基于非概率理论模型的在役RC桥梁可靠性研究
|
摘要
由于传统的概率可靠性分析方法存在一定的局限性,工程界急需发展一种信息需求量小、计算成本较低、结果偏于保守的结构可靠性分析方法。自上世纪90年代Ben-Haim和Elishakoff提出非概率可靠性概念,历经10余年的发展,国内外学者在该领域做了大量的研究工作,使之成为一个热门研究方向。特别是基于区间模型的非概率可靠性理论,理论上已趋完善,已经应用于岩土、机械及航空领域。但由于算法的不足,制约了该理论在工程界的广泛应用。基于此,本文对区间模型非概率可靠性指标的算法进行了研究,特别注重与有限元数值分析相结合,为非概率可靠性理论的应用奠定基础。
在役桥梁的可靠性评估一直以来是学者和管理部门关心的问题,针对现有评估方法的不足,本文将非概率可靠性评估体系引入到在役桥梁中,实施了基于承载能力极限状态和正常使用极限状态的非概率可靠性评估。在此基础上,完成了在役桥梁的寿命预测及阶段寿命预测。最后,探讨了在役桥梁反优化研究问题,为在役桥梁的维修加固提供技术支撑,并采用概率与非概率混合分析方法对在役桥梁进行了深入探讨。
总之,本文首先解决了非概率可靠性指标的算法问题,在此基础上,将非概率可靠性理论引入到在役桥梁中进行运用,具体的研究内容如下：
(1)针对目前非概率可靠性指标的常规算法及一维优化算法的不足,提出了改进一维优化算法,通过实例验证,证明了改进一维优化算法的有效性及可行性。改进一维优化算法主要针对极限状态方程明确的非概率可靠性指标求解,但当极限状态方程没有解析表达式时,则无法进行求解。因此,文章探讨了基于有限元的非概率响应面法,该方法能有效的求解极限状态方程不明确的结构的非概率可靠性指标。本文还提出了基于有限元的失效点寻优法,该法不需大量的有限元计算,迭代次数少,可广泛应用于结构的非概率可靠性评估。
(2)桥梁是一个复杂的结构,所含的变量较多,若将每个变量按区间变量考虑进行非概率可靠性指标的求解,其工作量极大,甚至有时无法计算。同时,若对桥梁的每个变量都进行测量,需耗大量的人力和财力。因此,为非概率可靠性评估的变量选择提供依据,指导现场测量,同时,判断各参数的单调性,为基于有限元的非概率可靠性指标的算法提供支撑,本文对参数进行了灵敏性分析。
(3)将非概率可靠性理论引入到在役桥梁中,分别建立在役桥梁的承载能力极限状态和正常使用极限状态的非概率可靠性评估,并在实桥中加以运用。
(4)综合考虑混凝土碳化的时变性、混凝土强度的时变性、钢筋锈蚀的时变性、锈蚀后钢筋强度的时变性及汽车人群荷载的时变性,建立了基于承载能力极限状态的动态非概率可靠性模型,并进行寿命预测。同时,基于正常使用极限状态的裂缝宽度及挠度控制,提出了阶段寿命预测的概念。最后,对一实桥分别进行了整体和阶段寿命预测。
(5)对在役桥梁的反优化设计进行了研究,分别建立了限载及加固方案两种反优化模型,通过实例验证了所提方法的有效性。概率与非概率相结合的混合模型将是未来研究的重点方向,在总结现有的混合模型的基础上,本文提出了一种新的概率-非概率混合模型,并对一实桥进行了混合模型的可靠性指标求解,通过比较分析,结果显示本文所提方法具有一定的实用性。
For limitation of traditional probabilistic reliability analysis, a kind of structural reliability analysis with small requirement for information, low calculating costs and conservative results has become an urgent need in the engineering field. Since the 90s of last century, with the concept of non-probabilistic reliability proposed by Ben-Haim and Elishakoff, after 10 years of development, scholars have done a lot of research work in this field and turned it into a hot research direction. Especially for the non-probabilistic reliability theory based on interval model, which has been perfected theoretically and applied in geotechnical, mechanical and aerospace fields. However, for lack of algorithm, wild application of this theory in the engineering filed was limited. On basis of this, the dissertation makes researches on algorithm of interval model non-probabilistic reliability. In addition, by combining finite element analysis, it lays foundation for application of non-probabilistic reliability theory.
Reliability evaluation of existing bridge has long been a concern of scholars and management departments; based on inadequacies of existing assessment methods, this dissertation introduces non-probabilistic reliability evaluation system into existing bridge and makes non-probabilistic reliability evaluation based on limit state of carrying capacity and normal condition. On this basis, prediction for life and stage life of existing bridge is completed. Lastly, the dissertation discusses anti-optimization problem of existing bridges and provides technical supports for maintenance and strengthening of them; in addition, it also makes in-depth discussion for existing bridge by applying probabilistic and non-probabilistic analysis.
In short, the dissertation first solves algorithmic problems of non-probabilistic reliability index. On this basis, it introduces non-probabilistic analysis into existing bridge. Specific researches are as follows:
(1) Due to inadequacies of conventional algorithm and one-dimensional optimization algorithm of non-probabilistic reliability index, it proposes to improve one-dimensional optimization algorithm; through verification of samples, it proves effectiveness and feasibility on improvement of one-dimensional optimization algorithm. Improvement of one-dimensional optimization algorithm mainly aims at solving non-probabilistic reliability index with explicit limit state equation; however, if analytical expression is not available, then, it is unable to have solution. Therefore, this dissertation explores non-probabilistic response surface analysis (RSA) based on finite element; the method can effectively solve non-probabilistic reliability index with undefined structure in limit state equation. The dissertation also proposes Failure Point Optimization Method based on finite element, which is of no needs for large numbers of FEM (Finite Element Method) and iterations. The method can wildly be applied in structural evaluation of non-probabilistic reliability.
(2) Bridge is a complex structure with relatively large variables. If we want to find solution of non-probabilistic reliability index with consideration of each variable as interval variable, the workload will be very large and sometimes it even could not be calculated. Simultaneously, large human and financial resources would be consumed if each variable of bridge were to be measured. Therefore, it provides evidence for variables of non-probabilistic reliability evaluation and guidance to on-site measurement. At one time, by judging monotonicity of each parameter and providing supports for algorithm of non-probabilistic reliability index based on finite element, the dissertation makes sensitivity analysis for parameters.
(3) Introducing non-probabilistic reliability theory into existing bridge, establishing non-probabilistic reliability evaluations for limit state of carrying capacity and normal condition of existing bridge, and apply them in real bridges.
