The current macrostability safety assessment for primary river dike trajectories in the Netherlands is applied to approach the failure probability of a dike during high water events. However, in the current schematization process that is described in the Wettelijk Beoordelings Instrumentarium (WBI) to assess the macrostability, aleatory and epistemic uncertainties are approached ’sufficiently safe’ by applying design values based on expert judgement via a semi-probabilistic assessment. Several primary river dike sections in the Alblasserwaard do not suffice the current safety standard set for the failure mechanism macrostability. The region is composed of a highly complex subsurface with large spatial variation, resulting in large schematization uncertainties for the macrostability assessment of the primary dike trajectories of the Alblasserwaard. With the recent development of full-probabilistic analysis possibilities in software such as D-Stability, it becomes possible to consider uncertainties as a stochastic variable in the macrostability safety assessment. Including schematization uncertainties within the macrostability safety assessment will improve the approximation of the failure probability of the primary dike trajectory. The largest schematization uncertainties in the macrostability safety assessment are currently considered to be the schematization of the subsurface in a vertical soil profile and the uncertainties in the schematization process of the pore water pressures in the dike during high water events. These uncertainties will be included in the calculation process to investigate the influence on the expected reliability of the primary dikes in the Alblasserwaard region. The subsurface schematization uncertainties are investigated by using soil scenarios to investigate the influence of local subsurface schematization in the vertical soil profile. The simplification of the soil profile and position of the soil layers is considered. The pore water pressures are separated into three components: the hydraulic head in the aquifer, the intrusion length, and the phreatic line. Each component will be included as a stochastic variable in the stability analysis. Fragility curves can be applied to describe the distribution function for each pore water pressure component, where the combined fragility curve will provide the combined failure probability and reliability index that includes the schematization uncertainty of the pore water pressures considered.
The soil scenarios can be applied to include schematization uncertainties of the subsurface in the macrostability safety assessment. The analysis showed that the simplification of the subsurface schematization only has a minor influence on the reliability index and failure probability of the case study dike cross-section Kortenhoevendijk. The schematization uncertainties of the pore water pressures can be considered in the macrostability safety assessment by combining the fragility curves of each component describing the pore water pressures underneath the dike. Results of the pore water pressure analysis are that failure probability is improved significantly for case study Kortenhoevendijk by a factor 1000 and case study Bergstoep by a factor 10. The approach to consider schematization uncertainties in the macrostability safety assessment via a full-probabilistic analysis can be used for dike sections prone to the uplift mechanism. This approach provides insight into the influence of schematization uncertainties on the failure probability of a dike cross-section. Including the pore water pressure schematization uncertainties in the macrostability safety assessment can have a significant impact on the outcome of the assessment. Including these uncertainties can make the difference between deciding whether a dike trajectory needs reinforcement, or deciding that reinforcement is not necessary.

The safety of the Dutch dikes depends on various failure mechanisms. Macro stability, a geotechnical failure mechanism, is highly affected by differences in soil strength because the sliding plane propagates through areas of least resistance. The variations in soil properties in space, also known as spatial
variability, are caused by geological processes and determine the locations of weaker zones in a dike. This highlights the importance of incorporating spatial variability into dike stability assessments.

The approach to incorporate spatial variability in the official Dutch assessment framework (WBI2017) relies on various assumptions. It assumes complete local variance reduction and neglects that the failure mechanism propagates through the weaker zones, leading to a mean strength reduction. Moreover, it assumes a default value for the ratio between local and regional variance α = 0.75, which lacks empirical evidence. Another method to incorporate spatial variability in stability calculations is the Random Finite Element Method (RFEM). This probabilistic technique models strong and weak zones through random fields. However, the main drawback is its considerable computation time, making it impractical to use for the assessment of the hundreds of kilometers of dikes in the Netherlands.

To address these issues, this research answers the question: What is an effective approach for incorporating spatial variability in soil into dike stability calculations? The study is divided into two parts: a data analysis and the creation of an RFEM model.

The first part investigated national and regional spatial correlations using variograms. The study found that the local spatial variance cannot be analyzed with variograms based on the national dataset. This is because the variograms average the variance in the local data due to their large scale. Investigating
local variability requires local data with a high enough density and accuracy in the research area.

