Bearing capacity

This page follows the load from where it acts on the base to the factor of safety; the units, conventions and assumptions are in Method and assumptions.

Eccentricity and effective area

The moments move the resultant off centre by $e_B = |M_B|/V$ and $e_L = |M_L|/V$. The bearing capacity uses Meyerhof’s effective area, the part of the base centred on the resultant. Its sides are $B - 2e_B$ and $L - 2e_L$: $B'$ is the shorter of them and $L'$ the longer, whichever the user called the width. A strip has only $B' = B - 2e_B$. The shape factors use $B'/L'$ and the $\gamma$ term uses $B'$, while the depth factors and the zone that sets $\gamma'$ use the shorter side of the base, $B$.

Contact pressure

The contact pressure is linear over the whole base, with $q_{max}$ and $q_{min}$ at opposite corners (at the edges, for a strip) and $A = BL$, or $B$ per metre for a strip. Once $6e_B/B + 6e_L/L$ exceeds 1, $q_{min}$ is negative: that corner would be in tension, which the soil can’t take, so it lifts off. Rischio doesn’t redistribute the pressure; it keeps the negative value to show how far out the load is. With one eccentricity this is the middle third: $e_B > B/6$ or $e_L > L/6$ makes $q_{min}$ negative and fails its eccentricity check.

When $e_B \ge B/2$ or $e_L \ge L/2$ the resultant falls outside the base and the footing overturns: there is no contact pressure or $q_u$, and the bearing check fails with $FS = 0$.

Ultimate bearing capacity

$q_u$ comes from the general bearing capacity equation (Meyerhof), with the factors as Das & Sivakugan give them and the $c$ and $\phi$ of the bearing stratum. The base of a retaining wall takes it from the same equation, as a strip:

$$q_u = c N_c F_{cs} F_{cd} F_{ci} + q N_q F_{qs} F_{qd} F_{qi} + \tfrac{1}{2} \gamma' B' N_\gamma F_{\gamma s} F_{\gamma d} F_{\gamma i}$$

The overburden $q$ is the effective vertical stress at the base: the weight of the strata above, with $\gamma$ above the water table and $\gamma_{sat}$ below, less the pore pressure. $\gamma'$ is the thickness-weighted average between $D_f$ and $D_f + B$, with $\gamma$ above the water table and $\gamma_{sat} - \gamma_w$ below. With one stratum these are the textbook’s three cases: the water table above the base, within $B$ below it, or deeper.

Bearing capacity factors

Reissner, Prandtl and Vesic; with $\phi = 0$, the undrained case, $N_c = 5.14$, $N_q = 1$ and $N_\gamma = 0$:

$$N_q = \tan^2\left(45^\circ + \frac{\phi}{2}\right) e^{\pi \tan\phi}, \qquad N_c = \frac{N_q - 1}{\tan\phi}, \qquad N_\gamma = 2\,(N_q + 1)\tan\phi$$

Shape factors

De Beer’s, with the effective dimensions $B'$ and $L'$; for a strip $B'/L' = 0$ and all three are 1:

$$F_{cs} = 1 + \frac{B'}{L'}\,\frac{N_q}{N_c}, \qquad F_{qs} = 1 + \frac{B'}{L'}\tan\phi, \qquad F_{\gamma s} = 1 - 0.4\,\frac{B'}{L'}$$

Depth factors

Hansen’s, with the full width $B$, the shorter side of the base, for $D_f/B \le 1$:

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

With $\phi = 0$, $F_{cd} = 1 + 0.4\,D_f/B$ and $F_{qd} = F_{\gamma d} = 1$. For $D_f/B > 1$, $D_f/B$ is replaced by $\tan^{-1}(D_f/B)$, in radians.

Inclination factors

Meyerhof’s, and Hanna and Meyerhof’s, with the inclination of the load $\beta = \tan^{-1}(|H|/V)$ in degrees, the same whichever way the load leans. Once $\beta \ge \phi$ the $\gamma$ term is gone, $F_{\gamma i} = 0$ (the square would bring it back); with $\phi = 0$, $F_{\gamma i} = 1$:

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

Factor of safety

The factor of safety is $FS = q_u/q_{max}$, against 3.0 by default. An eccentric load enters both terms: $q_u$ through $B'$ and $L'$, and $q_{max}$ through the moments.