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The scale of fluctuation (SOF) is the critical parameter to describe the soil spatial variability, which significantly influences the embedded geostructures. Due to the limited data in the horizontal direction, horizontal SOF estimation is relatively challenging and not well studied yet. This paper aims to develop an efficient convolutional neural network (CNN)-based approach for estimating the horizontal SOF by coupling random field and limited CPT data. Two or three columns (i.e. pseudo-CPT) were selected from the simulated 2D random field with prescribed SOF at the same spacing and combined into a two-dimensional matrix as input data to train the CNN model, namely CNN2 and CNN3 models. The dataset of CNN2 and CNN3 models contains 196,670 and 149,420 samples. Results on the training and testing datasets show that the trained CNN model has a good estimation performance as the mean squared error value is less than 0.1 and the correlation coefficient value is larger than 0.99. The effectiveness of trained CNN models was further verified by the new simulated CPT data with untrained SOF from the random field and CPT data from real site in Hollywood, South Carolina. The excellent agreement indicates that the trained CNN model has the ability to capture the horizontal SOF for limited CPT data from the actual project. Finally, the collected CPT data from the Shanghai site was applied for application. The COV of the estimated results of CNN3 and CNN2 models for the Shanghai site is 0.09 and 0.40, indicating the estimation performance of the CNN3 model has less variability than the CNN2 model. The proposed method provides the potential to characterize the soil spatial variability using very limited CPT data.

It is important to be able to construct a site-specific multivariate probability density function (PDF) of soil parameters based on limited and incomplete site-specific investigation data a...

This study investigates the possibility of representing the effective Young’s modulus (Eeff) for a footing problem supported on a spatially variable medium - the Young’s modulus actually “felt” by the footing - using a spatial average. The Eeff is simulated by a homogenization procedure that matches the responses between a random finite element analysis (RFEA) and a homogeneous finite element analysis. Emphasis is placed on whether the spatial average can well represent the numerical value of Eeff in each spatially varying realization, not just the statistics of Eeff within an ensemble (a weaker requirement). It is found that the conventional spatial averaging model that treats all soil regions equally important cannot satisfactorily represent Eeff. Extensive numerical results show that the concept of “mobilization” is essential: highly mobilized soil regions close to the footing should be given larger weights than non-mobilized remote regions. Moreover, the non-uniform weights can be prescribed prior to RFEA, that is, they do not depend on the specific response corresponding to a specific random field realization. The “prescribed mobilization” for the spatially variable Young’s modulus can be contrasted with the “emergent” mobilized shear strength in a spatially variable medium that results from the emergent nature of the critical failure path – it cannot be predicted prior to random finite element analysis. A key contribution of this paper is the development of a simple method based on the “pseudo incremental energy” to estimate the non-uniform weights for the spatial averaging using a single run of a homogeneous finite element analysis.

This paper develops a Bayesian approach for model comparison and characterization of the bivariate distribution of c′ and ϕ′ using limited site-specific data. The copula approach is presented to model the bivariate distribution of c′ and ϕ′. The Bayesian model comparison method is developed to select the most probable bivariate distribution model of c′ and ϕ′. The most probable model is used to characterize the joint probability density function (PDF) of c′ and ϕ′ under the Bayesian framework. The developed approach is illustrated and validated using real data of c′ and ϕ′ for clays from the core wall of Xiaolangdi rockfill dam in China.

A multivariate probability distribution model for seven parameters of coarse-grained soils is constructed based on the SAND/7/2794 database that was compiled by the authors. It is shown that the multivariate probability distribution captures the correlation behaviors in the database among the seven parameters. This multivariate distribution model serves as a prior distribution model in the Bayesian analysis and can be updated into the posterior distribution of the design soil parameter when multivariate site-specific information is available. It is shown that this Bayesian analysis is conceptually similar to what is routinely carried out in practice, which utilizes information from comparable sites to supplement limited site-specific information. The resulting posterior distribution from Bayesian analysis merely combines different uncertainties associated with different sources of “correlated” information in a more consistent way. In this paper, the parameters for the posterior distribution of the design soil parameter are summarized into engineer-friendly tables ( Tables 9 and 10 ) so that engineers do not need to conduct the actual Bayesian analysis. Caution should be taken in extrapolating the results of this paper to cases that are not covered by SAND/7/2794, because the resulting posterior distribution can be misleading. This caveat applies to conventional regression equations as well.

