Seismic design codes are currently moving from a force-based design approach to a performance-based design approach. For example, in a performance-based design approach it could be specified how many lanes must be available during the lifetime of a bridge given a certain earthquake intensity.The problem with this approach is that it is not specified what the probability must be that the performance criterion is satisfied. This raises the question whether the design codes are acceptably safe or not. Focus is laid on the Canadian Highway Bridge Design Code (CHBDC), in which a total resistance factor approach is used. Because the total resistance factor in the CHBDC is a multiplicative factor, lower resistance factors lead to stronger foundation designs. The goal of this thesis is to calibrate the design procedure in the CHBDC for geotechnical systems under seismic loading, by finding a relationship between resistance factors and the lifetime probabilities of failure of said systems. The resistance factor can then be fine-tuned to a lifetime probability of failure that is consistent with the lifetime probability of failure targeted in static design. As an example problem, the bearing capacity of a shallow foundation on a clay with a pseudo-dynamic earthquake load is tested. The research question that is answered in this thesis is: "What should the resistance factors for geotechnical seismic design be in order to achieve a target lifetime probability of failure that is consistent with static design targets?'' Not every possible combination of soil strengths and forces on the superstructure can be taken into account, and therefore the random finite element method is used in a Monte Carlo simulation. Thousands of realization sare performed for each resistance factor, design return period, and "actual''return period that the designed foundations are tested against. By seeing how many realizations of the Monte Carlo simulation fail given a certain earthquake intensity, the conditional probability of failure given that earthquake intensity can be estimated. The total lifetime probability of failure can then be estimated from the conditional probabilities of failure with the total probability theorem. As part of a parametric study, the lifetime probabilities of failure are estimated for six different scenarios, each of which has different sources of uncertainty.  The resulting lifetime probabilities of failure are interpolated in order to find a resistance factor that targets a lifetime probability of failure consistent with static design targets. Currently, the resistance factor that the CHBDC recommends for geotechnical systems under seismic loading are defined as the static resistance factor for that geotechnical system incremented with 0.20, meaning that compared to static design, weaker foundations are designed for seismic load cases. The resistance factor found in this thesis is closer to the resistance factor for static design than to the resistance factor for seismic design. It should therefore be considered to lower the seismic resistance factor to the value of the static resistance factor so that a sufficient lifetime reliability can be targeted.

In most current dike assessments only the stationary water levels are investigated in the assessment of the stability of the inner slope, while there are differences for all kind of dikes between the stationary and transient pore water pressures and therefore in the stability. This results in a conservative probability of failure, while determination of a probability of failure should not be conservative but should be as realistic as possible. When time dependency is included in a calculation, an average flood duration is used, while the flood duration is highly variable. The following research question is defined to address the problem: “What is the influence of the flood duration on slope stability and in what degree affects the flood duration the design?” The degree of influence of time dependency on the pore water pressures and slope stability depends on dike characteristics, flood wave characteristics and the delay in failure. The basis for answering the research question is the software SEEP/W to model the time dependent pore water pressures and the software SLOPE/W to calculate the safety factor for the stability of the inner slope. In the research theoretical dike are used and there is focused on the flood waves in the Rhine and Meuse. A correlation analysis is performed to get insight in the contribution of different flood wave shape variables to the safety factor. And a probabilistic analysis is performed using transient and stationary water levels to know the differences in probability of failure between taking the shape of a flood wave into account or not. In both probabilistic analyses is varied in the permeability and the strength of the material; the shape of the flood waves is varied in the transient analysis. In this way the contribution of the flood to the probability of failure can be quantified. Dike characteristics The differences in pore water pressure are especially large for dikes that consist of an impermeable material such as clay. When only the subsoil consists of clay, larger differences are expected than when only the dike body consist of clay. However, large differences in pore water pressures do not necessary lead to large differences in the safety factor. The largest differences in safety factor are obtained when uplifting of the hinterland takes place during the stationary state and/ or during the passage of a flood wave. A transient calculation is therefore most useful for dikes with an aquifer and a thin (thinner than 5 m) weak (low POP values) hinterland. Flood wave characteristics The differences in safety factor during a permanent water level and the passage of a flood wave are large when no stationary conditions are reached during the passage of a flood wave. This is the case for high and short flood waves. Both in the Rhine and Meuse, the amount of short waves (< 7days) is high, which increases the influence of a time dependent calculation. Also, the importance of a time dependent calculation increases when the response to the increased pore water pressures is delayed caused by the permeability of the material. The influence of the height of a flood wave on the stability increases when the soil is permeable. Delay in failure Time dependency causes failure of the embankment to not occur simultaneously with the maximum wave height. The flood wave is decisive for the dike failure, but the permeability and the strength of the dike determines the moment of failure. Influence on design Taking time dependency into account leads to higher safety factors and lower probabilities of failure with exception for dikes that consist completely out of sand. For these types of dikes, the probability of failure and safety factors are the same order of magnitude. This could affect the design, because the dikes are safer when time dependency is considered. The strength of the material is the largest contributor to the distribution of safety factors and therefore to the probability of failure (60-95%). Whereas the contribution of the permeability to the probability of failure is small (2-12%), the variation in the height and duration of a flood wave contribute for 2-20% to the probability of failure. In a permeable dike this contribution is mainly determined by the height of a flood wave, while in an impermeable dike the duration of a flood wave is of importance. Considering the influence of time in stability probabilities of failure, this research proved that probabilities of failure taking the duration into account differ significantly from stationary calculations. It is therefore useful to take time dependency into account when determining the correct safety factor for impermeable dikes, but it is not useful in determining the correct safety factor for permeable dikes, because a stationary calculation is sufficient. In clay dikes it is useful to take the variation in height and duration into account, while for a sand dike it is sufficient to only consider the variation in height of a flood wave. When the variation of the duration of a flood wave is not considered, it is recommended to use a representative duration of a flood wave; that results in the same total probability of failure as when the variation of the duration is included. At Lobith the duration of the representative flood wave varies from 13 - 16 days. At Borgharen the representative duration varies between the 10 – 11 days for different dike types.

