小样本岩土参数下考虑矩估计偏差的土质边坡可靠度分析
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摘要
以样本平均值和样本方差为表征的矩估计值在小样本岩土参数下具有较大不确定性,直接用于可靠度分析可能导致结果出现较大误差。根据抽样获得的同一小样本X可能来自不同总体W~*的事件发生概率关系,基于正态总体均值、方差与相应矩估计值的偏差分别服从t及χ~2分布的基本规律,依据条件概率原理构建了由W~*组成的岩土参数虚拟总体空间Ω_(POP)及对应发生概率空间Ω_(OCC)。以简单土坡为对象,采用MonteCarlo法获得了不同强度参数总体W~*下的失效概率空间Ω_(FAI),应用全概率公式建立了考虑样本矩估计偏差影响的边坡稳定可靠指标β_T计算方法,分析了多批次小样本抽样下β_T空间分布特征。研究表明:以小样本岩土参数矩估计偏差发生概率为权重的矩偏权值法能有效收窄β_T分布区间,减小参数估计误差传递至β_T空间的不确定性,且β_T空间平均值收敛于真值,表现出良好的有效性和无偏性;针对实际勘察工作中的单批次小样本取样特点,矩偏权值法能较好地降低数据在偏差偶然过大情况下造成的可靠性计算结果失真程度。
The moment estimations, such as mean and variance of samples, are quite uncertain in the context of limited geotechnical parameters, which may lead to a large bias in reliability analysis. The same small sample X may come from different populations W~*, and the biases between the population mean and variance and the corresponding moment satisfy the law of t and χ~2 distribution under assumption of normal distribution. Based on conditional probability principle, virtual population space Ω_(POP) and corresponding occurrence probability space Ω_(OCC) composed of W~* are constructed. The Monte-Carlo simulation methods is used to obtain the failure probability space Ω_(FAI) for a simple soil slope. Underlying the full probability formula, a momentary weighting method, denoted by MWM, is proposed to calculate reliability index β_T. The numerical examples show that MWM could effectively narrow the β_T range, and simultaneously reduce the uncertainty caused by parameter estimation error. The mean value of β_T converges to the true value, showing good validity and unbiasedness. Aiming at the characteristics of single batch sampling in actual investigation work, MWM can reduce the distortion degree of reliability analysis in large occasional moment estimation bias cases.
The moment estimations, such as mean and variance of samples, are quite uncertain in the context of limited geotechnical parameters, which may lead to a large bias in reliability analysis. The same small sample X may come from different populations W~*, and the biases between the population mean and variance and the corresponding moment satisfy the law of t and χ~2 distribution under assumption of normal distribution. Based on conditional probability principle, virtual population space Ω_(POP) and corresponding occurrence probability space Ω_(OCC) composed of W~* are constructed. The Monte-Carlo simulation methods is used to obtain the failure probability space Ω_(FAI) for a simple soil slope. Underlying the full probability formula, a momentary weighting method, denoted by MWM, is proposed to calculate reliability index β_T. The numerical examples show that MWM could effectively narrow the β_T range, and simultaneously reduce the uncertainty caused by parameter estimation error. The mean value of β_T converges to the true value, showing good validity and unbiasedness. Aiming at the characteristics of single batch sampling in actual investigation work, MWM can reduce the distortion degree of reliability analysis in large occasional moment estimation bias cases.
引文
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[11]CHING J Y,PHOON K K.Correlations among some clay parameters-the multivariate distribution[J].Canadian Geotechnical Journal,2014,51(6):686-704.
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[14]LI D Q,YANG Z Y,CAO Z J,et al.System reliability analysis of slope stability using generalized subset simulation[J].Applied Mathematical Modelling,2017,46:650-664.
[15]GRECO V R.Variability and correlation of strength parameters inferred from direct shear tests[J].Geotechnical and Geological Engineering,2016,34(2):585-603.
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[17]ROH G,HONG H P.Calibration of informationsensitive partial factors for assessing earth slopes[J].Journal of Geoengineering,2009,4(3):93-102.
[18]NELSON J D,BRAND E W,MOH Z C.A probabilistic approach to the correction of soil strength[C]//Proceeding of the First International Conference on Applications of Statistics and Probability to Soil and Structural Engineering.Bangkok:Asian Institute of Technology,1971:358-370.
[19]HOOPER J A,BUTLER F G.Some numerical results concerning the shear strength of London clay[J].Géotechnique,1966,16(4):282-304.
[20]陈立宏,陈祖煜,刘金梅.土体抗剪强度指标的概率分布类型研究[J].岩土力学,2005,26(1):37-40.CHEN Li-hong,CHEN Zu-yu,LIU Jin-mei.Probability distribution of soil strength[J].Rock and Soil Mechanics,2005,26(1):37-40.
[21]MATSUO M,KURODA K.Probabilistic approach to design of embankments[J].Soils and Foundations,2008,14(2):1-17.
