The Foundation of Foundation Design
Karl Terzaghi's 1943 bearing capacity equation remains the cornerstone of shallow foundation design. When a footing pushes into soil, three distinct resistance mechanisms activate: cohesion along the failure surface (cN_c), the weight of overburden soil resisting upward displacement (γD_fN_q), and the self-weight of the soil wedge beneath the footing (0.5γBN_γ). The simulation visualizes these failure zones — an active wedge directly beneath the footing, radial shear zones on each side, and passive Rankine zones pushed upward at the surface.
Bearing Capacity Factors
The dimensionless factors N_c, N_q, and N_γ encode the geometry of the failure surface as a function of the soil's internal friction angle φ. They grow exponentially with φ: at φ = 20°, N_q ≈ 6.4, but at φ = 40°, N_q ≈ 64.2 — a tenfold increase. This explains why dense sands and gravels support far heavier structures than loose or soft soils. For purely cohesive soils (φ = 0), only the N_c term contributes, giving q_u = 5.7c + γD_f.
Factor of Safety & Allowable Capacity
The ultimate bearing capacity represents imminent shear failure — a catastrophic event where the foundation punches into the ground. Design never approaches this limit. Dividing q_u by a factor of safety (typically FS = 3) yields the allowable bearing pressure q_a. In practice, settlement often controls design before bearing capacity is reached, especially for footings on sand where elastic compression and particle rearrangement limit tolerable pressure to values well below q_a.
Beyond Terzaghi
Modern geotechnical practice extends Terzaghi's framework with shape factors (for square and circular footings), depth factors, load inclination corrections, and layered-soil modifications developed by Meyerhof, Hansen, and Vesic. Finite element analysis can model complex geometries and nonlinear soil behavior. Yet Terzaghi's equation remains the essential first estimate, the benchmark against which all numerical results are compared, and the formula every civil engineering student learns by heart.