This paper aims to investigate the impact of sample size on geotechnical probabilistic model identification. First, the copula approach is presented to model the bivariate distribution of geotechnical parameters. Thereafter, the AIC scores are adopted to identify the best-fit marginal distribution and copula. Second, the variation of AIC scores because of small sample size is investigated using simulated data. Finally, the impact of the variation of AIC scores on identification of the best-fit marginal distribution and copula is examined. The minimum sample sizes for geotechnical data are also suggested to obtain a correct identification of the probabilistic models. The results indicate that the AIC scores estimated from a small sample exhibit large variation. The variation of the AIC scores has a significant impact on probabilistic model identification. The marginal distributions and copulas have a low percentage of correct identification when sample size is small. The percentages of correct identification for the marginal distributions and copulas increase with increasing sample size. The correlation coefficient between geotechnical parameters has a much larger impact on probabilistic model identification than the COV of geotechnical parameters. The suggested minimum sample sizes for geotechnical data are useful for guiding practical geotechnical site investigation.

The purpose of this study is to investigate whether the effective Young’s modulus (Eeff) for a spatial variable soil mass can be strongly correlated to any spatial average. The spatially variable Young’s modulus of the soil mass is modeled as a stationary lognormal random field, and the Eeff of the soil mass is simulated by random field finite element analysis. Spatial averages are calculated from the input random field. If a strong correlation exists, it is possible to replace a random field analysis by a simpler random variable analysis. Two classes of problems are considered: a soil cube subjected to displacement-controlled compression and a footing problem. For the soil cube problem, Eeff is found to be strongly correlated to a suitable spatial average. However, for the footing problem, only the statistics (mean and standard deviation) of Eeff can be well approximated by a suitable spatial average, but the correlation is not strong. It is possible that the two classes of problems behave differently because the finite elements in the soil cube are mobilized uniformly, whereas those in the footing problem are mobilized non-uniformly. This leads to a weighted spatial average model that applies a different weight on the log modulus of each finite element over the domain being averaged.

Although spatially varying quantities in real life (e.g., mechanical properties of soils) often contain a linear or nonlinear trend, stationary random fields with zero trend are often used ...

The scale of fluctuation (SOF) of a spatially variable soil property has been known to be the most important parameter that characterizes the effect of spatial averaging, whereas the type o...

Inherent spatial variability (ISV) of design soil properties (e.g., effective friction angle φ ′) can be incorporated into probability-based geotechnical analyses and designs using random field models. Defining a random field model includes determination of random field parameters (i.e., mean μ , standard deviation σ , and scale of fluctuation λ ) and the correlation function that specifies the spatial correlation of the concerned design soil property (e.g., φ ′) at different locations. This is, however, a challenging task at a given site due to a lack of direct test data of design soil properties and various uncertainties (e.g., transformation uncertainty) arising during site investigation. This paper develops Bayesian approaches for probabilistic characterization of the ISV of φ ′ using indirect test data (i.e., cone penetration test (CPT) data) and prior knowledge, which identify random field parameters of φ ′ through Markov Chain Monte Carlo Simulation (MCMCS) and, simultaneously, make use of Gaussian copula to select the most probable correlation function M ⁎ among a pool of candidate correlation functions based on MCMCS samples. The proposed Bayesian approaches account, rationally and transparently, for the transformation uncertainty associated with the transformation model between φ ′ and CPT data. The proposed approaches are illustrated and validated using real-life and simulated CPT data. Results show that the proposed approaches properly identify the random field model (including μ , σ , λ , and M ⁎ ) of φ ′ using project-specific CPT data, and the random field parameters of φ ′ depend on the correlation function used to interpret CPT data. In addition, the suitability of MCMCS in Bayesian probabilistic characterization of soil properties is highlighted, particularly for the cases with a limited number of test data.

An efficient analytical method for quantifying the correlation between performance functions of different slope failure modes in spatially variable soils is proposed, and its performance in slope system reliability analysis is investigated. First, a new correlation coefficient (NCC) is proposed to evaluate the correlation among slope failure modes considering spatial variability. For comparison and verification, the simulation-based correlation coefficient (SCC) is also presented. Second, appying these two types of correlation coefficients, the effects of soil spatial variability on the representative slip surfaces (RSSs) and the system probability of slope failure are investigated using different system reliability methods, including a probabilistic network evaluation technique, a risk aggregation approach, and a bimodal bounds method. A single-layered cohesive slope is investigated to illustrate the validity of the proposed NCC. The results indicate that the proposed NCC can efficiently and accurately quantify the correlation among slope failure modes considering soil spatial variability. The number of RSSs indicated by the NCC is in good agreement with the number obtained using the SCC. The system failure probabilities of slope stability obtained with the SCC and the NCC using a risk aggregation approach are generally comparable. Also, the system reliability bounds of slope stability obtained using the NCC are relatively close together and comparable to those obtained using the SCC. Thus, the NCC shows good performance when evaluating the correlation among slope failure modes, and was effectively applied to analyze a single-layered cohesive slope considering soil spatial variability.

This paper shows that the transformation uncertainty varies in space and can be decomposed into (a) a slowly fluctuating component contributed by the systematic bias for a local site and (b) a rapidly fluctuating component partly contributed by the measurement error. Based on a piezocone database, the slowly fluctuating component is found to have a vertical scale of fluctuation (SOF) of 17–60 m, larger than the vertical SOFs for most soil properties and characteristic dimensions for many geotechnical structures. This paper provides an empirical basis to justify the assumption that the transformation uncertainty is fully correlated in space.

