Critical Analysis of the Equations of Statics in the Bending Theories of Composite Plates Obtained on the Basis of Variational Principles of Elasticity Theory 1. General Theories of High Order

In a geometrically linear formulation based on the use of variational principles of elasticity theory and the generalized Galerkin method, three approaches to constructing equations of static bending of orthotropic plates with a constant thickness are considered. The displacements of plate points are approximated by polynomials in the transverse coordinate with unknown coefficients of expansion. The application of variational principles leads to the first two approaches for constructing these equations. In the first approach, the force boundary conditions on faces of the plates are not taken into account, but the resulting two-dimensional equations and the corresponding boundary conditions on their edges are consistent, since, according to these equations, any subarea of the plates and the entire structures as a whole are in equilibrium. The two-dimensional equations of statics and boundary conditions on edges of the plates obtained within this approach can also be obtained by the generalized Galerkin method. The simplest theory using this approach is the Reissner theory. The second approach takes into account the force boundary conditions on faces of the plates. The latter circumstance leads to the necessity for using dependent variations of kinematic variables or introduction of undetermined Lagrange multipliers. It is shown that the resulting two-dimensional static equations and boundary conditions on plate edges are inconsistent, since any arbitrary subarea of the plate and the entire structure as a whole are in the nonequilibrium state. The two-dimensional statics equations and boundary conditions constructed in this approach cannot be obtained by the generalized Galerkin method. The simplest theories using this approach are the Reddy–Nemirovskii and the Margueere–Timoshenko–Naghdi theories. The third approach takes into account the force boundary conditions on faces of the plates, but the two-dimensional equations and the corresponding boundary conditions on the edges are obtained by the generalized Galerkin method. The homogeneous polynomials in the transverse coordinate are used as weight functions. The two-dimensional equations and boundary conditions obtained in this approach are consistent. The simplest theory that uses this approach is the Ambartsumyan theory. The relevance of the study is determined by the fact that the choice of a simple, but adequate, theory of bending can be of fundamental importance in solving optimization problems for com