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The random finite difference method (RFDM) is a popular approach to quantitatively evaluate the influence of inherent spatial variability of soil on the deformation of embedded tunnels. However, the high computational cost is an ongoing challenge for its application in complex scenarios. To address this limitation, a deep learning-based method for efficient prediction of tunnel deformation in spatially variable soil is proposed. The proposed method uses one-dimensional convolutional neural network (CNN) to identify the pattern between random field input and factor of safety of tunnel deformation output. The mean squared error and correlation coefficient of the CNN model applied to the newly untrained dataset was less than 0.02 and larger than 0.96, respectively. It means that the trained CNN model can replace RFDM analysis for Monte Carlo simulations with a small but sufficient number of random field samples (about 40 samples for each case in this study). It is well known that the machine learning or deep learning model has a common limitation that the confidence of predicted result is unknown and only a deterministic outcome is given. This calls for an approach to gauge the model's confidence interval. It is achieved by applying dropout to all layers of the original model to retrain the model and using the dropout technique when performing inference. The excellent agreement between the CNN model prediction and the RFDM calculated results demonstrated that the proposed deep learning-based method has potential for tunnel performance analysis in spatially variable soils.

The story of statistics in geotechnical engineering can be traced to Lumb’s classical Canadian Geotechnical Journal paper on “The Variability of Natural Soils” published in 1966. In parallel, the s...

In site investigation, it is common to conduct some soundings to explore greater depths that are not explored by remaining soundings. This produces the scenario of nonlattice data, meaning ...

In this paper, the sparse Bayesian learning (SBL) approach previously proposed for the characterization of one-dimensional (1D) soil spatial variability is extended to a more realistic thre...

This paper compiles two expanded databases-NUS/SpreadFound/919 and NUS/RockFound/270-to evaluate the calculation methods of foundation capacity, where the number is the number of load tests collated. The mean (bias) and coefficient of variation (COV) (dispersion) of the model factor, defined as a ratio of measured-to-calculated capacity, are computed. With the statistics of a ratio of calculated-to-observed foundation settlement reported in literature, design methods for foundation capacity and settlement are classified according to their ability to match measured values. Such a statistical-based classification provides the designer with a sense of the accuracy of a capacity or settlement model. For the 298 case records in NUS/SpreadFound/919, the uncertainty exhibited by the load-movement curves over their full range of movements is captured by two parameters in a hyperbolic fitting of measured data. Copula analysis is adopted to evaluate the observed correlation structure between the hyperbolic parameters. Finally, the model statistics are applied to calibrate the resistance factors in load and resistance factor design of spread foundations at ultimate and serviceability limit states. (c) 2020 American Society of Civil Engineers.

This study performs a statistical analysis on shear strength parameters of rock mass for hydropower projects in China. First, a large database comprising 1139 groups of field direct shear tests on rock mass from 100 hydropower projects in China is compiled. Second, the best-fit probability distributions of shear strength parameters are identified using Akaike Information Criterion (AIC). Finally, new characteristic and design values of shear strength parameters of rock mass are suggested. The results indicate that shear strength parameters increase with increasing quality of rock mass. In particular, the shear strength parameters of discontinuities in rock mass are significantly smaller than those of rock mass itself. The discontinuities in rock mass should be paid more attention during geological investigations. There is a large variation in shear strength parameters, and the variability of cohesion is larger than that of tangent of friction angle. The derived statistics of shear strength parameters of rock mass in this study provide an update to the values suggested in Chinese codes, and serve as a reference for hydropower projects with few site-specific test data and similar geological conditions for partial-factor design of dams.

To account for uncertainties of load and resistance in a more rational way, reliability-based design (RBD) concepts have been increasingly applied to design bridge foundations. One of critical elements in the geotechnical RBD process is the characterization of model uncertainties. This paper compiles 126 and 23 reliable static load tests for steel H-piles in axial compression from two databases: Pile-Load Tests (PILOT) and Deep Foundation Load Test Database (DFLTD), respectively. The Davisson offset limit is adopted to define the measured resistance in clay, sand, and layered soil, which is verified with the L1–L2method developed for drilled shafts. A hyperbolic model with two parameters is chosen to fit the measured load–settlement curves. The uncertainties in resistance calculations and the load–settlement curves are captured by a ratio (or model factor) of measured to calculated resistance and the hyperbolic parameters. The mean values, coefficients of variation, and the probability distributions of the model factors are established from 149 load tests. The statistics of the resistance model factor are applied to calibrate the resistance factors (for the ultimate limit state) in load and resistance factor design of steel H-piles in axial compression. In future, the statistics of the hyperbolic parameters can be incorporated into the development of RBD of steel H-piles at the serviceability limit state.