The mobilized shear strength (τfm) of a soil is the most important design parameter governing ultimate limit states. The authors have proposed some models and equations to characterize several useful aspects of the mean, variance, and probability density function of τfm for a two dimensional (2D) spatially variable clay specimen. This paper demonstrates that the models and equations developed for 2D can be extended to three-dimensional (3D) cases. An important class of 3D cases are examined in detail: two horizontal scales of fluctuation (SOF) are equal, and both are larger than the vertical SOF. A key observation for 3D cases is that the averaging effect over the potential slip planes is significantly stronger than that for 2D cases. The upshot is that the phenomenon of the critical SOF (e.g., SOF producing the lowest mean value of τfm) is less clear for 3D cases.

In this paper, the uplift behavior of helical anchors in clay is investigated using the finite element limit analysis. According to the observations on the failure mechanism, the factors that influence the model uncertainty of the cylindrical shear method are examined. The finite element limit analysis is used to remove the correlation between the model factor M for the cylindrical shear method, which is defined as a ratio between the measured capacities Qu,m and the capacities Qu,c calculated from the cylindrical shear method, with the input parameters. A regression equation f is then proposed. Using the load tests database including 78 laboratory model-scale tests and 25 field full-scale tests, the model factor M’ = Qu,m/[fQu,c] is represented as a lognormal random variable with mean = 0.92 and coefficient of variation (cov) = 0.16. Finally, the accuracy of the modified cylindrical shear method (i.e., fQu,c) is examined using 19 large-deformation analyses.

This paper reviews previous studies on developments and applications of response surface methods (RSMs) in different slope reliability problems. Based on the review, four types of soil slope reliability analysis problems are identified from the literature, including single-layered soil slope reliability problem ignoring spatial variability, single-layered soil slope reliability problem considering spatial variability, multiple-layered soil slope reliability problem ignoring spatial variability, and multiple-layered soil slope reliability problem considering spatial variability, which are referred to as “Type I–IV problems” in this study. Then, the computational efficiency and accuracy of four commonly-used RSMs (namely single quadratic polynomial-based response surface method (SQRSM), single stochastic response surface method (SSRSM), multiple quadratic polynomial-based response surface method (MQRSM), and multiple stochastic response surface method (MSRSM)) are systematically compared for cohesive and c–ϕ slopes, and their feasibility and validity in the four types of slope reliability problems are discussed. Based on the comparison, some suggestions for selecting relatively appropriate RSMs in slope reliability analysis are provided: (1) SQRSM is suggested as a suitable method for the single-layered soil slope reliability problem ignoring spatial variability (i.e., Type I problem); (2) MQRSM is applicable to the multiple-layered soil slope reliability problem ignoring spatial variability (i.e., Type III problem); and (3) MSRSM is suggested to solve slope reliability problems (including single-layered and multiple-layered slopes) considering spatial variability (i.e., Type II and IV problems).

Because information collected in a site investigation is limited, it is not possible to obtain actual values for the mean, standard deviation, and scale of fluctuation for a soil property of interest. The deviation between the estimated values and the actual values is called the statistical uncertainty. There are at least two schools of thought on how to model the statistical uncertainty: frequentist thought and Bayesian thought. The purpose of this paper is to discuss their philosophical difference, to show how to quantify the statistical uncertainty based on these two distinct schools of thought, and to compare their performances. To quantify the statistical uncertainty, the confidence interval will be used for the frequentist school of thought, whereas the posterior probability distribution will be used for the Bayesian school of thought. Examples will be presented to compare the performances of these two schools of thought in terms of their consistencies. The results show that, in general, the Bayesian thought performs better in terms of consistency. In particular, the Markov chain Monte Carlo method is recommended when the amount of information available is very limited.

The cone penetration test (CPT) is a method widely used for site characterization as measurements are continuous, fast and economical compared to laboratory testing. However unlike laboratory testing that quantifies soil properties directly, readings obtained from CPT are indirect and needs to be interpreted. Hence many correlations have been proposed to infer a wide range of soil properties from CPT readings. Existing correlations have discovered separate relationships between normalized cone tip resistance (Qt) and effective friction angles (ϕ’), as well as Qt and overconsolidation ratio (OCR). This study employs the press-replace method (PRM), a novel simplified numerical technique, to perform systematic and extensive undrained investigations of CPT in a modified Cam-clay (MCC) soil model. It is the first comprehensive numerical study with an advanced soil model that considers rough cone-soil interactions. The PRM numerical approach overcomes the need to assume the CPT process as analogous to spherical or cylindrical cavity expansions, therefore produces results that are more authentic to real CPT. From the large database of numerical results, an equation is formulated that provides a framework on how Qt is related to both ϕ’ and OCR, rather than separately to ϕ’ or OCR as described in previous correlations. The proposed equation is then simplified so that ϕ’ and OCR can be easily estimated from Qt. Numerical results and predictions from the new equations are then compared to soil behaviour charts, other correlations and field data from different sites.

