Linear eigenvalue buckling analysis

Push a ruler end to end and it becomes easier to bend sideways; pull it and it becomes stiffer, like a tensioned string. The axial force changes the sideways stiffness even though the material has not changed. A finite element model captures this with a second stiffness matrix, the stress stiffness or geometric stiffness K_σ, built from the stresses the applied load causes.

Tensile stresses make K_σ add to the normal elastic stiffness K; compressive stresses make it subtract. Internal pressure therefore stiffens a shell, while vacuum, axial compression and bending compression soften it.

The model first runs a linear static analysis at the applied loads to get the stresses, and from them K_σ. It then scales those loads by a factor λ and asks at what values of λ the combined stiffness becomes singular, meaning a non-zero displacement shape φ exists that needs no extra load:

Each value of λ that satisfies this is an eigenvalue, and the shape φ that goes with it is a buckling mode. The mode gives only the shape, not the size: its amplitude is arbitrary, and the plotted displacements mean nothing in millimetres.

A pin-ended column is the simplest check of the idea. Its eigenvalues are the Euler loads P_n = n²·π²EI/L², with mode shapes of one, two and three half-waves. A good model of a column reproduces these values closely, which is a common first test of an eigenvalue set-up.

The lowest multiplier, λ1, is the first load at which the ideal structure loses stability. If λ1 is 4, the ideal structure would buckle at four times the applied load. Its mode shape shows where and how: lobes between two rings, dimples near a support, or a wrinkle in a skirt next to an opening.

For a column the multipliers are well separated. For thin shells they are often bunched together, several modes within a few per cent of each other. That clustering is itself a warning: it usually goes with strong sensitivity to imperfections, because a small dent can mix nearby modes and trigger buckling early.

Check each important mode for realism. A mode confined to one node where a point load is applied, or to a region of poor mesh, is usually a modelling artefact. A negative eigenvalue means the structure would buckle if the loads were reversed, which may or may not be a real case.

• The geometry is perfect: no out-of-roundness, dents or weld sinkage.
• The material stays elastic, so yielding before buckling is ignored.
• The shape barely changes before buckling, so the stress stiffness comes from a linear static solution.
• The loads keep their direction as the structure deflects.

Real shells break the first two assumptions. The eigenvalue therefore overestimates their capacity, by a wide margin for thin cylinders in axial compression, where test results fall well below the classical value. Treat λ1 as an upper bound, not as the buckling load.

Codes therefore reduce λ1 by a capacity-reduction factor below 1, which covers imperfections and depends on the shell shape, the kind of stress and the slenderness, and then divide by a design margin. The result must be at least 1. Together they typically ask for an ideal multiplier well above 1, and several times 1 for cylinders in axial compression. Some codes apply the reduction to the stress at the buckling location rather than to λ directly, and add a check where buckling and yielding interact.

For thin shells, or when the linear route gives a poor margin, a nonlinear check gives a more realistic capacity. The first eigenmode is scaled to an imperfection of the size the fabrication tolerances allow, the metal is allowed to yield, large displacements are switched on and the load is increased step by step until the structure can carry no more. The eigenvalue analysis is still useful there: it supplies the imperfection shape.