A thin flat plate is subjected to the following stresses:
𝜎xx = 160 MPa; 𝜎yy = 40 MPa; 𝜏xy = 80 MPa.
Factor of safety is defined as the ratio of the yield stress to the applied stress. The
yield stress of the material under uniaxial tensile load is 250 MPa. The factor of
safety for the plate assuming that material failure is governed by the von Mises
criterion is _____ (rounded off to two decimal places).
Official Answer Key:
1.25
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.59)**
**1. Intuitive Physical Setup & Given Parameters**:
Plane stress state in a thin flat plate:
- $\sigma_{xx} = 160\text{ MPa}$
- $\sigma_{yy} = 40\text{ MPa}$
- $\tau_{xy} = 80\text{ MPa}$
- Material uniaxial yield stress: $\sigma_y = 250\text{ MPa}$.
We need to determine the Factor of Safety (FOS) using the **von Mises failure criterion** (rounded off to two decimal places).
**2. Step-by-Step Failure Theory Derivation**:
1. **Von Mises Equivalent Stress ($\sigma_{\text{vm}}$) for Plane Stress**:
$\sigma_{\text{vm}} = \sqrt{\sigma_{xx}^2 - \sigma_{xx}\sigma_{yy} + \sigma_{yy}^2 + 3\tau_{xy}^2}$
2. **Substitute Stress Components**:
- $\sigma_{xx}^2 = 160^2 = 25600$
- $-\sigma_{xx}\sigma_{yy} = -(160)(40) = -6400$
- $\sigma_{yy}^2 = 40^2 = 1600$
- $3\tau_{xy}^2 = 3(80^2) = 3(6400) = 19200$
3. **Sum Under the Radical**:
$\sigma_{\text{vm}}^2 = 25600 - 6400 + 1600 + 19200 = 40,000\text{ MPa}^2$
$\sigma_{\text{vm}} = \sqrt{40,000} = 200\text{ MPa}$
4. **Compute Factor of Safety**:
$\text{FOS} = \frac{\sigma_y}{\sigma_{\text{vm}}} = \frac{250\text{ MPa}}{200\text{ MPa}} = 1.25$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $\text{FOS} = 1.25$.
- **Rounding Requirement**: Two decimal places $\implies \mathbf{1.25}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **1.25** (Acceptable Range: **1.23 to 1.27**)
- **Exam Takeaway**: In plane stress, $\sigma_{\text{vm}} = \sqrt{\sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau_{xy}^2}$. Here $\sqrt{25600 - 6400 + 1600 + 19200} = 200\text{ MPa}$, giving $\text{FOS} = 250/200 = 1.25$.
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