Bivariate Distribution Of Shear Strength Parameters Of Soils And Rocks Using Copula

Geotechnical engineering is an indispensable branch of water resources and hydropower engineering,including all kinds of geotechnical structures,like slopes,foundations,underground caverns,dam etc...In essence,the reliability analysis of geotechnical structures is to explore the effect induced by uncertain factors,e.g.from geotechnical parameter uncertainty,model uncertainty,test error uncertainty etc...Evaluating the uncertainty of parameter probabilistic model is a critical step in geotechnical reliability analysis and risk assessment,which shows that to achieve a thorough study of parameter probabilistic model is a meaningful research.As we know,the common shear strength parameters,cohesion and inner frictional angle,which are correlated by each other,play an important role in geotechnical stability and deformation analysis.It is an imperative way to construct the probabilistic model of shear strength parameters in geotechnical reliability analysis.However,with the limitation of economy and technology in practical engineering,the number of test data is usually not large enough to accurately determine the probabilistic model of shear strength parameters.Thus,to determine the probabilistic model of shear strength parameters with limited test data is a challenge task in geotechnical reliability research.As the bivariate distribution under incomplete probability information is not unique,seeking a approach to minimize the model selection uncertainty is the second challenge task.Following,the problem of how to use the constructed probability model in practical analysis and design should be further developed in geotechnical reliability.To address the above key issues,this thesis aims to apply the copula theory to geotechnical engineering and mainly explore the modeling of the bivariate probability distribution for the correlated geotechnical parameters,cohesion and inner frictional angle.This thesis seeks to address the above challenges from three levels.Firstly,copula theory is introduced to model the bivariate distribution of cohesion and inner frictional angle.Secondly,to explore the method to minimize the model selection uncertainty induced by limited test data,the Bayesian approach is proposed for selection the most probable bivariate distribution of cohesion and inner frictional angle.What's more,the posteriori distribution of shear strength parameters will be given under the frame of Bayesian theory.Finally,the bivariate probability distribution is applied in practical engineering analysis and design.Simultaneously,the effect on geotecnical reliability analysis and design results induced by different model selection will be explored.The implementation details and some conclusions are listed as follows.(1)The thesis first presents the background and significance of probabilistic modeling correlated geotechnical parameters:cohesion and inner frictional angle.A literature review of methods for probabilistic modeling of correlated parameters is conducted.The limitation of the existing method and the key issues to be further addressed are outlined.What's more,an overview of copula application in area of geotechnics is mainly given.(2)The copula theory is introduced systematically.Six parts of copula information are given:the definition and basic rules of copula,the commonly used dependence measures,the fitted copula functions for negative correlation between cohesion and inner frictional angle,parameter estimation methods,copula model selection methods and procedures of generating simulated copula data.(3)The modeling method of joint bivariate probability distributions is proposed using copula.29 sets of geotechnical shear strength parameters with negative correlation are compiled.According to comparison with bivariate normal distribution,the copula based bivariate distribution is verified to perform better on fitting the given test data and be with extensive application scope.The marginal model and copula function have effects on bivariate modeling.The suggested method provide an effective tool to solve the problem when modeling the correlated and nonnormal parameters.(4)Under the condition of limited test data,Bayesian approach is proposed for copula selection of shear strength parameters of soils and rocks.Compared with least square method of Euclidean distance and Akaike Information Criterion,the Bayesian approach performs better in identification accuracy,especially in the case of small number of data.The sample size,correlation,the type of the true copula and prior information of shear strength parameters have a significant impact on the accuracy of the Bayesian copula selection approach.Finally,the proposed Bayesian approach is applied to the 29 sets of test data and the best-fit copulas for all sets of test data are identified separately.The identification results show that the commonly adopted Gaussian Copula for characterizing the dependence structure between shear strength parameters does not always provide the best fit to the shear strength data,which further proves the importance and necessity of copula theory in geotechnical parameter model.(5)Bayesian approach for model comparison of bivariate distribution of cohesion and inner frictional angle using limited site-specific data are developed in the thesis.After selecting the most probable bivariate distribution model from a set of candidates,Bayesian theory is further used to characterize the joint distribution,with the result of posterior joint probability density function(PDF).The Markov chain Monte Carlo(MCMC)simulation is used to generate a large number of equivalent sample pairs from the posterior joint distribution.The differences between one-step Bayesian selection approach and two-step Bayesian selection approach are compared through two aspects:29 sets of site-specific data and vast set of simulated data.The results advice that two-stepped Bayesian selection approach with empirical distribution being used in second step performances better whether for efficiency or accuracy,which gives the engineer more confidence in model selection procedure.(6)The implementation procedure of geotechnical system reliability,incorporating copula based parameter distribution is developed.The effect on system reliability of geotechnical structures are studied from two perspectives,including effect procedure and mechanism.The system reliability of a retaining wall and a rock wedge slope is presented,where the results suggest the copula selection for shear strength parameters has a significant effect on geotechnical system reliability.These results provide a new insight on the frequently used assumption that the misusing of Gaussian Copula will underestimate the failure probability in geotechnical engineering.In order to get a more reasonable estimation of reliability,the geotechnical engineering issues are advised to be treated as system problem with the reasonable bivariate probabilistic model.(7)The implementation procedure of geotechnical reliability-based design,incorporating copula based parameter distribution is developed.The reliability-based design of a infinite slope and a square foundation is presented,where the results suggest the copula selection for shear strength parameters has a significant effect on geotechnical reliability-based design.The reason of difference between reliability-based design results is analyzed combined with geotechnical system reliability results.The target failure probability,the variability of shear strength parameters and correlation are founded be three influencing factors of the reliability-based design results.In term of engineering safety,No.16 Copula is advised when facing little data or limitation of knowledge.