曲线预应力混凝土箱梁分析方法及程序设计
|
摘要
曲线预应力混凝土箱梁桥以其经济、美观、实用的特点在我国的桥梁建设中已经被广泛的采用。然而由于曲线预应力混凝土箱梁桥在力学性能、荷载、构造和施工工艺等方面的复杂性,及计算程序的易用性,目前仍有许多课题急待解决。本文课题研究的主要目的是针对目前曲线梁桥分析的复杂性,选用适当的计算理论,编制简洁易用友好的有限元程序,解决曲线梁在有限元分析中的一系列问题。
采梁格法对曲线预应力混凝土箱梁桥进行有限元分析,用每结点六个自由度的空间直杆单元来分析梁格的纵横构件,并推导空间直梁单元的单元刚度矩阵、坐标转换矩阵和整体刚度矩阵,并对边界条件的处理进行了讨论,对钢束几何要素的计算公式进行推导,分析了各种预应力损失,推导出预应力等效荷载的表达式,并通过等效荷载计算预应力束的总效应和二次内力。
对网格划分和梁格刚度的取值进行研究,提出了梁格法分析曲线预应力混凝土箱梁桥时划分网格数目的确定方法:用梁格法分析曲线梁桥时,当每根纵向梁格构件所对应的圆心角在4~5度之间时,若再细分网格对结果的影响已经不大,此时可以认为结果已经收敛,用一算例来说明上部结构梁格截面特性的取值。
在Fortran PowerStation4.0集成开发环境下采用Fortran90语言将上述内容编制成程序,并采用Visual Basic 6.0语言为程序编制了友好的交互性输入界面。最后通过实桥算例验证了本文理论分析的正确性和程序的精确性。
Curved prestressed-concrete box-girder bridge, with its economic, aesthetic and practical features, has been widely adopted in China bridge construction. But because of the complexity in mechanical properties, loadings, structure, construction technology, and other aspects,and easy using in calculating program,many issues are still there to be resolved. The main problem researched in this paper is to workout a friendly finite element program,under the complexity in curved girder bridge analysis,which is using proper calculating theories,to solve a series of the problem in curved girder bridge finite element analyse.
In this paper, grillage analogy has been used in finite element analysis on curved beam bridge, three-dimensional straight-lever unit, which has six degrees of freedom for each node, is used to analyse the vertical and horizontal structure of the grillage, element stiffness matrix, coordinate transformation matrix and overall stiffness matrix for straight beam element is derived,the handling of boundary conditions is discussed, and the equivalence load of the prestressing force has been envolved, then the gross effect of prestressed reinforcement and secondary internal force is calculated through equivalent load.
To research the value of meshed grid and beam stiffness, the method of confirming the value of meshing grid and beam stiffness when analysising the prestressed concrete box-girder bridge by grillage was educed: while meshing the girder by grillage, when each vertical beam grid components corresponding to the central angle in four to five degrees, if the breakdown of the results of the grid had little impact, the results were convergence. An example used to illustrate the structure of the upper beam cross-section of the grid value.
The above-mentioned content is programming under Fortran90 language in Fortran PowerStation4.0 integrated development environment, static analysis module is obtained through straight-lever grillage finite-element method for superstructure of curved beam bridge, Visual Basic 6.0 language is used to programming a friendly interactive input interface
Finally, the correctness of this theoretical analysis and the accuracy of this program are verified through the calculation example.
