For a structural element subjected to plane stress with principal stresses $\sigma_1$ and $\sigma_2$ (where $\sigma_1 > \sigma_2$), which of the following statements is/are TRUE?

(A) The maximum in-plane shear stress is given by $\tau_{\max, in-plane} = \frac{\sigma_1 - \sigma_2}{2}$

(B) The normal stress on the planes of maximum in-plane shear stress is $\sigma_{avg} = \frac{\sigma_1 + \sigma_2}{2}$

(C) The planes of maximum shear stress are oriented at $45^\circ$ to the principal planes

(D) The shear stress on the principal planes is strictly non-zero

Answer Key:

A,B,C

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Concept & Domain Context:** * This question evaluates **Stress and Strain Tensors & Generalized Hooke's Law** under *Strength of Materials & Elasticity* in GATE AE. 2. **Stage 2: Mathematical Derivation:** * By definition of Mohr's circle of stress, principal planes have zero shear stress ($\tau = 0$), so D is false. Radius of Mohr's circle is $R = \frac{\sigma_1 - \sigma_2}{2} = \tau_{max}$. Center is at $(\frac{\sigma_1 + \sigma_2}{2}, 0)$. The angle in physical space is half of $2\theta = 90^\circ$, giving $45^\circ$. 3. **Stage 3: Distractor Elimination Forensics:** * The evaluated correct answer is derived using standard physical equations. * Distractor values correspond to typical unit conversion omissions or inverted formulas. 4. **Stage 4: Common Student Trap:** * *Trap:* Assuming principal planes carry non-zero shear stress. *Reference: T.H.G. Megson, Aircraft Structures for Engineering Students (Butterworth-Heinemann).*

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