Let's solve the problem step by step: --- ## **Problem Restatement** **Given:** - Unstiffened fuselage skin panel - Dimensions: \( a = 1.2\,\text{m} \), \( b = .6\,\text{m} \) - Thickness: \( t = 1.8\,\text{mm} = .0018\,\text{m} \) - Compressive stress to withstand: \( \sigma = 80\,\text{MPa} \) - Material: Al-2024 - Young’s modulus: \( E = 73\,\text{GPa} = 73 \times 10^9\,\text{Pa} \) - Poisson’s ratio: \( \nu = .33 \) **To check:** Will the panel buckle under the given stress? If yes, suggest possible solutions. --- ## **Step 1: Calculate the Critical Buckling Stress** For a rectangular plate with simply supported edges, the **critical buckling stress** is: \[ \sigma_{cr} = \frac{k \pi^2 E}{12(1-\nu^2)} \left(\frac{t}{b}\right)^2 \] Where: - \( k \) = Buckling coefficient (for a long plate, simply supported, \( k \approx 4 \)) - \( t \) = thickness - \( b \) = **shorter side** (so \( b = .6\,\text{m} \)) --- ### **Step 2: Plug in the Values** #### **Calculate \( 1-\nu^2 \):** \[ 1 - \nu^2 = 1 - (.33)^2 = 1 - .1089 = .8911 \] #### **Calculate \( \left(\frac{t}{b}\right)^2 \):** \[ \frac{t}{b} = \frac{.0018}{.6} = .003 \] \[ \left(\frac{t}{b}\right)^2 = (.003)^2 = .000009 \] #### **Plug in all values:** \[ \sigma_{cr} = \frac{4 \times \pi^2 \times 73 \times 10^9}{12 \times .8911} \times .000009 \] --- ### **Step 3: Calculate Numerator and Denominator** #### **Calculate \( 4 \pi^2 \):** \[ 4 \pi^2 = 4 \times (3.1416)^2 = 4 \times 9.8696 = 39.4784 \] #### **Numerator:** \[ 39.4784 \times 73 \times 10^9 = 2,883.9232 \times 10^9 \] #### **Denominator:** \[ 12 \times .8911 = 10.6932 \] #### **Fraction:** \[ \frac{2,883.9232 \times 10^9}{10.6932} = 269.76 \times 10^9 \] #### **Multiply by \( .000009 \):** \[ 269.76 \times 10^9 \times .000009 = 2.42784 \times 10^6~\text{Pa} = 2.43~\text{MPa} \] --- ## **Step 4: Compare Critical Stress with Applied Stress** - **Applied stress:** \( 80\,\text{MPa} \) - **Critical buckling stress:** \( 2.43\,\text{MPa} \) **Conclusion:** \[ 80\,\text{MPa} > 2.43\,\text{MPa} \] The panel **will buckle** under the given compressive stress. --- ## **Step 5: Possible Solutions if Buckling Occurs** 1. **Increase Thickness (\( t \)):** - Buckling stress increases with \( t^2 \). - Make the panel thicker. 2. **Reduce Unstiffened Panel Width (\( b \)):** - Buckling stress increases as \( 1/b^2 \). - Add longitudinal stiffeners to reduce the effective width. 3. **Use Stiffeners:** - Add stringers or frames to subdivide the panel, reducing the effective width for buckling. 4. **Change Material:** - Use a material with higher Young's modulus \( E \). --- ## **Final Answers** ### **1. Will the panel buckle?** **Yes, the panel will buckle.** ### **2. Critical buckling stress:** \[ \boxed{2.43\,\text{MPa}} \] ### **3. Suggestions if buckling occurs:** - Increase panel thickness. - Add stiffeners to reduce the effective width. - Use a material with a higher Young's modulus. --- Let me know if you want to see calculations for a different stiffener arrangement or to estimate a new required thickness!