采梁格法对曲线预应力混凝土箱梁桥进行有限元分析,用每结点六个自由度的空间直杆单元来分析梁格的纵横构件,并推导空间直梁单元的单元刚度矩阵、坐标转换矩阵和整体刚度矩阵,并对边界条件的处理进行了讨论,对钢束几何要素的计算公式进行推导,分析了各种预应力损失,推导出预应力等效荷载的表达式,并通过等效荷载计算预应力束的总效应和二次内力。
对网格划分和梁格刚度的取值进行研究,提出了梁格法分析曲线预应力混凝土箱梁桥时划分网格数目的确定方法:用梁格法分析曲线梁桥时,当每根纵向梁格构件所对应的圆心角在4~5度之间时,若再细分网格对结果的影响已经不大,此时可以认为结果已经收敛,用一算例来说明上部结构梁格截面特性的取值。
在Fortran PowerStation4.0集成开发环境下采用Fortran90语言将上述内容编制成程序,并采用Visual Basic 6.0语言为程序编制了友好的交互性输入界面。最后通过实桥算例验证了本文理论分析的正确性和程序的精确性。
Curved prestressed-concrete box-girder bridge, with its economic, aesthetic and practical features, has been widely adopted in China bridge construction. But because of the complexity in mechanical properties, loadings, structure, construction technology, and other aspects,and easy using in calculating program,many issues are still there to be resolved. The main problem researched in this paper is to workout a friendly finite element program,under the complexity in curved girder bridge analysis,which is using proper calculating theories,to solve a series of the problem in curved girder bridge finite element analyse.
In this paper, grillage analogy has been used in finite element analysis on curved beam bridge, three-dimensional straight-lever unit, which has six degrees of freedom for each node, is used to analyse the vertical and horizontal structure of the grillage, element stiffness matrix, coordinate transformation matrix and overall stiffness matrix for straight beam element is derived,the handling of boundary conditions is discussed, and the equivalence load of the prestressing force has been envolved, then the gross effect of prestressed reinforcement and secondary internal force is calculated through equivalent load.
To research the value of meshed grid and beam stiffness, the method of confirming the value of meshing grid and beam stiffness when analysising the prestressed concrete box-girder bridge by grillage was educed: while meshing the girder by grillage, when each vertical beam grid components corresponding to the central angle in four to five degrees, if the breakdown of the results of the grid had little impact, the results were convergence. An example used to illustrate the structure of the upper beam cross-section of the grid value.
