mmme2053 note on elastic instability (buckling)

uni/mmme/2053_mechanics_of_solids/elastic_instability.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2053 // Elastic Instability (Buckling)
+tags: [ elastic_instability, buckling ]
+uuid: b8b2cff7-8106-4968-bab5-f4cffcf8b5a0
+lecture_slides: [ ./lecture_slides/MMME2053-EI L1 Slides.pdf, ./lecture_slides/MMME2053-EI L2 Slides.pdf ]
+lecture_notes: [ ./lecture_notes/Elastic Instability (Buckling) Notes.pdf ]
+exercise_sheets: [ ./exercise_sheets/Elastic Instability (Buckling) Exercise Sheet.pdf, ./exercise_sheets/Elastic Instability (Buckling) Exercise Sheet Solutions.pdf ]
+worked_examples: [ ./worked_examples/MMME2053-EI WE1 Slides.pdf ]
+---
+
+# Notes from Lecture Slides (2)
+
+> In contrast to the classical cases considered here, actual compression members are seldom truly pinned or
+> completely fixed against rotation at the ends. Because of this uncertainty regarding the fixity of the ends,
+> struts or columns are often assumed to be pin-ended. This procedure is conservative.
+>
+> The above equations are not applicable in the inelastic range, i.e. for $\sigma > \sigma_y$ , and must be modified.
+>
+> The critical load formulae for struts or columns are remarkable in that they do not contain any strength
+> property of the material and yet they determine the load carrying capacity of the member. The only material
+> property required is the elastic modulus, $E$, which is a measure of the stiffness of the strut.
+
+# Stability of Equilibrium
+
+![(a) is a stable equilibrium (it will return to equilibrium if it deviates) whereas (b) is an unstable equilibrium (it will not return to equilibrium if it deviates)](./images/stable_unstable_equilibria.png)
+
+# Critical Buckling Load on a Strut
+
+Critical buckling load is given by:
+
+$$P_c = \frac{\pi^2EI}{L_\text{eff}^2}$$
+
+where $L_\text{eff}$ is the effective length:
+
+- Free-fixed -> $L_\text{eff} = 2l$
+- Hinged-hinged -> $L_\text{eff} = l$
+- Fixed-hinged -> $L_\text{eff} = 0.7l$
+- fixed-fixed -> $L_\text{eff} = 0.5l$
+
+where $l = 0.5L$
+
+Derivations detailed in lecture slides (1, pp. 8-21).
+
+# Compression of Rods/Columns
+
+Derivations detailed in lecture slides (2, pp. 3-5).
+
+Buckling will occur if
+
+$$\sigma = \frac{\pi^2E}{\frac{L^2}{K^2}}$$
+
+where $k$ is the radius of gyration and $\frac{L}{K}$ is the slenderness ratio.
+
+Plastic collapse will occur if $\sigma = \sigma_y$.
+
+This can be represented diagrammatically:
+
+![](./images/bucking_vs_plastic_collapse.png)

uni/mmme/2053_mechanics_of_solids/thick_walled_cylinders.md

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+---
+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
+
+$$\sigma_r = \sigma_\theta = A$$