Bearing Capacity of Shallow Foundations

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.

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.

It is important to distinguish between the total pressure at the base of the footing and the new pressure added by the structure.

• 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.

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.

The precise mathematical derivations for Terzaghi's dimensionless bearing capacity factors ($N_c$, $N_q$, $N_\gamma$) depend strictly on the effective friction angle ($\phi'$) of the soil.

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.

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.

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.

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:

The bearing capacity $q_u$ is calculated using $B'$ and $L'$, and the total allowable load is $Q_{allow} = (q_u / FS) \times B' \times L'$.

The location of the groundwater table critically affects the unit weight ($\gamma$) terms in the bearing capacity equations.

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.