Mechanics & Structures University Engineering

Thick-Walled Cylinders

Chief Noter

July 22, 2021
3 min read
Failure Strain Stress

The Lamé Equations
Internal Pressure Only
External Pressure Only
Combination of Both
Summary

The equilibrium of stresses in cylindrical polar coordinates is given as:

$\frac{\partial\sigma_r}{\partial r}+\frac{\sigma_r-\sigma_\theta}{r}+F=0$

$F$ , the body force, points radially outwards. It generally equals zero.

The Lamé Equations

These describe the variation in radial and hoop stresses throughout the thickness of the wall:

$\sigma_r=A-\frac{B}{r^2} \qquad \sigma_theta=A+\frac{B}{r^2}$

$A$ and $B$ are constants that can be found from the boundary consitions:

The ratio between the inner and outer radii is $k$ . The larger the value, the more inaccurate the thin-walled assumption is.

$k=\frac{r_o}{r_i}$

Internal Pressure Only

$@r=r_i, \sigma_r=P_i \qquad @r=r_o, \sigma_r=0$

When there is only an internal pressure, $P_i$ , the coefficients of the Lamé equations are:

$A=\frac{P_i}{k^2-1} \qqaud B=\frac{P_i r_o^2}{k^2-1}$

Therefore, the radial and hoop stresses are:

$sigma_r=\frac{P_i}{k^2-1}(1-\frac{r_o^2}{r^2})$

$sigma_\theta=\frac{P_i}{k^2-1}(1+\frac{r_o^2}{r^2})$

External Pressure Only

$@r=r_i, \sigma_r=0 \qquad @r=r_o, \sigma_r=P_o$

When there is only an external pressure, $P_o$ , (like a submarine) the coefficients of the Lamé equations are:

$A=\frac{P_o k^2}{k^2-1} \qqaud B=\frac{P_o r_o^2}{k^2-1}$

Therefore, the radial and hoop stresses are:

$sigma_r=\frac{-P_o}{k^2-1}(k^2-\frac{r_o^2}{r^2})$

$sigma_\theta=\frac{-P_o}{k^2-1}(k^2+\frac{r_o^2}{r^2})$

Internal & External Pressure

$@r=r_i, \sigma_r=P_i \qquad @r=r_o, \sigma_r=P_o$

These problems can be solved using superposition.

The Lamé equations describe the variation in radial and hoop stresses through the cylinder wall:
$A=\frac{P_i}{k^2-1} \qqaud B=\frac{P_i r_o^2}{k^2-1}$
For internal pressure only:
$sigma_r=\frac{P_i}{k^2-1}(1-\frac{r_o^2}{r^2})$

$sigma_\theta=\frac{P_i}{k^2-1}(1+\frac{r_o^2}{r^2})$
For external pressure only:
$sigma_r=\frac{-P_o}{k^2-1}(k^2-\frac{r_o^2}{r^2})$

$sigma_\theta=\frac{-P_o}{k^2-1}(k^2+\frac{r_o^2}{r^2})$
For internal and external pressure:
$@r=r_i, \sigma_r=P_i \qquad @r=r_o, \sigma_r=P_o$

© 2021 EngineeringNotes.net. All Rights Reserved.