Failure Modes in Bearing Capacity

A foundation fails in bearing when the applied load exceeds the soil's ability to resist without shear failure below the footing. Three modes are recognized:

  • General shear failure — a well-defined failure surface develops to the ground surface; heaving occurs on both sides. Typical of dense sands and stiff clays (relative density Dr > 67%).
  • Local shear failure — failure surface develops only partially; some heaving at the surface. Occurs in medium-dense sands and medium clays.
  • Punching shear failure — foundation punches through without a clear failure plane; occurs in loose sands and soft clays or under high embedment ratios.

Terzaghi's original equations apply to general shear; for local shear, Terzaghi suggested using 2c/3 and tan⁻¹(2tanφ/3) in place of c and φ.

Terzaghi's Bearing Capacity Equation

For a strip footing (B/L → 0) with general shear failure:

q_ult = c·Nc + q·Nq + 0.5·γ·B·Nγ

where c is cohesion (kPa), q = γ·Df is the overburden pressure at foundation depth Df, γ is unit weight of soil (kN/m³), B is footing width (m), and Nc, Nq, Nγ are dimensionless bearing capacity factors that depend solely on the friction angle φ':

  • Nq = e^(π tanφ) · tan²(45 + φ/2)
  • Nc = (Nq − 1) · cot φ (for φ > 0); Nc = 5.14 for φ = 0 (Skempton)
  • Nγ = 2(Nq + 1)tanφ (Meyerhof form) or 1.5(Nq − 1)tanφ (Hansen form)

Common tabulated values: at φ = 30°, Nc ≈ 30.1, Nq ≈ 18.4, Nγ ≈ 22.4. At φ = 35°, Nc ≈ 46.1, Nq ≈ 33.3, Nγ ≈ 48.0.

Meyerhof's Generalized Equation

Meyerhof extended Terzaghi to handle rectangular footings, inclined loads, and embedment above the footing level by introducing shape (s), depth (d), and inclination (i) correction factors:

q_ult = c·Nc·sc·dc·ic + q·Nq·sq·dq·iq + 0.5·γ·B·Nγ·sγ·dγ·iγ

Key correction factors (Meyerhof, also used in AASHTO LRFD Bridge Design Specifications):

  • Shape: sc = 1 + 0.2·(B/L)·Kp; sγ = 1 − 0.1·(B/L)·Kp (φ > 10°); where Kp = tan²(45 + φ/2)
  • Depth: dc = 1 + 0.2·(Df/B)·√Kp; dq = dγ = 1 + 0.1·(Df/B)·√Kp (φ > 10°)
  • Inclination: ic = iq = (1 − α/90°)²; iγ = (1 − α/φ)² where α is load inclination angle

Hansen's and Vesic's equations use slightly different factor formulations and are also accepted; AASHTO LRFD Article 10.6 specifically references them for bridge foundations.

Net vs Gross Bearing Pressure

The gross ultimate bearing capacity q_ult is the total pressure the soil can support. The net ultimate capacity q_net = q_ult − q subtracts the in-situ overburden stress that was already in equilibrium before construction. For ASD design, the allowable bearing pressure is:

q_allow = q_net / FS + q

Typical ASD factors of safety for bearing capacity range from FS = 2.5 to 3.0 for permanent loads. A FS of 2.0 may be acceptable when load is well-known and soil is well-characterized; FS = 3.0 is used when uncertainty is high.

LRFD Approach (IBC / AASHTO)

Under LRFD, the factored load effect must not exceed the factored resistance:

ΣηiγiQi ≤ φ·Rn

For geotechnical bearing resistance, AASHTO LRFD Article 10.5.5.2 specifies resistance factors φ_b ranging from 0.45 to 0.50 for spread footings on sand (SPT-based methods) and 0.35 to 0.40 for clay (Su-based). IBC Table 1806.2 lists presumptive allowable bearing pressures by soil description for cases where site-specific analysis is not performed — values range from 1,500 psf (75 kPa) for clay to 12,000 psf (575 kPa) for crystalline bedrock.

Water Table Effects

The water table directly affects the effective unit weight used in bearing capacity calculations. Three cases govern:

  • Water table at or above the base of the footing (depth Dw ≤ Df): use γ' = γsat − γw ≈ 9–10 kN/m³ for both the q term and the Nγ term.
  • Water table within B below the footing base (Df < Dw ≤ Df + B): use a linearly interpolated average unit weight for the Nγ term.
  • Water table deeper than B below the footing: no reduction needed for bearing capacity (though settlement analysis may still require buoyancy correction).

Worked Example

A 2.0 m × 2.0 m square footing is founded at Df = 1.2 m in a dense sand with φ' = 33°, c' = 0, γ = 18 kN/m³. Water table is deep. Determine q_allow using FS = 3.0.

Step 1 — Bearing capacity factors at φ = 33°: Nq = 26.3, Nc = 38.6 (not needed for c = 0), Nγ = 32.2.

Step 2 — Shape factors (B/L = 1.0, Kp = tan²(61.5°) = 3.33): sγ = 1 − 0.1(1.0)(3.33) = 0.667; sq = 1 + 0.2(1.0)(3.33) = 1.667.

Step 3 — Depth factors (Df/B = 0.6, √Kp = 1.826): dq = dγ = 1 + 0.1(0.6)(1.826) = 1.110.

Step 4 — q_ult = q·Nq·sq·dq + 0.5·γ·B·Nγ·sγ·dγ = (18 × 1.2)(26.3)(1.667)(1.110) + 0.5(18)(2.0)(32.2)(0.667)(1.110) = 21.6 × 48.72 + 18 × 23.82 = 1,052 + 429 = 1,481 kPa.

Step 5 — q_net = 1,481 − 21.6 = 1,459 kPa; q_allow = 1,459/3.0 + 21.6 = 507 kPa (≈ 10,600 psf).

This is well above typical column loads, confirming that settlement, not bearing capacity, usually governs design in dense sand. See Consolidation and Settlement Analysis for the next step.