This study introduces a force-based higher-order shear deformable beam finite element model that incorporates a rational shear stress distribution, designed for the precise analysis of functionally graded sandwich beams. Unlike conventional higher-order shear beam finite elements that regard generalized displacements as unknown fields, this model considers the distributions of stress resultants along the beam axis as the unknown fields. The specific forms of these stress resultants and the generalized displacements are analytically determined, based on the differential equilibrium equations of the higher-order shear beam. This approach effectively circumvents numerical errors that can arise from finite element discretization. Furthermore, the model introduces a stress equilibrium equation to accurately depict the distribution of transverse shear stress across the beam thickness. A corrected shear stiffness, which takes into account rational shear stress, is derived and incorporated into the proposed beam element. Numerical examples underscore the accuracy and efficacy of the proposed higher-order beam element model in the static analysis of functionally graded sandwich beams, particularly in terms of true transverse shear stress distribution.

翻译：本研究提出了一种基于力的高阶剪切变形梁有限元模型，该模型引入了合理的剪应力分布，专门用于功能梯度夹层梁的精确分析。与传统将广义位移视为未知场的高阶剪切梁有限元不同，本模型将沿梁轴方向的应力合力分布视为未知场。基于高阶剪切梁的微分平衡方程，这些应力合力及广义位移的具体形式通过解析方法确定，从而有效规避了有限元离散可能引入的数值误差。此外，该模型引入了一个应力平衡方程，以精确描述横剪应力沿梁厚度的分布。推导了考虑合理剪应力的修正剪切刚度，并将其纳入所提出的梁单元中。数值算例验证了所提出的高阶梁单元模型在功能梯度夹层梁静力分析中的精度与有效性，尤其是在真实横剪应力分布方面的表现。