Terzaghi's bearing capacity theory

Terzaghi's theory (1943) gives the ultimate bearing capacity of a shallow footing by considering the equilibrium of a failure wedge. It is the most widely used classical theory and extends Prandtl's and Reissner's work by including soil weight and a rough base.

Assumptions

1. The footing is a long strip (plane strain), and $D_f \le B$ (shallow foundation).
2. The base is rough, so the soil wedge directly below it stays elastic and moves down with the footing.
3. The soil is homogeneous and isotropic, and obeys Mohr–Coulomb's law $\tau = c + \sigma\tan\phi$.
4. The shear strength of the soil above the base level is ignored. It is replaced by a uniform surcharge $q = \gamma D_f$.
5. Failure occurs in general shear (for local shear, strength is reduced to two-thirds).
6. The load is vertical and central.

Failure mechanism

   q_u  ____________
       |  I  |
  II  /\    /\  II
 III /  \  /  \ III

Under the footing, zone I is an elastic (triangular) wedge with base angles $\phi$. Zone II is a radial shear zone bounded by a logarithmic spiral. Zone III is a passive Rankine zone.

Derivation (outline)

Considering vertical equilibrium of the wedge: $q_u B + W = 2P_p + 2C\sin\phi$, where $C = \dfrac{cB}{2\cos\phi}$ is the cohesive force on each inclined face, $W$ the weight of the wedge, and $P_p$ the passive resistance from zones II and III (made of three parts, due to $c$, $q$ and $\gamma$). This gives

$$q_u = cN_c + \gamma D_f N_q + 0.5\gamma B N_\gamma \quad (\text{strip})$$

For other shapes: square $q_u = 1.3cN_c + \gamma D_fN_q + 0.4\gamma BN_\gamma$; circular $q_u = 1.3cN_c + \gamma D_fN_q + 0.3\gamma BN_\gamma$. $N_c, N_q, N_\gamma$ depend only on $\phi$ (e.g. for $\phi = 35^\circ$: $57.8, 41.4, 42.4$). For local shear use $c' = \tfrac{2}{3}c$, $\tan\phi' = \tfrac{2}{3}\tan\phi$.