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Ultimate Bearing Capacity

Ultimate bearing capacity, abbreviated as \( q_u \), Is defined as the maximum pressure per unit area that soil can sustain at the base of a foundation before shear failure occurs in the supporting soil mass.  When the applied stress from a structure exceeds this limit, the soil beneath the foundation fails and the foundation may experience excessive settlement or collapse.  In practical design, engineers do not allow structures to approach this limit, instead, the allowable bearing capacity is used, which is the ultimate bearing capacity divided by a factor of safety. 

Ultimate Bearing Capacity formula

\( q_u \;=\; ( c \cdot N_c) + (\gamma \cdot D_f \cdot N_q) + (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma}) \)     (Ultimate Bearing Capacity)

\( c \;=\;  \dfrac{  q_u  - (\gamma \cdot D_f \cdot N_q)  - (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma})  }{  N_c } \)

\( N_c \;=\;  \dfrac{  q_u  - (\gamma \cdot D_f \cdot N_q)  - (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma})  }{  c } \)

\( \gamma \;=\;  \dfrac{  q_u - c \cdot N_c  }{  ( D_f \cdot N_q) + (0.5 \cdot  W_f \cdot N_{\gamma})  } \)

\( D_f \;=\;  \dfrac{  q_u  - ( c \cdot N_c )  - (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma})  }{  \gamma \cdot N_q } \)

\( N_q \;=\;  \dfrac{  q_u  - ( c \cdot N_c )  - (0.5 \cdot \gamma \cdot W_f \cdot N_{\gamma})  }{  \gamma \cdot D_f } \)

\( W_f \;=\;  \dfrac{  2 \cdot  [ \; q_u  - ( c \cdot N_c )  - ( \gamma \cdot D_f \cdot N_q )\; ] }{  \gamma \cdot N_{\gamma} } \)

\( N_{\gamma} \;=\;  \dfrac{  2 \cdot  [ \; q_u  - ( c \cdot N_c )  - ( \gamma \cdot D_f \cdot N_q )\; ] }{  \gamma \cdot W_f } \)
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 \(dimensionless\) \(dimensionless\)
Bearing Capacity 2
Ultimate bearing capacity is created by the shear strength and confinement of the soil beneath a foundation.  The soil resists the applied load through a combination of cohesion, friction between soil particles, and the confining pressure created by the overlying soil and foundation geometry.  The magnitude of the ultimate bearing capacity therefore depends on several measurable parameters, including soil cohesion, internal friction angle, unit weight of soil, foundation width, and depth of embedment.  These factors collectively determine the stress distribution and the shape of the shear failure surfaces that develop beneath the foundation.
 
When the applied stress reaches the ultimate bearing capacity, the soil fails along shear zones that extend outward and upward from the footing. In geotechnical theory, three primary failure modes are recognized: general shear failure, local shear failure, and punching shear failure, depending on soil density and stiffness.

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