本文應用混合分層(Layerwise, LW)高階剪切變形理論(Higher-order shear deformation theory, HSDT)進行受軸向載重之功能性梯度簡支梁的振動頻率及臨界載重分析。文中假設 FG 梁的材料性質隨厚度座標變化，FG 梁之有效材料性質可用二相材料混合法則(The rule of mixtures)來進行計算。數值範例結果顯示與文獻中提供精確解相吻合。本文亦使用類神經網路進行受軸向載重 FG 簡支梁振動頻率之預測和材料性質之辨識。在訓練類神經網路的過程中，預先依混合分層高階剪切變形理論產生訓練組及測試組，前者用於訓練神經網路，後者用於測試類神經網路精確度。經過適當訓練後，類神經網路所預測之振動頻率有相當不錯的精確度，而運算時間相較混合分層高階剪切變形理論解法大幅節省，亦達成本理論解法無法實現之材料辨識。

This paper uses Layerwise (LW) Higher-order Shear Deformation Theory (HSDT) to analyze the vibration frequency and critical load of a functionally graded simply supported beam under axial load. It is assumed that the material properties of the FG beam change with the thickness coordinates, and the effective material properties of the FG beam can be calculated by the rule of mixtures of two phases. The numerical example results show that they are in agreement with the exact solutions provided in the literature. This paper also uses neural network to predict the vibration frequency of the FG simply supported beam under axial load and identify the material properties. In the process of training neural networks, a training group and a test group are generated in advance according to the mixed layered high-order shear deformation theory. The former is used to train the neural network, and the latter is used to test the accuracy of the neural network. After proper training, the vibration frequency predicted by the neural network has quite good accuracy, and the calculation time is greatly saved compared with the mixed layered high-order shear deformation theoretical solution, which also achieves the material identification that can not be achieved by the theoretical solution.

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