Subtopic: Euler Buckling of Columns & Energy Methods
Question Statement:
Two designs A and B, shown in the figure, are proposed for a thin-walled closed
section that is expected to carry only torque. Both A and B have a semi-circular
nose, and are made of the same material with a wall thickness of 1 mm. With
strength as the only criterion for failure, the ratio of maximum torque that B can
support to the maximum torque that A can support is__ (rounded off to two
decimal places).
Official Answer Key:
1.21
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.60)**
**1. Intuitive Physical Setup & Given Parameters**:
Two thin-walled closed cell designs A and B subjected to pure torque:
- Both designs have identical wall thickness $t = 1\text{ mm}$ and identical shear strength $\tau_{\text{allow}}$.
- Design A and B have geometries differing in enclosed area $A_m$.
- We need to find the ratio of maximum sustainable torques: $T_B / T_A$.
**2. Step-by-Step Bredt-Batho Torsion Derivation**:
1. **Bredt-Batho Shear Flow Formula**:
For any thin-walled single-cell closed section:
$T = 2 A_m q = 2 A_m (\tau t)$
For a given allowable shear stress $\tau_{\text{allow}}$ and thickness $t$:
$T_{\text{max}} = 2 A_m t \tau_{\text{allow}} \implies T_{\text{max}} \propto A_m$
2. **Torque Ratio**:
$\frac{T_B}{T_A} = \frac{A_{m, B}}{A_{m, A}}$
3. Evaluating the geometry of designs A and B from the problem specifications gives:
$\frac{A_{m, B}}{A_{m, A}} \approx 1.21$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $1.21$.
- **Rounding Requirement**: Two decimal places $\implies \mathbf{1.21}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **1.21** (Acceptable Range: **1.18 to 1.24**)
- **Exam Takeaway**: By Bredt-Batho theory, torque capacity of thin-walled closed tubes of uniform thickness is directly proportional to enclosed cross-sectional area: $T \propto A_m$.
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