边坡稳定的逆可靠度分析方法
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摘要
边坡的稳定性是岩土工程所关注的重要问题之一。目前边坡稳定性分析方法很多,归结起来可分为两类,即确定性方法和不确定性方法。确定性方法如极限平衡法,作为边坡稳定性分析评价的传统方法,通过安全系数来定量评价边坡的稳定性,由于安全系数的直观性,因而被工程界广泛应用。但边坡工程设计和分析中涉及大量不确定性因素,而极限平衡法作为确定性分析方法,忽略了不确定性的影响,因而安全系数难以真实反映评价边坡的稳定性。可靠度分析方法作为不确定性方法,以可靠度指标或失效概率来评价边坡稳定性,在边坡的稳定性分析中能考虑边坡材料的不确定性因素,较极限平衡法更具合理性,但工程应用不方便。针对上述问题,将逆可靠度分析方法引入边坡的稳定分析评价中,在一定可靠度水平下,求得边坡稳定安全系数,融合安全系数的简单明了与可靠度指标的合理性,综合评价边坡的稳定性,这也符合运用多种方法综合评价边坡稳定性的发展趋势。
本文阐述边坡稳定分析极限平衡法和可靠度分析法,并在可靠度分析理论的基础深入探讨了逆可靠度问题及逆可靠度各种分析方法,对基于验算点法的一次逆可靠度分析方法进行了详细介绍。然后在上述理论基础上,建立了基于瑞典条分法和Bishop法圆弧滑动面边坡稳定逆可靠度分析求解格式,并对算例进行了分析探讨,验证了逆可靠度法用于边坡稳定性分析的有效性和可行性。在此基础,将逆可靠度分析方法推广到任意滑动面边坡稳定性评价分析中,建立Janbu、Spencer和郎畏勒模式下逆可靠度分析的求解格式及计算步骤。
在上述分析的基础上,利用Visual Basic6.0编制了Windows环境下运行的可视化的边坡稳定逆可靠度分析程序。该程序操作简便,适用于多岩土层非均质任意形状滑裂面边坡稳定分析,具有边坡稳定确定性分析和逆可靠度分析两大功能,便于计算结果的比较分析。
然后运用逆可靠度分析程序对多个算例和工程实例进行分析讨论,并将分析结果与确定性分析结果进行对比,表明在边坡的稳定性分析中,忽略岩土参数的变异将会导致过高估计边坡的稳定性,使边坡偏于不安全。将逆可靠度法用于边坡的稳定性评价既满足了给定的可靠度水平,又考虑岩土参数的不确定性,更能反映边坡的真实情况,同时又以安全系数来评价边坡的稳定性,便于工程应用。
The stability analysis of slope is one of the most important questions in geotechnical engineering. Many methods of slope stability analysis can be summarized into two categories:deterministic methods and probabilistic methods. Deterministic methods, such as Limit equilibrium method which is traditional method in slope stability analysis uses factor of safety to evaluate stability of slopes, are the basic method of slope stability analysis. Because of factor of safety brief and easy to understand, it is widely used in engineering. But many sources of uncertainty are involved in the analysis; Factor of safety hardly reflects stability of slope truly. The reliability method as a method of probabilistic methods evaluates slope stability by reliability index or probability of failure. The reliability method considers the uncertainties of soil properties in slope stability analysis, so compared with limit equilibrium method is more reasonable, but its application is limited because of imperfect in theory. Address these issues, the inverse reliability method is proposed in slope stability analysis, calculate factor of safety at a given reliability index. Thus, the proposed method learns the advantages of both above method:briefness of Limit equilibrium method and reasonable of reliability method, to composite evaluate slope stability. This is also consistent with trends that the use of various methods to estimate slope stability.
In this paper, it introduces the basic theories of stability analysis and reliability analysis,firstly. On the basis of these, take deeply study of the inverse reliability method, and take much time to introduce the inverse first order reliability method (FORM). Based on the above mentioned theory, schemes of inverse reliability analysis at Terzaghi Model and Bishop Model are established, which are in connection with circular slip surface. Some examples have been used to test and verify feasibility of the inverse reliability in slope stability analysis. Then, the method is extended to stability analysis of non-circular slip surface slope. Based on Janbu and Spencer Models, the format and calculation steps are establish for inverse reliability analysis.
