Bearing Capacity Calculator - Terzaghi Meyerhof

Ultimate bearing capacity is the contact pressure at which the soil beneath a footing fails in shear along a continuous slip surface rather than merely settling. The three terms of the equation are independent contributions superposed: c times Nc is the resistance from cohesion, gamma times Df times Nq is the stabilizing effect of the overburden surcharge alongside the footing, and 0.5 times gamma times B times Ng is the resistance mobilized by the self weight of the soil in the failure wedge below. The surcharge factor follows from plasticity theory as Nq = e raised to (pi tan phi) times tan squared of (45 + phi/2), with Nc = (Nq - 1) cot phi. For an undrained clay with phi equal to zero, Nc reduces to the exact Prandtl value of 2 + pi, or 5.14. Meyerhof 1963 expression Ng = (Nq - 1) tan(1.4 phi) is one of several competing forms, and Vesic and Hansen alternatives can differ by 20 to 30 percent for the same friction angle. The factors climb steeply with friction angle, from Nq of 6.4 at 20 degrees to 18.4 at 30 degrees and 64.2 at 40 degrees, so a 5 degree error in phi can change the answer by a factor of two. A global factor of safety of 3.0 on ultimate capacity is the traditional convention for permanent structures, with 2.0 to 2.5 sometimes accepted for temporary works or exceptionally well characterized ground. Allowable pressures in practice run from roughly 50 to 100 kPa for soft clays up to 300 to 600 kPa for dense sands. Several assumptions constrain the result: it describes an infinitely long strip footing on level, homogeneous ground failing in general shear, without shape, depth or load inclination corrections. Loose or soft soils fail in local or punching shear instead and require reduced strength parameters. Groundwater rising into the failure zone reduces the effective unit weight to roughly half its bulk value. For footings on sand wider than about a metre, tolerable settlement rather than shear failure normally governs the design.

The mathematical foundation predates soil mechanics as a discipline. Ludwig Prandtl published a plasticity solution in 1920 for the indentation of a rigid punch into a weightless cohesive-frictional half space, producing the characteristic slip-line field of an active wedge, a radial shear fan and a passive wedge. Hans Reissner extended it in 1924 to include a surcharge acting at the level of the punch base, which supplied the Nq term. Neither solution could accommodate soil self weight, because the governing equations lose their closed form once gravity acts within the failing mass. Karl Terzaghi, whose 1925 book Erdbaumechanik established soil mechanics as a quantitative field, closed that gap in his 1943 Theoretical Soil Mechanics by superposing an approximate self-weight term onto the Prandtl and Reissner solution, yielding the familiar three-term equation and the tabulated factors still taught today. George Geoffrey Meyerhof extended the framework in a 1951 Geotechnique paper and refined it in the Canadian Geotechnical Journal in 1963, introducing shape, depth and load inclination factors and allowing the failure surface to extend upward through the soil above the footing base, which Terzaghi had treated only as a surcharge. Jorgen Brinch Hansen in 1970 and Aleksandar Vesic in 1973 published further correction schemes and alternative Ng expressions that remain in common use. Modern European practice under Eurocode 7 keeps the same bearing capacity equation but replaces the single global factor of safety with partial factors applied separately to loads and to material strengths.