mmme2053 note on elastic instability (buckling)

uni/mmme/2053_mechanics_of_solids/elastic_instability.md

@@ -9,3 +9,52 @@ 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)