Based on the above analysis, a program using Visual Basic 6.0 language has been developed and it can run in Windows environment. The program can use the inverse reliability method and Limit equilibrium method to evaluate stability of slope with multilayer and arbitrary slip surface. It facilitates comparison of results.
Then, several examples are discussed by the program of the inverse reliability analysis, and the results will be compared with the deterministic analysis, which show that slope stability analysis, ignoring the variability of geotechnical parameters, will lead to overestimation of the slope stability, so that slope is unsafe. The inverse reliability method developed to slope stability evaluation not only meet a given reliability level, but also take into account the uncertainty of geotechnical parameters, to better reflect the real situation of slope. At the same time, it is convenient to application and evaluates the slope stability by safety factor in engineering project.
本文阐述边坡稳定分析极限平衡法和可靠度分析法,并在可靠度分析理论的基础深入探讨了逆可靠度问题及逆可靠度各种分析方法,对基于验算点法的一次逆可靠度分析方法进行了详细介绍。然后在上述理论基础上,建立了基于瑞典条分法和Bishop法圆弧滑动面边坡稳定逆可靠度分析求解格式,并对算例进行了分析探讨,验证了逆可靠度法用于边坡稳定性分析的有效性和可行性。在此基础,将逆可靠度分析方法推广到任意滑动面边坡稳定性评价分析中,建立Janbu、Spencer和郎畏勒模式下逆可靠度分析的求解格式及计算步骤。
在上述分析的基础上,利用Visual Basic6.0编制了Windows环境下运行的可视化的边坡稳定逆可靠度分析程序。该程序操作简便,适用于多岩土层非均质任意形状滑裂面边坡稳定分析,具有边坡稳定确定性分析和逆可靠度分析两大功能,便于计算结果的比较分析。
然后运用逆可靠度分析程序对多个算例和工程实例进行分析讨论,并将分析结果与确定性分析结果进行对比,表明在边坡的稳定性分析中,忽略岩土参数的变异将会导致过高估计边坡的稳定性,使边坡偏于不安全。将逆可靠度法用于边坡的稳定性评价既满足了给定的可靠度水平,又考虑岩土参数的不确定性,更能反映边坡的真实情况,同时又以安全系数来评价边坡的稳定性,便于工程应用。
The stability analysis of slope is one of the most important questions in geotechnical engineering. Many methods of slope stability analysis can be summarized into two categories:deterministic methods and probabilistic methods. Deterministic methods, such as Limit equilibrium method which is traditional method in slope stability analysis uses factor of safety to evaluate stability of slopes, are the basic method of slope stability analysis. Because of factor of safety brief and easy to understand, it is widely used in engineering. But many sources of uncertainty are involved in the analysis; Factor of safety hardly reflects stability of slope truly. The reliability method as a method of probabilistic methods evaluates slope stability by reliability index or probability of failure. The reliability method considers the uncertainties of soil properties in slope stability analysis, so compared with limit equilibrium method is more reasonable, but its application is limited because of imperfect in theory. Address these issues, the inverse reliability method is proposed in slope stability analysis, calculate factor of safety at a given reliability index. Thus, the proposed method learns the advantages of both above method:briefness of Limit equilibrium method and reasonable of reliability method, to composite evaluate slope stability. This is also consistent with trends that the use of various methods to estimate slope stability.
In this paper, it introduces the basic theories of stability analysis and reliability analysis,firstly. On the basis of these, take deeply study of the inverse reliability method, and take much time to introduce the inverse first order reliability method (FORM). Based on the above mentioned theory, schemes of inverse reliability analysis at Terzaghi Model and Bishop Model are established, which are in connection with circular slip surface. Some examples have been used to test and verify feasibility of the inverse reliability in slope stability analysis. Then, the method is extended to stability analysis of non-circular slip surface slope. Based on Janbu and Spencer Models, the format and calculation steps are establish for inverse reliability analysis.
Based on the above analysis, a program using Visual Basic 6.0 language has been developed and it can run in Windows environment. The program can use the inverse reliability method and Limit equilibrium method to evaluate stability of slope with multilayer and arbitrary slip surface. It facilitates comparison of results.
