GATE AE 2013 • Q18 MCQ • 1 Mark Aerospace Structures

Aerospace Structures

Thin-Walled Structures

Subtopic: Shear Flow in Open and Closed Thin-Walled Beams

Question Statement:

The stiffened cross-section of a long slender uniform structural member is idealized as shown in the figure below. The lumped areas at $A, B, C$ and $D$ have equal cross-sectional area of $3\text{ cm}^2$. The webs $AB, BC, CD$ and $DA$ are each $5\text{ mm}$ thick. The structural member is subjected to a twisting moment of $10\text{ kNm}$. The magnitudes of the shear flow in the webs, $q_{AB}, q_{BC}, q_{CD},$ and $q_{DA}$ in $\text{kN/m}$ are, respectively:

Options:

  • (A) 20, 20, 20, 20
  • (B) 0, 0, 50, 50
  • (C) 40, 40, 0, 0
  • (D) 50, 50, 50, 50

Official Answer Key:

D

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2013 Q.18)** **1. Torsion of Thin-Walled Closed Section (Bredt-Batho Theory)**: For a closed single-cell thin-walled structural tube subjected to a pure twisting moment $T$: $$T = 2 A_m q$$ where: - $T = 10\text{ kNm} = 10 \times 10^3\text{ N}\cdot\text{m}$ - $A_m$ is the mean area enclosed by the median line of the thin walls - $q$ is the shear flow circulating continuously along the closed perimeter. **2. Constant Shear Flow Property**: In a single-cell thin-walled tube under pure torsion without axial restraint or internal shear webs, the shear flow $q$ is constant throughout the entire closed wall perimeter: $$q_{AB} = q_{BC} = q_{CD} = q_{DA} = q$$ Evaluating for the cross-sectional geometry gives equal magnitude of $50\text{ kN/m}$ across all four webs. **3. Official Verification**: - **Official Answer Key**: **D** (50, 50, 50, 50)
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