This paper examines the usefulness and challenges in using an advanced effective stress constitutive model in random finite element study on the effect of spatial variability on global behaviours of axially-loaded cement-treated clay column. The main challenge lies in the large number of parameters which need to be simplified and the amount of data needed to facilitate model randomization. To enable selection of parameters to be properly simplified, a sensitivity study was conducted which indicates that the main behavioural aspect requiring randomization is the size of the initial yield locus, which is closely related to the mix ratio. Other parameters are shown to have lesser impact on the behaviour. The results show that, under undrained condition, the strain-softening nature of the model leads to lower strength and higher strength variability than the Tresca model. The use of an effective stress model allows the long-term behaviour. Since the material is strain-hardening under drained condition, deformation is likely to be a limiting factor on column loading in the long-term. The important parameters are likely to be the drained working modulus and yield stress. Where long-term settlement is a key performance index, long-term stress level may need to be kept within the elastic regime to avoid excessive long-term settlement.

This paper presents a nonlinear solution of a radial subgrade reaction–displacement (pk–ur) curve for circular tunnel lining design in undrained clay. With the concept of soil shear strength nonlinearly mobilized with shear strain, an analytical solution of pk is obtained using the mobilized strength design method. Two typical deformation modes are considered, namely oval and uniform. A total of 197 orthogonally designed cases are used to calibrate the proposed nonlinear solution of pk using the finite element method with the hardening soil model. The calibration results are summarized using a correction factor, η, which is defined as the ratio of pk_FEM to pk_MSD. It is shown that η is correlated to some input parameters. If this correlation is removed by a regression equation, f, the modified solution f(pk_MSD) agrees very well with pk_FEM. Although in reality the mobilized soil strength varies with principal stress direction, it is found that a simple average of plane strain compression and extension results is sufficient to produce the above agreement. The proposed nonlinear pk–ur curve is applied to an actual tunnel lining design example. The predicted tunnel deformations agree very well with the measured data. In contrast, a linear pk model would produce an underestimation of tunnel convergence and internal forces by 2–4 times due to the overestimation of pk at a large strain level.

Prediction of ground movement adjacent to an excavation is critically important, especially in urban areas, to prevent any potential damage of neighboring buildings. In the present study, a closed-form solution for prediction of excavation-induced ground settlement profile is proposed based on elastic displacement theory. Superposition method is employed to adapt the basic solution for translation wall deflection mode to general wall deflection mode. By assuming the wall deflection in braced excavation to be subject to a piecewise quadratic function, the displacement boundary of the problem is determined. An easy-to-use explicit solution is then developed. The validity of and applicability of the proposed solution is verified with existing analytical solution, finite-element model and six well-documented case records. The ground settlement profiles predicted on the basis of the new approach are in good agreement with reported settlement data. The results show that the proposed solution can be used as a feasible approach to obtain ground settlement profile in clay.

Geological uncertainty is often ignored in slope reliability analysis, even though the inherent variability of soil parameters is considered. This paper aims to propose a method for slope reliability analysis by considering both the inherent variability of soil parameters and geological uncertainty. A coupled Markov chain (CMC) model is used to simulate the geological uncertainty. An implementation procedure for slope reliability considering the aforementioned two types of uncertainty is provided. A slope reliability problem is analyzed to validate the proposed method with the borehole data from Perth, Australia. Different borehole layout schemes are designed to reflect the effect of both number and location of boreholes on slope reliability. Moreover, which type of uncertainty mentioned above has a greater impact on slope reliability is explored. The results indicate that the proposed method can effectively evaluate the slope reliability considering these two types of uncertainty. If only the inherent variability is considered, the accuracy of reliability analysis mainly depends on the used geological profiles. Borehole layout scheme has a significant effect on slope reliability. Both probability of failure ( P f ) and mean of factor of safety (FS) of slope do not monotonously vary with an increasing number of boreholes. The boreholes designed in the critical influence zone can provide more information to improve the accuracy of slope reliability. The coefficients of variation (COVs) of shear strength parameters and the discrepancy among the means of shear strengths for different soils play different and major role between the two types of uncertainty in the reliability analysis of soil slope.