Then, several examples are discussed by the program of the inverse reliability analysis, and the results will be compared with the deterministic analysis, which show that slope stability analysis, ignoring the variability of geotechnical parameters, will lead to overestimation of the slope stability, so that slope is unsafe. The inverse reliability method developed to slope stability evaluation not only meet a given reliability level, but also take into account the uncertainty of geotechnical parameters, to better reflect the real situation of slope. At the same time, it is convenient to application and evaluates the slope stability by safety factor in engineering project.
引文
[1]陈祖煜.土质边坡稳定性分析-原理·方法·程序.北京：中国水利水电出版社,2003,1-376
[2]谢定义,姚仰平,党发宁.高等土力学.北京：高等教育出版社,2007,268-396
[3]贡金鑫.工程结构可靠度计算方法.大连：大连理工大学出版社,2003,1-173
[4]陈祖煜.土质边坡稳定分析程序STAB2005使用平台.北京：中国水利水电科学研究院,2005,11-28
[5]张鲁渝,郑颖人,时卫民.边坡稳定分析中关于不平衡推力法的讨论.岩石力学与工程学报,2005,24(1)：177-182
[6]时卫民,郑颖人,唐伯明等.边坡稳定不平衡推力法的精度分析及其使用条件.岩土工程学报,2004,26(3)：313-317
[7]苏爱军,冯明权.滑坡稳定性传递系数计算法的改进.地质灾害与环境保护,2002,9(3)：51-55
[8]Hovland H J. Three dimensional slope stability analysis method. Journal of Geotechnical the Engineering Division, ASCE,1997,103(9):971-986
[9]蒋臻蔚.三维剩余推力法初探-黑河右坝肩变形体稳定性评价例析：[西安工程学院硕士学位论文].西安：西安工程学院,2000,24-38
[10]杜建成,黄大寿,胡定.边坡稳定的三维极限平衡分析法.四川大学学报(工程科学版),2001,33(4)：9-12
[11]陈祖煜,弥宏亮,汪小刚.边坡稳定三维分析的极限平衡方法.岩土工程学报,2001,23(5)：525-529
[12]弥宏亮,陈祖煜,张发明等.边坡稳定三维极限分析方法及工程应用.岩土力学,2002,23(5)：649-653
[13]Wei W B, Cheng Y M, Li L. Three-dimensional slope failure analysis by the strength reduction and limit equilibrium methods. Computers and Geotechnics, 2009,36(1-2):70-80
[14]张均锋,丁桦.边坡稳定性分析的三维极限平衡法及应用.岩石力学与工程学报,2005,24(3)：365-370
[15]Griffiths D V, Lane P A. Slope stability analysis by finite elements. Geotechnique,1999,49(3):387-403
[16]宋二祥.土工结构安全系数的有限元计算.岩土工程学报,1997,19(2)：1-7
[17]郑颖人,赵尚毅,张鲁渝.用有限元强度折减法进行边坡稳定分析.中国工程科 学,2002,10(4)：57-61
[18]郑颖人,赵尚毅.用有限元强度折减法在土坡及岩坡中的应用.岩土工程学报,2004,23(19)：3379-3381
[19]郑颖人,赵尚毅.岩土工程极限分析有限元法及其应用.土木工程学报,2005,38(1)：91-99
[20]赵尚毅,郑颖人,张玉芳.有限元强度折减法中边坡失稳的判据探讨.岩土力学,2005,26(2)：332-336
[21]门玉明.土坡稳定性的极限分析方法.西安地质学院学报,1996,18(2)：58-63
[22]Donald I B, Chen Zuyu. Slope stability by an upper bound plasticity method. Canadian Geotechnical Journal,1997,34(11):853-862
[23]李国英,沈珠江.下限原理有限单元法及其在土工问题中的应用.岩土工程学报,1997,19(5)：84-89
