Operations, Logistics & Engineering
Brinch Hansen (1970) Shape & Depth Factors Formula: Rectangular Footing Bearing Capacity
Upgrade 2D strip-footing bearing capacity to real 3D rectangular and square pads (B <= L) embedded at depth D_f using Jørgen Brinch Hansen’s (1970) shape multipliers (s_q, s_gamma, s_c) and depth factor d_q.
Why 2D strip-footing equations mispredict square and rectangular column pads—and how Brinch Hansen fixes 3D geometry!
Prandtl-Terzaghi strip formulas assume an infinitely long wall footing (B / L -> 0), ignoring shear resistance on the two end faces of a 3D rectangular column pad (width B <= length L) and shearing resistance through overburden soil above foundation depth D_f! Danish geotechnical pioneer Jørgen Brinch Hansen (1970, Danish Geotechnical Institute Bulletin No. 28) formulated the universal shape and depth multipliers underpinning Eurocode 7 and Canadian Foundation Engineering Manuals!
For a rectangular pad (0 < B <= L) on drained soil with friction angle phi' > 0: Surcharge Shape Factor s_q = 1 + (B / L) * TAN(phi') (or 1 + (B / L) * SIN(phi') in EC7; Hansen 1970 uses TAN(phi') in surcharge!), Self-Weight Shape Factor s_gamma = MAX(0.6, 1 - 0.4 * (B / L)) (note s_gamma = 0.60 < 1 for a square pad B = L!), Cohesion Shape Factor s_c = 1 + (B / L) * (N_q / N_c), and Surcharge Depth Factor d_q = 1 + 2 * TAN(phi') * (1 - SIN(phi'))^2 * IF(D_f / B <= 1, D_f / B, ATAN(D_f / B)) (switching smoothly to ATAN(D_f/B) in radians when D_f > B so deep piers never blow up linearly!)!
| Brinch Hansen (1970) 3D Multiplier (Square Pad B=2 m, L=2 m, D_f=1.5 m, phi'=32°) | Formula | Factor Value |
|---|---|---|
| Surcharge Shape Factor s_q | 1 + (B / L) * TAN(phi') | 1.6249 (+62.5% 3D End Confinement!) |
| Self-Weight Shape Factor s_gamma | 1 - 0.4 * (B / L) | 0.6000 (-40.0% 3D Wedge Reduction) |
| Embedment Depth Factor d_q (D_f / B = 0.75 <= 1) | 1 + 2 * TAN(phi') * (1 - SIN(phi'))^2 * (D_f/B) | 1.2071 (+20.7% Overburden Shear) |
Brinch Hansen (1970) Surcharge Depth Factor d_q (Handles Both Shallow D_f/B <= 1 & Deep ATAN(D_f/B))
Evaluates 1 + 2 * TAN(phi') * (1 - SIN(phi'))^2 * k, where k = D_f / B for D_f <= B and k = ATAN(D_f / B) (radians) for D_f > B.
=IF(OR(B2 <= 0, B2 >= 50, B3 <= 0, B4 < B3, B5 < 0), "Invalid Pad Geometry (Requires 0 < B <= L & 0 < phi < 50)", ROUND(1 + 2 * TAN(RADIANS(B2)) * ((1 - SIN(RADIANS(B2))) ^ 2) * IF(B5 / B3 <= 1, B5 / B3, ATAN(B5 / B3)), 4))Combined Surcharge Multiplier s_q * d_q for 3D Rectangular Pad (B <= L)
Multiplies Brinch Hansen’s surcharge shape factor s_q = 1 + (B / L) * TAN(phi') by depth factor d_q.
=IF(OR(B2 <= 0, B2 >= 50, B3 <= 0, B4 < B3, B5 < 0), "Invalid Pad Geometry (Requires 0 < B <= L & 0 < phi < 50)", ROUND((1 + (B3 / B4) * TAN(RADIANS(B2))) * (1 + 2 * TAN(RADIANS(B2)) * ((1 - SIN(RADIANS(B2))) ^ 2) * IF(B5 / B3 <= 1, B5 / B3, ATAN(B5 / B3))), 4))Try the example
Use the original values or change them to test your understanding.