Abstract

The present paper investigates the 3 D free vibration behavior of curved metallic and composite beams via a novel beam theory. The refined beam theory is constructed within the framework of the Carrera Unified Formulation (CUF), which expands 3 D displacement fields as 1 D generalized displacement unknowns over the cross-section. As a novelty, a set of improved hierarchical Legendre polynomials called the improved hierarchical Legendre expansion (IHLE) is used to describe the cross-sectional deformation. In this way, displacements at shared sides between piles can be interpolated by Lagrange polynomials, while displacements at the rest of the cross-section remain to be defined by hierarchical Legendre polynomials. Such determined cross-sectional kinematics not only retain the hierarchical properties of HLE in part but also facilitate the implementation of the Layer-Wise approach without paying much attention to the order that expansion terms appear over the cross-section. Due to the absence of the mesh generation and the convenience of collocation techniques, the differential quadrature based-meshless method is employed for the approximate solution of strong form governing equations derived by the principle of virtual displacements. Several numerical cases, including curved beams with various material properties and boundary conditions, are proposed to illustrate the optimized computational efficiency of this novel model over the 3 D finite element method and consistent convergence properties over the previous CUF-HLE model.

摘要

本文通过一种新的梁理论研究了弯曲金属梁和复合梁的 3D 自由振动行为。细化梁理论是在 Carrera 统一公式 (CUF) 的框架内构建的，该公式将 3D 位移场扩展为横截面上的 1D 广义位移未知数。作为一种新颖性，一组改进的层次勒让德多项式称为改进的层次勒让德展开 (IHLE) 用于描述横截面变形。这样，桩之间共享边的位移可以通过拉格朗日多项式进行插值，而横截面其余部分的位移仍由分层勒让德多项式定义。这种确定的横截面运动学不仅部分保留了 HLE 的分层特性，而且还促进了 Layer-Wise 方法的实施，而无需过多关注扩展项出现在横截面上的顺序。由于没有网格生成和搭配技术的便利性，采用基于微分求积的无网格方法来近似解由虚位移原理导出的强形式控制方程。提出了几个数值案例，包括具有各种材料特性和边界条件的弯曲梁，以说明这种新型模型在 3D 有限元方法上的优化计算效率以及与以前的 CUF-HLE 模型一致的收敛特性。