岩土体结构失稳概率的二次二阶矩评估方法
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摘要
常规二次二阶矩可靠性方法只适用于功能函数偏导数(一阶和二阶)能够简便地由解析法求解的工程,使得其难以解决复杂岩土体结构的稳定可靠度问题.针对这一局限,基于数值差分原理,导出功能函数在验算点处各阶导数近似表达式,结合随机变量在X空间和Y空间的变换,构建了基于差分方法的梯度矢量数值求解工具;以该工具置换Breitung二次二阶矩方法中的梯度计算准则,形成一种适用于任意形式功能函数的二次二阶矩可靠度算法,消除了经典方法的局限;利用所提方法解决了功能函数为隐式和未知形式的边坡可靠度问题,展示了该方法解决复杂岩土体结构概率稳定性问题的能力,并同时在基准Y空间内建议了具有普遍意义的合适步长系数值0.01.
Conventional second-order second-moment reliability method(SORM) is only applicable to the engineering where the first and second order partial derivative of performance functions can be easily calculated by analytic method,which results in the difficulty in solving reliability problems of complex geotechnical engineering structures.Aiming at above limitation,firstly,approximate calculation expressions for partial derivatives on the checking point are deduced based on finite difference principle. Combine with conversion method of random variables between X space and Y space,solving method for gradient vectors is formed. Secondly,calculation criterion of gradient in Breitung′ s SORM is substituted by the above solving method,and then,an improved SORM which is suitable for arbitrary form of performance function is proposed. The limitation of the conventional SORM is eliminated. Finally,reliability problem of two slopes which performance functions are implicit and indefinable are solved by using the proposed method,and it shows the accuracy and availability of the proposed method to deal with complex structures reliability problem. Meanwhile,the value of step length coefficient c = 0.01 in the benchmark space(Yspace) which possess the universal meaning is also achieved.
Conventional second-order second-moment reliability method(SORM) is only applicable to the engineering where the first and second order partial derivative of performance functions can be easily calculated by analytic method,which results in the difficulty in solving reliability problems of complex geotechnical engineering structures.Aiming at above limitation,firstly,approximate calculation expressions for partial derivatives on the checking point are deduced based on finite difference principle. Combine with conversion method of random variables between X space and Y space,solving method for gradient vectors is formed. Secondly,calculation criterion of gradient in Breitung′ s SORM is substituted by the above solving method,and then,an improved SORM which is suitable for arbitrary form of performance function is proposed. The limitation of the conventional SORM is eliminated. Finally,reliability problem of two slopes which performance functions are implicit and indefinable are solved by using the proposed method,and it shows the accuracy and availability of the proposed method to deal with complex structures reliability problem. Meanwhile,the value of step length coefficient c = 0.01 in the benchmark space(Yspace) which possess the universal meaning is also achieved.
引文
[1] GB 50153-2008工程结构可靠性设计统一标准[S].北京:中国建筑工业出版社,2008:57—59.GB 50153-2008 Unified standard for reliability design of engineering structures[S]. Beijing:China Architecture&Building Press,2008:57—59.(In Chinese)
[2] GB 50199-2013水利水电工程结构可靠性设计统一标准[S].北京:中国计划出版社,2013:41—43.GB 50199-2013 Unified standard for reliability design of hydraulic engineering structures[S]. Beijing:China Planning Press,2013:41—43.(In Chinese)
[3] ISO 2394 General principles on reliability for structures[S].Geneva:International Standards Organization,2015:40.
[4] HASOFER A M,LIND N C.Exact and invariant second moment code format[J]. Journal of Engineering Mechanics Division,1974,100(1):111—121.
[5] BREITUNG K.Asymptotic approximation for multinormal integrals[J].Journal of Engineering Mechanics,1984,110(3):357—366.
[6] ROSENBLATT M.Remarks on a multivariate transformation[J].Annals of Mathematical Statistics,1952,23(3):470—472.
