Ultimate Bearing Capacity formula |
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\( q_u \;=\; ( c \cdot N_c) + (\gamma \cdot D_f \cdot N_q) + (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma}) \) (Ultimate Bearing Capacity) \( c \;=\; \dfrac{ q_u - (\gamma \cdot D_f \cdot N_q) - (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma}) }{ N_c } \) \( N_c \;=\; \dfrac{ q_u - (\gamma \cdot D_f \cdot N_q) - (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma}) }{ c } \) \( \gamma \;=\; \dfrac{ q_u - c \cdot N_c }{ ( D_f \cdot N_q) + (0.5 \cdot W_f \cdot N_{\gamma}) } \) \( D_f \;=\; \dfrac{ q_u - ( c \cdot N_c ) - (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma}) }{ \gamma \cdot N_q } \) \( N_q \;=\; \dfrac{ q_u - ( c \cdot N_c ) - (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma}) }{ \gamma \cdot D_f } \) \( W_f \;=\; \dfrac{ 2 \cdot [ \; q_u - ( c \cdot N_c ) - ( \gamma \cdot D_f \cdot N_q )\; ] }{ \gamma \cdot N_{\gamma} } \) \( N_{\gamma} \;=\; \dfrac{ 2 \cdot [ \; q_u - ( c \cdot N_c ) - ( \gamma \cdot D_f \cdot N_q )\; ] }{ \gamma \cdot W_f } \) |
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| Symbol | English | Metric |
| \( q_u \) = Ultimate Bearing Capacity | \(lbf\;/\;in^2\) | \(Pa\) |
| \( c \) = Cohesion (Internal Molecular Attraction) | \(lbf\;/\;in^2\) | \(Pa\) |
| \( N_c \) = Shape Factor | \(dimensionless\) | \(dimensionless\) |
| \( D_f \) = Foundation Depth | \(ft\) | \(m\) |
| \( N_q \) = Depth Factor | \(dimensionless\) | \(dimensionless\) |
| \( \gamma \) (Greek symbol gamma) = Unit Weight of Soil | \(lbf\) | \(N\) |
| \( W_f \) = Foundation Width | \(ft\) | \(m\) |
| \( N_{\gamma} \) = Inclination Factor | \(dimensionless\) | \(dimensionless\) |
Ultimate bearing capacity, abbreviated as \( q_u \), is the maximum contact stress that soil or rock can sustain at the base of a foundation immediately prior to shear failure. It represents a limit state condition in which the applied load causes the supporting soil mass to reach its shear strength along a defined failure surface. At this point, any additional load produces large settlements with little or no increase in resistance. Ultimate bearing capacity is therefore a strength-based parameter and does not include any factor of safety.
For shallow foundations, the ultimate bearing capacity for a strip footing under vertical, concentric loading on homogeneous soil, is expressed as the sum of three components: a cohesion term, an overburden pressure term, and a soil unit weight term. These components incorporate dimensionless bearing capacity factors that depend solely on the soil’s angle of internal friction. The equation represents equilibrium at failure and assumes a continuous footing, level ground surface, and general shear failure conditions.
Ultimate bearing capacity is distinct from allowable bearing capacity. It corresponds strictly to the stress at shear failure and does not account for uncertainties in soil properties, construction tolerances, load variability, or serviceability requirements. In design practice, ultimate bearing capacity is reduced by an appropriate factor of safety to obtain an allowable value, or it is incorporated into reliability-based or limit state design frameworks.
Failure associated with ultimate bearing capacity may occur as general shear, local shear, or punching shear, depending on soil density or consistency and footing embedment conditions. Dense sands and stiff clays typically exhibit well-defined failure surfaces characteristic of general shear failure, while loose sands and soft clays may experience progressive or punching-type failures. In summary, ultimate bearing capacity is the theoretical maximum stress that a soil mass can support beneath a foundation before shear failure occurs. It is a fundamental geotechnical parameter derived from soil shear strength and foundation geometry and forms the basis for safe and serviceable foundation design.
