Practice of Bayesian Probability Theory in Geotechnical Engineering

Different types of uncertainties are involved in the geotechnical engineering, for example, uncertainty of concept, uncertainty of material classification, soil parameters, constitutive or empirical models, boundary conditions, other uncertainty caused by measurement error, and so on. All these uncertainties affect the evaluation of the soil or structure performance, and they are required to be considered and analyzed for reliable evaluation results. According to the type of uncertainty, different analysis methods have been utilized to evaluate the uncertainty, for example, stochastic theory, fuzzy mathematics, theory of gray systems, artificial neural networks, genetic algorithms, and so on. In some cases, these methods are coupled to solve the uncertain problems. Among these methods, the probabilistic method in stochastic theory has been widely applied to many aspects of geotechnical engineering, for example, slope analysis, liquefaction analysis, reliability analysis of reinforced embankment, seepage analysis, soil–water characteristic curve, reducing uncertainties in empirical correlations, and so on.

This chapter presents a new estimation model for soil–water characteristic curve (SWCC) of granular soils with the effect of initial density. The proposed model, namely the modified Fredlund and Xing (MFX) model, is based on the Fredlund and Xing equation without increasing the number of parameters. The uncertainty of measurement data is analyzed by using the Bayesian method and applied in the estimation of the relative permeability function (k_r). Different levels of confidence intervals of SWCC are obtained by using the Bayesian approach. It is found that the proposed method can better predict the k_r values with smaller uncertainty for the soil with different initial dry densities than the Fredlund and Xing equation. The new MFX model is readily adopted for engineers to quickly estimate the SWCC and relative permeability of granular soil under different initial dry densities.

Following Chap. 2, the work is extended to predict SWCC of a soil with different initial densities by using Bayesian method, for both coarse-grained and fine-grained soil. An adjustment parameter (β) is introduced to express the relationships between the matric suctions of two soil samples. The parameter β is a function of the initial void ratio, matric suction or volumetric water content. Bayesian model class selection is adopted to determine the optimal predictive models of β. The optimal models of β are validated by comparing the estimated matric suction and measured data. The comparisons show that the newly proposed method produces more accurate SWCCs than the other models. Furthermore, the influence of the model parameters of β on the predicted matric suction and SWCC is evaluated using Latin hypercube sampling.

Many researchers have proposed various constitutive models for the purpose of capturing the complex physical mechanisms governing the creep behavior of soft soils. However, the more complex the model, the larger the number of associated uncertain parameters it has, and the less robust it is against modeling/measurement error. In this chapter, the Bayesian model class selection approach is applied to select the most plausible/suitable model describing the creep behavior of soft soil using laboratory measurements. In total, nine 1-D time-dependent constitutive models for the analysis of creep of clay are chosen for the assessment. Consolidated data from the intact samples of Vanttila clay and reconstituted samples of Hong Kong Marine Clay were adopted in the case study. All unknown model parameters are identified simultaneously by adopting the transitional Markov Chain Monte Carlo (TMCMC) method, and their uncertainty is quantified through the posterior probability density functions (PDFs). Engineers can adopt the present probabilistic method to determine the most suitable model and its associated model parameters for a given soft soil for the prediction of long-term creep behavior. The strategy also provides uncertainty evaluation of the model prediction based on the given data.

Following the work presented in the previous chapter, we further investigate the effect of loading duration on the uncertainty of the creep behavior prediction for clay analysis. Consolidation data from intact samples of Vanttila clay was used in the analysis. The results show that sufficient creep test data is crucial for reliable models and the associated parameters. The appropriate amount of test data used for updating can be determined by the Bayesian probabilistic approach, and this can avoid conducting an unnecessary long-term consolidation test. The proposed method can quantify the uncertainty of model prediction to facilitate better judgment by engineers for the loading duration and model uncertainty in their predictions.