[7]贡金鑫.工程结构可靠度计算方法[M].大连:大连理工大学出版社,2003:82—86.GONG J X.Computational methods for reliability of engineering structures[M].Dalian:Dalian University of Technology Press,2003:82—86.(In Chinese)
[8]赵洪波,冯夏庭.支持向量机函数拟合的边坡稳定性估计[J].岩石力学与工程学报,2003,22(2):241—245.ZHAO H B,FENG X T.Application of support vector machines function regression in the evaluation stability of slope[J].Chinese Journal of Rock Mechanics and Engineering,2003,22(2):241—245.(In Chinese)
[9]黄靓,易伟健,汪优.岩土工程可靠度分析的改进响应面法研究[J].岩土力学,2008,29(2):370—374.HUANG L,YI W J,WANG Y.Improvement study on response surface method for reliability analysis in geotechnical engineering[J].Rock and Soil Mechanics,2008,29(2):370—374.(In Chinese)
[10]祁小辉,李典庆,曹子君,等.考虑地层变异的边坡稳定不确定性分析[J].岩土力学,2017,38(5):1—12.QI X H,LI D Q,CAO Z J,et al.Uncertainty analysis of slope stability considering geologic uncertainty[J]. Rock and Soil Mechanics,2017,38(5):1—12.(In Chinese)
[11]周芬,郭奥飞,杜运兴.边坡稳定可靠度分析的新型四阶矩法[J].湖南大学学报(自然科学版),2016,43(5):113—119.ZHOU F,GUO A F,DU Y X.A new method of four order moments for reliability analysis on slope stability[J].Journal of Hunan University(Natural Sciences),2016,43(5):113—119.(In Chinese)
[12]李典庆,肖特,曹子君,等.基于极限平衡法和有限元法的边坡协同式可靠度分析[J].岩土工程学报,2016,38(6):1004—1013.LI D Q,XIAO T,CAO Z J,et al.Auxiliary slope reliability analysis using limit equilibrium method and finite element method[J].Chinese Journal of Geotechnical Engineering,2016,38(6):1004—1013.(In Chinese)
[13]朱勇,冯夏庭,周辉,等.岩土强度参数空间中的极限状态面与系统可靠度分析[J].岩石力学与工程学报,2016,35(S2):3473—3480.ZHU Y,FENG X T,ZHOU H,et al.Limit state surface in strength parameter space and system reliability analysis[J].Chinese Journal of Rock Mechanics and Engineering,2016,35(S2):3473—3480.(In Chinese)
[14]舒苏旬,龚文惠.边坡稳定分析的神经网络改进模糊点估计法[J].岩土力学,2015,36(7):2111—2116.SHU S X,GONG W H.An improved fuzzy point estimate method for slope stability ananlysis based on neural network[J].Rock and Soil Mechanics,2015,36(7):2111—2116.(In Chinese)
[15]吴振君,王水林,葛修润.约束随机场下的边坡可靠度随机有限元分析方法[J].岩土力学,2009,30(10):3086—3092.WU Z J,WANG S L,GE X R.Slope reliability analysis by random FEM under constraint random field[J]. Rock and Soil Mechanics,2009,30(10):3086—3092.(In Chinese)
[16]陈昌富,杨宇.边坡稳定性分析水平条分法及其进化计算[J].湖南大学学报(自然科学版),2004,31(3):72—75.CHEN C F,YANG Y.Horizontal slice method of slope stability analysis based on evolutionary algorithm[J].Journal of Hunan University(Natural Sciences),2004,31(3):72—75.(In Chinese)
[17] TAN X H,WANG X E,HU N,et al.Two methods for predicting reliability index and critical probabilistic slip surface of soil slopes[J].Geotechnical and Geological Engineering,2016,34(5):1283—1292.
[18] JI J,JAYANTHA K K.Efficient reliability method for implicit limit state surface with correlated non-Gaussian variables[J].International Journal for Numerical and Analytical Methods in Geomechanics,2015,39(17):1898—1911.
[19] SU Y H,LI X,XIE Z Y.Probabilistic evaluation for the implicit limit state function of stability of a highway tunnel in China[J].Tunnelling and Underground Space Technology,2011,26(2):422—434.
[20] XU B,LOW B K.Probabilistic stability analyses of embankments based on finite-element method[J]. Journal of Geotechnical and Geoenvironmental Engineering,2006,132(11):1444—1454.
[2] GB 50199-2013水利水电工程结构可靠性设计统一标准[S].北京:中国计划出版社,2013:41—43.GB 50199-2013 Unified standard for reliability design of hydraulic engineering structures[S]. Beijing:China Planning Press,2013:41—43.(In Chinese)
[3] ISO 2394 General principles on reliability for structures[S].Geneva:International Standards Organization,2015:40.
[4] HASOFER A M,LIND N C.Exact and invariant second moment code format[J]. Journal of Engineering Mechanics Division,1974,100(1):111—121.
[5] BREITUNG K.Asymptotic approximation for multinormal integrals[J].Journal of Engineering Mechanics,1984,110(3):357—366.
