Buckling Energy of Beam/section

[Su Solberg]

Hi guys, I wonder whether we can roughly estimate the energy used to buckle a hollow tube (by calculation)?
I was asked to calculate the backstay of a cane to ensure it can stand for the rebound force due to abnormal operation.

Thanks in advance.

 

[Mapes]

It is easiest to calculate the minimum energy required to achieve buckling. The critical load $F$ for buckling a long, slender object is $F=\pi^2EI/L_\mathrm{eff}^2$ (where $L_\mathrm{eff}$ is the effective length for the loading condition you're interested in). The axial deflection $\delta$ from an axial load is, as usual, $\delta=FL/AE$, and the moment of inertia for a hollow tube is $I=\pi(r_\mathrm{o}^4-r_\mathrm{i}^4)/2$. So the strain energy from axial loading up to the level of the critical buckling load is

$$\frac{1}{2}F\delta=\frac{1}{2}\left(\frac{\pi^4E^2I^2}{L_\mathrm{eff}^4}\right)\left(\frac{L}{AE}\right)=\frac{\pi^5E(r_\mathrm{o}^4-r_\mathrm{i}^4)(r_\mathrm{o}^2+r_\mathrm{i}^2)L}{8L_\mathrm{eff}^4}$$

which may be of some use.