In the second part, the inclusion of spatial variability was studied for a case study dike using RFEM, which is part of dike trajectory 34-2, located between Willemstad and Noordschans. The research highlighted two differences between assumptions made by RFEM and the WBI2017 method: (1) the
inclusion of statistical uncertainty and (2) the use of different stress components in calculating the undrained shear stress. The results of the different methods can only be compared if these differences are accounted for. Furthermore, the study found that using a probabilistic calculation with α = 0.8 better fits the results of the realistic RFEM model of the case study dike, particularly in the lower tail of the distribution of the results, compared to the default value of α = 0.75. Therefore, it can be concluded that α = 0.8 leads to a more realistic approximation of the probability of failure of this cross-section.

To investigate the importance of this finding, an assessment was carried out following the guidelines of WBI2017 but with α = 0.8. This showed that the probability of failure for dike trajectory 34-2 was reduced by 39.72% but that the safety category of the dike trajectory (for macro stability) remains unchanged.

Therefore, the answer to the research question is that when considering the computational requirements of RFEM, it is more effective to keep using the WBI2017 approach of implementing spatial variability into the input parameters of dike stability calculations with α = 0.75.

These findings are relevant as they validate the use of the current method with the available data. The study used different approaches for incorporating spatial variability in stability calculations and provides valuable insights for future research.

In the years to come, the Netherlands will face a substantial challenge as over 1,500 kilometers of aging quay walls and sheet pile walls approach the end of their technical lifespan. Infrastructure managers anticipate that the necessary replacements will necessitate investments amounting to billions of euros. Moreover, this task carries a significant environmental footprint, notably in terms of CO2 emissions. The construction work required for these replacements will also result in disruptions and reduced accessibility, inconveniencing users.
This study addresses two pivotal aspects. Firstly, it focuses on enhancing the design aspects of new structures and optimizing costs, with a specific focus exploring how these enhancements can ease the financial challenges faced by infrastructure managers. Secondly, it investigates the safety of existing structures and explores ways to maximize their loadbearing capacity while maintaining safety standards. The expected outcomes of this study promise improved design aspects, cost-efficiency, and enhanced safety measures.
Quay walls can fail due to various mechanisms. This research investigates three primary causes: yielding of soil, yielding of quay wall and anchor yielding. Quay walls illustrate the complexities of soil-structure interaction. To address this, models were developed in both Plaxis and D-Sheet Piling. D-Sheet Piling was the preferred choice due to its computational speed. The reliability analysis was conducted with Probabilistic Toolkit. Considering the calculation methods, First Order Reliability Method (FORM) was employed, emphasizing in efficient computational results in contrast to the Monte-Carlo approach.
In the first aspect, the partial factors were recalculated and compared them with the existing EC partial factor approach. To optimize the current design methodology, the retaining height of the structure was adjusted based on its reliability index. Additionally, the maximum anchor force required was re-evaluated for the structure. This procedure has been conducted for two scenarios, considering and not considering model uncertainty.
Furthermore, an analysis was conducted to understand how altering the retaining height can lead to reduced steel usage, subsequently impacting costs and CO2 emissions. In the second aspect, it was pursued to enhance the structure’s performance by introducing a factor "n" across four distinct scenarios: 1. Simultaneously increasing all loads. 2. Increasing the surcharge loads on the terrain. 3. Increasing the bollard load. 4. Raising the final excavation level in front of the quay wall. While this study aligns with the extensive body of research in the field of civil engineering, It seeks to offer a new and sustainable approach on understanding quay wall design, focusing specifically on the designers’ viewpoint. Through the exploration of innovative design frameworks and approaches, this research seeks to make a valuable contribution to the long-term sustainability of quay wall structures. It aims to redefine our approach to accessibility and safety in these crucial structures. The comprehensive investigations conducted throughout this study provide an enhanced comprehension of quay wall design, reliability, and the optimization of performance.

