本文將Reissner混合變分原理和Eringen非局部彈性力學連結，發展非局部Timoshenko梁有限元素法，並將其應用於具彈性支承之多層奈米碳管幾何非線性撓曲分析。具自由端、簡支承和固定端不同組合邊界條件的多層奈米碳管在其最外層受到載重，使用He等人和Ru的模型來分別模擬在任兩層之間和僅在相鄰兩層之間的凡德瓦爾交互作用力，且使用Pasternak基礎模型來模擬多層奈米碳管和其周圍彈性介質之間的交互作用，應用Galerkin方法來推導Timoshenko梁幾何非線性有限元素分析相應的弱型式數學方程，其中考慮von Kármán幾何非線性效應，以Eringen非局部彈性理論來評估微小尺度效應，而多層奈米碳管之廣義位移和力分量的有限元素解則藉由直接迭代過程來得到。本有限元素法數值解顯示其收斂非常快速，且本有限元素法收斂解和文獻中使用基於強型式數學方程的微分擬合法所得到的解非常吻合。

In conjunction with the Reissner mixed variational theorem (RMVT) and nonlocal Timoshenko beam theory (TBT), we develop a finite element method for the geometrically nonlinear bending analysis of a multi-walled carbon nanotube (CNT) resting on an elastic foundation. The multi-walled carbon nanotube is subjected to mechanical loads on its outermost surface, with combinations of free, simply-supported and clamped edge conditions. The van der Waals (vdW) interactive forces between all pairs of walls and between the adjacent walls only are estimated using the He et al. and Ru models, respectively, and the interaction between the multi-walled carbon nanotube and its surrounding medium is simulated using the Pasternak foundation model. A weak-form formulation for the geometrically nonlinear finite element (FE) analysis is derived using the Galerkin approach, in which the von Kármán geometrical nonlinearity (VKGN) is considered. The Eringen nonlocal elasticity theory (ENET) is used to account for the small length scale effect. The finite element solutions of the generalized displacement and force components induced in the multi-walled carbon nanotube are obtained using an iterative process. Numerical implementation shows that the finite element solutions converge rapidly, and their convergent solutions closely agree with those obtained using the differential quadrature (DQ) method, based on a strong-form formulation, available in the literature.

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