| There are various uncertainties in geotechnical engineering. Reliability-based methods provide an effective and practical tool for quantifying these uncertainties. In a reliability analysis of geotechnical structures, geotechnical parameters are usually correlated with each other. For example, shear strength parameters (cohesion and friction angle) often exhibit a strong negative correlation. Two hyperbolic curve-fitting parameters underlying load-displacement curves of piles are correlated with each other. These parameters typically follow nonnormal distributions. To achieve a rigorous evaluation of geotechnical reliability, the joint cumulative distribution function (CDF) or probability density function (PDF) of these parameters should be known. In practice, geotechnical data is often of a small sample size. This is because small sample size is a main feature of geotechnical data. Based on the limited data sets, only the marginal distributions and covariance matrix of geotechnical parameters can be determined, which poses a problem of incomplete probability information. Based on this incomplete probability information, the joint CDF or PDF of the geotechnical parameters cannot be determined uniquely when the true joint probability distribution is a multivariate nonnormal distribution. Therefore, characterization and simulation of multivariate distributions based on incomplete probability information remains an outstanding challenge. In addition, there are large statistical uncertainties in the constructed multivariate distribution of geotechnical parameters derived from a small sample. Thus, the reliability analyses are significantly affected by the uncertainties in the constructed multivariate distribution, which should be properly considered to achieve a more meaningful estimate of the geotechnical reliability. The key issue is to characterize the uncertainties in the constructed multivariate distribution based on the limited data. Finally, the performance function in a reliability analysis of geotechnical structures is usually implicit, which causes difficulty in solving the reliability index or probability of failure. Therefore, the method to solve the reliability problems involving an implicit function accurately and efficiently needs to be further developed.To address the above three key issues, this thesis aims to introduce the copula theory into geotechnical engineering and explore uncertainty modeling of correlated geotechnical parameters using copulas and its effect on geotechnical reliability. Specifically, the problems studied in this thesis include:(1) the modeling of the bivariate and multivariate distribution of two or multiple geotechnical parameters using copulas,(2) the impact of copula selection on geotechnical reliability under incomplete probability information,(3) the approaches for evaluating the slope reliability under incomplete probability information,(4) the methods for identifying the best-fit marginal distributions and copulas with a small sample, and (5) the improved knowledge-based clustered partitioning (KCP) approach for solving the slope reliability problems involving correlated nonnormal variables and implicit performance functions. The implementation details and some conclusions are listed as follows.(1) This thesis first presents the background and significance of uncertainty modeling of correlated geotechnical parameters using copulas and its effect on geotechnical reliability. An overview of the existing methods for uncertainty modeling of geotechnical parameters is given. The limitations underlying the existing methods, the key issues to be addressed and future research topics are outlined. The applications of the copula theory in the field of finance and hydrology are briefly reviewed. Detailed review of the applications of the copula theory in the field of geotechnical engineering is presented.(2) The copula theory is introduced. The definition of copulas and the commonly used dependence measures are given. Seven bivariate copulas belonging to Elliptical, Plackett and Archimedean families are presented. The copula functions, copula density functions, parameter estimation method, identification of the best-fit copula and simulation algorithms associated with the aforementioned bivariate copulas are given. Moreover, six multivariate Elliptical and Archimedean copulas are also presented. The estimation of copula parameters using maximum likelihood method and the simulation algorithms are introduced.(3) The modeling of joint probability distributions of geotechnical parameters and the evaluation of geotechnical reliability under incomplete probability information remain a challenge that has not been studied extensively. Therefore, a copula-based method is proposed to model the joint probability distributions of correlated geotechnical parameters with given marginal distributions and covariance. The impact of copulas for modeling the dependence structures of geotechnical parameters on geotechnical reliability is investigated. The results indicate that the copula selection has a significant impact on geotechnical reliability. The probabilities of failure produced by different copulas differ considerably. Such a difference increases with decreasing probability of failure.(4) A method for the probabilistic analysis of load-displacement curves of single pile using copulas is proposed. The effect of copulas for modeling the bivariate distributions of the two curve-fitting parameters on serviceability limit state (SLS) reliability of single pile is studied. Two dependence measures, namely, the Pearson linear correlation coefficient and the Kendall rank correlation coefficient are employed to determine the copula parameters underlying the Gaussian copula. The differences between these two methods are identified. A copula-based procedure for generating the load-displacement curves of single pile is presented. The results indicate that the proposed copula-based method provides a more general and flexible way of modeling and simulating the bivariate probability distribution of the two curve-fitting parameters in isolation from their marginal probability distributions, which facilitates the generation of load-displacement curves and thus results in more accurate reliability results for SLS of single pile.(5) Evaluation of the slope reliability under incomplete probability information is a challenging problem. Three copula-based methods for reliability analysis of slopes under incomplete probability information are proposed. The procedure for modeling the bivariate distribution of shear strength parameters using copulas is introduced. A concept of nominal factor of safety for slope stability analysis is defined. The dispersion of probability of slope failure under incomplete probability information is derived. The results indicate that the dispersion of probability of slope failure increases with increasing nominal factor of slope safety. The proposed three copula-based methods can narrow the dispersion of probability of slope failure effectively, which can improve the reliability estimation of slopes efficiently. These methods provide an effective tool for reliability analysis of slopes under incomplete probability information.(6) Identification of the best-fit joint probability distribution underlying geotechnical parameters with a small sample size is a challenging problem. A bootstrap method for the identification of the best-fit joint probability distribution underlying correlated geotechnical parameters with a small sample size is proposed. The Akaike Information Criterion (AIC) is adopted for identifying the best-fit marginal distribution and copula. The influence of the sample size of geotechnical parameters on identification accuracy is investigated. The SLS reliability analysis of single pile considering the statistical uncertainty in the bivariate distribution of two curve-fitting parameters is conducted. The proposed bootstrap method can effectively consider the variation of the AIC values of the fitted marginal distributions and copulas. Furthermore, it can account for the possibilities of each candidate marginal distribution and copula being the best-fit marginal distribution and copula. The bootstrap method provides a general and practical tool for the identification of the best-fit joint probability distribution underlying geotechnical parameters with a small sample size.(7) The procedure for modeling the multivariate distribution of multiple geotechnical parameters using copulas is presented. This multivariate distribution is then employed to update the marginal distributions and bivariate distributions of multivariate geotechnical parameters based on the measurements from other sources. The results indicate that copulas provide a general and flexible way of modeling and simulating the multivariate distribution of multiple geotechnical parameters in isolation from their marginal distributions. The multivariate distribution can effectively update the marginal distributions and bivariate distributions of multivariate geotechnical parameters based on measurements from other sources, which provides a practical tool for multi-source information fusion and reuse.(8) A new reliability method, knowledge-based clustered partitioning (KCP) method is proposed to solve the reliability problems involving correlated nonnormal geotechnical parameters and implicit performance function. The multivariate Gaussian copula is adopted to model the multivariate distribution of multiple correlated geotechnical parameters, which facilitates the generation of correlated nonnormal geotechnical parameters and reliability computation using the KCP method. To remove the limitations of the KCP method with the binary step length, the KCP method with changing step length is proposed. The results indicate that the proposed KCP method can evaluate the reliability of rock slope stability involving correlated nonnormal geotechnical parameters and implicit performance function accurately and efficiently. Furthermore, the global optimization solutions can be determined using the proposed KCP method. |
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