Stress-Based Finite Element Methods for Dynamics Analysis of Euler-Bernoulli Beams with Various Boundary Conditions

Abstract In this research, two stress-based finite element methods including the curvature-based finite element method (CFE) and the curvature-derivative-based finite element method (CDFE) are developed for dynamics analysis of Euler-Bernoulli beams with different boundary conditions. In CFE, the curvature distribution of the Euler-Bernoulli beams is approximated by its nodal curvatures then the displacement distribution is obtained by its integration. In CDFE, the displacement distribution is approximated in terms of nodal curvature derivatives by integration of the curvature derivative distribution. In the introduced methods, compared with displacement-based finite element method (DFE), not only the required number of degrees of freedom is reduced, but also the continuity of stress at nodal points is satisfied. In this paper, the natural frequencies of beams with different type of boundary conditions are obtained using both CFE and CDFE methods. Furthermore, some numerical examples for the static and dynamic response of some beams are solved and compared with those obtained by DFE method.

摘要 本研究针对不同边界条件下的欧拉-伯努利梁（Euler-Bernoulli beams）动力学分析，开发了两类基于应力的有限元法，分别为基于曲率的有限元法（curvature-based finite element method, CFE）与基于曲率导数的有限元法（curvature-derivative-based finite element method, CDFE）。在基于曲率的有限元法中，通过节点曲率近似欧拉-伯努利梁的曲率分布，随后通过积分得到位移分布；而在基于曲率导数的有限元法中，则通过对曲率导数分布进行积分，以节点曲率导数近似位移分布。相较于基于位移的有限元法（displacement-based finite element method, DFE），本文提出的两种方法不仅缩减了所需的自由度数量，同时满足了节点处的应力连续性要求。本文采用CFE与CDFE两种方法，求解了不同边界条件下梁的固有频率。此外，针对若干梁的静力学与动力学响应开展了数值算例分析，并将所得结果与基于位移的有限元法的计算结果进行了对比。