[24]Vanmarcke E H. Reliability of earth slopes. Journal of the Geotechnical Engineering Division, ASCE 1977,103(11):1227-1246
[25]Revilla J, Castillo E. The calculus of variations applied to stability of slopes. Geoteehnique,1977,27(1):1-11
[26]马忠政,祁红卫,侯学渊.边坡稳定验算中全面搜索的一种新方法.岩土力学,2000,21(3)：256-259
[27]肖专文,张奇志.遗传进化算法在边坡稳定性分析中的应用.岩土工程学报,1998,20(1)：44-46
[28]Li K S, Lumb P. Probabilistic design of slopes. Canadian Geotechnical Journal 1987;24(4):520-535
[29]高大钊.土力学可靠性分析原理.北京：中国建筑工业出版社,1989,55-120
[30]祝玉学.露天矿边坡设计的可靠性分析方法.水文地质,1980,47(5)：42-46
[31]陈昌富,王贻荪,邹银生.边坡可靠性分析分步混合遗传算法.土木工程学报,2003,36(2)：72-77
[32]何满潮,苏永华,景海河.块状岩体的稳定可靠性分析模型及其应用.岩石力学与工程学报,2002,21(3)：343-348
[33]叶万军.基于可靠度理论的黄土高边坡优化设计：[长安大学硕士论文].长安：长安大学,2003,27-34
[34]黄志全,李华晔,马莎等.岩石边坡块状结构岩体稳定性分析和可靠性评价.岩石力学与工程学报,2004,23(24)：4200-4205
[35]罗文强,王亮清,龚珏.正态分布下边坡稳定性二元指标体系研究.岩石力学与工程学报,2005,24(13)：2288-2292
[36]Wong F S. Slope reliability and response method. Journal of Geotechnical Engineering,1985,111(1):32-53
[37]Faravelli L. A response surface approach for reliability analysis. Journal of Engineering Mechanics,ASCE,1989,115(12):2763-2781
[38]Schueller G I, et al. On efficient computational scheme to calculate structural failure probability. Probability Engineering Mechanics,1989,4(1):65-73
[39]Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety,1990,7(1):57-66
[40]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
[41]陆有忠,杨有贞,张会林.边坡工程可靠性的支持向量机估计.岩石力学与工程学报,2005,24(1)：149-154
[42]杨坤,周创兵,张听等.边坡块状结构岩体模糊随机可靠性分析.岩石力学与工程学报,2006,25(2)：407-413
[43]夏元友.基于神经网络的岩质边坡稳定性评估系统研究.自然灾害学报1996,5(1)：98-104
[44]薛新华,张我华,刘红军.基于SOFM神经网络的边坡稳定性评价.岩土力学,2008,29(8)：2236-2240
[45]苏永华,伍文国,李志勇等.极限平衡模式下的边坡稳定概率小样本最优拟合算法.岩石力学与工程学报,2009,28(1)：173-180
[46]李俊,彭振斌,周斌.利用差分法计算基于Bishop分析模式的边坡稳定可靠度.矿冶工程,2008,28(6)：11-14
[47]苏永华,赵明华,张月英等.利用差分法计算基于Spencer分析模式的边坡稳定可靠度.岩石力学与工程学报,2006,25(增1)：2751-2756
[48]Kiureghian A D, Zhang Y, Li C C. Inverse Reliability Problem. Journal of Engineering Mechanics,1994,120 (5):1154-1159
[49]Winterstein SR, et al. Environmental contours for extreme response:inverse FORM with omission factors. In:Proceedings of the ICOSSAR-93.1993
[50]Li H, Foschi R O. An inverse reliability method and its application.Structural Safety 1998,20(3):257-270
[51]Li H, Foschi RO. Response:"an inverse reliability method and its application". Structural Safety,2000,22(5):103-106.
