Foundations & Retaining Walls

8% of exam

Bearing capacity and settlement of spread footings, combined footings and mats, pile and drilled-shaft capacity, and retaining-wall stability against sliding, overturning, and bearing.

5 concepts

Exam areas

E. Shallow foundations

Shallow Foundation Bearing Capacity & Settlement

The general bearing-capacity equation with bearing-capacity, shape, depth and groundwater factors; net vs. gross allowable pressure with a factor of safety; and elastic plus consolidation settlement.

Every spread footing has to satisfy two completely separate limit states, and the exam loves to see whether you keep them apart. The first is a STRENGTH check: will the soil shear and the footing punch through (bearing capacity)? The second is a SERVICEABILITY check: how much will the footing settle, and is that settlement tolerable? A footing can pass one and fail the other — a wide footing on dense sand has enormous bearing capacity but may still settle too much, while a narrow footing on soft clay may have ample factor of safety against settlement-controlled service load yet not enough against a shear failure. The NCEES PE Civil Reference Handbook gives you the bearing-capacity machinery in §3.4 Bearing Capacity, the elastic-settlement machinery in §3.5 Foundation Settlement (specifically the elastic method §3.5.2), and the consolidation machinery —

C_c

C_r

e_0

p_c

, and the NC/OC split — in §3.2 Consolidation; on exam day those sections, plus their factor tables, are where you live for this topic.

The general bearing-capacity equation

Ultimate bearing pressure is the sum of three contributions — cohesion, surcharge (the soil above the footing base), and the self-weight of the failure wedge below the base. Each is scaled by a dimensionless bearing-capacity factor that depends only on the friction angle

\phi

, read from the handbook's Bearing Capacity Factors table (

N_c

N_q

N_\gamma

). For a strip footing the bare three-term form applies; for square or rectangular footings each term carries a shape factor

s_c

s_q

s_\gamma

. Here

q = \gamma_a D_f

is the total surcharge at the base,

\gamma

is the unit weight of the soil beneath the base, and

B_f

is the footing width.

q_{ult} = c\,N_c\,s_c + q\,N_q\,s_q + \tfrac{1}{2}\,\gamma\,B_f\,N_\gamma\,s_\gamma

Reading the factors — and the groundwater correction

The factors grow steeply with

\phi

: at

\phi = 0

(saturated clay, undrained)

N_c = 5.14

N_q = 1

N_\gamma = 0

, so for a clay the equation collapses to

q_{ult} = 5.14\,c_u\,s_c + q

. At

\phi = 30^\circ

the factors jump to

N_c = 30.1

N_q = 18.4

N_\gamma = 22.4

. The AASHTO shape factors the handbook tabulates are

s_c = 1 + (B_f/L_f)(N_q/N_c)

s_\gamma = 1 - 0.4(B_f/L_f)

, and

s_q = 1 + (B_f/L_f)\tan\phi

. If the water table rises into the failure zone you must also apply the groundwater correction factors

C_{wq}

and

C_{w\gamma}

(and use buoyant

\gamma'

for any term below the table), because submergence roughly halves the effective unit weight and therefore the

N_\gamma

and surcharge contributions.

s_c = 1 + \frac{B_f}{L_f}\frac{N_q}{N_c}, \quad s_\gamma = 1 - 0.4\frac{B_f}{L_f}, \quad s_q = 1 + \frac{B_f}{L_f}\tan\phi

Gross, net, and allowable bearing

The factor of safety is applied to the NET bearing capacity, not the gross. The net ultimate

q_{net} = q_{ult} - q

removes the surcharge that was already there before the footing existed (the soil you excavated is replaced by the footing). Divide

q_{net}

by the factor of safety — usually

FS = 3

for bearing — to get the net allowable bearing pressure, the value you compare against the net applied service pressure (

P/A

minus the overburden that the column load replaces). The cardinal rule is to keep the two sides of the check on the SAME basis: a net allowable goes against a net applied, and a gross allowable goes against a gross applied. Mixing the bases is the classic error, and the two ways of mixing them fail in opposite directions. Comparing a GROSS allowable (

q_{ult}/FS

, the larger capacity) against a NET applied pressure (the smaller demand) under-designs — you require a bigger number to beat a smaller one, crediting yourself capacity you do not actually have. The reverse mismatch — a NET allowable (the smaller capacity) against a GROSS applied pressure (the larger demand) — is merely over-conservative: you require the smaller number to beat the larger one, which oversizes the footing rather than endangering it. Separately, applying

