Asymptotic post-buckling analysis by Koiter's method with a general purpose finite element code
Many structures are sensitive to initial imperfections, sometimes leading to a great decrease in buckling load. Koiter showed that the effect of initial imperfections is largely determined by the initial post-buckling behavior of the perfect structure. The present work seeks to implement Koiter’s method of asymptotic post-buckling analysis on a finite element program Engineering Analysis Language (EAL).
EAL is based on engineering strain measures. It is shown via examples that the predicted post-buckling behavior of a structure for engineering strain measure is approximately the same as that for Green’s strain measure provided the strains are small. To characterize the post-buckling behavior by Koiter’s method in the finite element form, the linear and incremental stiffness matrices are required. These matrices comprise the tangent stiffness matrix. As EAL uses the modified Newton-Raphson procedure to solve nonlinear structures, it calculates the tangent stiffness. The first and second order incremental stiffnesses are extracted by partial differentiation of the tangent stiffness using a second order central difference scheme. The linear stiffness is directly given by the EAL processor ”K”. These stiffnesses are then used to get the post-buckling load-displacement behavior close to the bifurcation point. Numerical results for the initial post-buckling behavior are obtained for truss and frame structures using the Koiter’s analysis procedure on EAL. It is compared to the nonlinear load-displacement behavior of the structures with small initial imperfections. The post-buckling load-displacement behavior for a knee frame is also compared to the behavior obtained experimentally by Roorda  and analytically by Koiter . The asymptotic analysis procedure has given good asymptotic post-buckling results.