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:

        \[@r=r_i, \sigma_r=P_i \qquad @r=r_o, \sigma_r=P_o\]