Current studies have been focused on constitutive models selection using optimization methods, or simple formulas and models selection using Bayesian methods. In contrast, this chapter presents the Bayesian selection for advanced soil models accounting for soil uncertainty. Six representative sand model classes from elastic perfect plastic to critical-state-based are considered. Triaxial tests on Hostun sand are selected as training and testing data. The Bayesian model class selection is performed using an enhanced transitional Markov chain Monte Carlo method, whereby the generalization ability for each model is simultaneously evaluated. The most plausible/suitable model in terms of predictive ability, generalization ability and model complexity is selected.

Parametric identification using Bayesian approach is appealing in geotechnical engineering compared to that using deterministic method, because the soil uncertainty can be taken into account. To date, this approach has been verified only for certain conventional simple constitutive models. This chapter presents an enhanced version of the differential evolution transitional Markov chain Monte Carlo (DE-TMCMC) method and a competitive Bayesian parameter identification approach for use in advanced soil models. To realize the intended computational savings, a parallel computing implementation of DE-TMCMC is achieved using the single program/multiple data (SPMD) technique in MATLAB, with comparison of the proposed DE-TMCMC and the original TMCMC for identification of parameters of a critical-state-based sand model from laboratory tests in terms of robustness and effectiveness based on multiple independent calculations. Results indicate that DE-TMCMC is highly robust and efficient in identifying the parameters of advanced soil models. Finally, Bayesian parameter identification is applied in conjunction with DE-TMCMC to identify parameters of an elasto-viscoplastic model from two in situ pressuremeter tests. All results demonstrate the excellent ability of the enhanced Bayesian parameter identification approach to identify soil model parameters from both laboratory and in situ tests.

Pullout shear resistance of grouted soil nails is affected by many factors, such as overburden pressure, grouting pressure, soil dilation and degree of saturation of soil. Due to the complexity of pullout mechanism, these factors have not been well incorporated in current design methods. In this chapter, Bayesian analysis is performed to investigate the relative importance of several key factors and to build a new design formula to estimate maximum pullout shear stress of grouted soil nails. By using a series of laboratory soil nail pullout test data, Bayesian inference is performed to select the predictive formula with suitable complexity and to identify its parameters. It has been found that the most important factors are the degree of saturation and the product of grouting pressure and overburden pressure. It is shown that the proposed optimal model exhibits significantly stronger correlation with the measurements than the existing effective stress method for the laboratory and field tests.

In this chapter, the physical and chemical properties of natural fibers (i.e., natural moisture content, specific gravity, breaking tensile strength, breaking strain, cellulose, hemicellulose, lignin and ash) were examined for their influences in the soil reinforcement. A total of 11 factors including soil moisture content, soil density and fiber percentage were evaluated by using the Bayesian nonparametric general regression (BNGR) method. The robustness of the BNGR algorithm was validated using k-fold cross-validation. A parametric study was carried out to unveil the effects of these input variables. The results indicated that a higher fiber percentage or larger soil density induces higher reinforcement effect. Besides, the cellulose in a natural fiber is found to have a significant effect on the unconfined compressive strength (UCS) followed by the ash. With respect to the physical properties, the breaking strain is found to be highly correlated with the UCS, while the effects of fiber natural moisture content, specific gravity and breaking tensile strength are less significant parameters.

In this chapter, we present an efficient Bayesian back-analysis procedure for braced excavations using wall deflection data at multiple points. A response surface method is adopted to efficiently evaluate the wall response. Deflection data for 49 wall sections from 11 case histories are collected to characterize the model error of the finite element method for evaluating the deflections at various points. A braced excavation project in Hangzhou, China is chosen to illustrate the effectiveness of the proposed procedure. The results indicate that the soil parameters could be updated more significantly for the updating that uses the deflection data at multiple points than that only uses the maximum deflection data. The predicted deflections from the updated parameters agree fairly well with the field observations. The main significance of the proposed procedure is that it improves the updating efficiency of the soil parameters without adding monitoring effort compared with the traditional method that uses the maximum deflection data.