Bearing capacity in geotechnical engineering refers to the capacity of soil to support the loads applied to the ground. It is the maximum average contact pressure between the foundation and the soil that should not produce shear failure in the soil.

Derivation of Bearing Capacity Expression

The most commonly used method for deriving the bearing capacity of shallow foundations is Terzaghi’s Bearing Capacity Theory.

Here’s a step-by-step derivation:

1. Assumptions

The soil is homogeneous and isotropic.
The failure surface is well-defined.
The footing is continuous and has a rough base.
The soil above the base of the foundation is replaced by a uniform surcharge.

2. Failure Mechanism:

The soil under the footing is divided into three zones:

Zone I: A relatively undeformed wedge of soil directly under the footing.
Zone II: Transition zones forming log spiral fans.
Zone III: Passive Rankine zones extending outward and upward from the edges of the footing.

3. Ultimate Bearing Capacity Equation:

The ultimate bearing capacity ((q_u)) is given by the sum of three components:

Cohesion component: (cN_c)
Depth component: (γD_f N_q)
Width component: (0.5γBNγ)

The general form of Terzaghi’s bearing capacity equation for a strip footing is:
qu​=cNc​+γDf​Nq​+0.5γBNγ​
Where:

(c) = Cohesion of the soil
(γ) = Unit weight of the soil
(D_f) = Depth of the footing
(B) = Width of the footing
(N_c, N_q, N_γ) = Bearing capacity factors, which are functions of the soil’s internal friction angle ((φ))

4. Bearing Capacity Factors:

These factors are derived from the theory of plasticity and depend on the internal friction angle ((φ)):

(N_c = (N_q – 1) / tan(ϕ)
(N_q = e^(π * tan(φ)) * tan(2 * (45 degrees + φ/2))
(N_γ = 2(N_q + 1) \tan(φ))

Example Calculation

For a strip footing with the following parameters:

Cohesion ((c)) = 25 kPa
Unit weight ((γ)) = 18 kN/m³
Depth of footing ((D_f)) = 1.5 m
Width of footing ((B)) = 2 m
Internal friction angle ((φ)) = 30°

Using the bearing capacity factors for (φ = 30°):

(N_c ≈ 37.2)
(N_q ≈ 22.5)
(N_γ ≈ 19.7)

The ultimate bearing capacity ((q_u)) is:
qu​=(25×37.2)+(18×1.5×22.5)+(0.5×18×2×19.7)
qu​=930+607.5+354.6=1892.1 kPa

This derivation and example illustrate how the bearing capacity of soil is calculated using Terzaghi’s method

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