Exact finite element formulation of quasi-3D beam based on analytical internal force fields for accurate static analysis of FG sandwich beams

Abstract

This paper presents a novel exact finite element formulation of quasi-3D beam for high-fidelity analysis of functionally graded sandwich beams. Unlike conventional displacement-based elements that rely on approximate interpolation functions or existing exact finite element methods requiring closed-form solutions of generalized displacements, the proposed approach constructs an exact beam element from analytical expressions of internal forces. The method innovatively integrates force-based beam element formulation with the exact finite element framework through a systematic implementation procedure. Firstly, analytical expressions for internal forces are derived by applying differential equilibrium equations, geometric relations, and constitutive equations. These expressions are then used to obtain integral forms of generalized displacements expressed in terms of internal force parameters. Subsequently, nodal generalized displacements are defined, and their relationship with internal force parameters is established via the consistency condition. Finally, exact shape functions are derived from interpolation definitions, and the exact stiffness matrix and equilibrium equations are formulated using the principle of virtual work. To further enhance solution accuracy, modified cross-sectional stiffness matrices accounting for equilibrium-based stress distributions are incorporated. Comprehensive numerical examples demonstrate that the proposed element achieves superior solution accuracy and computational efficiency compared to conventional methods, validating its potential for precise structural analysis of complex beam structures with material gradients.