A single-cell thin-walled closed tube of enclosed area $A_m = 0.20\text{ m}^2$ and perimeter $S = 2.0\text{ m}$ has a uniform wall thickness $t = 2.0\text{ mm}$ and shear modulus $G = 25\text{ GPa}$. If a twisting torque $T = 10\text{ kNm}$ is applied, the rate of twist $\frac{d\theta}{dz}$ (in $\text{rad/m}$) according to Bredt-Batho theory is closest to:

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### 1. Strategy & Theoretical Principles From Bredt-Batho theory for pure torsion of a single-cell thin-walled closed tube: $$\frac{d\theta}{dz} = \frac{T}{4 A_m^2 G} \oint \frac{ds}{t}$$ For uniform wall thickness $t$ around perimeter $S = \oint ds$: $$\frac{d\theta}{dz} = \frac{T S}{4 A_m^2 G t}$$ ### 2. Analytical Formulation & Calculation Given: - Torque $T = 10\text{ kNm} = 10{,}000\text{ Nm}$ - Enclosed area $A_m = 0.20\text{ m}^2 \implies A_m^2 = 0.04\text{ m}^4$ - Perimeter $S = 2.0\text{ m}$ - Thickness $t = 2.0\text{ mm} = 0.002\text{ m}$ - Shear modulus $G = 25\text{ GPa} = 25 \times 10^9\text{ Pa}$ 1. Denominator calculation: $$4 A_m^2 G t = 4 \times 0.04 \times (25 \times 10^9) \times 0.002$$ $$4 \times 0.04 = 0.16$$ $$0.16 \times 25 \times 10^9 = 4.0 \times 10^9$$ $$4.0 \times 10^9 \times 0.002 = 8.0 \times 10^6\text{ N}\cdot\text{m}^2$$ 2. Numerator calculation: $$T S = 10{,}000 \times 2.0 = 20{,}000\text{ N}\cdot\text{m}^2$$ 3. Rate of twist: $$\frac{d\theta}{dz} = \frac{20{,}000}{8.0 \times 10^6} = \frac{2}{800} = \frac{1}{400} = 0.0025\text{ rad/m}$$ Wait, let's verify arithmetic: $$\frac{d\theta}{dz} = \frac{20000}{8000000} = 0.0025\text{ rad/m}$$ Let's recheck: If $T = 20\text{ kNm}$, then $d\theta/dz = 0.0050\text{ rad/m}$. With $T = 10\text{ kNm}$, $d\theta/dz = 0.0025\text{ rad/m}$. Let's keep $T = 10\text{ kNm}$ and set correctAnswer to $0.0025$ (tolerance $0.0024-0.0026$)! ### 3. Distractor Forensics - Candidate omitting factor of 4 in Bredt-Batho denominator gets $0.010\text{ rad/m}$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Note the factor of 4 multiplying $A_m^2$ in the denominator ($4 A_m^2 G$). - **Standard Reference:** Megson, T.H.G., *Aircraft Structures for Engineering Students*, Chapter 18.

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