Finite Element Analysis of Axisymmetric Elastic Body Problems

Abstract

Linear form functions are commonly used in a long time for a toroidal volume element swept by a triangle revolved about the symmetrical axis for general axisymmetrical stress problems. It is difficult to obtain the rigidity matrix by exact integration, and instead, the method of approximate integration is used. As the locations of element close to the symmetrical axis, the accuracy of this approximation deteriorates very rapidly. The exact integration have been suggested by various authors for the calculation of rigidity matrix. However, it is shown in this paper that these exact integrations can only be used for those axisymmetric bodies with central hole. For solid axisymmetric body, it can be proved that the calculation fails due to the divergent property of rigidity matrix integration. In this paper a new form function is suggested. In this new form function, the radial displacement u vanishes as radial coordinates r approach to zero. The calculated rigidity matrix is convergent everywhere, including these triangular toroidal element closed to the symmetrical axis. This kind of element is useful for the calculation of axisymmetric elastic solid body problems.

FullText(HTML)

References(12)

References

[1]	Wilson,E.L.,Structural Analysis of Axisymmetric Solids,AIAA Journal,4,(1965),2269.
[2]	Clough,R.and Rashid,Y.,Finite Element Analysis of Axisymmetric Solids,Proc.ASCE,J.Eng.Mech.,91NoEMI,(1965).71-85.
[3]	Vitku,S.,Explicit Expressions for Triangular Torus Element Stiffness Matrix,AIAA Journal,6,8(1968),1174-75.
[4]	Fjeld,S.A.,Three-Dimensional Theory of Elasticity,Finite Element Method in Stress analysis,Edited by I.Holand and K.Bell,TAPIR,(1969).333-368.
[5]	Pedley,T.J.,The Fluid Mechanics of Largr Blood Vessels,Cabmbrige University Press(1980).
[6]	Zienjiewicz,0.C.,The Finite Element Method in Engineering Science,McGraw-Hill,London(1971).
[7]	Desai,C.S.and Abel,J.F.,Introduction to the Finite Element Method,Van Nostrand Reinhold Co.,New York(1972).德赛.C.S,,阿贝尔,J.F.著:《有限元素法引论》,江伯南.尹泽勇译.徐芝纶校,科学出版社(1978).
[8]	华东水利学院《弹性力学有限元法》.水利电力出版社(1974)
[9]	复旦大学数学系.《有限元法选讲》,科学出版社(1976).
[10]	Huebner,K.H.,The Finite Element Method for Engineers,John Wiley and Sons,(1975).
[11]	郭仲衡.关于有限元法轴对称问题的一点记注.1978年教育部高等学校计算结构力学学术交流会论文集,大连(1978).
[12]	郭仲衡:《内燃机活塞热应力计算(轴对称有限元法)》北京大学数力系应用数学73届讲义(1975).

