# 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; 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.

## Introduction

A foundation is the part of a structure which transmits the weight of the structure to the ground. All structures constructed on land are supported on foundations. A foundation is a connecting link between the structure proper and the ground which supports it.

## General bearing failure

A general bearing failure occurs when the load on the footing causes large movement of the soil on a shear failure surface which extends away from the footing and up to the soil surface. Calculation of the capacity of the footing in general bearing is based on the size of the footing and the soil properties. The basic method was developed by Terzaghi, with modifications and additional factors by Meyerhof and Vesić. The general shear failure case is the one normally analyzed. Prevention against other failure modes is accounted for implicitly in settlement calculations.[1] There are many different methods for computing when this failure will occur.

## Terzaghi's Bearing Capacity Theory

Karl von Terzaghi was the first to present a comprehensive theory for the evaluation of the ultimate bearing capacity of rough shallow foundations. This theory states that a foundation is shallow if its depth is less than or equal to its width.[2] Later investigations, however, have suggested that foundations with a depth, measured from the ground surface, equal to 3 to 4 times their width may be defined as shallow foundations.[2]

Terzaghi developed a method for determining bearing capacity for the general shear failure case in 1943. The equations, which take into account soil cohesion, soil friction, embedment, surcharge, and self-weight, are given below.[2]

For square foundations:

${\displaystyle q_{ult}=1.3c'N_{c}+\sigma '_{zD}N_{q}+0.4\gamma 'BN_{\gamma }\ }$

For continuous foundations:

${\displaystyle q_{ult}=c'N_{c}+\sigma '_{zD}N_{q}+0.5\gamma 'BN_{\gamma }\ }$

For circular foundations:

${\displaystyle q_{ult}=1.3c'N_{c}+\sigma '_{zD}N_{q}+0.3\gamma 'BN_{\gamma }\ }$

where

${\displaystyle N_{q}={\frac {e^{2\pi \left(0.75-\phi '/360\right)\tan \phi '}}{2\cos ^{2}\left(45+\phi '/2\right)}}}$
${\displaystyle N_{c}=5.14\ }$ for φ' = 0
${\displaystyle N_{c}={\frac {N_{q}-1}{\tan \phi '}}}$ for φ' > 0
${\displaystyle N_{\gamma }={\frac {\tan \phi '}{2}}\left({\frac {K_{p\gamma }}{\cos ^{2}\phi '}}-1\right)}$
c′ is the effective cohesion.
σzD′ is the vertical effective stress at the depth the foundation is laid.
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%.[1]
${\displaystyle N_{\gamma }={\frac {2\left(N_{q}+1\right)\tan \phi '}{1+0.4\sin 4\phi '}}}$

For foundations that exhibit the local shear failure mode in soils, Terzaghi suggested the following modifications to the previous equations. The equations are given below.

For square foundations:

${\displaystyle q_{ult}=0.867c'N'_{c}+\sigma '_{zD}N'_{q}+0.4\gamma 'BN'_{\gamma }\ }$

For continuous foundations:

${\displaystyle q_{ult}={\frac {2}{3}}c'N'_{c}+\sigma '_{zD}N'_{q}+0.5\gamma 'BN'_{\gamma }\ }$

For circular foundations:

${\displaystyle q_{ult}=0.867c'N'_{c}+\sigma '_{zD}N'_{q}+0.3\gamma 'BN'_{\gamma }\ }$

${\displaystyle N'_{c},N'_{q}andN'_{y}}$, the modified bearing capacity factors, can be calculated by using the bearing capacity factors equations(for ${\displaystyle N_{c},N_{q},andN_{y}}$, respectively) by replacing the effective internal angle of friction${\displaystyle (\phi ')}$ by a value equal to ${\displaystyle :tan^{-1}\,({\frac {2}{3}}tan\phi ')}$ [2]

## Meyerhof's Bearing Capacity theory

In 1951, Meyerhof published a bearing capacity theory which could be applied to rough shallow and deep foundations.[3] Meyerhof (1951, 1963) proposed a bearing-capacity equation similar to that of Terzaghi's but included a shape factor s-q with the depth term Nq. He also included depth factors and inclination factors.

## Factor of safety

Calculating the gross allowable-load bearing capacity of shallow foundations requires the application of a factor of safety (FS) to the gross ultimate bearing capacity, or:

${\displaystyle q_{all}={\frac {q_{ult}}{FS}}}$ [2]