-(\sum M_R-\sum M_o)/\sum V$.

Why this works. A cantilever wall does not resist earth pressure by its own weight; it resists it by the weight of the soil it carries on its heel. Taking the active thrust on a vertical plane through the back of the heel is what makes that soil countable, because the plane defines a free body in which the soil column and the concrete act together. Rankine's theory is legitimate on that plane provided the plane is vertical and the soil between it and the stem moves with the wall, which is the case when the back face of the stem is vertical, as here. The vertical component of the thrust then falls at the outermost point of the base and does double duty, adding both to the resisting moment and to the normal force available for friction.

Common pitfall. Two errors recur. The first is applying Coulomb's coefficient on the battered front face or on the inclined back of the stem while still counting the whole soil column over the heel as resisting weight — the two choices belong to different free bodies and combining them double-counts. The second is forgetting that with a sloping backfill the thrust is inclined, so that $P_a$ rather than $P_h$ must be used with the $H'/3$ lever arm only for the horizontal part, while the vertical part is transferred to the weight table with a lever arm of the full base width. Dropping $P_v$ altogether here would reduce the overturning factor of safety from 3.16 to about 2.9 and the sliding factor from 1.37 to 1.31.

Source. Das, Principles of Foundation Engineering, 9th ed., §8.3–8.6 and Fig. 8.2 (proportioning) and Eq. (8.5) ($K_a$ for a sloping backfill); Das, Principles of Geotechnical Engineering, 9th ed., §13.6; CFEM 4th ed., §26 (earth retaining structures).

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