This paper aims to propose a bootstrap method for characterizing the uncertainty in probabilistic models and its effect on geotechnical reliability. First, the copula theory is briefly introduced. Second, both the uncertainties in parameters and type of the best-fit marginal distributions and copulas are characterized by the bootstrap method. Finally, four load-test datasets of load-settlement curves of piles are used to illustrate the proposed method. The serviceability limit state reliability analysis of piles is presented to illustrate the practical application of the proposed method. The results indicate that the bootstrap method can effectively characterize the uncertainty in probabilistic models derived from a small sample. Through bootstrapping, the uncertainties in both the parameters and type of the specified probabilistic models are simultaneously incorporated into geotechnical reliability analyses. The probability of failure of piles is represented by a confidence interval at a specified confidence level instead of a single fixed probability, which enables the engineers to make a more informed decision.

This paper aims to propose a bootstrap method for characterizing the effect of uncertainty in shear strength parameters on slope reliability. The procedure for a traditional slope reliability analysis with fixed distributions of shear strength parameters is presented first. Then, the variations of the mean and standard deviation of shear strength parameters and the Akaike Information Criterion values associated with various distributions are studied to characterize the uncertainties in distribution parameters and types of shear strength parameters. The reliability of an infinite slope is presented to demonstrate the validity of the proposed method. The results indicate that the bootstrap method can effectively model the uncertain probability distributions of shear strength parameters. The uncertain distributions of shear strength parameters have a significant influence on slope reliability. With the bootstrap method, the slope reliability index is represented by a confidence interval instead of a single fixed index. The confidence interval increases with increasing factor of slope safety. Considering both the uncertainties in distribution parameters and distribution types of shear strength parameters leads to a higher variation and a wider confidence interval of reliability index.

In this study, we show that existing simplified reliability-based design methods, including the partial factor method and the quantile-based method (quantile value method; QVM), are not robust over design scenarios with variable degree of redundancy. This variable degree of redundancy is quite common in geotechnical design. For instance, a pile group has more redundancy than a single pile. Less obviously, a pile embedded in multiple soil layers has more redundancy than one embedded in a single soil layer. Implementing a fixed set of partial factors or a fixed quantile will not produce designs with uniform reliability indices. This paper shows that the degree of redundancy can be effectively quantified by the concept of an “effective random dimension” (ERD), and it proposes a practical method of estimating ERD for reliability-based design. It is shown by numerical examples that by incorporating ERD, the ERD-QVM outperforms the original QVM in achieving a more uniform reliability level across different design scenarios.

Slope reliability under incomplete probability information is a challenging problem. In this study, three copula-based approaches are proposed to evaluate slope reliability under incomplete probability information. The Nataf distribution and copula models for characterizing the bivariate distribution of shear strength parameters are briefly introduced. Then, both global and local dispersion factors are defined to characterize the dispersion in probability of slope failure. Two illustrative examples are presented to demonstrate the validity of the proposed approaches. The results indicate that the probabilities of slope failure associated with different copulas differ considerably. The commonly used Nataf distribution or Gaussian copula produces only one of the various possible solutions of probability of slope failure. The probability of slope failure under incomplete probability information exhibits large dispersion. Both global and local dispersion factors increase with decreasing probability of slope failure, especially for small coefficients of variation and strongly negative correlations underlying shear strength parameters. The proposed three copula-based approaches can effectively reduce the dispersion in probability of slope failure and significantly improve the estimate of probability of slope failure. In comparison with the Nataf distribution, the copula-based approaches result in a more reasonable estimate of slope reliability.

This paper presents a critical evaluation of the model factor M = qu,m/qu,c for Eurocode 7 calculating the bearing capacity of circular footings on dense sand, where qu,m = measured capacity and qu,c = Eurocode 7 calculated capacity. Regression analysis is required to remove the dependency of M on the input parameters. Because the input parameters cannot be varied systematically in load tests, previous studies showed that finite-element limit analysis (FELA) can be used as an alternative to load tests for regression. This is further verified from the model factor MFELA = qu,m/qu,FELA with a mean of 1 and a coefficient of variation (cov) of 0.1, where qu,FELA = FELA predicted capacity. A correction factor (Ms = qu,FELA/qu,c) is next defined, which can be decomposed as a product of a systematic part f and a residual part η (i.e., Ms = fη), which is modeled as a lognormal random variable with mean = 1 and cov = 0.11. Finally, a new model factor (M′ = qu,m/q′u,c = qu,m/fqu,c) is defined. The model sta...

