This study proposes a new numerical-and-analytical method for solving geometrically nonlinear problems of bending of complex-shaped plates made of functionally graded materials developed. The problem was formulated within the framework of a refined higher-order theory considering the quadratic law of distribution of transverse tangential stresses along the plate thickness. To linearize the nonlinear problem, we used the method of continuous continuation in the parameter associated with the external load. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear plate bending (displacements, strains, and stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta–Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometrically linear deformation. The right-hand sides of the differential equations, at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme, were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution as a formula. This solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. Test problems are solved for a homogeneous rigidly fixed and functionally graded hinged square plate subjected to a uniformly distributed load of varying intensity. The results for deflections and stresses obtained by the developed method are compared with the solutions obtained by radial basis functions. The problem of bending of a functionally graded plate of complex shape is solved. The influence of the gradient properties of the material and geometric shape on the stress-strain state is investigated.

本研究提出了一种新的数值分析方法，用于解决由功能梯度材料制成的复杂形状板弯曲的几何非线性问题。该问题是在考虑横向切向应力沿板厚度分布的二次定律的精炼高阶理论框架内制定的。为了使非线性问题线性化，我们在与外部载荷相关的参数中使用了连续连续的方法。对于线性化问题的变分公式，构建了拉格朗日泛函，并在运动学上可能的位移速度下定义。为了找到非线性板弯曲问题的主要未知数（位移、应变和应力），我们对常微分方程组的柯西问题进行了公式化。柯西问题通过具有自动步骤选择的龙格-库塔-默森方法解决。初始条件是通过几何线性变形问题的求解找到的。对应于龙格-库塔-默森格式的负载参数固定值时，微分方程的右侧是通过拉格朗日泛函变分问题的解获得的。变分问题采用Ritz方法结合R函数方法求解。后者使得可以将近似解呈现为公式。该解结构恰好满足全部（一般结构）或部分（部分结构）的边界条件。解决了均匀刚性固定且功能分级的铰接方板承受不同强度的均匀分布载荷的测试问题。将通过所开发的方法获得的挠度和应力的结果与通过径向基函数获得的解进行比较。解决了形状复杂的功能梯度板的弯曲问题。研究了材料的梯度特性和几何形状对应力应变状态的影响。