An idealized thin-walled aircraft wing section consists of two concentrated booms separated by a vertical distance $h = 0.50\text{ m}$. Boom 1 (upper) and Boom 2 (lower) each have cross-sectional area $B_1 = B_2 = 400\text{ mm}^2$. The skin between the booms is thin and carries only shear stress (zero direct bending capability). If a vertical bending moment $M_x = 20\text{ kNm}$ is applied to the section, the direct tensile/compressive stress $\sigma$ in each boom (in $\text{MPa}$) is:

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### 1. Strategy & Theoretical Principles In the idealized boom-skin structural idealization of aircraft structures: - Booms carry all direct axial/bending stresses ($B_i$). - Thin skin panels carry only shear stresses. For a symmetric two-boom section with vertical separation $h$, the booms form an internal couple of moment arm $h$: $$M_x = P \cdot h$$ where $P = \sigma B$ is the axial force in each boom. ### 2. Analytical Formulation & Calculation Given: - $M_x = 20\text{ kNm} = 20{,}000\text{ Nm}$ - Separation $h = 0.50\text{ m}$ - Boom area $B = 400\text{ mm}^2 = 400 \times 10^{-6}\text{ m}^2 = 4.0 \times 10^{-4}\text{ m}^2$ 1. Boom axial force $P$: $$P = \frac{M_x}{h} = \frac{20{,}000\text{ Nm}}{0.50\text{ m}} = 40{,}000\text{ N} = 40\text{ kN}$$ 2. Direct stress $\sigma$: $$\sigma = \frac{P}{B} = \frac{40{,}000\text{ N}}{400 \times 10^{-6}\text{ m}^2} = 100 \times 10^6\text{ Pa} = 100.0\text{ MPa}$$ (Upper boom in compression $-100\text{ MPa}$, lower boom in tension $+100\text{ MPa}$). ### 3. Distractor Forensics - Candidate evaluating using beam equation $\sigma = M y / I_{xx}$ with $I_{xx} = 2 B (h/2)^2 = 2 \times (400 \times 10^{-6}) \times (0.25)^2 = 5.0 \times 10^{-5}\text{ m}^4$: $\sigma = \frac{20000 \times 0.25}{5.0 \times 10^{-5}} = 100\text{ MPa}$, perfectly confirming the result. ### 4. Exam Trap & Standard Reference - **Trap Alert:** In boom idealizations, do not include the skin area unless effective boom areas with $t b/6$ are specified. - **Standard Reference:** Megson, T.H.G., *Aircraft Structures for Engineering Students*, Chapter 20 (Structural Idealization).

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