Abstract

In this paper, an accurate yet computationally efficient beam model based on hierarchical Legendre expansion functions is developed for the analysis of non-uniform or restrained torsion problems of Functionally Graded (FG) beams with solid or thin-walled section. The mechanical properties of the FG beams studied in this paper, such as Young’s modulus and shear modulus, are assumed to continuously vary along either the length or thickness direction following a power-law distribution. The proposed beam model is based on the assumption that the beam’s cross-section is infinitely rigid in its own plane. However, the longitudinal displacement field over the beam’s cross-section is enriched in an element-wise manner by the unknown longitudinal displacement parameters multiplying with hierarchical Legendre expansion functions. The proposed modeling methodology has two novel aspects: First, it allows the torsional or twisting angle, which is a priori defined as an unknown kinematic variable, to be directly captured without a post-processing recovery step, even if there exists a strong flexual-torsional coupling within the beam; second, the longitudinal warping response of the beam, triggered by stretching, bending, twisting, or the coupling between them, can be captured without the pre-determination of warping modes and at a lower level of DOFs. The strong-form governing equations of non-uniform or restrained torsion problems of the FG beam are derived based on the principle of minimum potential energy and is directly solved by the high-quality general-purpose ordinary differential equation (ODE) solver, i.e. COLSYS ODE solver. Also, a more efficient Rayleigh-Ritz energy method is applied to provide the weak-form solutions. The resulting beam model is suitable to a more general cross-section, such as solid section, branched open section, or half open-half closed section. The accuracy and efficiency of the proposed beam model are validated extensively by comparing with the previously published results in literature. Effects of the power-law index, taper ratio, and section-type on the torsional response of FG beams with material gradation along length or thickness direction are studied with various numerical examples.

摘要

在本文中，基于分层Legendre展开函数的一种精确而计算有效的梁模型被开发出来，用于分析具有实心或薄壁截面的功能梯度（FG）梁的非均匀或约束扭转问题。本文研究的FG梁的机械性能，例如杨氏模量和剪切模量，沿幂律分布沿长度或厚度方向连续变化。提出的梁模型基于以下假设：梁的横截面在其自己的平面中是无限刚性的。但是，通过将未知的纵向位移参数乘以分层的Legendre展开函数，可以以元素方式丰富梁横截面上的纵向位移场。先验定义为未知的运动学变量，即使光束中存在强烈的挠曲-扭转耦合，也可以直接捕获而无需进行后期处理恢复步骤；第二，可以在不预先确定翘曲模态且自由度较低的情况下捕获由拉伸，弯曲，扭曲或光束之间的耦合引起的梁的纵向翘曲响应。基于最小势能原理，推导了FG梁不均匀或约束扭转问题的强形式控制方程，并直接由高质量的通用常微分方程（ODE）求解器进行求解，即COLSYS ODE求解器。而且，更有效的瑞利-里兹能量方法被用来提供弱形式的解决方案。所得的梁模型适用于更一般的横截面，例如实心部分，分支开放部分或半开放半封闭部分。通过与文献中先前发表的结果进行比较，所提出的光束模型的准确性和效率得到了广泛的验证。通过各种数值实例研究了幂律指数，锥度比和截面类型对材料梯度沿长度或厚度方向的FG梁的扭转响应的影响。