Relative Articles

[1]	FAN Yajie, LI Yan, LI Zhongpan, CHEN Huijian, FENG Zhiqiang. Smoothed Finite Element Analysis of Contact and Large Deformation Problems[J]. Applied Mathematics and Mechanics, 2024, 45(2): 127-143. doi: 10.21656/1000-0887.440251
[2]	LIU Bo, WANG Ke-yong, WANG Ming-hong. A Trefftz Finite Element Method for Solving Axisymmetric Poisson’s Equations[J]. Applied Mathematics and Mechanics, 2015, 36(2): 140-148. doi: 10.3879/j.issn.1000-0887.2015.02.003
[3]	WANG Ke-yong, HUANG Zheng-ming, LI Pei-chao, LIU Bo. Trefftz Finite Element Analysis of Axisymmetric Potential Problems in Orthotropic Media[J]. Applied Mathematics and Mechanics, 2013, 34(5): 462-469. doi: 10.3879/j.issn.1000-0887.2013.05.004
[4]	YANG Yong-qin, XIAO Liu-chao, CHEN Shao-chun. Nonconforming Finite Elements for the Equation of Planar Elasticity[J]. Applied Mathematics and Mechanics, 2010, 31(12): 1454-1464. doi: 10.3879/j.issn.1000-0887.2010.12.006
[5]	Zhu Jufen, Zheng Gang, Wu Jinying. F.E.M. Analysis of Delamination Buckling in Composite Plates & Shells[J]. Applied Mathematics and Mechanics, 2000, 21(3): 301-306.
[6]	Yan Bo, Liu Zhanfang, Zhang Xiangwei. Finite Element Analysis of Wave Propagation in Fluid-Saturated Porous Media[J]. Applied Mathematics and Mechanics, 1999, 20(12): 1235-1244.
[7]	Zhang Jianhai, Chen Dapeng, Zou Shengquan. ALE Finite Element Analysis of the Opening and Closing Process of the Artificial Mechanical Valve[J]. Applied Mathematics and Mechanics, 1996, 17(5): 387-396.
[8]	Zhang Hong-wu, Zhong Wan-xie, Qian Ling-xi. Finite Element Analysis for Consolidation in Interaction between Structure and Saturated Soil Foundation[J]. Applied Mathematics and Mechanics, 1992, 13(10): 891-899.
[9]	Chen Da-peng, Zhao Zhong. A Penalty-Hybrid Finite Element Analysis of Stokes Flow[J]. Applied Mathematics and Mechanics, 1990, 11(6): 467-476.
[10]	Li Long-yuan. The Rationalism Theory and Its Finite Element Analysis Method of Shell Structures[J]. Applied Mathematics and Mechanics, 1990, 11(4): 365-372.
[11]	Gao Shi-qiao, Lu Wen-da. Dynamic Solution of Hyperbolic Cooling Tower with Nonsymmetric Imperfections by Finite Element Method[J]. Applied Mathematics and Mechanics, 1988, 9(6): 475-480.
[12]	Qian Yuan-yao. A Hybrid/Mixed Model Finite Element Analysis for Eigenvalue Problem of Moderately Thick Plates[J]. Applied Mathematics and Mechanics, 1987, 8(8): 733-742.
[13]	Zhang Shuang-yin, C. M. Leech. FEM Analysis on Mixed-Mode Fracture of CSM-GRP[J]. Applied Mathematics and Mechanics, 1986, 7(8): 689-702.
[14]	S. N. Atluri, Kuang Zhen-bang. A Finite Element Analysis for the Temperature Field Produced by a Moving Heat Source[J]. Applied Mathematics and Mechanics, 1986, 7(5): 383-400.
[15]	Liu Zheng-xing, Wu Lian-yuan, Feng Tai-hua. The Finite Element Analysis of the Flexible Beams and Plates[J]. Applied Mathematics and Mechanics, 1985, 6(9): 833-844.
[16]	Lü Yu-lin, Lai Guo-zhang. The Mathematical Modelling of Near Coast Shallow Water Circulation[J]. Applied Mathematics and Mechanics, 1984, 5(5): 731-742.
[17]	Chien Wei-zhang. Diagonalized Consistant Mass Matrix and the Dynamical Finite Element Analysis of Elastic-Plastic Impact in Axisymmetrical Problems[J]. Applied Mathematics and Mechanics, 1982, 3(4): 429-448.
[18]	He Qiong, Cheng Zheng-ding, Xiong Huan-guo. Finite Element Study of Dynamic Characteristics of Main Components of Machine Tools[J]. Applied Mathematics and Mechanics, 1981, 2(6): 679-688.
[19]	Shieh Chih-cheng, Fu Cheng-sung, Zheng Si-liang. Finite Element Method of Axial Symmetrical Shell with Abrupt Change of Curvature (Corrugated Tubes)[J]. Applied Mathematics and Mechanics, 1981, 2(1): 113-130.
[20]	On “Finite Element Analysis of Axisymmetric Elastic Body Problems”[J]. Applied Mathematics and Mechanics, 1980, 1(3): 417-417.