Terzaghi vs Meyerhof vs Hansen vs Vesic — Bearing Capacity Compared

The general bearing-capacity equation has three terms — cohesion, surcharge, and self-weight — each multiplied by a dimensionless factor (N_c, N_q, N_γ) that depends on the friction angle φ:

q_u = c·N_c·s_c·d_c·i_c + q·N_q·s_q·d_q·i_q + ½·γ·B·N_γ·s_γ·d_γ·i_γ

N_c and N_q are essentially identical across all four methods — they come from Prandtl-Reissner's 1920 derivation. The fight is over N_γ. At φ = 30°:

Method	N_c	N_q	N_γ
Terzaghi (1943)	30.14	18.40	15.67
Meyerhof (1963)	30.14	18.40	15.67
Hansen (1970)	30.14	18.40	14.39
Vesic (1973)	30.14	18.40	22.40

Vesic's N_γ is ~50 % higher than Hansen's at the same friction angle. For a granular-soil bearing-capacity calculation dominated by the third term, that's a 50 % spread in your final q_u.

Try it — change the parameters live

The failure mechanism below is rendered from the same formulas that GeoEq uses in ge.bearing_capacity(). Drag the sliders to see how the three failure zones — elastic wedge, radial shear (log-spiral), and Rankine passive — change with the friction angle. The four ultimate-capacity values on the right are recomputed live for each method.

Why they disagree

Terzaghi (1943) assumed the failure surface goes vertical from the footing edge — a conservative simplification.
Meyerhof (1963) extended the failure surface above footing level, capturing more shear resistance — slightly higher capacity.
Hansen (1970) revised Meyerhof's surcharge geometry and got a smaller N_γ coefficient.
Vesic (1973) introduced a different mathematical form, giving the largest values.

The same footing, all four methods, in eight lines

import geoeq as ge

# 2 m square footing at 1 m depth on dense sand (phi=35)
for m in ("terzaghi", "meyerhof", "hansen", "vesic"):
    res = ge.bearing_capacity(
        c=0, gamma=18, Df=1, B=2, L=2,
        phi=35, method=m,
    )
    q_all = ge.bearing_allowable(res["q_u"], FS=3)
    print(f"{m:9s}  q_u = {res['q_u']:6.0f} kPa, q_allow = {q_all:.0f} kPa")

# terzaghi   q_u =   1180 kPa, q_allow = 393 kPa
# meyerhof   q_u =   1352 kPa, q_allow = 451 kPa
# hansen     q_u =   1248 kPa, q_allow = 416 kPa
# vesic      q_u =   1538 kPa, q_allow = 513 kPa

That's a 30 % spread on a single footing — enough to take a borderline design and either pass or fail it depending on which textbook you opened.

Bearing capacity vs footing width for four methods — Ultimate bearing capacity vs footing width for c=0, φ=30°, D_f=1 m. Generated by `ge.bearing_capacity_plot()`.

Which one to use, in practice

There is no universally "right" answer. The four methods reflect four honest interpretations of the same physics; the spread is the literature's honest uncertainty.

Hansen is the working compromise

Most practising engineers in North America and Europe default to Hansen. It gives values between Terzaghi (conservative) and Vesic (permissive), and its shape/depth/inclination factor set is the most rigorous of the four.

When to be more conservative

Unfamiliar soil deposits — pick Terzaghi's N_γ and a high FS.
Foundations on slopes — none of the standard methods handle slope inclination beyond ~10°; switch to Bowles' charts or numerical methods.
Sensitive to settlement, not collapse — bearing capacity is rarely the controlling check. Use ge.settlement_schmertmann() or 1-D consolidation instead.

When you can be less conservative

Well-characterised dense sand, multiple CPTs, careful FS — Vesic is defensible.
Eccentric loading — use Hansen's effective-width method (B' = B - 2e).

The honest verdict

Don't pick a method to get the answer you want — pick it to match the quality of your soil data and the consequences of failure. Run all four with ge.bearing_capacity(), plot them with ge.bearing_capacity_plot(), and let the spread inform your factor of safety. A 30 % spread between methods means your FS should include that uncertainty.

Terzaghi vs Meyerhof vs Hansen vs Vesic — which bearing-capacity method should you trust?