5.3 Foundations, Retaining Walls & Slope Stability

Bearing Capacity of Shallow Foundations

A foundation must satisfy two limits: strength (it must not punch through the soil) and serviceability (it must not settle excessively). Strength is governed by Terzaghi's bearing-capacity equation for a strip footing:

q_ult = c·Nc + q·Nq + 0.5·γ·B·Nγ

• c = cohesion, q = γ·D_f = effective surcharge from soil above footing base (D_f = depth), γ = soil unit weight, B = footing width.
• Nc, Nq, Nγ = bearing-capacity factors, functions of φ only (tabulated in the FE Reference Handbook). For φ = 0: Nc = 5.14, Nq = 1, Nγ = 0.

Shape factors adjust the strip result for square/circular footings. The allowable bearing capacity applies a factor of safety:

q_all = q_ult / FS, with FS = 2.5–3.0 typical. A net vs. gross distinction matters: net q_ult subtracts the overburden q already present before construction.

Shallow vs. Deep Foundations & Settlement

Shallow foundations (spread/strip footings, mats) transfer load near the surface when competent soil is shallow. Deep foundations (driven piles, drilled shafts) carry load to deeper strata when surface soils are weak or settlement-prone.

Pile capacity is the sum of end bearing plus skin friction:

Q_ult = Q_p + Q_s = q_p·A_p + Σ(f_s·A_s)

where q_p = unit point resistance, A_p = pile tip area, f_s = unit skin friction, A_s = shaft surface area.

Settlement has two parts: immediate (elastic) settlement in sands and time-dependent consolidation settlement in clays (Section 5.2). Footings on sand are usually settlement-controlled in serviceability, while bearing failure controls in soft clay. A trap: applying Terzaghi to a deep pile — bearing-capacity factors and the surcharge term differ for embedded foundations.

Lateral Earth Pressure & Retaining-Wall Stability

Rankine theory gives horizontal earth pressure for a smooth vertical wall with level backfill. The active state (wall moves away from soil) and passive state (wall pushed into soil) use:

The active pressure at depth z is σ_a = Ka·γ·z (a triangular distribution); the resultant P_a = ½·Ka·γ·H² acts at H/3 above the base.

Gravity/cantilever retaining walls are checked for three modes, each with its own factor of safety:

• Overturning FS = ΣM_resist/ΣM_overturn ≥ 1.5–2.0.
• Sliding FS = ΣF_resist/ΣF_driving ≥ 1.5.
• Bearing — base pressure ≤ q_all (no uplift; resultant in middle third).

Slope Stability, Compaction & a Worked Example

Slope stability is expressed as a factor of safety FS = resisting/driving forces (or moments). For an infinite slope of cohesionless soil, FS = tanφ/tanβ (β = slope angle). Rotational failures in clay use method-of-slices or stability charts; FS ≥ 1.5 is typical for permanent slopes.

Compaction densifies fill to raise strength and reduce settlement. The Proctor test (standard or modified) compacts soil at varying water contents to find the maximum dry unit weight γ_d,max at the optimum moisture content. Field quality is reported as relative compaction = (field γ_d / lab γ_d,max) × 100%, commonly specified ≥ 95%.

Worked earth-pressure example: A 12-ft cantilever wall retains dry sand with γ = 115 lb/ft³, φ = 30°.

• Ka = tan²(45° − 15°) = tan²(30°) = 0.333.
• P_a = ½·Ka·γ·H² = 0.5 × 0.333 × 115 × 12² = 2758 lb per ft of wall, acting at H/3 = 4 ft above the base.

Surcharge, Water & Practical Stability Checks

Real walls rarely see dry, level backfill only. Two additions appear constantly on the FE Civil exam:

• Uniform surcharge q on the backfill adds a rectangular pressure Ka·q over the full height; its resultant Ka·q·H acts at H/2 (not H/3 like the soil triangle).
• Water behind the wall adds full hydrostatic pressure γ_w·z that does not get reduced by Ka — water pushes equally in all directions. Effective backfill weight below the water table uses buoyant γ'. Adequate drainage (weep holes) is the standard fix to avoid this large force.

For the bearing check, the resultant of vertical loads should land within the middle third of the base so the entire footing stays in compression (no uplift at the heel). When it does, base pressures are q = (ΣV/B)(1 ± 6e/B), with eccentricity e.

Neglecting passive resistance at the toe is conservative; neglecting surcharge or water is unconservative and a frequent point-loser.

** A footing can pass bearing capacity yet fail serviceability through excessive or differential settlement, which cracks superstructures; tolerable angular distortion is often limited to about 1/500. Elastic settlement of footings on sand uses δ = q·B·(1−ν²)·I/Es (Es = soil modulus, ν = Poisson's ratio, I = influence factor). Separately, an overall (global) slope-stability failure can pass beneath a wall or footing along a deep slip surface even when the structural element itself is adequate — checked by limit-equilibrium slices.

Finally, compaction links back to all of this: a fill placed at optimum moisture and ≥95% relative compaction has the higher strength, lower compressibility, and lower permeability that the bearing, settlement, and earth-pressure equations all assume.