[52]Roberto Minguez, Enrique Castillo, Ali S.Hadi. Solving the inverse reliability problem using decomposition techniques. Structural Safety,2005,27(1):1-23
[53]Cheng Jin, Zhang Jie, Cai C.S, et al. A new approach for solving inverse reliability problems with implicit response functions. Engineering Structures, 2007,29(1):71-79
[54]Cheng Jin, Zhang Jie et al. A new approach for solving inverse reliability problems with implicit response functions. Engineering Structures,2007,29(8) 71-79
[55]程进,肖汝诚.基于逆可靠度法的大跨悬索桥主缆安全系数评估.中国公路学报,2007,20(1)：58-61
[56]Cornell C A. A probability based structural code. Journal of the American concrete institute,1969,66(12):974-985
[57]Hasofer A M, Lind N C. Exact and invariant second-moment code format. Journal of the engineering mechanics,1974,100(1):111-121
[58]赵国藩,金伟良,贡金鑫.结构可靠度理论.北京：中国建筑工业出版社,2000
[59]Box G E P, Wilson K B. On the experimental attainment of optimum condition. Journal of the Royal Stat Society. Series B.1951,13(1):1-34
[60]Fiessler B, et al. Quadratic limit states in structural reliability. Journal of the Engineering Mechanics Division, ASCE,1979,105(4):661-676
[61]Kiureghian A D, et al.Second-order reliability approximation. Journal of Engineering Mechanics,1984,110(8):1208-1225
[62]Tvedt L. The distribution of quadratic forms in normal space-application to structural reliability, Journal of Engineering Mechanics.1990,116(6):1183-1197
[63]Breitung K. Asymptotic approximations for multinormal integrals. Journal of the Engineering Mechanics, ASCE,1984,110(3):357-366
[64]李云贵,张保和.结构可靠度的渐近分析方法.大连理工大学学报,1994,34(4)：442-447
[65]Christensen P T, Murotsu Y. Application of Structural Systems Reliability Theory. Society for Industrial and Applied Mathematics,1987,29(4):674-675
[66]Liu P L, Kiureghian A D. Optimization algorithms for structural reliability. Structural Safety,1991,9(3):161-177
[67]Reid S G. Specification of design criteria based on probabilistic measures of design performance. Structural Safety,2002,24 (2):333-345
[68]门玉明.土坡稳定性的极限分析方法.西安地质学院学报,1996,18(2)：59-63
[69]陈祖煜.土力学经典问题的极限分析上、下限解.岩土工程学报,2002,24(1)：1-11
[70]李小强,白世伟,李铀.最小势能方法在二维边坡稳定分析中的应用.岩土力学,2004,25(6)：909-912
[71]陈佳,彭社琴,刘根亮.基于极限分析上限法的危岩体崩塌稳定性分析.河南 理工大学学报：自然科学版,2009,28(2)：180-184
[72]刘明,熊彩霞,马金荣.模糊划分矩阵在岩土参数概率分布中的应用.西安工程学院学报,2002,24(4)：38-41
[73]Hassan A M, Wolff T F. Search algorithm for minimum reliability index of earth slopes. Journal of Geotechnical and Geoenvironmental Engineering:1999, 125(4):301-308.
[2]谢定义,姚仰平,党发宁.高等土力学.北京：高等教育出版社,2007,268-396
[3]贡金鑫.工程结构可靠度计算方法.大连：大连理工大学出版社,2003,1-173
[4]陈祖煜.土质边坡稳定分析程序STAB2005使用平台.北京：中国水利水电科学研究院,2005,11-28
[5]张鲁渝,郑颖人,时卫民.边坡稳定分析中关于不平衡推力法的讨论.岩石力学与工程学报,2005,24(1)：177-182
[6]时卫民,郑颖人,唐伯明等.边坡稳定不平衡推力法的精度分析及其使用条件.岩土工程学报,2004,26(3)：313-317
[7]苏爱军,冯明权.滑坡稳定性传递系数计算法的改进.地质灾害与环境保护,2002,9(3)：51-55
[8]Hovland H J. Three dimensional slope stability analysis method. Journal of Geotechnical the Engineering Division, ASCE,1997,103(9):971-986
[9]蒋臻蔚.三维剩余推力法初探-黑河右坝肩变形体稳定性评价例析：[西安工程学院硕士学位论文].西安：西安工程学院,2000,24-38
[10]杜建成,黄大寿,胡定.边坡稳定的三维极限平衡分析法.四川大学学报(工程科学版),2001,33(4)：9-12
[11]陈祖煜,弥宏亮,汪小刚.边坡稳定三维分析的极限平衡方法.岩土工程学报,2001,23(5)：525-529
[12]弥宏亮,陈祖煜,张发明等.边坡稳定三维极限分析方法及工程应用.岩土力学,2002,23(5)：649-653
