Bearing capacity of shallow foundations

Terzaghi bearing capacityallowable bearing pressurebearing capacity factorsfooting designNc Nq Ngamma

The Terzaghi bearing-capacity equation, its Nc and Nq factors, and the net allowable pressure a footing may actually be designed to.

Bearing Capacity Factor Nq

N_q = e^{\pi\tan\phi}\,\tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Prandtl–Reissner surcharge bearing capacity factor Nq from the friction angle, the value tabulated by Meyerhof, Hansen and Vesic.

Bearing Capacity Factor Nc

N_c = (N_q - 1)\cot\phi

Prandtl's cohesion bearing capacity factor Nc derived from Nq and the friction angle, tending to 5.14 as the friction angle goes to zero.

Terzaghi Ultimate Bearing Capacity (Strip Footing)

q_u = c\,N_c + q\,N_q + \tfrac{1}{2}\,\gamma\,B\,N_\gamma

Terzaghi's three-term ultimate bearing capacity of a shallow strip footing, summing the cohesion, surcharge and footing-width contributions.

Net Allowable Bearing Pressure

q_{all} = \frac{q_u - q}{FS}

Net allowable bearing pressure for a footing, the ultimate capacity less the existing overburden, divided by the chosen factor of safety.

How they fit together

Terzaghi's equation adds three contributions that resist a footing punching into the ground: the soil's cohesion, the surcharge of whatever soil sits beside the footing at its founding depth, and the self-weight of the soil in the failure wedge below it. The N factors are pure functions of the friction angle φ, and they climb steeply — a few degrees of extra φ can change Nq by half again, which is why a sloppy φ makes everything downstream of it decorative.

Read which term dominates and you know what the design hinges on. In saturated clay under undrained loading φ is near zero, Nq collapses to 1 and Nγ to 0, so cohesion carries essentially the whole capacity and footing width barely matters. In sand there is no cohesion, so capacity comes entirely from surcharge and self-weight, and both depth and width buy you real capacity. The recurring error is confusing gross and net: the allowable pressure must subtract the overburden that was there before you dug, or you charge the soil twice for weight it was already carrying. And the factor of safety, usually 3, is not padding — it is doing double duty, because a footing that satisfies bearing capacity can still fail on settlement, which this equation says nothing whatever about and which governs most real shallow foundations.