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, 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，通过微分平衡方程获得横向剪应力沿梁厚度的分布，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元不同，本文所提出的梁单元模型将内力作为未知场，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，通过引入几何关系和本构方程，用内力表示广义位移，并基于边界平衡条件和单元内相容性条件构建梁单元方程组。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准确性和有效性，特别是在真实横向剪应力分布方面。