[13]Wei W B, Cheng Y M, Li L. Three-dimensional slope failure analysis by the strength reduction and limit equilibrium methods. Computers and Geotechnics, 2009,36(1-2):70-80
[14]张均锋,丁桦.边坡稳定性分析的三维极限平衡法及应用.岩石力学与工程学报,2005,24(3)：365-370
[15]Griffiths D V, Lane P A. Slope stability analysis by finite elements. Geotechnique,1999,49(3):387-403
[16]宋二祥.土工结构安全系数的有限元计算.岩土工程学报,1997,19(2)：1-7
[17]郑颖人,赵尚毅,张鲁渝.用有限元强度折减法进行边坡稳定分析.中国工程科 学,2002,10(4)：57-61
[18]郑颖人,赵尚毅.用有限元强度折减法在土坡及岩坡中的应用.岩土工程学报,2004,23(19)：3379-3381
[19]郑颖人,赵尚毅.岩土工程极限分析有限元法及其应用.土木工程学报,2005,38(1)：91-99
[20]赵尚毅,郑颖人,张玉芳.有限元强度折减法中边坡失稳的判据探讨.岩土力学,2005,26(2)：332-336
[21]门玉明.土坡稳定性的极限分析方法.西安地质学院学报,1996,18(2)：58-63
[22]Donald I B, Chen Zuyu. Slope stability by an upper bound plasticity method. Canadian Geotechnical Journal,1997,34(11):853-862
[23]李国英,沈珠江.下限原理有限单元法及其在土工问题中的应用.岩土工程学报,1997,19(5)：84-89
[24]Vanmarcke E H. Reliability of earth slopes. Journal of the Geotechnical Engineering Division, ASCE 1977,103(11):1227-1246
[25]Revilla J, Castillo E. The calculus of variations applied to stability of slopes. Geoteehnique,1977,27(1):1-11
[26]马忠政,祁红卫,侯学渊.边坡稳定验算中全面搜索的一种新方法.岩土力学,2000,21(3)：256-259
[27]肖专文,张奇志.遗传进化算法在边坡稳定性分析中的应用.岩土工程学报,1998,20(1)：44-46
[28]Li K S, Lumb P. Probabilistic design of slopes. Canadian Geotechnical Journal 1987;24(4):520-535
[29]高大钊.土力学可靠性分析原理.北京：中国建筑工业出版社,1989,55-120
[30]祝玉学.露天矿边坡设计的可靠性分析方法.水文地质,1980,47(5)：42-46
[31]陈昌富,王贻荪,邹银生.边坡可靠性分析分步混合遗传算法.土木工程学报,2003,36(2)：72-77
[32]何满潮,苏永华,景海河.块状岩体的稳定可靠性分析模型及其应用.岩石力学与工程学报,2002,21(3)：343-348
[33]叶万军.基于可靠度理论的黄土高边坡优化设计：[长安大学硕士论文].长安：长安大学,2003,27-34
[34]黄志全,李华晔,马莎等.岩石边坡块状结构岩体稳定性分析和可靠性评价.岩石力学与工程学报,2004,23(24)：4200-4205
[35]罗文强,王亮清,龚珏.正态分布下边坡稳定性二元指标体系研究.岩石力学与工程学报,2005,24(13)：2288-2292
[36]Wong F S. Slope reliability and response method. Journal of Geotechnical Engineering,1985,111(1):32-53
[37]Faravelli L. A response surface approach for reliability analysis. Journal of Engineering Mechanics,ASCE,1989,115(12):2763-2781
[38]Schueller G I, et al. On efficient computational scheme to calculate structural failure probability. Probability Engineering Mechanics,1989,4(1):65-73
[39]Bucher C G, Bourgund U. A fast and efficient response surface approach for structural reliability problems. Structural Safety,1990,7(1):57-66
[40]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
[41]陆有忠,杨有贞,张会林.边坡工程可靠性的支持向量机估计.岩石力学与工程学报,2005,24(1)：149-154
[42]杨坤,周创兵,张听等.边坡块状结构岩体模糊随机可靠性分析.岩石力学与工程学报,2006,25(2)：407-413
[43]夏元友.基于神经网络的岩质边坡稳定性评估系统研究.自然灾害学报1996,5(1)：98-104
[44]薛新华,张我华,刘红军.基于SOFM神经网络的边坡稳定性评价.岩土力学,2008,29(8)：2236-2240
[45]苏永华,伍文国,李志勇等.极限平衡模式下的边坡稳定概率小样本最优拟合算法.岩石力学与工程学报,2009,28(1)：173-180
[46]李俊,彭振斌,周斌.利用差分法计算基于Bishop分析模式的边坡稳定可靠度.矿冶工程,2008,28(6)：11-14
[47]苏永华,赵明华,张月英等.利用差分法计算基于Spencer分析模式的边坡稳定可靠度.岩石力学与工程学报,2006,25(增1)：2751-2756
[48]Kiureghian A D, Zhang Y, Li C C. Inverse Reliability Problem. Journal of Engineering Mechanics,1994,120 (5):1154-1159
[49]Winterstein SR, et al. Environmental contours for extreme response:inverse FORM with omission factors. In:Proceedings of the ICOSSAR-93.1993
[50]Li H, Foschi R O. An inverse reliability method and its application.Structural Safety 1998,20(3):257-270
[51]Li H, Foschi RO. Response:"an inverse reliability method and its application". Structural Safety,2000,22(5):103-106.