[22]罗冲,殷坤龙,陈丽霞,等.万州区滑坡滑带土抗剪强度参数概率分布拟合及其优化[J].岩石力学与工程学报,2005,24(9):1588-1593.LUO Chong,YIN Kun-long,CHEN Li-xia,et al.Probability distribution fitting and optimization of shear strength parameters in sliding zone along horizontal-stratum landslides in Wanzhou city[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(9):1588-1593.
[23]陈炜韬,王玉锁,王明年,等.黏土质隧道围岩抗剪强度参数的概率分布及优化实例[J].岩石力学与工程学报,2006,25(增刊2):3782-3787.CHEN Wei-tao,WANG Yu-suo,WANG Ming-nian,et al.Probability distribution and optimizing example of shear strength parameters of surrounding rock in cohesive soil tunnel[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(Supp.2):3782-3787.
[24]苏卫卫,黄宏伟,张洁.上海软黏土抗剪强度指标概率分布类型研究[J].地下空间与工程学报,2012,8(增刊2):1695-1699.SU Wei-wei,HUANG Hong-wei,ZHANG Jie.Risk analysis of the underground passage through rail transit construction scheme[J].Chinese Journal of Underground Space and Engineering,2012,8(Supp.2):1695-1699.
[25]国家铁路局.TB 10001-2016铁路路基设计规范[S].北京:中国铁道出版社,2017.National Railway Administration of the People’s Republic of China.TB 10001-2016 Code for design of railway earth structure[S].Beijing:China Railway Publishing Uouse,2017.
[26]JIANG S H,LI D Q,CAO Z J,et al.Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation[J].Journal of Geotechnical and Geoenvironmental Engineering,2015,141(2):1-13.
[27]WANG Y,CAO Z J,AU S K.Practical reliability analysis of slope stability by advanced Monte Carlo simulations in a spreadsheet[J].Canadian Geotechnical Journal,2010,48(1):162-172.
[28]TOBUTT D C.Monte Carlo simulation methods for slope stability[J].Computers and Geosciences,1982,8(2):199-208.
[29]WANG Y,CAO Z J,LI D Q.Practical reliability analysis of slope stability by advanced Monte Carlo simulations in a spreadsheet[J].Canadian Geotechnical Journal,2011,48(1):162-172.
[30]MOOD A,GRAYBILL F,BOES D.Introduction to the theory of statistics,3rd Edition[M].New York:Mc Graw Hill,1974.
[31]李亮,张丙强.c,?相关性对边坡整体稳定性的影响[J].铁道科学与工程学报,2004,1(1):62-68.LI Liang,ZHNAG Bing-qiang.Influence of correlations of c and?on slope stability[J].Journal of Railway Science and Engineering,2004,1(1):62-68.
[2]AKBAS S O,KULHAWY F H.Characterization and estimation of geotechnical variability in Ankara clay:a case history[J].Geotechnical and Geological Engineering,2010,28(5):619-631.
[3]LIU P L,KIUREGHIAN A D.Multivariate distribution models with prescribed marginals and covariances[J].Probabilistic Engineering Mechanics,1986,1(2):105-112.
[4]唐小松,李典庆,周创兵,等.不完备概率信息条件下边坡可靠度分析方法[J].岩土工程学报,2013,35(6):1027-1034.TANG Xiao-song,LI Dian-qing,ZHOU Chuang-bing,et al.Reliability analysis of slopes with incomplete probability information[J].Chinese Journal of Geotechnical Engineering,2013,35(6):1027-1034.
[5]TANG W H,YUCEMEN M S,ANG A H S.Probability-based short term design of soil slopes[J].Canadian Geotechnical Journal,1976,13(13):201-215.
[6]骆飞,罗强,蒋良潍,等.小样本岩土参数的Bootstrap估计及边坡稳定分析[J].岩石力学与工程学报,2017,36(2):370-379.LUO Fei,LUO Qiang,JIANG Liang-wei,et al.Bootstrap estimation for geotechnical parameters of small samples and slope stability analysis[J].Chinese Journal of Rock Mechanics and Engineering,2017,36(2):370-379.
[7]LUMB P.The variability of natural soils[J].Canadian Geotechnical Journal,1966,3(2):74-97.
[8]HYNES M E,VANMARCKE E H.Subjective reliability assessment in geotechnical engineering:I-95embankment case study[R].Massachusetts:Department of Civil Engineering,Massachusetts Institute of Technology,1975.
[9]唐小松,李典庆,曹子君,等.有限数据条件下边坡可靠度分析的Bootstrap方法[J].岩土力学,2016,37(3):893-901.TANG Xiao-song,LI Dian-qing,CAO Zi-jun,et al.ABootstrap method for analyzing slope reliability based on limited shear-strength parameter data[J].Rock and Soil Mechanics,2016,37(3):893-901.
[10]李典庆,唐小松,周创兵.基于Copula理论的岩土体参数不确定性表征与可靠度分析[M].北京:科学出版社,2015.LI Dian-qing,TANG Xiao-song,ZHOU Chuang-bing.Uncertainty representation and reliability analysis of rock and soil parameters based on Copula theory[M].Beijing:Science Press,2015.