(4) By considering time varying of concrete carbonation, concrete strength, steel corrosion, corroded steel strength and loads of cars and crowds synthetically, the dissertation establishes dynamic non-probabilistic reliability model based on carrying capacity limit state and predicts its life. Simultaneously, based on controls to crack width and deflection of normal limit state, it proposes the concept of stage life prediction. Finally, the dissertation makes prediction to overall and stage life of real bridge respectively.
(5) By making researches to anti-optimization design of existing bridge, it establishes two types of anti-optimization models respectively:load limitation and reinforcement; and verifies effectiveness of the proposed methods via samples. Hybrid model combined by probability and non-probability will be the focus of future researches; based on summarization of existing hybrid model, the dissertation puts forward a kind of new probability and non-probability hybrid model and makes solution for reliability index of hybrid model. Results of coMParative analysis show that methods proposed in this dissertation have a certain amount of practicability.
在役桥梁的可靠性评估一直以来是学者和管理部门关心的问题,针对现有评估方法的不足,本文将非概率可靠性评估体系引入到在役桥梁中,实施了基于承载能力极限状态和正常使用极限状态的非概率可靠性评估。在此基础上,完成了在役桥梁的寿命预测及阶段寿命预测。最后,探讨了在役桥梁反优化研究问题,为在役桥梁的维修加固提供技术支撑,并采用概率与非概率混合分析方法对在役桥梁进行了深入探讨。
总之,本文首先解决了非概率可靠性指标的算法问题,在此基础上,将非概率可靠性理论引入到在役桥梁中进行运用,具体的研究内容如下：
(1)针对目前非概率可靠性指标的常规算法及一维优化算法的不足,提出了改进一维优化算法,通过实例验证,证明了改进一维优化算法的有效性及可行性。改进一维优化算法主要针对极限状态方程明确的非概率可靠性指标求解,但当极限状态方程没有解析表达式时,则无法进行求解。因此,文章探讨了基于有限元的非概率响应面法,该方法能有效的求解极限状态方程不明确的结构的非概率可靠性指标。本文还提出了基于有限元的失效点寻优法,该法不需大量的有限元计算,迭代次数少,可广泛应用于结构的非概率可靠性评估。
(2)桥梁是一个复杂的结构,所含的变量较多,若将每个变量按区间变量考虑进行非概率可靠性指标的求解,其工作量极大,甚至有时无法计算。同时,若对桥梁的每个变量都进行测量,需耗大量的人力和财力。因此,为非概率可靠性评估的变量选择提供依据,指导现场测量,同时,判断各参数的单调性,为基于有限元的非概率可靠性指标的算法提供支撑,本文对参数进行了灵敏性分析。
(3)将非概率可靠性理论引入到在役桥梁中,分别建立在役桥梁的承载能力极限状态和正常使用极限状态的非概率可靠性评估,并在实桥中加以运用。
(4)综合考虑混凝土碳化的时变性、混凝土强度的时变性、钢筋锈蚀的时变性、锈蚀后钢筋强度的时变性及汽车人群荷载的时变性,建立了基于承载能力极限状态的动态非概率可靠性模型,并进行寿命预测。同时,基于正常使用极限状态的裂缝宽度及挠度控制,提出了阶段寿命预测的概念。最后,对一实桥分别进行了整体和阶段寿命预测。
(5)对在役桥梁的反优化设计进行了研究,分别建立了限载及加固方案两种反优化模型,通过实例验证了所提方法的有效性。概率与非概率相结合的混合模型将是未来研究的重点方向,在总结现有的混合模型的基础上,本文提出了一种新的概率-非概率混合模型,并对一实桥进行了混合模型的可靠性指标求解,通过比较分析,结果显示本文所提方法具有一定的实用性。
For limitation of traditional probabilistic reliability analysis, a kind of structural reliability analysis with small requirement for information, low calculating costs and conservative results has become an urgent need in the engineering field. Since the 90s of last century, with the concept of non-probabilistic reliability proposed by Ben-Haim and Elishakoff, after 10 years of development, scholars have done a lot of research work in this field and turned it into a hot research direction. Especially for the non-probabilistic reliability theory based on interval model, which has been perfected theoretically and applied in geotechnical, mechanical and aerospace fields. However, for lack of algorithm, wild application of this theory in the engineering filed was limited. On basis of this, the dissertation makes researches on algorithm of interval model non-probabilistic reliability. In addition, by combining finite element analysis, it lays foundation for application of non-probabilistic reliability theory.
Reliability evaluation of existing bridge has long been a concern of scholars and management departments; based on inadequacies of existing assessment methods, this dissertation introduces non-probabilistic reliability evaluation system into existing bridge and makes non-probabilistic reliability evaluation based on limit state of carrying capacity and normal condition. On this basis, prediction for life and stage life of existing bridge is completed. Lastly, the dissertation discusses anti-optimization problem of existing bridges and provides technical supports for maintenance and strengthening of them; in addition, it also makes in-depth discussion for existing bridge by applying probabilistic and non-probabilistic analysis.
In short, the dissertation first solves algorithmic problems of non-probabilistic reliability index. On this basis, it introduces non-probabilistic analysis into existing bridge. Specific researches are as follows:
(1) Due to inadequacies of conventional algorithm and one-dimensional optimization algorithm of non-probabilistic reliability index, it proposes to improve one-dimensional optimization algorithm; through verification of samples, it proves effectiveness and feasibility on improvement of one-dimensional optimization algorithm. Improvement of one-dimensional optimization algorithm mainly aims at solving non-probabilistic reliability index with explicit limit state equation; however, if analytical expression is not available, then, it is unable to have solution. Therefore, this dissertation explores non-probabilistic response surface analysis (RSA) based on finite element; the method can effectively solve non-probabilistic reliability index with undefined structure in limit state equation. The dissertation also proposes Failure Point Optimization Method based on finite element, which is of no needs for large numbers of FEM (Finite Element Method) and iterations. The method can wildly be applied in structural evaluation of non-probabilistic reliability.
(2) Bridge is a complex structure with relatively large variables. If we want to find solution of non-probabilistic reliability index with consideration of each variable as interval variable, the workload will be very large and sometimes it even could not be calculated. Simultaneously, large human and financial resources would be consumed if each variable of bridge were to be measured. Therefore, it provides evidence for variables of non-probabilistic reliability evaluation and guidance to on-site measurement. At one time, by judging monotonicity of each parameter and providing supports for algorithm of non-probabilistic reliability index based on finite element, the dissertation makes sensitivity analysis for parameters.
(3) Introducing non-probabilistic reliability theory into existing bridge, establishing non-probabilistic reliability evaluations for limit state of carrying capacity and normal condition of existing bridge, and apply them in real bridges.
