Bending of bidirectional functionally graded nanobeams under mechanical loads and magnetic force was investigated. The nanobeam is assumed to be resting on the Winkler–Pasternak foundation. Eringen’s nonlocal elasticity theory and Timoshenko beam model are utilized to describe the mechanical behavior of the nanobeam. Material properties of the functionally graded beam are assumed to vary in the thickness and length of the nanobeam. Hamilton’s principle is employed to derive the governing equation and related boundary conditions. These equations are solved using the generalized differential quadrature method. The obtained results are compared with the results presented in other studies, to ensure the validity and versatility of this method. This comparison shows a good agreement between the results. Results are presented and discussed for different values of functionally graded materials indices, different aspect ratios, and different boundary conditions. The effect of the magnetic field and elastic foundation on buckling load has also been studied. The difference in nanobeam behavior for different values of the size-effect parameter is clearly shown.

中文翻译：

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双向 FGM Timoshenko 纳米束在机械和磁力作用下的弯曲分析并停留在 Winkler-Pasternak 基础上

研究了双向功能梯度纳米束在机械载荷和磁力下的弯曲。假设纳米束位于 Winkler-Pasternak 基础上。Eringen 的非局部弹性理论和 Timoshenko 梁模型用于描述纳米梁的机械行为。假设功能梯度梁的材料特性在纳米梁的厚度和长度上有所不同。汉密尔顿原理用于推导控制方程和相关的边界条件。使用广义微分求积法求解这些方程。将获得的结果与其他研究中的结果进行比较，以确保该方法的有效性和多功能性。这种比较表明结果之间的一致性很好。针对不同的功能梯度材料指数值、不同的纵横比和不同的边界条件，给出并讨论了结果。还研究了磁场和弹性地基对屈曲载荷的影响。清楚地显示了不同尺寸效应参数值的纳米束行为的差异。