[11]CHING J Y,PHOON K K.Correlations among some clay parameters-the multivariate distribution[J].Canadian Geotechnical Journal,2014,51(6):686-704.
[12]中华人民共和国国家标准编写组.GB50021-2001岩土工程勘察规范[S].北京:中国建筑工业出版社,2009.The National Standards Compilation Group of People’s Republic of China.GB50021-2001 Code for investigation of geotechnical engineering[S].Beijing:China Architecture and Building Press,2009.
[13]PHOON K K.Reliability-based design in geotechnical engineering:computations and applications[M].Boca Raton:CRC Press,2008.
[14]LI D Q,YANG Z Y,CAO Z J,et al.System reliability analysis of slope stability using generalized subset simulation[J].Applied Mathematical Modelling,2017,46:650-664.
[15]GRECO V R.Variability and correlation of strength parameters inferred from direct shear tests[J].Geotechnical and Geological Engineering,2016,34(2):585-603.
[16]高大钊.土力学可靠性原理[M].北京:中国建筑工业出版社,1989.GAO Da-zhao.Principles of soil mechanics reliability[M].Beijing:China Architecture and Building Press,1989.
[17]ROH G,HONG H P.Calibration of informationsensitive partial factors for assessing earth slopes[J].Journal of Geoengineering,2009,4(3):93-102.
[18]NELSON J D,BRAND E W,MOH Z C.A probabilistic approach to the correction of soil strength[C]//Proceeding of the First International Conference on Applications of Statistics and Probability to Soil and Structural Engineering.Bangkok:Asian Institute of Technology,1971:358-370.
[19]HOOPER J A,BUTLER F G.Some numerical results concerning the shear strength of London clay[J].Géotechnique,1966,16(4):282-304.
[20]陈立宏,陈祖煜,刘金梅.土体抗剪强度指标的概率分布类型研究[J].岩土力学,2005,26(1):37-40.CHEN Li-hong,CHEN Zu-yu,LIU Jin-mei.Probability distribution of soil strength[J].Rock and Soil Mechanics,2005,26(1):37-40.
[21]MATSUO M,KURODA K.Probabilistic approach to design of embankments[J].Soils and Foundations,2008,14(2):1-17.
[22]罗冲,殷坤龙,陈丽霞,等.万州区滑坡滑带土抗剪强度参数概率分布拟合及其优化[J].岩石力学与工程学报,2005,24(9):1588-1593.LUO Chong,YIN Kun-long,CHEN Li-xia,et al.Probability distribution fitting and optimization of shear strength parameters in sliding zone along horizontal-stratum landslides in Wanzhou city[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(9):1588-1593.
[23]陈炜韬,王玉锁,王明年,等.黏土质隧道围岩抗剪强度参数的概率分布及优化实例[J].岩石力学与工程学报,2006,25(增刊2):3782-3787.CHEN Wei-tao,WANG Yu-suo,WANG Ming-nian,et al.Probability distribution and optimizing example of shear strength parameters of surrounding rock in cohesive soil tunnel[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(Supp.2):3782-3787.
[24]苏卫卫,黄宏伟,张洁.上海软黏土抗剪强度指标概率分布类型研究[J].地下空间与工程学报,2012,8(增刊2):1695-1699.SU Wei-wei,HUANG Hong-wei,ZHANG Jie.Risk analysis of the underground passage through rail transit construction scheme[J].Chinese Journal of Underground Space and Engineering,2012,8(Supp.2):1695-1699.
[25]国家铁路局.TB 10001-2016铁路路基设计规范[S].北京:中国铁道出版社,2017.National Railway Administration of the People’s Republic of China.TB 10001-2016 Code for design of railway earth structure[S].Beijing:China Railway Publishing Uouse,2017.
[26]JIANG S H,LI D Q,CAO Z J,et al.Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation[J].Journal of Geotechnical and Geoenvironmental Engineering,2015,141(2):1-13.
[27]WANG Y,CAO Z J,AU S K.Practical reliability analysis of slope stability by advanced Monte Carlo simulations in a spreadsheet[J].Canadian Geotechnical Journal,2010,48(1):162-172.
[28]TOBUTT D C.Monte Carlo simulation methods for slope stability[J].Computers and Geosciences,1982,8(2):199-208.
[29]WANG Y,CAO Z J,LI D Q.Practical reliability analysis of slope stability by advanced Monte Carlo simulations in a spreadsheet[J].Canadian Geotechnical Journal,2011,48(1):162-172.
[30]MOOD A,GRAYBILL F,BOES D.Introduction to the theory of statistics,3rd Edition[M].New York:Mc Graw Hill,1974.
[31]李亮,张丙强.c,?相关性对边坡整体稳定性的影响[J].铁道科学与工程学报,2004,1(1):62-68.LI Liang,ZHNAG Bing-qiang.Influence of correlations of c and?on slope stability[J].Journal of Railway Science and Engineering,2004,1(1):62-68.
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