混凝土索塔开裂行为的宏细观力学分析
|
摘要
最近几十年，斜拉桥在我国取得了飞速的发展，其中混凝土斜拉桥占据较大比重。在外荷载、温度荷载以及混凝土收缩徐变等因素作用下，混凝土斜拉桥索塔承受着拉索传递的巨大压力，据不完全统计，全国有75%的索塔发生开裂现象，严重影响斜拉桥的寿命甚至威胁人们的生命安全，因而索塔开裂的行为引起越来越多研究人员的关注。本文引入分形损伤理论对斜拉桥高大索塔进行安全性评价，提出了基于裂缝分布状态的高大索塔安全状态评价模式。通过理论分析、模型试验与数值模拟方法，分别从宏观和细观的角度，研究了裂缝的开裂、扩展及结构破坏过程，建立了一套以分形损伤为理论支撑的用于评价索塔开裂后安全状态的新方法，研究了索塔开裂后裂缝分布分形维数与荷载等级之间的关系，为后期实桥应用提供了理论与实践指导。本文的主要研究内容和结论如下：
①在研究智能移动视频扫描装置监测高大塔柱裂缝技术的基础上，针对各种滤波算法适用范围和裂缝图像特点的研究，提出中值滤波法，将进行裂纹图像的二值化与提取方法用于裂缝图像分割，并采用图像拼接技术、裂缝矢量化技术及裂缝图像模拟重现技术，实现了裂缝监测信息处理。
②把三维裂缝求解问题转换为两个平面的裂缝求解问题，即等效成一个无限大板中心裂纹受到无穷远处的单向拉伸力作用和一个无限大板在单边裂缝受到拉伸荷载作用，采用双悬臂梁理论计算裂缝宽度与深度的关系，推导了裂缝深度的计算公式，为研究三维裂缝问题提供了新思路；根据裂纹分形模型建立混凝土开展裂缝的应力强度因子表达式，结合断裂力学理论，推导了应力强度因子在直线型裂纹和裂纹不规则扩展这两种假设下的关系。
③根据斜拉桥索塔的特点，基于子模型技术，采用FRANC3D软件，模拟了索塔裂缝的萌生以及扩展的过程，并计算了索塔结构的应力强度因子与分形维数之间的关系、分形维数与荷载等级之间的关系；建立weibull随机概率分布与材料裂缝分布之间的关系，结合断裂损伤力学与分形理论来评价混凝土索塔表面开裂的严重程度。
④建立了基于离散元的颗粒流微观数值模型，采用Fish语言编写程序，模拟混凝土材料的细观破坏过程，并选取了三种不同半径、不同数量团簇研究混凝土开裂性能，当团簇数量越多，对材料性质的提高越显著；当团簇半径越大、结构体系越稳定，则性质增强效果好。团簇材料的破损机理在于随着团簇半径的增加，基质所占比例减小，基质中更容易产生微裂纹，结构表面更容易破坏。
⑤基于多尺度均匀化理论，采用有限元法建立了混凝土细观-宏观等效力学性能的多尺度模型，并以索塔表面为例建立有限元模型，对混凝土材料的宏观整体与细观局部化效应进行分析。
⑥完成了索塔节段试验模型，验证了上述理论和方法的可靠性，研究了索塔开裂后裂缝分布分形维数与荷载等级之间的关系，为后期实桥应用提供了理论与试验基础。
Cable-stayed bridge got rapidly developed in designing and constructing duringthese decades. Concrete cable-stayed bridge occupy larger proportion. Tall cable towers’cracking behavior has aroused more and more attention from the researchers. Under theouter load, temperature load, concrete shrinkage and creep effect, tower of concretecable-stayed bridge suffers huge pressure transmitted by cables. According toincomplete statistics, tower fracture occurs in80%of the Chinese cable bridges, itseriously affects the cable-stayed bridge’s safety and even people's life. This articleintroduced the fractal damage and the fractal fracture mechanics into safety assessmentof cable-stayed bridge’s cable tower. Then, tall tower’ safety state evaluation model isput forward based on fracture distribution in this paper Beginning and expansion offractures and the structure failure process are analyzed from the macroscopic andmesoscopic point of view through theoretical analysis. This paper set up a new methodsupported by the theory of fractal damage for evaluation of cable tower safety state aftercracking occurs. The relationship between the load level and fractal dimension offracture distribution after the cable tower cracking occurs is researched and it providestheoretical and experimental basis to further application to real bridge. The followingmainly innovative work has been done in this paper.
①Tall tower’s crack intelligent mobile video scanning device monitoringtechnology is put forward Median filtering method is put forward according to range ofall kinds of filter algorithm applicable and the crack image characteristics. Crackimage’s binarization and extraction method are used to crack image segmentation.Crack monitoring information processing is realized by image mosaicing, crack vectorquantization and cracks image’s simulation technology.
②The solution of the three-dimension crack is equivalent to solutions of twotwo-dimension crack. One is equivalent to solution of center crack of infinite-sized plateunder the uni-axial tensile force from infinity distance, the other is equivalent tosolution of unilateral crack of infinite-sized plate under the tensile load. Relationshipbetween crack height and depth is calculated with double cantilever beam theory, itprovides a brand new method to research three-dimension crack problems. Stressintensity factor expression of tall cable tower’s concrete crack is established based on crack fractal model. The relationship between stress intensity factors under linear crackassumption and irregular crack extension assumption is deduced combined with fracturemechanics theory.
③Process of the initiationand propagation of cable tower’s cracks is simulatedbased on the submodel technique and FRANC3D software combined with thecharacteristics of the cable-stayed bridge’s tall tower. The relationships between stressintensity factor of the cable tower structure and the fractal dimension, fractal dimensionand the bearing capacity is calculated out. The relationship between weibull randomprobability distribution and the material cracks’ distribution is established. The severityof the cable tower of concrete surface cracking is evaluated combined with fracturedamage mechanics and fractal theory.
④In this paper, the micro particle flow numerical model based on discrete elementis established and the mesoscopic damage process of concrete is simulated with Fishsoftware. Concrete cracking performance is researched with selecting three kinds ofconcrete particle flow with different radius and number of clusters. As a result, thematerial properties improves significantly while the number of clusters increases Thelarger clusters radius is, the greater the stability of the structure system and the betternature of the enhancement effect would be. Mechanism of clusters material’s damagecould be concluded as following. With the increasing of radius of clusters, proportion ofmatrix would decreases, and the matrix of micro cracks occurs more, at the same time,structure surface damages more easily.
⑤Based on the theory of multi-scale homogenization. The finite element method isadopted to establish the concrete mesoscopic-macro equivalent mechanical propertiesof multi-scale model. At the same time, as an example, a finite element model of thesurface of the cable tower is established for macro and micro effect analysis to concrete.
⑥Cable tower segment model test was done to verify the reliability of thetheoretical system. The relationship between the load level and fractal dimension offracture distribution after the cable tower cracking occurs is researched and it providestheoretical and experimental basis to further application to real bridge.
