Development of classical bearing capacity theories

Classical bearing capacity theories were developed step by step. Each author added a term that the previous theory neglected.

1. Rankine (1857) and Bell. Based on Rankine earth pressure. The soil below the footing is in active state and the soil beside it in passive state. For cohesionless soil $q_u = \gamma D_f\left(\dfrac{1+\sin\phi}{1-\sin\phi}\right)^2$. Bell (1915) extended it to $c$–$\phi$ soil: $q_u = \gamma D_f N_\phi^2 + 2c\sqrt{N_\phi}(N_\phi + 1)$ with $N_\phi = \tan^2(45^\circ + \phi/2)$. Pauker's method is a similar approach for cohesionless soil. These ignore footing width and base friction.
2. Prandtl (1921). Studied punching of a rigid die into a weightless $c$–$\phi$ medium. Found a failure surface with a wedge, a log-spiral fan and a passive zone, giving $q_u = cN_c$ with $N_c = (N_q - 1)\cot\phi$ and $N_q = e^{\pi\tan\phi}\tan^2(45^\circ + \phi/2)$. The footing is on the surface and soil is weightless.
3. Reissner (1924). Extended Prandtl to include the surcharge $q = \gamma D_f$, adding the $N_q$ term.
4. Terzaghi (1943). Added the weight of soil ($N_\gamma$ term), a rough base and the effect of shape (square, circular), and gave $q_u = cN_c + qN_q + 0.5\gamma BN_\gamma$.
5. Meyerhof (1951, 1963). Included the shear strength of soil above the base, and gave shape, depth and inclination factors.
6. Hansen (1970) and Vesic (1973). Further refined the factors for shape, depth, inclination of load, base and ground tilt, with improved values of $N_\gamma$.

So the progression is: Rankine (weight and surcharge, approximate) → Prandtl (cohesion) → Reissner (surcharge) → Terzaghi (soil weight, practical form) → Meyerhof, Hansen, Vesic (general equation with correction factors).