This study proposes a set of simple equations for the mean value and variance of the mobilized shear strength for spatially variable soil masses subjected to uniform stress states. These equations are fairly effective in explaining the complicated behaviors for the mobilized shear strengths, regardless of stress states (e.g., compression or shear), spatial variability patterns (e.g., isotropic or anisotropic), and inherent mean and variance of the random field. Two mechanisms that affect the behaviors of the mobilized shear strength are identified: (1) the averaging effect along the potential slip curves, and (2) the emergent feature of a critical slip curve. The emergence is associated with the slip curve with the minimum averaged strength. In any realization of the random field, it is not possible to know a priori the location of the minimum average; hence, it would not coincide with a prescribed average. It is shown that the well-known phenomenon of critical scale of fluctuation is the result o...

In this paper, the formulation of a lower-bound limit analysis for axisymmetric problems by means of finite elements leads to an optimization problem with a large number of variables and constraints. For the Mohr-Coulomb criterion, it is shown that these axisymmetric problems can be solved by second-order cone programming (SOCP). First, a brief introduction to SOCP is given and how axisymmetric lower-bound limit analysis can be formulated in this way is described. Through the use of an efficient toolbox (MOSEK or SDPT3), large-scale SOCP problems can be solved in minutes on a desktop computer. The method is then applied to estimate the collapse load of circular footings and uplift capacity of single or multiplate circular anchors. By comparing the present analysis with the results reported in the literature, it is shown that the results obtained from the proposed method are accurate and computationally more efficient than the numerical lower-bound limit analysis incorporated with linear programming.

Recent studies have shown that the sample path smoothness in soil spatial variability can have a significant effect on the failure probability of geotechnical problems. The purpose of the current study is to propose a procedure that can identify the sample path smoothness based on site investigation data. It is shown that two factors determine whether or not the sample path smoothness can be identified: the type of auto-correlation function (ACF) model and the parameter estimation method. In order to identify the sample path smoothness, a non-classical two-parameter ACF model, such as the powered exponential (PE) model and Whittle-Matern (WM) model, must be adopted together with the maximum likelihood (ML) method. The method of moments (MM) is incapable of identifying the sample path smoothness regardless of the ACF model type, classical or otherwise, although it is effective in identifying the scale of fluctuation (SOF). Between the two non-classical ACF models, the WM model is more flexible because it covers a wider range of sample path smoothness than the PE model. Neither the PE model nor the WM model is able to model the “hole effect” (non-monotonic auto-correlation). The development of a sufficiently flexible non-classical model that can simultaneously identify SOF, sample path smoothness, and hole effect remains an open research question.

Various empirical equations have been proposed to estimate the unconfined compressive strength (qu) of cement-treated soils based on the proportions of water, cement, and clay or based on measured early strength. Most equations provide only a point estimate of qu and no measure of uncertainties associated with the estimation. As the design’s safety level is governed by the uncertainties involved, it is essential to quantify uncertainties associated with the estimation. The coefficient of variation (COV) is taken as a measure of uncertainties. It is straightforward to estimate the mean and COV of qu based on a single parameter. Similar estimation based on multiple parameters, which may be intercorrelated, is not as straightforward. This paper aims to integrate multiple parameters (water-to-cement ratio, cement content, and 7-day strength) that may contribute to the estimation of 91-day strength using a multivariate normal distribution. The multivariate normal model can be used to derive the mean, C...

The adequacy of two approximate methods based on incomplete information, namely method P and method S, for constructing multivariate distributions with given marginal distributions and covariance has not been studied systematically. This article aims to study the errors of the method P and method S. First, the method P and method S as well as the exact method are presented. Second, the performance of the two approximate methods is evaluated based on their abilities to match exact solutions for system probabilities of failure. Finally, an illustrative example of a parallel system is investigated to demonstrate the errors associated with the two methods. The results indicate that the errors in system probabilities of failure for the two methods highly depend on the level of system probability of failure, the performance function underlying the system, and the degree of correlation. Such errors increase greatly with decreasing system probabilities of failure. When the target system probability of failure is ...

A copula-based method is presented to investigate the impact of copulas for modeling bivariate distributions on system reliability under incomplete probability information. First, the copula theory for modeling bivariate distributions as well as the tail dependence of copulas are briefly introduced. Then, a general parallel system reliability problem is formulated. Thereafter, the system reliability bounds of the parallel systems are generalized in the copula framework. Finally, an illustrative example is presented to demonstrate the proposed method. The results indicate that the system probability of failure of a parallel system under incomplete probability information cannot be determined uniquely. The system probabilities of failure produced by different copulas differ considerably. Such a relative difference in the system probabilities of failure associated with different copulas increases greatly with decreasing component probability of failure. The maximum ratio of the system probabilities of failure for the other copulas to those for the Gaussian copula can happen at an intermediate correlation. The tail dependence of copulas has a significant influence on parallel system reliability. The copula approach provides new insight into the system reliability bounds in a general way. The Gaussian copula, commonly used to describe the dependence structure among variables in practice, produces only one of the many possible solutions of the system reliability and the calculated probability of failure may be severely biased.

