Question 3 of 9: Earth Pressure and the Relative Movement of the Retaining Structure
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. Professional Engineers Ontario /
Engineers Canada National Examinations, May 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.
B. M. Das, Principles of Foundation Engineering, 9th ed. — bearing
capacity (Ch. 3), stress increase in a soil mass (Ch. 6), retaining walls (Ch. 8), pile
foundations (Ch. 11), subsurface exploration (Ch. 2).
B. M. Das, Principles of Geotechnical Engineering, 9th ed. —
consolidation (Ch. 11), shear strength (Ch. 12), lateral earth pressure (Ch. 13).
Canadian Geotechnical Society, Canadian Foundation Engineering Manual
(CFEM), 4th ed. — the governing Canadian practice document for site investigation,
bearing resistance, deep foundations and earth-retaining structures.
R. F. Craig, Craig's Soil Mechanics, 9th ed. — earth pressure theory,
effective stress and slope stability.
D. P. Coduto, Foundation Design: Principles and Practices, 3rd ed. —
in-situ testing, settlement of shallow and deep foundations.
M. J. Tomlinson & J. Woodward, Pile Design and Construction Practice,
6th ed. — shaft adhesion in clay, pile-group behaviour, 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 — bearing capacity factors $N_c$, $N_q$, $N_{\gamma}$ from
Das, Principles of Foundation Engineering, 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. Assumed: unit weight of water
$\gamma_w = 9.81$ kN/m$^3$; general shear failure; the sand extends at least $2B$ below
the base.
Q7 — adhesion factor $\alpha = 1.0$ for soft clay
($c_u \le 50$ kPa) from NAVFAC DM-7.2 Fig. 1 and Tomlinson & Woodward Table 4.6;
$\lambda = 0.24$ at an embedded length of 12 m from Vijayvergiya & Focht (1972) as
tabulated by Das, Table 11.7; bearing factor $N_c^{*}=9$ for $L/D \ge 4$ (Skempton).
Assumed: pile spacing $s = 3d = 1.5$ m centre to centre, driven closed-end concrete
piles, clay $\gamma_{sat} = 17$ kN/m$^3$ with the water table at ground level.
Q8 — compression index from the Skempton correlation
$C_c = 0.009\,(LL-10)$, Das Principles of Geotechnical Engineering Eq. (11.42);
$2{:}1$ stress distribution, Das Eq. (6.31); Boussinesq rectangular influence factor
(Das Table 6.6) used as the cross-check.
Q9 — Coulomb active pressure coefficient, Das
Principles of Foundation Engineering Eq. (8.13) with the wall friction angle
prescribed by the question, $\delta = 0.6\phi' = 18^{\circ}$. Assumed: unit weight of
reinforced concrete $\gamma_c = 24$ kN/m$^3$ (CSA A23.3 nominal); passive resistance in
front of the toe neglected; the backfill surface is horizontal and carries no surcharge.
Question 3: Earth Pressure and the Relative Movement of the Retaining Structure (7 marks)
Earth pressure is not a load that the ground applies independently of the structure; it
is the reaction to the deformation that the structure imposes on the ground. A
soil element behind a wall starts in its at-rest condition, in which the horizontal
effective stress is $\sigma'_h = K_0\sigma'_v$ with $K_0 \approx 1-\sin\phi'$ for a
normally consolidated soil (about 0.5 for $\phi' = 30^{\circ}$). That state exists only
while there is no lateral strain. Once the wall moves, the soil is sheared, and the
horizontal stress migrates toward whichever failure state the movement is driving it to.
Mobilised lateral earth pressure coefficient against the relative
movement between wall and backfill. The active state needs a movement roughly an order of
magnitude smaller than the passive state.
The active side. If the wall yields away from the backfill, the
soil expands laterally, the horizontal stress falls, and the Mohr circle grows until it
touches the failure envelope. The horizontal stress can fall no further: it has reached the
active value $\sigma'_a = K_a\sigma'_v - 2c'\sqrt{K_a}$. The remarkable feature is how
little movement is needed — a rotation of about $0.001H$ at the top for a dense sand
and $0.004H$ for a loose sand, that is 10 to 40 mm on a 10 m wall. Any conventional
cantilever or gravity wall is flexible enough to deliver this, which is why design for the
active state is normal. A wall restrained against movement — a basement wall propped
by the floor slabs, a bridge abutment held by the deck, a rigid box culvert — never
develops the active state and must be designed for $K_0$, roughly 50 per cent more thrust.
The passive side. If the wall is pushed into the soil, the
element is compressed laterally, $\sigma'_h$ rises above $\sigma'_v$, and the ground
resists by mobilising a much larger failure wedge that has to be lifted. The limiting value
$\sigma'_p = K_p\sigma'_v + 2c'\sqrt{K_p}$ is very large ($K_p = 3.0$ against
$K_a = 0.33$ for $\phi' = 30^{\circ}$, a factor of nine), but it requires a movement one
to two orders of magnitude greater than the active case, of order $0.01H$ to $0.05H$ in
dense sand and up to $0.1H$ in loose sand.
An example of passive pressure being generated. Consider the anchor
block, or deadman, of a tied-back sheet pile quay wall. The dredged sheet pile wall pulls
on the tie rod; the tie rod pulls the anchor block toward the water; the block therefore
presses against the soil behind it, and it is precisely that soil which must supply
the resistance. The block is pushed into the ground, so the soil behind it is compressed and
mobilises passive pressure over its buried height, while the soil in front of it drops to
the active value. The net available anchor force per metre run is
$P_p - P_a = \tfrac{1}{2}\gamma H^2 (K_p - K_a)$. Because the block must move perhaps
$0.02H$ to develop full $K_p$, and because that movement is transmitted straight back to
the top of the sheet pile as extra deflection, designers routinely apply a factor of safety
of 1.5 to 2.0 to $K_p$ — not because the ultimate value is uncertain but because the
serviceability of the wall cannot tolerate the displacement needed to reach it. The same
mechanism embeds the toe of every cantilever sheet pile wall, resists sliding at the shear
key beneath a gravity wall, and takes the thermal expansion thrust of an integral bridge
abutment into its backfill.