①在研究智能移动视频扫描装置监测高大塔柱裂缝技术的基础上，针对各种滤波算法适用范围和裂缝图像特点的研究，提出中值滤波法，将进行裂纹图像的二值化与提取方法用于裂缝图像分割，并采用图像拼接技术、裂缝矢量化技术及裂缝图像模拟重现技术，实现了裂缝监测信息处理。
②把三维裂缝求解问题转换为两个平面的裂缝求解问题，即等效成一个无限大板中心裂纹受到无穷远处的单向拉伸力作用和一个无限大板在单边裂缝受到拉伸荷载作用，采用双悬臂梁理论计算裂缝宽度与深度的关系，推导了裂缝深度的计算公式，为研究三维裂缝问题提供了新思路；根据裂纹分形模型建立混凝土开展裂缝的应力强度因子表达式，结合断裂力学理论，推导了应力强度因子在直线型裂纹和裂纹不规则扩展这两种假设下的关系。
③根据斜拉桥索塔的特点，基于子模型技术，采用FRANC3D软件，模拟了索塔裂缝的萌生以及扩展的过程，并计算了索塔结构的应力强度因子与分形维数之间的关系、分形维数与荷载等级之间的关系；建立weibull随机概率分布与材料裂缝分布之间的关系，结合断裂损伤力学与分形理论来评价混凝土索塔表面开裂的严重程度。
④建立了基于离散元的颗粒流微观数值模型，采用Fish语言编写程序，模拟混凝土材料的细观破坏过程，并选取了三种不同半径、不同数量团簇研究混凝土开裂性能，当团簇数量越多，对材料性质的提高越显著；当团簇半径越大、结构体系越稳定，则性质增强效果好。团簇材料的破损机理在于随着团簇半径的增加，基质所占比例减小，基质中更容易产生微裂纹，结构表面更容易破坏。
⑤基于多尺度均匀化理论，采用有限元法建立了混凝土细观-宏观等效力学性能的多尺度模型，并以索塔表面为例建立有限元模型，对混凝土材料的宏观整体与细观局部化效应进行分析。
⑥完成了索塔节段试验模型，验证了上述理论和方法的可靠性，研究了索塔开裂后裂缝分布分形维数与荷载等级之间的关系，为后期实桥应用提供了理论与试验基础。
Cable-stayed bridge got rapidly developed in designing and constructing duringthese decades. Concrete cable-stayed bridge occupy larger proportion. Tall cable towers’cracking behavior has aroused more and more attention from the researchers. Under theouter load, temperature load, concrete shrinkage and creep effect, tower of concretecable-stayed bridge suffers huge pressure transmitted by cables. According toincomplete statistics, tower fracture occurs in80%of the Chinese cable bridges, itseriously affects the cable-stayed bridge’s safety and even people's life. This articleintroduced the fractal damage and the fractal fracture mechanics into safety assessmentof cable-stayed bridge’s cable tower. Then, tall tower’ safety state evaluation model isput forward based on fracture distribution in this paper Beginning and expansion offractures and the structure failure process are analyzed from the macroscopic andmesoscopic point of view through theoretical analysis. This paper set up a new methodsupported by the theory of fractal damage for evaluation of cable tower safety state aftercracking occurs. The relationship between the load level and fractal dimension offracture distribution after the cable tower cracking occurs is researched and it providestheoretical and experimental basis to further application to real bridge. The followingmainly innovative work has been done in this paper.
①Tall tower’s crack intelligent mobile video scanning device monitoringtechnology is put forward Median filtering method is put forward according to range ofall kinds of filter algorithm applicable and the crack image characteristics. Crackimage’s binarization and extraction method are used to crack image segmentation.Crack monitoring information processing is realized by image mosaicing, crack vectorquantization and cracks image’s simulation technology.
②The solution of the three-dimension crack is equivalent to solutions of twotwo-dimension crack. One is equivalent to solution of center crack of infinite-sized plateunder the uni-axial tensile force from infinity distance, the other is equivalent tosolution of unilateral crack of infinite-sized plate under the tensile load. Relationshipbetween crack height and depth is calculated with double cantilever beam theory, itprovides a brand new method to research three-dimension crack problems. Stressintensity factor expression of tall cable tower’s concrete crack is established based on crack fractal model. The relationship between stress intensity factors under linear crackassumption and irregular crack extension assumption is deduced combined with fracturemechanics theory.
③Process of the initiationand propagation of cable tower’s cracks is simulatedbased on the submodel technique and FRANC3D software combined with thecharacteristics of the cable-stayed bridge’s tall tower. The relationships between stressintensity factor of the cable tower structure and the fractal dimension, fractal dimensionand the bearing capacity is calculated out. The relationship between weibull randomprobability distribution and the material cracks’ distribution is established. The severityof the cable tower of concrete surface cracking is evaluated combined with fracturedamage mechanics and fractal theory.
④In this paper, the micro particle flow numerical model based on discrete elementis established and the mesoscopic damage process of concrete is simulated with Fishsoftware. Concrete cracking performance is researched with selecting three kinds ofconcrete particle flow with different radius and number of clusters. As a result, thematerial properties improves significantly while the number of clusters increases Thelarger clusters radius is, the greater the stability of the structure system and the betternature of the enhancement effect would be. Mechanism of clusters material’s damagecould be concluded as following. With the increasing of radius of clusters, proportion ofmatrix would decreases, and the matrix of micro cracks occurs more, at the same time,structure surface damages more easily.
⑤Based on the theory of multi-scale homogenization. The finite element method isadopted to establish the concrete mesoscopic-macro equivalent mechanical propertiesof multi-scale model. At the same time, as an example, a finite element model of thesurface of the cable tower is established for macro and micro effect analysis to concrete.
⑥Cable tower segment model test was done to verify the reliability of thetheoretical system. The relationship between the load level and fractal dimension offracture distribution after the cable tower cracking occurs is researched and it providestheoretical and experimental basis to further application to real bridge.
引文
[1]李俊如,高建光,王耀辉.超声波检测混凝土裂缝及裂缝成因分析[J].岩土力学,2001,22(3):291-295.
[2]黄政宇,纪学灵,黄靓.超声波检测混凝土缺陷成像的定性分析法[J].湖南大学学报35(8),2008(8):5-8.
[3]王登科.超声波检测钻孔桩混凝土质量的缺陷分析和判断[J].公路.1995(9):29-31.
[4] ASTM Standard C1383-98a.“Standard Test Method for Measuring the P-Wave Speed and theThickness of Concrete Plates Using the Impact-Echo Method”. published by the AMERICANSOCIETY FOR TESTING AND MATERIALS (AST M).1-12.
[5]王彬.基于声发射技术的预应力混凝土损伤检测理论及应用[D]:[硕士学位论文].镇江:江苏大学,2006.
[6]陈兵,姚武,吴科如.声发射技术在混凝土研究中的应用[J].无损检测,2000,22(9):387-390.
[7] Chen H L, Cheng C T, Chen S E. Determination of fracture parameters of mortar and concretebeams by using acoustic emission[J]. Materials Evaluation,1992,50(7):888-894.
[8] Ohtsu M. Rate Process Analysis of Acoustic Emission Activity in Core Test of Concrete.Concrete Library, SJCE,1992(20):143-153.
[9] S T Dai, J F Labuz. Damage and failure analysis of brittle materials by acoustic emission.Journal of Materials in Civil Engineering, ASCE,1997,9(4):200-205.
[10] Otsuka K, Date H, Kurita T. Fracture process zone in concrete tension specimens by X-ray andAE techniques. Fracture mechanics of concrete structures. Proceedings FRAMCOS, Germany:AEDIFICATIO Publishers.1998.151-170.
[11]李佳.混凝土静态轴拉声发射试验相关参数研究.[D]:[硕士学位论文].南京：河海大学，2008.
[12] Shigeishi M, Colombob S, Broughtonb K J, etal. Acoustic emission to assess and monitor theintegrity of bridges. Construction and Building Materials,2001,15(1):35-49.
[13]丁幼亮,邓扬,李爱群.声发射技术在桥梁结构健康监测中的应用研究进展[J].防灾减灾工程学报，2010,3(30):341-351.
