悬索结构的非线性有限元分析
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摘要
悬索结构具有经济、结构形式多样、布置灵活、能满足大跨度的需要等特点,近年来,它越来越多地应用到桥梁和房屋建筑结构中。随着悬索结构体系及其荷载形式越来越复杂,不少学者提出了各种计算分析方法。
本文首先介绍了悬索结构的分析方法,其通常分为解析法和有限元法两大类。解析法只适用于结构形式和荷载形式都简单的情况;目前针对悬索结构的分析一般都采用有限元法,它适用于任何结构形式和荷载形式的悬索结构。因为有限元法中关于单元模型的选用直接影响到计算结果,本文介绍了目前主要的索单元模型,主要有两节点单元模型(包括两节点直线杆单元模型、两节点抛物线索单元模型、两节点悬链线索单元模型)和多节点等参索单元模型(包括三节点等参索单元模型、五节点等参索单元模型)。在较详细地介绍了各种单元模型的位移模式和形函数后,对各种单元模型进行了比较。
在广泛阅读相关文献的基础上,本文比较了已有的单元模型的优缺点,提出了一种四节点等参曲线索单元有限元模型,并采用三次多项式作为位移插值函数,推导了形函数的具体表达式,基于Lagrangian 轴向应变的定义建立了单元的几何关系。接着基于U.L 坐标描述,从虚功原理出发推导出单元的平衡方程和单元切线刚度矩阵,引入边界处理条件集成了整体结构的平衡方程和总切线刚度矩阵,并采用Newton—Raphson 法和增量法相结合的方法求解非线性方程组。本文利用Fortran 95 标准语言编制了基于上述理论推导的有限元程序,将程序运用到小垂度单索结构、大垂度单索结构和索网结构的算例分析中,并将得到的计算结果与其他参考文献的结果比较。比较结果表明,本文的方法收敛速度快,计算精度高且计算用时少,可供悬索结构分析、设计时采用。
Currently, the suspend structure is widely applied to bridge and house construction which can meet the needs of span with its multiple forms and flexibility. As the increasing complex of suspends structure and the forms of load, multiple methods of analysis have been proposed.
This paper firstly introduce methods of suspend structure analysis, usually classifying into Analysis and Finite Element. The Analysis only applies to those simple structure and load, while Finite Element, the common method for suspends structure, is frequently adoptive to any structure and load forms, as of which the element model option gives directly impact on the calculation results. This paper then introduces the main cable units model, focus on two-node element model (including two-node straight-line element, two-node parabolic cable element model and two-node catenary element model) and multiple-nod isoparametric element model (including three-node isoparametric element model and five-node isoparametric element model). Finally this paper makes a comparison among multiple element models based on exposition of sorts of displacement models and form functions.
On the basis of the advantages and disadvantages of existed element model, this paper proposes four-node isoparametric cable finite element model through widely absorbed in relative literature. The paper, applying cubic multinomial to displacement function, deducts the expression of form function and geometrical relation of elements on Lagrangian axis. Then the element equation and unit tangent rigid matrix are deducted on the basis of description of UL coordiate and principle of imaginary, the balance equation and overall tangent rigid matrix are integrated via introduction of terminal condition, and the nonlinear equation group is finally parsed on the basis of combination method of Newton-Raphson and increment. Bases on Fortran 95 standard language and the above theory, the paper compiles the finite element program, applies the program to the analysis of small dip cable structure, large dip cable structure and cable network structure, and then compares this results with other reference literature.
The result indicates the quickness, preciseness and effectiveness of the above method, which is available for cable structure analysis and cable design.
本文首先介绍了悬索结构的分析方法,其通常分为解析法和有限元法两大类。解析法只适用于结构形式和荷载形式都简单的情况;目前针对悬索结构的分析一般都采用有限元法,它适用于任何结构形式和荷载形式的悬索结构。因为有限元法中关于单元模型的选用直接影响到计算结果,本文介绍了目前主要的索单元模型,主要有两节点单元模型(包括两节点直线杆单元模型、两节点抛物线索单元模型、两节点悬链线索单元模型)和多节点等参索单元模型(包括三节点等参索单元模型、五节点等参索单元模型)。在较详细地介绍了各种单元模型的位移模式和形函数后,对各种单元模型进行了比较。
在广泛阅读相关文献的基础上,本文比较了已有的单元模型的优缺点,提出了一种四节点等参曲线索单元有限元模型,并采用三次多项式作为位移插值函数,推导了形函数的具体表达式,基于Lagrangian 轴向应变的定义建立了单元的几何关系。接着基于U.L 坐标描述,从虚功原理出发推导出单元的平衡方程和单元切线刚度矩阵,引入边界处理条件集成了整体结构的平衡方程和总切线刚度矩阵,并采用Newton—Raphson 法和增量法相结合的方法求解非线性方程组。本文利用Fortran 95 标准语言编制了基于上述理论推导的有限元程序,将程序运用到小垂度单索结构、大垂度单索结构和索网结构的算例分析中,并将得到的计算结果与其他参考文献的结果比较。比较结果表明,本文的方法收敛速度快,计算精度高且计算用时少,可供悬索结构分析、设计时采用。
Currently, the suspend structure is widely applied to bridge and house construction which can meet the needs of span with its multiple forms and flexibility. As the increasing complex of suspends structure and the forms of load, multiple methods of analysis have been proposed.
