本文發展基於虛位移原理(the principle of virtual displacement, PVD)之有限圓柱層殼元素法，應用於具兩端簡支承之功能性梯度材料中空圓柱殼及多層複合材料中空圓柱殼承受軸、圍壓及組合載重時的三維線性挫屈分析，其材料性質依組成材料之體積比沿著厚度方向呈現冪級數形式變化。文中，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，且每個主要變數的相對階數可以自由選擇，在形狀函數厚度方向上的假設可以分成線性、二次或是三次多項式函數分布。PVD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理論解進行比較。

The unified formulations of finite cylindrical layer methods (FCLMs) based on the principle of virtual displacement (PVD) are developed for the three-dimensional (3D) linear buckling analysis of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders and laminated composite ones under combined axial compression and external pressure. The material properties of the FGM layer are assumed to obey the power-law distributions of the volume fraction of the constituents through the thickness coordinate. In these formulations, the cylinder is divided into a number of finite cylindrical layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the primary variables of each individual layer, respectively, as well as the related order of each primary variable can be freely chosen, such as the layerwise linear, quadratic or cubic function distribution through the thickness coordinate. The accuracy and convergence of the PVD-based FCLMs developed in this article are assessed by comparing their solutions with the exact 3D solutions available in the
literature.

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