(4) By considering time varying of concrete carbonation, concrete strength, steel corrosion, corroded steel strength and loads of cars and crowds synthetically, the dissertation establishes dynamic non-probabilistic reliability model based on carrying capacity limit state and predicts its life. Simultaneously, based on controls to crack width and deflection of normal limit state, it proposes the concept of stage life prediction. Finally, the dissertation makes prediction to overall and stage life of real bridge respectively.
(5) By making researches to anti-optimization design of existing bridge, it establishes two types of anti-optimization models respectively:load limitation and reinforcement; and verifies effectiveness of the proposed methods via samples. Hybrid model combined by probability and non-probability will be the focus of future researches; based on summarization of existing hybrid model, the dissertation puts forward a kind of new probability and non-probability hybrid model and makes solution for reliability index of hybrid model. Results of coMParative analysis show that methods proposed in this dissertation have a certain amount of practicability.
引文
[1]陈化新,梅巍.浅议旧桥加固技术.今日科苑,2009(18)：109-110
[2]Elishakoff I. Essay on uncertainties in elastic and viscoelastic structures:form A. M. Freudenthal's criticisms to modern convex modeling. Computers and Strutures, 1995,56(6):871-895
[3]吕震宙,冯蕴雯.结构可靠性问题研究的若干进展.力学进展,2000,30(1)：21-28
[4]孙晓燕.服役期及加固后的钢筋混凝土桥梁可靠性研究：[博士学位论文].大连：大连理工大学,2004
[5]李桂青,李秋胜.工程结构时变可靠度理论及其应用.北京：科学出版社,2001
[6]杨伟军,张建仁,梁兴文.基于动态可靠度的服役桥梁维修加固策略.中国公路学报,2002,15(3)：49-52
[7]孙晓燕,黄承逵,赵国藩等.基于动态可靠度和经济优化相结合的服役桥梁维修加固风险决策.工程力学,2004,21(5)：5-10
[8]Allm I. M., Maslehuddia M., Saricimen H. Al-mana A I. Influence of atmospheric corrosion on the mechanical properties of reinforcing stee. Constr. And Build. Mat., 1994,8(1):35-41
[9]Dimitri V Val, Robert E. Melchers reliability of deteriorating RC slab bridges. Struct. Engrg.,ASCE,1996,123(12):1638-1644
[10]Michael P. Enright, Dan M. Frangopol. Service-life predication of deteriorating concrete bridges. Journal of Structural Engineering,1998,124(3):309-316
[11]Geidl V., Saunders S. Calculation of reliability for timevarying loads and resistance. Structural Safety,1986,4(4):285-292
[12]Amimesh Dey, Sankaran Mahadevan. Reliability estimation with time-variant loads and resistances. Journal of Structural Engineering,2000,126(5):612-620
[13]Stewart M. G, Rosowsky D. V. Time-dependent reliability of deteriorating reinforced concrete buidges. Structural Safety, Amsterdam,1998(20):91-109
[14]王光远.结构服役期间的动态可靠度及其维修理论初探.哈尔滨建筑大学学报,1990(2)：22-25
[15]郝海霞,张建仁.钢筋混凝土板梁桥退化的可靠性分析.国外桥梁,2000(4)：43-48
[16]万世明,赵善锐,黄广胜.桥梁桩基结构系统桩身构件可靠性分析.西南交通大学学报,2000(8)：366-370
[17]Brown C. B. A fuzzy safety measure. J Eng Mech Div,1979(105):855-872
[18]Furuta H. Fuzzy logic and its contribution to reliability analysis. In:Rackwitz R, Angusti G, Borri A(eds.) Reliability and Optimization of Structural Systems. Chapman & Hall, London,1995:61-75
[19]Cai K. Y., Wen C. Y., Zhang M. L. Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context. Fuzzy sets and Systems,1991,42(2): 1454-172
[20]Cremona C., Gao Y. The possibilistic reliability theory:theoretical aspects and applications. Structural Safety,1997,19(21):173-201
[21]Ben-Haim Y. A non-probabilistic concept of reliability. Structural Safety,1994, 14(4):227-245
[22]Ben-Haim Y. A non-probabilistic measure of reliability of linear systems based on expansion of convex models. Structural Safety,1995,17(2):91-109
[23]Elishakoff I. Discussion on a non-probabilistic concept of reliability. Structural Safety,1995,17(3):195-199
[24]Qiu Z. P., Elishakoff I. Anti-optimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput Methods Appl Mech Eng,1998,152(3-4):361-372
[25]Qiu Z. P., Elishakoff I. Anti-optimization technique-a generalization of interval analysis for nonprobabilistic treatment of uncertainty. Chaos Solitons Fractals, 2001,12(9):1747-1759
[26]Lombardi M., Haftka R. T. Anti-optimization technique for structural design under load uncertainties. Comput Methods Appl Mech Eng,1998,157(1-2):19-31
[27]Ganzerli S., Pantelides C. P. Optimum structural design via convex model superposition. Comput Struct,2000(74):639-647
[28]郭书祥,吕震宙,冯元生.基于区间分析的结构非概率可靠性模型.计算力学学报,2001,18(1)：56-62
[29]Au FTK, Cheng Y. S., Tham L. G., et al. Robust design of structures using convex models. Comput Struct,2003,81(28-29):2611-2619
[30]郭书祥,吕震宙.结构的非概率可靠性方法和概率可靠性方法的比较.应用力学学报,2003,20(3)：107-110
[31]Qiu Z. P., Mueller P. C., Frommer A. The new non-probabilistic criterion of failure for dynamical systems based on convex models. Mathematical and Computer Modelling,2004,40(1-2):201-215
[32]邱志平,陈山奇,王晓军.结构非概率鲁棒可靠性准则.计算力学学报,2004,21(1)：1-6
[33]王晓军,邱志平,武哲.结构非概率集合可靠性模型.力学学报,2007,39(5)：641-646
[34]Wang X. J., Qiu Z. P., Elishakoff I. Non-probabilistic Set2theoretic Model for Structural Safety Measure. Acta Mechanica,2008,198(122):51-64