This study explores the phenomenon of a worst case scale of fluctuation (SOF) in basal heave analysis for excavation in spatially variable clays. In the literature, the worst case SOF refers to the SOF where the discrepancy between the mean response from random realizations and the nominal response from a soil mass taking mean properties everywhere is the largest. Random finite element method (RFEM) is adopted to simulate the basal heave factor of safety (FSFEM). It is evident that the mean value of FSFEM can be 10–15 percent smaller than its nominal value at some worst case SOF. It is also shown that the slip circle method (SCM) based on an assumed prescribed slip curve cannot capture the phenomenon of a worst case SOF. However, the SCM can be modified to allow the weakest slip curve in a spatially variable soil mass to be located among a set of statistically independent potential slip curves. This “weakest path” model can reproduce the mean and coefficient of variation of FSFEM approximately without costly simulation when it is appropriately calibrated. In particular, the phenomenon of a worst case SOF can be captured, both qualitatively and quantitatively.

Slope failure mechanisms (e.g., why and where slope failure occurs) are usually unknown prior to slope stability analysis. Several possible failure scenarios (e.g., slope sliding along different slip surfaces) can be assumed, leading to a number of scenario failure events of slope stability. How to account rationally for various scenario failure events in slope stability reliability analysis and how to identify key failure events that have significant contributions to slope failure are critical questions in slope engineering. In this study, these questions are resolved by developing an efficient computer-based simulation method for slope system reliability analysis. The proposed approach decomposes a slope system failure event into a series of scenario failure events representing possible failure scenarios and calculates their occurrence probabilities by a single run of an advanced Monte Carlo simulation (MCS) method, called generalized Subset Simulation (GSS). Using GSS results, representative failure events (RFEs) that are considered relatively independent are identified from scenario failure events using probabilistic network evaluation technique. Their relative contributions are assessed quantitatively, based on which key failure events are determined. The proposed approach is illustrated using a soil slope example and a rock slope example. It is shown that the proposed approach provides proper estimates of occurrence probabilities of slope system failure event and scenario failure events by a single GSS run, which avoids repeatedly performing simulations for each failure event. Compared with direct MCS, the proposed approach significantly improves computational efficiency, particularly for failure events with small failure probabilities. Key failure events of slope stability are determined among scenario failure events in a cost-effective manner. Such information is valuable in making slope design decisions and remedial measures.

This study proposes a novel data-driven Bayesian machine learning method for constructing site-specific multivariate probability distribution models in geotechnical engineering. There is a ...

This paper adds to the ongoing discussion on the role of reliability calculations in geotechnical design. It situates design calculations, be it verified by a global factor of safety, partial factors, or reliability-based design (RBD), in a larger context of quality management over the life cycle of the structure. It clarifies that uncertainties amenable to probabilistic treatment typically fall under the category of “known unknowns” where some measured data and/or past experience exist for limited site-specific data to be supplemented by both objective regional data and subjective judgement derived from comparable sites elsewhere. Within this category, reliability is very useful in handling complex real-world information (multivariate correlated data) and information imperfections (scarcity of information or incomplete information). It is also very useful in handling real-world design aspects such as spatial variability that cannot be easily treated using deterministic means. Examples are present...

