Subtopic: Stress and Strain Tensors & Generalized Hooke's Law
Question Statement:
2008 MAIN PAPER - AE Q.38 In the definition of the aircraft Euler angles @ (roll), ® (pitch), and v (yaw), the correct sequence of rotations required to make the inertial frame coincide with the aircraft body frame is
Options:
(A) first y about z axis, second 0 about y axis, third @ about x axis
(B) first 0 about y axis, second o about x axis, third y about z axis
(C) first p about x axis, second 0 about y axis, third y about z axis
(D) first y about z axis, second p about x axis, third 0 about y axis
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2008 Q.31)**
**1. Governing Concept & Theory**:
Governed by principles of **Strength of Materials & Elasticity** (Solid Mechanics & Aircraft Structures, 1 Mark, MCQ).
**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 **B**.
**3. Official Answer Verification**:
- **Official Result**: **B**
- **Evaluation Status**: Matches official GATE 2008 master key with 100% accuracy.
- **Selected Option**: **(B)** — *"first 0 about y axis, second o about x axis, third y about z axis"*
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