Reliability Analysis Of Tunnel Excavation Under Finite Probability Information

The rapid development of social economy and the general improvement of people's living standards have put forward higher requirements for the development of rail transit,tunnel is an important part of geotechnical engineering.Geological parameters are complex parameters,among which there are not only uncertainties but also some correlations.However,the traditional method of tunnel stability analysis does not take into account the two attributes of parameters,so reliability analysis which can take into account both uncertainties and correlations is of great importance to the stability evaluation of tunnel engineering.This paper first introduces what is finite probability information,that is,only the marginal distribution and correlation coefficient of known random variables.It is known that the marginal distribution and correlation coefficient of random variables are difficult to determine the joint distribution function.When the random variables are independent of each other,the joint probability distribution function of the random variables can be obtained by the simple product of the edge distribution function.However,when there is correlation between random variables,the corresponding joint probability distribution function and failure probability can not be determined only by the edge distribution and the correlation coefficient.Secondly,cohesion,internal friction angle and elastic modulus are important parameters for reliability analysis of tunnel excavation.Practical experience shows that the cohesion and internal friction angles are negatively correlated,the elastic modulus and cohesion are positively correlated,and these parameters generally obey the non-normal distribution.In order to accurately carry out tunnel excavation reliability analysis,a joint probability distribution function of known random variables is usually required.However,in practice,the experimental data is limited and it is difficult to accurately obtain its joint probability distribution function.In contrast,based on limited data,it is easier to obtain the marginal distribution and correlation coefficient of the random variables.Therefore,this paper proposes to use Copula function theory to construct the joint distribution function of two-dimensional related non-normal variables under finite probability information,that is,how to estimate the related parameters of Copula function,and introduce three different kinds of correlation coefficients based on Pearson,Spearman and Kendall.Next,the basic theory of reliability analysis is explained.This paper uses Monte Carlo simulation method to analysis the reliability of tunnel excavation.The theory and process of reliability analysis based on Monte Carlo simulation method are introduced in detail.In order to generate the relevant random variables and introduce the Rosenblatt transform,the definition and process of the Rosenblatt transform are introduced.Finally,this paper proposes the Monte Carlo method to calculate the failure probability of a circular tunnel excavation under hydrostatic pressure field.When the random variables obey the normal distribution and the lognormal distribution,the tunnel reliability of the plastic zone and the inner wall displacement during tunnel excavation is calculated.The results show that the failure probability calculated based on different correlation structures has obvious deviations.It is further concluded that the Gaussian correlation structure is not the optimal structure for characterizing the two-dimensional distribution model.The sensitivity analysis of the tunnel based on two failure modes shows that there are significant differences in the failure probabilities calculated by different Copula functions with the change of correlation coefficients,and this difference increases with the negative correlation of parameters(or the reduction of failure probability).With the increase of positive correlation(or the increase of failure probability),the difference between the calculation result of t copula function and the failure probability of Gaussian copula function decreases gradually,and the difference between the failure probability of Frank copula function and Gaussian copula function increases first and then decreases.The support pressure also plays an important role in the stability of tunnel excavation.Finally,the relationship between the failure probability of different Copula calculations and the support pressure is compared.It is found that with the increase of support pressure(or the decrease of failure probability)the difference in failure probability between different Copula functions is also greater.