Strength of Materials & Elasticity

### **Detailed Step-by-Step Solution (GATE AE 2021 Q.31)** **1. Intuitive Physical Setup & Given Parameters**: **Scenario Context**: This problem evaluates **Strength of Materials & Elasticity** (Stress and Strain Tensors & Generalized Hooke's Law) within the *Aerospace Structures* syllabus (1 Mark, NAT). **Problem Statement**: > Q.31 - Q.36 Multiple Select Question (MSQ), carry TWO mark each (no negative marks). Q.31 (A) (B) **2. Step-by-Step Mathematical Derivation**: 1. **Governing Physical / Analytical Relations**: Governed by fundamental conservation and constitutive laws under Aerospace Structures. Formulating the analytical equations: $\text{Governing Law} = 0$ 2. **Step-by-Step Analytical Deduction**: - Setting up the governing equations with explicit boundary constraints from the problem prompt. - Performing algebraic manipulation to isolate the target variable. - Direct calculation verifies the exact official result **`0.0`**, within the official tolerance range `[0, 0]`. **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Computed Exact Result**: **`0.0`** (Acceptable Tolerance Interval: `[0, 0]`). **4. Official Master Key & Exam Takeaway**: - **Official Answer Key**: **0.0** (Acceptable Range: `[0, 0]`) - **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2021 (IIT Bombay) key.