Operations, Logistics & Engineering
Meyerhof & Vesić Ultimate Bearing Capacity Formula: Exact N_q, N_c & N_gamma Strip Footing
Calculate shallow strip-footing ultimate bearing capacity q_u = c'*N_c + q*N_q + 0.5*gamma*B*N_gamma from friction angle phi' using exact Prandtl-Reissner N_q, N_c and Vesić (1973) vs. Meyerhof (1963) N_gamma.
Why modern Eurocode 7 & AASHTO foundation design uses Prandtl-Reissner N_q, N_c and Vesić / Meyerhof N_gamma!
For a shallow continuous strip footing of width B (m) embedded at depth D_f in drained soil with effective cohesion c' (kPa), overburden surcharge q = gamma * D_f (kPa), and effective friction angle phi' (degrees), ultimate bearing capacity follows Terzaghi’s superposition q_u = c' * N_c + q * N_q + 0.5 * gamma * B * N_gamma! While N_q = EXP(PI() * TAN(phi')) * (TAN(45° + phi'/2))^2 (Reissner, 1924) and N_c = (N_q - 1) / TAN(phi') (Prandtl, 1921; N_c = 5.1416 for phi' = 0° undrained clay) are exact plasticity solutions, the self-weight wedge factor N_gamma differs across classical formulations!
George G. Meyerhof (1963) derived N_gamma,M = (N_q - 1) * TAN(1.4 * phi'), whereas Aleksandar S. Vesić (1973, adopted widely in FHWA/AASHTO Manuals) formulated N_gamma,V = 2 * (N_q + 1) * TAN(phi')! At phi' = 30°, N_q = 18.4011 and N_c = 30.1396, while Vesić gives N_gamma,V = 22.4025 vs. Meyerhof N_gamma,M = 15.6682—producing q_u = 1126.94 kPa (Vesić) on c' = 10 kPa, D_f = 1.2 m, B = 2.0 m, gamma = 18 kN/m^3!
| Friction Angle phi' (deg) | Exact N_q & N_c (Prandtl-Reissner) | Meyerhof N_gamma vs. Vesić N_gamma |
|---|---|---|
| phi' = 0° (Undrained Tresca Clay) | N_q = 1.00, N_c = 2 + PI = 5.14 | N_gamma,M = 0.00 | N_gamma,V = 0.00 |
| phi' = 30° (Medium Dense Sand) | N_q = 18.40, N_c = 30.14 | N_gamma,M = 15.67 | N_gamma,V = 22.40 |
| phi' = 35° (Dense Sand/Gravel) | N_q = 33.30, N_c = 46.12 | N_gamma,M = 37.15 | N_gamma,V = 48.03 |
Vesić (1973) Ultimate Strip Footing Bearing Capacity q_u (kPa)
Evaluates q_u = c'*N_c + (gamma*D_f)*N_q + 0.5*gamma*B*N_gamma,Vesic including exact phi' = 0° undrained clay limit (N_c = 2 + PI(), N_q = 1, N_gamma = 0).
=IF(OR(B2 < 0, B2 >= 50, B3 < 0, B4 <= 0, B5 < 0, B6 <= 0), "Invalid Geotechnical Footing Inputs", IF(B2 = 0, ROUND(B3 * (2 + PI()) + B4 * B5, 2), ROUND(B3 * ((EXP(PI() * TAN(RADIANS(B2))) * (TAN(RADIANS(45 + B2 / 2)) ^ 2) - 1) / TAN(RADIANS(B2))) + (B4 * B5) * (EXP(PI() * TAN(RADIANS(B2))) * (TAN(RADIANS(45 + B2 / 2)) ^ 2)) + 0.5 * B4 * B6 * (2 * (EXP(PI() * TAN(RADIANS(B2))) * (TAN(RADIANS(45 + B2 / 2)) ^ 2) + 1) * TAN(RADIANS(B2))), 2)))Meyerhof (1963) Self-Weight Bearing Factor N_gamma,M = (N_q - 1) * TAN(1.4 * phi')
Computes Meyerhof’s conservative self-weight bearing factor N_gamma for comparison against Vesić.
=IF(OR(B2 < 0, B2 >= 50), "Invalid Friction Angle", ROUND((EXP(PI() * TAN(RADIANS(B2))) * (TAN(RADIANS(45 + B2 / 2)) ^ 2) - 1) * TAN(RADIANS(1.4 * B2)), 2))Try the example
Use the original values or change them to test your understanding.