Vesic Bearing Capacity

\[ q_u = cN_c s_c d_c i_c + qN_q s_q d_q i_q + 0.5 \gamma B N_{\gamma} s_{\gamma} d_{\gamma} i_{\gamma} \]

Bearing capacity factors

• \(N_c = \cot(\phi) \left(N_q - 1\right)\)
• \(N_q = \tan^2\left(45 + \frac{\phi}{2}\right) \cdot e^{\pi \tan(\phi)}\)
• \(N_{\gamma} = 2(N_q + 1) \tan(\phi)\)

Shape factors

Shape factors for strip footing

• \(s_c = 1.0\)
• \(s_q = 1.0\)
• \(s_{\gamma} = 1.0\)

Shape factors for rectangular footing

• \(s_c = 1 + \dfrac{B}{L} \cdot \dfrac{N_q}{N_c}\)
• \(s_q = 1 + \dfrac{B}{L} \cdot \tan(\phi)\)
• \(s_{\gamma} = 1.0 - 0.4 \dfrac{B}{L}\)

Shape factors for square or circular footing

• \(s_c = 1 + \dfrac{N_q}{N_c}\)
• \(s_q = 1 + \tan(\phi)\)
• \(s_{\gamma} = 0.6\)

Depth factors

These equations were provided by Hansen (1970)

• For \(\dfrac{D_f}{B} \le 1\)

  • For \(\phi = 0^{\circ}\)

    • \(d_c = 1 + 0.4 \dfrac{D_f}{B}\)
    • \(d_q = 1\)
    • \(d_{\gamma} = 1\)

  • For $\phi \gt 0^{\circ} $

    • \(d_c = d_q - \dfrac{1 - dq}{N_c \tan(\phi)}\)
    • \(d_q = 1 + 2 \tan(\phi) (1 - \sin(\phi)^2)(\dfrac{D_f}{B})\)
    • \(d_{\gamma} = 1\)
  • For \(\dfrac{D_f}{B} \gt 1\)
  • For \(\phi = 0\)
    • \(d_c = 1 + 0.4 \cdot \underbrace{\tan^{-1}\left(\dfrac{D_f}{B}\right)}_{\text{radians}}\)
    • \(d_q = 1\)
    • \(d_{\gamma} = 1\)
  • For \(\phi \gt 0^{\circ}\)
    • \(d_c = d_q - \dfrac{1 - dq}{N_c \tan(\phi)}\)
    • \(d_q = 1 + 2 \tan(\phi) (1 - \sin(\phi)^2) \cdot \underbrace{\tan^{-1}\left(\dfrac{D_f}{B}\right)}_{\text{radians}}\)
    • \(d_{\gamma} = 1\)

Inclination factors

These equations were provided by Meyerhof (1963); Hanna and Meyerhof (1981)

• \(i_c = i_q = \left (1 - \dfrac{\beta^{\circ}}{90^{\circ}}\right)^2\)
• \(i_{\gamma} = \left(1 - \dfrac{\beta}{\phi} \right)^2\)

where \(\beta = \text{inclination of the load on the foundation with respect to the vertical}\)

\(\beta\) should be in the same units as \(\phi\) (degrees or radians) for \(i_{\gamma}\).