Elastic instability

There are a lot of ways to study this kind of instability. One of them is to use the method of incremetal deformations based on superposing a small perturbation on an equilibrium solution. 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. Assuming a clockwise angular deflection θ {displaystyle heta } , the clockwise moment exerted by the force becomes M F = F L sin ⁡ θ {displaystyle M_{F}=FLsin heta } . The moment equilibrium equation is given by F L sin ⁡ θ = k θ θ {displaystyle FLsin heta =k_{ heta } heta } where k θ {displaystyle k_{ heta }} is the spring constant of the angular spring (Nm/radian). Assuming θ {displaystyle heta } is small enough, implementing the Taylor expansion of the sine function and keeping the two first terms yields F L ( θ − 1 6 θ 3 ) ≈ k θ θ {displaystyle FL{Bigg (} heta -{frac {1}{6}} heta ^{3}{Bigg )}approx k_{ heta } heta } which has three solutions, the trivial θ = 0 {displaystyle heta =0} , and θ ≈ ± 6 ( 1 − k θ F L ) {displaystyle heta approx pm {sqrt {6{Bigg (}1-{frac {k_{ heta }}{FL}}{Bigg )}}}}