Generalized Formulation for the Behavior of Geometrically Curved and Twisted Three-Dimensional Timoshenko Beams and Its Isogeometric Analysis Implementation
Generalized Formulation for the Behavior of Geometrically Curved and Twisted Three-Dimensional Timoshenko Beams and Its Isogeometric Analysis Implementation
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几何弯曲和扭曲三维Timoshenko 梁行为的广义公式及其等几何分析实现
DOI：
10.1115/1.4054438
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Cusatis, Gianluca
中科院分区：
文献类型：
--
作者：
Yin, Hao;Lale, Erol;Cusatis, Gianluca
本文提出了一种新的几何弯曲和扭曲的三维梁的控制方程的推导。梁的运动学模型推导严格通过采用梁的轴的参数化描述，使用当地的Frenet-Serret参考系统，并引入到经典的，一阶应变与位移关系的柯西连续体的梁的横截面平面性的约束。所得到的梁运动学模型包括由梁轴曲线的雅可比矩阵的逆组成的乘法项。这一项不包括在经典的梁公式在文献中;它的贡献完全消失的直梁和可以忽略不计，只有弯曲和扭曲的梁与细长的几何形状。此外，为了简化对复杂梁几何形状的描述，控制方程是根据梁轴在梁截面内的一般位置导出的。最后，本研究追求的曲梁制定的概念框架内的等几何分析，它允许精确描述的梁的几何形状的数值实现。这避免了应力锁定问题和相应的收敛问题时遇到的经典直梁有限元离散的几何形状的弯曲和扭曲的梁。最后，本文给出了几个数值例子的解决方案，以证明所提出的理论公式和数值实现的准确性和有效性。
This article presents a novel derivation for the governing equations of geometrically curved and twisted three-dimensional Timoshenko beams. The kinematic model of the beam was derived rigorously by adopting a parametric description of the axis of the beam, using the local Frenet–Serret reference system, and introducing the constraint of the beam cross ection planarity into the classical, first-order strain versus displacement relations for Cauchy’s continua. The resulting beam kinematic model includes a multiplicative term consisting of the inverse of the Jacobian of the beam axis curve. This term is not included in classical beam formulations available in the literature; its contribution vanishes exactly for straight beams and is negligible only for curved and twisted beams with slender geometry. Furthermore, to simplify the description of complex beam geometries, the governing equations were derived with reference to a generic position of the beam axis within the beam cross section. Finally, this study pursued the numerical implementation of the curved beam formulation within the conceptual framework of isogeometric analysis, which allows the exact description of the beam geometry. This avoids stress locking issues and the corresponding convergence problems encountered when classical straight beam finite elements are used to discretize the geometry of curved and twisted beams. Finally, this article presents the solution of several numerical examples to demonstrate the accuracy and effectiveness of the proposed theoretical formulation and numerical implementation.