[14] Sang-Hoon Kim, Jung-Ju Lee and Il-Bum Kwon, Monitoring of Beam Deflection Subjected toBending Load Using Brillouin Distributed Optical Fiber Sensors, Proceedings of SPIE-TheInternational Society for Optical Engineering,2001:63-69.
[15] Chai, J, Wei, S.M, Chang, X.T, Liu, J.X. Monitoring deformation and damage on rockstructures with distributed fiber optical sensing[J], International Journal of Rock Mechanicsand Mining Sciences,2004,41(5):1-6.
[16] Ou Jinping, Hou Shuang, Seismic Damage Identification using multi-line distributed fiberoptic sensor system, Proceedings of SPIE,2005:1003-1008.
[17] Roger G. Duncan, Brooks A. Childers, Dawn K. Gifford, Don E. Pettit, Andrew W. Hickson,Timothy L. Brown, Distributed sensing technique for test article damage detection andMonitoring, Proceedings of SPIE[C],2003:367-375.
[18]杨建良,向清,黄德修.机敏材料与结构中的准分布式光纤传感阵列[J],应用光学,2004,18(4),:33-36.
[19]刘浩吾,谢玲玲.桥梁裂缝监测的光纤传感网络[J],桥梁建设,2003(2),:78-81.
[20]涂亚庆,刘兴长.光纤智能结构[M].北京:高等教育出版社,2005.
[21] Marianne Bérubé Dufour, Dominique Derome, Radu Zmeureanu. Analysis of thermogramsfor the estimation of dimensions of cracks in building envelope. Infrared Physics&Technology,2009(52):70–78.
[22] F. Bakhtiari-Nejad, A. Khorram, M. Rezaeian.Analytical estimation of natural frequencies andmode shapes of a beam having two cracks. International Journal of Mechanical Sciences2014:193–202.
[23] Zhe Cheng, NiaoqingHun, XiaofeiZhang. Crack level estimation approach for planetarygearbox based on simulation signal and GRA. Journal of Sound and Vibration,2012:5853–5863.
[24] Mohammad H.F. Dado, Omar A. Shpli. Crack parameter estimation in structures using finiteelement modeling. International Journal of Solids and Structures,2003:5389–5406.
[25] Ahmed A. Elshafey, Nabil Dawood, H. Marzouk, M. Haddara.Crack width in concrete usingartificial neural networks. Engineering Structures,2013:676–686.
[26] Masayuki Kamaya. Estimation of elastic-plastic fracture toughness by numerical simulationbased on a stress-based criterion for ductile crack initiation. International Journal of PressureVessels and Piping,2013:1-7.
[27] Martin Noel, Khaled Soudki. Estimation of the crack width and deformation of FRP-reinforcedconcrete flexural members with and without transverse shear reinforcement. EngineeringStructures2014:393–398.
[28] Minhhuy Le, Jinyi Lee, Jongwoo Jun, Jungmin Kim.Estimation of sizes of cracks on pipes innuclear power plants using dipole moment and finite element methods. NDT&EInternational,2013:56–63.
[29] Gang Li, Shuanhai He, Yongfeng Ju, Kai Du. Long-distance precision inspection method forbridge cracks with image processing. Automation in Construction,2013.
[30] Ahmed A. Elshafey, Nabil Dawood, H. Marzouk, M. Haddara. Predicting of crack spacing forconcrete by using neural networks. Engineering Failure Analysis2013,31:344–359.
[31] J.H. Espina-Herna′ndez, E.Ramrez-Pacheco. Rapid estimation of artificial near-side crackdimensions in aluminium using a GMR-based eddy current sensor. NDT&E International,2012,51:94–100.
[32] M. Hadj Meliani, Z. Azari, G. Pluvinage, Yu.G. Matvienko. The effective T-stress estimationand crack paths emanating from U-notches. Engineering Fracture Mechanics,2010.77:1682–1692.
[33] Ahmed A. Elshafey, Nabil Dawood. Crack width in concrete using artificial neural networks.Engineering Structures,2013,52:676–686.
[34] Ahmed A. Elshafey. Nabil Dawood.Predicting of crack spacing for concrete by using neuralnetworks. Engineering Failure Analysis,2013,31:344–359.
[35] K.G. Papakonstantinou, M. ShinozukaProbabilistic.model for steel corrosion in reinforcedconcrete structures of large dimensions considering crack effects. Engineering Structures,2013,57:306–326.
[36] Martin No l, Khaled Soudki Estimation of the crack width and deformation of FRP-reinforcedconcrete flexural members with and without transverse shear reinforcement. EngineeringStructures,2014,59:393–398.
[37] J. Bach, H.W. H ppel.Crack initiation mechanisms in AA6082fatigued in the VHCF-regime.International Journal of Fatigue,2014,60:23–27.
[38] Hugo C. Biscaia, Carlos Chastre. A smeared crack analysis of reinforced concrete T-beamsstrengthened with GFRP composites. Engineering Structures,2013,56:1346–1361.
[39] Michael Scharnwebern, Wolfgang Tirschler.Mechanisms of short crack propagation inaustenitic–ferritic duplex steel. Materials Science&Engineering A,2014,595:269–283.
[40] Marta S owik, Piotr Smarzewski.Study of the scale effect on diagonal crack propagation inconcrete beams. Computational Materials Science,2012,64:216–220.
[41] A.K.M. Farid Uddin, Kenichiro Numata. Mechanisms of crack propagation due to corrosion ofreinforcement in concrete by AE-SiGMA and BEM. Construction and Building Materials,2004,18:181–188.
[42]黄贻凤.桥梁混凝土开裂理论及数值分析[D]:[硕士学位论文].武汉:武汉理工大学,2005.
[43]杨海科,李波.浅谈简支梁桥裂缝开展特点及裂缝与承载力的关系[J].东北公路.2003,26(3):101-104.
[44]宋一凡,崔军,赵小星等.钢筋混凝土T型梁桥裂缝特征参数与结构评估试验研究[J].交通运输工程学报,2001,1(3):46-49.
[45]贺拴海，宋一凡，赵小星等.钢筋混凝土梁式结构裂缝特征与损伤评估方法试验研究[J].土木工程学报,2003,36(2):6-9.
[46]周敉，贺拴海，袁万城.RC梁桥承载力的振动测试评估方法[J].交通运输工程学报,2006,6(3):62-67.
[47]赵煜，任伟，李春风等.预应力混凝土简支箱梁裂缝损伤参数影响分析[J].长安大学学报(自然科学版),2010,30(2):58-63.
[48]赵煜，贺拴海，李春风等.在役预应力混凝土箱梁开裂后承载力评估[J].同济大学学报(自然科学版),2010,38(9):1271-1275.
[49]赵煜，周博，贺拴海等.基于裂缝特征的PC简支梁损伤刚度评估方法[J].长安大学学报(自然科学版).2011,31(4):39-44.
[50] Alemdar Bayraktar, Abdurrahman Sahin. Numerical damage assessment of Haghia Sophia belltower by nonlinear FE modeling. Applied Mathematical Modeling,2010,34:92–121.
[51] Anna Anzani, Luiga Binda.A multilevel approach for the damage assessment of Historicmasonry towers. Journal of Cultural Heritage,2010,11:459-470.
[52] Eduardo Nuno Brito Santos Júlio.Structural assessment of the tower of the University ofCoimbra by modal identification. Engineering Structures,2008.30:3468-3477.
[53] Fernando Pe a, Paulo.B. Lourenco.Numerical models for the seismic assessment of an oldmasonry tower. Engineering Structures,2010,32:1466-1478.