This paper firstly introduce methods of suspend structure analysis, usually classifying into Analysis and Finite Element. The Analysis only applies to those simple structure and load, while Finite Element, the common method for suspends structure, is frequently adoptive to any structure and load forms, as of which the element model option gives directly impact on the calculation results. This paper then introduces the main cable units model, focus on two-node element model (including two-node straight-line element, two-node parabolic cable element model and two-node catenary element model) and multiple-nod isoparametric element model (including three-node isoparametric element model and five-node isoparametric element model). Finally this paper makes a comparison among multiple element models based on exposition of sorts of displacement models and form functions.
On the basis of the advantages and disadvantages of existed element model, this paper proposes four-node isoparametric cable finite element model through widely absorbed in relative literature. The paper, applying cubic multinomial to displacement function, deducts the expression of form function and geometrical relation of elements on Lagrangian axis. Then the element equation and unit tangent rigid matrix are deducted on the basis of description of UL coordiate and principle of imaginary, the balance equation and overall tangent rigid matrix are integrated via introduction of terminal condition, and the nonlinear equation group is finally parsed on the basis of combination method of Newton-Raphson and increment. Bases on Fortran 95 standard language and the above theory, the paper compiles the finite element program, applies the program to the analysis of small dip cable structure, large dip cable structure and cable network structure, and then compares this results with other reference literature.
The result indicates the quickness, preciseness and effectiveness of the above method, which is available for cable structure analysis and cable design.
引文
[1] 沈世钊,徐崇宝,赵臣.悬索结构设计.北京:中国建筑工业出版社,1997.67-159
[2] 刘北辰.工程计算力学理论与应用.北京:机械工业出版社,1994.435-466
[3] Tang Jianmin, Shen Zuyan, Qian Ruojun. A nonlinear finite element method with five-node element for analysis of cable structures [J]. Proceedings of the LASS International Symposium, 1995,2:929-935
[4] 唐建民,沈祖炎,钱若军.索穹顶结构非线性分析的曲线索单元有限元法[J].同济大学学报,1996,24(1):6-10
[5] 单圣涤,李飞云,陈洁余,朱祖楞.悬索曲线理论及其应用.湖南:湖南科学技术出版社,1983.1-35
[6] Prem Krishna. Cable suspended roofs. New York: McGraw-Hill, 1978. 44-56
[7] Irvine,H.M..Cable Structures. MIT Press, Cambridge, Massachusetts,1981.
