Material and geometric nonlinear buckling of simply supported columns

This study investigated the material and geometric nonlinear buckling of simply supported columns. The column body material followed Ludwick's constitutive law. Rectangular and elliptical cross sections were considered, and corresponding generalized moments of inertia (GMIs) were explicitly formulated. By applying the GMIs, the governing differential equations and boundary conditions of post-buckling columns were derived based on the Bernoulli-Euler beam theory, and the buckling loads and elastica were computed using numerical methods. To derive the elastica of buckling columns, the differential equations were integrated using the Runge-Kutta method, and the eigenvalues of the buckling load were determined using the bisection method. In our numerical experiments, the GMI formulae were applied, and parametric studies were conducted to analyze post-buckling columns in terms of buckling load, equilibrium path, elastica, and nonlinear stress along the cross section.