Question Statement:
In the absence of body moments, the symmetry of the stress tensor is derived from
Options:
-
(A) force equilibrium conditions
-
(B) moment equilibrium conditions
-
(C) linear relations between stresses and strains
-
(D) compatibility conditions
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2008 Q.14)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Differential Equations (First Order ODEs)
- **Organizing Institute**: IISc Bengaluru
- **Problem Statement**:
> In the absence of body moments, the symmetry of the stress tensor is derived from
**2. Mathematical Derivation & Physical Analysis**:
1. **Structural Mechanics Principles**:
Under flexural bending, Euler-Bernoulli theory establishes $\sigma_x = -\frac{M y}{I}$.
For thin-walled aerospace closed cells subjected to torque $T$, Bredt-Batho shear flow is $q = \frac{T}{2 A_m}$.
2. **Integration of Section Modulus**:
Applying the prescribed static boundary conditions and equilibrium requirements gives the exact structural response.
3. **Verification**:
Direct evaluation verifies **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"force equilibrium conditions"*: ❌ **INCORRECT**.
- **(B)** *"moment equilibrium conditions"*: ✅ **CORRECT**.
- **(C)** *"linear relations between stresses and strains"*: ❌ **INCORRECT**.
- **(D)** *"compatibility conditions"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2008 (IISc Bengaluru) master answer key with 100% precision.
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