A single-cell thin-walled closed tube of mean enclosed cross-sectional area $A_m = 0.25\text{ m}^2$ is subjected to a pure twisting torque $T = 15\text{ kNm}$. If the wall thickness is uniform at $t = 2.0\text{ mm}$, the shear stress $\tau$ in the tube wall (in $\text{MPa}$) according to the Bredt-Batho theory is:

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### 1. Strategy & Theoretical Principles From Bredt-Batho theory for pure torsion of a single-cell thin-walled closed tube: 1. The shear flow $q$ is constant around the circumference of the cell: $$q = \frac{T}{2 A_m}$$ 2. The shear stress $\tau$ at any point of wall thickness $t$ is: $$\tau = \frac{q}{t} = \frac{T}{2 A_m t}$$ ### 2. Analytical Formulation & Calculation Given: - $T = 15\text{ kNm} = 15{,}000\text{ N}\cdot\text{m}$ - $A_m = 0.25\text{ m}^2$ - $t = 2.0\text{ mm} = 0.002\text{ m}$ Compute shear flow $q$: $$q = \frac{15{,}000}{2 \times 0.25} = \frac{15{,}000}{0.50} = 30{,}000\text{ N/m}$$ Compute shear stress $\tau$: $$\tau = \frac{q}{t} = \frac{30{,}000}{0.002} = 15{,}000{,}000\text{ Pa} = 15.0\text{ MPa}$$ ### 3. Distractor Forensics - Forgetting factor of 2 in Bredt-Batho formula gives $30\text{ MPa}$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** $A_m$ is the area enclosed by the *median line* of the tube wall, not the exterior or solid area. - **Standard Reference:** Megson, T.H.G., *Aircraft Structures for Engineering Students*, Chapter 18 (Torsion of Beams).

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