Understanding Shallow Foundation Bearing Capacity

Shallow foundations such as spread footings, mat slabs, and strip footings transfer structural loads into soil at relatively small depths. Their performance hinges on the ability of the supporting soil to resist shearing along potential failure surfaces beneath the footing. In the early twentieth century, Karl Terzaghi synthesized empirical observations and limit equilibrium analyses into a practical set of equations for estimating the ultimate bearing capacity of footings resting on homogeneous soil. These equations remain a cornerstone of geotechnical engineering practice. The calculator above applies Terzaghi’s general shear failure model, which assumes the soil behaves as a weighty, frictional continuum characterized by cohesion c, unit weight γ, and internal friction angle φ.

The ultimate bearing pressure q_ult represents the maximum load per unit area the soil can sustain before a failure mechanism develops. For a strip footing of width B at depth D_f, Terzaghi expressed the ultimate pressure as the sum of cohesion, overburden, and footing width contributions, each multiplied by dimensionless bearing capacity factors. The equation is often written as

$$q_{\mathrm{ult}}=cN{c}_{}+\gamma D{f}_{}N{q}_{}+0.5\gamma BN{\gamma }_{}$$

The factors N_c, N_q, and N_γ depend solely on the soil’s friction angle. Their functional forms derive from limit equilibrium solutions for a strip footing and are given by

$$N{q}_{}=e^{\pi tan\phi }[tan(\frac{\pi }{4}+\frac{\phi }{2}\left)\right]{\left\{}^2$$

In applying these formulas, care must be taken with units. The soil unit weight γ should be expressed in kilonewtons per cubic meter or pounds per cubic foot consistent with other terms. Cohesion c is likewise taken as a unit stress. The width B and depth D_f appear linearly, indicating that wider or deeper foundations increase bearing capacity, though with diminishing returns. For typical building footings, factors of safety between two and three are used to reduce q_ult to an allowable bearing pressure q_all = q_ult/FS. The allowable value is compared against the applied contact stress from structural loads to assess adequacy.

Table 1 lists indicative ranges of soil parameters for common materials. Cohesion and friction angle vary dramatically depending on mineral composition, density, and moisture content. Clays derive shear strength primarily from cohesion, whereas sands rely on frictional resistance. Silty or mixed soils fall between these extremes. Unit weight reflects not only the specific gravity of solids but also the degree of saturation. Engineers often obtain these parameters from laboratory tests on undisturbed samples or from field tests such as the standard penetration test.

Soil Type	c (kPa)	φ (deg)	γ (kN/m³)
Soft Clay	15 – 25	0 – 5	16 – 18
Stiff Clay	50 – 100	10 – 20	18 – 20
Loose Sand	0	28 – 32	15 – 17
Dense Sand	0	34 – 40	17 – 20
Sandy Gravel	0	36 – 42	19 – 21

Beyond material properties, footing geometry also influences performance. The Terzaghi equation was derived for strip footings with a length much greater than its width. For square or circular footings, shape factors modify the contribution of each term. Similarly, inclined or eccentric loads reduce bearing capacity relative to the concentric case. These refinements are omitted here for clarity, but the explanation highlights their existence. Engineers designing heavily loaded structures should consult comprehensive geotechnical texts that treat these effects rigorously.

The failure mode assumed in the Terzaghi model is a classic general shear failure, featuring a well-defined wedge of soil beneath the footing and radial shear surfaces reaching the ground surface. In loose sands or soft clays, however, local shear or punching shear failures may occur, resulting in lower ultimate loads and higher settlements. Some design methodologies apply correction factors when these alternative mechanisms govern. Additionally, the presence of groundwater near the footing reduces effective stress and therefore bearing capacity. Accounting for buoyant unit weight in the equation provides a conservative estimate when water tables are high.

Settlement considerations often govern design before shear failure is approached. Even if a footing can theoretically carry a large load, excessive settlement can damage superstructures or utility connections. Consolidation of clay layers may continue for years after construction, while granular soils experience immediate settlement. Geotechnical engineers routinely perform settlement analyses using elastic theory or empirical correlations to ensure the predicted movement remains within acceptable limits. The bearing capacity computation presented here represents only one component of a complete foundation design.

When comparing alternative footing dimensions or soil improvement techniques, the calculator provides a rapid way to gauge relative benefits. Increasing width from 2 to 3 meters, for instance, raises the third term in the equation by 50 percent, yet the overall improvement in allowable pressure may be less if cohesion and overburden contributions dominate. Soil stabilization methods such as compaction, grouting, or replacement can substantially boost friction angle or unit weight, yielding greater increases in capacity than enlarging the footing. Designers must weigh these options against cost, constructability, and schedule constraints.

The simplicity of the Terzaghi bearing capacity formula belies the complexity of soil behavior. Soils are heterogeneous, anisotropic, and sensitive to moisture changes. Despite these challenges, the equation continues to serve as a reliable starting point for preliminary design and educational exercises. By experimenting with different input values in this calculator, students and practitioners can develop an intuition for how soil parameters influence foundation performance. They can also appreciate the importance of field investigations in providing defensible data. As with all geotechnical calculations, the results should be tempered with caution, validated against local experience, and supplemented with professional engineering judgment.