At a critical point on an aircraft aluminum fuselage skin, the state of plane stress is given by normal stresses $\sigma_x = 80\text{ MPa}$, $\sigma_y = 20\text{ MPa}$, and shear stress $\tau_{xy} = 40\text{ MPa}$. Using Mohr's circle analysis, what is the maximum in-plane shear stress $\tau_{max}$ (in MPa)?

Answer Key:

50.0

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Mohr's Circle Radius:** The maximum in-plane shear stress equals the radius $R$ of Mohr's circle of stress. 2. **Stage 2: Mathematical Formulation:** * $\tau_{max} = R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$. 3. **Stage 3: Substitution:** * Center offset: $\frac{\sigma_x - \sigma_y}{2} = \frac{80 - 20}{2} = 30\text{ MPa}$. * Shear term: $\tau_{xy} = 40\text{ MPa}$. * $\tau_{max} = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50.0\text{ MPa}$. 4. **Stage 4: Principal Stresses for Cross-Check:** * $\sigma_{avg} = \frac{80 + 20}{2} = 50\text{ MPa}$. * $\sigma_1 = 50 + 50 = 100\text{ MPa}$, $\sigma_2 = 50 - 50 = 0\text{ MPa}$. * $\tau_{max} = \frac{\sigma_1 - \sigma_2}{2} = \frac{100 - 0}{2} = 50.0\text{ MPa}$. 5. **Stage 5: Official Reference:** James M. Gere and Barry J. Goodno, *Mechanics of Materials*, Chapter 7.

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