Let's break down the question and solve it step-by-step for each part. **Given Data:** - Inner cylinder inside diameter, \( d_i = 120 \) mm - Contact diameter, \( d_c = 200 \) mm - Outer cylinder outside diameter, \( d_o = 260 \) mm - Radial stress at contact diameter, \( p = 25 \) MPa - Young's modulus, \( E = 200 \) GPa = \( 200 \times 10^3 \) MPa - Poisson's ratio, \( \nu = .3 \) ## 2.1 Minimum Hoop Stress in the Inner Cylinder ### Step 1: Formula for Hoop Stress (at inside and contact diameters) For a thick cylinder: \[ \sigma_\theta = \frac{A}{r^2} + B \] where \(A\) and \(B\) are constants found using boundary conditions. #### For the inner cylinder: - At \( r_1 = 60 \) mm, \( r_2 = 100 \) mm (radii) - At inside (\( r = 60 \) mm): \( p_1 = \) - At contact (\( r = 100 \) mm): \( p_2 = 25 \) MPa \[ A = \frac{p_1 r_1^2 - p_2 r_2^2}{r_2^2 - r_1^2} \] \[ B = \frac{(p_2 - p_1) r_1^2 r_2^2}{r_2^2 - r_1^2} \] #### Step 2: Calculation - \( r_1 = 60 \) mm, \( r_2 = 100 \) mm, \( p_1 = \), \( p_2 = 25 \) MPa \[ A = \frac{ \cdot 60^2 - 25 \cdot 100^2}{100^2 - 60^2} = \frac{-25 \cdot 10000}{10000 - 360} = \frac{-250000}{640} = -39.06 \] \[ B = \frac{(25 - ) \cdot 60^2 \cdot 100^2}{10000 - 360} = \frac{25 \cdot 360 \cdot 10000}{640} = \frac{25 \cdot 36,000,000}{640} = \frac{900,000,000}{640} = 140625 \] Hoop stress at the **inside** (\( r = 60 \) mm): \[ \sigma_{\theta,\text{min}} = \frac{A}{r^2} + B = \frac{-39.06}{60^2} + 140625 \] \[ = \frac{-39.06}{360} + 140625 \approx -.01085 + 140625 \approx 140625 \text{ MPa} \] Clearly, this value is unreasonably high (likely due to unit inconsistency). **Be careful with units.** Let’s recalculate \(A\) and \(B\) in MPa, mm²: \[ A = \frac{-250000}{640} = -39.06 \quad (\text{units: MPa}) \] \[ B = \frac{90000000}{640} = 14062.5 \quad (\text{units: MPa}) \] Hoop stress at \( r = 60 \) mm: \[ \sigma_{\theta} = \frac{-39.06}{60^2} + 14062.5 = -.01085 + 14062.5 \approx 14062.5 \text{ MPa} \] **This is still too high; let's double-check:** - For a typical thick cylinder, the correct formula for hoop stress at the inner wall is: \[ \sigma_{\theta, \text{min}} = p_2 \frac{r_2^2 + r_1^2}{r_2^2 - r_1^2} \] \[ = 25 \cdot \frac{100^2 + 60^2}{100^2 - 60^2} = 25 \cdot \frac{10000 + 360}{10000 - 360} = 25 \cdot \frac{13600}{640} = 25 \cdot 2.125 = 53.13 \text{ MPa} \] ### **Final Answer 2.1:** \[ \boxed{53.1 \ \text{MPa}} \] is the minimum hoop stress in the inner cylinder. --- ## 2.2 Maximum Hoop Stress in the Outer Cylinder ### Step 1: Use the same formula for the outer cylinder - Inner radius: \( r_2 = 100 \) mm (contact surface) - Outer radius: \( r_3 = 130 \) mm - Internal pressure (\( p_1 \)) at \( r_2 \): 25 MPa - External pressure (\( p_2 \)) at \( r_3 \): MPa \[ \sigma_{\theta, \text{max}} = p_1 \frac{r_3^2 + r_2^2}{r_3^2 - r_2^2} \] \[ = 25 \cdot \frac{130^2 + 100^2}{130^2 - 100^2} = 25 \cdot \frac{16900 + 10000}{16900 - 10000} = 25 \cdot \frac{26900}{690} = 25 \cdot 3.8986 = 97.5 \text{ MPa} \] ### **Final Answer 2.2:** \[ \boxed{97.5 \ \text{MPa}} \] is the maximum hoop stress in the outer cylinder. --- ## 2.3 Change in Diameter of the Inner Cylinder at the Contact Diameter ### Step 1: Radial displacement due to pressure Radial displacement at \( r = r_2 \) for a thick cylinder: \[ \Delta r = \frac{1 + \nu}{E} \cdot \frac{r_2^2 p}{r_2^2 - r_1^2} \cdot r_2 \] - \( \nu = .3 \) - \( E = 200 \times 10^3 \) MPa - \( r_2 = 100 \) mm - \( p = 25 \) MPa - \( r_1 = 60 \) mm \[ \Delta r = \frac{1 + .3}{200000} \cdot \frac{100^2 \cdot 25}{10000 - 360} \cdot 100 \] \[ = \frac{1.3}{200000} \cdot \frac{250000}{640} \cdot 100 \] \[ = .0000065 \cdot 39.0625 \cdot 100 = .0002539 \cdot 100 = .02539 \text{ mm} \] ### Step 2: Change in **diameter** (multiply by 2): \[ \Delta d = 2 \times \Delta r = 2 \times .02539 = .0508 \text{ mm} \] ### **Final Answer 2.3:** \[ \boxed{.051 \ \text{mm}} \] is the change in diameter of the inner cylinder at the contact diameter. --- ## **Summary Table** | Part | Value | |------|--------------------------| | 2.1 | 53.1 MPa (min, inner) | | 2.2 | 97.5 MPa (max, outer) | | 2.3 | .051 mm (diameter incr) | --- **Let me know if you need detailed derivation steps or further explanations.**