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.

The reliability assessment of hydraulic structures, in primary water defences before 2017, is characterised by exceedance probabilities and implicit uncertainty. In January of 2017, the new water law enforces a change in the assessment of primary water defences. The water law requests a probability of flooding when assessing the safety requirements. The studies of VNK1 and 2 are based on this new safety approach and provide background information and a basis for the new water law. A failure probability and explicit uncertainties provide an increase of reliability in the safety judgement of primary water defences and more specifically hydraulic structures. The aim of a more reliable assessment procedure is to some extent limited by the tools prescribed by the water law. The assessment tool Riskeer does not allow the engineers to have transparency in calculations results, which prevents insight into the model behaviour and contribution to the failure probability. To investigate the mechanics contributing to the overall failure probability for hydraulic structures, analysis of the limit states, failure mechanisms and Riskeer software were performed. The results of the analysis were incorporated into a reference model to validate the suspected mechanics. The input for the reference model was a sensitivity analysis and case study, which cover the assessment tracks of Non-Closure and Strength and Stability for hydraulic structures. The aim of this reference model was not to replace the Riskeer software but describe the mechanics in the correct way. Analysis of the schematisation manual and previous assessment software (Ring toets) provided the incorporated failure mechanisms and the specific model that was used to obtain the limit state. Additionally, a fault tree defined how the different failure mechanisms are related. The knowledge of the fault trees combined with sensitivity analysis provided information about the largest contributors to the overall failure probability, regarding resistance and solicitation variables. The dominant solicitation factor in all failure mechanisms was the water level difference. The resistance factor was dominated by the failure probability of Non-Closure in the gate and the strength of a gate element. In the comparison of the reference model with the Riskeer software, the results were similar but not accurate enough to replace one with another. The reference model was significantly influenced by the approximation in the hydraulic boundary condition, which is formed by assuming a distribution for the water level, wave height and wave period. In the testing of the Riskeer software, the software results showed a difference between two levels of the fault tree. The lower level describes the fault tree including all mechanisms and sub-mechanisms. The top level shows the top probability of the fault tree on which extra simulations of scenarios are executed. The Riskeer software showed the top level as the final result, however, the reference model only describes the lower level. The extra simulations work in a conservative capacity and cannot be assessed by the reference model. The main processes in the fault trees were validated by the reference model, as is shown by their characteristics in the sensitivity analysis. Together with the temporary calculation results, a rough validation of the input parameters was feasible in the lower level. In contrast to the top level, where numerous scenarios were simulated without any transparency to what these values entail. The sensitivity analysis also showed the difference of the two levels over the different variables, which is a (almost constant) significant difference.

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.