[54] Angelo D’Ambrisi, Valentina Mariani. Seismic assessment of a historical masonry tower withnonlinear static and dynamic analyses tuned on ambient vibration tests. EngineeringStructures,2012,36:210-219.
[55] Tian-Hong Pan a, Shyan-Shu Shieh.Statistical multi-model approach for performanceassessment of cooling tower. Energy Conversion and Management,2011,52:1377-1385.
[56] C. Gentile, A. Saisi. Ambient vibration testing of historic masonry towers for structuralidentification and damage assessment. Construction and Building Materials,2007,21:1311–1321.
[57] Raid Karoumi.Some Modeling Aspects in the Nonlinear Finite Element Analysis of CableSupport Bridges[J].Computer﹠Structures,1999.
[58]雒应,叶亚丽,杨炳成.斜拉桥变形观测方法及精度分析[J].测绘科学,2006,31(5):67-68.
[59]赵国藩,金伟良,贡金鑫.结构可靠度理论[M]:第1版.北京：中国建筑工业出版社，2000.
[60]沈惠申,高峰.斜拉桥索塔的可靠性分析[J],中国公路学报,1994,(7):40-43.
[61]马保林,尚鑫,徐岳等.混凝土斜拉桥结构健康状态评价模型[J].长安大学学报(自然科学版),2006,26(2):47-50.
[62]胡雄,吉祥,陈兆能等.拉索桥梁安全性与耐久性评价的专家系统设计[J].应用力学学报,1998,15(4):122-126.
[63]张建仁,刘扬.遗传算法和人工神经网络在斜拉桥可靠度分析中的应用[J].土木工程学报,2001,34(1):7-13.
[64]李杨海，鲍卫刚.公路桥梁结构可靠度与概率极限状态设计[M].北京:人民交通出版社,1997.
[65] Lajtal，E.Z.，Brittle fracture in compression.Int，J.Fracture，1974,10:525-536.
[66] Costin，I.S，A microcrack model for the deformation and failure of brittle rock.J.Geophys，1983,88:9485-9492.
[67] Botsus，J.Kunin，B.On self-similarity of crack layer.Int.J.Fracture，1987,35:51-56.
[68]谢和平.岩石、混凝土损伤力学[M].徐州:中国矿业大学出版社,1990.
[69] Mam，I.G.A modified Griffith criterion for the evolution of damage with afractal distribution ofcrack lengths：application to seismic event，Geophys.J.Int.，1991,107:353-362.
[70]谢和平、高峰.岩石损伤过程中的分形描述[J].岩石力学与工程学报,1991(10):55-67.
[71] Bronerg,H.Swedish solid mechanics.Report,1974.
[72] Lemaitre,J.A continuous damage mechanics model for ductile fracture，J.Eng.Material&Tech,1985,107:83-95.
[73] Weibull W.Proc Roy Swedish Inst Eng Res,1939:151-153.
[74] Gumbel E.Statistics of Extremes.Columbia University Press, New York,1958.
[75] Bolotin V V. Statistical Methods in structural Mechanics.Publishing House for Buildingconstruction,Mexico,1965.
[76] Batdorf S B. Fundamentals of the Statistical Theory of Fracture.Edited by R C Batdt，FractMech of Cer,1977.
[77] Jayatilaka A,de S. Fracture of Engineering for Brittle Materials. Applied Science PublisherLTD，London,1979.
[78] Provan J W.概率断裂力学和可靠性[M].北京:航空工业出版社,1989.
[79]高峰,谢和平,赵鹏. Weibull模量和岩石强度的分形性质[J]，科学通报，1993,38（5）:1427-1430.
[80] Mandelbrot， B.B， et al. Fractal character of fracture surfaces of metals.Nature,1984,308:721-723.
[81] Pande,C.S.et al.Fractal characterization of fracture surface.Acta Metal.1987,35(7):1633-1637.
[82] Mecholsky.J.J.&Mackin.T.J.Fractal analysis of fracture in Ocala chert.J. Mat.SCi. Letters,1988,7:1145-1147.
[83] Cahn,R.Fractal dimension and fracture.Nature,1989,338:201-202.
[84] Chen,C.T&Runt,J.Fractal analysis of polystyrene fracture surfaces.Polymer Communications,1989,30:334-335.
[85] Peng.G&Tian.D.The fractal nature of a fracture surface.J，of Physis Math and Genral，1990,23:3257-3261.
[86] Saouma V E, Barton C C, Gamaleldin N A. Fractal characterization of fracture surfaces inconcrete[J]. Engineering Fracture Mechanics,1990,35(1):47-53.
[87] Saouma，V.C&Barton，C.C.Fractals，fracture and size effect in concrete.J of EngineeringMechanics ASCE,1994,120（4）:835-854.
[88] Carpmteri,A.Fractal nature of material microstructure and size effects on apparent mechanicalproperties,Mechanics of Materials,1994,18:89-101.
[89]董连科.分形理论及其应用[M].沈阳:辽宁科学技术出版社,1991.
[90] Wang J. Morphology and mechanical behavior of rock joints: Doctoral Thesis. Gliwice:Silesian Technical University,1994.
[91]洪起超.工程断裂力学基础[M].上海:上海交通大学出版社,1987.
[92]徐世良,殷福新,沈威等.水工地下有压隧洞衬砌结构限裂设计研究[R].大连:大连理工大学,2008.254-307.
[93]谢和平.脆性材料裂纹扩展的分形运动学[J].力学学报.1994,26(6):757-762.
[94] Mandelbrot B B. The fractal geometry of nature[M]. New York: W H Freman,1982.
[95]谢和平.动态裂纹扩展中的分形效应[J].力学学报.1994,27(1):18-27.
[96]陈明宪.斜拉桥建造技术[M]:第一版.北京:人民交通出版社.2005
[97]陈骑彪.斜拉桥索塔混凝土防止开裂的控制方法研究[D]:[硕士学位论文].重庆:重庆交通大学,2010
[98]甘应朋.夷陵大桥斜拉桥主塔下塔柱裂缝分析与处理[J].交通科技.2007(3):51-53
[99]经柏林,李绍林.荆州长江公路大桥北汊北塔下塔柱裂缝控制研究[J].华东公路.2000:26-27.
[100]白兴荣.斜拉桥病害及成因分析[D]:[硕士学位论文].重庆:重庆交通大学土木建筑学院,2008.
[101]黎春林,程华才.铜陵大桥主塔温度裂缝成因分析[J].铜陵学院报,2006(5):76-78.
[102]Al-Gadhib, A.H., Baluch, M.H., Shaalan, A. Khan, A.R. Damage model for monotonic andfatigue response of high strength concrete. International Journal of Damage Mechanics,2000,9.
[103]郑颖人.广义塑性力学理论[J].岩土力学,2000,21(2):188―192.
[104]郑颖人,孔亮.塑性力学中的分量理论-广义塑性力学[J].岩土工程学报,2000,22(3):269-274.
[105]冀晓东,宋玉普,刘建.混凝土冻融损伤本构模型研究[J].计算力学学报,2011,28(3):461-467.
[106]商怀帅,宋玉普,覃丽坤,于长江冻融循环后在三向受压荷载下混凝土性能的试验研究[J].水利学报,2006,37(7).
[107]丁发兴,余志武.基于损伤泊松比的混凝土多轴强度准则[J].固体力学学报,2007,28(1).
[108]王文标,黄晨光,段祝平.失效准则在广义Mohr空间的表示及应用[J].力学学报,2005,37(3).
[109]俞茂宏,刘继明.论岩土材料屈服准则的基本特性和创新[J].岩石力学与工程学报,2007,26(9).
