Support Buckling

From Qwiki

Consider a cylindrical support with radius $R$ and length $l$ , being loaded by a mass $m$ . As the cylindrical support bends (see figure), the inner and outer surfaces will, respectively, be elongated and contracted by an amount

$\delta l = (r_c \pm R)\theta - l \approx \pm R \theta$

where the subtended angle $θ$ may be approximated by

$\theta \approx \frac{\delta x}{l} \ .$

The elastic energy associated with this bending may be approximated by

$\delta E_{el} \approx V\delta U \approx\frac{E V}{2}\left(\frac{\delta l}{l}\right)^2 = \frac{\pi R^4 E}{2 l^3}\delta x^2$

where $δ U$ is the elastic energy density and $E$ is the elastic modulus of the supporting material.

As the support bends, the loading mass changes height by and amount

$\delta y \approx -l(1-\cos{\theta}) \approx -\frac{l\theta^2}{2} \approx -\frac{\delta x^2}{2 l}$

so that the released potential energy is

$\delta E_{gr} = mg \delta y \approx -\frac{mg}{2l}\delta x^2 \ .$

Balancing this drop in gravitational energy with the gain in elastic energy yields the critical relationship between the radius and length such that the rod is on the verge of buckling:

$\delta E_{el} = -\delta E_{gr} \quad \Longrightarrow \quad \frac{R^2}{l} \sim \left(\frac{mg}{\pi E}\right)^{1/2}$

So that for the support to be stable under load, the aspect ratio is constrained by:

$\frac{R^2}{l} > \left(\frac{mg}{\pi E}\right)^{1/2}.$

In reality, $R 2 / l$ will have to be even larger than this, as the above expression for the stored elastic energy is an overestimate. A more accurate estimate of the stored elastic energy, obtained by integrating the energy density over the support volume, yield