[35]Jiang C., Han X., Liu G R. Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval. Computer Methods in Applied Mechanics and Engineering,2007(196):4791-4800
[36]Jiang C., Han X., Liu G R. A nonlinear interval number programming method for uncertain optimization problems. European Journal of Operational Research,2008, 188(1):1-13
[37]乔心州,仇原鹰,孔宪光.一种基于椭球凸集的结构非概率可靠性模型.工程力学,2009,26(11)：203-208
[38]曹鸿钧,段宝岩.基于凸集合模型的非概率可靠性研究.计算力学学报,2005,22(5)：546-578
[39]Moore R., Yang C. Interval analysis. Technical Document, Lockheed Missiles an Space Division, Number LMSD285875,1959
[40]Moore R. E. Interval Analysis. New York:Englewood, Prentice-Hall,1966
[41]沈祖和.区间分析方法及其应用.应用数学和计算数学,1983(2)：34-58
[42]Rao S. S., Berkel L. Analysis of uncertain structural systems using interval analysis. AIAA J,1997,35(4):727-735
[43]吕震宙,冯蕴雯,岳珠峰.改进的区间截断法及基于区间分析的非概率可靠性分析方法.计算力学学报,2002,19(3)：260-264
[44]赵明华,蒋冲,曹文贵.基于区间理论的挡土墙稳定性非概率可靠性分析.岩土工程学报,2008,30(4)：468-472
[45]Markov S. An iterative method for algebraic solution to interval equations. Applied Numerical Mathematics,1999(30):225-239
[46]郭书祥,吕震宙.区间有限元静力控制方程的一种迭代解法.西北工业大学学报,2002,20(1)：20-23
[47]Dessombz O., Thouverez F., Laine J. P., et al. Analysis of mechanical systems using interval computations applied to finite element methods. Journal of sound and Vibration,2001,239(5):949-968
[48]Qiu Z., Elishakoff I. Anti-optimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput. Methods Appl. Mech. Engrg.,1998,152(3-4):361-372
[49]Su-Huan Chen, Xiao-Wei Yang. Interval finite element method for beam struectures. Finite Elements in Analysis and Design,2000,34(1):75-88
[50]Dessombz O., Thouverez F., Lai Nedandl J-P. Ejd Zed Quel. Analysis of Mechanical Systems Using Interval Computations Applied to Finite Element Methods. Journal of Sound and vibration,2001,239(5):949-965
[51]Sergey P., Shary. Interval Guass-Seidel Method for Generalized Solution Sets to Interval Linear Systems. Reliable Computing,2001,7(2):141-155
[52]Stewart Mcwilliam. Anti-optimization of uncertain structures using interval analysis. Computers and Structures,2001,79(4):421-430
[53]邱志平,顾元宪.有界不确定参数结构位移范围的区间摄动法.应用力学学报,1999,16(1)：1-10
[54]邱志平.不确定参数结构静力响应和特征值间题的区间分析方法：[博士学位论文].吉林：吉林工业大学,1994
[55]苏静波.工程结构不确定性区间分析方法及其应用研究：[博士学位论文].南京：河海大学,2006
[56]郭书祥,吕震宙.区间运算和静力区间有限元.应用数学和力学,2001,22(12)：1249-1254
[57]Koylouglu U., Cakmak A., Nielsen S. Interval algebra to deal with pattern loading and structural uncertainties. Journal of Engineering Mechanics,1995,121(11): 1149-1157
[58]刘世君.岩石力学反演分析研究及工程应用：[博士学位论文].南京：河海大学,2003
[59]喻和平.区间分析理论及其在边坡工程中的应用：[博士学位论文].南京：河海大学,2006
[60]Wu J., Zhao Y. Q., Chen S. H. An improved interval analysis method for uncertain structures. Structural Engineering and Mechanics,2005,20(6):713-726
[61]苏静波,邵国建.基于区间分析的工程结构不确定性研究现状与进展.力学进展,2005,35(3)：338-344
[62]郭书祥,张陵,李颖.结构非概率可靠性指标的求解方法.计算力学学报,2005,22(2)：227-231
[63]Luo Y. J., Kang Z., Luo Z., et al. Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Structural and Multidisciplinary Optimization,2009,39(3):297-310
[64]罗阳军,亢战.超椭球模型下结构非概率可靠性指标的迭代算法.计算力学学报,2008,25(6)：747-752
[65]马超,吕震宙.隐式极限状态方程的非概率可靠性分析.机械强度,2009,31(1)：45-50
[66]江涛,陈建军,张弛江.区间模型非概率可靠性指标的仿射算法.机械强度,2007,29(2)：251-255
[67]江涛,陈建军,张建国等.基于区间摸型的响应面法.机械设计与研究,2005,21(6)：12-16
[68]Jiang Tao, Chen Jian-Jun, Xu Ya-Lan. A semi-analytic method for calculating non-probabilistic reliability index based on interval models. Applied Mathematical Modelling,2007,31(7):1362-1370
[69]赵明华,蒋冲,曹文贵.基于区间理论的挡土墙稳定性非概率可靠性分析.岩土工程学报,2007,40(11)：467-472
[70]张勇,赖国伟,程睿等.高拱坝的非概率可靠性分析.中国农村水利水电,2008(5)：62-65
[71]胡举喜,邱志平.基于区间分析方法的智能桁架结构振动特性分析.航空学报,2007,28(14)：877-880
[72]蒋冲,赵明华,曹文贵.基于区间分析的岩土结构非概率可靠性分析.湖南大学学报(自然科学版),2008,35(3)：11-14
[73]曹文贵,张永杰.基于区间组合法的边坡稳定非概率模糊可靠性分析方法.土木工程学报,2007,40(11)：64-69
[74]张永杰,曹文贵,赵明华.基于承载力和沉降的条形基础稳定性区间非概率可靠性综合分析方法.岩石力学与工程学报,2009,29(4)：805-812
[75]佘玮杰.桥梁结构可靠性研究综述.交通世界,2008(21)：168-169
[76]郭书祥,吕震宙.结构可靠性分析的概率-非概率混合模型.机械强度,2002,24(4)：524-526
[77]王军,邱志平.结构的概率-非概率混合可靠性模型.航空学报,2009,30(8)：1398-1404
[78]钟玉湘,程远胜.同时存在两类不确定性量的最轻结构设计方法.华中科技大学学报(自然科学版),2004,32(3)：43-45
[79]程远胜,钟玉湘,曾广武.基于概率和非概率混合模型的结构鲁棒设计方法.计算力学学报,2005,22(4)：501-505
[80]宋政吉,王慧,王立.相容非概率信息的可靠性理论与方法研究.航天器工程,2007,16(4)：29-35
[81]孙海龙,姚卫星.概率体系和非概率体系下结构元件的可靠性分析方法回顾.机械科学与技术,2007,26(2)：140-146
[82]王向阳,陈建桥.复合材料层合板结构的概率与非概率模型.华中科技大学学报(自然科学版),2006,34(12)：71-73
[83]Xiaoping Du, Agus Sudjianto, Beiqing Huang. Reliability-Based Design With the Mixture of Random and Interval Variables. Transactions of the ASME, Design Engineering Technical Conferences and Computers and Information in Engineering Conference(DETC2003),2003:1067-1076
[84]Du Xiaoping, Sudjianto Agus, Huang Beiqing. Reliability-based design with the mixture of random and interval variables. Journal of Mechanical Design.2005, 127(6):1068-1076
[85]Yangjun Luo, Zhan Kang, Alex Li. Structural reliability assessment based on probability and convex set mixed model. Computers and Structures,2009, 87(21-22):1408-1415