This thesis discusses the characterization of the undrained shear strength, S_u from the net cone resistance, q_net of clay in Dutch sites using Hierarchical Bayesian Modelling (HBM). The performance of the HBM is compared with the current practice methods of the site characterization which propose either the use of only site–specific observations (unpooled models) or the whole data simultaneously (pooled models). HBM can incorporate information from multiple sources such as prior knowledge of the engineers and behaviour met in the examined and the neighbouring sites. The use of different sources of information has been proposed by Eurocode-7 without providing a formal / mathematical procedure.
Literature studies have highlighted the potential benefits of incorporating the HBM into the characterization of the geotechnical parameter values. Therefore, this thesis aims to assess whether HBM can enhance the geotechnical decision-making by precisely quantifying the uncertainty in the geotechnical parameter values and making more accurate predictions of them. The impact of using input from the HBM results in a reliability analysis of a dike slope is examined as well.
First, a considerable number of paired q_net–S_u measurements is collected, and subsequently is divided into groups. Different statistical models are employed to describe this collected data. Two components characterize a statistical model; the functional form which is the relationship between S_u and q_net and the pooling family (pooled, unpooled and HBM), the method followed to train the statistical model parameters. The statistical models are applied in a comparative study to select the fittest one and to compare the behaviour of the HBM to the other pooling families. The comparative study is performed by applying the Bayesian Data Analysis (BDA) whose applicability is ensured by applying it in an artificial example using artificial data.
The first result of the BDA with real data is the comparison of the HBM with the current practice pooled and unpooled models suggesting the ln⁡〖S_u 〗-ln⁡〖q_net 〗 HBM as the fittest model. The HBM estimations for the statistical model parameters fall between the current practice’s methods and they experience lower uncertainty by borrowing information from the neighbouring sites to make site-specific estimations. Between the current practice and the HBM, the latter predicts the S_u with lower uncertainty.
The reliability analysis using input from the HBM yields different reliability indices than those proposed by the current practice models. This situation combined with the choice of the HBM after following the BDA workflow propose that the HBM can lead to safer and more economic design.
Overall, the use of the HBM for predicting the S_u from q_net with grouped data can be beneficial for the engineering practice. First, the HBM reduces the uncertainty of the statistical model parameters without inheriting extreme values and provides more certain prediction for the S_u accounting for the prior engineering knowledge and the behaviour met in neighbouring sites. Additionally, performing reliability analysis of a dike slope exhibits that the use of HBM derived values can suggest safer and more economic design over the standard approach.

The Water Authority Rivierenland is responsible for periodically assessing the safety of the dike trajectories in the Alblasserwaard region. In 2012, the safety assessment for inner slope macro-stability was performed based on stress dependent design values for the shear strength, based on the cell test
collection of the Water Authority. The most recent assessment on the inner slope macro-stability of the Lekdijk has shown a significant deviation from the previous assessment. According to these results, large scale reinforcement works are requested. Before starting any additional soil investigation, the water authority is interested in investigating the sensitivity of the schematization of the shear strength. The transition from the cell test collection to the triaxial and direct simple shear test collection, and therefore the transition from using the Mohr-Coulomb calculation model to the SHANSEP formulation, is expected to have the most impact on the outcome of the macro-stability safety assessment. The influence of the two test collections on the macro-stability safety assessment is analyzed by using a D-Geo-Stability model from the previous safety assessment for one cross-section of the Lekdijk, and transferring the model to D-Stability. The D-Stability model with the cell test collection parameters and the Mohr-Coulomb shear strength calculation model can be adjusted to the triaxial and direct simple shear test collection with SHANSEP shear strength calculation model. The transition from drained to undrained modelling results in a decrease of the shear strength of the soil around the failure surface. Therefore, the transition from the previous to the new test collection has resulted in a lower safety factor in the macro-stability analysis, having a negative impact on the overall macro-stability safety assessment.

Using an expensive-to-evaluate numerical model, such as a finite element method (FEM) model, is deemed unavoidable in solving modern geotechnical engineering problems. At the same time, the application of reliability analysis in dealing with uncertainties (e.g. soil properties) is increasing rapidly. This could pose a time-wise problem for an FEM model since reliability analysis normally takes much more than only one realization (function call) of the model. It becomes a bigger problem when a design optimization process is taking place. More often than not, design optimization is performed by a ”trial-and-error” method in practice, which the process itself would even take longer just to give engineers the ”sense” of achieving an optimal design (in terms of safety and economy). Therefore, the actual optimality of the design is not systematically proven and quantified. This research proposes a novel reliability-based design optimization (RBDO) method by combining existing theories regarding active-learning Kriging-based Monte Carlo Simulation (AK-MCS) and (1+1)-Covariance Matrix Adaptation evolution scheme ((1+1)-CMA-ES). To achieve accuracy and efficiency, the method consists of four enrichment stages. These enrichment stages ensure the method accurately and efficiently predicts the optimal design combination by considering the reliability constraint. The chosen case study is the reinforcement design of the Starnmeer polder dyke in the Netherlands, which is simulated as an FEM model. Within a limited number of function calls, the proposed RBDO method could accurately predict the optimal dimensions of the dyke that delivers the targeted reliability index. The reliable performance of the proposed method is further demonstrated by solving three analytical optimization problems.

