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Modal Buckling Analysis

The problem in this case is to determine critical buckling loads (Eulerian buckling load) of the structure. The assumption is that the geometric (or differential) stiffness matrix is a linear function of applied load. The aim of the buckling analysis is to calculate the factor that can be applied to load before the structure buckles. At buckling the determinant of the sum of the elastic stiffness and the critical geometric stiffness is zero.

∣K+λKg,crit∣=0\left| \mathbf{K} + \lambda\mathbf{K}_{g,crit} \right| = 0

Using the assumption of differential stiffness a linear function of loads gives

Kg,crit=λKg\mathbf{K}_{g,crit} = \lambda\mathbf{K}_{g}

so the equation is

∣K+λKg∣=0\left| \mathbf{K} + \lambda\mathbf{K}_{g} \right| = 0

and the eigenvalue problem is then

KΦ=−ΛKgΦ\mathbf{K}\boldsymbol{\varPhi} = - \Lambda\mathbf{K}_{g}\boldsymbol{\varPhi}