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GATE 2022 CE (CE2) – Question 18

Geotechnical Engineering · Shear strength, Mohr circle, stress-strain characteristics, stress paths · 1 mark · Multiple choice

The figure shows stresses with $\sigma_z>\sigma_x$, compression positive. Let principal stresses be $\sigma_1,\sigma_3$. Find the angle between the major principal stress plane and the horizontal.

Soil stress element with normal stresses sigma-z and sigma-x and shear stress tau-zx; compression is positive. See the attached source image.
  1. $\tan^{-1}[\tau_{zx}/(\sigma_1-\sigma_x)]$
  2. $\tan^{-1}[\tau_{zx}/(\sigma_3-\sigma_x)]$
  3. $\tan^{-1}[\tau_{zx}/(\sigma_1+\sigma_x)]$
  4. $\tan^{-1}[\tau_{zx}/(\sigma_1+\sigma_3)]$

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Correct answer: (A) $\tan^{-1}[\tau_{zx}/(\sigma_1-\sigma_x)]$

Explanation

For the major principal direction, the eigenvector equation gives $(\sigma_1-\sigma_x)n_x=\tau_{zx}n_z$. The normal to the major principal plane makes angle θ with the vertical, so $\tan\theta=n_x/n_z$.

Hence $\theta=\tan^{-1}[\tau_{zx}/(\sigma_1-\sigma_x)]$, **A**. Using σ₃ instead selects the other principal direction.