Climatic conditions in uence peak discharges in rivers and change sea levels; therefore, attention to the safety of dikes is of ever growing importance. Macro instability is one of the dike failure mechanisms that can inundate the hinterland. Soil heterogeneity plays an important role in assessing dike safety, especially for slope stability, because it is a major source of uncertainty. To assess a dike network for safety, numerical simulations for a full probabilistic analysis can be computationally expensive. Therefore, this study investigates how to build a state-of-the-art data-driven framework from a numerical model to predict the safety margins from the macro stability of dikes. Inputs and outputs of tens of thousands D-Stability simulations were used to create a training dataset. The most relevant features were selected based on global sensitivity analysis and the representation of soil heterogeneity in the framework. The maximisation of Shannon's information entropy and the generation of the training dataset was achieved by employing a smart sampling strategy for the input parameters. The sampling strategy consists of a Latin hypercube optimised uncorrelated uniform distributed dataset combined with a correlated dataset for optimal training eciency. The uncertainty due to soil heterogeneity is represented by a Gaussian random eld with a trend. This trend is commonly determined from a geotechnical cone penetration test. With a CPT, it also is possible to nd the vertical scale of uctuation, which is parametrised by the correlation length of the uctuations in soil strength. The second-order Markov correlation function is used to represent the correlation of the random elds. The Gaussian random eld is later mapped onto 16 stacked horizontal layers to model the heterogeneous soil properties. The surrogate model consists of an ensemble of thirteen machine learning models. The most important model is a multi-layer perceptron feed forward articial neural network. The other models are histogram based gradient boosting regression trees. Random search and Bayesian optimisation are used as hyperparametrisation techniques to optimise the prediction capability is of the individual ML algorithms. Weights for each model are determined based on optimisation for error reduction for maximum performance. The surrogate predicts the factor of safety (FOS) as well as the coordinates of the slop failure circles and line of depth from the Uplift-Van method. The surrogate model ensemble that predicted FOS is quite accurate with respect to the numerical FOS of D-Stability, and yet the prediction of the failure plane is still slightly worse. A case study was used to demonstrate the performance of the framework. Despite the uncertainty of the subsoil, due to the soil heterogeneity, the surrogate was able to accurately predict the failure probability. However, the prediction of the far end circle coordinates showed lower performance due to propagating errors. Concluding, application of the framework is possible for dike reinforcement optimisation, risk-based dike safety assessment, length effect, and effcient Monte Carlo simulations.

As a consequence of the difficulty to obtain complete direct information about the subsurface, and the importance of heterogeneity of soil properties in geotechnical design, a desire exists in the geotechnical community for a smarter use of the available data in common practice. The objective of this thesis is to provide a potential solution for this, by implementing the Random Finite Element Method (RFEM) in PLAXIS, a well-known geotechnical calculation software package, allowing users to stochastically model engineering problems while accounting for the spatial variability of soil properties.

As a result of this project, practitioners can use the upside of stochastic analysis using RFEM to derive new insights about the behavior of their system, such as potential soil-related asymmetries, all while obtaining a better picture of the risks associated with a certain design. As with any technological advancement, responsible use of the framework is required to ensure the upside is attained and computational concerns do not hinder the use of the model, feature which still requires further research and improvement.

This research focuses on reliability-based, Bayesian updating of grout anchors’ bearing capacity. Firstly, grouted anchors are commonly used in practice, even though large uncertainties surround their bearing capacity. Secondly, every anchor must be tested on their working load for quality assurance. Because of the combination of those two factors, Bayesian updating on grout anchors yields large potential. The measurement data can be used in the design process, to reduce uncertainties. The main goal of this research was to establish how reliability-based updating can be applied in the context of grout anchor bearing capacity. The motivation to use such techniques is to increase the efficiency of existing structures, like in this case study for quay walls, or to adapt the design during construction, effectively cutting costs in both cases.

This research was conducted using measurement data from the HHTT quay wall at the port of Rotterdam, applying a recently developed analytical anchor displacement model, and the Bayesian updating technique ’aBUS-SuS’. The combination of both of these methods in this context is a novelty approach to the subject. The first step was to derive the soil parameters necessary to reproduce the measurements to prove firstly, that the model is capable to capture anchor behavior, secondly to get a better understanding of how the model functions in this context, and thirdly to establish a baseline of the parameters necessary for the estimation of prior distributions.
The second step was the application of the Bayesian updating algorithm. There, the prior distributions, which are needed as an input, are generated using the results from the first step. The resulting posterior distributions are then related to failure limits of the critical soil parameters that were derived according to a common anchor failure criterion. This gives an indication of the anchor’s utilization and disposition to failure.

