Field data of armour damage in mound breakwaters is scarce, and experimental testing methods usually neglect the influence of pre-existing damage on subsequent damage increments. This study proposes a probabilistic framework, based on a dataset of 44 cumulative damage experiments, to estimate long-term damage progression in a full-scale, non-overtopped, cube-armoured mound breakwater located in the depth-induced wave breaking zone. A Gaussian copula-based Bayesian network is constructed to model the multivariate relationships between existing armour damage (S_e), the increment of armour damage (ΔS_e), the dimensionless water depth at the toe of the structure (h_s/H_m0), wave steepness (H_m0/L_0p), and the stability number (N_s). Each variable is modelled with a univariate parametric distribution, enabling inference of probabilities for values not directly observed in the experimental dataset. Model validation includes testing the Gaussian copula assumption, which is deemed a reasonable model, and assessing the defined graph with satisfactory results. The model is conditionalized using a historical wave dataset to generate synthetic damage curves, which are subsequently used to quantify a gamma process to model the survivability of the structure along its design life. A case study of a hypothetical breakwater with D_n=2 m close to the port of Tarragona (Spain) illustrates the methodology. According to the results of the model, the probability of observing a dimensionless damage S_e>5 corresponding to Initiation of Destruction after 10 years is 0.29. Overall, the obtained results are deemed conservative; this could be caused by the use of data from 2D experiments which do not take into account oblique wave attack. However, the approach is adaptable to other datasets with additional variables, breakwaters in different conditions or with other configurations, and can also be used in combination with simulations of synthetic wave data, making it relevant under changing climate conditions.

Extreme sea level events pose significant risks to coastal regions, with non-tidal residuals (NTRs) being a primary driver in low-lying areas like the Netherlands, where shallow seas amplify their impact. This study investigates the spatial patterns of NTRs along the Dutch coast using time series clustering on historical NTR hydrographs. The design of hydraulic boundary conditions divides the Netherlands into three coastal regions. To evaluate whether this division sufficiently captures regional variability, three clustering scenarios (k = 3, k = 4, and k = 5) were explored. The analysis identified k = 5 as the optimal configuration based on the Davies-Bouldin index. This result emphasized the importance of fine-scale approaches to understanding regional spatial variations in NTR dynamics. Regional bathymetry and tide-surge interactions were explored as drivers of these spatial patterns. Southern stations near river systems and deeper waters displayed characteristics distinct from northern stations in the Wadden Sea, which are influenced by shallow tidal flats. Analysis of the M2 tidal constituent and the timing of NTR maxima relative to high tides underscored the role of tidal dynamics in shaping spatial clusters. Future research will focus on integrating spatio-temporal patterns and environmental drivers into clustering methodologies, providing deeper insights for coastal risk management and adaptation strategies.

This study explores the statistical dependence between wind speed and surge height along the Dutch coast using a large synthetic dataset. Storms were clustered based on wind direction, tidal offset, wind rotation, tidal peak, surge and wind exceedance duration, resulting in 16 clusters per wind direction and per location. Apart from wind direction, comparing clusters revealed a limited impact of clustering based on these storm characteristics on the choice of the best-fitting copula model, suggesting sub-clustering may not be necessary for accurately representing the statistical dependence between extreme wind speeds and surge heights. The BB8 copula generally provided the best fit to the data. However, the observed upper tail dependence did not decrease to zero, particularly for western to northern wind directions, indicating non-negligible dependence in joint extremes of wind speed and surge height. Therefore, applying the BB8 copula (or any other copula model without upper tail dependence) may lead to underestimation of the flood risk, when applied in probabilistic analyses. The findings from this study provide valuable insights for refining hydraulic load models for reliability assessments and design of flood defenses.

Extreme storms over the North Sea drive coastal flood risk in the Netherlands, causing high waves and extreme sea levels. Designing flood defenses requires accurate statistical extrapolation of hydraulic load conditions with return periods of 1,000 years or more. This is a challenging task given limited observational data. This study uses a large, simulated dataset (~9,000 years) to explore the statistical dependence between extreme wind speed u and surge height s. Storms were clustered using several techniques. Self-organizing maps (SOM) effectively captured physical relationships, such as the influence of wind direction and tidal offset on storm dynamics, however variability in statistical dependence between u and s for different clusters was better represented using manual clusters. Copula models were fitted to the cluster data, with the BB8 copula outperforming others. This study illustrates the potential of machine learning to identify patterns in large datasets while emphasizing the relevance of manual clustering approaches for revealing nuanced statistical dependencies critical to flood risk assessment.