[86]Rui Ge, Jianqiao Chen, Junhong Wei. Reliability-based design of composites under the mixed uncertainties and the optimization algorithm. Acta Mechanica Solida Sinica,2008,21(1):19-27
[87]Penmetsa RC, Grandhi RV. Efficient estimation of structural reliability for problems with uncertain intervals. Comput Struct,2002,80(12):1103-1112
[88]Yangjun Luo, Zhan Kang, Alex Li. Structural reliability assessment based on probability and convex set mixed model. Computers and Structures,2009, 87(21-22):1408-1415
[89]Zhan Kang, Yangjun Luo. Reliability-based structural optimization with probability and convex set hybrid models. Structural and Multidisciplinary Optimization. DOI:10.1007/s00158-009-0461-6
[90]Starks S. A., Kreinovich V, Longpre Ceberio M., et al. Towards Combining Probabilistic and Interval Uncertainty in Engineering Calculations.2004, Proc. of the NSF Workshop on Reliable Engineering Computing, Savannah, GA,15-17 September,2004:193-213
[91]Kreinovich V., Beck J., Ferregut C., et al. Monte-Carlo-Type Techniques for Processing Interval Uncertainty, and Their Engineering Applications. Proc. of the NSF Workshop on Reliable Engineering Computing, Savannah, GA,15-17 September,2004:139-160
[92]江涛.结构系统非概率可靠性算法研究：[博士学位论文].西安：西安电子科技大学,2006
[93]Box G. E. P., Wilson K. B. On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society,1951, Series B:13
[94]Fiessler B., Neumann H. J., Rackwitz R. Quadratic limit states in structural reliability. J Engng Mech ASCE,1979,105 (4):661-676
[95]Melchers R. E. Radial importance sampling for structural reliability. J Engng Mech, ASCE,1990,116(1):189-203
[96]Nowak A. S., Collins K. R. Reliability of Structures. Boston:McGraw-Hill,2000
[97]Carlier J., Li Y., Lutton J. L. Reliability evaluation of large telecommunication networks. Discrete Applied Mathematics,1997,76(1/3):61-80
[98]Olsson A., Sandberg G, Dahlblom O. On latin hypercube sampling for structural reliability analysis. Structural Safety,2003,25 (1):47-68
[99]Melchers R. E., Ahammed M., Middleton C. FORM for discontinuous and truncated probability density functions. Structural Safety,2003,25(3):305-313
[100]Faravelli. Response surface approach for reliability analysis. J Engng Mech ASCE, 1989,115(12):2763-2781
[101]Bucher C. G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety,1990,7(1):57-66
[102]Maymon G Probability of failure of structural without a closed form failure function. Compu Struct,1993,49(2):301-313
[103]Bayer V., Schueller G I. Discussion on:A new look at the response surface approach for reliability. Structural Safety,1994,16(3):227-234
[104]Liu Y. W., Moses F. A sequential response surface method and its application in the reliability analysis of aircraft structural system. Structural Safety,1994,16(1): 39-46
[105]Guan X. L., Melchers R. E. Multitangent-plane surface method for reliability calculation. J Eng Mech,1997,123(10):996-1002
[106]Guan X. L., Melchers R. E. Effect of response surface parameter variation on structural reliability estimates. Structural Safety,2001,23(4):429-444
[107]Das P. K., Zheng Y. Cumulative formation of response surface and its use in reliability analysis. Probabilistic Engineering Mechanics,2000,15(4):309-315
[108]Sankaran Mahadevan et al. Multiple linearization method for nonlinear reliability analysis. J. Engng. Mech. ASCE,2001,127(11):1165-1173
[109]Gupta S., Manohar C. S. An improved response surface method for the determination of failure probability and importance measures. Structural Safety, 2004,26 (2):123-139
[110]Kaymaz I., McMahon C. A. A response surface method based on weighted regression for structural reliability analysis. Probabilistic Engineering Mechanics, 2005,20(1):11-17
[111]Deng J., Gu D. S., Li X. B., et al. Structural reliability analysis for implicit performance function using artificial neural network. Structural Safety,2005,27(1): 25-48
[112]刘成立,吕震宙.非线性隐式极限状态方程失效概率计算的组合响应面法.工程力学,2006,23(2)：29-33
[113]许英,姜华峰,马国山.基于响应面法的边坡稳定可靠性分析.水运工程,2009(11)：55-58
[114]桂劲松,康海贵.结构可靠度分析的响应面法及其Matlab实现.计算力学学报,2004,21(6)：683-687
[115]苏永华,赵明华,蒋德松等.响应面方法在边坡稳定可靠度分析中的应用.岩石力学与工程学报,2006,25(7)：1417-1424
[116]Rajashekhar M. R., Ellingwood B. R. A new look at the response safe approach for reliability analysis. Structural Safety,1993,12(3):205-220
[117]程进.基于响应面法的几何非线性结构概率响应分析.同济大学学报(自然科学版).2006,34(9)：1148-1151
[118]中华人民共和国交通部.公路工程结构可靠度设计统一标准(GB/T 50283-1999). 北京：中国计划出版社,1999
[119]刘玉玲.考虑抗力随时间变化的桥梁可靠度评估研究：[博士学位论文].合肥：合肥工业大学,2009
[120]陈旭勇,樊建平,邓泽贤等.基于动态可靠性加固拱桥剩余寿命预测.公路交通科技,2008,25(10)：84-87
[121]中交公路规划设计院.公路钢筋混凝土及预应力混凝土桥涵设计规范(JTGD62——2004).北京：人民交通出版社,2004
[122]郭书祥,吕震宙.结构体系的非概率可靠性分析方法.计算力学学报,2002,19(3)：332-335
[123]Elishakoff I., Haftka R. T., Fang J. Structural design under bounded uncertainty-optimization with anti-optimization. Computers & Structures,1994, 57(6):1401-1405
[124]Lombardi M. Optimization of uncertain structures using non-probabilistic models. Computers & Structures,1998,67(1-3):99-103
