The GATE Grind

GATE 2023 CE (CE2) – Question 41

Geotechnical Engineering · Shear strength, Mohr circle, stress-strain characteristics, stress paths · 2 marks · Multiple choice

In the given figure, Point O indicates the stress point of a soil element at initial non-hydrostatic stress condition. For the stress path (OP), which of the following loading conditions is correct?

p = (σᵥ + σₕ)/2 on horizontal axis, q = (σᵥ − σₕ)/2 on vertical axis; OP increases p and q with unit slope.
  1. $\sigma_v$ is increasing and $\sigma_h$ is constant
  2. $\sigma_v$ is constant and $\sigma_h$ is increasing
  3. $\sigma_v$ is increasing and $\sigma_h$ is decreasing
  4. $\sigma_v$ is decreasing and $\sigma_h$ is increasing

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $\sigma_v$ is increasing and $\sigma_h$ is constant

Explanation

The axes of the stress path diagram are the mean stress $p=\dfrac{\sigma_v+\sigma_h}2$ and the shear stress $q=\dfrac{\sigma_v-\sigma_h}2$ (these are the stress-path coordinates).

**Step 1: invert the definitions.**
$$\sigma_v=p+q,\qquad\sigma_h=p-q .$$

**Step 2: slope of the path OP.** The line has unit positive slope, so a step along it changes both coordinates equally: $dq=dp>0$.

**Step 3: changes in the stresses.**
$$d\sigma_v=dp+dq=2\,dp>0,\qquad d\sigma_h=dp-dq=0 .$$

So the vertical stress **increases** and the horizontal stress stays **constant**: this is the loading path of an axial compression test with constant cell pressure (option A).