Bearing Capacity Factor Nq

$N_q$ answers a precise question: how many times the surcharge beside a footing can the footing itself carry, on friction alone? Ludwig Prandtl solved the plastic wedge problem for metals in 1920 and Hans Reissner extended it to a surcharged surface in 1924, giving this closed form. At φ = 30° it evaluates to $e^{\pi \tan 30^\circ} \times \tan^2(60^\circ)$ = 6.134 × 3.000 = 18.40, and at 35° to 33.30 — the numbers printed in Das's Table 3.3 and in every foundation code appendix.

The trap is which $N_q$. Terzaghi's own 1943 factor uses a different failure surface and gives 22.46 at 30°, over 20% higher; Meyerhof, Hansen and Vesic all adopted the Prandtl–Reissner value above, which is what modern practice and this page use. Mixing a Terzaghi $N_q$ with a Vesic $N_\gamma$ produces a bearing capacity that belongs to no theory at all. The second trap is the exponential: $N_q$ roughly doubles for every 5° of friction angle, so a φ estimated 3° too high from an SPT correlation inflates capacity by 40%. That sensitivity is why site investigation is cheaper than optimism.