The above-mentioned content is programming under Fortran90 language in Fortran PowerStation4.0 integrated development environment, static analysis module is obtained through straight-lever grillage finite-element method for superstructure of curved beam bridge, Visual Basic 6.0 language is used to programming a friendly interactive input interface
Finally, the correctness of this theoretical analysis and the accuracy of this program are verified through the calculation example.
引文
[1]邵容光,夏淦.混凝土弯梁桥[M].北京:人民交通出版社, 1996
[2] Timoshenko S P, Gere. Theory of Elastic Stability, 2nd Edn [J]. NewYork: McGraw Hill Book Co. Inc. 1961
[3] Federhofer K, Dynamik des Bogentragers und Kreisringes[M].Wien:Springer,1950
[4] Yang Y B, Kuo S R, Effect of curvature on stability of curved beams[M].J Engrg Struct Div.1987,113:1l85～1202
[5]姚玲森.曲线梁[M].北京:人民交通出版杜, 1989
[6] Morris D L. Curved beamed stiffness coefficients[M].J Engrg Struct Div,ASCE,1967,94: 1165~1174
[7] Chandhuri S K, Shore S, Thin walled curved beam finite element[J].J Engrg Mech Div,ASCE,1977,103: 921~937
[8]严琼,陈玉萍.曲线梁桥计算方法及有限元法在曲梁中的应用[J].山西交通科技, 2005.1(2)58～59
[9] McManus, P. F. Nasir, G. A and Culver, C. G. Horizontally curved girders state-of-the-art[J], J.S.D.ASCE, 1969, 95(5) :853-870
[10] Vlasov V Z .Thin-walled elastic beam .2nd Edn [M]. Washington. D. C., National Science Foundation, 1961
[11] Dabrowski R. Curved thin-walled girder, theory and analysis[M]. New York, Springer, 1968
[12]包世华,周坚.薄壁杆件结构力学[M].北京:中国建筑工业出版社, 2006
[13]李乔.薄壁曲箱梁的空间分析理论[D].成都:西南交通大学, 1988
[14]李国豪.桥梁与结构理论研究[M].上海:上海科技文献出版社, 1983
[15]李国豪.大曲率薄壁箱梁的扭转和弯曲[J].土木工程学报, 1987, 2(1) :65-75
[16]李明昭.断面可变形的矩形箱式薄壁圆弧曲杆净力分析法[J].同济大学学报, 1982 (2) :26-34
[17]黄剑源.薄壁结构的扭转分析[M].北京:中国铁道出版社, 1998
[18]倪元增,钱寅泉.弹性薄壁梁桥分析[M].北京:人民交通出版社, 1998
[19]钟万勰,徐新生,张洪武.弹性曲梁问题的直接法[J].工程力学, 1996, 13(4) :1-8
[20]罗旗帜.计算曲线梁桥的传递矩阵法[J].中南公路工程, 1990(3) :32-41
[21] Lavelle F H, Boick J S. A program to analysis curved girder bridge[EB]. Engineering Bulletin No 8 University of Rhode Island. Kingston RI, 1965, 25
[22] Sawko F. Computer analysis of grillages curved in plan[C]. International Association for Bridge and Structural Engineering, 1967, 27
[23] Lavelle F H, Grieg R A, wemmer H R. CUGAR 1 Curved girder analysis revision 1[EB]. Engineering Bulletin No 14 University of Rhode Island. Kingston R I. 1971
[24] Lavelle F H, Laska R J. CUGAR2 User's Manual[R]. University of Rhode Island. Kingston R I, 1975
