# Step-by Solution: Thickalled Cylinder Under Internal Pressure Given: - Inner diameter, \( d_i = 75 \) mm \(\implies r_i = 37.5 \) mm - Outer diameter, \( d_o = 100 \) mm \(\implies r_o = 50 \) mm - Internal pressure, \( p_i = 15 \) MPa - External pressure, \( p_o = \) MPa (open to atmosphere) We'll use Lame's equations for thick-walled cylinders. --- ## 1. **Lame's Equations** For a thick-walled cylinder under internal pressure: - **Radial Stress:** \( \sigma_r(r) = A - \frac{B}{r^2} \) - **Hoop (Circumferential) Stress:** \( \sigma_\theta(r) = A + \frac{B}{r^2} \) Where \( A \) and \( B \) are found from boundary conditions: - At \( r = r_i \), \( \sigma_r(r_i) = -p_i \) (negative is inward pressure) - At \( r = r_o \), \( \sigma_r(r_o) = -p_o = \) --- ## 2. **Determine Constants \( A \) and \( B \)** ### **Boundary Conditions:** \[ \begin{align*} \sigma_r(r_i) &= A - \frac{B}{r_i^2} = -p_i \\ \sigma_r(r_o) &= A - \frac{B}{r_o^2} = \\ \end{align*} \] ### **Solving:** Subtract second from first: \[ \left( A - \frac{B}{r_i^2} \right) - \left( A - \frac{B}{r_o^2} \right) = -p_i - \] \[ -\frac{B}{r_i^2} + \frac{B}{r_o^2} = -p_i \] \[ B \left( \frac{1}{r_o^2} - \frac{1}{r_i^2} \right) = -p_i \] \[ B = \frac{-p_i}{(\frac{1}{r_o^2} - \frac{1}{r_i^2})} \] \[ A = \frac{B}{r_o^2} \] --- ## 3. **Calculate \( A \) and \( B \)** ### **Convert mm to meters:** - \( r_i = .0375 \) m - \( r_o = .050 \) m ### **Compute \( B \):** \[ \frac{1}{r_o^2} - \frac{1}{r_i^2} = \frac{1}{(.05)^2} - \frac{1}{(.0375)^2} \] \[ = \frac{1}{.0025} - \frac{1}{.00140625} \] \[ = 400 - 711.11 = -311.11~\text{m}^{-2} \] \[ B = \frac{-15 \times 10^6}{-311.11} = 48,211~\text{N/m} \] ### **Compute \( A \):** \[ A = \frac{B}{r_o^2} = \frac{48,211}{.0025} = 19,284,444~\text{N/m}^2 = 19.28~\text{MPa} \] --- ## 4. **Find the Maximum Stresses** ### **(a) Maximum Hoop (Circumferential) Stress** Maximum hoop stress occurs at the inner wall (\( r = r_i \)): \[ \sigma_\theta(r_i) = A + \frac{B}{r_i^2} \] \[ \frac{B}{r_i^2} = \frac{48,211}{(.0375)^2} = \frac{48,211}{.00140625} = 34,302,857~\text{N/m}^2 = 34.30~\text{MPa} \] \[ \sigma_\theta(r_i) = 19.28 + 34.30 = 53.58~\text{MPa} \] --- ### **Maximum Shear Stress** Maximum shear stress at any point: \[ \tau_{\max} = \frac{1}{2} (\sigma_\theta - \sigma_r) \] At the inner wall (\( r = r_i \)), \( \sigma_r = -p_i = -15~\text{MPa} \): \[ \tau_{\max,\; r_i} = \frac{1}{2}(53.58 - (-15)) = \frac{1}{2}(68.58) = 34.29~\text{MPa} \] --- ### **(b) Maximum Radial Stress** Radial stress is maximum (in magnitude) at the inner wall: \[ \sigma_r(r_i) = -p_i = -15~\text{MPa} \] --- ### **(c) Maximum Longitudinal Stress (\( \sigma_z \))** For a closed-end cylinder: \[ \sigma_z = \frac{p_i r_i^2}{r_o^2 - r_i^2} \] \[ r_i^2 = (.0375)^2 = .00140625 \] \[ r_o^2 - r_i^2 = .0025 - .00140625 = .00109375 \] \[ \sigma_z = \frac{15 \times 10^6 \times .00140625}{.00109375} = \frac{21,093.75}{.00109375} = 19,285,714~\text{N/m}^2 = 19.29~\text{MPa} \] --- ## 5. **Thin-Walled Approximation** For a thin-walled cylinder (\( t \ll r \)), use: - Wall thickness \( t = r_o - r_i = .0125 \) m - Mean radius \( r_m = (r_i + r_o)/2 = .04375 \) m ### **Hoop Stress (Thin wall):** \[ \sigma_\theta = \frac{p_i r_m}{t} = \frac{15 \times 10^6 \times .04375}{.0125} \] \[ = \frac{656,250}{.0125} = 52,500,000~\text{N/m}^2 = 52.5~\text{MPa} \] ### **Longitudinal Stress (Thin wall):** \[ \sigma_z = \frac{p_i r_m}{2t} = \frac{52.5}{2} = 26.25~\text{MPa} \] --- ## 6. **Summary Table: Thick vs. Thin Wall** | Stress Type | Thick-walled (MPa) | Thin-walled (MPa) | |---------------------|-------------------|-------------------| | Max Hoop | 53.6 | 52.5 | | Max Shear | 34.3 | 33.75* | | Max Radial | -15 | -15 | | Longitudinal | 19.3 | 26.25 | \*Max shear (thin wall) \( = \frac{1}{2}(\sigma_\theta - \sigma_r) = \frac{1}{2}(52.5 - (-15)) = 33.75 \) MPa. --- ## 7. **Conclusions** - **Maximum hoop (circumferential) stress** in the thick wall is **53.6 MPa**, slightly higher than the thin-wall estimate (**52.5 MPa**). - **Maximum shear stress** is **34.3 MPa** (thick wall), slightly higher than the thin-wall estimate (**33.75 MPa**). - **Maximum radial stress** is always at the inner surface, and equals the internal pressure (\(-15~\text{MPa}\)). - **Longitudinal stress** is lower for the thick-wall solution (**19.3 MPa**) compared to the thin-wall estimate (**26.25 MPa**). --- ## **Final Answers** ### **(a) Maximum shear (circumferential) stress:** \[ \boxed{53.6~\text{MPa} \text{ (hoop/circumferential, at inner wall)}} \] ### **(b) Maximum radial stress:** \[ \boxed{-15~\text{MPa} \text{ (at inner wall)}} \] ### **(c) Maximum longitudinal stress:** \[ \boxed{19.3~\text{MPa}} \] **The thin-walled approximation underestimates the maximum circumferential and shear stress slightly, and overestimates the longitudinal stress compared to the more accurate thick-wall solution.**