Hansen's Bearing Capacity Theory - Engineering Projects

Hansen 1970 proposed a general bearing capacity equation. This equation is widely used because the equation can be used for both shallow as well as deep foundation. Full scale test on footings has indicated that the Hansen equation gives better correlation than the Terzaghi’s equation. Terzaghi’s equation is known to give conservative results. However, it is still in wide use for its simplicity. The proposed form of the equation is: $Q_{ult} = \dfrac{1}{2}B\gamma N_{\gamma}s_{\gamma}d_{\gamma}i_{\gamma}g_{\gamma}b_{\gamma} + qN_qs_qd_qi_qg_qb_q + cN_cs_cd_ci_cg_cb_c$ ———- (1)

Table 1 Shape, Depth, load Inclination, Ground and Base Inclination factors. Use term with prime factor when $\phi = 0$

Shape Factors

Depth Factors

Inclination Factors

Other Factors

Ground factors

(base on slope)

$s_c' = 0.2\dfrac{B}{L}$

$s_c = 1+ \dfrac{N_q}{N_c}\times \dfrac{B}{L}$

S_c = 1 for strip

d_c‘ = 0.4 k

d_c = 1 + 0.4k

$k = \dfrac{D}{B}$ for $\dfrac{D}{B}\geq 1,$ ,

$k = tan^{-1}\dfrac{D}{B}$ for $[\dfrac{D}{B}>1]$

k in radians

$i_c' = 0.5-0.5\sqrt{1-\dfrac{H}{A_fc_a}}$

$i_c = i_q-\dfrac{1-i_q}{N_q-1}$

$g_c' = \dfrac{\beta^0}{147^0}$

$g_c' = 1-\dfrac{\beta^0}{147^0}$

$s_q = 1 + \dfrac{B}{L}sin\phi$

$d_q = 1 + 2 tan\phi \times (1 - sin\phi)^2k$

k defined above

$i_q = (1-\dfrac{0.5H}{V + A_fc_acon\phi})^5$

$g_q = g\gamma$

= $(1-0.5 tan\beta)^5$

S_q = 1-0.4B/L

$d_{\gamma} = 1$ for all $\phi$

$i_{\gamma} = (1-\dfrac{0.7H}{V + A_fc_acon\phi})^5$

$i_{\gamma} = (1-\dfrac{0.7-\dfrac{\eta ^0}{(450^0)^5}}{V + A_fc_acot\phi})^{\alpha 2}$

Base factors

(tilted base)

$b_c' = \dfrac{\eta^0}{147^0}$ $b_c = 1 - \dfrac{\eta^0}{147^0}$ $b_q = e^{-2\eta tan \phi}$ $b_{\gamma} = e^{-2.7 \eta tan}$

$\eta$ in radians

Where s, d, i, g, b are the shape, depth, inclination and ground factors. For pure cohesive soil the above equation takes the form of:

$Q_{ult} = cN_C(1 + s_c + d_c - i_c - g_c -b_c) + q$ ———- (2)

The bearing capacity factors are given by:

$N_c = (N_q-1) cot\phi$ $N_q = (e^{\pi tan\phi})tan^2(45^0 + \dfrac{\phi}{2})$ $N_{\gamma} = 1.5(N_q - 1) tan\phi$

Hansen’s shape, depth and other factors are given in Table 4.1 below. Hansen’s equation also takes into consideration of base tilting and footings on slopes. When the values used in the inclination equations has the horizontal load component H parallel to B, one should use B’ with the $N_{\gamma}$ term in the bearing capacity equation and if H is parallel to L use L’ with $N_{\gamma}$ . For a footing on clay with $\phi = 0$ compute i_c using H parallel to B and/or L as appropriate and note that it is a subtractive constant in the modified bearing capacity equation. When the base is tilted, the component H and V are perpendicular and parallel to the base respectively as compared with when it is horizontal. For footing on slopes g_i factors are used to reduce the bearing capacity.