[6] ROSENBLATT M.Remarks on a multivariate transformation[J].Annals of Mathematical Statistics,1952,23(3):470—472.
[7]贡金鑫.工程结构可靠度计算方法[M].大连:大连理工大学出版社,2003:82—86.GONG J X.Computational methods for reliability of engineering structures[M].Dalian:Dalian University of Technology Press,2003:82—86.(In Chinese)
[8]赵洪波,冯夏庭.支持向量机函数拟合的边坡稳定性估计[J].岩石力学与工程学报,2003,22(2):241—245.ZHAO H B,FENG X T.Application of support vector machines function regression in the evaluation stability of slope[J].Chinese Journal of Rock Mechanics and Engineering,2003,22(2):241—245.(In Chinese)
[9]黄靓,易伟健,汪优.岩土工程可靠度分析的改进响应面法研究[J].岩土力学,2008,29(2):370—374.HUANG L,YI W J,WANG Y.Improvement study on response surface method for reliability analysis in geotechnical engineering[J].Rock and Soil Mechanics,2008,29(2):370—374.(In Chinese)
[10]祁小辉,李典庆,曹子君,等.考虑地层变异的边坡稳定不确定性分析[J].岩土力学,2017,38(5):1—12.QI X H,LI D Q,CAO Z J,et al.Uncertainty analysis of slope stability considering geologic uncertainty[J]. Rock and Soil Mechanics,2017,38(5):1—12.(In Chinese)
[11]周芬,郭奥飞,杜运兴.边坡稳定可靠度分析的新型四阶矩法[J].湖南大学学报(自然科学版),2016,43(5):113—119.ZHOU F,GUO A F,DU Y X.A new method of four order moments for reliability analysis on slope stability[J].Journal of Hunan University(Natural Sciences),2016,43(5):113—119.(In Chinese)
[12]李典庆,肖特,曹子君,等.基于极限平衡法和有限元法的边坡协同式可靠度分析[J].岩土工程学报,2016,38(6):1004—1013.LI D Q,XIAO T,CAO Z J,et al.Auxiliary slope reliability analysis using limit equilibrium method and finite element method[J].Chinese Journal of Geotechnical Engineering,2016,38(6):1004—1013.(In Chinese)
[13]朱勇,冯夏庭,周辉,等.岩土强度参数空间中的极限状态面与系统可靠度分析[J].岩石力学与工程学报,2016,35(S2):3473—3480.ZHU Y,FENG X T,ZHOU H,et al.Limit state surface in strength parameter space and system reliability analysis[J].Chinese Journal of Rock Mechanics and Engineering,2016,35(S2):3473—3480.(In Chinese)
[14]舒苏旬,龚文惠.边坡稳定分析的神经网络改进模糊点估计法[J].岩土力学,2015,36(7):2111—2116.SHU S X,GONG W H.An improved fuzzy point estimate method for slope stability ananlysis based on neural network[J].Rock and Soil Mechanics,2015,36(7):2111—2116.(In Chinese)
[15]吴振君,王水林,葛修润.约束随机场下的边坡可靠度随机有限元分析方法[J].岩土力学,2009,30(10):3086—3092.WU Z J,WANG S L,GE X R.Slope reliability analysis by random FEM under constraint random field[J]. Rock and Soil Mechanics,2009,30(10):3086—3092.(In Chinese)
[16]陈昌富,杨宇.边坡稳定性分析水平条分法及其进化计算[J].湖南大学学报(自然科学版),2004,31(3):72—75.CHEN C F,YANG Y.Horizontal slice method of slope stability analysis based on evolutionary algorithm[J].Journal of Hunan University(Natural Sciences),2004,31(3):72—75.(In Chinese)
[17] TAN X H,WANG X E,HU N,et al.Two methods for predicting reliability index and critical probabilistic slip surface of soil slopes[J].Geotechnical and Geological Engineering,2016,34(5):1283—1292.
[18] JI J,JAYANTHA K K.Efficient reliability method for implicit limit state surface with correlated non-Gaussian variables[J].International Journal for Numerical and Analytical Methods in Geomechanics,2015,39(17):1898—1911.
[19] SU Y H,LI X,XIE Z Y.Probabilistic evaluation for the implicit limit state function of stability of a highway tunnel in China[J].Tunnelling and Underground Space Technology,2011,26(2):422—434.
[20] XU B,LOW B K.Probabilistic stability analyses of embankments based on finite-element method[J]. Journal of Geotechnical and Geoenvironmental Engineering,2006,132(11):1444—1454.