The results exhibited failure conditions for those anchors that were at failure, verifying that the novelty approach can be applied to the problem. Another finding was that depending on the soil parameter, conclusions can be drawn about the likeliness of the failure type. This means how brittle or elastic the failure might be.

The analyses showed that the studied procedure has potential and the anchor’s proneness to failure can be assessed. The underlying analytical model is suitable for the applied type of Bayesian updating because the results are in the same range as for the calibration, but on average slightly lower in their mean magnitude. Thus, additional information is obtained, and the overall uncertainty is reduced. Furthermore, the analyses of the suitability tests gave a quantifiable indication of the anchors’ reliability under failure load that was in line with measured failure indication under working load.

Even though the uncertainty that surrounds the anchor-design was reduced, limitations to the framework arose. The analytical model requires uncommon soil parameters as an input. Those parameters need to be specifically determined in laboratory investigations and cannot be tested for on site. The model in- and output needs to match the measurements to the failure criterion of the anchors, and in anchor design, advanced analytical models, like the one used in this research are scarce. The measurement always have certain errors inherent. These errors can lead to diffuse and noisy posterior distributions. Furthermore, a minimum amount of measurements must be available in the first place, otherwise the algorithm does not converge properly towards a solution.

This is why it is recommended to get detailed insight of the soil parameters at question also under higher stresses and strains than the anticipated working loads. This helps to give better estimations of the prior distributions and their bounds, and also gives larger confidence in the posterior distributions as a result. Furthermore, this also helps to establish precise parametric limits corresponding to failure, for which the reliability can be evaluated.

A random field generator based on Local Average Subdivision (LAS) method is proposed in this study in order to achieve probabilistic soil classification and quantify the uncertainty of the generated most probable geological cross section. CPT data and Robertson’s soil classification chart (1990) are adopted to classify the soil. The sole application of LAS makes the random field unconditional, which has been improved to conditional random field generator by using Kriging interpolation. Both unconditional and conditional generator are tested in an illustrative example and the results indicate that the improvement from unconditional to conditional random field reduces the uncertainty of the most probable result of classifications and the classifications in the unconditional random field will converge if there are enough realizations. Additionally, the conditional random field generator is further applied in a case with three conducted CPTs, which build up a domain with very large scale of fluctuations. It’s found that the uncertainty of the generated most probable result of classifications is pretty low so it’s speculated that the proposed generator can be best applied in a large scale of fluctuation scenario. Another finding in the case study is that the proposed random field generator can be used to verify the reliability of conducted CPTs.

Reliability analysis is a rational method for dealing with uncertainties. It is increasingly used for the design and assessment of (civil) structures. Reliability analysis, possibly in combination with performance data, allows for instance to update (extend) the lifetime of existing quay wall structures. This gives economic advantages when assessing quay walls, figuring there are thousands of kilometres of quay walls worldwide. Soil-structure analysis (e.g. analysis of quay walls) is however complex and in general requires finite element (FE) models. Conventional reliability methods are mostly incapable of dealing with FE models because they have long (infeasible) calculation times e.g. Monte Carlo, or they cannot deal with the typical noisy and incomplete FE model output e.g. FORM. One way of dealing with these challenges is by using a metamodelling approach, i.e. build up and make use of a response surface on a limited number of model evaluations. One specific application which uses metamodelling is ERRAGA, which is an abbreviation for Efficient and Robust Reliability Analysis for Geotechnical Applications. This thesis investigates the potential of ERRAGA/metamodelling for reliability analysis of quay walls in engineering practice.\medskip

In this thesis ERRAGA is tested on two realistic case studies located in the port of Rotterdam. The first case is called the ‘Sleepbootkade’, a quay constructed from a combi wall with anchors. The second case study is called the HHTP-quay, a combi wall with relieving platform and MV-piles. In the first case focus is placed on getting to understand the details and workings of the ERRAGA method. In the second case focus has been put on the creation of the reliability model and the practical applicability of ERRAGA using the experiences of the first case. Two critical limit states are tested within the case studies: wall failure in bending and geotechnical failure. Especially the limit state geotechnical failure is challenging as relevant output parameters are known to be noisy and/or incomplete.\medskip

The main finding of this thesis is that reliability analysis of quay walls using ERRAGA in combination with a numerical model shows good potential. ERRAGA can generate reliable and accurate results within only hundreds of realizations, hereby overcoming the main drawbacks of existing reliability methods. Main drawback of the ERRAGA method seems to be its inherent complexity and ‘black box’ nature. For the user to employ the full potential of the method some more in depth knowledge and especially the user-defined settings are required. Based on the gained experiences in this thesis some first recommendations are presented for the use and application of ERRAGA in projects.