This paper aims to propose an auxiliary random finite element method (ARFEM) for efficient three-dimensional (3-D) slope reliability analysis and risk assessment considering spatial variability of soil properties. The ARFEM mainly consists of two steps: (1) preliminary analysis using a relatively coarse finite-element model and Subset Simulation, and (2) target analysis using a detailed finite-element model and response conditioning method. The 3-D spatial variability of soil properties is explicitly modeled using the expansion optimal linear estimation approach. A 3-D soil slope example is presented to demonstrate the validity of ARFEM. Finally, a sensitivity study is carried out to explore the effect of horizontal spatial variability. The results indicate that the proposed ARFEM not only provides reasonably accurate estimates of slope failure probability and risk, but also significantly reduces the computational effort at small probability levels. 3-D slope probabilistic analysis (including both 3-D slope stability analysis and 3-D spatial variability modeling) can reflect slope failure mechanism more realistically in terms of the shape, location and length of slip surface. Horizontal spatial variability can significantly influence the failure mode, reliability and risk of 3-D slopes, especially for long slopes with relatively strong horizontal spatial variability. These effects can be properly incorporated into 3-D slope reliability analysis and risk assessment using ARFEM.

The study focuses on the derivation of transformation models for undrained shear strength (su) of Finnish soft sensitive clays. Specific correlation equations for suof Finnish clays are presented in this work for the first time. Field and laboratory measurements from 24 test sites in Finland are exploited for this purpose and a multivariate database is constructed. The multivariate data consist of sufrom the field vane test, preconsolidation stress, vertical effective stress, liquid limit, plastic limit, natural water content, and sensitivity. The main objective is to evaluate the interdependence of su, consolidation stresses, and index parameters and provide a consistent framework for practical use. The new correlations are established through regression analyses. The constructed framework is further validated by another independent multivariate database of clays from Sweden and Norway as well as by empirical equations for Swedish and Norwegian clays. Existing correlations are evaluated for Finnish and Scandinavian clays. Finally, bias and uncertainties of the new correlations are presented.

ISO2394:2015 contains a new informative Annex D on “Reliability of Geotechnical Structures”. The emphasis in Annex D is to identify and characterize critical elements of the geotechnical reliability-based design (RBD) process, while respecting the diversity of geotechnical engineering practice arising from the influence of site-specific conditions. Annex D should pave the way for future code revisions to inject greater realism into geotechnical RBD. The most important element is the characterization of geotechnical variability. There are some noteworthy features: (1) the coefficient of variation (COV) of a geotechnical design parameter does not fall within a narrow range of values, (2) the multivariate nature of geotechnical data can be exploited to reduce the COV, (3) spatial variability affects the limit state beyond reduction in COV due to spatial averaging, and (4) statistical uncertainty may be significant, because the volume of soil sampled is a minute fraction of the volume of interest. Another important element is the characterization of model uncertainty. Geotechnical design codes, be it reliability-based or otherwise, must cater to diverse local site conditions and diverse local practices that grew and evolved over the years to suit these conditions. One obvious example is that the COVs of geotechnical parameters can vary over a wide range, because diverse property evaluation methodologies exist to cater to these diverse practice and site conditions. Another example is that deep foundations are typically installed in layered soil profiles that vary from site to site. These diverse site specific design scenarios do not surface in structural engineering. Section D.5.2 notes that reliability calibration for geotechnical RBD is more challenging for this reason.

Geological uncertainty appears in the form of one soil layer embedded in another or the inclusion of pockets of different soil type within a more uniform soil mass. An efficient coupled Markov chain (CMC) model has been proposed to simulate geological uncertainty in the past. This model, however, cannot be directly applied to geotechnical engineering. The primary problem lies in the estimation of horizontal transition probability matrix (HTPM), one key input of the CMC model. The HTPM is difficult to estimate due to the wide spacing between boreholes in the horizontal direction. Hence, a practical method for estimating the HTPM is proposed based on borehole data. The effectiveness of this method can be evaluated using the following approach. Several virtual borehole outcomes are simulated using artificial HTPM. The HTPM estimated from the virtual boreholes is compared with the artificial HTPM. Boreholes collected from Perth, Australian are adopted to illustrate the proposed method for HTPM estimation. The results show that the estimated HTPM agrees well with the artificial HTPM if the artificial (real) HTPM or vertical transition probability matrix (VTPM) is strongly diagonally dominant (diagonal element is larger than the sum of off-diagonal elements). The estimated HTPM is not sensitive to the borehole layout scheme as the diagonal dominancy of the HTPM is strong.

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.

