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Question 53
Question Statement:
The maximum normal stress in MN/m$^2$ for the thin-walled beam of square cross-section (side $b = 100$ mm, thickness $t = 2$ mm) under an applied bending moment of $M = 2$ kN-m is _________ (round off to the nearest integer).
Official Answer Key:
75.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.53)**
**1. Intuitive Physical Setup & Given Parameters**:
- Square box section: outer dimension $b = 100\text{ mm} = 0.1\text{ m}$.
- Wall thickness: $t = 2\text{ mm} = 0.002\text{ m}$.
- Applied bending moment: $M = 2\text{ kN m} = 2000\text{ N m}$.
**2. Step-by-Step Structural Derivation**:
1. **Area Moment of Inertia of Thin-Walled Square Tube**:
$I = \frac{2}{3} b^3 t = \frac{2}{3} (0.1)^3 (0.002) = \frac{2}{3} \times 10^{-3} \times 0.002 = \frac{4}{3} \times 10^{-6}\text{ m}^4 \approx 1.333 \times 10^{-6}\text{ m}^4$
2. **Maximum Bending Stress**:
$\sigma_{\max} = \frac{M y_{\max}}{I} = \frac{M (b/2)}{I}$
$\sigma_{\max} = \frac{2000 \times 0.05}{\frac{4}{3} \times 10^{-6}} = \frac{100}{1.333 \times 10^{-6}} = 75 \times 10^6\text{ N/m}^2 = 75.0\text{ MN/m}^2$
The maximum normal stress is exactly **75 MN/m$^2$**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: **75.0 MN/m$^2$** (Acceptable Range: [73, 77]).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **75.0**
- **Key Formula**: $\sigma = \frac{M y}{I}$ with $I = \frac{2}{3} b^3 t$ for thin square tube.
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