Mechanics of deformable solids Definition

Buckling

Builds on Elastic instability Bifurcation theory

Definition

Consider an elastic structure under a compressive load $\lambda \ge 0$ with total potential energy $\Pi(u, \lambda)$ . Suppose there is a fundamental branch of equilibria $u_0(\lambda)$ which is stable for small $\lambda$ . Examples are the straight state of a column and the flat state of a plate. The structure buckles at $\lambda_c$ if, at $\lambda = \lambda_c$ :

the fundamental branch loses stability, so that $\lambda_c$ is a critical load in the sense of elastic instability, and
a second branch of equilibria, different from $u_0$ , passes through $(u_0(\lambda_c), \lambda_c)$ .

The smallest such $\lambda_c$ is the buckling load, and the kernel of the second variation $D_u^2 \Pi(u_0(\lambda_c), \lambda_c)$ consists of the buckling modes.

Remarks

Example: a straight elastic column pinned at both ends, of length $L$ and bending stiffness $EI$ , under an axial compressive force $P$ . The straight state is an equilibrium for every $P$ . The linearized equation $EI w'' + P w = 0$ , $w(0) = w(L) = 0$ , has the nonzero solution $w = \sin(\pi x / L)$ exactly when $P = \pi^2 EI / L^2$ , which is the buckling load.

Non-example: the same column under tension, $P < 0$ . The equation $EI w'' + P w = 0$ with these end conditions has only the solution $w = 0$ , and the straight state stays stable, so no buckling occurs.

When the load reaches a maximum on a single equilibrium path without a bifurcation, as for a shallow arch, one speaks of snap-through or limit-point instability instead.

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Bifurcation theory concept
Elastic instability concept
Buckling definition

Tag	`HSR1BM0PEP`
URI	`m2://HSR1BM0PEP`
Kind	definition
Writing	W0 stub
MSC	74G60, 74K10
Defines	Buckling
Uses	Elastic instability, Bifurcation theory
Changed	3 November 2026

Buckling

Definition

Remarks

Reading path

Further reading

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