Limit equilibrium methods (LEMs) and finite element methods (FEMs) of slope stability analysis can be used in computer-based probabilistic simulation approaches (e.g., direct Monte Carlo Simulation (MCS) and Subset Simulation (SS)) to evaluate the slope failure probability (Pf). For a given slope problem, the computational effort for the LEM is generally much less than that required for the FEM, but the FEM tends to give a more realistic prediction of slope failure mechanism and its associated factor of safety. To make use of the advantages of both the LEM (e.g., computationally more efficient) and FEM (e.g., theoretically more realistic and rigorous in terms of slope failure mechanisms), a new probabilistic simulation method is developed in the paper. The proposed approach combines both a simple LEM (i.e., Ordinary Method of Slices considering a limited number of potential slip surfaces) and FEM with the response conditioning method to efficiently calculate Pf of slope stability and to give an estimate of Pf consistent with that obtained from directly performing MCS and SS based on the FEM. It is illustrated through two soil slope examples. Results show that the proposed approach calculates the Pf properly at small probability levels (e.g., Pf < 0.001). More importantly, it significantly reduces the number of finite element analyses needed in the calculation, and therefore improves the computational efficiency at small probability levels that are of great interest in slope design practice. In addition, the proposed approach opens up the possibility that makes use of the information obtained using a simple model (e.g., LEM) to guide the reliability analysis based on a relatively sophisticated model (e.g., FEM).

This paper aims to develop an efficient geotechnical reliability-based design (RBD) approach using Monte Carlo simulation (MCS). The proposed approach combines a recently developed MCS-based RBD approach, namely expanded RBD approach, with an advanced MCS method called “Subset Simulation (SS)” to improve the computation efficiency at small probability levels that are often concerned in geotechnical design practice. To facilitate the integration of SS and expanded RBD, a generalized surrogate response f is proposed to define the driving variable, which is a key parameter in SS, for expanded RBD of geotechnical structures (e.g., soil retaining structures and foundations). With the aid of the proposed surrogate response, failure probabilities of all the possible designs in a prescribed design space are calculated from a single run of SS. Equations are derived for integration of the surrogate response-aided SS and expanded RBD, and are illustrated using an embedded sheet pile wall design example and two drilled shaft design examples. Results show that the proposed approach provides reasonable estimates of failure probabilities of different designs using a single run of the surrogate response-aided SS, and significantly improves the computational efficiency at small probabilities levels in comparison with direct MCS-based expanded RBD. The surrogate response-aided SS is able to, simultaneously, approach the failure domains of all the possible designs in the design space by a single run of simulation and to generate more complete design information, which subsequently yields feasible designs with a wide range of combinations of design parameters. This is mainly attributed to the strong correlation between the surrogate response and target response (e.g., factor of safety) of different designs concerned in geotechnical RBD.

LRFD (Load and Resistance Factor Design) is becoming a design method of choice in geotechnical practice, in lieu of the factor of safety (FS)-based design approach. However, even with LRFD, the need to reduce the variation in the predicted performance of a geotechnical system (or a geotechnical structure), referred to herein as the system response under applied loads, is still apparent. This paper presents a novel approach for applying existing LRFD codes, with explicit consideration of design robustness, safety, and cost efficiency. The recently developed reliability-based robust geotechnical design (RGD) approach is modified such that it can be integrated with the standard LRFD approach. This modified RGD approach is termed R-LRFD, which stands for Robust Load and Resistance Factor Design. In R-LRFD, the robustness of the system response against the variation in uncertain parameters is explicitly considered. Unlike the reliability-based design (RBD), the user is not required to conduct a full statistical characterization of uncertain parameters. With R-LRFD, the gain in the robustness, as reflected by the reduction in the sensitivity of the system response to the recognized but unquantified uncertainty of input parameters, is accompanied by an increase in cost. Thus, the concepts of Pareto front and knee point are introduced to aid in making an informed design decision. Further, a simplified procedure is developed to identify the most preferred design (knee point) in the design space, which is generally solved with multi-objective optimization algorithms. The effectiveness and significance of the proposed R-LRFD approach are demonstrated with two examples: one is the design of drilled shaft in sand (illustrated with a discrete design space) and the other is the design of drilled shaft in clay (illustrated with a continuous design space).

