Question

The vertical stress under the corner of a uniformly loaded rectangular area of size a, b at depth z and m=a/z, n=b/z is given by ___________

(a) \(σ_z=\frac{2q’}{πz}\frac{1}{\left[1+(\frac{x}{z})^2\right]^2}\)

(b) \(σ_z=\frac{q}{4π} \left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}\right] \)

(c) \(σ_z=\frac{q}{4π} \left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}* \frac{m^2+n^2+2}{m^2+n^2+1}+tan^{-1}⁡\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1} \right] \)

(d) \(σ_z=q\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right] \)

This question was posed to me in class test.

Asked question is from Stress Distribution in chapter Stress Distribution of Geotechnical Engineering I

Answer 1

Correct option is (c) \(σ_z=\frac{q}{4π} \left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}* \frac{m^2+n^2+2}{m^2+n^2+1}+tan^{-1}⁡\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1} \right] \)

The best I can explain: The vertical stress under the corner of a uniformly loaded rectangular area of size a, b at depth z and m=a/z, n=b/z is given by,

\(σ_z=\frac{q}{4π} \left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}* \frac{m^2+n^2+2}{m^2+n^2+1}+tan^{-1}⁡\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1} \right] \)

m and n are interchangeable terms. The above form of solution is after Newmark(1935).