FS = 3

q_{ult}

instead of

q_{net}

is conservative but wrong.

q_{net} = q_{ult} - \gamma_a D_f, \qquad q_{all,net} = \frac{q_{net}}{FS}

Elastic (immediate) settlement

On sands and stiff overconsolidated clays the bulk of settlement happens essentially as the load is applied — it is elastic and immediate. The handbook's elastic method (§3.5.2) gives the surface settlement from the stress increase

\Delta p

at the base, the footing width

B_f

, the soil's Young's modulus

E_m

and Poisson's ratio

\nu

, and a shape-and-rigidity factor

C_d

tabulated by footing shape and the point of interest (center, corner, average). Picking

C_d

for the wrong point — corner versus center — is a classic slip; the center of a flexible footing settles roughly twice the corner.

\delta_v = C_d\,\Delta p\,B_f\,\frac{1 - \nu^2}{E_m}

Consolidation settlement of clay

Under a saturated clay, water must squeeze out before the soil compresses, so settlement is time-dependent and can take years. For a normally consolidated clay the primary consolidation settlement uses the compression index

C_c

, the initial void ratio

e_0

, the layer thickness

H_0

, the initial effective overburden

p_o

, and the final stress

p_f = p_o + \Delta p

. If the clay is overconsolidated and the final stress stays below the preconsolidation pressure

p_c

, use the much smaller recompression index

C_r

; if

\Delta p

pushes

p_f

past

p_c

, split the calculation into a recompression part (from

p_o

p_c

using

C_r

) and a virgin part (from

p_c

p_f

using

C_c

). Always compute

\Delta p

at mid-depth of the clay layer, not at the footing base.

S_c = \frac{C_c}{1 + e_0}\,H_0\,\log_{10}\!\frac{p_f}{p_o} \quad (\text{normally consolidated})

Exam strategy

First decide which limit state the question is testing — a bearing question gives you

\phi

c

\gamma

D_f

B

and asks for

q_{ult}

or an allowable load; a settlement question gives you

C_c

e_0

, stresses or

E

and

\nu

. For bearing, read

N_c

N_q

N_\gamma

from the table at your

\phi

(do not interpolate the wrong row), apply shape factors only for square/rectangular footings, and apply

FS

q_{net}

. Keep both sides of the bearing check on the same basis — net allowable against net applied — because a gross allowable against a net applied is the dangerous, under-designing mismatch. For settlement, keep elastic (sand, immediate) separate from consolidation (clay, time-dependent), evaluate

\Delta p

at the layer mid-depth, and watch whether the clay is NC or OC — using

C_c

on an overconsolidated clay overstates settlement by a factor of three or more. Units: keep pressures in ksf or psf consistently, and remember

\log_{10}

(not

\ln

) in the consolidation equation.

Key equations

General bearing capacity (strip)

q_{ult} = c\,N_c + q\,N_q + \tfrac{1}{2}\gamma B_f N_\gamma

Ultimate bearing pressure of a concentrically loaded strip footing. $q = \gamma_a D_f$ surcharge; $N_c, N_q, N_\gamma$ from the handbook table at $\phi$ .

Bearing capacity with shape factors

q_{ult} = c\,N_c s_c + q\,N_q s_q + \tfrac{1}{2}\gamma B_f N_\gamma s_\gamma

Square/rectangular footing form; multiply each term by its AASHTO shape factor.

Shape factors (AASHTO)

s_c = 1 + \tfrac{B_f}{L_f}\tfrac{N_q}{N_c},\; s_q = 1 + \tfrac{B_f}{L_f}\tan\phi,\; s_\gamma = 1 - 0.4\tfrac{B_f}{L_f}

$B_f/L_f = 1$ for a square footing, $\to 0$ for a strip. Per handbook §3.4.2.1.

Net allowable bearing

q_{all,net} = \dfrac{q_{ult} - \gamma_a D_f}{FS}

Apply $FS$ (typically 3) to the NET ultimate. Compare against net applied service pressure — same basis on both sides.