Finite-element analysis of many soil-structure interaction problems involves material zones of widely differing stiffnesses. The large relative differences in material stiffnesses usually result in an ill-conditioned system for which practical preconditioners become ineffec- tive for the iterative solution of such systems. The study suggests that a block diagonal preconditioner can exploit such differences. However, the theoretical preconditioner is expensive for practical use. Less costly block diagonal preconditioners are numerically evaluated for various inexact forms of the blocks in conjunction with conjugate gradient solver. The study includes analysis of two representative soil-structure in- teraction problems and proposes two simplified block diagonal preconditioners that effectively mitigate such material ill conditionings. DOI: 10.1061/(ASCE)GM.1943-5622.0000197. © 2013 American Society of Civil Engineers. CE Database subject headings: Stiffness; Soil-structure interactions; Preconditions; Blocks; Finite element method; Material stiffness. Author keywords: Stiffness ratio; Soil-structure interaction problem; Block diagonal preconditioner; Preconditioned conjugate gradient.

The spatial averaged shear strength is associated with a prescribed finite size spatial domain. It is not intended to cover the mobilized strength along a slip curve arising as a solution to a boundary value problem in a spatially variable medium. Nonetheless, the concept of strength in soil mechanics is fundamentally related to the mobilized strength along a slip curve. In this study, a plane strain soil specimen 12.8 m wide by 48 m high is subjected to undrained compression and shear via finite element analysis (FEA). The yield stress recorded before FEA fails to converge is defined as the mobilized strength. This mobilized strength is compared with spatial average over the entire domain and the line average along the critical slip curve. Numerical results show that the statistics of the mobilized strength is close to the statistics produced by the minimum of line averages along potential slip curves. There are two important details associated with this proposed minimum line average mechanism. First, the orientations of the critical slip curves are primarily controlled by mechanics, rather than spatial variation, but the vertical positions are fairly random, depending on the realizations of the random fields. In other words, the orientations of the slip curves are close to those produced in a homogeneous medium. Spatial variation perturbs the slip curves in two aspects: (1) orientation and (2) regularity of the curve. Both aspects are secondary. As such, potential slip curves can be viewed as roughly parallel to the critical slip curve. Second, the line average for the critical slip curve is the minimum value over the line averages for the potential slip curves. Based on this relatively simple line average mechanism, it is possible to compare the statistics of spatial average with those of the mobilized strength. The spatial variability scenarios under which spatial average is approximately applicable are identified. It is important to note that only simple and uniform stress states are studied in this paper.

SUMMARY Non-associated flow rule is essential when the popular Mohr–Coulomb model is used to model nonlinear behavior of soil. The global tangent stiffness matrix in nonlinear finite element analysis becomes nonsymmetric when this non-associated flow rule is applied. Efficient solution of this large-scale nonsymmetric linear system is of practical importance. The standard Krylov solver for a non-symmetric solver is Bi-CGSTAB. The Induced Dimension Reduction [IDR(s)] solver was proposed in the scientific computing literature relatively recently. Numerical studies of a drained strip footing problem on homogenous soil layer show that IDR(s=6) is more efficient than Bi-CGSTAB when the preconditioner is the incomplete factorization with zero fill-in of global stiffness matrix Kep (ILU(0)-Kep). Iteration time is reduced by 40% by using IDR(s=6) with ILU(0)-Kep. To further reduce computational cost, the global stiffness matrix Kep is divided into two parts. The first part is the linear elastic stiffness matrix Ke, which is formed only once at the beginning of solution step. The second part is a low-rank matrix Δ, which is re-formed at each Newton–Raphson iteration. Numerical studies show that IDR(s=6) with this ILU(0)-Ke preconditioner is more time effective than IDR(s=6) with ILU(0)-Kep when the percentage of yielded Gauss points in the mesh is less than 15%. The total computation time is reduced by 60% when all the recommended optimizing methods are used. Copyright © 2013 John Wiley & Sons, Ltd.

Because most geotechnical analyses may involve elastoplastic geomaterials, a robust solution scheme is of critical importance to the efficiency of the entire finite-element (FE) computation. To accelerate the Krylov subspace iterative methods, some preconditioning techniques have been developed based on the factorization of elastic stiffness. However, the idea of constructing a preconditioner based on elastic stiffness is heuristic. In this study, a class of Neumann expansion preconditioners constructed from the constant (elastic) partition and varying (plastic) partition of the elastoplastic stiffness matrix was proposed. Based on two numerical examples, the performances of truncated Neumann expansion preconditioners were examined with associated and nonassociated soil plasticity considered, respectively. It is interesting to note that the convergence behaviors of truncated Neumann expansion preconditioners closely depended on the approximation to the elastic stiffness part as well as the truncat...