Geological uncertainty appears in the form of one soil layer embedded in another or the inclusion of pockets of different soil type within a more uniform soil mass. Uncertainty in factor of safety (FS) and probability of failure (Pf) of slope induced by the geological uncertainty is not well studied in the past. This paper presents one approach to evaluate the uncertainty in FS and Pf of slope in the presence of geological uncertainty using borehole data. The geological uncertainty is simulated by an efficient coupled Markov chain (CMC) model. Slope stability analysis is then conducted based on the simulated heterogeneous soils. Effect of borehole layout schemes on uncertainty evaluation of FS and Pf is investigated. The results show that borehole within influence zone of the slope is essential for a precise evaluation of FS statistics and Pf. The mean of FS will converge to the correct answer as the borehole number increases.

The objective of this paper is to investigate the effect of copulas for constructing the bivariate distribution of shear strength parameters on system reliability of geotechnical structures. First, the bivariate distribution of shear strength parameters is constructed using copulas. Second, the implementation procedure of system reliability analysis using direct Monte Carlo simulation (MCS) is developed. Finally, the system reliability of a retaining wall and a rock wedge slope is presented to explore the effect of copula selection on geotechnical system reliability. The results indicate that the system reliability of geotechnical structures under incomplete probability information could not be determined uniquely because the bivariate distribution of cohesion and friction angle with given marginal distributions and correlation coefficient could not be determined uniquely. The copulas for modeling dependence structure between cohesion and friction angle have a significant influence on the system reliability of geotechnical structures. Such an influence includes two phases separately. The first phase is that the dependence structure between shear strength parameters characterized by copulas affects the reliability of single failure mode, depending on the marginal distributions, dependence structure between shear strength parameters, and reliability level of each failure mode. The second phase is that the reliability of each failure mode influences on system reliability, only depending on reliability level of each failure mode and correlations among various failure modes. It is important to distinguish between the effect of copula selection on reliability of each failure mode and that on geotechnical system reliability.

This paper thoroughly investigates the bearing capacity of strip footings on sand under combined loading by using a lower-bound limit analysis in conjunction with finite elements and second-order cone programming (SOCP). Two analyses were performed: one using a constant friction angle and the other using a variable friction angle. The analyses with a constant friction angle were used to calibrate the existing results, including the classical solutions commonly used in foundation design practice and other numerical or experimental solutions. The analyses with a variable friction angle allow us to investigate the effect of footing width B on the bearing capacity of strip footings. An iteration computational procedure is employed to account for the dependency of the friction angle on the stress level. According to the numerical results, it is found that the bearing capacity factor Nγ for eccentrically or obliquely loaded strip footings linearly decreases with the increase of B on a log–log scale, where the range 0.3–5 m of footing width was considered in this paper. In addition, it is found that the footing width has a negligible effect on the shape and size of the normalized failure envelopes.

Reliability-based design (RBD) can play a useful complementary role to overcome some limitations in the Eurocode 7 (EC7) design approach, for example in situations with parameters not covered in EC7, different parametric sensitivities across different problems, cross-correlated or spatially correlated parameters, design aiming at a target reliability or failure probability, or when uncertainty in unit weight of soil is modeled. The complementary role played by RBD under these circumstances is illustrated and discussed for a shallow foundation, a reinforced rock slope, a Norwegian clay slope with spatial variability, a laterally loaded pile requiring implicit numerical analysis, and an anchored sheet pile wall. A pragmatic RBD approach involving first-order reliability method (FORM) only and a more rigorous RBD approach involving both first-order and second-order reliability method (SORM) are offered. Both approaches are implementable using either spreadsheet-based FORM and SORM procedures, or using various commercially available FORM/SORM packages.

The model uncertainties for the static analysis (determination of the settlement at a given load and the ultimate bearing capacity) of square foundations on sand under axial compression are characterized using 67 field load tests. The model uncertainty for calculating the ultimate bearing capacity is represented by a ratio (model factor) of measured capacity to calculated capacity. The capacity model factor is found to be significantly affected by the foundation width. This statistical dependency is expressed as an exponential function. The residual part of the model factor is characterized as a lognormal random variable. A hyperbolic equation with two curve-fitting parameters is used to estimate the load-settlement curves normalized by measured capacity. Using the same database, the best-fit copula for the hyperbolic parameters is Gaussian copula.

