Stratification identification and spatial interpolation play a fundamental role in geotechnical site characterization. A unified approach is needed to perform these two tasks simultaneously to reduce overall uncertainty in site characterization. This paper explores the applicability of the Mixture of Gaussian Processes (MoGP) to address this gap, with a specific focus on characterizing and completing missing CPT data. The investigation encompasses both synthetic and real-world field CPT datasets and includes a comparison of the MoGP's interpolation accuracy with the use of a single GP for entire datasets. Additionally, the study examines the sensitivity of the model's performance with respect to the number of training data points. Although the interpolation performance of the MoGP model is promising with synthetic data, limitations appear in its application to real-site CPT data. @en

Clay rocks are multiphase porous media having a complex structure and behaviour characterised by heterogeneity, damage and viscosity, existing on a wide range of scales. The mesoscopic scale of mineral inclusions embedded in a clay matrix has an important role in the mechanisms of deformation under mechanical loading by cracking and creeping. This study introduces a micromechanical approach to model the time-dependent mechanical behaviour of clay rocks. A heterogeneous clay rock is represented at the mesoscopic scale as a composite material consisting of rigid elastic mineral inclusions (quartz, calcite and pyrite) embedded in a clay matrix. To describe the damageable rock behaviour and its failure modes at the small scale, interfaces between different mineral phases and within the clay matrix are considered. Viscous effects are incorporated inside the clay aggregates, with intergranular microfractures propagating in the clay matrix, in order to investigate their contribution to the creep behaviour of clay rock at the macroscale. The mesostructure of the clay rock is represented in digital 2D Representative Elementary Areas (REAs). The overall mesoscale behaviour of the clay rock under mechanical solicitation is numerically obtained from the REA by computational homogenisation within a two-scale finite element squared framework. Then, the model is validated at mesoscale against experimental data. The variability of the material response and the time evolution of the mineral interfacial damage state are investigated in relation to the small-scale properties and failure, while considering mesostructure variability. The results can give some valuable insights into creep behaviour of the clay rock from a small-scale perspective.

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Uncertainty is inevitable in the characterisation of a geotechnical site, especially due to the inherently heterogeneous nature of the ground. In this paper, a method for characterising a subsurface with limited cone penetration test (CPT) data is proposed. The method is based on integrating predictions of CPT parameters with a probabilistic approach for subsoil classification at the CPTs. The predicted stratigraphy is able to capture the spatial variability of soil measured via CPTs and takes account of the uncertainties that arise from transforming CPT measurements into soil units as well as errors in the measurements themselves. The applicability of the proposed method is demonstrated for a site in the Netherlands. The results show that the proposed approach can identify the most likely classification in the domain with good accuracy. Furthermore, the significance of considering the uncertainties in predicting the most likely classification is illustrated via finite element stability analyses of a slope cut-out in the domain.

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Geological uncertainty can significantly influence the computed response of a geotechnical structure. For example, ignoring the presence of a weak soil layer embedded within a stronger layer and assuming a deterministic stratigraphic boundary can significantly underestimate the probability of failure. In this paper, the coupled Markov chain method has been used for modelling this form of uncertainty. A strategy for estimating the horizontal transition probability matrix with limited data has been proposed, which is one of the biggest challenges with using this method. In particular, different sampling intervals in the vertical and horizontal directions have been considered in estimating the matrix for simulating realistic field situations. The applicability of the proposed method has been demonstrated using a set of CPTs in the Netherlands. The results highlight a problem that arises due to the coupling algorithm used in this method.@en

Soil liquefaction is investigated considering a saturated soil deposit and by implementing standard techniques of random field theory to distribute initial void ratio values and assess liquefaction risk. The soil domain is represented in a 2-dimensional (2D) random finite element model for the dynamic analysis of coupled behavior. Multiple Monte Carlo realizations are subjected to a base acceleration, while cyclic and small strain soil behaviours are achieved through a hypoplastic constitutive model. This investigation demonstrates that 2D stochastic simulations converge to 2D deterministic simulations when small standard deviations and/or small scales of fluctuation are used. However, large standard deviations combined with relatively large scales of fluctuation may cause significant uncertainty in the response of the soil deposit. Finally, common techniques employed to assess soil liquefaction are evaluated based on the results of the deterministic and random field analyses.

