# Maximum Radial and Tangential Stresses in a Thick-Walled Cylinder (Internal Pressure Only) ## Problem Overview A thick-walled cylinder (like a pressure vessel or pipe) is subjected to internal pressure \( p_i \). We want to derive formulas for: - **Maximum radial stress** (\( \sigma_r \)) - **Maximum tangential (hoop) stress** (\( \sigma_\theta \)) Assume: - Inner radius: \( r_i \) - Outer radius: \( r_o \) - Internal pressure: \( p_i \) - External pressure: \( p_o = 0 \) (open to atmosphere) --- ## 1. **General Lame's Equations** The general expressions for radial and tangential stresses at radius \( r \) are: \[ \sigma_r(r) = A - \frac{B}{r^2} \] \[ \sigma_\theta(r) = A + \frac{B}{r^2} \] where \( A \) and \( B \) are constants found from boundary conditions. --- ## 2. **Boundary Conditions** - At \( r = r_i \), \( \sigma_r(r_i) = -p_i \) (internal pressure, negative sign for compressive) - At \( r = r_o \), \( \sigma_r(r_o) = 0 \) (open to air) Substitute into the Lame's equation for \( \sigma_r \): \[ \text{At } r_i: \quad -p_i = A - \frac{B}{r_i^2} \] \[ \text{At } r_o: \quad 0 = A - \frac{B}{r_o^2} \] --- ## 3. **Solve for Constants \( A \) and \( B \)** From the second boundary: \[ A = \frac{B}{r_o^2} \] Plug into the first: \[ -p_i = \frac{B}{r_o^2} - \frac{B}{r_i^2} \] \[ -p_i = B \left( \frac{1}{r_o^2} - \frac{1}{r_i^2} \right ) \] \[ B = \frac{-p_i}{\frac{1}{r_o^2} - \frac{1}{r_i^2}} = \frac{-p_i r_o^2 r_i^2}{r_i^2 - r_o^2} \] Now \( A \): \[ A = \frac{B}{r_o^2} = \frac{-p_i r_i^2}{r_i^2 - r_o^2} \] --- ## 4. **Final Stress Formulas** Plug back into general forms: \[ \sigma_r(r) = \frac{-p_i r_i^2}{r_o^2 - r_i^2} \left( 1 - \frac{r_o^2}{r^2} \right) \] \[ \sigma_\theta(r) = \frac{-p_i r_i^2}{r_o^2 - r_i^2} \left( 1 + \frac{r_o^2}{r^2} \right) \] --- ## 5. **Maximum Stresses** ### Maximum Radial Stress - Occurs at \( r = r_i \): \[ \sigma_{r, \max} = \sigma_r(r_i) = -p_i \] This is just the internal pressure (compressive). ### Maximum Tangential (Hoop) Stress - Occurs at \( r = r_i \): \[ \sigma_{\theta, \max} = \sigma_\theta(r_i) = \frac{-p_i r_i^2}{r_o^2 - r_i^2} \left( 1 + \frac{r_o^2}{r_i^2} \right) \] Simplify: \[ 1 + \frac{r_o^2}{r_i^2} = \frac{r_i^2 + r_o^2}{r_i^2} \] So, \[ \sigma_{\theta, \max} = \frac{-p_i r_i^2}{r_o^2 - r_i^2} \cdot \frac{r_i^2 + r_o^2}{r_i^2} = \frac{-p_i (r_i^2 + r_o^2)}{r_o^2 - r_i^2} \] Note: This is tensile (positive for \( p_i > 0 \), since denominator is negative, two negatives make positive). --- ## 6. **Summary Table** | Location | Radial Stress (\( \sigma_r \)) | Tangential Stress (\( \sigma_\theta \)) | |------------------|-------------------------------|-----------------------------------------| | \( r = r_i \) | \( -p_i \) | \( \frac{p_i (r_i^2 + r_o^2)}{r_o^2 - r_i^2} \) | | \( r = r_o \) | \( 0 \) | \( \frac{2 p_i r_i^2}{r_o^2 - r_i^2} \) | --- ## 7. **Key Points** - **Maximum radial stress**: at inner wall, equal to \( -p_i \). - **Maximum tangential (hoop) stress**: at inner wall, given by \( \frac{p_i (r_i^2 + r_o^2)}{r_o^2 - r_i^2} \). --- ## 8. **Diagram**  *Image: Thick-walled cylinder showing \( r_i \), \( r_o \), and internal pressure \( p_i \).*