notes/uni/mmme/2053_mechanics_of_solids/thick_walled_cylinders.md

---
author: Akbar Rahman
date: \today
title: MMME2053 // Thick Walled Cylinders
tags: [ thick_walled_cylinders ]
uuid: b53973dc-2c57-4e37-8409-96875125f4de
lecture_slides: [ ./lecture_slides/MMME2053_TC1_Intro.pdf, ./lecture_slides/MMME2053_TC2.pdf, ./lecture_slides/MMME2053_TC3.pdf ]
lecture_notes: [ ./lecture_notes/MMME2053_TC_Notes.pdf ]
exercise_sheets: [ ./exercise_sheets/Thick Cylinders Exercise Sheet.pdf, ./exercise_sheets/Thick Walled Cylinders Exercise Sheet Solutions.pdf ]
worked_examples: [ ./worked_examples/MMME2053_TC_WE1.pdf, ./worked_examples/MMME2053_TC_WE2.pdf, ./worked_examples/MMME2053_TC_WE3.pdf ]
---

# Lame's Equations

Derivation in lecture slides 2 (pp. 3-11)

$$\sigma_h = A + \frac{B}{r^2}$$

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

where $A$ and $B$ are *Lame's constants* (constants of integration).

Note that $\sigma_r$ does not vary with radius, $r$.

## Obtaining Lame's Constants

The constants can be obtained by using the boundary conditions of the problem:

At the inner radius ($r = R_i$) the pressure is only opposing the fluid inside:

$$\sigma_r= -p_i$$

At the outer radius ($r = R_o$) the pressure is only opposing the fluid outside (e.g. atmospheric
pressure):

$$\sigma_r = -p_o$$

Therefore:

\begin{align*}
-p_i &= C - \frac{D}{R_i^2}
-p_o &= C - \frac{D}{R_o^2}
\end{align*}

where $C$ and $D$ are constants which can be determined.

## Cylinder with Closed Ends

$$\sigma_z = \frac{R_i^2p_i - R_o^2p_o}{R_o^2-R_i^2}$$

## Cylinder with Pistons

No axial load is transferred to the cylinder.

$$\sigma_z = 0$$

## Solid Cylinder

$$\sigma_r = \sigma_\theta = A$$