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Free-field site response analysis is a standard technique used to predict soil deposit dynamic response and liquefaction susceptibility. Such analyses are typically carried out by implementing periodic boundaries to guarantee the same speed of the dynamic waves travelling across them. However, when using random fields to consider the impact of soil spatial variability there is the possibility of an inconsistency with periodic boundaries. This is due to the generation of non-identical properties at the lateral boundaries on using traditional random fields. To overcome this inconsistency, this paper proposes periodic random fields to model spatial variability by matching the periodicity at the boundaries. To investigate the significance of using the proposed approach, a heterogeneous soil deposit subjected to earthquake loading is analysed using the random finite element method. The results show that, for certain values of the horizontal scale of fluctuation, ensuring consistency at the lateral boundaries could result in less conservative predictions of the extent of the liquefied areas. @en

Natural soil deposits may possess a highly anisotropic nature. The fabric anisotropy of soils which is induced during the soil formation process can lead to severe variation in field scale responses. Although the influence of fabric on the response of sands is well known and several advanced constitutive models have been developed to account for it, most of the studies incorporating anisotropy have focused on element test simulations while practical boundary value problem simulations are usually omitted. In this paper, the undrained response and liquefaction resistance of anisotropic sand deposits with different inherent fabric anisotropies are numerically investigated through element test simulations and one-dimensional nonlinear effective stress site response analyses. A novel semi-micromechanical constitutive model accounting for the effect of fabric anisotropy on sand liquefaction has been incorporated into a fully coupled dynamic in-house code employing the u-p formulation. The proposed numerical framework shows that, in both element test simulations and site response analyses, the fabric effects stemming from both the inherent and induced anisotropies can significantly influence the liquefaction resistance of sands.@en

The stability of six regional dyke cross-sections in the Netherlands was re-assessed using the random finite element method (RFEM), which explicitly accounts for the spatial variability of strength parameters. The RFEM assessments of the cross-sections were shown to result in significantly narrower response distributions than those obtained by ignoring the spatial variability, and therefore would result in more economical designs. Given the complexity of RFEM for applications in daily engineering practice, the results obtained from the re-assessments of the six dyke cross-sections were used to propose partial factors that can be used in practice to achieve the desired reliability levels for regional dykes. When applied in a conventional semi-probabilistic assessment of a dyke cross-section, these partial factors would result in the same level of reliability as would have been obtained by carrying out an RFEM analysis of the same cross-section.

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This research focuses on investigating the relative performance of a range of machine learning algorithms, namely the artificial neural network, support vector machine, Gaussian process regression, random forest, and XGBoost, for predicting the undrained shear strength from cone penetration test data. This is to assess how machine learning could help us lower the need for laboratory test data. The training dataset compiles 526 data from 12 regions and the testing dataset consists of 20 data from a polder located close to Leiden in the Netherlands. In addition, k-fold and group k-fold cross-validation strategies are both applied to validate the models. The poor performance of the models during group k-fold cross-validation suggests that, while machine learning techniques can perform well when site-specific data are included during training, they struggle to generalize without site-specific data. This highlights the difficulty of capturing soil heterogeneity and suggests that either machine learning methods should be trained on specific sites for which some data are already available, or much larger training datasets are needed.@en

To protect embankments along German inland waterways against local slope sliding failure caused by ship-induced water level drawdown, they are mainly secured by bank revetments. Often, large embankment sections are designed on the basis of a limited number of field and laboratory tests. Thus, uncertainties arise with regard to the mechanical and hydraulic ground properties. Current design standards account for these uncertainties by conservative design assumptions and empirical knowledge. This paper investigates the effects of vertically non-homogeneous ground properties on the required armour layer thickness using 1D random fields and an infinite slope model, which was modified to account for ship-induced drawdowns. Within the limitations of the infinite slope assumptions, the effects of a spatially variable friction angle and hydraulic conductivity are investigated and compared to deterministic benchmark cases. The investigations show that the level of safety obtained with the deterministic design depends strongly on the choice of the characteristic values. Particularly, the hydraulic conductivity determines the reliability of the design. In some cases, the 5 % quantile of the hydraulic conductivity does not yield a conservative estimate of the required armour layer thickness. In the case of the effective friction angle, the 5 % quantile may overestimate the required armour layer thickness for permeable soils. For less permeable soils, the 5 % quantile meets the solution of the random field analyses. For the combination of random effective friction angle and random hydraulic conductivity, all investigated benchmark studies seem to ensure engineering safety, but on different reliability levels. Based on these findings, recommendations regarding site exploration and choice of characteristic values of hydraulic conductivity and effective friction angle are provided.@en

1D soil column techniques are widely used to evaluate the potential of liquefaction in a system of soil layers. This approach generally leads to large inaccuracies since (1) soil layers are hardly homogeneous and perfectly horizontal and (2) horizontal effects are neglected. To demonstrate the limitation of 1D strategies and the need for 2D simulations, a series of benchmark problems are proposed and studied considering a fully coupled RFEM framework with small strain effects to account for cyclic behavior. First, a 1D simulation of a homogeneous material is tested against similar 1D problems including the spatial variation of soil properties (in this case void ratio). Then, a 2D domain is analyzed using the void ratio distribution obtained from combining the 1D columns. This investigation demonstrates that, by combining the effects of the horizontal direction and the spatial distribution of the soil properties, liquefaction triggering, spatial spreading and propagation extent may change significantly.@en