[110]钱七虎,戚承志.岩石、岩体的动力强度与动力破坏准则[J].同济大学学报(自然科学版),2008,36(12).
[111]冷飞,林皋,杜荣强.基于热力学定律的混凝土本构模型研究[J].工程力学,2008,25(2).
[112]韦立德,杨春和,徐卫亚.统计渗流模型和统计损伤本构模型研究[J].岩土力学,2004,25(10).
[113]罗建辉,尚守平,李丽娟.破坏准则的描述新方法[J].湖南大学学报,1999,26(3).
[114]蓝章礼.桥梁健康监测系统的数据采集[D]:[博士学位论文].重庆:重庆大学,2008.
[115]徐兵.基于图像处理技术的桥梁病害检查和裂缝测量研究[D]:[硕士学位论文].陕西:长安大学,2009.
[116]Zhangli Lan, Xiaofan Yang, Jianting Zhou, et al. Novel system for measuring bridgedeflections based on laser and video. Chinese Journal of Scientific Instrument,2009,11:029
[117]Itasca Consulting Group, Inc. PFC (Particle Flow Code), Version3.0. Minneapolis:ICG,2004.
[118]肖辉，刘忠，李春月.混凝土单轴压缩破坏试验的二维离散元数值模拟[J].建筑与结构设计,2010(12):67-70.
[119]周健,屈俊童,贾敏才.混凝土框架倒塌全过程的颗粒流数值模拟[J].地震研究,2005,128(13):288-292.
[120]郑瑞杰.混凝土框架连续性倒塌的3维离散元数值模拟[J].工业建筑,2009,39(增刊):426-430.
[121]许尚杰,尹小涛.基于颗粒流的混凝土材料数值实验研究[J].实验力学,2009,24(3):251-257.
[122]刘庭金,朱合华,莫海鸿.非均质混凝土破坏过程的细观数值试验[J].岩石力学与工程学报,2005,24(22):4120-4132.
[123]尹小涛.岩土材料工程性质数值试验研究[D]:[博士学位论文].武汉:中国科学院武汉岩土力学研究所.
[124]Curtin W A, Millerr R E. Atomistic continuum coupling in computational materials science.Modeling Simulate Material Science Engineer,2003,11(3):33-68.
[125]冯淼林,吴长春.基于三维均匀化方法的复合材料本构数值模拟[J].中国科学技术大学学报,2000,30(6):693-699.
[126]Budiansky. On the elastic moduli of some heterogeneous materials. Joural of Mechanics,Physics and Solids,1965,13:223-227.
[127]Hill R. A self-consistent mechanics of composite materials. Joural of Mechanics,Physics andSolids,1965,12:213-222.
[128]Mori T, Tanaka K. Average stress in matrix and average elastic energy of material withmisfitting inclusions. Acta Metal,1973,21:571-574.
[129]方岱宁,周储伟.有限元计算细观力学对复合材料力学行为的数值分析[J].力学进展,1998,(28):173-188.
[130]Bensoussan A, Lions J L, Papanicolaou G. Asymptotic analysis for periodic structures.Amsterdam: North Holland,1978.
[131]Sanchez-Palencia E. Non-homogeneous media and vibration theory. Berlin: Springer Verlag,1980.
[132]Lions L J. Some methods in the mathematical analysis of system and their control.Beijing:Seince Press,1981.
[133]Guedes J M, Kikuchi N. Preprocessing and postprocessing for materials based on thehomogenization method with adaptive finite element methods. Computer Methods in AppliedMechanics and Engineering,1990,83(2):143-198.
[134]Bendsor M P, Kikuchi N. Generating optimal topologies in structural design using ahomogenization method. Computer Methods in Applied Mechanics and Engineering,1988,71:197-224.
[135]Hassani B,Hinton E. Review of homogenization and topology optimization I-homogenizationtheory for media with periodic structure. Computers and Structures,1998,69(6):707-717.
[136]Hassani B, Hinton E. Review of homogenization and topology optimization II-analytical andnumerical solution of homogenization equations. Computers and Structures,1998,69(6):719-738.
[137]Hassani B, Hinton E. Review of homogenization and topology optimization III-topologyoptimization using optimality criteria. Computers and Structures,1998,69(6):739-756.
[138]Cook R D, Malkus D S, Plesha M E. Concepts and application of finite element analysis. NewYork: Wiley,1989.
[139]Sigmund O. Materials with prescribed constitutive parameters: an inverse homogenizationproblem. Technical Report470.The Technical University of Denmark,1993.
[140]Hassani B. Direct method to derive the boundary conditions of the homogenization equationfor symmetric cells. Communications in Numerical Methods in Engineering,1996,12(3):185.
[141]Hassani B. Finite element solution of the homogenization equation for symmetric cells ofperiodicity. Technical Report CR/887/95.Department of Civil Engineering,1995.
[142]刘兴法．混凝土结构的温度应力分析[M].北京：人民交通出版社，1991．
[143]F凯尔别克．太阳辐射对桥梁结构的影响[M].刘兴法，译．北京：中国铁道出版社，1981．
[144]王效通.预应力混凝土箱梁温度场计算的有限元法[J].西南交通大学学报,1985(3)：52-61．
[145]Prokopski G. Fracture toughness of concretes at high temperature[J]. Journal of MaterialsScience.1995,30:1609-1612.
[146]Abdel-Fattah H., Hamoush S.A. Variation of the fracture toughness of concrete withtemperature[J]. Construction and Building Materials.1997,11(2):105-108.
[147]刘涛,杨凤鹏.精通ANSYS[M].北京:清华大学出版社,2001．
[148]樊启武,钱永久.采用间接耦合法分析斜拉桥索塔瞬态温度应力场[J].石家庄铁道学院学报(自然科学版),2008,21(1):18-22.
[149]曹发辉．斜拉桥施工控制中温度影响研究[D]．[硕士学位论文].成都：西南交通大学土木工程学院，2004．
[150]罗建群,等.水工混凝土建筑物老化病害及防治[M].北京:中国农业科技出版社,1995.47-51.
[151]李安贵,等.模糊数学及其原理[M].北京:冶金工业出版社,2006.251-252.
[152]李天科,等.土石坝老化评价方法研究[J].中国农村水利水电,2007(4):100-102.
[153]朱丽楠等.模糊数学理论在泵站工程老化评价中的应用[J].武汉大学学报,2003,36(4):32-35.
[154]张济忠.分形[M].北京:清华大学出版社,1995.
[155]谢和平.分形岩石力学导论.[M].
[2]黄政宇,纪学灵,黄靓.超声波检测混凝土缺陷成像的定性分析法[J].湖南大学学报35(8),2008(8):5-8.
[3]王登科.超声波检测钻孔桩混凝土质量的缺陷分析和判断[J].公路.1995(9):29-31.
[4] ASTM Standard C1383-98a.“Standard Test Method for Measuring the P-Wave Speed and theThickness of Concrete Plates Using the Impact-Echo Method”. published by the AMERICANSOCIETY FOR TESTING AND MATERIALS (AST M).1-12.
[5]王彬.基于声发射技术的预应力混凝土损伤检测理论及应用[D]:[硕士学位论文].镇江:江苏大学,2006.
[6]陈兵,姚武,吴科如.声发射技术在混凝土研究中的应用[J].无损检测,2000,22(9):387-390.
[7] Chen H L, Cheng C T, Chen S E. Determination of fracture parameters of mortar and concretebeams by using acoustic emission[J]. Materials Evaluation,1992,50(7):888-894.
