Let's break this problem into clear steps: --- ## **Step 1: Problem Restatement** - **Given:** - Two steel tubes, diameters: 50 mm (inner), 55 mm (outer of inner tube), 60 mm (outer tube). - Diametral interference: .062 mm. - Applied torque: \( T = 900 \) N·m. - Applied bending moment: \( M = 675 \) N·m. - No slipping between cylinders. - Yield strength: \( S_y = 415 \) MPa. - **Required:** - **Radial component** of stress at inner member's outer radius (i.e., at \( r = 27.5 \) mm). - **Tangential (hoop) component** of stress at the same location. --- ## **Step 2: Identify Stress Sources** There are three sources of stress at the outer radius of the inner member: 1. **Interference (shrink fit) pressure** — causes radial and tangential stresses. 2. **Torque** — causes shear stress (not needed here as question asks for normal stresses). 3. **Bending moment** — causes normal axial stress (not needed for radial/tangential at this radius). We'll **focus on the interference fit for radial and tangential stresses** at the outer radius of the inner member. --- ## **Step 3: Stresses Due to Interference Fit** For a **thick-walled cylinder** subjected to internal/external pressure (here, the pressure is from the shrink fit): ### **Radial Stress (\(\sigma_r\))** At \( r = b \) (outer radius of inner tube): \[ \sigma_r = -p \] Where \( p \) is the contact pressure due to the interference fit. ### **Tangential (Hoop) Stress (\(\sigma_\theta\))** At \( r = b \): \[ \sigma_\theta = \frac{a^2 + b^2}{b^2 - a^2} p \] Where: - \( a = \) inner radius of inner tube = \(25 \) mm - \( b = \) outer radius of inner tube = \(27.5 \) mm --- ## **Step 4: Find Contact Pressure \( p \)** The **contact pressure** \( p \) for interference fit is: \[ \delta = \delta_1 + \delta_2 \] Where: - \( \delta \) = total radial interference = \( \frac{.062}{2} = .031 \) mm (*since diametral interference is split between two radii*) - \( \delta_1 \) = expansion of inner member's outer radius - \( \delta_2 \) = contraction of outer member's inner radius The formulas: \[ \delta_1 = \frac{p b}{E} \frac{b^2 + a^2}{b^2 - a^2} \] \[ \delta_2 = \frac{p b}{E} \frac{c^2 + b^2}{c^2 - b^2} \] Where: - \( b = 27.5 \) mm = .0275 m - \( a = 25 \) mm = .025 m - \( c = 30 \) mm = .03 m - \( E \) (steel) ≈ \(210,000 \) MPa Combine: \[ \delta = \frac{p b}{E} \left[ \frac{b^2 + a^2}{b^2 - a^2} + \frac{c^2 + b^2}{c^2 - b^2} \right] \] Solve for \( p \): \[ p = \frac{\delta E}{b \left( \frac{b^2 + a^2}{b^2 - a^2} + \frac{c^2 + b^2}{c^2 - b^2} \right) } \] Plug in values: - \( \delta = .031 \) mm \( = .000031 \) m - \( E = 210,000 \) MPa \( = 210 \times 10^9 \) Pa - \( b = .0275 \) m - \( a = .025 \) m - \( c = .03 \) m Calculate each term: \[ b^2 + a^2 = (.0275)^2 + (.025)^2 = .00075625 + .000625 = .00138125 \] \[ b^2 - a^2 = (.0275)^2 - (.025)^2 = .00075625 - .000625 = .00013125 \] \[ \frac{b^2 + a^2}{b^2 - a^2} = \frac{.00138125}{.00013125} \approx 10.53 \] \[ c^2 + b^2 = (.03)^2 + (.0275)^2 = .0009 + .00075625 = .00165625 \] \[ c^2 - b^2 = (.03)^2 - (.0275)^2 = .0009 - .00075625 = .00014375 \] \[ \frac{c^2 + b^2}{c^2 - b^2} = \frac{.00165625}{.00014375} \approx 11.53 \] So the denominator is \( 10.53 + 11.53 = 22.06 \). Now plug in: \[ p = \frac{.000031 \times 210 \times 10^9}{.0275 \times 22.06} \] \[ = \frac{6,510,000}{.60665} \] \[ \approx 10,740,000 \text{ Pa } = 10.74 \text{ MPa} \] --- ## **Step 5: Calculate Stresses** ### **Radial Stress at \( r = b \):** \[ \sigma_r = -p = -10.74 \text{ MPa} \] ### **Tangential (Hoop) Stress at \( r = b \):** \[ \sigma_\theta = \frac{a^2 + b^2}{b^2 - a^2} p = 10.53 \times 10.74 = 113.1 \text{ MPa} \] --- ## **Step 6: Final Answers** \[ \boxed{ \begin{aligned} \text{The radial component of stress at the inner member's outer radius is} & : \quad -10.7 \ \text{MPa} \\ \text{The tangential component of stress at the inner member's outer radius is} & : \quad 113 \ \text{MPa} \\ \end{aligned} } \] (Rounded to one decimal place) --- ### **Summary Table** | Stress Component | Value | |------------------------|------------| | Radial (\(\sigma_r\)) | \(-10.7\) MPa | | Tangential (\(\sigma_\theta\)) | \(113\) MPa | --- **Let me know if you want to include the effects of torque or bending, or need help with any other part!**