本文使用基於虛位移原理(the principle of virtual displacements, PVDs)之有限圓柱層殼元素法分析一兩端為簡支承之功能性材料三明治中空圓柱殼的擬三維靜態撓曲及自然振動行為，其材料性質假設功能性材料的材料參數是以體積比沿著厚度方向呈現冪級數形式變化。本文理論公式中，圓柱將被切割成數個有限層殼元素，其中每層的曲面位移和橫向位移分別以三角函數和拉格朗治多項式進行內插模擬。由於本文是以h-refinement程序對有限層殼元素法所得之數值解進行收斂性探討，所以對於位移分量展開的相對階數將保持為變數，且可以自由選擇線性、二次或是三次元素理論進行分析運算。本文中，基於PVD之有限圓柱層殼元素法的收斂速度及準確率也將與現有文獻中的三維解進行比較。

A unified formulation of finite cylindrical layer methods (FCLMs) based on the principle of virtual displacements (PVDs) is developed for the (3D) bending and free vibration analyses of simply-supported, functionally graded material (FGM) sandwich circular hollow cylinders, in which the material properties of the FGM layer are assumed to obey the power-law distributions of the volume fractions of the constituents through the thickness coordinate. In this formulation, the cylinder is divided into a number of cylindrical finite layers, where the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the displacement components of each individual layer, respectively. Because an h-refinement is adopted in this article to yield the convergent solutions, the relative orders used for expansion of the displacement components remain variable, and can be freely chosen as linear, quadratic and cubic ones. The accuracy and convergence rate of a variety of PVD-based FCLMs developed in this article are assessed by comparing their solutions with the available 3D ones.

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