[8] Ohtsu M. Rate Process Analysis of Acoustic Emission Activity in Core Test of Concrete.Concrete Library, SJCE,1992(20):143-153.
[9] S T Dai, J F Labuz. Damage and failure analysis of brittle materials by acoustic emission.Journal of Materials in Civil Engineering, ASCE,1997,9(4):200-205.
[10] Otsuka K, Date H, Kurita T. Fracture process zone in concrete tension specimens by X-ray andAE techniques. Fracture mechanics of concrete structures. Proceedings FRAMCOS, Germany:AEDIFICATIO Publishers.1998.151-170.
[11]李佳.混凝土静态轴拉声发射试验相关参数研究.[D]:[硕士学位论文].南京：河海大学，2008.
[12] Shigeishi M, Colombob S, Broughtonb K J, etal. Acoustic emission to assess and monitor theintegrity of bridges. Construction and Building Materials,2001,15(1):35-49.
[13]丁幼亮,邓扬,李爱群.声发射技术在桥梁结构健康监测中的应用研究进展[J].防灾减灾工程学报，2010,3(30):341-351.
[14] Sang-Hoon Kim, Jung-Ju Lee and Il-Bum Kwon, Monitoring of Beam Deflection Subjected toBending Load Using Brillouin Distributed Optical Fiber Sensors, Proceedings of SPIE-TheInternational Society for Optical Engineering,2001:63-69.
[15] Chai, J, Wei, S.M, Chang, X.T, Liu, J.X. Monitoring deformation and damage on rockstructures with distributed fiber optical sensing[J], International Journal of Rock Mechanicsand Mining Sciences,2004,41(5):1-6.
[16] Ou Jinping, Hou Shuang, Seismic Damage Identification using multi-line distributed fiberoptic sensor system, Proceedings of SPIE,2005:1003-1008.
[17] Roger G. Duncan, Brooks A. Childers, Dawn K. Gifford, Don E. Pettit, Andrew W. Hickson,Timothy L. Brown, Distributed sensing technique for test article damage detection andMonitoring, Proceedings of SPIE[C],2003:367-375.
[18]杨建良,向清,黄德修.机敏材料与结构中的准分布式光纤传感阵列[J],应用光学,2004,18(4),:33-36.
[19]刘浩吾,谢玲玲.桥梁裂缝监测的光纤传感网络[J],桥梁建设,2003(2),:78-81.
[20]涂亚庆,刘兴长.光纤智能结构[M].北京:高等教育出版社,2005.
[21] Marianne Bérubé Dufour, Dominique Derome, Radu Zmeureanu. Analysis of thermogramsfor the estimation of dimensions of cracks in building envelope. Infrared Physics&Technology,2009(52):70–78.
[22] F. Bakhtiari-Nejad, A. Khorram, M. Rezaeian.Analytical estimation of natural frequencies andmode shapes of a beam having two cracks. International Journal of Mechanical Sciences2014:193–202.
[23] Zhe Cheng, NiaoqingHun, XiaofeiZhang. Crack level estimation approach for planetarygearbox based on simulation signal and GRA. Journal of Sound and Vibration,2012:5853–5863.
[24] Mohammad H.F. Dado, Omar A. Shpli. Crack parameter estimation in structures using finiteelement modeling. International Journal of Solids and Structures,2003:5389–5406.
[25] Ahmed A. Elshafey, Nabil Dawood, H. Marzouk, M. Haddara.Crack width in concrete usingartificial neural networks. Engineering Structures,2013:676–686.
[26] Masayuki Kamaya. Estimation of elastic-plastic fracture toughness by numerical simulationbased on a stress-based criterion for ductile crack initiation. International Journal of PressureVessels and Piping,2013:1-7.
[27] Martin Noel, Khaled Soudki. Estimation of the crack width and deformation of FRP-reinforcedconcrete flexural members with and without transverse shear reinforcement. EngineeringStructures2014:393–398.
[28] Minhhuy Le, Jinyi Lee, Jongwoo Jun, Jungmin Kim.Estimation of sizes of cracks on pipes innuclear power plants using dipole moment and finite element methods. NDT&EInternational,2013:56–63.
[29] Gang Li, Shuanhai He, Yongfeng Ju, Kai Du. Long-distance precision inspection method forbridge cracks with image processing. Automation in Construction,2013.
[30] Ahmed A. Elshafey, Nabil Dawood, H. Marzouk, M. Haddara. Predicting of crack spacing forconcrete by using neural networks. Engineering Failure Analysis2013,31:344–359.
[31] J.H. Espina-Herna′ndez, E.Ramrez-Pacheco. Rapid estimation of artificial near-side crackdimensions in aluminium using a GMR-based eddy current sensor. NDT&E International,2012,51:94–100.
[32] M. Hadj Meliani, Z. Azari, G. Pluvinage, Yu.G. Matvienko. The effective T-stress estimationand crack paths emanating from U-notches. Engineering Fracture Mechanics,2010.77:1682–1692.
[33] Ahmed A. Elshafey, Nabil Dawood. Crack width in concrete using artificial neural networks.Engineering Structures,2013,52:676–686.
[34] Ahmed A. Elshafey. Nabil Dawood.Predicting of crack spacing for concrete by using neuralnetworks. Engineering Failure Analysis,2013,31:344–359.
[35] K.G. Papakonstantinou, M. ShinozukaProbabilistic.model for steel corrosion in reinforcedconcrete structures of large dimensions considering crack effects. Engineering Structures,2013,57:306–326.
[36] Martin No l, Khaled Soudki Estimation of the crack width and deformation of FRP-reinforcedconcrete flexural members with and without transverse shear reinforcement. EngineeringStructures,2014,59:393–398.
[37] J. Bach, H.W. H ppel.Crack initiation mechanisms in AA6082fatigued in the VHCF-regime.International Journal of Fatigue,2014,60:23–27.
[38] Hugo C. Biscaia, Carlos Chastre. A smeared crack analysis of reinforced concrete T-beamsstrengthened with GFRP composites. Engineering Structures,2013,56:1346–1361.
[39] Michael Scharnwebern, Wolfgang Tirschler.Mechanisms of short crack propagation inaustenitic–ferritic duplex steel. Materials Science&Engineering A,2014,595:269–283.
[40] Marta S owik, Piotr Smarzewski.Study of the scale effect on diagonal crack propagation inconcrete beams. Computational Materials Science,2012,64:216–220.
[41] A.K.M. Farid Uddin, Kenichiro Numata. Mechanisms of crack propagation due to corrosion ofreinforcement in concrete by AE-SiGMA and BEM. Construction and Building Materials,2004,18:181–188.
[42]黄贻凤.桥梁混凝土开裂理论及数值分析[D]:[硕士学位论文].武汉:武汉理工大学,2005.
[43]杨海科,李波.浅谈简支梁桥裂缝开展特点及裂缝与承载力的关系[J].东北公路.2003,26(3):101-104.
[44]宋一凡,崔军,赵小星等.钢筋混凝土T型梁桥裂缝特征参数与结构评估试验研究[J].交通运输工程学报,2001,1(3):46-49.
[45]贺拴海，宋一凡，赵小星等.钢筋混凝土梁式结构裂缝特征与损伤评估方法试验研究[J].土木工程学报,2003,36(2):6-9.
[46]周敉，贺拴海，袁万城.RC梁桥承载力的振动测试评估方法[J].交通运输工程学报,2006,6(3):62-67.
[47]赵煜，任伟，李春风等.预应力混凝土简支箱梁裂缝损伤参数影响分析[J].长安大学学报(自然科学版),2010,30(2):58-63.
