本文主旨為根據一階剪變形板理論(first-order shear deformation plate theory)和高階剪變形板理論(higher-order shear deformation plate theory)之基礎架構，分析簡支撐矩形功能梯度板的自然振動頻率、模態與挫屈負荷。首先利用板理論之位移場及應變與位移的關係，求得應力場之合應力與合力矩，進而建立功能梯度板的本構方程式(constitutive equations)。接著使用漢米爾頓原理 (Hamilton’s principle)與有限元素的近似解來來推導功能梯度板之有限元素運動方程式。本文並根據級數解(series solution)之理論，藉由假設滿足邊界條件之奈維爾解(Navier’s solution)來建立功能梯度板之級數解運動方程式。

利用本文之有限元素分析模式與級數解模式，與求解特徵值與特徵向量的方法，可以求得功能梯度板之自然振動頻率、振動模態與挫屈負荷。本文針對陶瓷、功能梯度材料與金屬等不同材料的組合，探討各種參數與起始應力對功能梯度板之自然振動頻率與挫屈負荷的影響，結果並與文獻比較以驗證本文分析模式的準確性。

Based on the first-order and higher-order shear deformation plate theories, the present study develops a finite element model and two series-solution models to analyze the natural frequencies, vibration modes and buckling loads of simply-supported rectangular plates of functionally graded materials(FGMs). First, this study uses the displacement fields and the stress-strain relations to develop the constitutive equations of stress and moment resultants. Next, substituting the constitutive equations into the Hamilton’s principle, this study obtains the present finite element model for the FGM plates. Also this study applies the Navier’s solution to obtain the equations of FGM plates for free vibration and buckling analyses.

The present finite element model and the series solution models are used to find the natural frequencies, vibration modes and buckling loads of the FGM plates. Considering different composition of ceramic, FGM and metal, this study also investigates the effects of various parameters, such as aspect ratio, the thickness of FGM and initial stress, on the natural frequencies and buckling loads of FGM plates. The present results are compared with that in literature to verify the accuracy of the present models.

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