In this paper, a force-based beam finite element model based on a modified higher-order shear deformation theory is proposed for the accurate analysis of functionally graded beams. In the modified higher-order shear deformation theory, the distribution of transverse shear stress across the beam's thickness is obtained from the differential equilibrium equation on stress, and a modified shear stiffness is derived to take the effect of transverse shear stress distribution into consideration. In the proposed beam element model, unlike traditional beam finite elements that regard generalized displacements as unknown fields, the internal forces are considered as the unknown fields, and they are predefined by using the closed-form solutions of the differential equilibrium equations of higher-order shear beam. Then, the generalized displacements are expressed by the internal forces with the introduction of geometric relations and constitutive equations, and the equation system of the beam element is constructed based on the equilibrium conditions at the boundaries and the compatibility condition within the 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：本文提出了一种基于修正高阶剪切变形理论的力法梁有限元模型，用于功能梯度梁的精确分析。在修正高阶剪切变形理论中，横向剪应力沿梁厚度的分布由应力微分平衡方程求得，并推导了修正剪切刚度以考虑横向剪应力分布的影响。与将广义位移视为未知场的传统梁有限单元不同，所提出的梁单元模型将内力视为未知场，并利用高阶剪切梁微分平衡方程的闭式解预先定义内力。随后，通过引入几何关系与本构方程，将广义位移表达为内力的函数，并基于边界平衡条件与单元内部协调条件构建梁单元方程系统。数值算例表明，所提出的高阶梁单元模型在功能梯度夹层梁的静力分析中具有精度与有效性，尤其是在真实横向剪应力分布方面。