[8] W C Knudson. Static and dynamic analysis of cable net structures [D].Doctoral dissertation, University of Califormia, Berkely, Califormia, 1971
[9] H J Ernst. Der E-Modul von Seilen unter Beruksichtigung des Durchhanges [J]. Der Bauingenieur, 1965, 40(2):52-55
[10] Burley E, Harvey R C. Behavior of Tension Structures Subjected to Uniformly Distributed Cable Loadings. Paper Presented at International Conference on Tension Roof Structures. Polytechnic of Central London ,1974.8～10
[11] 唐建民.柔性结构非线性分析的杆单元有限元法.中南工学院学报,1996,10(1):50-55
[12] 胡松,何艳丽,王肇民.大挠度索结构的非线性有限元分析[J].工程力学,2000,17(2):36-43
[13] 唐建民,卓家寿. 悬索结构大位移分析改进的两节点索单元模型. 河海大学学报,1999,27(4)
[14] 袁行飞,董石麟. 两节点曲线索单元非线性分析. 工程力学,1999,16(4)
[15] 杨孟刚,陈政清. 两节点曲线索单元精细分析的非线性有限元法. 工程力学,2003,20(1)
[16] H B Jayaraman. A curved element for the analysis of cable structure [J]. Computer and structures.1981,14: 325-333
[17] 张其林.预应力结构非线性分析的索单元理论[J].工程力学,1993,10(4):93-101
[18] 彭卫,孙炳楠,唐锦春. 一种用于索结构分析的悬链线单元. 应用数学和力学,1999,20(5)
[19] 王春江,董石麟,王人鹏,钱若军. 一种考虑初始垂度影响的非线性索单元. 力学季刊,2002,23(3)
[20] 聂建国,陈必磊,肖建春. 多跨连续长索在支座处存在滑移的非线性静力分析. 计算力学学报,2003,20(3)
[21] A.H Peyrot & A. M. Goulois. Analysis of flexible transmission lines[J]. Structural division ASCE, 1978.763-779
[22] 杨孟刚,陈政清.基于UL 列式的两节点悬链线索元非线性有限元分析[J].土木工程学报,2003,36(8):63-68
[23] 屈本宁,刘北辰.索-梁混合有限元模型及其在索桥分析中的应用.计算结构力学及其应用,1990,7(4):93-100
[24] 唐建民,何署廷,段春平.悬索结构非线性有限元分析.河海大小学报,1998,26(6):45-49
[25] Tang Jianmin, Shen Zuyan and Qian Ruojun. A nonlinear finite element method with five-node curved element for analysis of cable structures[J]. Proceedings of IASS International Symposium, 1995, 2:929-935
[26] 舒赣平,范圣刚,王建峰,刘辉,陈绍礼. 悬索及悬索-框架结构的几何非线性分析. 工业建筑,2002,32(5)
[27] Mason J. Variational, Incremental and Energy Method in Solid Mechanics and Shell Theory, Elsevier Scientific Publishing Company, 1980
[28] 杜国华,毛昌时,司徒妙龄.桥梁结构分析.上海:同济大学出版社,1994.230-235
[29] 丁皓江,何福保,谢贻权,徐兴.弹性和塑性力学中的有限单元法.北京:机械工业出版社,1989.165-181
[30] 龚景海,邱国志.空间结构计算机辅助设计.北京:中国建筑工业出版社,2002.173-195
[31] 刘正兴,孙雁,王国庆.计算固体力学.上海:上海交通大学出版社,2000.230-274
[32] 王勖成.有限单元法.北京:清华大学出版社,2003.617-660
[33] 殷有泉.固体力学非线性有限元引论.北京:北京大学出版社,清华大学出版社,1987
[34] 项海帆,姚玲森.高等桥梁结构理论.北京:人民交通出版社,2001.237-262
[35] 彭国伦.Fortran 95 程序设计.北京:中国电力出版社,2002
[36] 张其林.索与膜结构.上海:同济大学出版社,2002.1-104
[37] Forster B. Cable and membrane roofs—a historical survey. Structural Engineering Review ,1994,6(3-4):145-174
[38] Saitoh M. Role of String: aesthetics and technology of tension structure. Kobe: IABSE Symposium, 1998:699-710
[39] [日]崛高夫,村山茂明著.张育民,李世超等译.悬索理论及其应用.北京:中国林业出版社,1992.3-107
[40] [美]彼莱奇科,[美]廖荣锦, [美]默然著.庄茁等译.连续体和结构的非线性有限元. 北京:清华大学出版社,2002
[41] 浙江大学建筑工程学院,浙江大学建筑设计研究院.空间结构.北京:中国计划出版社,2003
[2] 刘北辰.工程计算力学理论与应用.北京:机械工业出版社,1994.435-466
[3] Tang Jianmin, Shen Zuyan, Qian Ruojun. A nonlinear finite element method with five-node element for analysis of cable structures [J]. Proceedings of the LASS International Symposium, 1995,2:929-935
[4] 唐建民,沈祖炎,钱若军.索穹顶结构非线性分析的曲线索单元有限元法[J].同济大学学报,1996,24(1):6-10
[5] 单圣涤,李飞云,陈洁余,朱祖楞.悬索曲线理论及其应用.湖南:湖南科学技术出版社,1983.1-35
[6] Prem Krishna. Cable suspended roofs. New York: McGraw-Hill, 1978. 44-56
[7] Irvine,H.M..Cable Structures. MIT Press, Cambridge, Massachusetts,1981.
