Numerical Evaluation of Terzaghi's Nγ

The major difficulty in obtaining Terzaghi's bearing capacity factor Nγ 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 Pγ. Terzaghi solved the problem graphically, by trial and error, and provided a plot of ϕ versus Nγ, probably based on the results of a few trial calculations. This note now provides the explicit mathematical expression for ΔPγ/Δφ and results of the numerical solution of ΔPγ/ΔφO for ϕ=0° to 53° that include Nγ values and geometric details of log spirals for each Nγ calculation. The results show that Terzaghi's Nγ values for ϕ=0° to 44° and the numerical results agree well. His Nγ value for ϕ=48°, however, appears much higher (780 versus 650), probably due to the sensitivity of Nγ to high γ values.