Hansen's Bearing-Capacity Equations

For footings that support eccentric loads.

Hansen (1970) proposed bearing-capacity equations that are an extension of the earlier Meyerhof work. Hansen's equations allow any D/B and thus can be used for shallow footings and deep bases.

The spreadsheet is based on Chapter 4-3 of Foundation Analysis and Design, Ed. 5 by Joseph E. Bowles (permission granted).

The equations for vertical load with moments on a horizontal base are:
PHI>0: q-ult = c*Nc*d-c*s-c + q-bar*Nq*d-q*s-q + 0.5*gamma*B'*Ngam*d-gam*s-gam
PHI=0: q-ult = 5.14*su*(1 + s'c + d'c) + q-bar
where:
c = soil cohesion in kPa
su = undrained shear strength in kPa
Nc = cohesion multiplier = (Nq - 1)*cot(PHI)
d-c = depth factor for cohesion = 1 + 0.4*k
d'c = depth factor for PHI = 0, d'c = 0.4*k
where k = D/B for D/B <= 1 else k = arctan(D/B)
s-c = shape factor for cohesion = 1 + (Nq/Nc)*(B'/L')
s'c = shape factor for PHI = 0, s'c = 0.2*B'/L'
q-bar = overburden pressure in kPa = gamma*D
Nq = overburden multiplier = e^(PI*tan(PHI))*Kp
where Kp = passive earth pressure coefficient = tan(PI/4+PHI/2)^2
PHI = angle of internal friction in radians (entered in degrees)
B = least lateral base dimension in m
L = the other base dimension in m
D = depth to base in m
B' = B modified for eccentricity = B - 2*EB
L' = L modified for eccentricity = L - 2*EL
EB = MB/V must be <= B/6
EL = ML/V must be <= L/6

Programmer's note: Sometimes it happens that B' is greater than L', which distorts the shape factors that are a function of B'/L'. Consequently, the program uses the smaller of B-2*EB and L-2*EL for B' and the larger of these two values for L'.

V = vertical load in kN
MB, ML = moments in B and L respectively
gamma = unit weight of soil in kN/m3
d-q = 1 + k*2*tan(PHI)*(1-sin(PHI))^2
s-q = 1 + (B'/L')*sin(PHI)
Ngam = wedge weight multiplier = 1.5*(Nq - 1)*tan(PHI)
d-gam = 1.0 for all PHI
s-gam = 1 - 0.4*B'/L' >= 0.6
Input values are obtained from tests of soil samples.

You should do at least one calculation manually, for two reasons:
1. to check the spreadsheet so you can have confidence in its results; and
2. to appreciate how much time the spreadsheet saves.

Please support the ongoing maintenance of this site. Donate $7 and I shall send you the Excel spreadsheet so you can satisfy yourself that the program uses the equations correctly.

Hansen's Bearing-Capacity Equations
not on slope
Given:
unit weight of soil, gamma				kN/m3
angle of internal friction, PHI				degrees
				kPa
depth of base, D				m
width of base, B				m
length of base, L				m
vertical load, V				kN
moment in plane of width, MB				kNm
moment in plane of length, ML				kNm
required safety factor, SF
Then:
excentricity in width, EB				m
excentricity in length, EL				m
effective width, B'				m
effective length, L'				m
passive earth pressure coefficient, Kp
multiplier k				radians
shape factor for soil, s-q
shape factor for gamma, s-gam
depth factor for soil, d-q
depth factor for gamma, d-gam		1.00
effective overburden pressure, q-bar				kPa
overburden multiplier, Nq
cohesion multiplier, Nc
wedge weight multiplier, Ngam
ultimate bearing capacity, q-ult				kPa
safe bearing capacity				kPa
actual bearing stress				kPa