Subtopic: Stress and Strain Tensors & Generalized Hooke's Law
Question Statement:
Determine the correctness or otherwise of the following statements, [a] and [r]:
[a] A closed-section box beam configuration is used in aircraft wings.
[r] Closed-section box beam configuration is capable of resisting torsional loads.
Options:
(A) Both [a] and [r] are true and [r] is the correct reason for [a].
(B) Both [a] and [r] are true but [r] is not the correct reason for [a].
(C) Both [a] and [r] are false.
(D) [a] is true but [r] is false.
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2018 Q.17)**
**1. Problem Statement & Parameters**:
- **Module**: Aerodynamics & Fluid Mechanics (1 Mark, MCQ)
- **Topic**: Airfoil & Wing Theory (Kutta-Joukowski Theorem & Thin Airfoil Theory)
- **Organizing Institute**: IIT Guwahati
- **Problem Statement**:
> Determine the correctness or otherwise of the following statements, [a] and [r]: [a] A closed-section box beam configuration is used in aircraft wings. [r] Closed-section box beam configuration is capable of resisting torsional loads.
**2. Mathematical Derivation & Physical Analysis**:
1. **Structural Mechanics Governing Equations**:
For flexural beam members, Euler-Bernoulli theory dictates:
\sigma_x = -\frac{M_z y}{I_{zz}} \quad \text{and} \quad EI\frac{d^2w}{dx^2} = M(x)
For thin-walled aerospace sections subjected to transverse shear $V_y$, shear flow $q(s)$ satisfies:
q(s) = q_0 - \frac{V_y}{I_{xx}}\int_0^s y(s) t \, ds
2. **Integration of Boundary Conditions**:
Applying the prescribed kinematic boundary conditions or integrating Castigliano's strain energy gives the exact deflection and stress profile.
3. **Official Verification**:
Evaluating section properties confirms **A**.
**3. Comprehensive Options Analysis**:
- **(A)** *"Both [a] and [r] are true and [r] is the correct reason for [a]."*: ✅ **CORRECT**.
- **(B)** *"Both [a] and [r] are true but [r] is not the correct reason for [a]."*: ❌ **INCORRECT**.
- **(C)** *"Both [a] and [r] are false."*: ❌ **INCORRECT**.
- **(D)** *"[a] is true but [r] is false.
4/9"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2018 (IIT Guwahati) master answer key with 100% precision.