A kriging-based metamodelling approach for analysing the structural reliability of a sheetpile wall in a dyke is formulated. This specific problem is characterised by high target reliabilities ((Formula presented.)) in combination with a noisy and incomplete numerical model response. Starting from the original formulation of active learning kriging-based Monte Carlo simulation (AK-MCS), a robust two-stage metamodel framework is formulated in combination with adaptive multiple importance sampling, Gaussian process classification and kernel enhancements. Learning functions and convergence criteria are revised to maintain consistency with the metamodel enhancements. The developed metamodel is applied in the reliability analysis of a soil-structure interaction problem involving a sheetpile wall in a dyke body, which is representative of a class of problems encountered in engineering practice. Low dimensional example studies demonstrate the workings of the model and give insight into the model response. Full probabilistic analyses are then performed to estimate the probabilities of structural failure in a reliability updating context. The results show that after several necessary enhancements of the classical formulations, metamodelling approaches can be used successfully in combination with noisy and incomplete computational models as are often encountered in geotechnical engineering practice.

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A reliability-based analysis framework, accounting for uncertainty arising from the spatial variability of soil properties, has been validated for the controlled, well-instrumented slope failure of an historic dyke in the Netherlands. Using soil property statistics derived from the results of laboratory and cone penetration test (CPT) data for the different soil layers at the site, the dyke was analysed for the initial (i.e. operating) conditions, as well as for the later stage of the test leading up to failure. The computed probabilities of failure and back-figured factors of safety were consistent with the point at which failure occurred in the test, as was the range of possible failure mechanisms. The uncertainty in the stability assessment was reduced by considering the spatial nature of the soil variability, and by conditioning analyses to CPT measurement data. It is shown that the reliability-based approach enables more informed stability assessments that could make the difference between a dyke being assessed as safe or requiring costly improvement.@en

À l’échelle du laboratoire, les roches argileuses sont des matériaux hétérogènes dont le comportement thermo-hydromécanique est en grande partie contrôlé par la microstructure. Le choix du nombre et de la taille des échantillons à étudier en laboratoire est déterminant pour appréhender la variabilité des propriétés de la roche argileuse à petite échelle. Cet article présente une méthode statistique permettant de préciser la surface (ou le volume) et le nombre d’échantillons à prendre en compte pour qu’une propriété p choisie caractérisant la microstructure, soit statistiquement représentative. Initialement établie dans un cas général par Kanit et al. (2003. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40(13–14): 3647–3679), cette méthode consiste à partitionner un échantillon de propriété moyenne connue, en sous-échantillons de surface D × D afin de calculer l’écart-type et l’erreur relative de la mesure de p en fonction de D. Cette méthode permet ainsi de définir des surfaces élémentaires représentatives de p en tenant compte de l’erreur relative par rapport à . La méthode est d’abord présentée dans des cas généraux en 2D et 3D, et un exemple type est ensuite développé en 2D pour caractériser la fraction argileuse d’une lamine sédimentaire de Bowland (Royaume-Uni). La fraction surfacique argileuse est choisie comme propriété p, à partir d’une image grand-champ en microscopie électronique à balayage. La méthode est applicable en 2D et 3D sur les matériaux finement divisés autant sur les roches que sur les sols argileux, tant que l’échantillon considéré contient suffisamment d’éléments figurés (inclusions rigides ou pores dans une matrice par exemple) pour permettre l’utilisation des statistiques. L’apport principal visé pour la communauté des ingénieurs est dans la mesure du possible un meilleur ciblage de la quantité d’échantillons à prélever en forage pour mieux évaluer la variabilité des paramètres macroscopiques des roches argileuses. Les limites de la méthode sont ensuite discutées.

