Question

The stress component in x-direction on a horizontal plane in Cartesian coordinates for horizontal line load is ___________

(a) \(σ_x=\frac{2Q}{xzsinθcos⁡θ} \)

(b) \(σ_x=\frac{2Qxz^2}{π(x^2+z^2)^2} \)

(c) \(σ_x=\frac{2Qx^3}{π(x^2+z^2)^2} \)

(d) \(σ_x=\frac{2Qx^2 z}{π(x^2+z^2)^2} \)

The question was posed to me during an interview.

Origin of the question is Stress Distribution in chapter Stress Distribution of Geotechnical Engineering I

Answer 1

Right choice is (c) \(σ_x=\frac{2Qx^3}{π(x^2+z^2)^2} \)

Explanation: On a horizontal plane, the relation between the stress component in x-direction in Cartesian coordinates and polar coordinates is,

\(σ_x=σ_r  sin^2⁡θ \,where\, σ_r=K \frac{Q cos⁡θ}{r}\)

And \(K=\frac{2}{2α+sin2α}\)

∴ \(σ_x=\frac{2Qx^3}{π(x^2+z^2)^2}. \)