本文基於廣義的Hamilton原理與Timoshenko梁理論，發展適用於幾何非線性自由振動分析之混合有限元素法。文中考慮具彈性基礎之功能性梯度材料梁，其邊界條件為簡支承、固定端及自由端等不同邊界條件之組合，功能性梯度材料梁由兩相材料所組成，其材料性質是根據組成體積分率的冪次函數分佈，沿厚度方向逐漸且平滑地變化，並且使用複合材料之二相材料混合原理評估梁的有效材料特性。本混合有限元素法的弱形式數學方程式則是利用變分法推導求得，其中也考慮了馮卡門幾何非線性效應。數值範例中利用迭代程序獲得梁的非線性振幅-頻率關係的有限元素解。結果顯示，本有限元素解可迅速達到收斂，且其收斂解與文獻中之精確解相當吻合。

Based on the generalized Hamilton’s principle combined with the Timoshenko beam theory, the authors develop a mixed finite element method for the geometrically nonlinear free vibration analyses of a functionally graded beam resting on an elastic medium and with combinations of simply-supported, clamped and free edge conditions. The functionally graded beam is composed of a two-phase material, the material properties of which gradually and smoothly vary through the thickness direction according to the power-law distributions of the volume fractions of the constituents, and the effective material properties of the beam are estimated using the rule of mixtures. A weak-form formulation for the mixed finite element analysis is derived using the variational approach, in which the von Kármán geometrical nonlinearity is considered. The finite element solutions of the nonlinear amplitude-frequency relations of the beam are obtained using an iterative process. The results show that the finite element solutions converge rapidly, and the convergent solutions closely agree with the accurate ones available in the literature.

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