One‐Dimensional Finite Elements With Arbitrary Cross‐Sectional Displacement Fields

This paper introduces an unprecedented unified approach for developing structural theories with an arbitrary kinematic variable over the beam cross-section. Each of the three displacement variables can be analyzed using an independent expansion function. Both the order of the expansion and the number of terms in each field can be any. That is, the same order does not necessarily correspond to the same number of unknown variables. This method permits starting from a general model and write classical and known higher-order beam theories without any restrain. In this paper, the structural theories are built by using the polynomial expansion of the cross-sectional variables. The Carrera unified formulation (CUF) is employed to describe the cross-sectional kinematics. The finite element method (FEM) is employed to discretize the structure along the beam axis, utilizing Lagrange-based elements. The governing equations and related FE arrays for linear analysis are derived using the principle of virtual displacements. Both compact and thin-walled beams are examined to highlight the importance of each term of the three considered expansions. Various loading conditions, including bending, torsion, torsion-bending, and different beam slenderness ratios, are considered. The selected case studies are drawn from existing literature. The accuracy of the models presented is assessed for both displacements and stress components. The results demonstrate that the choice of the most suitable model closely depends on the specific parameters of the individual problem. That is, each structural problem has its own “best” computational models in terms of accuracy versus degree of freedom.