[25] Lavelle F H CUGAR3 Algorithm[R]. Consortium of University Research Teams Report No 56 University of Rhode Island. Kingston R I, 1976
[26]高岛春生.曲线梁桥[M].北京:中国建筑工业出版社, 1979
[27] C.P.Heins.结构杆件的弯曲与扭转[M].北京:人民交通出版社. 1981
[28]谢旭,黄剑源.薄壁断面曲线高架桥的空间结构分析法[J].土木工程学报, 1993, 26(6) :3-12
[29]黄剑源,谢旭.薄壁曲线杆系结构空间分析的刚度法[J].土木工程学报, 1994, 27(5) :3-19
[30]孙广华.曲线梁桥计算[M].北京:人民交通出版社, 1995
[31]刑志成.弯斜桥计算理论与实用计算[M].北京:人民交通出版社, 1994
[32]李惠生,张罗溪.曲线桥梁结构分析[M].北京:中国铁道出版社, 1992
[33]贺栓海,张翔.弹性曲板的有限条—传递矩阵法分析及其在弯桥上的应用[J].西安公路学院学报, 1989, 7(1) :70-81
[34]蔺锡九,陈德荣,李长明.等截面箱形连续弯梁桥的空间分析有限条法[J].公路交通科技, 1984, 2 :11-16
[35]成晓.有限条法分析连续弯箱梁桥[D].南京:南京工学院, 1987
[36]赵振铭.多跨弯曲连续箱梁有限条法分析中凝聚矩阵应用的探讨[J].福州大学学报, 1998, 26(1) :83-87
[37]刑志成.弯斜桥计算理论与实用计算[M].北京:人民交通出版社, 1994
[38] Lyons L P R. A general mite element system with special reference to the analysis of celluar structures[D]. Imperial College of Science and Technology, London, England, 1977
[39] Sawko, F. and Merriman, P.A. Finite Element Analysis of Bxidge Curved in Plan[C] Developments in Bridge Design and Collstruction, Crosby Lockwood & Son Ltd.,1971 : 334-346
[40] Jirousek J, Bouberguig A. A macro-element analysis of prestressed curved box-girder bridges[J]. Computers & Structures, 1979, 10(3) :467-482
[41] Adet Fam, Cart Turkstra .A finite element scheme for box bridge analysis[J]. Computers & Structures, 1975, 5(2) :179-186
[42] Iph K, Aneja A M., Frederic Roll M. Model analysis of curved box beam highway bridge[J]. Journal of Structural Division. ASCE, 1971, 12 :2861-2878
[43]杨冰.预应力混凝土和钢筋混凝土曲线箱梁的非线性有限元分析和试验[D].上海:同济大学, 1987
[44]项贻强.斜弯桥梁的空间分析理论及其试验研究[D].杭州:浙江大学, 2000
[45]徐艳秋,许克宾,王丽.薄壁弯梁桥空间有限元分析[J].土木工程学报, 2003, 36(2) :58-62
[46]李国豪,石洞, C.P.Heins.曲线桥地震分析的有限单元法[J].同济大学学报, 1984, 23(1) :1-21
[47]邵容光,孙广华,夏淦.弯梁桥内力及变形纵向影响线的有限单元法计算[J].华东公路. 1984(4) :8-12
[48]张叔辉.曲桥分析的薄壁箱梁单元[J].土木工程学报, 1984, 17(2) :1-13
[49]谢旭,黄剑源.薄壁箱梁桥约束扭转下翘曲、畸变和剪力滞效应的空间分析[J].土木工程学报, 1995, 28(4) :3-14
[50]王拥军.直、曲线独立柱支承连续箱梁桥的空间分析[C].中国公路学会桥梁和结构工程学桥梁学术讨论会论文集.北京:人民交通出版社, 1997
[51]王荣辉,刘光栋,邱波.薄壁曲线箱梁空间计算的曲板板梁单元法[J].中国公路学报, 2000, 13(1) :73-76
[52]韦成龙,曾庆元.薄壁曲线箱梁考虑翘曲、畸变和剪滞效应的空间分析[J].土木工程学报, 2000, 33(6) :81-87
[53]钟新谷,马平,曾庆元.多室箱形梁非线性有限元分析[J].土木工程学报, 1999, 32(6) :32-39
[54]吴亚平,赖远明,朱元林等.考虑剪滞效应的薄壁曲梁有限单元法[J].工程力学. 2002, 19(4) :85-89
[55]李欣然,卢树圣,戴公连.曲线薄壁箱梁结构的梁段单元分析[J].海南大学学报自然科学版, 1998, 16(4) :300-304