The Dutch Water board revised their guidelines for the safety assessment of dikes in 2017. A major change for the assessment of macro-stability is the use of the SHANSEP method to estimate the strength of impermeable cohesive layers. The failure probabilities are estimated with a deterministic analysis using limit equilibrium methods. The use of design values and safety factors to account for uncertainty is basic and proven to be conservative leading to over engineering.

In this thesis, it is investigated how the SHANSEP method can be incorporated to the more advanced Random Finite Element Method. It is found that three random fields for SHANSEP parameters S,m and POP are required. The random fields do not show particular trends in mean or standard deviation. A random field generator is coded in Python. a simple version of the in-house FEM is modified to read the generated random fields. This code is used to test various geotechnical assumptions.

A final version of the assumptions is coded into a more advanced version of the simulator to do the comparison. The output of the FEM code are the FOS and failure mechanism of a single evaluation with a combination of three random fields. A mean and standard deviation of the FOS results are calculated. The probability of failure is estimated by the area under the probability density function of a lognormal distribution for values below unity. The probability of failure of the deterministic case is estimated using the First Order Second Moment method.

The results show that the probability of failure is overestimated in a FOSM analysis by one order of magnitude compared to the most conservative RFEM simulation. It is expected that this difference is even higher for the more conservative deterministic approach the Dutch guidelines prescribe. The slip surfaces of RFEM were found to be similar to their deterministic counterpart. The RFEM slip surfaces went through local weak zones in random fields.

It is recommended to Dutch policy makers to investigate the use the random finite element method. Although conservatism is preferable in safety assessments, an conservatism of this significance compared to the RFEM approach is unnecessarily costly.

Since 2017, Dutch flood defences are assessed according to new safety standards. These standards are based on flooding probabilities and rely on several assumptions and approximations. There are concerns that the combination of these assumptions leads to conservative results. Recently computed probabilities of failure are often much higher than expected by dike managers and the outcomes of former assessment methods. This conservative bias results in a large and expensive reinforcement task in the coming years which can be reduced by improving the current assessment procedure.  One of the reasons for the current conservatism is the assumption of mutual independence of dike sections and failure mechanisms. Currently, the different elements are assessed independently, while failure mechanisms and failure at different dike sections are likely to occur during the same extreme load event. Furthermore, correlations in space and between different parameters are present within the subsoil characteristics. Neglecting these correlations results in rather high estimations of the failure probabilities.  The aim of this thesis is to investigate how correlations affect the reliability assessment of a dike trajectory. To achieve this, an integral, full probabilistic model is developed that enables simultaneous assessment of dike sections and failure mechanisms while accounting for uncertainties and (spatial) correlations within the model input. The model is based on Monte Carlo simulation. The failure probability of a dike trajectory is computed by counting failure if one or more limit state function 푍푗,푘 for failure mechanism 푗 of dike section 푘 returns a negative realisation. Correlations between the model input parameters are provided by means of a Gaussian copula. A particular aspect of the model is the implementation of metamodeling for the assessment of macrostability. This failure mechanism cannot be described by an analytical limit state function that is easily implemented in the Monte Carlo framework. Therefore, metamodels are created by means of Gaussian process regression. This method makes it possible to assess macrostability within an integral, full-probabilistic framework that is able to include interdependencies between e.g. macrostability and piping, within acceptable computation costs. The model is applied to a case study of dike trajectory 43-4, which is located along the Waal between Sprok and Sterreschans, in the east of the Netherlands.  The effects of different plausible correlations have been studied. This research shows that including certain correlations can significantly reduce the assessed failure probabilities, by a factor ten or more in some situations. However, the impact of correlation strongly depends on the situation. The most significant reduction can be achieved for cases in which (1) parameters that play a dominant role in failure of the corresponding mechanism are correlated; (2) the failure probabilities of the different elements are similar, i.e. for a flood defence where failure is not dominated by one dike section or one failure mechanism; and (3) the failure probabilities of the corresponding elements are smaller than approximately 10−3. The model forms a solid, flexible basis that can easily be adapted or extended to improve the understanding about interactions between failure mechanisms, even though some aspects are simplified or neglected. All in all, the conservatism in current safety assessments can be partly solved by considering the interdependencies between mechanisms and dike sections and by approaching a dike trajectory as an integral system.