It is common to construct a consistent multivariate distribution from non-normal marginals and Pearson product–moment correlations using the well known translation approach. A practical variant of this approach is to match the Spearman rank correlations of the measured data, rather than the Pearson correlations. In this paper, the performance of these translation methods is evaluated based on their abilities to match the following exact solutions from one benchmark bivariate example where the joint distribution is known: (1) high order joint moments, (2) joint probability density functions (PDFs), and (3) probabilities of failure. It is not surprising to find significant errors in the joint moments and PDFs. However, it is interesting to observe that the Pearson and Spearman methods produce very similar results and neither method is consistently more accurate or more conservative than the other in terms of probabilities of failure. In addition, the maximum error in the probability of failure may not be associated with a large correlation. It can happen at an intermediate correlation.

Biot's symmetric indefinite linear systems of equations are commonly encountered in finite-element computations of geotechnical problems. The development of efficient solution methods for Biot's linear systems of equations is of practical importance to geotechnical software packages. In conjunction with the Krylov-subspace iterative method symmetric quasi-minimal residual (SQMR), some zero-level fill-in incomplete factorization preconditioning techniques including a symmetric successive overrelaxation (SSOR) type method and several zero-level incomplete LU [ILU(0)] methods are investigated and compared for Biot's symmetric indefinite linear systems of equations. Numerical experiments are carried out based on three practical geotechnical problems. Numerical results indicate that ILU(0) preconditioners are classical and generally efficient when adequately stabilized. However, the tunnel problem provides a counterexample demonstrating that ILU(0) preconditioners cannot be fully stabilized by prelimin...

Determining the joint probability distribution of correlated non-normal geotechnical parameters based on incomplete statistical data is a challenging problem. This paper proposes a Gaussian copula-based method for modelling the joint probability distribution of bivariate uncertain data. First, the concepts of Pearson and Kendall correlation coefficients are presented, and the copula theory is briefly introduced. Thereafter, a Pearson method and a Kendall method are developed to determine the copula parameter underlying Gaussian copula. Second, these two methods are compared in computational efficiency, applicability, and capability of fitting data. Finally, four load-test datasets of load-displacement curves of piles are used to illustrate the proposed method. The results indicate that the proposed Gaussian copula-based method can not only characterize the correlation between geotechnical parameters, but also construct the joint probability distribution function of correlated non-normal geotechnical parameters in a more general way. It can serve as a general tool to construct the joint probability distribution of correlated geotechnical parameters based on incomplete data. The Gaussian copula using the Kendall method is superior to that using the Pearson method, which should be recommended for modelling and simulating the joint probability distribution of correlated geotechnical parameters. There exists a strong negative correlation between the two parameters underlying load-displacement curves. Neglecting such correlation will not capture the scatter in the measured load-displacement curves. These results substantially extend the application of the copula theory to multivariate simulation in geotechnical engineering.

Geotechnical design parameters are typically estimated based on the transformations from site investigation results. In general, one expects the transformation uncertainty to change depending on the number and type of sites in the database. This study tries to address the following two issues pertaining to the transformation uncertainty: (a) how transformation uncertainties change with the number and type of sites in the database and (b) whether transformation uncertainties will eventually fall within a narrow range when a “generic” transformation is developed from a sufficiently large database. This study also attempts to propose a framework to establish such a generic transformation and quantify its uncertainty. This framework is demonstrated by the transformation between piezocone CPTU data and undrained shear strengths ( S u ) of clays. It was found that the CPTU– S u transformation and its uncertainty is site or region-dependent, and the “local” transformation equation from one site may not be applicable to another site, both in terms of the mean trend (which is well known) as well as the coefficient of variation (c.o.v.). An approach is proposed to develop the generic CPTU– S u transformation equations that can be applied for downstream reliability analysis or design in the absence of local data. Sensitivity analysis shows that it requires data from at least 15 sites with the accompanying implicit assumption that sufficient geographical coverage typically implies sufficient geologic diversity to reliably build such “generic” transformation equations.