[56]段海娟,赵人达,周益云.曲箱梁考虑剪力滞效应的弯扭耦合分析[J].中国铁道科学, 2002, 23(1) :34-37
[57]谢智敏,曹卫力.弯梁桥常见问题分析[J].广西交通科技, 2003, 28(2):49～51
[58]扬党旗.华强立交A匝道独柱曲线梁桥病害分析[J].桥梁建设, 2003(2):58～61
[59]张元凯.弯箱梁梁格实用计算分析[D].上海同济大学桥梁系, 2000
[60]王卫锋.某立交曲线匝道桥事故分析[J].华南理工大学学报, 1999, 27(11):104～107
[61]王爱梅,孙广华.曲线梁桥预应力钢索侧向崩出的预防[J].江苏交通科技, 2000(5):17～19
[62]肖恩源.预应力连续弯桥墩顶开裂的探讨[J].公路, 2002(4):32～34
[63]肖汝诚.桥梁结构分析及程序系统[M].北京:人民交通出版社, 2002
[64]赵巧燕.多跨混凝土弯梁桥若干问题研究[D].北京工业大学, 2003
[65]王觉民.曲线梁桥的损伤诊断与加固施工[D].湖南大学, 2002
[66]范立础.桥梁工程[M].北京:人民交通出版社, 2001
[67]罗喜恒.预应力混凝土曲线箱梁桥的空间分析[D].同济大学桥梁工程系, 1994
[68]杨勇.曲线薄壁箱梁的空间受力分析[J].长沙铁道学院学报, 2001, 18(2):17～21
[69]李欣然,戴公连.曲线薄壁箱梁结构的梁段单元分析[J].海南大学学报自然科学版, 1998, 16(4):300～304
[70]刘磊,许克宾.结构分析中的曲梁单元[J].北方交通大学学报, 2002, 26(4):20～23
[71]赵红华,钱若军.考虑多种因素共同作用空间梁分析模型[J].太原理工大学学报, 2003, 34(4):418～420
[72]李士辉,陈晓军.浅谈弯桥的形式和受力特性[J].黑龙江交通科技, 1997(1):35～39
[73]韩皓,李彬.支座预偏心对曲线箱梁桥受力影响研究[J].华东公路, 1999(4):142～147
[74] Waldron P. Equivalent Beam Analysis of Thin-walled Beam Structures[J]. Computer &Structures. ASCE, 1985(7):60～64
[75] Di-Gen, Song. Alexander C S. Shear-Log Analysis and Box Beams[J].Journal of Structural Engineering, 1994, 116(5):59～64
[76]蔡峰.曲线梁桥结构分析研究与程序设计[D].长安大学, 2002
[77]李国平.桥梁预应力混凝土技术及设计原理[M].北京:人民交通出版社, 2003
[78]中交公路规划设计院.公路桥涵设计通用规范JTG D60-2004[S].北京:人民交通出版社, 2004
[79]王伟生.曲梁中预应力等代荷载的计算[J].桥梁建设, 1997(1):78～83
[80]盛飞,曹干珍.预应力混凝土曲线梁桥设计控制荷载的探讨[J].广东公路交通, 2001(3):15～19
[2] Timoshenko S P, Gere. Theory of Elastic Stability, 2nd Edn [J]. NewYork: McGraw Hill Book Co. Inc. 1961
[3] Federhofer K, Dynamik des Bogentragers und Kreisringes[M].Wien:Springer,1950
[4] Yang Y B, Kuo S R, Effect of curvature on stability of curved beams[M].J Engrg Struct Div.1987,113:1l85～1202
[5]姚玲森.曲线梁[M].北京:人民交通出版杜, 1989
[6] Morris D L. Curved beamed stiffness coefficients[M].J Engrg Struct Div,ASCE,1967,94: 1165~1174
[7] Chandhuri S K, Shore S, Thin walled curved beam finite element[J].J Engrg Mech Div,ASCE,1977,103: 921~937
[8]严琼,陈玉萍.曲线梁桥计算方法及有限元法在曲梁中的应用[J].山西交通科技, 2005.1(2)58～59
[9] McManus, P. F. Nasir, G. A and Culver, C. G. Horizontally curved girders state-of-the-art[J], J.S.D.ASCE, 1969, 95(5) :853-870
[10] Vlasov V Z .Thin-walled elastic beam .2nd Edn [M]. Washington. D. C., National Science Foundation, 1961
[11] Dabrowski R. Curved thin-walled girder, theory and analysis[M]. New York, Springer, 1968
