Numerical Evaluation of Terzaghi's $N_{\gamma }$

The major difficulty in obtaining Terzaghi's bearing capacity factor $N_{\gamma }$ as a function of friction angle ϕ is finding the log‐spiral failure surface (angle φ) defining the radial shear zone that provides minimum passive pressure due to self weight $P_{\gamma }$. Terzaghi solved the problem graphically, by trial and error, and provided a plot of ϕ versus $N_{\gamma }$, probably based on the results of a few trial calculations. This note now provides the explicit mathematical expression for $\Delta P\gamma /\Delta \phi$ and results of the numerical solution of $\Delta P\gamma /\Delta \phi O$ for $\varphi =0°$ to 53° that include $N_{\gamma }$ values and geometric details of log spirals for each $N_{\gamma }$ calculation. The results show that Terzaghi's $N_{\gamma }$ values for $\varphi =0°$ to 44° and the numerical results agree well. His $N_{\gamma }$ value for $\varphi =48°$, however, appears much higher (780 versus 650), probably due to the sensitivity of $N_{\gamma }$ to high γ values.