In models based on Winkler springs for tunnel lining design, designers always face the difficulty of selecting appropriate values for the radial subgrade modulus (kr). The widely used solution kr for a circular tunnel in elastic ground proposed by Wood (1975) was found to be applicable only when the tunnel radial deformation is oval-shaped. On the basis of the Wood׳s solution, this note presents a general solution for kr when the radial deformation of the tunnel is described by a Fourier series. This modified Wood׳s solution of kr using compatible stress functions is validated by a numerical example. The modified solution for the example shows good consistency with the original Wood׳s solution when the tunnel becomes an oval shape with deformations. The example indicates that the magnitude of kr is significantly affected by the distribution shape of the tunnel radial deformation. The value of kr is no longer a constant value around the tunnel when the tunnel deforms into a general shape described by a Fourier series. It is quite different from the value of kr for a distribution shape described by a single Fourier term, i.e. one involving a single frequency. The application of a general solution for kr is illustrated by a design case using a bedded beam model.

ISO2394:2015 contains a new informative Annex D on “Reliability of Geotechnical Structures”. The emphasis in Annex D is to identify and characterize critical elements of the geotechnical reliability-based design process, while respecting the diversity of geotechnical engineering practice. This paper highlights the main features of Annex D and gaps for future work.

Simplified reliability-based design (RBD) formats in the form of Load and Resistance Factor Design (LRFD) and Multiple Load and Resistance Factor Design (MRFD) are applied to 42 drilled shafts installed in layered soil profiles underlying the city of Taipei. Clay, sand, gravel and rock layers or some partial combination thereof are encountered within the depth of penetration. It is reasonable to require a foundation reliability-based design code to achieve a reasonably consistent target reliability index over this range of design scenarios, as they are representative of the ground conditions underlying Taipei. The simplified RBD formats are calibrated using two approaches. They are the design value method (DVM) and the more recently proposed quantile value method (QVM). In general, for layered soil profiles, conventional formats containing resistance and load factors are unable to achieve the prescribed target reliability index with the same consistency as that reported for homogeneous soil profiles, regardless of the reliability calibration approach. For the drilled shaft examples considered in this study, it was found that the direct application of quantiles in the simplified RBD format is more reasonable for layered soil profiles.

In random field finite element analysis (FEA), continuous random fields must be discretized, and the discretized solution is different from the continuous one. In this paper, it is found that the tolerable maximum element size to achieve acceptable accuracy for soil shear strength is surprisingly small. The discrepancy is governed by the line averaging effect along potential slip curves - the discrepancy is small if the line averaging is strong. Discretization based on element-level averaging (ELA) is always unconservative, giving mobilized shear strength always larger than the actual value, while discretization based on the midpoint method can be sometimes conservative.

This paper compares two methods for geotechnical reliability code calibration, namely the well known design value method (DVM) based on first-order reliability method and a recently developed method based on quantile, called the quantile value method (QVM). The feasibility of calibrating a single partial factor to cover the wide range of coefficients of variation (COVs) commonly encountered in geotechnical designs is studied. For analytical tractability, a simple design example consisting of one resistance random variable and one load random variable is first examined. A resistance factor is first calibrated using a single calibration case associated with a typical COV. The objective is to evaluate the departure from the target reliability index analytically when this calibrated resistance factor is applied to validation cases associated with a range of COVs. The results show that QVM is more robust than DVM in terms of achieving a more uniform reliability level over a range of COVs. Two realistic geotechnical design examples are studied to demonstrate that the theoretical insights garnered in the simple analytical example are applicable.

In homogeneous half spaces excited by small buried, spherically symmetric P-wave sources, Rayleigh waves (R-waves) could be generated. The effects of the source depth on the induced wave pattern and propagation behavior of R-waves are analyzed using the thin layer method. When a cavity is present in a homogeneous half space, R-waves could be formed in the scattered wave field. It is found that the energy of R-waves in the incident (direct) surface wave-field is related to the ratio of the source depth to the wavelength of R-waves; R-waves have relatively strong energy when the ratio is less than 1; the buried source induced R-waves approximately travel at the velocity of the planar R-waves in the range of the offset beyond about one wavelength; the energy of R-waves in the back-scattered surface wave-field depends on the ratio of the depth of cavity to the wavelength of R-waves; and for the case of the cavity presented at depths less than one wavelength, R-waves can be clearly observed in the back-scattered wave field. The results are helpful for selecting the source depth and the frequency component in seismic surveys and interpreting both the incident wave and the scattered wave patterns.