Clay (undrained) bearing

q_{ult} = 5.14\,c_u\,s_c + \gamma_a D_f

$\phi = 0$ case: $N_c = 5.14$ , $N_q = 1$ , $N_\gamma = 0$ . $c_u$ = undrained shear strength.

Allowable column load

Q_{all} = q_{all,net}\,A = q_{all,net}\,B_f L_f

Net allowable pressure times footing plan area; the service load the footing can carry.

Elastic settlement

\delta_v = C_d\,\Delta p\,B_f\,\dfrac{1 - \nu^2}{E_m}

Immediate settlement (sand, stiff clay). $C_d$ from the shape/rigidity table; $E_m$ = soil modulus, $\nu$ = Poisson's ratio.

Consolidation settlement (NC)

S_c = \dfrac{C_c}{1 + e_0}H_0\log_{10}\dfrac{p_f}{p_o}

Normally consolidated clay. $C_c$ compression index, $e_0$ initial void ratio, $H_0$ layer thickness, $p_f = p_o + \Delta p$ .

Consolidation settlement (OC, below pc)

S_c = \dfrac{C_r}{1 + e_0}H_0\log_{10}\dfrac{p_f}{p_o}

Overconsolidated clay with $p_f \le p_c$ . Uses the recompression index $C_r \ll C_c$ .

Consolidation settlement (OC, crossing pc)

S_c = \dfrac{C_r}{1+e_0}H_0\log\dfrac{p_c}{p_o} + \dfrac{C_c}{1+e_0}H_0\log\dfrac{p_f}{p_c}

When $\Delta p$ pushes the stress past the preconsolidation pressure $p_c$ ; recompression then virgin.

Stress increase (2:1 method)

\Delta p = \dfrac{P}{(B_f + z)(L_f + z)}

Approximate vertical stress at depth $z$ below the base; evaluate at clay mid-depth for settlement.

Worked examples

Net allowable load on a square footing in sand

Problem. A $7\,\text{ft}\times7\,\text{ft}$ square footing bears at $D_f = 5\,\text{ft}$ in a cohesionless sand with $\phi = 32^\circ$ , $c = 0$ , and total unit weight $\gamma = 120\,\text{pcf}$ . The water table is deep. Using $N_q = 23.2$ , $N_\gamma = 30.2$ and a factor of safety of 3, find the net allowable bearing pressure and the allowable column service load.

Solution. Shape factors (square,

B_f/L_f = 1

s_q = 1 + \tan 32^\circ = 1 + 0.625 = 1.625

;

s_\gamma = 1 - 0.4(1) = 0.600

. Surcharge:

q = \gamma_a D_f = 120(5) = 600\,\text{psf}

. Ultimate (the

c

term drops out):

q_{ult} = q N_q s_q + \tfrac{1}{2}\gamma B_f N_\gamma s_\gamma = 600(23.2)(1.625) + 0.5(120)(7)(30.2)(0.600)

= 22{,}620 + 7{,}610 = 30{,}230\,\text{psf} = 30.2\,\text{ksf}

. Net:

q_{net} = 30{,}230 - 600 = 29{,}630\,\text{psf} = 29.6\,\text{ksf}

. Net allowable:

q_{all,net} = 29{,}630/3 = 9{,}880\,\text{psf} = 9.88\,\text{ksf}

. Allowable load:

Q_{all} = 9.88(7\times7) = 484\,\text{kips}

. **Answer:

q_{all,net} = 9.88\,\text{ksf}

Q_{all} \approx 484\,\text{kips}

.** Sanity check: the

N_\gamma

term (

7.6\,\text{ksf}

) is much smaller than the surcharge term — expected for a shallow, wide footing; and

FS

acts on the net, leaving a healthy allowable.

q_{ult} = 600(23.2)(1.625) + \tfrac{1}{2}(120)(7)(30.2)(0.600) = 30.2\ \text{ksf}

Primary consolidation settlement of a clay layer

Problem. A $10\,\text{ft}$ thick normally consolidated clay layer has $C_c = 0.30$ and $e_0 = 0.90$ . At the layer mid-depth the initial effective overburden is $p_o = 2.0\,\text{ksf}$ , and a new footing increases the stress at mid-depth by $\Delta p = 1.5\,\text{ksf}$ . Compute the primary consolidation settlement.

