The GATE Grind

GATE 2025 CE (CE2) – Question 39

Solid Mechanics · Transformation of stress, Mohr circle · 2 marks · Multiple choice

Let the state of stress at a point in a body be the difference of two plane states of stress shown in the figure. Consider all the possible planes perpendicular to the x-y plane and passing through that point. The magnitude of the maximum compressive stress on any such plane is $k\sigma_0$, where $k$ is equal to _____ (round off to one decimal place).

the first state is an element with a tensile stress $3\sigma_0$ on its two faces normal to the x axis. The second state is an element turned by 45°, with a tensile stress $3\sigma_0$ on its two faces whose normal is at 45° to the axes (along the direction up-left to down-right).
  1. 3.0
  2. 2.1
  3. 1.7
  4. 1.5

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) 2.1

Explanation

The first state is uniaxial tension $3\sigma_0$ along x: $\sigma_x = 3\sigma_0$, $\sigma_y = 0$, $\tau_{xy} = 0$. The second state is a uniaxial tension $3\sigma_0$ along a direction at 45° to the axes, with the components $\sigma_x = \sigma_y = 1.5\sigma_0$ and $\tau_{xy} = -1.5\sigma_0$. The difference is $\sigma_x = 1.5\sigma_0$, $\sigma_y = -1.5\sigma_0$ and $\tau_{xy} = 1.5\sigma_0$. The centre of the Mohr circle is 0 and its radius is $\sqrt{1.5^2 + 1.5^2}\sigma_0 = 2.12\sigma_0$, so the maximum compressive stress is $2.1\sigma_0$.