An analytical solution for dynamic instability and vibration analysis of structural members with open and closed sections

Due to the excellent bending resistance characteristics, open- and closed-section members are widely used in engineering practice. However, the interactions between plates of different sections have a significant effect on the mechanical behavior of members. Therefore, taking those interactions into consideration is a critical step in establishing the analytical model to investigate the static and dynamic behavior of the structures. This investigation proposes a spring plate model to analyze the vibration, static and dynamic buckling of members, in which the deformation of the spring plate is described by coupling polynomials and trigonometric series. The results show that this class of functions can accurately characterize the restraint of plates. The rotational restraint stiffness for single-plate and double-plates constraints are accurately obtained. Subsequently, analytical solutions for vibration and buckling problem are obtained by the Rayleigh–Ritz method. The dynamic buckling region of the members is further obtained using the Bolotin theory. When the frequency of the periodic load is in this region, the dynamic buckling of the structure is triggered. The presented model is provided to be an effective tool for analyzing the vibration and static and dynamic instability of thin-walled structures by comparing the existing results with finite element method (FEM) results. At the same time, understanding the vibration and dynamic behaviors is beneficial for design safely in terms of structural stability, load bearing capacity, and seismic resistance.