At the laboratory scale, clayrocks are heterogeneous materials and their thermo-hydromechanical behaviour is generally controlled by the microstructure. Choosing the number and size of samples is therefore a key point to both decrease and understand microstructure variability in the laboratory, at small scales. This paper focuses on a statistical method, which consists in dividing numerically an initial sample characterized by a chosen property , in sub-samples of surface D × D (or volume) characterized by a property of pi. The variance and relative error of p are calculated as a function of D to determine representative elementary areas (or volumes) of p, which take into account their relative error. A case study is presented on a clay lamina of Bowland Shale (UK). The clay fraction is taken in example to represent p, on a large field view of microstructure acquired under scanning electron microscopy. The method is availabe in 2D and 3D, on clay rocks and soils. The sample chosen has to contain a significant number of image’s particles to use statistics. The main advantage of this method is to target the volume of samples to analyze, in order to characterize the microstructure of such heterogeneous rock at the borehole scale, and to evaluate the variability of macroscopic parameters. Limitations of the method are then discussed.@en

Various simplified approaches are used to calculate the characteristic values of shear strength properties, which have then been used in deterministic stability analyses of a dyke cross-section. The calculated factors of safety are compared with the 5-percentile ‘system response’ of the dyke cross-section, calculated using the more exhaustive random finite-element method (RFEM), which is consistent with the requirements of Eurocode 7. The simplified methods accounting for variance reduction due to averaging of property values mostly give factors of safety within 10% of the RFEM solution, whereas the factor of safety based on the 5-percentile material properties is significantly over-conservative.@en

In the context of underground exploitation, the behaviour of rocks near galleries and tunnels conditions their stability. Underground drilling generates deformations, damage, fracturing, and significant modification of flow characteristics in the surrounding rock. However, the influence of small-scale characteristics and behaviour on the rock deformations and damage at engineering scale remains a complex issue. Consequently, the multiscale behaviour of a clay rock is modelled starting from the large scale of the excavation damaged zone around galleries and then enriching the approach by considering microstructural characteristics from the scale of mineral inclusions. Lastly, a double-scale numerical framework is considered. It allows to relate small- to large-scale rock behaviour in terms of deformations and material rupture. In fact, the development of damage and cracking at microscale allows to predict large-scale fracturing. The developed method focuses on a claystone in the particular context of long-term management of high-level nuclear wastes by deep geological repository. The results highlight the possibilities of double-scale computing in the prediction of the behaviour of underground engineering structures.

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This paper studies the effect of (not) accounting for the systematic uncertainty in the characterization of soil parameters in spatially variable soils and the influence this has on the calculated probability of failure. Not fully accounting for systematic uncertainty is considered as an incomplete characterization of the uncertainty in site investigation. An example slope in a spatially variable soil is modeled, using a combination of subset simulation with the random finite element method. The relationship between uncertainty in the mean shear strength and probability of slope failure is then used to demonstrate that even small levels of systematic uncertainty can have a significant effect on the calculated probability of failure, which potentially increases by several orders of magnitude @en

The Dutch dyke network includes 14,000 km of regional dykes that are regularly assessed in order to reduce the risk of flooding. The current strategy for maintenance and/or upgrading of existing dykes is based on safety assessments using partial factors and reliability-based characteristic values of material properties, interpreted from Eurocode 7 (EC7). In this paper, an historic dyke in the Netherlands has been analysed for stability using two methods: (a) based only on the point statistics of the material properties, as is widely adopted in Dutch practice based on a simplified interpretation of EC7, and (b) based on both the point and spatial statistics of the material properties, using a probabilistic approach in line with the full requirements of EC7. The results of safety assessments by the two methods show that a consideration of the spatial variability leads to a narrower range of possible responses, and thereby to a higher computed factor of safety at the target reliability of 95%. Moreover, for the dyke section considered, it results in much reduced remedial action compared to that suggested as a consequence of using the simplified approach, and thereby to a more economic design and reduced environmental impact.@en

A case study involving the assessment and re-design of an existing dyke, founded on a layered soil, has compared deterministic analysis based on 5-percentile property values and a reliability-based random finite element analysis consistent with the requirements of Eurocode 7. The results show that a consideration of the spatial nature of soil variability generally leads to higher computed factors of safety and, for those dyke sections requiring remedial action, to more economic designs. Back-figured characteristic values are shown to be considerably higher than the 5-percentile soil properties; hence, a reduction in over-conservatism is achieved.@en

Accounting for spatial variability in probabilistic slope stability analysis using the random finite element method (RFEM) typically leads to a distribution of calculated factors of safety as well as a distribution of resulting depths of the sliding body. Factor of safety (or its corresponding probability of failure) and sliding depth are weakly correlated when looking at the results of classical Monte Carlo simulations, but the mean (i.e. the expected) depth of the sliding surface as a function of the factor of safety shows a clear trend. This trend becomes more prominent when looking at the weak tail of the factor of safety distribution in more detail. In this paper, subset simulation is combined with RFEM to address the weak tail in an efficient manner and to simulate slope failure events at low probability levels, thereby demonstrating the tendency to shallow modes of failure in such cases. The mechanism behind the tendency to shallow modes of failure is then further investigated by simplification of the problem to limit equilibrium. Slide-specific variance reduction through spatial averaging is demonstrated as the mechanism behind the tendency to a shallow sliding surface as the mode of improbable slope failure. @en