Question

Which of the following statement(s) is/are true about the state of stress in a plane?

• A. Maximum or major principal stress is algebraically the largest direct stress at a point.
• B. The magnitude of minor principal stress cannot be greater than the magnitude of major principal stress.
• C. The planes of maximum shear stress are inclined at \(90^\circ\) to the principal axes.
• D. The normal stresses along the planes of maximum shear stress are equal.

Hint:

Use Mohr's circle: principal planes at \(\tau=0\); max shear occurs \(45^\circ\) from them with the same normal stress \((\sigma_1+\sigma_2)/2\) on both shear planes.

Answer 1

The Correct Option is A, B, D

Step 1: Principal stresses.
For plane stress, rotation to principal planes yields \(\tau=0\); the normal stresses there are the principal stresses \(\sigma_1\) (major) and \(\sigma_2\) (minor) with \(\sigma_1 \ge \sigma_2\) algebraically. \(\Rightarrow\) (A) is true. By definition \(|\sigma_2| \le |\sigma_1|\) is guaranteed for ordered principal values (if both tensile or both compressive, the inequality is obvious; with opposite signs, \(\sigma_1\) is the algebraic maximum). \(\Rightarrow\) (B) is true.

Step 2: Orientation of maximum shear planes.
Mohr's circle gives planes of maximum shear at \(45^\circ\) to the principal planes (i.e., \(2\theta=90^\circ\) on Mohr's circle). \(\Rightarrow\) (C) is false (it is \(45^\circ\), not \(90^\circ\)).

Step 3: Normal stress on the max–shear planes.
On the two orthogonal planes of maximum shear, the normal stress equals the circle center \(\sigma_{\text{avg}}=\dfrac{\sigma_1+\sigma_2}{2}\) on both planes; hence they are equal to each other. \(\Rightarrow\) (D) is true.

Final Answer:
\[ \boxed{(A),\ (B),\ (D)} \]