Soil Bearing Capacity Calculator

Ultimate bearing capacity is the contact pressure at which soil beneath a footing develops a continuous shear failure surface and the foundation punches or rotates into the ground. Terzaghi expressed it as three additive contributions: strength from cohesion mobilized along the failure surface, written c times Nc; strength from the surcharge of soil sitting beside and above the footing base, gamma times Df times Nq; and strength from the weight of the soil inside the failing wedge itself, 0.5 times gamma times B times Ngamma. The N factors depend on the friction angle alone and grow very rapidly with it. Nq is 1.0 at phi equal to zero, about 18 at 30 degrees, and above 60 at 40 degrees, which is why dense sand supports several times what loose sand will carry at the same footing size. Shape corrections apply because a square or circular footing fails in three dimensions rather than plane strain: the cohesion term is raised by roughly 30 percent and the width term reduced to a 0.4 coefficient. Design normally applies a factor of safety near 3 to the ultimate value, so allowable pressures land between about 1,500 psf for soft clay and 8,000 psf for dense sand or gravel. Two caveats dominate practice. Settlement rather than shear usually governs footings on sand, so the allowable pressure must also be checked against tolerable movement, commonly one inch total. And the theory assumes general shear failure; loose or soft deposits fail locally and require reduced strength parameters.

The bearing capacity equations descend from plasticity theory originally developed for metals. Ludwig Prandtl published a solution in 1920 for the indentation of a rigid punch into a weightless plastic half space, producing the active wedge, the radial shear fan, and the passive zone that still appear in every bearing capacity diagram. Hans Reissner extended the solution in 1924 to include a surcharge next to the loaded strip, which yields the Nq term directly. Karl Terzaghi, who founded soil mechanics with his 1925 book Erdbaumechanik, combined those results with soil self weight and published the familiar three-term equation in Theoretical Soil Mechanics in 1943, adding the 1.3 and 0.4 shape coefficients for square footings from experimental work. Because self weight makes the problem mathematically non-exact, Ngamma has no closed-form solution and several competing approximations coexist to this day. George Meyerhof, in papers running from 1951 through the 1960s, generalized the method with shape, depth, and load inclination factors suited to real footings rather than idealized strips. Aleksandar Vesic proposed the widely used approximation Ngamma equals 2 times the quantity Nq plus 1, times tan phi, in 1973. These variants are consolidated in modern references such as the AASHTO LRFD bridge design specifications and FHWA foundation manuals, where calibrated resistance factors have largely replaced the single global factor of safety.