本文基於Reissner混合變分原理，依序發展三種不同元素(有限層狀元素、有限矩形柱狀元素與有限環形柱狀元素)之有限層狀與柱狀方法，將其應用於具不同邊界組合之功能性材料積層板與殼，分析其靜力與自由振動物理問題。在研究初期，發展基於Reissner混合變分原理與虛位移原理之有限層狀元素法，將其應用於雙邊皆為簡支承之功能性材料積層板之動態與靜態問題分析上；此功能性材料在厚度方向之假設可為自然對數亦或是呈冪級數方式分佈。在分析時，將該板切割成數個獨立之層狀元素，且在元素內x-y平面使用雙傅立葉函數將元素內之變數展開，在厚度方向之變數則採用Lagrange多項式函數做模擬；其模擬的函數可自由選擇一階線性或二階拋物線函數
。在後續範例中，將與文獻內之三維解析解做比較，以討論各面內與側向主變數之函數階數選擇對此數值方法精確度之影響。
後續將發展有限層狀元素為有限矩形與環形柱狀元素，將其應用在具變化邊界之功能性積層板與中空圓柱殼靜力問題上；另外亦將發展有限矩形元素法於奈米碳管加勁材料積層板之自然振動分析上，該板之其中一對邊界為簡支承，另一對則可為簡支承、固定端與自由端之組合。在奈米碳管之功能性材料設計上共有四種基本分佈形式，各自代表奈米碳管纖維加勁材料在該板厚度方向之分佈形態，分別為均勻分佈、V形分佈、菱形分佈與X形分佈。在分析時，將該板與中空圓柱殼切割成有限個矩形柱狀與環形柱狀元素，在元素之y方向或θ方向使用單傅立葉函數將元素內之變數展開，而在x-ζ平面則採用Lagrange多項式函數加以模擬
；在x-ζ平面之Lagrange多項式階數將影響此數值方法之精度，此處共有三種元素可供選擇，分別為線性元素(L4)與二次元素(Q8與Q9)。在後續範例中，雙邊皆為簡支承之板殼物理問題將與文獻內之三維解析解做比較，以討論有限矩形柱狀與環形柱狀元素之適用性；另外在變化邊界之範例則與商用軟體ANSYS(12.0版)所得之數值解做比較，以確定本法之精確度。

關鍵字: Reissner混合變分原理、虛位移原理、有限層狀法、有限柱狀法、功能性梯度材料、奈米碳管、撓曲、振動、板、圓柱

SUMMARY
The Reissner mixed variational theorem (RMVT)- based finite layer methods (FLMs), finite rectangular prism methods (FRPMs) and finite cylindrical prism methods (FCPMs), are developed for the three-dimensional (3D) analysis of multilayered functionally graded material (FGM) plates/ circular hollow cylinders with various boundary conditions and under mechanical loads. In these formulations, the structures is divided into a number of finite rectangular/cylindrical prisms, in which the trigonometric functions and Lagrange polynomials are used to interpolate the circumferential direction and the axial–radial surface variations of the primary field variables of each individual prism, respectively.

The number of nodes of the nodal surface of each prism can be set at four for linear FRPMs/ FCPMs, and eight and nine for quadratic ones. These quadratic FRPM/ FCPM solutions of simply supported, multilayered composite cylinders and sandwiched FGM ones obtained in this way are in excellent agreement with the exact 3D solutions available in the literature, and those solutions for the plates/ cylinders with combinations of various edge conditions closely agree with the solutions obtained using the ANSYS commercial software.

Keywords: Reissner mixed variational theorem; Finite layer methods; Finite prism methods; Functionally graded materials; Carbon nanotubes

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