Bearing Capacity - Theory & Concepts

Bearing Capacity

The bearing capacity of a soil is the maximum contact pressure that the soil can practically support without failing in shear. This is the primary safety criterion for the design of shallow foundations (

D_f \le B

Terzaghi's Bearing Capacity Theory

Karl Terzaghi (1943) proposed the first comprehensive, widely adopted theory for the ultimate bearing capacity (

q_u

) of a continuous, rough strip footing.

General Equation (Strip Footing)

Terzaghi's Ultimate Bearing Capacity

Classic three-term formula for the ultimate bearing capacity of a continuous strip footing on a semi-infinite soil mass.

q_{u} = c N_c + q N_q + 0.5 \gamma B N_\gamma

Variables

Symbol	Description	Unit
$q_{u}$	Ultimate bearing capacity	-
$c$	Cohesion of the soil	-
$N_{c}$	Bearing capacity factor for cohesion	-
$q$	Effective overburden pressure at foundation base (\gamma D_f)	-
$N_{q}$	Bearing capacity factor for surcharge	-
$\gamma$	Unit weight of soil below foundation	-
$B$	Width of the strip footing	-
$N_\gamma$	Bearing capacity factor for soil weight	-

$c N_c$ : The cohesion term. Accounts for shear strength derived from soil cohesion.
$q N_q$ : The surcharge term. Accounts for the stabilizing weight of soil above the base.
$0.5 \gamma B N_\gamma$ : The density (wedge) term. Accounts for resistance provided by soil weight below the base.
$N_c, N_q, N_\gamma$ are strictly mathematical functions of the internal friction angle ( $\phi$ ).

Shape Factors

Because Terzaghi's original equation was explicitly derived for an infinitely long strip footing, empirical shape factors are applied to modify the equation for other common foundation geometries:

Square Footing ( $B \times B$ ):

Bearing Capacity (Square Footing)

Terzaghi's bearing capacity equation modified with empirical shape factors for a square footing.

q_u = 1.3 c N_c + q N_q + 0.4 \gamma B N_\gamma

Variables

Symbol	Description	Unit
$q_{u}$	Ultimate bearing capacity	-
$c$	Cohesion	-
$N_c, N_q, N_\gamma$	Bearing capacity factors	-
$q$	Overburden pressure	-
$\gamma$	Unit weight of soil	-
$B$	Width of the square footing	-

Circular Footing (Diameter $B$ ):

Bearing Capacity (Circular Footing)

Terzaghi's bearing capacity equation modified with empirical shape factors for a circular footing.

q_u = 1.3 c N_c + q N_q + 0.3 \gamma B N_\gamma

Variables

Symbol	Description	Unit
$q_{u}$	Ultimate bearing capacity	-
$c$	Cohesion	-
$N_c, N_q, N_\gamma$	Bearing capacity factors	-
$q$	Overburden pressure	-
$\gamma$	Unit weight of soil	-
$B$	Diameter of the circular footing	-

Allowable Bearing Capacity

To account for inherent soil variability and uncertainties in load estimation, the design load must provide an adequate Factor of Safety ( $FS$ ) against catastrophic shear failure.

Allowable Bearing Capacity

Safe design bearing capacity obtained by dividing the ultimate capacity by a factor of safety; typically FS = 3.0 for permanent structures.

q_{all} = \frac{q_u}{FS}

Variables

Symbol	Description	Unit
$q_{all}$	Allowable bearing capacity	-
$q_{u}$	Ultimate bearing capacity	-
$F S$	Factor of Safety	-

Typical FS: 3.0 is the standard for permanent structures.

Net Allowable Bearing Capacity ( $q_{all(net)}$ ): Represents the additional pressure the soil can safely take above the original in-situ overburden pressure.

Net Allowable Bearing Capacity

Allowable bearing capacity based on the net stress increase above the original overburden pressure; more accurate for foundation design.

q_{all(net)} = \frac{q_u - q}{FS}

Variables

Symbol	Description	Unit
$q_{all(net)}$	Net allowable bearing capacity	-
$q_{u}$	Ultimate bearing capacity	-
$q$	Original in-situ overburden pressure (\gamma D_f)	-
$F S$	Factor of Safety	-

Interactive Bearing Capacity Lab

Visualize how adjusting foundation dimensions (

B, D_f

) and critical soil properties (

c, \phi

) influences the ultimate bearing capacity curve.

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.

Effect of the Groundwater Table

The position of the groundwater table critically affects bearing capacity by altering the effective unit weight of the soil, thereby reducing its resisting weight.

Important

Water does not physically "lubricate" sand; rather, it introduces buoyancy (Archimedes' principle) that physically lifts the soil particles, reducing inter-particle friction (effective stress).

Water Table Correction Scenarios

Case 1: Water table at or above the ground surface ( $D_w = 0$ ): The soil is completely submerged. Use the buoyant unit weight ( $\gamma' = \gamma_{sat} - \gamma_w$ ) in all terms (for both $q$ and the wedge term). This effectively halves the bearing capacity.
Case 2: Water table exactly at foundation level ( $D_w = D_f$ ): Use the moist/dry unit weight ( $\gamma$ ) to calculate the surcharge term ( $q$ ). However, use the buoyant unit weight ( $\gamma'$ ) for the wedge term ( $0.5 \gamma' B N_\gamma$ ) since the failure wedge is fully submerged.
Case 3: Water table at depth $D_w \ge D_f + B$ : The water is too deep to intersect the theoretical failure surface. No correction is needed. Use the moist/dry unit weight ( $\gamma$ ) in all terms.
Intermediate Case ( $D_f < D_w < D_f + B$ ): Linearly interpolate an effective unit weight for the wedge term based on the exact depth of the water.

