GATE 2023 AE – Question 26
Which of the following statement(s) is/are true about the state of stress in a plane?
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Correct answer: (A) Maximum or major principal stress is algebraically the largest direct stress at a point.; (D) The normal stresses along the planes of maximum shear stress are equal.
Explanation
For a plane stress state, the principal stresses are $\sigma_{1,2}=\dfrac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}$.
- **A. The major principal stress is the algebraically largest direct stress at the point.** True. $\sigma_1$ is the maximum value of the normal stress over all planes. ✓
- **B. The minor principal stress cannot be larger in magnitude than the major one.** False. If $\sigma_1=50$ MPa and $\sigma_2=-100$ MPa, the minor principal stress has the larger magnitude.
- **C. The planes of maximum shear are 90° from the principal axes.** False. They are inclined at **45°** to the principal planes.
- **D. The normal stresses on the planes of maximum shear are equal.** True. On both planes the normal stress equals the average $\dfrac{\sigma_1+\sigma_2}{2}$. ✓
Answer **A and D**.