# Bearing capacity

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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 will 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.

## Shallow foundations

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

### General shear failure

A general shear failure is one where the pressure exerted by the foundation causes shearing along a curved surface beginning below the footing and extending away from it, so that soil below the footing is displaced downwards, and soil adjacent to the footing is lifted.

The general shear failure case is the one normally analyzed. Prevention against other failure modes is normally accounted for implicitly in settlement calculations.[1] There are many different methods for computing when this failure will occur.

#### Terzaghi Method

Karl Terzaghi developed a method for determining bearing capacity for the general shear failure case for shallow foundations in 1943. Others (Hansen, Meyerhof, and Vesic) have made adjustments to Terzaghi's equations based on experimental and empirical data. Other research has found that the Terzaghi equations do not produce reliable results for deep foundations.

Terzaghi's equations are given below.

For square foundations:

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

For continuous strip 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

${\displaystyle N_{q}={\frac {e^{2\pi \left(0.75-\phi '/360\right)\tan \phi '}}{2\cos ^{2}\left(45+\phi '/2\right)}}}$
${\displaystyle N_{c}=5.7\ }$ 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 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%. [1]
${\displaystyle N_{\gamma }={\frac {2\left(N_{q}+1\right)\tan \phi '}{1+0.4\sin 4\phi '}}}$

### Punching shear failure

Punching shear failure occurs in loose soils when the block of soil immediately below the footing compresses significantly (or punches into a very soft layer below, as in the case of fills on Bay Mud or similar soils), with the shear failure occurring at the footing perimeter.

Various empirical equations have been proposed to analyse punching shear failure, though often the possibility of punching shear is eliminated by densifying the soil or using deep foundations instead of shallow foundations.

## Deep foundations

Bearing capacity of deep foundations depends on the structure of the foundation as well as the soil properties. An isolated shaft may provide support from end-bearing at the tip of the shaft and/or from friction along the sides of the shaft. Piles or piers in groups may have reduced capacity as the stresses each shaft introduces in the soil causes stresses on other shafts. Additionally, a large group of piers or piles may cause group effects for end bearing.

### End-bearing

Pile capacity for end-bearing can be calculated using equations similar to the Terzaghi equations given above, though for deep foundations, many of the factors and relationships do not continue to apply. For piles, settlement considerations are more likely than bearing failure to limit allowable capacity.

### Skin friction

Pile capacity from skin friction is calculated based on the surface area of the pile, the shear strength of the soil, and various reduction factors to account for the method of pile construction and adhesion between the pile material and the soil. Skin friction can be considered in combination with end bearing, if the movement required to mobilize both sources of support is similar.

### Group effects

Pile or pier groups may have a bearing capacity different from the sum of the individually-calculated bearing capacities, as the stresses induced in the soil by the shafts may reduce the capacity of the shafts, and because the group of foundations may act as a single unit, including the shafts and the soil between them, creating the equivalent of a large block foundation.