> 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
# 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 $\sigma = \sigma_y$.
