Bearing Capacity of Shallow Foundations

Bearing Capacity of Shallow Foundations

Modes of failure, Terzaghi's and general bearing capacity equations.

Overview

Bearing capacity is the maximum contact pressure that a foundation can exert on the underlying soil without causing shear failure. For shallow foundations (where the depth

D_f

is typically less than or equal to the width

B

), determining the ultimate bearing capacity (

q_u

) and selecting an appropriate allowable bearing capacity (

q_{allow}

) is the cornerstone of foundation design.

Modes of Shear Failure

When a shallow foundation is loaded progressively to failure, the soil beneath it typically exhibits one of three distinct modes of shear failure, depending largely on the relative density or consistency of the soil.

Procedure

STEP-BY-STEP

General Shear Failure: Characterized by a sudden, catastrophic failure accompanied by a well-defined failure surface extending up to the ground surface. There is noticeable, significant heaving of the soil on one or both sides of the footing. This mode is typical for relatively dense sands, stiff clays, and overconsolidated soils.
Local Shear Failure: The failure surface develops progressively but does not fully reach the ground surface before the foundation begins to undergo significant settlement. Heaving is slight and occurs only in the immediate vicinity of the footing. This is characteristic of soils with medium density or medium stiffness.
Punching Shear Failure: The foundation effectively "punches" downward into the soil with a roughly vertical shear surface. There is little to no heaving at the ground surface, but massive downward settlement occurs. This mode is typical for very loose sands, soft clays, and highly compressible soils.

Gross vs. Net Bearing Capacity

Pressure Definitions

Gross Ultimate Bearing Capacity ( $q_{u,gross}$ ): The absolute maximum total pressure (including the weight of the structure, foundation, and soil surcharge) that can be applied to the soil before catastrophic shear failure occurs.
Net Ultimate Bearing Capacity ( $q_{u,net}$ ): The maximum additional pressure (above the existing overburden pressure $q$ ) that the foundation can support before shear failure. This represents the actual structural load capacity.
$q_{u,net} = q_{u,gross} - q$
Where $q = \gamma \cdot D_f$ (the effective stress of the excavated soil).

Terzaghi's Bearing Capacity Equation (1943)

Karl Terzaghi developed the first comprehensive theory for the ultimate bearing capacity of shallow rough foundations. His theory assumed a general shear failure mode in a homogeneous soil under a continuous (strip) footing.

Terzaghi's Equation

The ultimate bearing capacity (

q_u

) of a continuous (strip) footing is given by:

q_u = c' N_c + q N_q + 0.5 \gamma B N_\gamma

Where:

$c'$ = effective cohesion of the soil
$q$ = effective overburden pressure at the foundation base level ( $q = \gamma \cdot D_f$ )
$\gamma$ = unit weight of the soil below the foundation level
$B$ = width of the footing
$N_c, N_q, N_\gamma$ = Terzaghi's dimensionless bearing capacity factors, which depend strictly on the effective friction angle ( $\phi'$ ) of the soil.

Terzaghi's N Factors

The precise mathematical derivations for Terzaghi's factors are:

N_q = \frac{e^{2(3\pi/4 - \phi'/2)\tan \phi'}}{2 \cos^2(45^\circ + \phi'/2)}

N_c = \cot \phi' (N_q - 1)

N_\gamma \approx \frac{1}{2} \left( \frac{K_{p\gamma}}{\cos^2 \phi'} - 1 \right) \tan \phi'

Note: Because

N_\gamma

is empirically complex to define exactly, multiple approximations exist. The exact values are almost universally obtained from standard tables based on

\phi'

Modifications for Foundation Shape

Terzaghi modified his original strip footing equation empirically to account for square and circular footings.

Square Footing: $q_u = 1.3 c' N_c + q N_q + 0.4 \gamma B N_\gamma$
Circular Footing: $q_u = 1.3 c' N_c + q N_q + 0.3 \gamma B N_\gamma$ (where $B$ is the diameter)

Skempton's Bearing Capacity for Clays

Undrained Conditions ( $\phi = 0$ )

For foundations resting on saturated cohesive soils (clays) subjected to rapid loading, the undrained condition (

\phi = 0

c = s_u

) governs. A.W. Skempton observed that for clays, Terzaghi's

N_c = 5.14

(or

5.7

) is only valid for surface footings. Skempton proposed a modified equation where

N_c

increases with depth and varies with footing shape:

q_{u,net} = s_u \cdot N_{c,skempton}

Where

N_{c,skempton}

starts at 5.14 for a surface strip footing and increases up to a maximum of 9.0 for deep square or circular footings (e.g.,

D_f/B > 4

The General Bearing Capacity Equation (Meyerhof, 1963)

While Terzaghi's equation is foundational, it has limitations. It assumes a strip footing, a rough base, no shear resistance in the soil above the foundation base (

D_f

), and vertical concentric loading.

