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# Bearing capacity

In geotechnical engineering, bearing capacity is the capacity of soil to support the loads applied to the ground. The bearing capacity of soil is the maximum average contact pressure between the foundation and the soil which should not produce shear failure in the soil. Ultimate bearing capacity is the theoretical maximum pressure which can be supported without failure; while allowable bearing capacity is the ultimate bearing capacity divided by a factor of safety. Sometimes, on soft soil sites, large settlements may occur under loaded foundations without actual shear failure occurring; in such cases, the allowable bearing capacity is based on the maximum allowable settlement.

There are three modes of failure that limit bearing capacity: general shear failure, local shear failure, and punching shear failure.

## General shear failure

The general shear failure case is the one normally analyzed. Prevention against other failure modes is accounted for implicitly in settlement calculations. There are many different methods for computing when this failure will occur.

### Terzaghi

Karl von Terzaghi developed a method for determining bearing capacity for the general shear failure case in 1943. The equations are given below.

For square foundations:

$q_\left\{ult\right\} = 1.3 c\text{'} N_c + sigma \text{'}_\left\{zD\right\} N_q + 0.4 gamma \text{'} B N_gamma$

For continuous foundations:

$q_\left\{ult\right\} = c\text{'} N_c + sigma \text{'}_\left\{zD\right\} N_q + 0.5 gamma \text{'} B N_gamma$

For circular foundations:

$q_\left\{ult\right\} = 1.3 c\text{'} N_c + sigma \text{'}_\left\{zD\right\} N_q + 0.3 gamma \text{'} B N_gamma$

where

$N_q = frac\left\{ e ^\left\{ 2 pi left\left(0.75 - phi \text{'}/360 right\right) tan phi \text{'} \right\} \right\}\left\{2 cos ^2 left\left(45 + phi \text{'}/2 right\right) \right\}$
$N_c = 5.7$ for φ' = 0
$N_c = frac\left\{ N_q - 1 \right\}\left\{ tan phi \text{'}\right\}$ for φ' > 0
$N_gamma = frac\left\{ tan phi \text{'} \right\}\left\{2\right\} left\left(frac\left\{ K_\left\{p gamma\right\} \right\}\left\{ cos ^2 phi \text{'} \right\} - 1 right\right)$
c′ is the effective cohesion.
σzD′ is the vertical effective stress at the depth the foundation is lain.
γ''′ is the effective unit weight when saturated or the total unit weight when not fully saturated.
B is the width or the diameter of the foundation.
φ′ is the effective internal angle of friction.
K is obtained graphically. Simplifications have been made to eliminate the need for K. One such was done by Coduto, given below, and it is accurate to within 10%.
$N_gamma = frac\left\{ 2 left\left(N_q + 1 right\right) tan phi \text{'} \right\}\left\{1 + 0.4 sin 4 phi \text{'} \right\}$