The objective of this contribution is the computation of the Airy stress function for functionally graded beam-type structures subjected to transverse and shear loads. For simplification, the material parameters are kept constant in the axial direction and vary only in the thickness direction. The proposed method can be easily extended to material varying in the axial and thickness direction. In the first part an iterative procedure is applied for the determination of the stress function by means of Boley’s method. This method was successfully applied by Boley for two-dimensional (2D) isotropic plates under plane stress conditions in order to compute the stress distribution and the displacement field. In the second part, a shear loaded cantilever made of isotropic, functionally graded material is studied in order to verify our theory with finite element results. It is assumed that the Young’s modulus varies exponentially in the transverse direction and the Poisson ratio is constant. Stresses and displacements are analytically determined by applying our derived theory. Results are compared to a 2D finite element analysis performed with the commercial software ABAQUS. It is found that the analytical and numerical results are in perfect agreement.

中文翻译：

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将 Boley 方法扩展到功能梯度梁

该贡献的目的是计算承受横向和剪切载荷的功能梯度梁型结构的艾里应力函数。为简化起见，材料参数在轴向保持不变，仅在厚度方向变化。所提出的方法可以很容易地扩展到在轴向和厚度方向上变化的材料。在第一部分中，迭代程序用于通过 Boley 方法确定应力函数。Boley 成功地将该方法应用于平面应力条件下的二维 (2D) 各向同性板，以计算应力分布和位移场。在第二部分中，由各向同性制成的剪切载荷悬臂，研究了功能梯度材料，以用有限元结果验证我们的理论。假设杨氏模量在横向上呈指数变化并且泊松比是恒定的。应力和位移是通过应用我们的推导理论来分析确定的。将结果与使用商业软件 ABAQUS 执行的二维有限元分析进行比较。发现解析结果和数值结果完全一致。