[125]Michael P. Enright, Dan M. Frangopol. Service-life predication of deteriorating concrete bridges. Journal of Structural Engineering,1998,124(3):309-317
[126]赵冬兵.在役钢筋混凝土桥梁承载能力退化数值模拟和维修需求预测：[博士学位论文].上海：同济大学,2005
[127]Smolczyk HG Proc. RILEM Symp. Tsting of Concrete,1962:485-489
[128]Lesage-de-Contenay C. Deterrioration and repair. Bahrain Proc,1995:467-483
[129]牛狄涛,陈亦奇,于澍.混凝土结构的碳化模式与碳化寿命分析.西安建筑科技大学学报,1995,27(4)：365-369
[130]牛荻涛,王庆霖.一般大气环境下混凝土强度经时变化模型.工业建筑,1995,25(6)：36-38
[131]牛荻涛.混凝土结构耐久性与寿命预测.北京：科学出版社,2003
[132]张建仁,刘扬,许福友等.结构可靠度理论及其在桥梁工程中的应用.北京：人民交通出版社,2003：35-38
[2]Elishakoff I. Essay on uncertainties in elastic and viscoelastic structures:form A. M. Freudenthal's criticisms to modern convex modeling. Computers and Strutures, 1995,56(6):871-895
[3]吕震宙,冯蕴雯.结构可靠性问题研究的若干进展.力学进展,2000,30(1)：21-28
[4]孙晓燕.服役期及加固后的钢筋混凝土桥梁可靠性研究：[博士学位论文].大连：大连理工大学,2004
[5]李桂青,李秋胜.工程结构时变可靠度理论及其应用.北京：科学出版社,2001
[6]杨伟军,张建仁,梁兴文.基于动态可靠度的服役桥梁维修加固策略.中国公路学报,2002,15(3)：49-52
[7]孙晓燕,黄承逵,赵国藩等.基于动态可靠度和经济优化相结合的服役桥梁维修加固风险决策.工程力学,2004,21(5)：5-10
[8]Allm I. M., Maslehuddia M., Saricimen H. Al-mana A I. Influence of atmospheric corrosion on the mechanical properties of reinforcing stee. Constr. And Build. Mat., 1994,8(1):35-41
[9]Dimitri V Val, Robert E. Melchers reliability of deteriorating RC slab bridges. Struct. Engrg.,ASCE,1996,123(12):1638-1644
[10]Michael P. Enright, Dan M. Frangopol. Service-life predication of deteriorating concrete bridges. Journal of Structural Engineering,1998,124(3):309-316
[11]Geidl V., Saunders S. Calculation of reliability for timevarying loads and resistance. Structural Safety,1986,4(4):285-292
[12]Amimesh Dey, Sankaran Mahadevan. Reliability estimation with time-variant loads and resistances. Journal of Structural Engineering,2000,126(5):612-620
[13]Stewart M. G, Rosowsky D. V. Time-dependent reliability of deteriorating reinforced concrete buidges. Structural Safety, Amsterdam,1998(20):91-109
[14]王光远.结构服役期间的动态可靠度及其维修理论初探.哈尔滨建筑大学学报,1990(2)：22-25
[15]郝海霞,张建仁.钢筋混凝土板梁桥退化的可靠性分析.国外桥梁,2000(4)：43-48
[16]万世明,赵善锐,黄广胜.桥梁桩基结构系统桩身构件可靠性分析.西南交通大学学报,2000(8)：366-370
[17]Brown C. B. A fuzzy safety measure. J Eng Mech Div,1979(105):855-872
[18]Furuta H. Fuzzy logic and its contribution to reliability analysis. In:Rackwitz R, Angusti G, Borri A(eds.) Reliability and Optimization of Structural Systems. Chapman & Hall, London,1995:61-75
[19]Cai K. Y., Wen C. Y., Zhang M. L. Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context. Fuzzy sets and Systems,1991,42(2): 1454-172
[20]Cremona C., Gao Y. The possibilistic reliability theory:theoretical aspects and applications. Structural Safety,1997,19(21):173-201
[21]Ben-Haim Y. A non-probabilistic concept of reliability. Structural Safety,1994, 14(4):227-245
[22]Ben-Haim Y. A non-probabilistic measure of reliability of linear systems based on expansion of convex models. Structural Safety,1995,17(2):91-109
[23]Elishakoff I. Discussion on a non-probabilistic concept of reliability. Structural Safety,1995,17(3):195-199
[24]Qiu Z. P., Elishakoff I. Anti-optimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput Methods Appl Mech Eng,1998,152(3-4):361-372
[25]Qiu Z. P., Elishakoff I. Anti-optimization technique-a generalization of interval analysis for nonprobabilistic treatment of uncertainty. Chaos Solitons Fractals, 2001,12(9):1747-1759
[26]Lombardi M., Haftka R. T. Anti-optimization technique for structural design under load uncertainties. Comput Methods Appl Mech Eng,1998,157(1-2):19-31
[27]Ganzerli S., Pantelides C. P. Optimum structural design via convex model superposition. Comput Struct,2000(74):639-647
[28]郭书祥,吕震宙,冯元生.基于区间分析的结构非概率可靠性模型.计算力学学报,2001,18(1)：56-62
[29]Au FTK, Cheng Y. S., Tham L. G., et al. Robust design of structures using convex models. Comput Struct,2003,81(28-29):2611-2619
[30]郭书祥,吕震宙.结构的非概率可靠性方法和概率可靠性方法的比较.应用力学学报,2003,20(3)：107-110
[31]Qiu Z. P., Mueller P. C., Frommer A. The new non-probabilistic criterion of failure for dynamical systems based on convex models. Mathematical and Computer Modelling,2004,40(1-2):201-215
[32]邱志平,陈山奇,王晓军.结构非概率鲁棒可靠性准则.计算力学学报,2004,21(1)：1-6
[33]王晓军,邱志平,武哲.结构非概率集合可靠性模型.力学学报,2007,39(5)：641-646
[34]Wang X. J., Qiu Z. P., Elishakoff I. Non-probabilistic Set2theoretic Model for Structural Safety Measure. Acta Mechanica,2008,198(122):51-64
[35]Jiang C., Han X., Liu G R. Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval. Computer Methods in Applied Mechanics and Engineering,2007(196):4791-4800
[36]Jiang C., Han X., Liu G R. A nonlinear interval number programming method for uncertain optimization problems. European Journal of Operational Research,2008, 188(1):1-13
[37]乔心州,仇原鹰,孔宪光.一种基于椭球凸集的结构非概率可靠性模型.工程力学,2009,26(11)：203-208
[38]曹鸿钧,段宝岩.基于凸集合模型的非概率可靠性研究.计算力学学报,2005,22(5)：546-578
[39]Moore R., Yang C. Interval analysis. Technical Document, Lockheed Missiles an Space Division, Number LMSD285875,1959
