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} = ½ BÎ³ NÎ³ sÎ³ dÎ³ iÎ³ gv bÎ³ + qN_{q} s_{q} d_{q} i_{q} g_{q} b_{q} + cN_{c} s_{c} d_{c} i_{c} g_{c} b_{c} ———- (1)

**Table 1 Shape, Depth, load Inclination, Ground and Base Inclination factors. Use term with prime factor when É¸ = 0**

Shape Factors |
Depth Factors |
Inclination Factors |
Other Factors
(base on slope) |

S_{c}‘ = 0.2 B/L
S
S |
d_{c}‘ = 0.4 k
d k = D/B for D/Bâ‰¤ 1, k = tan k in radians |
i_{c}‘ = 0.5 – 0.5√(1-H/A_{f}c_{a})
i |
g_{c}‘ = Î²^{0}/147^{0}
g |

S_{q} = 1 + B/L sinÉ¸ |
d_{q} = 1+2tanÉ¸Ã—(1-sinÉ¸)^{2}k
k defined above |
i_{q} = [1-(0.5H)/(V+ A_{f}c_{a} cotÉ¸)]^{5} |
g_{q} = gÎ³
= (1-0.5 tanÎ²) |

S_{q} = 1-0.4B/L |
dÎ³ = 1 for all É¸ | iÎ³=[1-(0.7H)/(V+ A_{f}c_{a} cotÉ¸)]^{5}
iÎ³ = [1-{0.7H –(Î· /{V+ A |
Base factors
(tilted base) b b b bÎ³ = e^(-2.7Î· tanÉ¸) Î· 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) cotÉ¸

N_{q} = [e^(Ï€tanÉ¸)]tan^{2} (45^{0} – É¸/2)

NÎ³ = 1.5(N_{q} – 1)tanÉ¸

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 NÎ³ term in the bearing capacity equation and if H is parallel to L use L’ with NÎ³. For a footing on clay with É¸=O 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.

Note:

i_{q}, iÎ³ > 0

A_{f} = Effective footing area B^{‘}Ã—L^{‘} for eccentric loading

C_{a} = Adhesion to base = cohesion or a reduced value

D = Depth of footing (Used with B and not B’)

e_{B}, e_{L}: Eccentricity of load with respect to center of the footing area

H = Horizontal component of the footing load with H â‰¤ VtanÎ´ + c_{aA}f

V = Total vertical load on footing

Î’ = Slope of ground away from base with downward (+)

Î” = Friction angle between base and soil, Î´ = É¸ for concrete on soil

Î· = Tilt angle of base from horizontal with (+) upwards as usual case

**General Case**

- Do not use s
_{i}in combination with i_{i} - Can use s
_{i}in combination with d_{i}, g_{i}, and b_{i} - For L/ B less than or equal to 2 use É¸
_{u} - For L/B> 2 use É¸
_{ps}= 1.5 É¸_{u }– 17 - For É¸ <1 34
^{0}Use É¸_{ps}= É¸_{u}