Question 1 of 9: Limitations of the Terzaghi bearing-capacity equation

Question 1: Limitations of the Terzaghi bearing-capacity equation (7 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.

Terzaghi (1943) produced the first rational bearing-capacity theory for shallow foundations, and its three-term form — $q_u = c N_c + q N_q + 0.5 \gamma B N_{\gamma}$ — survives essentially unchanged inside the modern general equation. What changed is everything that multiplies those three terms. The general (Meyerhof–Hansen–Vesic) equation writes $q_u = c N_c F_{cs} F_{cd} F_{ci} + q N_q F_{qs} F_{qd} F_{qi} + 0.5 \gamma B N_{\gamma} F_{\gamma s} F_{\gamma d} F_{\gamma i}$, and the limitations of the Terzaghi form are exactly the physical effects those factors represent.

Shape. Terzaghi derived a plane-strain (strip) solution and then patched square and circular footings in with two hard-coded empirical constants (1.3 c N_c and 0.4 gamma B N_γ for a square). There is nothing for a rectangle of arbitrary B/L, which is the commonest footing shape in practice; the general equation carries continuous shape factors in B/L.

Depth. Terzaghi replaces the soil above founding level by an equivalent surcharge $q = \gamma D_f$ and explicitly discards its shear strength. That is safe but wasteful, and it becomes badly wrong for deeper footings: the equation is only intended for $D_f \le B$, i.e. genuinely shallow foundations. The general equation restores the discarded shear through depth factors F_cd, F_qd, which for a footing at D_f/B = 2 raise the cohesion term by more than 40 per cent.

Load inclination and eccentricity. Terzaghi assumes a vertical, concentric load. Real abutments, retaining-wall bases and braced-frame columns deliver inclined and eccentric resultants. The general equation handles inclination through F_ci, F_qi, F_γi and eccentricity through Meyerhof's effective-area rule ($B' = B - 2e$), neither of which exists in the Terzaghi form.

Failure mode. Terzaghi's derivation is a general-shear mechanism: a rigid elastic wedge, a radial shear zone and a Rankine passive zone, all mobilising peak strength simultaneously. Loose sands and soft normally consolidated clays fail in local or punching shear instead, so the theory over-predicts capacity unless the strength parameters are artificially reduced (Terzaghi's own crude fix was to use $\tan\phi^{*} = \tfrac{2}{3}\tan\phi^{\prime}$ and $c^{*} = \tfrac{2}{3}c^{\prime}$). Vesic's rigidity-index correction replaces that guess with a calculation.

Geometry of the ground. Terzaghi assumes a horizontal ground surface, a horizontal footing base and a homogeneous half-space. Footings on or near a slope, on inclined bases, or on layered profiles are outside its scope; the general equation has ground-slope and base-tilt factors, and layered-soil rules exist alongside it.

Two further practical points are worth making. First, the two equations do not even share the same factors: Terzaghi's N_c = 5.7 for phi = 0, against 5.14 in the general equation, so the Terzaghi value is about 11 per cent higher for undrained clay. Second, neither equation says anything about settlement, and for footings on sand or on stiff clay it is almost always settlement, not bearing capacity, that governs the design.