[40]Moore R. E. Interval Analysis. New York:Englewood, Prentice-Hall,1966
[41]沈祖和.区间分析方法及其应用.应用数学和计算数学,1983(2)：34-58
[42]Rao S. S., Berkel L. Analysis of uncertain structural systems using interval analysis. AIAA J,1997,35(4):727-735
[43]吕震宙,冯蕴雯,岳珠峰.改进的区间截断法及基于区间分析的非概率可靠性分析方法.计算力学学报,2002,19(3)：260-264
[44]赵明华,蒋冲,曹文贵.基于区间理论的挡土墙稳定性非概率可靠性分析.岩土工程学报,2008,30(4)：468-472
[45]Markov S. An iterative method for algebraic solution to interval equations. Applied Numerical Mathematics,1999(30):225-239
[46]郭书祥,吕震宙.区间有限元静力控制方程的一种迭代解法.西北工业大学学报,2002,20(1)：20-23
[47]Dessombz O., Thouverez F., Laine J. P., et al. Analysis of mechanical systems using interval computations applied to finite element methods. Journal of sound and Vibration,2001,239(5):949-968
[48]Qiu Z., Elishakoff I. Anti-optimization of structures with large uncertain-but-non-random parameters via interval analysis. Comput. Methods Appl. Mech. Engrg.,1998,152(3-4):361-372
[49]Su-Huan Chen, Xiao-Wei Yang. Interval finite element method for beam struectures. Finite Elements in Analysis and Design,2000,34(1):75-88
[50]Dessombz O., Thouverez F., Lai Nedandl J-P. Ejd Zed Quel. Analysis of Mechanical Systems Using Interval Computations Applied to Finite Element Methods. Journal of Sound and vibration,2001,239(5):949-965
[51]Sergey P., Shary. Interval Guass-Seidel Method for Generalized Solution Sets to Interval Linear Systems. Reliable Computing,2001,7(2):141-155
[52]Stewart Mcwilliam. Anti-optimization of uncertain structures using interval analysis. Computers and Structures,2001,79(4):421-430
[53]邱志平,顾元宪.有界不确定参数结构位移范围的区间摄动法.应用力学学报,1999,16(1)：1-10
[54]邱志平.不确定参数结构静力响应和特征值间题的区间分析方法：[博士学位论文].吉林：吉林工业大学,1994
[55]苏静波.工程结构不确定性区间分析方法及其应用研究：[博士学位论文].南京：河海大学,2006
[56]郭书祥,吕震宙.区间运算和静力区间有限元.应用数学和力学,2001,22(12)：1249-1254
[57]Koylouglu U., Cakmak A., Nielsen S. Interval algebra to deal with pattern loading and structural uncertainties. Journal of Engineering Mechanics,1995,121(11): 1149-1157
[58]刘世君.岩石力学反演分析研究及工程应用：[博士学位论文].南京：河海大学,2003
[59]喻和平.区间分析理论及其在边坡工程中的应用：[博士学位论文].南京：河海大学,2006
[60]Wu J., Zhao Y. Q., Chen S. H. An improved interval analysis method for uncertain structures. Structural Engineering and Mechanics,2005,20(6):713-726
[61]苏静波,邵国建.基于区间分析的工程结构不确定性研究现状与进展.力学进展,2005,35(3)：338-344
[62]郭书祥,张陵,李颖.结构非概率可靠性指标的求解方法.计算力学学报,2005,22(2)：227-231
[63]Luo Y. J., Kang Z., Luo Z., et al. Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Structural and Multidisciplinary Optimization,2009,39(3):297-310
[64]罗阳军,亢战.超椭球模型下结构非概率可靠性指标的迭代算法.计算力学学报,2008,25(6)：747-752
[65]马超,吕震宙.隐式极限状态方程的非概率可靠性分析.机械强度,2009,31(1)：45-50
[66]江涛,陈建军,张弛江.区间模型非概率可靠性指标的仿射算法.机械强度,2007,29(2)：251-255
[67]江涛,陈建军,张建国等.基于区间摸型的响应面法.机械设计与研究,2005,21(6)：12-16
[68]Jiang Tao, Chen Jian-Jun, Xu Ya-Lan. A semi-analytic method for calculating non-probabilistic reliability index based on interval models. Applied Mathematical Modelling,2007,31(7):1362-1370
[69]赵明华,蒋冲,曹文贵.基于区间理论的挡土墙稳定性非概率可靠性分析.岩土工程学报,2007,40(11)：467-472
[70]张勇,赖国伟,程睿等.高拱坝的非概率可靠性分析.中国农村水利水电,2008(5)：62-65
[71]胡举喜,邱志平.基于区间分析方法的智能桁架结构振动特性分析.航空学报,2007,28(14)：877-880
[72]蒋冲,赵明华,曹文贵.基于区间分析的岩土结构非概率可靠性分析.湖南大学学报(自然科学版),2008,35(3)：11-14
[73]曹文贵,张永杰.基于区间组合法的边坡稳定非概率模糊可靠性分析方法.土木工程学报,2007,40(11)：64-69
[74]张永杰,曹文贵,赵明华.基于承载力和沉降的条形基础稳定性区间非概率可靠性综合分析方法.岩石力学与工程学报,2009,29(4)：805-812
[75]佘玮杰.桥梁结构可靠性研究综述.交通世界,2008(21)：168-169
[76]郭书祥,吕震宙.结构可靠性分析的概率-非概率混合模型.机械强度,2002,24(4)：524-526
[77]王军,邱志平.结构的概率-非概率混合可靠性模型.航空学报,2009,30(8)：1398-1404
[78]钟玉湘,程远胜.同时存在两类不确定性量的最轻结构设计方法.华中科技大学学报(自然科学版),2004,32(3)：43-45
[79]程远胜,钟玉湘,曾广武.基于概率和非概率混合模型的结构鲁棒设计方法.计算力学学报,2005,22(4)：501-505
[80]宋政吉,王慧,王立.相容非概率信息的可靠性理论与方法研究.航天器工程,2007,16(4)：29-35
[81]孙海龙,姚卫星.概率体系和非概率体系下结构元件的可靠性分析方法回顾.机械科学与技术,2007,26(2)：140-146
[82]王向阳,陈建桥.复合材料层合板结构的概率与非概率模型.华中科技大学学报(自然科学版),2006,34(12)：71-73
[83]Xiaoping Du, Agus Sudjianto, Beiqing Huang. Reliability-Based Design With the Mixture of Random and Interval Variables. Transactions of the ASME, Design Engineering Technical Conferences and Computers and Information in Engineering Conference(DETC2003),2003:1067-1076
[84]Du Xiaoping, Sudjianto Agus, Huang Beiqing. Reliability-based design with the mixture of random and interval variables. Journal of Mechanical Design.2005, 127(6):1068-1076
[85]Yangjun Luo, Zhan Kang, Alex Li. Structural reliability assessment based on probability and convex set mixed model. Computers and Structures,2009, 87(21-22):1408-1415
[86]Rui Ge, Jianqiao Chen, Junhong Wei. Reliability-based design of composites under the mixed uncertainties and the optimization algorithm. Acta Mechanica Solida Sinica,2008,21(1):19-27
[87]Penmetsa RC, Grandhi RV. Efficient estimation of structural reliability for problems with uncertain intervals. Comput Struct,2002,80(12):1103-1112
[88]Yangjun Luo, Zhan Kang, Alex Li. Structural reliability assessment based on probability and convex set mixed model. Computers and Structures,2009, 87(21-22):1408-1415