Soil properties are spatially variable due to the natural deposition process. Because of this inherent spatial variability, a slope can actually fail along any potential slip surface. A single value of Factor of safety cannot account for this variation dominated the slope stability problem. Probabilistic analysis considering the spatial variability is a reasonable method to quantify the risk of the slope stability problem. Thus, in order to better simulate this variation, the theory of random field has been widely used in the slope stability problem. However, statistical outcomes derived from the probabilistic analysis will be influenced by how the random fields are generated and how the random field values are assigned to each potential slip surface. In order to investigate the extent of this influence, this study proposed a new probabilistic slope stability analysis method and compared it with the other two methods in terms of accuracy and efficiency. In this report, three different probabilistic slope stability analysis methods are presented. These methods combined the traditional limit equilibrium method of slices with random fields, which can account for the inherent spatial variability of soil properties. An exponential decaying function is used to describe the correlation structure of this spatial variation. This correlation structure is further expressed by the form of covariance matrix. Since the covariance matrix is a symmetric positive definite matrix, Cholesky decomposition is used to decompose it into the product of two triangular matrices. Because the triangular matrix is more computationally efficient, the two-dimensional random field is generated by multiplying a normal random number vector with the lower triangular matrix derived from Cholesky decomposition.

The application of reliability analysis in geotechnical engineering is relatively new compared to the other sections of civil engineering such as structural engineering and hydraulic engineering. However, due to its increases use in recent years, reliability analysis is planned to be included extensively in the upcoming Eurocode 7 (EN 1997). This research aims to compare the accuracy and efficiency between the applications of 22 selected reliability methods in 9 selected geotechnical engineering problems with various number of independent variables and modes of failure. The accuracy of the reliability methods are determined based on the Probability of Failure (Pf) errors, while the efficiency is based on the number of realizations (N) each method needs. The Monte Carlo Simulation is found to be the most accurate method despite its shortcomings in efficiency (ranked as the least efficient). Moreover, the FOSM method is found to be the most efficient despite its serious shortcoming in accuracy where it is also ranked as the most inaccurate. However, putting both accuracy and efficiency into account, the AK-MCS 0 order is proven to be the best method when applied to the discussed geotechnical engineering problems. The research also points out the necessity to perform multiple reliability methods for each geotechnical engineering problem.

The deterministic approach for interpreting CPT soil profiles poses the serious limitation of not taking data uncertainty into account. Therefore, a Bayesian model was developed by Wang et al. (2013) that, for a given CPT profile, determines the most probable number of soil layers and most probable soil layer thicknesses by simulating and comparing multiple ‘model classes’ with different complexities. In this study, this proposed model is implemented into the Python coding environment after which the functionality is verified by conducting a case study on a 23 푚 CPT profile from the Groningen area (NE Netherlands). For the given CPT profile, the model distinguishes 6 separate soil layers from which the position and thickness are in agreement with the deterministic analysis and the available borehole data. However, the case study suggests that the model fails to correctly identify the most probable soil types for CPT measurements within the vicinity of the edges of the Robertson chart. This is most-likely related to a “cut-off”-effect of the joint Gaussian distribution describing the uncertainty of a single datapoint. A subsequent study on the integration of the statistical parameters within the model is therefore required. Additionally, the code includes several optimizing strategies, but remains time consuming for very complex model classes. Further optimization is suggested to achieve greater model precision and efficiency.

In the past few decades, the use of Bayes' theorem in geotechnical engineering has increased significantly. A good amount of work is done on implementing this theorem for improving the knowledge on soil properties, identification of soil strata, and quantifying different uncertainties involved in predicting soil properties. However, the results of using this approach on a geotechnical structure have not been mentioned in the past literature. Hence, in the current work, Bayes' approach is used to predict the soil property parameters and further these parameters are used in slope stability analysis of a basic slope. It is aimed to investigate if the uncertainties in such analysis can be quantified. For this, a simple Bayes' model is developed to predict the input parameters of the soil properties. This model is developed using the PyMC3 package in python which is specially developed to solve Bayes' problems. Along with Bayes' statistics, normal/frequentist statistics is also used to derive the soil properties from two different data sets (3000 data points and 24 data points). To conduct the reliability studies of a slope, a Finite element model developed at TU Delft is used. The results of slope stability analysis obtained using both approaches are further compared. Results clearly show that when uncertainty is involved Bayes' approach encapsulates its effect on calculated FOS whereas the frequentist approach doesn't. This difference in both approaches is observed due to consideration of all the possible values of input parameters by Bayes' approach. With the frequentist approach, a single input parameter is predicted neglecting all the other possible inputs. Hence, it is concluded that using the Bayes' approach better and more reliable results over the frequentist approach by providing more insight about the uncertainties involved in the analysis.

