Application of Variational Method to Stability Analysis of Cantilever Vertical Plates with Bimodular Effect

Abstract

In the design of cantilevered balconies of buildings, many stability problems exist concerning vertical plates, in which reaching a critical load plays an important role during the stability analysis of the plate. At the same time, the concrete forming vertical plate, as a typical brittle material, has larger compressive strength but lower tensile strength, which means the tensile and compression properties of concrete are different. However, due to the complexities of such analyses, this difference has not been considered. In this study, the variational method is used to analyze stability problems of cantilever vertical plates with bimodular effect, in which different loading conditions and plate shapes are also taken into account. For the effective implementation of a variational method, the bending strain energy based on bimodular theory is established first, and critical loads of four stability problems are obtained. The results indicate that the bimodular effect, as well as different loading types and plate shapes, have influences on the final critical loads, resulting in varying degrees of buckling. In particular, if the average value of the tensile modulus and compressive modulus remain unchanged, the introduction of the bimodular effect will weaken, to some extent, the bending stiffness of the plate. Among the four stability problems, a rectangular plate with its top and bottom loaded is most likely to buckle; next is a rectangular plate with its top loaded, followed by a triangular plate with its bottom loaded. A rectangular plate with its bottom loaded is least likely to buckle. This work may serve as a theoretical reference for the refined analysis of vertical plates. Plates are made of concrete or similar material whose bimodular effect is relatively obvious and cannot be ignored arbitrarily; otherwise the greater inaccuracies will be encountered in building designs.

Application of the Variational Method to the Large Deformation Problem of Thin Cylindrical Shells with Different Moduli in Tension and Compression

“…It is easy to see that they are quite close. In fact, in the previous study [ 32 ], the variational method was successfully used in the analysis of stability of cantilever vertical plates with different moduli. In addition, the recent study [ 33 ] also showed that, for the large deformation of thin shallow spherical shells, the perturbation solution agrees well with the variational solution.…”

Variational Solution and Numerical Simulation of Bimodular Functionally Graded Thin Circular Plates under Large Deformation

“…Due to the fact that the displacement variational equation itself represents the equilibrium equation and stress boundary conditions, it naturally avoids the consideration for the equilibrium condition of thin plates, while in the perturbation method, the establishment of an equation of equilibrium is necessary and somewhat complicated. Recent studies [31,32] also indicated that the variational method can be successfully used in the analysis of plates and shells. First, Xue et al [31] adopted the variational method to obtain the critical loads of cantilever vertical plates with different moduli.…”

“…Recent studies [31,32] also indicated that the variational method can be successfully used in the analysis of plates and shells. First, Xue et al [31] adopted the variational method to obtain the critical loads of cantilever vertical plates with different moduli. Thereafter, He et al [32] also used the variational method to solve bimodular thin shells under large deformation.…”