Meyerhof developed a more versatile "General Bearing Capacity Equation" that incorporates shape (

s

), depth (

d

), and load inclination (

i

) factors, making it applicable to a wider range of realistic design scenarios.

q_u = c' N_c s_c d_c i_c + q N_q s_q d_q i_q + 0.5 \gamma B N_\gamma s_\gamma d_\gamma i_\gamma

Where:

$s_c, s_q, s_\gamma$ = Shape factors (accounting for rectangular, square, or circular shapes)
$d_c, d_q, d_\gamma$ = Depth factors (accounting for the shearing resistance of the soil above the base level)
$i_c, i_q, i_\gamma$ = Inclination factors (accounting for non-vertical loads)
$N_c, N_q, N_\gamma$ = Meyerhof's bearing capacity factors (note: these values differ slightly from Terzaghi's factors for the same $\phi'$ , particularly $N_\gamma$ ).

Vesic's Compressibility Factors

Accounting for Soil Compressibility

Both Terzaghi and Meyerhof assume the soil fails in general shear (a rigid-plastic behavior). However, very loose sands or highly compressible clays often fail in local or punching shear. A.S. Vesic introduced compressibility factors (

C_c, C_q, C_\gamma

) to modify the general equation when the soil's rigidity index (

I_r = \frac{G}{c + q \tan \phi}

) falls below a critical threshold. This allows a unified equation to handle all failure modes.

Eccentric Loading

Effective Area Method (Meyerhof)

When a foundation is subjected to a vertical load (

Q

) and a moment (

M

), the load acts eccentrically with an eccentricity

e = M/Q

. This causes non-uniform pressure. To apply the general bearing capacity equation, Meyerhof proposed the effective area concept. The foundation is assumed to have reduced, effective dimensions (

B'

and

L'

), where the load acts perfectly in the center of the effective area:

B' = B - 2e_B

L' = L - 2e_L

The bearing capacity

q_u

is calculated using

B'

and

L'

, and the total allowable load is

Q_{allow} = (q_u / FS) \times B' \times L'

Effect of Groundwater Table

The location of the groundwater table critically affects the unit weight (

\gamma

) terms in the bearing capacity equations.

If the water table is at or above the ground surface, the submerged unit weight ( $\gamma' = \gamma_{sat} - \gamma_w$ ) must be used for both the overburden term ( $q$ ) and the width term ( $0.5 \gamma B N_\gamma$ ).
If the water table is exactly at the foundation base level ( $D_f$ ), use the dry/moist unit weight for the overburden term ( $q$ ), but the submerged unit weight ( $\gamma'$ ) for the width term.
If the water table is far below the foundation ( $> B$ below the base), no correction is needed; use the moist/dry unit weight throughout.

Bearing Capacity on Layered Soils

Strong over Weak vs. Weak over Strong

Natural soil deposits are rarely homogeneous. When a footing rests on a stronger soil layer underlain by a weaker layer (e.g., dense sand over soft clay), the failure surface might punch through the top layer into the weaker layer below. The ultimate bearing capacity must be checked against both failure within the top layer alone, and a "punching shear" failure extending into the weaker bottom layer. Specialized empirical methods (like Meyerhof's method for layered soils) are required to calculate the composite capacity.

Key Takeaways

Shallow foundations generally fail in general, local, or punching shear modes, largely dictated by soil density.
The Net Ultimate Bearing Capacity represents the actual structural load capacity beyond the existing overburden pressure of the soil.
Terzaghi's fundamental equation calculates ultimate bearing capacity based on soil cohesion, overburden pressure, and footing width, but assumes a continuous strip footing and vertical loads.
Skempton's method provides a depth-dependent $N_c$ factor specifically tailored for saturated clays under undrained conditions.
Meyerhof's General Bearing Capacity Equation improves upon Terzaghi by incorporating shape, depth, and load inclination factors for more complex, realistic scenarios.
Eccentric loads are handled by reducing the foundation dimensions to an "effective area" where the load acts concentrically.
The presence of a high groundwater table significantly reduces bearing capacity by lowering the effective unit weight of the soil beneath and above the footing.
A substantial Factor of Safety (typically 2.5 to 3.0) is applied to the ultimate capacity to determine the safe allowable bearing capacity for design.

Bearing Capacity Simulator (Square Footing)

Width, B (m): 2

Depth, Df (m): 1

Friction Angle,

\\phi

(deg): 30

Cohesion, c (kPa): 10

Ultimate Bearing Capacity

0.0 kPa

Calculated using Terzaghi's formula for square footing.

The red dashed lines represent the potential shear failure surface in the soil. As $\\phi$ increases, the failure surface extends further outward, mobilizing more soil resistance.

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Overview

Modes of Shear Failure

Procedure

Gross vs. Net Bearing Capacity

Pressure Definitions

Terzaghi's Bearing Capacity Equation (1943)

Terzaghi's Equation

Terzaghi's N Factors

Modifications for Foundation Shape

Skempton's Bearing Capacity for Clays

Undrained Conditions (ϕ=0\phi = 0ϕ=0)

The General Bearing Capacity Equation (Meyerhof, 1963)

Vesic's Compressibility Factors

Accounting for Soil Compressibility

Eccentric Loading

Effective Area Method (Meyerhof)

Effect of Groundwater Table

Bearing Capacity on Layered Soils

Strong over Weak vs. Weak over Strong

Bearing Capacity Simulator (Square Footing)

Undrained Conditions ( $\phi = 0$ )