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GATE 2023 CE (CE1) – Question 41

Geotechnical Engineering · Stress distribution in soils, Boussinesq theory, pressure bulbs · 2 marks · Multiple choice

Consider that a force P is acting on the surface of a half-space (Boussinesq's problem). The expression for the vertical stress ($\sigma_z$) at any point (r, z), within the half-space is given as, $\sigma_z = \frac{3P}{2\pi}\frac{z^3}{(r^2 + z^2)^{5/2}}$ where, r is the radial distance, and z is the depth with downward direction taken as positive. At any given r, there is a variation of $\sigma_z$ along z, and at a specific z, the value of $\sigma_z$ will be maximum. What is the locus of the maximum $\sigma_z$?

  1. $z^2 = \frac{3}{2}r^2$
  2. $z^3 = \frac{3}{2}r^2$
  3. $z^2 = \frac{5}{2}r^2$
  4. $z^3 = \frac{5}{2}r^2$

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Correct answer: (A) $z^2 = \frac{3}{2}r^2$

Explanation

At a fixed $r$, set $\frac{d}{dz}\left[z^3(r^2 + z^2)^{-5/2}\right] = 0$: $3z^2(r^2 + z^2) - 5z^4 = 0$, so $3r^2 + 3z^2 = 5z^2$ and $z^2 = \frac{3}{2}r^2$.