Discrete tomography

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, therefore, a connecting link between the structure proper and the ground which supports it.

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'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. 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(Das, 2007).

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

For square foundations:

$q_{ult}=1.3c^{\prime }{N}_c+\sigma {\text{'}}_{zD}{N}_q+0.4{\gamma }^{\prime }B{N}_{\gamma }$

For continuous foundations:

$q_{ult}=c^{\prime }{N}_c+\sigma {\text{'}}_{zD}{N}_q+0.5{\gamma }^{\prime }B{N}_{\gamma }$

For circular foundations:

$q_{ult}=1.3c^{\prime }{N}_c+\sigma {\text{'}}_{zD}{N}_q+0.3{\gamma }^{\prime }B{N}_{\gamma }$

where

$N_q=\frac{e^{2\pi (0.75-{\varphi }^{\prime }/360)\tan {\varphi }^{\prime }}}{2{\cos }^2(45+{\varphi }^{\prime }/2)}$

$N_c=5.7$ for φ' = 0

$N_c=\frac{N_q-1}{\tan {\varphi }^{\prime }}$ for φ' > 0

$N_{\gamma }=\frac{\tan {\varphi }^{\prime }}{2}(\frac{K_{p\gamma }}{{\cos }^2{\varphi }^{\prime }}-1)$

c′ is the effective cohesion.

σ_zD′ 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_pγ is obtained graphically. Simplifications have been made to eliminate the need for K_pγ. One such was done by Coduto, given below, and it is accurate to within 10%.

$N_{\gamma }=\frac{2(N_q+1)\tan {\varphi }^{\prime }}{1+0.4\sin 4{\varphi }^{\prime }}$

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:

$q_{ult}=0.867c^{\prime }N{\text{'}}_c+\sigma {\text{'}}_{zD}N{\text{'}}_q+0.4{\gamma }^{\prime }BN{\text{'}}_{\gamma }$

For continuous foundations:

$q_{ult}=\frac{2}{3}{c}^{\prime }N{\text{'}}_c+\sigma {\text{'}}_{zD}N{\text{'}}_q+0.5{\gamma }^{\prime }BN{\text{'}}_{\gamma }$

For circular foundations:

$q_{ult}=0.867c^{\prime }N{\text{'}}_c+\sigma {\text{'}}_{zD}N{\text{'}}_q+0.3{\gamma }^{\prime }BN{\text{'}}_{\gamma }$

$N{\text{'}}_c,N{\text{'}}_{q}andN{\text{'}}_y$, the modified bearing capacity factors, can be calculated by using the bearing capacity factors equations(for $N_c,N_q,and{N}_y$, respectively) by replacing the effective internal angle of friction$({\varphi }^{\prime })$ by a value equal to $:ta{n}^{-1}(\frac{2}{3}tan{\varphi }^{\prime })$

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:

$q_{all}=\frac{q_{ult}}{FS}$

Meyerhofs's Bearing Capacity theory

In 1951, Meyerhof published a bearing capacity theory which could be applied to rough shallow and deep foundations.

See also

• Geotechnical engineering
• Foundation (architecture)
• Soil mechanics

References

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External links

• Bearing Capacity Factors related to the angle of internal friction