Governing Equations:
Section 5.4.1.2.1 of BS 8004:2015
The following expression may be used to calculate the ultimate bearing resistance of
a spread foundation (Rv) on coarse soils using effective stress parameters:
\(\frac{R'_v}{A'} = c' N_c b_c s_c i_c d_c g_c r_c + q' N_q b_q s_q i_q d_q g_q r_q + 0.5 \gamma' B' N_\gamma b_\gamma s_\gamma i_\gamma d_\gamma g_\gamma r_\gamma \)
where:
c′, q′, and γ′ are as defined in BS EN 1997-1:2004+A1:2013, D.4;
B′ is defined in BS EN 1997-1:2004+A1:2013, D.4; and
the various coefficients Nc, bc, sc, etc. are defined below, after Poulos et al. (2001)
The bearing coefficients in the above equation should be calculated from:
\[ N_q = e^{\pi \tan \varphi} \tan^2\left(45^\circ + \frac{\varphi}{2} \right) \]
\[ N_\gamma = \begin{cases} a \cdot e^{b \varphi} & \text{for } \varphi > 0^\circ \\ 0 & \text{for } \varphi = 0^\circ \end{cases} \]
\[ N_c = (N_q - 1) \cot \varphi \]
The shape factors should be calculated from:
\[ s_q = 1 + \left(\frac{B}{L}\right) \tan \varphi \]
\[ s_\gamma = 1 - 0.4 \left(\frac{B}{L}\right) \]
\[ s_c = 1 + \left(\frac{B}{L}\right) \left(\frac{N_q}{N_c}\right) \]
The load inclination factors should be calculated from:
\[ i_q = \left[ 1 - \left( \frac{H}{V + A' c' \cot \varphi} \right)^m \right] \]
\[ i_\gamma = \left[ 1 - \left( \frac{H}{V + A' c' \cot \varphi} \right)^{m + 1} \right] \]
\[ i_c = \begin{cases} i_q - \left( \frac{1 - i_q}{N_c \tan \varphi} \right) & \text{for } \varphi > 0^\circ \\ 1 - \left( \frac{mH}{c' N_c A'} \right) & \text{for } \varphi = 0^\circ \end{cases} \]
The base inclination (also known as "foundation tilt") factors should be calculated from:
\[ b_q \approx b_\gamma \]
\[ b_\gamma = \left( 1 - \alpha \tan \varphi \right)^2 \]
\[ b_c = \begin{cases} b_q - \left( \frac{1 - b_q}{N_c \tan \varphi} \right) & \text{for } \varphi > 0^\circ \\ 1 - \left( \frac{2\alpha}{\pi + 2} \right) & \text{for } \varphi = 0^\circ \end{cases} \]
The ground inclination (also known as "surface tilt") factors should be calculated from:
\[ g_q = \begin{cases} \left( 1 - \tan \omega \right)^2 & \text{for } \varphi > 0^\circ \\ 1 & \text{for } \varphi = 0^\circ \end{cases} \]
\[ g_\gamma \approx g_q \]
\[ g_c = \begin{cases} g_q - \left( \frac{1 - g_q}{N_c \tan \varphi} \right) & \text{for } \varphi > 0^\circ \\ 1 - \left( \frac{2\omega}{\pi + 2} \right) & \text{for } \varphi = 0^\circ \end{cases} \]
The depth factors should be calculated from:
\[ d_q = 1 + 2 \tan \varphi (1 - \sin \varphi)^2 \tan^{-1} \left( \frac{D}{B} \right) \]
\[ d_\gamma = 1 \]
\[ d_c = \begin{cases} d_q - \left( \frac{1 - d_q}{N_c \tan \varphi} \right) & \text{for } \varphi > 0^\circ \\ 1 + 0.33 \tan^{-1} \left( \frac{D}{B} \right) & \text{for } \varphi = 0^\circ \end{cases} \]
The rigidity factors should be calculated from:
\[ r_q = \exp \left[ \left( -4.4 + 0.6 \frac{B}{L} \right) \tan \varphi + \frac{3.07 \sin \varphi \log_{10} 2 I_r}{1 + \sin \varphi} \right] \]
\[ r_\gamma = r_q \]
\[ r_c = \begin{cases} r_q - \left( \frac{1 - r_q}{N_c \tan \varphi} \right) & \text{for } \varphi > 0^\circ \\ 0.32 + 0.12 \left( \frac{B}{L} \right) + 0.60 \log_{10} I_r & \text{for } \varphi = 0^\circ \end{cases} \]
where
φ is angle of shearing resistance of the soil;
a is 0.0663 for a smooth foundation or 0.1054 for a rough foundation;
b is 9.3 or 9.6 for a smooth or rough foundation, respectively (when φ is entered in radians);
alternatively, b = 0.162 or 0.168, respectively (when φ is entered in degrees);
B (also known as W) is the breadth or width of the foundation on plan;
L is the length of the foundation on plan;
D is the depth to the underside of the foundation;
A′ is the effective area of the foundation;
H is the horizontal force applied to the foundation;
V is the vertical force applied to the foundation;
m is (2 + B/L) / (1 + B/L) for loading in the direction of B or (2 + L/B) / (1 + L/B) for loading in the direction of L;
α is the inclination of the underside of the footing from the horizontal;
ω is the inclination of the ground surface below the horizontal in the direction away from the foundation;
Ir is G / (c′ + σ′v tan φ);
G is the soil’s shear modulus of elasticity;
c′ is the soil’s effective cohesion; and
σ′v is the vertical effective stress on the foundation.