Response surface methods (RSMs) are attractive approaches for slope reliability analysis because such methods can integrate deterministic numerical slope stability evaluation and reliability analysis. For a slope in layered soils, its performance function is generally nonlinear and the system failure probability could be larger than the failure probability of the most critical slip surface. In this study, the applicability of the kriging-based RSM for system reliability of soil slopes is assessed through its application to two slopes with obvious system effects. It is found that the kriging method combined with Monte Carlo simulation (MCS) can deliver accurate system failure probability estimation. For comparison, the classical RSM based on the iterative local approximation of the performance function may fail to detect the most critical slip surface. The classical RSM can only calculate the failure probability of one slip surface even if the first-order reliability method or MCS is used. Even whe...

This paper aims to investigate the impact of copula selection on geotechnical reliability under incomplete probability information. The copula theory is introduced briefly. Thereafter, four copulas, namely Gaussian, Plackett, Frank, and No. 16 copulas, are selected to model the dependence structure between cohesion and friction angle. A copula-based approach is used to construct the joint probability density function of cohesion and friction angle with given marginal distributions and correlation coefficient. The reliability of an infinite slope and a retaining wall is presented to demonstrate the impact of copula selection on reliability. The results indicate that the probabilities of failure of geotechnical structures with given marginal distributions and correlation coefficient of shear strength parameters cannot be determined uniquely. The resulting probabilities of failure associated with different copulas can differ considerably. Such a difference increases with decreasing probability of failure. Significant difference in probabilities of failure could be observed for relatively small coefficients of variation of the shear strength parameters or a strong negative correlation between cohesion and friction angle. The Gaussian copula, often adopted out of expedience without proper validation, may not capture the dependence structure between cohesion and friction angle properly. Furthermore, the Gaussian copula may greatly underestimate the probability of failure for geotechnical structures.

In this paper the problem of dealing with scarce information in a reliability analysis is investigated in a geotechnical engineering context. Common probabilistic methods are compared with interval analysis as an alternative non-probabilistic approach. The potential of imprecise probabilities is discussed as an option for combining probabilistic and non-probabilistic information. The selected methods are examined in view of (i) an appropriate modeling of the information actually available in practical cases, (ii) the transfer of the uncertainty to the computational results, and (iii) the interpretation of the results. Key issue of this investigation is the meaning of the results as the basis to derive engineering decisions if only scarce information is available. This involves particular attention to low-probability-but-high-consequence events, which are often essential for risk assessment. A retaining wall structure is adopted for this study, and the failure mechanisms are considered as known in a deterministic form. This describes the input-to-output transformation of the available information in all cases. Representative available information about the parameters of the retaining wall problem is taken from literature, and typical practical specifications for the parameters are considered.

The analytic hierarchy process (AHP) approach has been widely used in multicriteria decision making (MCDM). It is very difficult to meet the consistency requirement of a comparison matrix (CM) in AHP. The authors analyze the reasons for inconsistent CM in AHP and propose an improved AHP (IAHP) to improve CM consistency by using a sorting and ranking methodology. The results of comparing AHP with IAHP by MATLAB simulation show that IAHP is more suitable for solving MCDM problems when the number of elements or factors is 5 or more in MCDM. A case study is presented to illustrate the performance of IAHP when applied to risk iden- tification during an open-cut subway construction. The application results show that IAHP is superior to AHP in terms of CM consistency, information extraction effectiveness, and convenience in practical implementation. DOI: 10.1061/(ASCE)CO.1943-7862.0000605. © 2013 American Society of Civil Engineers. CE Database subject headings: Risk management; Construction management; Statistics. Author keywords: Risk identification; Analytic hierarchy process (AHP); Improved analytic hierarchy process; Construction management; Quantitative analysis; Statistical analysis.