[52]Roberto Minguez, Enrique Castillo, Ali S.Hadi. Solving the inverse reliability problem using decomposition techniques. Structural Safety,2005,27(1):1-23
[53]Cheng Jin, Zhang Jie, Cai C.S, et al. A new approach for solving inverse reliability problems with implicit response functions. Engineering Structures, 2007,29(1):71-79
[54]Cheng Jin, Zhang Jie et al. A new approach for solving inverse reliability problems with implicit response functions. Engineering Structures,2007,29(8) 71-79
[55]程进,肖汝诚.基于逆可靠度法的大跨悬索桥主缆安全系数评估.中国公路学报,2007,20(1)：58-61
[56]Cornell C A. A probability based structural code. Journal of the American concrete institute,1969,66(12):974-985
[57]Hasofer A M, Lind N C. Exact and invariant second-moment code format. Journal of the engineering mechanics,1974,100(1):111-121
[58]赵国藩,金伟良,贡金鑫.结构可靠度理论.北京：中国建筑工业出版社,2000
[59]Box G E P, Wilson K B. On the experimental attainment of optimum condition. Journal of the Royal Stat Society. Series B.1951,13(1):1-34
[60]Fiessler B, et al. Quadratic limit states in structural reliability. Journal of the Engineering Mechanics Division, ASCE,1979,105(4):661-676
[61]Kiureghian A D, et al.Second-order reliability approximation. Journal of Engineering Mechanics,1984,110(8):1208-1225
[62]Tvedt L. The distribution of quadratic forms in normal space-application to structural reliability, Journal of Engineering Mechanics.1990,116(6):1183-1197
[63]Breitung K. Asymptotic approximations for multinormal integrals. Journal of the Engineering Mechanics, ASCE,1984,110(3):357-366
[64]李云贵,张保和.结构可靠度的渐近分析方法.大连理工大学学报,1994,34(4)：442-447
[65]Christensen P T, Murotsu Y. Application of Structural Systems Reliability Theory. Society for Industrial and Applied Mathematics,1987,29(4):674-675
[66]Liu P L, Kiureghian A D. Optimization algorithms for structural reliability. Structural Safety,1991,9(3):161-177
[67]Reid S G. Specification of design criteria based on probabilistic measures of design performance. Structural Safety,2002,24 (2):333-345
[68]门玉明.土坡稳定性的极限分析方法.西安地质学院学报,1996,18(2)：59-63
[69]陈祖煜.土力学经典问题的极限分析上、下限解.岩土工程学报,2002,24(1)：1-11
[70]李小强,白世伟,李铀.最小势能方法在二维边坡稳定分析中的应用.岩土力学,2004,25(6)：909-912
[71]陈佳,彭社琴,刘根亮.基于极限分析上限法的危岩体崩塌稳定性分析.河南 理工大学学报：自然科学版,2009,28(2)：180-184
[72]刘明,熊彩霞,马金荣.模糊划分矩阵在岩土参数概率分布中的应用.西安工程学院学报,2002,24(4)：38-41
[73]Hassan A M, Wolff T F. Search algorithm for minimum reliability index of earth slopes. Journal of Geotechnical and Geoenvironmental Engineering:1999, 125(4):301-308.