SUMMARY The finite element (FE) simulation of large-scale soil–structure interaction problems (e.g. piled-raft, tunnelling, and excavation) typically involves structural and geomaterials with significant differences in stiffness and permeability. The symmetric quasi-minimal residual solver coupled with recently developed generalized Jacobi, modified symmetric successive over-relaxation (MSSOR), or standard incomplete LU factorization (ILU) preconditioners can be ineffective for this class of problems. Inexact block diagonal preconditioners that are inexpensive approximations of the theoretical form are systematically evaluated for mitigating the coupled adverse effects because of such heterogeneous material properties (stiffness and permeability) and because of the percentage of the structural component in the system in piled-raft foundations. Such mitigation led the proposed preconditioners to offer a significant saving in runtime (up to more than 10 times faster) in comparison with generalized Jacobi, modified symmetric successive over-relaxation, and ILU preconditioners in simulating piled-raft foundations. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY This paper aims to propose a procedure for modeling the joint probability distribution of bivariate uncertain data with a nonlinear dependence structure. First, the concept of dependence measures is briefly introduced. Then, both the Akaike Information Criterion and the Bayesian Information Criterion are adopted for identifying the best-fit copula. Thereafter, simulation of copulas and bivariate distributions based on Monte Carlo simulation are presented. Practical application for serviceability limit state reliability analysis of piles is conducted. Finally, four load–test datasets of load–displacement curves of piles are used to illustrate the proposed procedure. The results indicate that the proposed copula-based procedure can model and simulate the bivariate probability distribution of two curve-fitting parameters underlying the load–displacement models of piles in a more general way. The simulated load–displacement curves using the proposed procedure are found to be in good agreement with the measured results. In most cases, the Gaussian copula, often adopted out of expedience without proper validation, is not the best-fit copula for modeling the dependence structure underlying two curve-fitting parameters. The conditional probability density functions obtained from the Gaussian copula differ considerably from those obtained from the best-fit copula. The probabilities of failure associated with the Gaussian copula are significantly smaller than the reference solutions, which are very unconservative for pile safety assessment. If the strong negative correlation between the two curve-fitting parameters is ignored, the scatter in the measured load–displacement curves cannot be simulated properly, and the probabilities of failure will be highly overestimated. Copyright © 2011 John Wiley & Sons, Ltd.

This paper proposes a quantile-based approach for calibrating reliability-based partial factors that is based on an equivalence principle between the design quantiles for the random variables and the target reliability. The potential advantage is to allow a single design quantile to maintain a more uniform reliability over a wider range of design parameters. The proposed approach does not require the capacity to be lumped as a single lognormal random variable in the Load and Resistance Factor Design (LRFD) format nor does it require tedious segment by segment optimization of the resistance factors in the Multiple Resistance Factor Design (MRFD) format. The applicability, usefulness, and limitation of the proposed approach are illustrated using three examples. The results show that the proposed approach is able to maintain a uniform reliability over a wider range of design parameters with a single design quantile, which cannot be easily attained by other calibration methods such as the First-Order Reliability Method.

Simulation of Markov chain samples using the Metropolis–Hastings algorithm is useful for reliability estimation. Subset simulation is an example of the reliability estimation method utilizing this algorithm. The efficiency of the simulation is governed by the correlation between the simulated Markov chain samples. The objective of this study is to propose a modified Metropolis–Hastings algorithm with reduced chain correlation. The modified algorithm differs from the original in terms of the transition probability. It has been verified that the modified algorithm satisfies the reversibility condition and therefore the simulated samples follow the target distribution for the correct theoretical reasons. When applied to subset simulation, the modified algorithm produces a more accurate estimate of failure probability as indicated by a lower coefficient of variation and a lower mean square error. The advantage is more significant for small failure probability. Examples of soil slope with spatially variable properties were presented to demonstrate the applicability of the proposed modification to reliability estimation of engineering problems. It was found that the modified algorithm produces a more accurate estimator over the range of random dimensions studied.

This study addresses the complexity of slope reliability problems based on limit equilibrium methods (LEMs). The main focus is on the existence of multiple failure modes that poses difficulty to many LEM-based slope reliability methods. In particular, when weak seams are present, the failure modes associated with those seams may be difficult to detect. A systematic way of searching the failure modes is proposed, and its robustness over slopes with or without weak seams is demonstrated. It is found that in the presence of weak seams, assuming circular slip surfaces may cause underestimation of slope failure probability. The conclusion of the study promotes the use of finite elements as the stability method for reliability evaluation because it is not necessary to search for failure surfaces in finite-element stability analysis.

Direct measurement of the soil-water characteristic curve (SWCC) is costly and time consuming. A first-order estimate from statistical generalization of experimental data belonging to soils with similar textural and structural properties is useful. A simple approach is to fit the data with a nonlinear function and to construct an appropriate probability model of the curve-fitting parameters. This approach is illustrated using sandy clay loam, loam, loamy sand, clay, and silty clay data in Unsaturated Soil Database. This paper demonstrates that a lognormal random vector is suitable to model the curve-fitting parameters of the SWCC. Other probability models using normal, gamma, Johnson, and other distributions do not provide better fit than the proposed lognormal model. The engineering impact of adopting a probabilistic SWCC is briefly discussed by studying the uncertainty of unsaturated shear strength due to the uncertainty of SWCC.

Several popular convergence criteria which are frequently used in practical finite element computations are investigated for two kinds of systems: the symmetric positive definite linear system and the symmetric indefinite system involving two distinct variables (displacement and pore fluid pressure). For the first system, the relative residual norm and the relative improvement norm are satisfactory as long as boundary fixities are handled appropriately. For the second system, the relative improvement norm must be adopted with greater care. It was further shown numerically that decoupled relative residual norms can be attractive alternates to the current global stopping criterion.