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Hansen Model The Hansen model considers tilted bases and slopes in addition to foundation shape and eccentricity, load inclination, and foundation depth. Assumptions are based on an extension of Meyerhof’s work to include tilting of the base and construction on a slope. Any D/B ratio may be used permitting bearing capacity analysis of both shallow and deep foundations. Bearing capacity factors N_c and N_q are the same as Meyerhof’s factors. Correction factors Z_c , Z_Y , and Z_q in the basic soil bearing stress Equation are

Cohesion: Z_c = Z_cs Z_ci Z_cd Zc_B Z_cdelta
Wedge: Z_Y = Z_Ys Z_Yi Z_Yd Z_YB Z_Ydelta
Surcharge: Z_q = Z_qs Z_qi Z_qd Z_qB Z_qdelta

Subscripts s, i, d, B, and delta indicate shape with eccentricity, inclined loading, foundation depth, ground slope, and base tilt, respectively. The Table below illustrates the Hansen dimensionless bearing capacity and correction factors for solution of the basic soil bearing stress Equation.

Bearing Capacity

Φ	Nc - Cohesion	NY-Wedge	Nq-Surcharge
Φ=0	5.14	0.0	1.0
Φ>0	(Nq - 1) COT(Φ)	1.5(Nq - 1) TAN(Φ)	NΦeπTAN(Φ)

Foundation Shape With Eccentricity

Φ	Zes - Cohesion	ZYs-Wedge	Zqs-Surcharge
	Strip = 1.0
Φ=0	0.2B'/W'	1.0	1.0
Φ>0	1+NqB'/NcW'	1-0.4B'/W'	1+B'/W' TAN(Φ)

Inclined Loading

Φ	Zci - Cohesion	ZYi-Wedge	Zqi-Surcharge
Φ=0	(1 - [ 1 - T/Aecb]0.5)/0.5	delta = 0 [1 - 0.7T/(Q + AecaCOT(Φ))]0.5	[1 - 0.5T/(Q + AecaCOT(Φ))]0.5
Φ>0	Zqi - 1 - Zqi/(Nq - 1)	delta > 0 [1-(0.7-delta/450)T/(Q + AecaCOT(Φ))]0.5	[1 - 0.5T/(Q + AecaCOT(Φ))]0.5

Foundation Depth

Φ	Zcd - Cohesion	ZYd-Wedge	Zqd-Surcharge
Φ=0	0.4k	1.0	1.0
Φ>0	1 + 0.4k	1.0	1 + 2TAN(Φ)(1-SIN(Φ))2k

Foundation on Slope b

Φ	ZcB - Cohesion	ZYB-Wedge	ZqB-Surcharge
Φ=0	1 - b/147.3	[1.0 - 0.5TAN(b)]0.5	[1.0 - 0.5TAN(b)]0.5
Φ>0	ZqB - (1 - ZqB/147.3)	[1.0 - 0.5TAN(b)]0.5	[1.0 - 0.5TAN(b)]0.5

Tilted Foundation delta

Φ	Zcdelta - Cohesion	ZYdelta-Wedge	Zqdelta-Surcharge
Φ=0	1 - delta/147	e -.0478TAN(Φ)	e -.0358TAN(Φ)
Φ>0	Zqdelta - ( 1 - Zqdelta/147.3)	e -.0478TAN(Φ)	e -.0358TAN(Φ)

Notes:
Eccentricity and Inclined Loading Factors are not to be used simultaniously
Factors not used are equal to 1.0
Φ = Angle of internal friction angle in degrees
N_Φ = TAN²(45 + Φ/2)
B' = effective width of foundation - B - 2e_B in feet
W' = effective lateral width of foundation - W - 2e_W in feet
B = Foundation width in feet
W = foundation lateral width in feet
D = foundation depth in feet
Q = vertical load on foundation - qBW in kips
q = bearing pressure on foundations in ksf
T = horizontal load on foundation - positive - right in kips
R = resultant load on foundation (Q² + T²)^0.5 in kips
Θ = Angle of Resultant load with vertical axis - COS ^-1(Q/R) in degrees
e_B = eccentricity parrallel with B - MB/Q
e_W = eccentricity parrallel with W - MW/Q
MB = Moment parrallel with B - ft-kip
MW = Moment parrallel with W - ft-kip
Φ_a = Friction angle between base and soil in degrees
c_a = Adhesion of soil to base in ksf
c = soil cohesion or undrained shear strength
delta = base tilt from horizontal - upward positive - degrees
b = slpe of ground from base - downward positive - in degrees
k = D/B if D/B <=1 - or TAN^-1(D/B) if D/B > 1 - in radians
A_e = Effective area of foundation - B'W' - in ft²

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