Elastic instability explained

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 ( \theta - (1)/(6) \theta ^{3)} ≈ k _{\theta} \theta

which has three solutions, the trivial

\theta = 0 , and \theta ≈ \pm \sqrt{ 6 ( 1 - (k _{\theta})/(F L) ) }

which is imaginary (i.e. not physical) for

F L < k _{\theta} and real otherwise. This implies that for small compressive forces, the only equilibrium state is given by \theta = 0 , while if the force exceeds the value k _{\theta/L} 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

E _{spring} = \int k _{\theta} \theta d \theta = (1)/(2) k _{\theta} \theta ^{2}

and the work done by the force is simply the force multiplied by the vertical displacement of the beam end, which is

L ( 1 - \cos\theta) . Thus, E _{force} = \int{F d x = F L ( 1 - \cos \theta )}

The energy equilibrium condition

E _{spring} = E _{force} now yields F = k _{\theta} / L as before (besides from the trivial \theta = 0 ).

Stability of the solutions

Any solution

\theta is stable iff a small change in the deformation angle \Delta \theta results in a reaction moment trying to restore the original angle of deformation. The net clockwise moment acting on the beam is M(\theta) = F L \sin \theta - k _{\theta} \theta

An infinitesimal clockwise change of the deformation angle

\theta results in a moment M(\theta + \Delta \theta) = M + \Delta M = F L (\sin \theta + \Delta \theta \cos \theta ) - k _{\theta} (\theta + \Delta \theta)

which can be rewritten as

\Delta M = \Delta \theta (F L \cos \theta - k _{\theta)}

since

F L \sin \theta = k _{\theta} \theta due to the moment equilibrium condition. Now, a solution \theta is stable iff a clockwise change \Delta \theta > 0 results in a negative change of moment \Delta M < 0 and vice versa. Thus, the condition for stability becomes(\Delta M)/(\Delta \theta) = (d M)/(d \theta) = FL \cos \theta - k _{\theta} < 0

The solution

\theta = 0 is stable only for FL < k _{\theta} , which is expected. By expanding the cosine term in the equation, the approximate stability condition is obtained:|\theta| > \sqrt{ 2 ( 1 - (k _{\theta})/(F L) ) }

for

FL > k _{\theta} , 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

F L ( \sin \theta _{1} + \sin \theta _{2} ) = k _{\theta} \theta _{1} F L \sin \theta _{2} = k _{\theta} ( \theta _{2} - \theta _{1} )

where

\theta _{1} and \theta _{2} are the angles of the two beams. Linearizing by assuming these angles are small yields\begin{pmatrix} F L - k _{\theta} & F L \\ k _{\theta} & F L - k _{\theta \end{pmatrix} \begin{pmatrix} \theta} _{1} \\ \theta _{2 \end{pmatrix}} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}

The non-trivial solutions to the system is obtained by finding the roots of the determinant of the system matrix, i.e. for

(F L)/(k _{\theta}) = (3)/(2) \mp (\sqrt{ 5)/() } \approx \left\