本文主要是由梁理論建立FGM梁轉換為等效均質梁的觀念，再將沿著厚度方向分佈的楊氏模數函數轉換為等效楊氏模數的概念延伸至FGM板、FGM濺鍍板與FGM介面塗層板。首先利用梁曲率與彎矩關係推導FGM梁與等效均質梁之關係，再由尤拉立柱公式解出FGM梁之挫屈載重。而FGM板利用等效均質梁的觀念將其係數 轉換為等效均質板之楊氏模數與慣性矩。本文選擇三種楊氏模數分佈曲線，分別為冪次方函數(P-FGM)、S型函數(S-FGM)、指數函數(E-FGM)。考慮兩種邊界條件之FGM板:(一)兩對邊為簡支端，另兩對邊為自由端；(二)兩對邊為簡支端，另兩對邊為固定端，以有限元素軟體MARC分析等效均質板挫屈載重之數值解。為了證明等效均質板所求得之挫屈載重是正確的，將數值分析的結果與文獻之理論解交叉比對，分別探討材料指數p與材料梯度 的改變對挫屈載重的影響。本文可將FGM材料轉換為等效均質材料，簡化FGM材料繁複的計算，減少FGM材料與理論解誤差的累積。
研究結果顯示，FGM板之有限元素法數值解與理論值之挫屈載重誤差，和等效均質板之有限元素法分析與理論值之挫屈載重誤差皆在合理範圍內。楊氏模數分佈曲線以S-FGM材料分佈抵抗挫屈載重為最大，其次是E-FGM；而P-FGM材料最容易產生挫屈。在相同材料分佈曲線下，降低材料指數反而可以提昇構件之整體勁度。若邊界條件之束制條件增加，則整體抵抗挫屈的載重增加，就行為來說是較不容易產生挫屈。以FGM濺鍍板抵抗挫屈能力最佳，能有效緩解板在受軸壓所產生之應力集中的問題。

This article is based on Beam Theory and is shown to establish the concept of functionally graded beam transforms into equivalent homogeneous beam. Moreover, extending the concept of relationship between functionally graded material and equivalent homogeneous material to FGM plate, FGM coated plate and FGM undercoated plate to complete this paper.Using beam moment-curvature equation derived the relationship between functionally graded beam and equivalent homogeneous beam, and solved beam buckling solution by Euler’s Column Formula. Using concept of equivalent homogeneous beam for FGM plate material coefficient transforms into equivalent homogeneous plate. The Young’s modulus of the beam are assumed to be graded continuously in the direction z of thickness. The variation of the Young’s modulus follows a power-law, sigmoid function and exponential function distribution in terms of the volume fractions of functionally graded material. It’s assumed that the Poisson’s ratio of the beam remain constant. Two kinds of boundary conditions are considered. One is to subjected simple-support at the opposite of x-axis and another is cantilever-support support at the opposite of x-axis.To check the accuracy of equivalent homogeneous plate analytical buckling solution, the numerical buckling solutions by MARC program of finite element method are obtained and compare with the theoretical buckling solution of literature.
The result shows that the error of buckling load of finite element method is reasonable in FGM plates and equivalent homogeneous plates compare with theoretical value, it means FGM material can be replaced by equivalent homogeneous materials. Young’s modulus distribution curve of S-FGM is the best to resist the critical buckling load, second is E-FGM, the last is P-FGM. Reduce material index can improve the stiffness of member in the same material distribution curve. Add conditions of constraint can make critical buckling load increase under different boundary conditions. FGM coated plate is the best type to resist the critical buckling load. It’s the most effective plate type to relief the stress concentration problem in whole kinds of plates.

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