[12]包世华,周坚.薄壁杆件结构力学[M].北京:中国建筑工业出版社, 2006
[13]李乔.薄壁曲箱梁的空间分析理论[D].成都:西南交通大学, 1988
[14]李国豪.桥梁与结构理论研究[M].上海:上海科技文献出版社, 1983
[15]李国豪.大曲率薄壁箱梁的扭转和弯曲[J].土木工程学报, 1987, 2(1) :65-75
[16]李明昭.断面可变形的矩形箱式薄壁圆弧曲杆净力分析法[J].同济大学学报, 1982 (2) :26-34
[17]黄剑源.薄壁结构的扭转分析[M].北京:中国铁道出版社, 1998
[18]倪元增,钱寅泉.弹性薄壁梁桥分析[M].北京:人民交通出版社, 1998
[19]钟万勰,徐新生,张洪武.弹性曲梁问题的直接法[J].工程力学, 1996, 13(4) :1-8
[20]罗旗帜.计算曲线梁桥的传递矩阵法[J].中南公路工程, 1990(3) :32-41
[21] Lavelle F H, Boick J S. A program to analysis curved girder bridge[EB]. Engineering Bulletin No 8 University of Rhode Island. Kingston RI, 1965, 25
[22] Sawko F. Computer analysis of grillages curved in plan[C]. International Association for Bridge and Structural Engineering, 1967, 27
[23] Lavelle F H, Grieg R A, wemmer H R. CUGAR 1 Curved girder analysis revision 1[EB]. Engineering Bulletin No 14 University of Rhode Island. Kingston R I. 1971
[24] Lavelle F H, Laska R J. CUGAR2 User's Manual[R]. University of Rhode Island. Kingston R I, 1975
[25] Lavelle F H CUGAR3 Algorithm[R]. Consortium of University Research Teams Report No 56 University of Rhode Island. Kingston R I, 1976
[26]高岛春生.曲线梁桥[M].北京:中国建筑工业出版社, 1979
[27] C.P.Heins.结构杆件的弯曲与扭转[M].北京:人民交通出版社. 1981
[28]谢旭,黄剑源.薄壁断面曲线高架桥的空间结构分析法[J].土木工程学报, 1993, 26(6) :3-12
[29]黄剑源,谢旭.薄壁曲线杆系结构空间分析的刚度法[J].土木工程学报, 1994, 27(5) :3-19
[30]孙广华.曲线梁桥计算[M].北京:人民交通出版社, 1995
[31]刑志成.弯斜桥计算理论与实用计算[M].北京:人民交通出版社, 1994
[32]李惠生,张罗溪.曲线桥梁结构分析[M].北京:中国铁道出版社, 1992
[33]贺栓海,张翔.弹性曲板的有限条—传递矩阵法分析及其在弯桥上的应用[J].西安公路学院学报, 1989, 7(1) :70-81
[34]蔺锡九,陈德荣,李长明.等截面箱形连续弯梁桥的空间分析有限条法[J].公路交通科技, 1984, 2 :11-16
[35]成晓.有限条法分析连续弯箱梁桥[D].南京:南京工学院, 1987
[36]赵振铭.多跨弯曲连续箱梁有限条法分析中凝聚矩阵应用的探讨[J].福州大学学报, 1998, 26(1) :83-87
[37]刑志成.弯斜桥计算理论与实用计算[M].北京:人民交通出版社, 1994
[38] Lyons L P R. A general mite element system with special reference to the analysis of celluar structures[D]. Imperial College of Science and Technology, London, England, 1977
[39] Sawko, F. and Merriman, P.A. Finite Element Analysis of Bxidge Curved in Plan[C] Developments in Bridge Design and Collstruction, Crosby Lockwood & Son Ltd.,1971 : 334-346
[40] Jirousek J, Bouberguig A. A macro-element analysis of prestressed curved box-girder bridges[J]. Computers & Structures, 1979, 10(3) :467-482
[41] Adet Fam, Cart Turkstra .A finite element scheme for box bridge analysis[J]. Computers & Structures, 1975, 5(2) :179-186
