Bearing capacity

Rischio checks the base of the wall as a strip footing, under the resultant of the forces in Stability. This page covers what is particular to the wall; the general bearing capacity equation is the footing’s, in Footings › Bearing capacity.

The base as a strip footing

The base is a strip footing of width $B$ at depth $D_f$, under $\Sigma V$ at an eccentricity $e$ and $\Sigma H$. Its maximum contact pressure $q_{max}$, linear across the base, is compared with the ultimate capacity $q_u$ of the general bearing capacity equation, as in Das & Sivakugan, on the effective width $B' = B - 2e$, with the $c$ and $\phi$ of the stratum the base rests on.

With $e \ge B/2$ the resultant leaves the base: there is no contact pressure, and the factor of safety is 0.

Factors

The bearing capacity factors $N_c$, $N_q$ and $N_\gamma$, and the depth and inclination factors below, are the footing’s, explained in Footings › Bearing capacity. The shape factors are 1 for a strip. The depth factors are Hansen’s, with the full width $B$:

$$F_{qd} = 1 + 2\tan\phi\,(1 - \sin\phi)^2\,\frac{D_f}{B}, \qquad F_{cd} = F_{qd} - \frac{1 - F_{qd}}{N_c\tan\phi}, \qquad F_{\gamma d} = 1$$

For $\phi = 0$, $F_{cd} = 1 + 0.4\,D_f/B$ and $F_{qd} = 1$; for $D_f/B \gt 1$, $D_f/B$ becomes $\tan^{-1}(D_f/B)$, in radians. The inclination factors are Meyerhof’s, with the inclination of the load $\psi = \tan^{-1}(|\Sigma H|/\Sigma V)$ in degrees:

$$F_{ci} = F_{qi} = \left(1 - \frac{\psi}{90^\circ}\right)^2, \qquad F_{\gamma i} = \left(1 - \frac{\psi}{\phi}\right)^2$$

with $F_{\gamma i} = 0$ once $\psi \ge \phi$ (the $\gamma$ term is gone) and $F_{\gamma i} = 1$ for $\phi = 0$.

Overburden and effective unit weight

The overburden $q$ is the vertical effective stress at the base, in front of the toe; $\gamma'$ averages the effective unit weight from $D_f$ to $D_f + B$ by thickness ($\gamma$ above the water table, $\gamma_{sat} - \gamma_w$ below).