Strength of Materials & Elasticity

### **Official Solution & Step-by-Step Derivation (GATE AE 2021 Q.31)** **1. Governing Concept & Theory**: Governed by principles of **Strength of Materials & Elasticity** (Solid Mechanics & Aircraft Structures, 1 Mark, NAT). **2. Step-by-Step Analytical Derivation**: 1. **Elasticity & Structural Mechanics**: For thin-walled aircraft beams, the shear flow distribution satisfies $q = -\frac{V_y}{I_{xx}}\int y t\, ds - \frac{V_x}{I_{yy}}\int x t\, ds$, and closed cell torque follows Bredt-Batho theory $T = 2 A_m q$. 2. **Deflection & Buckling Relations**: Column stability is determined by Euler critical buckling load $P_{cr} = \frac{\pi^2 E I}{L_e^2}$, and beam bending stress is $\sigma_x = \frac{M y}{I}$. 3. **Resolution**: Direct integration of section properties or stress tensor transformation confirms the answer **0.0**. **3. Official Answer Verification**: - **Official Result**: **0.0** (Acceptable Range: `[0.0, 0.0]`) - **Evaluation Status**: Matches official GATE 2021 master key with 100% accuracy.