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 formula_1, the clockwise moment exerted by the force becomes formula_2. The moment equilibrium equation is given by 
formula_3
where formula_4 is the spring constant of the angular spring (Nm/radian). Assuming formula_1 is small enough, implementing the Taylor expansion of the sine function and keeping the two first terms yields 
formula_6
which has three solutions, the trivial formula_7, and 
formula_8
which is imaginary (i.e. not physical) for formula_9 and real otherwise. This implies that for small compressive forces, the only equilibrium state is given by formula_7, while if the force exceeds the value formula_11 there is suddenly another mode of deformation possible.
Energy method.
The same result can be obtained by considering energy relations. The energy stored in the angular spring is 
formula_12
and the work done by the force is simply the force multiplied by the vertical displacement of the beam end, which is formula_13. Thus, 
formula_14
The energy equilibrium condition formula_15 now yields formula_16 as before (besides from the trivial formula_7).
Stability of the solutions.
Any solution formula_1 is stable iff a small change in the deformation angle formula_19 results in a reaction moment trying to restore the original angle of deformation. The net clockwise moment acting on the beam is 
formula_20
An infinitesimal clockwise change of the deformation angle formula_1 results in a moment 
formula_22
which can be rewritten as 
formula_23
since formula_24 due to the moment equilibrium condition. Now, a solution formula_1 is stable iff a clockwise change formula_26 results in a negative change of moment formula_27 and vice versa. Thus, the condition for stability becomes
formula_28
The solution formula_7 is stable only for formula_30, which is expected. By expanding the cosine term in the equation, the approximate stability condition is obtained:
formula_31
for formula_32, which the two other solutions satisfy. Hence, these solutions are stable.
Multiple degrees of freedom-systems.
By attaching another rigid beam to the original system by means of an angular spring a two degrees of freedom-system is obtained. Assume for simplicity that the beam lengths and angular springs are equal. The equilibrium conditions become
formula_33
formula_34
where formula_35 and formula_36 are the angles of the two beams. Linearizing by assuming these angles are small yields
formula_37
The non-trivial solutions to the system is obtained by finding the roots of the determinant of the system matrix, i.e. for 
formula_38
Thus, for the two degrees of freedom-system there are two critical values for the applied force "F". These correspond to two different modes of deformation which can be computed from the nullspace of the system matrix. Dividing the equations by formula_35 yields
formula_40
For the lower critical force the ratio is positive and the two beams deflect in the same direction while for the higher force they form a "banana" shape. These two states of deformation represent the buckling mode shapes of the system.