General Bearing Capacity Equation

Meyerhof (1963) and Hansen (1970) extended Terzaghi's classical theory to account for more complex, real-world conditions including rectangular shapes, deep embedment, and inclined loading.

The General Equation

General Bearing Capacity Equation

Extended Meyerhof-Hansen formula that accounts for foundation shape, depth of embedment, and inclined loading through correction factors.

q_u = c N_c F_{cs} F_{cd} F_{ci} + q N_q F_{qs} F_{qd} F_{qi} + 0.5 \gamma B N_\gamma F_{\gamma s} F_{\gamma d} F_{\gamma i}

Variables

Symbol	Description	Unit
$q_{u}$	Ultimate bearing capacity	-
$c$	Cohesion	-
$N_c, N_q, N_\gamma$	Bearing capacity factors	-
$F_{cs}, F_{qs}, F_{\gamma s}$	Shape factors (accounting for B/L ratio)	-
$F_{cd}, F_{qd}, F_{\gamma d}$	Depth factors (accounting for D_f/B ratio)	-
$F_{ci}, F_{qi}, F_{\gamma i}$	Inclination factors (accounting for angled loads)	-
$q$	Effective overburden pressure	-
$\gamma$	Unit weight of soil	-
$B$	Width of foundation	-

Eccentric Loading (Meyerhof's Effective Area)

When a foundation is subjected to a bending moment in addition to a vertical load (eccentric loading), the soil pressure is not uniform. The bearing capacity is significantly reduced.

Meyerhof's Effective Area Method

To calculate bearing capacity under eccentric loading, Meyerhof proposed assuming the load is concentric on a smaller, fictitious effective area ( $B' \times L'$ ).

Calculate Eccentricity: $e = M / P$ .
Determine Effective Dimensions:
- $B' = B - 2e$ (if eccentricity is along the width).
- $L' = L - 2e$ (if eccentricity is along the length).
Calculate Ultimate Capacity ( $q_u'$ ): Use the General Bearing Capacity Equation substituting $B'$ for $B$ , and calculating shape factors based on $B'/L'$ .
Calculate Total Ultimate Load ( $Q_{ult}$ ): Multiply $q_u'$ by the effective area.

Total Ultimate Load (Eccentric)

Ultimate total load capacity of a foundation under eccentric loading using Meyerhof's effective area method.

Q_{ult} = q_u' \times (B' \times L')

Variables

Symbol	Description	Unit
$Q_{ult}$	Total ultimate load	-
$q_{u}^{'}$	Ultimate bearing capacity on effective area	-
$B^{'}$	Effective width (B - 2e)	-
$L^{'}$	Effective length (L - 2e)	-

Settlement Control

Bearing capacity strictly evaluates catastrophic shear failure (Ultimate Limit State). However, excessive Settlement (Serviceability Limit State) often governs the actual design, especially for very large footings resting on sand or clay.

Primary Types of Settlement

1. Immediate (Elastic) Settlement ( $S_e$ ): Occurs instantaneously as the load is applied, primarily due to elastic distortion of soil particles. Dominant in coarse-grained granular soils.

Elastic Settlement

Immediate settlement at the time of load application due to elastic distortion; governed by the soil's elastic modulus and Poisson's ratio.

S_e = q B \frac{1 - \nu^2}{E_s} I_s

Variables

Symbol	Description	Unit
$S_{e}$	Immediate (elastic) settlement	-
$q$	Net applied pressure	-
$B$	Width of footing	-
$ν \nu$	Poisson's ratio of the soil	-
$E_{s}$	Modulus of elasticity of the soil	-
$I_{s}$	Influence factor (depends on shape and rigidity)	-

2. Consolidation Settlement ( $S_c$ ): Time-dependent settlement caused by the slow, gradual expulsion of pore water from the voids in fine-grained cohesive soils (clays). (Detailed in the Compressibility chapter).

Key Takeaways

Bearing Capacity is defined as the maximum applied load per unit area that the soil can safely support without undergoing a catastrophic shear failure.
Terzaghi's Equation remains the foundational standard for analyzing shallow foundations, requiring specific modification via Shape Factors for non-strip geometries.
The Water Table acts as a severe destabilizing agent. It reduces the effective bearing capacity by roughly 50% if it rises to the surface due to the physical loss of effective soil weight (buoyancy).
Meyerhof's Effective Area Method ( $B' = B - 2e$ ) must be used to safely derate the bearing capacity of foundations subjected to eccentric loads (moments).
Factors of Safety (typically $FS = 3.0$ ) are universally applied to the ultimate capacity to derive the safe Allowable Bearing Capacity ( $q_{all}$ ) for structural design.
Practical foundation design is ultimately governed by the lower bounding value between Bearing Capacity (Shear Failure, ULS) and Settlement limits (Serviceability, SLS).

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