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

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