采用前期提出的独立覆盖流形法,提出梁的独立覆盖分析方法。该方法的特点是:与实体分析采用同一模式,可以应用完全多项式的覆盖函数进行梁的实体分析,或使多项式中的某些项不参与计算来模拟梁的基本假设,从而实现Timoshenko梁和Euler-Bernoulli梁的计算;仅要求近似场函数的C0连续性;避免了Timoshenko梁在求解细长梁时的剪切自锁;即使对于实体分析而言,通常情况下也不存在由于梁高远小于梁长而导致的数值病态。为体现梁结构的特点,以二维的矩形截面梁为例,给出了局部坐标系下的独立覆盖流形法公式,以及“先截面、后轴向”的积分方式,并用几个算例验证了方法的有效性。本思路可以直接推广到求解三维问题,为梁板壳分析提供全新的途径。
Abstract
Using Numerical Manifold Method (NMM) based on independent covers proposed in previous studies, a numerical method for beam analysis based on independent covers is presented. The solution process is almost the same with solid analysis: complete polynomials can be used as cover functions to analyze beams as solids; or, with some polynomial terms not involved in the computation, the fundamental beam assumptions, such as the assumption for Timoshenko beam, or the assumption for Euler-Bernoulli beam, are implemented. The approximate field functions with only C0 continuity are needed. In the meantime, the “shear locking” problem for Timoshenko beam is avoided when dealing with long beams with small cross-sections. Even for totally solid analysis, the numerical ill-conditioning problem due to very small ratio of height to length does not exist in general situations. With two-dimensional beams with rectangular cross-sections as a case study, the formulae of independent covers in the local coordinate system are given to reflect the features of beams, together with the integration approach of “firstly along the cross-section, and then along the axial direction”. Some numerical examples demonstrate the validity of the method. The idea of the paper can be directly expanded to solve three-dimensional problems, and to provide a new approach for analysis of beams, plates and shells.
关键词
数值流形方法 /
独立覆盖 /
Timoshenko梁 /
Euler-Bernoulli梁 /
梁板壳分析
Key words
Numerical Manifold Method (NMM) /
independent covers /
Timoshenko beam /
Euler-Bernoulli beam /
beam, plate and shell analysis
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参考文献
[1] ZIENKIEWICZ O C, TAYLOR R L.有限元方法[M].5版.庄 茁,岑 松,译.北京: 清华大学出版社, 2006.
[2] 王勖成.有限单元法[M].北京:清华大学出版社,2003.
[3] SHI G H. Manifold Method of Material Analysis[C]∥U.S. Army Research Office. Transactions of the Ninth Army Conference on Applied Mathematics and Computing. Minneapolis Minnesota, U. S. A. June 18-21, 1991:51-76.
[4] BABUSKA I, MELENK J M. The Partition of Unity Method[J]. International Journal for Numerical Methods in Engineering, 1997, 40(4):727-758.
[5] ZHENG H, LIU Z J, GE X R. Numerical Manifold Space of Hermitian Form and Application to Kirchhoff’s Thin Plate Problems[J]. International Journal for Numerical Methods in Engineering, 2013, 95(9): 721-739.
[6] 祁勇峰, 苏海东, 崔建华.部分重叠覆盖的数值流形方法初步研究[J].raybet体育在线
院报, 2013,30(1):65-70.
[7] SU Hai-dong, QI Yong-feng, GONG Ya-qi, et al. Preliminary Research of Numerical Manifold Method Based on Partly Overlapping Rectangular Covers[C]∥DDA Commission of International Society for Rock Mechanics. Proceedings of the 11th International Conference on Analysis of Discontinuous Deformation (ICADD11). Fukuoka, Japan, August 27-29, 2013, London :Taylor & Francis Group, 2013: 341-347.
[8] 苏海东, 祁勇峰, 龚亚琦,等. 任意形状覆盖的数值流形方法初步研究[J]. raybet体育在线
院报, 2013, 30(12): 91-96.
[9] 苏海东,颉志强,龚亚琦, 等.基于独立覆盖的流形法收敛性及覆盖网格特性[J]. raybet体育在线
院报,2016,33(2):131-136.
[10]苏海东,颉志强.独立覆盖流形法的本质边界条件施加方法[J]. raybet体育在线
院报,2017,34(12):140-146.
[11]苏海东,祁勇峰,龚亚琦.裂纹尖端解析解与周边数值解联合求解应力强度因子[J].raybet体育在线
院报, 2013,30(6):83-89.
[12]申向东.材料力学[M].北京:中国水利水电出版社,2012.
[13]林绍忠. 用预处理共轭梯度法求解有限元方程组及程序设计[J]. 河海大学学报, 1998, 26(3): 112-115.
基金
中央级公益性科研院所基本科研业务费项目(CKSF2015033/CL,CKSF2016022/C)
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