---
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)