Asymptotic-Numerical Method to Analyze the Postbuckling Behavior, Imperfection-Sensitivity, and Mode Interaction in Frames

The paper presents the formulation and illustrates the application of an asymptotic-numerical (semianalytical) method to analyze the geometrically nonlinear behavior of plane frames. The method adopts an “internally constrained” beam model and involves two distinct procedures: (1) an asymptotic analysis, which employs a perturbation technique to establish a sequence of systems of equilibrium differential equations and boundary conditions, and (2) the successive numerical solution of such systems, by means of the finite element method. This method can be applied to investigate the behavior of frames with arbitrarily complex configurations (member number and orientation) and leads to the determination of analytical expressions which provide: (1) the initial postbuckling behavior of perfect frames and (2) the nonlinear equilibrium paths of frames containing small initial imperfections or acted by primary bending moments, including the influence of eventual buckling mode interaction phenomena. In order to validate and illustrate the application and potential of the proposed method, several numerical results are presented, concerning (1) four validation examples (Euler column and three simple frames—two or three members), for which there exist some (perfect frame) analytical and numerical asymptotic results reported in the literature; (2) a single-bay pitched-roof frame with partially restrained column bases; and (3) a three-bay frame with two leaning columns. These results comprise (1) the initial postbuckling behavior of perfect frames (individual and coupled buckling modes) and (2) geometrically nonlinear equilibrium paths describing the behavior of frames containing initial geometrical imperfections or primary bending moments. In the latter case, most of the semianalytical results are compared with fully numerical values, yielded by finite element analyses performed in the commercial code ABAQUS.