[42] Iph K, Aneja A M., Frederic Roll M. Model analysis of curved box beam highway bridge[J]. Journal of Structural Division. ASCE, 1971, 12 :2861-2878
[43]杨冰.预应力混凝土和钢筋混凝土曲线箱梁的非线性有限元分析和试验[D].上海:同济大学, 1987
[44]项贻强.斜弯桥梁的空间分析理论及其试验研究[D].杭州:浙江大学, 2000
[45]徐艳秋,许克宾,王丽.薄壁弯梁桥空间有限元分析[J].土木工程学报, 2003, 36(2) :58-62
[46]李国豪,石洞, C.P.Heins.曲线桥地震分析的有限单元法[J].同济大学学报, 1984, 23(1) :1-21
[47]邵容光,孙广华,夏淦.弯梁桥内力及变形纵向影响线的有限单元法计算[J].华东公路. 1984(4) :8-12
[48]张叔辉.曲桥分析的薄壁箱梁单元[J].土木工程学报, 1984, 17(2) :1-13
[49]谢旭,黄剑源.薄壁箱梁桥约束扭转下翘曲、畸变和剪力滞效应的空间分析[J].土木工程学报, 1995, 28(4) :3-14
[50]王拥军.直、曲线独立柱支承连续箱梁桥的空间分析[C].中国公路学会桥梁和结构工程学桥梁学术讨论会论文集.北京:人民交通出版社, 1997
[51]王荣辉,刘光栋,邱波.薄壁曲线箱梁空间计算的曲板板梁单元法[J].中国公路学报, 2000, 13(1) :73-76
[52]韦成龙,曾庆元.薄壁曲线箱梁考虑翘曲、畸变和剪滞效应的空间分析[J].土木工程学报, 2000, 33(6) :81-87
[53]钟新谷,马平,曾庆元.多室箱形梁非线性有限元分析[J].土木工程学报, 1999, 32(6) :32-39
[54]吴亚平,赖远明,朱元林等.考虑剪滞效应的薄壁曲梁有限单元法[J].工程力学. 2002, 19(4) :85-89
[55]李欣然,卢树圣,戴公连.曲线薄壁箱梁结构的梁段单元分析[J].海南大学学报自然科学版, 1998, 16(4) :300-304
[56]段海娟,赵人达,周益云.曲箱梁考虑剪力滞效应的弯扭耦合分析[J].中国铁道科学, 2002, 23(1) :34-37
[57]谢智敏,曹卫力.弯梁桥常见问题分析[J].广西交通科技, 2003, 28(2):49～51
[58]扬党旗.华强立交A匝道独柱曲线梁桥病害分析[J].桥梁建设, 2003(2):58～61
[59]张元凯.弯箱梁梁格实用计算分析[D].上海同济大学桥梁系, 2000
[60]王卫锋.某立交曲线匝道桥事故分析[J].华南理工大学学报, 1999, 27(11):104～107
[61]王爱梅,孙广华.曲线梁桥预应力钢索侧向崩出的预防[J].江苏交通科技, 2000(5):17～19
[62]肖恩源.预应力连续弯桥墩顶开裂的探讨[J].公路, 2002(4):32～34
[63]肖汝诚.桥梁结构分析及程序系统[M].北京:人民交通出版社, 2002
[64]赵巧燕.多跨混凝土弯梁桥若干问题研究[D].北京工业大学, 2003
[65]王觉民.曲线梁桥的损伤诊断与加固施工[D].湖南大学, 2002
[66]范立础.桥梁工程[M].北京:人民交通出版社, 2001
[67]罗喜恒.预应力混凝土曲线箱梁桥的空间分析[D].同济大学桥梁工程系, 1994
[68]杨勇.曲线薄壁箱梁的空间受力分析[J].长沙铁道学院学报, 2001, 18(2):17～21
[69]李欣然,戴公连.曲线薄壁箱梁结构的梁段单元分析[J].海南大学学报自然科学版, 1998, 16(4):300～304
[70]刘磊,许克宾.结构分析中的曲梁单元[J].北方交通大学学报, 2002, 26(4):20～23
[71]赵红华,钱若军.考虑多种因素共同作用空间梁分析模型[J].太原理工大学学报, 2003, 34(4):418～420
[72]李士辉,陈晓军.浅谈弯桥的形式和受力特性[J].黑龙江交通科技, 1997(1):35～39
[73]韩皓,李彬.支座预偏心对曲线箱梁桥受力影响研究[J].华东公路, 1999(4):142～147
[74] Waldron P. Equivalent Beam Analysis of Thin-walled Beam Structures[J]. Computer &Structures. ASCE, 1985(7):60～64
[75] Di-Gen, Song. Alexander C S. Shear-Log Analysis and Box Beams[J].Journal of Structural Engineering, 1994, 116(5):59～64
[76]蔡峰.曲线梁桥结构分析研究与程序设计[D].长安大学, 2002
[77]李国平.桥梁预应力混凝土技术及设计原理[M].北京:人民交通出版社, 2003
[78]中交公路规划设计院.公路桥涵设计通用规范JTG D60-2004[S].北京:人民交通出版社, 2004
[79]王伟生.曲梁中预应力等代荷载的计算[J].桥梁建设, 1997(1):78～83
[80]盛飞,曹干珍.预应力混凝土曲线梁桥设计控制荷载的探讨[J].广东公路交通, 2001(3):15～19