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 composite plates.

中文翻译：

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1. 高阶一般理论

在基于弹性理论变分原理和广义 Galerkin 方法的几何线性公式中，考虑了三种构造具有恒定厚度的正交各向异性板的静态弯曲方程的方法。板点的位移由具有未知膨胀系数的横向坐标中的多项式近似。变分原理的应用导致了构建这些方程的前两种方法。在第一种方法中，不考虑板面上的力边界条件，但产生的二维方程和它们边缘上的相应边界条件是一致的，因为根据这些方程，板的任何子区域并且整个结构作为一个整体处于平衡状态。用这种方法获得的板边缘静力学和边界条件的二维方程也可以通过广义伽辽金方法获得。使用这种方法的最简单的理论是 Reissner 理论。第二种方法考虑了板面上的力边界条件。后一种情况导致需要使用运动学变量的相关变化或引入未确定的拉格朗日乘子。结果表明，由于板的任意子区域和整个结构整体处于非平衡状态，所得到的二维静力方程和板边缘的边界条件不一致。这种方法构建的二维静力学方程和边界条件无法通过广义伽辽金方法获得。使用这种方法的最简单的理论是 Reddy-Nemirovskii 和 Margueere-Timoshenko-Naghdi 理论。第三种方法考虑了板面上的力边界条件，但二维方程和边上相应的边界条件是通过广义伽辽金方法获得的。横向坐标中的齐次多项式用作权重函数。这种方法得到的二维方程和边界条件是一致的。使用这种方法的最简单的理论是 Ambartsumyan 理论。研究的相关性取决于以下事实：选择一个简单但足够的，