[8] W C Knudson. Static and dynamic analysis of cable net structures [D].Doctoral dissertation, University of Califormia, Berkely, Califormia, 1971
[9] H J Ernst. Der E-Modul von Seilen unter Beruksichtigung des Durchhanges [J]. Der Bauingenieur, 1965, 40(2):52-55
[10] Burley E, Harvey R C. Behavior of Tension Structures Subjected to Uniformly Distributed Cable Loadings. Paper Presented at International Conference on Tension Roof Structures. Polytechnic of Central London ,1974.8～10
[11] 唐建民.柔性结构非线性分析的杆单元有限元法.中南工学院学报,1996,10(1):50-55
[12] 胡松,何艳丽,王肇民.大挠度索结构的非线性有限元分析[J].工程力学,2000,17(2):36-43
[13] 唐建民,卓家寿. 悬索结构大位移分析改进的两节点索单元模型. 河海大学学报,1999,27(4)
[14] 袁行飞,董石麟. 两节点曲线索单元非线性分析. 工程力学,1999,16(4)
[15] 杨孟刚,陈政清. 两节点曲线索单元精细分析的非线性有限元法. 工程力学,2003,20(1)
[16] H B Jayaraman. A curved element for the analysis of cable structure [J]. Computer and structures.1981,14: 325-333
[17] 张其林.预应力结构非线性分析的索单元理论[J].工程力学,1993,10(4):93-101
[18] 彭卫,孙炳楠,唐锦春. 一种用于索结构分析的悬链线单元. 应用数学和力学,1999,20(5)
[19] 王春江,董石麟,王人鹏,钱若军. 一种考虑初始垂度影响的非线性索单元. 力学季刊,2002,23(3)
[20] 聂建国,陈必磊,肖建春. 多跨连续长索在支座处存在滑移的非线性静力分析. 计算力学学报,2003,20(3)
[21] A.H Peyrot & A. M. Goulois. Analysis of flexible transmission lines[J]. Structural division ASCE, 1978.763-779
[22] 杨孟刚,陈政清.基于UL 列式的两节点悬链线索元非线性有限元分析[J].土木工程学报,2003,36(8):63-68
[23] 屈本宁,刘北辰.索-梁混合有限元模型及其在索桥分析中的应用.计算结构力学及其应用,1990,7(4):93-100
[24] 唐建民,何署廷,段春平.悬索结构非线性有限元分析.河海大小学报,1998,26(6):45-49
[25] Tang Jianmin, Shen Zuyan and Qian Ruojun. A nonlinear finite element method with five-node curved element for analysis of cable structures[J]. Proceedings of IASS International Symposium, 1995, 2:929-935
[26] 舒赣平,范圣刚,王建峰,刘辉,陈绍礼. 悬索及悬索-框架结构的几何非线性分析. 工业建筑,2002,32(5)
[27] Mason J. Variational, Incremental and Energy Method in Solid Mechanics and Shell Theory, Elsevier Scientific Publishing Company, 1980
[28] 杜国华,毛昌时,司徒妙龄.桥梁结构分析.上海:同济大学出版社,1994.230-235
[29] 丁皓江,何福保,谢贻权,徐兴.弹性和塑性力学中的有限单元法.北京:机械工业出版社,1989.165-181
[30] 龚景海,邱国志.空间结构计算机辅助设计.北京:中国建筑工业出版社,2002.173-195
[31] 刘正兴,孙雁,王国庆.计算固体力学.上海:上海交通大学出版社,2000.230-274
[32] 王勖成.有限单元法.北京:清华大学出版社,2003.617-660
[33] 殷有泉.固体力学非线性有限元引论.北京:北京大学出版社,清华大学出版社,1987
[34] 项海帆,姚玲森.高等桥梁结构理论.北京:人民交通出版社,2001.237-262
[35] 彭国伦.Fortran 95 程序设计.北京:中国电力出版社,2002
[36] 张其林.索与膜结构.上海:同济大学出版社,2002.1-104
[37] Forster B. Cable and membrane roofs—a historical survey. Structural Engineering Review ,1994,6(3-4):145-174
[38] Saitoh M. Role of String: aesthetics and technology of tension structure. Kobe: IABSE Symposium, 1998:699-710
[39] [日]崛高夫,村山茂明著.张育民,李世超等译.悬索理论及其应用.北京:中国林业出版社,1992.3-107
[40] [美]彼莱奇科,[美]廖荣锦, [美]默然著.庄茁等译.连续体和结构的非线性有限元. 北京:清华大学出版社,2002
[41] 浙江大学建筑工程学院,浙江大学建筑设计研究院.空间结构.北京:中国计划出版社,2003