Solution. Final stress:

p_f = p_o + \Delta p = 2.0 + 1.5 = 3.5\,\text{ksf}

. Layer thickness in inches:

H_0 = 10\,\text{ft} = 120\,\text{in}

. Settlement:

S_c = \dfrac{C_c}{1+e_0}H_0\log_{10}\dfrac{p_f}{p_o} = \dfrac{0.30}{1.90}(120)\log_{10}\dfrac{3.5}{2.0}

\log_{10}(1.75) = 0.2430

, so

S_c = 0.1579(120)(0.2430) = 4.60\,\text{in}

. **Answer:

S_c = 4.60\,\text{in}

.** Sanity check: the strain is

S_c/H_0 = 4.60/120 = 3.8\%

— a plausible primary-consolidation strain for a soft NC clay under a

75\%

stress increase. Had the clay been overconsolidated with

C_r \approx 0.05

, the settlement would have been roughly

C_r/C_c = 1/6

of this, about

0.77\,\text{in}

S_c = \frac{0.30}{1.90}(120)\log_{10}\frac{3.5}{2.0} = 4.60\ \text{in}

Common pitfalls

•Applying the factor of safety to the GROSS ultimate $q_{ult}$ instead of the net $q_{net} = q_{ult} - \gamma_a D_f$ . The surcharge was already there before the footing; subtract it first, then divide by $FS$ .
•Comparing a GROSS allowable ( $q_{ult}/FS$ ) against a NET applied pressure — this under-designs, because you credit yourself the larger capacity while charging only the smaller demand. Keep both sides on the same basis: net allowable vs net applied, or gross allowable vs gross applied.
•Forgetting the shape factors on a square or rectangular footing — or applying them to a strip footing where $B_f/L_f \to 0$ makes them unity. Strip = bare three-term form; square = each term times its $s$ factor.
•Reading the wrong $N$ row: $N_c, N_q, N_\gamma$ are tabulated by $\phi$ in degrees. At $\phi = 0$ , $N_c = 5.14$ and $N_\gamma = 0$ — the $\gamma$ term vanishes for undrained clay.
•Using $\ln$ instead of $\log_{10}$ in the consolidation equation. The base-10 log is required; $\ln$ overstates settlement by a factor of $2.303$ .
•Using the compression index $C_c$ on an overconsolidated clay whose final stress stays below $p_c$ . Use the much smaller recompression index $C_r$ , or split the calculation if $p_f$ crosses $p_c$ .
•Evaluating $\Delta p$ at the footing base rather than at the clay-layer mid-depth, which overstates the stress and the settlement; stress spreads and decays with depth.

References

NCEES PE Civil Reference Handbook — §3.4 Bearing Capacity (general equation, factor and shape-factor tables)
NCEES PE Civil Reference Handbook — §3.5.2 Foundation Settlement (elastic method)
NCEES PE Civil Reference Handbook — §3.2 Consolidation (compression/recompression indices, NC/OC settlement)
FHWA-NHI-06-089, Soils and Foundations, Vol. II — source of the bearing-capacity and settlement relations the handbook reproduces

Combined Footings, Mats & Eccentric Loads

Soil-pressure distribution under an eccentric or combined footing, the e ≤ B/6 kern keeping the base in compression, uniform vs. trapezoidal vs. reduced-area pressure, and mat foundations.

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F. Deep foundations

Deep Foundations: Pile & Drilled-Shaft Capacity

Axial capacity as skin friction plus end bearing, allowable capacity with a factor of safety, pile-group efficiency and block failure, and the downdrag (negative skin friction) penalty.

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G. Retaining walls

Retaining-Wall Stability

Factors of safety against sliding, overturning, and bearing for a gravity or cantilever wall, the middle-third resultant location, and the active earth-pressure driving force.

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Cantilever Retaining-Wall Design

Proportioning the stem, heel, and toe; designing the stem and base slab for earth-pressure moments and shears per ACI 318-14; and detailing the flexural reinforcement.

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