[48]赵煜，贺拴海，李春风等.在役预应力混凝土箱梁开裂后承载力评估[J].同济大学学报(自然科学版),2010,38(9):1271-1275.
[49]赵煜，周博，贺拴海等.基于裂缝特征的PC简支梁损伤刚度评估方法[J].长安大学学报(自然科学版).2011,31(4):39-44.
[50] Alemdar Bayraktar, Abdurrahman Sahin. Numerical damage assessment of Haghia Sophia belltower by nonlinear FE modeling. Applied Mathematical Modeling,2010,34:92–121.
[51] Anna Anzani, Luiga Binda.A multilevel approach for the damage assessment of Historicmasonry towers. Journal of Cultural Heritage,2010,11:459-470.
[52] Eduardo Nuno Brito Santos Júlio.Structural assessment of the tower of the University ofCoimbra by modal identification. Engineering Structures,2008.30:3468-3477.
[53] Fernando Pe a, Paulo.B. Lourenco.Numerical models for the seismic assessment of an oldmasonry tower. Engineering Structures,2010,32:1466-1478.
[54] Angelo D’Ambrisi, Valentina Mariani. Seismic assessment of a historical masonry tower withnonlinear static and dynamic analyses tuned on ambient vibration tests. EngineeringStructures,2012,36:210-219.
[55] Tian-Hong Pan a, Shyan-Shu Shieh.Statistical multi-model approach for performanceassessment of cooling tower. Energy Conversion and Management,2011,52:1377-1385.
[56] C. Gentile, A. Saisi. Ambient vibration testing of historic masonry towers for structuralidentification and damage assessment. Construction and Building Materials,2007,21:1311–1321.
[57] Raid Karoumi.Some Modeling Aspects in the Nonlinear Finite Element Analysis of CableSupport Bridges[J].Computer﹠Structures,1999.
[58]雒应,叶亚丽,杨炳成.斜拉桥变形观测方法及精度分析[J].测绘科学,2006,31(5):67-68.
[59]赵国藩,金伟良,贡金鑫.结构可靠度理论[M]:第1版.北京：中国建筑工业出版社，2000.
[60]沈惠申,高峰.斜拉桥索塔的可靠性分析[J],中国公路学报,1994,(7):40-43.
[61]马保林,尚鑫,徐岳等.混凝土斜拉桥结构健康状态评价模型[J].长安大学学报(自然科学版),2006,26(2):47-50.
[62]胡雄,吉祥,陈兆能等.拉索桥梁安全性与耐久性评价的专家系统设计[J].应用力学学报,1998,15(4):122-126.
[63]张建仁,刘扬.遗传算法和人工神经网络在斜拉桥可靠度分析中的应用[J].土木工程学报,2001,34(1):7-13.
[64]李杨海，鲍卫刚.公路桥梁结构可靠度与概率极限状态设计[M].北京:人民交通出版社,1997.
[65] Lajtal，E.Z.，Brittle fracture in compression.Int，J.Fracture，1974,10:525-536.
[66] Costin，I.S，A microcrack model for the deformation and failure of brittle rock.J.Geophys，1983,88:9485-9492.
[67] Botsus，J.Kunin，B.On self-similarity of crack layer.Int.J.Fracture，1987,35:51-56.
[68]谢和平.岩石、混凝土损伤力学[M].徐州:中国矿业大学出版社,1990.
[69] Mam，I.G.A modified Griffith criterion for the evolution of damage with afractal distribution ofcrack lengths：application to seismic event，Geophys.J.Int.，1991,107:353-362.
[70]谢和平、高峰.岩石损伤过程中的分形描述[J].岩石力学与工程学报,1991(10):55-67.
[71] Bronerg,H.Swedish solid mechanics.Report,1974.
[72] Lemaitre,J.A continuous damage mechanics model for ductile fracture，J.Eng.Material&Tech,1985,107:83-95.
[73] Weibull W.Proc Roy Swedish Inst Eng Res,1939:151-153.
[74] Gumbel E.Statistics of Extremes.Columbia University Press, New York,1958.
[75] Bolotin V V. Statistical Methods in structural Mechanics.Publishing House for Buildingconstruction,Mexico,1965.
[76] Batdorf S B. Fundamentals of the Statistical Theory of Fracture.Edited by R C Batdt，FractMech of Cer,1977.
[77] Jayatilaka A,de S. Fracture of Engineering for Brittle Materials. Applied Science PublisherLTD，London,1979.
[78] Provan J W.概率断裂力学和可靠性[M].北京:航空工业出版社,1989.
[79]高峰,谢和平,赵鹏. Weibull模量和岩石强度的分形性质[J]，科学通报，1993,38（5）:1427-1430.
[80] Mandelbrot， B.B， et al. Fractal character of fracture surfaces of metals.Nature,1984,308:721-723.
[81] Pande,C.S.et al.Fractal characterization of fracture surface.Acta Metal.1987,35(7):1633-1637.
[82] Mecholsky.J.J.&Mackin.T.J.Fractal analysis of fracture in Ocala chert.J. Mat.SCi. Letters,1988,7:1145-1147.
[83] Cahn,R.Fractal dimension and fracture.Nature,1989,338:201-202.
[84] Chen,C.T&Runt,J.Fractal analysis of polystyrene fracture surfaces.Polymer Communications,1989,30:334-335.
[85] Peng.G&Tian.D.The fractal nature of a fracture surface.J，of Physis Math and Genral，1990,23:3257-3261.
[86] Saouma V E, Barton C C, Gamaleldin N A. Fractal characterization of fracture surfaces inconcrete[J]. Engineering Fracture Mechanics,1990,35(1):47-53.
[87] Saouma，V.C&Barton，C.C.Fractals，fracture and size effect in concrete.J of EngineeringMechanics ASCE,1994,120（4）:835-854.
[88] Carpmteri,A.Fractal nature of material microstructure and size effects on apparent mechanicalproperties,Mechanics of Materials,1994,18:89-101.
[89]董连科.分形理论及其应用[M].沈阳:辽宁科学技术出版社,1991.
[90] Wang J. Morphology and mechanical behavior of rock joints: Doctoral Thesis. Gliwice:Silesian Technical University,1994.
[91]洪起超.工程断裂力学基础[M].上海:上海交通大学出版社,1987.
[92]徐世良,殷福新,沈威等.水工地下有压隧洞衬砌结构限裂设计研究[R].大连:大连理工大学,2008.254-307.
[93]谢和平.脆性材料裂纹扩展的分形运动学[J].力学学报.1994,26(6):757-762.
[94] Mandelbrot B B. The fractal geometry of nature[M]. New York: W H Freman,1982.
[95]谢和平.动态裂纹扩展中的分形效应[J].力学学报.1994,27(1):18-27.
[96]陈明宪.斜拉桥建造技术[M]:第一版.北京:人民交通出版社.2005
[97]陈骑彪.斜拉桥索塔混凝土防止开裂的控制方法研究[D]:[硕士学位论文].重庆:重庆交通大学,2010
[98]甘应朋.夷陵大桥斜拉桥主塔下塔柱裂缝分析与处理[J].交通科技.2007(3):51-53
[99]经柏林,李绍林.荆州长江公路大桥北汊北塔下塔柱裂缝控制研究[J].华东公路.2000:26-27.
[100]白兴荣.斜拉桥病害及成因分析[D]:[硕士学位论文].重庆:重庆交通大学土木建筑学院,2008.
[101]黎春林,程华才.铜陵大桥主塔温度裂缝成因分析[J].铜陵学院报,2006(5):76-78.