Levees are earthen structures that are designed to protect land from flooding and are commonly composed of impervious soils and built on sandy foundations. They are sensitive to an erosion process that is known as piping. After several years of investigation into piping, Sellmeijer proposed design rules by curve fitting results of a numerical model. The design rule are currently applied in the assessment of levees, despite being obtained under the assumption of uniform and homogenous subsoils. Consequently, the design rules seem to be fine for standard consultancy, but it is insufficient for heterogeneous and anisotropic. In this thesis, the effects of anisotropy and heterogeneity are analysed in a elementary and a realistic numerical model study in order to improve the design rule of Sellmeijer. In the elementary study, several aquifers compositions are created by adding an artificial element with a different permeability to the body or by varying the permeability of the body depending on direction. The realistic study is a modification of the elementary study since stratified aquifer compositions have been implemented. After each simulation, the critical head computed in the elementary or realistic composition is compared to the critical head computed with Sellmeijer design rules which requires one calculation value of the bulk permeability. To improve the design rules an alternative method to determine the bulk permeability is proposed that assumes that there is curved or radial flow towards the exit point instead of the Dupuit method which assumes that there is only horizontal flow. As a consequence, the bulk permeability is calculated as the geometric or logarithmic mean of the vertical and horizontal permeability. Based on both studies, it can be concluded that both anisotropy and heterogeneity in the aquifer greatly increase the critical head so that heterogeneous and anisotropic aquifers are significantly more resistant to piping. In some cases, the aquifer can withstand a 45% higher water level. Furthermore, Sellmeijer’s design rules more accurately approach the heterogeneous and anisotropic aquifer, if the bulk permeability is determined with the new approach.

Soil properties used to determine the stability of soil structures are variable in nature. Uncertainty in soil can be attributed to its inherent variability, as well as sources of error encountered while estimating the magnitude of its properties. The modelling of these uncertainties will produce more meaningful solutions when evaluating stability. This is especially relevant when quantifying the probability and risk associated with a rare event of failure.

An uncertainty framework is implemented in this report to evaluate improbable slope failure. A slope stability program using a modified subset simulation approach is expanded to account for cross-correlation between cohesion, friction angle and unit weight of soil. The mean of the three soil properties and their correlation coefficients are treated as random variables in the analysis. A parametric study is performed to evaluate the influence of the mean and correlation coefficients of these properties on the probability of failure. The influence of randomising these properties is also evaluated in the analysis. The implemented method is applied for a practical slope example, based on values reported in literature for the expected variability in the mentioned soil properties.

Results demonstrate that modelling the mean of C, phi and gamma as a random variable leads to a significant increase in the probability of failure for a slope. While treating the correlation coefficients as random in the analysis will lead to very little changes in the outcome, some generated correlation matrices may lead to a notable decrease in probability of failure. The stability of the slope is heavily influenced by the input parameters used in the analysis. Furthermore, a proper choice for coefficient of variation of each property and the horizontal and vertical scales of fluctuation is necessary to avoid inaccurate results in the analysis. Other inputs investigated include the type of distribution for each soil property and the range of possible values for their means. Different distribution types are tested in the analysis to identify which of these properly model the variability in the parameter.

By evaluating generated samples within each subset level, it is evident that a combination of low mean values for C and positive correlation between phi and gamma is required for failure at low probability of failure levels. Although the influence of set means of C, phi and gamma on the calculated probability of failure is similar, the same conclusion cannot be made when the means of the properties are random. At low probability of failure levels, the outcome is very sensitive to changes in the minimum possible value for C and less to changes in phi and gamma. Furthermore, it is demonstrated that the mode of failure may be overestimated when analysing stability by reducing the strength of the slope. Shallow failures are encountered when slope is failing under a strength reduction factor of 1, which is a more likely mode of failure in spatially variable soil.