Elastic Instability

Elastic instability is a form of instability occurring in elastic systems, such as buckling of beams and plates subject to large compressive loads.

There are a lot of ways to study this kind of instability. One of them is to use the method of incremental deformations based on superposing a small perturbation on an equilibrium solution.

Single degree of freedom-systems

Consider as a simple example a rigid beam of length ''L'', hinged in one end and free in the other, and having an angular spring attached to the hinged end. The beam is loaded in the free end by a force ''F'' acting in the compressive axial direction of the beam, see the figure to the right.

Moment equilibrium condition

Assuming a clockwise angular deflection $\theta$, the clockwise moment exerted by the force becomes $M_F = F L \sin\theta$. The moment equilibrium equation is given by

$F L \sin \theta = k_\theta \theta$

where $k_\theta$ is the spring constant of the angular spring (Nm/radian). Assuming $\theta$ is small enough, implementing the Taylor expansion of the sine function and keeping the two first terms yields

$F L \Bigg(\theta - \frac \theta^3\Bigg) \approx k_\theta \theta$

which has three solutions, the trivial $\theta = 0$, and

$\theta \approx \pm \sqrt$

which is