Question

Derivation of lami's theorem in thick pressure vessel

Answer 1

Stress for Thick Walled Cylinders using Lamé’s Equations

Thick Walled Cylinder Stress Calculator

Summary

Thick Wall Cylindrical Hoop Stress Calculator

Thick Wall Cylindrical Radial Stress Calculator

Thick Wall Cylindrical Axial Stress Calculator

Summary

Three of the primary mechanical stresses (not to be confused with ‘principle stresses’) that can be applied to a cylindrically shaped object are:

Hoop Stress

Radial Stress

Axial Stress

If the object/vessel has walls with a thickness greater than one-tenth of the overall diameter, then these objects can be assumed to be ‘thick-walled’. The general equations to calculate the stresses are:

Hoop Stress,

(1) \begin{align*}\sigma\x_h = A + \frac{B}{r^2}\end{align*}

Radial Stress,

(2) \begin{align*}\sigma\x_r = A - \frac{B}{r^2}\end{align*}

From a thick-walled cylinder, we get the boundary conditions:

\sigma\x_r = -p\x_o at r = r\x_o and \sigma\x_r = -p\x_i at r = r\x_i

Thick Walled Cylinder

Applying these boundary conditions to the above simultaneous equations gives us the following equations for the constants A & B:

(3) \begin{align*}A = \frac{p\x_i r\x_i ^2 - p\x_o r\x_o ^2}{r\x_o ^2 - r\x_i ^2}\end{align*}

(4) \begin{align*}B = \frac{(p\x_i - p\x_o) r\x_o ^2 r\x_i ^2}{r\x_o ^2 - r\x_i ^2}\end{align*}

Finally, solving the general equations with A & B gives Lamé’s equations:

Hoop Stress,

(5) \begin{align*}\sigma\x_h = \frac{p\x_i r\x_i ^2 - p\x_o r\x_o ^2}{r\x_o ^2 - r\x_i ^2} + \frac{(p\x_i - p\x_o) r\x_o ^2 r\x_i ^2}{(r\x_o ^2 - r\x_i ^2)r^2}\end{align*}

Radial Stress,

(6) \begin{align*}\sigma\x_r = \frac{p\x_i r\x_i ^2 - p\x_o r\x_o ^2}{r\x_o ^2 - r\x_i ^2} - \frac{(p\x_i - p\x_o) r\x_o ^2 r\x_i ^2}{(r\x_o ^2 - r\x_i ^2)r^2}\end{align*}

The axial stress for a closed-ended cylinder is calculated by means of the equilibrium, which reduces to:

Axial Stress,

(7) \begin{align*}\sigma\x_a = p\x_i \frac{r\x_i ^2}{r\x_o ^2 - r\x_i

ri = internal radius

ro = external radius

r = radius at point of interest (usually ri or ro)

Internal Pressure

Pa

External Pressure

Pa

Internal Radius

m

External Radius

m

Radius at Point of Interest

m

Result:

Pa

Thick Wall Cylinder Axial Stress Calculator

Answer 2

10.2 LAME’S THEOREM

Lame’s theorem gives the solution to thick cylinder problem. The theorem is based on the following assumptions:

Material of the cylinder is homogeneous and isotropic.

Plane sections of the cylinder perpendicular to the longitudinal axis remain plane under the pressure.

The second assumption implies that the longitudinal strain is same at all points, i.e., the strain is independent of the radius.