notes on thick walled cylinders

uni/mmme/2053_mechanics_of_solids/thick_walled_cylinders.md

@@ -9,3 +9,51 @@ 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
+
+$$\sigma_r = \sigma_\theta = A$$