[89]Zhan Kang, Yangjun Luo. Reliability-based structural optimization with probability and convex set hybrid models. Structural and Multidisciplinary Optimization. DOI:10.1007/s00158-009-0461-6
[90]Starks S. A., Kreinovich V, Longpre Ceberio M., et al. Towards Combining Probabilistic and Interval Uncertainty in Engineering Calculations.2004, Proc. of the NSF Workshop on Reliable Engineering Computing, Savannah, GA,15-17 September,2004:193-213
[91]Kreinovich V., Beck J., Ferregut C., et al. Monte-Carlo-Type Techniques for Processing Interval Uncertainty, and Their Engineering Applications. Proc. of the NSF Workshop on Reliable Engineering Computing, Savannah, GA,15-17 September,2004:139-160
[92]江涛.结构系统非概率可靠性算法研究：[博士学位论文].西安：西安电子科技大学,2006
[93]Box G. E. P., Wilson K. B. On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society,1951, Series B:13
[94]Fiessler B., Neumann H. J., Rackwitz R. Quadratic limit states in structural reliability. J Engng Mech ASCE,1979,105 (4):661-676
[95]Melchers R. E. Radial importance sampling for structural reliability. J Engng Mech, ASCE,1990,116(1):189-203
[96]Nowak A. S., Collins K. R. Reliability of Structures. Boston:McGraw-Hill,2000
[97]Carlier J., Li Y., Lutton J. L. Reliability evaluation of large telecommunication networks. Discrete Applied Mathematics,1997,76(1/3):61-80
[98]Olsson A., Sandberg G, Dahlblom O. On latin hypercube sampling for structural reliability analysis. Structural Safety,2003,25 (1):47-68
[99]Melchers R. E., Ahammed M., Middleton C. FORM for discontinuous and truncated probability density functions. Structural Safety,2003,25(3):305-313
[100]Faravelli. Response surface approach for reliability analysis. J Engng Mech ASCE, 1989,115(12):2763-2781
[101]Bucher C. G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety,1990,7(1):57-66
[102]Maymon G Probability of failure of structural without a closed form failure function. Compu Struct,1993,49(2):301-313
[103]Bayer V., Schueller G I. Discussion on:A new look at the response surface approach for reliability. Structural Safety,1994,16(3):227-234
[104]Liu Y. W., Moses F. A sequential response surface method and its application in the reliability analysis of aircraft structural system. Structural Safety,1994,16(1): 39-46
[105]Guan X. L., Melchers R. E. Multitangent-plane surface method for reliability calculation. J Eng Mech,1997,123(10):996-1002
[106]Guan X. L., Melchers R. E. Effect of response surface parameter variation on structural reliability estimates. Structural Safety,2001,23(4):429-444
[107]Das P. K., Zheng Y. Cumulative formation of response surface and its use in reliability analysis. Probabilistic Engineering Mechanics,2000,15(4):309-315
[108]Sankaran Mahadevan et al. Multiple linearization method for nonlinear reliability analysis. J. Engng. Mech. ASCE,2001,127(11):1165-1173
[109]Gupta S., Manohar C. S. An improved response surface method for the determination of failure probability and importance measures. Structural Safety, 2004,26 (2):123-139
[110]Kaymaz I., McMahon C. A. A response surface method based on weighted regression for structural reliability analysis. Probabilistic Engineering Mechanics, 2005,20(1):11-17
[111]Deng J., Gu D. S., Li X. B., et al. Structural reliability analysis for implicit performance function using artificial neural network. Structural Safety,2005,27(1): 25-48
[112]刘成立,吕震宙.非线性隐式极限状态方程失效概率计算的组合响应面法.工程力学,2006,23(2)：29-33
[113]许英,姜华峰,马国山.基于响应面法的边坡稳定可靠性分析.水运工程,2009(11)：55-58
[114]桂劲松,康海贵.结构可靠度分析的响应面法及其Matlab实现.计算力学学报,2004,21(6)：683-687
[115]苏永华,赵明华,蒋德松等.响应面方法在边坡稳定可靠度分析中的应用.岩石力学与工程学报,2006,25(7)：1417-1424
[116]Rajashekhar M. R., Ellingwood B. R. A new look at the response safe approach for reliability analysis. Structural Safety,1993,12(3):205-220
[117]程进.基于响应面法的几何非线性结构概率响应分析.同济大学学报(自然科学版).2006,34(9)：1148-1151
[118]中华人民共和国交通部.公路工程结构可靠度设计统一标准(GB/T 50283-1999). 北京：中国计划出版社,1999
[119]刘玉玲.考虑抗力随时间变化的桥梁可靠度评估研究：[博士学位论文].合肥：合肥工业大学,2009
[120]陈旭勇,樊建平,邓泽贤等.基于动态可靠性加固拱桥剩余寿命预测.公路交通科技,2008,25(10)：84-87
[121]中交公路规划设计院.公路钢筋混凝土及预应力混凝土桥涵设计规范(JTGD62——2004).北京：人民交通出版社,2004
[122]郭书祥,吕震宙.结构体系的非概率可靠性分析方法.计算力学学报,2002,19(3)：332-335
[123]Elishakoff I., Haftka R. T., Fang J. Structural design under bounded uncertainty-optimization with anti-optimization. Computers & Structures,1994, 57(6):1401-1405
[124]Lombardi M. Optimization of uncertain structures using non-probabilistic models. Computers & Structures,1998,67(1-3):99-103
[125]Michael P. Enright, Dan M. Frangopol. Service-life predication of deteriorating concrete bridges. Journal of Structural Engineering,1998,124(3):309-317
[126]赵冬兵.在役钢筋混凝土桥梁承载能力退化数值模拟和维修需求预测：[博士学位论文].上海：同济大学,2005
[127]Smolczyk HG Proc. RILEM Symp. Tsting of Concrete,1962:485-489
[128]Lesage-de-Contenay C. Deterrioration and repair. Bahrain Proc,1995:467-483
[129]牛狄涛,陈亦奇,于澍.混凝土结构的碳化模式与碳化寿命分析.西安建筑科技大学学报,1995,27(4)：365-369
[130]牛荻涛,王庆霖.一般大气环境下混凝土强度经时变化模型.工业建筑,1995,25(6)：36-38
[131]牛荻涛.混凝土结构耐久性与寿命预测.北京：科学出版社,2003
[132]张建仁,刘扬,许福友等.结构可靠度理论及其在桥梁工程中的应用.北京：人民交通出版社,2003：35-38