1.4 Bearing Capacity

Introduction to Bearing Capacity

Bearing capacity is the ability of the underlying soil to support the loads applied by a foundation structure without undergoing a shear failure. In construction engineering, evaluating bearing capacity is critical when designing shallow foundations (like spread footings and mat foundations) or when placing heavy construction equipment (like large crane mats) on the ground surface. If the applied pressure exceeds the soil's ultimate bearing capacity, the foundation will punch into the ground, and the soil will violently shear outward, causing catastrophic failure.

Terzaghi's Bearing Capacity Theory

Karl Terzaghi developed the foundational theory for calculating the ultimate bearing capacity ($q_{ult}$) of shallow continuous (strip) foundations. Terzaghi modeled a general shear failure mechanism where a wedge of soil beneath the footing is pushed downward, displacing adjacent soil zones laterally and upward until failure surfaces reach the ground surface.

The general Terzaghi equation for the ultimate bearing capacity of a continuous strip footing is:

$q_{ult} = c N_c + \gamma D_f N_q + 0.5 \gamma B N\gamma$

where:

• $c$ = Cohesion of the soil
• $\gamma$ = Unit weight of the soil
• $D_f$ = Depth of embedment of the foundation
• $B$ = Width of the foundation
• $N_c, N_q, N_\gamma$ = Terzaghi's bearing capacity factors (dimensionless)

The bearing capacity factors ($N_c, N_q, N_\gamma$) are purely a function of the soil's effective angle of internal friction ($\phi'$). These values are typically obtained from charts or tables provided in the reference handbook.

The equation is elegantly composed of three distinct terms that contribute to bearing capacity:

1. The Cohesion Term ($c N_c$): Represents the contribution of the soil's cohesive shear strength. For purely cohesionless soils (sands), $c=0$, and this term drops out.
2. The Surcharge Term ($\gamma D_f N_q$): Represents the resistance provided by the weight of the soil above the foundation base depth ($D_f$). The deeper the footing is embedded, the harder it is for the failure wedge to push soil up to the surface.
3. The Footing Dimension Term ($0.5 \gamma B N_\gamma$): Represents the resistance provided by the weight of the soil wedge directly beneath the footing. Wider footings engage a larger volume of soil, increasing capacity.

Factors Affecting Bearing Capacity

Several practical factors require modifications to the standard Terzaghi equation:

Footing Shape

Terzaghi's original equation applies to infinitely long continuous footings. For square, rectangular, or circular footings, empirical shape factors are applied to the terms. For example, for a square footing of width $B$:

$q_{ult} = 1.3 c N_c + \gamma D_f N_q + 0.4 \gamma B N\gamma$

Groundwater Table

The presence of groundwater severely impacts bearing capacity by reducing the effective unit weight of the soil, thereby reducing the surcharge and dimension terms.

• If the water table is at or above the ground surface, the buoyant unit weight ($\gamma' = \gamma_{sat} - \gamma_w$) must be used in both the surcharge and dimension terms.
• If the water table is exactly at the base of the footing, the surcharge term uses the total unit weight, but the dimension term must use the buoyant unit weight.
• If the water table is deep below the footing base (typically deeper than width $B$ below the base), the water table has no effect on bearing capacity.

Load Inclination and Eccentricity

If the applied load is not strictly vertical (inclined) or is applied off-center (eccentric), the bearing capacity is significantly reduced. Eccentric loads require the use of an "effective width" ($B' = B - 2e$), where $e$ is the eccentricity.

Ultimate vs. Allowable Bearing Capacity

The ultimate bearing capacity ($q_{ult}$) is the exact pressure that will cause a catastrophic shear failure. In engineering design, we never load a soil to its ultimate capacity. We use an allowable bearing capacity ($q_{all}$) by applying a generous Factor of Safety (FS).

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

In geotechnical engineering, the standard Factor of Safety against bearing capacity failure is typically 3.0. This high FS accounts for the inherent variability of soil properties, uncertainties in the analytical models, and limits settlement to acceptable levels.

Net Bearing Capacity: Sometimes, problems ask for the net allowable bearing capacity ($q_{net(all)}$). The net ultimate capacity is the ultimate capacity minus the original overburden pressure at the foundation depth ($\gamma D_f$).

$q_{net(all)} = \frac{q_{ult} - \gamma D_f}{FS}$

Worked Example: Square Footing Capacity

Problem: A 5 ft by 5 ft square footing is to be embedded 3 ft deep in a stiff cohesive soil (clay) with a unit weight of 120 pcf, a cohesion of 1500 psf, and a friction angle of $\phi = 0^\circ$. The groundwater table is located 20 ft below the ground surface. For $\phi = 0^\circ$, the bearing capacity factors are $N_c = 5.7$, $N_q = 1.0$, and $N_\gamma = 0$. Determine the ultimate bearing capacity and the allowable bearing capacity using a Factor of Safety of 3.0.

Solution:

1. Identify parameters and appropriate equation: Since the footing is square, use the modified Terzaghi equation for square footings.

   • $c = 1500 \text{ psf}$
   • $\gamma = 120 \text{ pcf}$
   • $D_f = 3 \text{ ft}$
   • $B = 5 \text{ ft}$
   • The water table is deep, so use total unit weight.
   • For $\phi = 0$, $N_\gamma = 0$, so the third term drops out.

2. Calculate $q_{ult}$: $q_{ult} = 1.3 c N_c + \gamma D_f N_q + 0.4 \gamma B N\gamma$ $q_{ult} = 1.3(1500)(5.7) + (120)(3)(1.0) + 0$ $q_{ult} = 11115 + 360 = 11475 \text{ psf}$

3. Calculate $q_{all}$: $q_{all} = \frac{q_{ult}}{FS} = \frac{11475}{3.0} = 3825 \text{ psf}$

Answer: The ultimate bearing capacity is 11,475 psf, and the allowable bearing capacity is 3,825 psf.

Exam Tips

• The bearing capacity of pure clay ($\phi=0$) relies entirely on cohesion and embedment depth. The width of the footing ($B$) does not affect the ultimate bearing capacity of a purely cohesive soil.
• When structural loads are given (e.g., Column Load = 100 kips), remember to divide by the foundation area ($B \times L$) to convert the load to an applied pressure before comparing it to the bearing capacity.
• Always scrutinize the water table location in bearing capacity problems. Using total unit weight instead of buoyant unit weight when the water table is high is a guaranteed path to a wrong answer.