[102]Al-Gadhib, A.H., Baluch, M.H., Shaalan, A. Khan, A.R. Damage model for monotonic andfatigue response of high strength concrete. International Journal of Damage Mechanics,2000,9.
[103]郑颖人.广义塑性力学理论[J].岩土力学,2000,21(2):188―192.
[104]郑颖人,孔亮.塑性力学中的分量理论-广义塑性力学[J].岩土工程学报,2000,22(3):269-274.
[105]冀晓东,宋玉普,刘建.混凝土冻融损伤本构模型研究[J].计算力学学报,2011,28(3):461-467.
[106]商怀帅,宋玉普,覃丽坤,于长江冻融循环后在三向受压荷载下混凝土性能的试验研究[J].水利学报,2006,37(7).
[107]丁发兴,余志武.基于损伤泊松比的混凝土多轴强度准则[J].固体力学学报,2007,28(1).
[108]王文标,黄晨光,段祝平.失效准则在广义Mohr空间的表示及应用[J].力学学报,2005,37(3).
[109]俞茂宏,刘继明.论岩土材料屈服准则的基本特性和创新[J].岩石力学与工程学报,2007,26(9).
[110]钱七虎,戚承志.岩石、岩体的动力强度与动力破坏准则[J].同济大学学报(自然科学版),2008,36(12).
[111]冷飞,林皋,杜荣强.基于热力学定律的混凝土本构模型研究[J].工程力学,2008,25(2).
[112]韦立德,杨春和,徐卫亚.统计渗流模型和统计损伤本构模型研究[J].岩土力学,2004,25(10).
[113]罗建辉,尚守平,李丽娟.破坏准则的描述新方法[J].湖南大学学报,1999,26(3).
[114]蓝章礼.桥梁健康监测系统的数据采集[D]:[博士学位论文].重庆:重庆大学,2008.
[115]徐兵.基于图像处理技术的桥梁病害检查和裂缝测量研究[D]:[硕士学位论文].陕西:长安大学,2009.
[116]Zhangli Lan, Xiaofan Yang, Jianting Zhou, et al. Novel system for measuring bridgedeflections based on laser and video. Chinese Journal of Scientific Instrument,2009,11:029
[117]Itasca Consulting Group, Inc. PFC (Particle Flow Code), Version3.0. Minneapolis:ICG,2004.
[118]肖辉，刘忠，李春月.混凝土单轴压缩破坏试验的二维离散元数值模拟[J].建筑与结构设计,2010(12):67-70.
[119]周健,屈俊童,贾敏才.混凝土框架倒塌全过程的颗粒流数值模拟[J].地震研究,2005,128(13):288-292.
[120]郑瑞杰.混凝土框架连续性倒塌的3维离散元数值模拟[J].工业建筑,2009,39(增刊):426-430.
[121]许尚杰,尹小涛.基于颗粒流的混凝土材料数值实验研究[J].实验力学,2009,24(3):251-257.
[122]刘庭金,朱合华,莫海鸿.非均质混凝土破坏过程的细观数值试验[J].岩石力学与工程学报,2005,24(22):4120-4132.
[123]尹小涛.岩土材料工程性质数值试验研究[D]:[博士学位论文].武汉:中国科学院武汉岩土力学研究所.
[124]Curtin W A, Millerr R E. Atomistic continuum coupling in computational materials science.Modeling Simulate Material Science Engineer,2003,11(3):33-68.
[125]冯淼林,吴长春.基于三维均匀化方法的复合材料本构数值模拟[J].中国科学技术大学学报,2000,30(6):693-699.
[126]Budiansky. On the elastic moduli of some heterogeneous materials. Joural of Mechanics,Physics and Solids,1965,13:223-227.
[127]Hill R. A self-consistent mechanics of composite materials. Joural of Mechanics,Physics andSolids,1965,12:213-222.
[128]Mori T, Tanaka K. Average stress in matrix and average elastic energy of material withmisfitting inclusions. Acta Metal,1973,21:571-574.
[129]方岱宁,周储伟.有限元计算细观力学对复合材料力学行为的数值分析[J].力学进展,1998,(28):173-188.
[130]Bensoussan A, Lions J L, Papanicolaou G. Asymptotic analysis for periodic structures.Amsterdam: North Holland,1978.
[131]Sanchez-Palencia E. Non-homogeneous media and vibration theory. Berlin: Springer Verlag,1980.
[132]Lions L J. Some methods in the mathematical analysis of system and their control.Beijing:Seince Press,1981.
[133]Guedes J M, Kikuchi N. Preprocessing and postprocessing for materials based on thehomogenization method with adaptive finite element methods. Computer Methods in AppliedMechanics and Engineering,1990,83(2):143-198.
[134]Bendsor M P, Kikuchi N. Generating optimal topologies in structural design using ahomogenization method. Computer Methods in Applied Mechanics and Engineering,1988,71:197-224.
[135]Hassani B,Hinton E. Review of homogenization and topology optimization I-homogenizationtheory for media with periodic structure. Computers and Structures,1998,69(6):707-717.
[136]Hassani B, Hinton E. Review of homogenization and topology optimization II-analytical andnumerical solution of homogenization equations. Computers and Structures,1998,69(6):719-738.
[137]Hassani B, Hinton E. Review of homogenization and topology optimization III-topologyoptimization using optimality criteria. Computers and Structures,1998,69(6):739-756.
[138]Cook R D, Malkus D S, Plesha M E. Concepts and application of finite element analysis. NewYork: Wiley,1989.
[139]Sigmund O. Materials with prescribed constitutive parameters: an inverse homogenizationproblem. Technical Report470.The Technical University of Denmark,1993.
[140]Hassani B. Direct method to derive the boundary conditions of the homogenization equationfor symmetric cells. Communications in Numerical Methods in Engineering,1996,12(3):185.
[141]Hassani B. Finite element solution of the homogenization equation for symmetric cells ofperiodicity. Technical Report CR/887/95.Department of Civil Engineering,1995.
[142]刘兴法．混凝土结构的温度应力分析[M].北京：人民交通出版社，1991．
[143]F凯尔别克．太阳辐射对桥梁结构的影响[M].刘兴法，译．北京：中国铁道出版社，1981．
[144]王效通.预应力混凝土箱梁温度场计算的有限元法[J].西南交通大学学报,1985(3)：52-61．
[145]Prokopski G. Fracture toughness of concretes at high temperature[J]. Journal of MaterialsScience.1995,30:1609-1612.
[146]Abdel-Fattah H., Hamoush S.A. Variation of the fracture toughness of concrete withtemperature[J]. Construction and Building Materials.1997,11(2):105-108.
[147]刘涛,杨凤鹏.精通ANSYS[M].北京:清华大学出版社,2001．
[148]樊启武,钱永久.采用间接耦合法分析斜拉桥索塔瞬态温度应力场[J].石家庄铁道学院学报(自然科学版),2008,21(1):18-22.
[149]曹发辉．斜拉桥施工控制中温度影响研究[D]．[硕士学位论文].成都：西南交通大学土木工程学院，2004．
[150]罗建群,等.水工混凝土建筑物老化病害及防治[M].北京:中国农业科技出版社,1995.47-51.
[151]李安贵,等.模糊数学及其原理[M].北京:冶金工业出版社,2006.251-252.
[152]李天科,等.土石坝老化评价方法研究[J].中国农村水利水电,2007(4):100-102.
[153]朱丽楠等.模糊数学理论在泵站工程老化评价中的应用[J].武汉大学学报,2003,36(4):32-35.
[154]张济忠.分形[M].北京:清华大学出版社,1995.
[155]谢和平.分形岩石力学导论.[M].