Abstract Multi-term extended Kantorovich method (MTEKM) is used to solve the bending problem of thin skew (parallelogram) functionally graded plate resting on the Winkler elastic foundation under uniformly distributed transverse load. Formulations are based on the classical plate theory (CPT) with the physical neutral surface is considered as the reference plane. Various configurations of clamped, simply supported and free edges are considered. By implementing the concept of Galerkin’s weighted residual method, the fourth-order partial differential governing equation and boundary conditions are converted into two sets of ordinary differential equations (ODE), which are then solved numerically using “Chebfun” numerical computation package. Convergence and accuracy of MTEKM are investigated. Results obtained with MTEKM are compared to finite element method (FEM) solutions. FEM has been implemented using ANSYS software, in which the plate is modeled with shell elements, while the elastic foundation is modeled as a pair of contact/target elements. In addition, the effects of both the Winkler foundation stiffness and material power index have been investigated. Applying MTEKM in bending analysis of thin skew plates offered more accurate results than the single-term EKM but with the cost of more computation time but still provides simplicity and rapid convergence. It is found that MTEKM well suits the bending problem of skew FGM plates resting on elastic foundation.

中文翻译：

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基于多项扩展Kantorovich方法的Winkler弹性地基薄FGM斜板弯曲分析

摘要 采用多项扩展Kantorovich方法(MTEKM)解决均匀分布的横向荷载作用下放置在Winkler弹性基础上的薄斜(平行四边形)功能梯度板的弯曲问题。公式基于经典板理论 (CPT)，将物理中性表面视为参考平面。考虑了夹紧边缘、简支边缘和自由边缘的各种配置。通过实施伽辽金加权残差法的概念，将四阶偏微分控制方程和边界条件转化为两组常微分方程（ODE），然后使用“Chebfun”数值计算包进行数值求解。研究了 MTEKM 的收敛性和准确性。将使用 MTEKM 获得的结果与有限元法 (FEM) 解决方案进行比较。FEM 已使用 ANSYS 软件实现，其中板用壳单元建模，而弹性基础建模为一对接触/目标单元。此外，还研究了 Winkler 基础刚度和材料功率指数的影响。在薄斜板的弯曲分析中应用 MTEKM 提供比单项 EKM 更准确的结果，但需要更多的计算时间，但仍然提供简单和快速收敛。结果表明，MTEKM 非常适合放置在弹性基础上的倾斜 FGM 板的弯曲问题。而弹性基础被建模为一对接触/目标元素。此外，还研究了 Winkler 基础刚度和材料功率指数的影响。在薄斜板的弯曲分析中应用 MTEKM 提供比单项 EKM 更准确的结果，但需要更多的计算时间，但仍然提供简单和快速收敛。结果表明，MTEKM 非常适合放置在弹性基础上的倾斜 FGM 板的弯曲问题。而弹性基础被建模为一对接触/目标元素。此外，还研究了 Winkler 基础刚度和材料功率指数的影响。在薄斜板的弯曲分析中应用 MTEKM 提供比单项 EKM 更准确的结果，但需要更多的计算时间，但仍然提供简单和快速收敛。结果表明，MTEKM 非常适合放置在弹性基础上的倾斜 FGM 板的弯曲问题。