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Question 52
Question Statement:
For the state of plane stress shown in the figure with $\sigma_x = 80$ MPa, $\sigma_y = -20$ MPa, and $\tau_{xy} = 40$ MPa, the minimum principal stress (in MPa) is _________ (round off to the nearest integer).
Official Answer Key:
-33.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.52)**
**1. Intuitive Physical Setup & Given Parameters**:
- Normal stresses: $\sigma_x = 80\text{ MPa}$, $\sigma_y = -20\text{ MPa}$.
- Shear stress: $\tau_{xy} = 40\text{ MPa}$.
**2. Step-by-Step Mathematical Derivation**:
1. **Principal Stress Equation**:
$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left( \frac{\sigma_x - \sigma_y}{2} \right)^2 + \tau_{xy}^2}$
2. **Evaluating Individual Terms**:
$\sigma_{avg} = \frac{\sigma_x + \sigma_y}{2} = \frac{80 + (-20)}{2} = \frac{60}{2} = 30\text{ MPa}$
$\frac{\sigma_x - \sigma_y}{2} = \frac{80 - (-20)}{2} = \frac{100}{2} = 50\text{ MPa}$
$R = \sqrt{50^2 + 40^2} = \sqrt{2500 + 1600} = \sqrt{4100} \approx 64.031\text{ MPa}$
3. **Minimum Principal Stress ($\sigma_2$)**:
$\sigma_2 = \sigma_{avg} - R = 30 - 64.031 = -34.031\text{ MPa}$
Rounding to the nearest integer:
$\sigma_2 \approx -34\text{ MPa}$
This falls right in the official acceptable range **[-35, -31] MPa**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Official Range**: **-35 to -31 MPa**.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **-33.0** (Acceptable Range: **-35 to -31**)
- **Key Formula**: $\sigma_{2} = \frac{\sigma_x + \sigma_y}{2} - \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$.
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