Hencky bar-grid model and Hencky bar-net model for buckling analysis of rectangular plates

This chapter presents a novel numerical framework for elastic buckling analysis of rectangular plates with rectangular cutouts by using two physical structural models, the Hencky bar-grid model (eHBM) and the Hencky bar-net model (HBM). The eHBM comprises rigid bar-grids joined by elastic primary axial, secondary axial, and torsional springs, while HBM consists of rigid bar-nets connected by elastic rotational and torsional springs. The in-plane displacements at each bar-grid joint of eHBM or the deflections at each bar-net joint of HBM can be computed by solving a set of algebraic equations obtained from minimizing the sum of strain energy of all elastic springs and the total potential energy of the external loads. The in-plane stresses can be calculated from the obtained in-plane displacements. The buckling analysis is carried out in two steps. In the first step, eHBM is used to determine the prebuckling in-plane stress distributions in a rectangular plate under any applied in-plane loads. In the second step the computed in-plane stress distribution is applied as loads at the joints of the HBM for the buckling analysis. Several buckling problems involving rectangular plates and cutouts under various in-plane load conditions and boundary conditions are solved to illustrate the convergence, accuracy, and validity of the eHBM-HBM approach. The approach features a monotonic convergence of the buckling solutions to the continuum plate solutions with respect to decreasing grid size from below. Finally, we show that the eHBM-HBM approach can be easily used to optimize the locations of cutouts in rectangular plates for maximum buckling load.