Effective resource planning in higher education requires anticipating student demand for courses, especially when dealing with elective programs. Monitoring student preference is a recurring topic in the literature; however, to the authors’ knowledge, no simple methods for estimating student preferences when choosing courses in higher education have been proposed. This study develops and explores the use of a simple questionnaire to capture patterns in student course preferences within a university context. The research is developed in the context of the nine Cross-Over modules offered as part of the curriculum of the master’s programs (MSc) of the Faculty of Civil Engineering and Geosciences of Delft University of Technology (The Netherlands). No prior registration is required far in advance for these courses, making an accurate estimation of student numbers critical for the planning and allocation of educational resources. The developed questionnaire is applied three times in two different academic years to the students’ choice of Cross-Over modules. The questionnaire was shared in 2021, with 225 responses out of 339 students, in 2022, with 159 responses out of 365 students, and in 2024, with 94 responses out of 272 students. Student enrollment in the academic year 2023/2024 is used to assess the performance of the questionnaire. The questionnaire is able to capture general preferences of the students, providing fair estimates of the number of students per course; larger differences are observed in courses with a lower number of students. In addition, some patterns were identified in student preferences: there is a relationship between the first and second choices, and students usually choose modules closer to their own disciplines. The developed questionnaire provides with a reasonable first estimation of the expected number of students in courses, allowing for better planning and allocation of educational resources beforehand.

Due to the inherent uncertainties in manufacturing properties and intrinsic variability of materials, the assumption of homogeneous input variables is generally not justified. As a result, stochastic forward problems have emerged as a tool to incorporate these uncertainties into numerical simulations, improving model prediction capability in structural analyses. Although most of the existing methods focus on the consideration of stochastic loading, the recently developed statFEM employs a Bayesian paradigm to incorporate data and propagate uncertainties from random physical properties in finite element models. This tool, however, is not developed for cases when Gaussian assumptions are inadequate. The present work provides a copula-based approach embedded into the statFEM methodology to propagate uncertainty from arbitrarily distributed physical properties. The random variables are defined in terms of a Gaussian Copula Process, where samples are drawn from a latent variable governed by a Gaussian Process and then brought respectively to copula and marginal spaces, producing random variables with the desired distribution while retaining the usually desired smooth Gaussian dependence in the spatial domain. The quality of the approximated results is then assessed in a simplified 1D Poisson problem by comparing with Monte Carlo sampling results for different random diffusion coefficients, demonstrating that the method is capable of providing good responses for non-Gaussian physical parameters.

Base isolation of high-rise buildings has growing popularity to limit peak floor accelerations under seismic loads; however, it may increase susceptibility to wind-induced vibrations due to the increase in fundamental vibration period. This study presents an equivalent coupled-two-beam (CTB) model incorporating base isolation (BI) and a tuned mass damper inerter (TMDI) to evaluate passive vibration control under lateral wind loads for various lateral resisting systems. A 144-meter-tall building was analyzed under along-wind and across-wind loads simulated as Gaussian processes, considering six isolator-damper configurations: (1) fixed-base (FB), (2) FB with a top TMDI (FB-TTMDI), (3) BI, (4) BI with a top TMDI (BI-TTMDI), (5) BI with a bottom TMDI (BI-BTMDI), and (6) BI with double TMDI (BI-DTMDI). TMDIs were compared to traditional tuned mass dampers (TMDs) to assess mass amplification under varying base isolator damping. Optimization strategies were explored to enhance vibration control: for FB-TTMDI, the TMDI placement minimized RMS accelerations, while for BI-TTMDI, it was optimized to reduce peak displacement. Finally, design guidelines are provided for ultimate and serviceability limit states. Results indicate hybrid control systems are most effective when lateral deformation resembles pure bending, making them suited for shear wall-frame and tubular systems.

Offshore floating structures are experiencing harsh environmental conditions risking their safety. Therefore, mooring lines are crucial for ensuring structures’ stability. Sudden increases in tensions after temporarily slack of the mooring line are called snap loads and are the most critical load states. These snap loads and their dependence to various factors are investigated in the present study. 12 study locations in the south-eastern North Sea are selected. For each location, wave and current variables are extracted from a three-dimensional large-scale numerical model covering the European Shelf. Mooring tensions at different rope positions are calculated via a Finite Element model for flexible mooring lines for different hydrodynamic conditions and used subsequently to obtain tension rates as indicator for snap loads. The dependence among 13 variables per study location is modelled via Gaussian copula-based Bayesian Networks (GCBN). This allows for spatial analysis of the relationships between hydrodynamic variables and tension rates, but also to determine the influence of hydrodynamic variables on expected tension rates. Furthermore, distributions of tension rates are obtained under specific constant hydrodynamic conditions. The results indicate that conditionalising on certain hydrodynamic variables can reduce the expected tension rates, as their marginal distributions are characterised by heavy tails. Still, mooring systems should be designed conservatively. However, once specific hydrodynamic information is available, uncertainties can be minimised, enhancing safety and reliability. Thus, accounting for the dependence among hydrodynamic variables and tension rates is crucial for improving the safety of structures under varying environmental conditions.

This study evaluates five scoring rules, or measures of statistical accuracy, for assessing uncertainty estimates from expert judgment studies and model forecasts. These rules — the Continuously Ranked Probability Score ((Formula presented.)), Kolmogorov-Smirnov ((Formula presented.)), Cramer-von Mises ((Formula presented.)), Anderson Darling ((Formula presented.)), and chi-square test — were applied to 6864 expert uncertainty estimates from 49 Classical Model (CM) studies. We compared their sensitivity to various biases and their ability to serve as performance-based weight for expert estimates. Additionally, the piecewise uniform and Metalog distribution were evaluated for their representation of expert estimates because four of the five rules require interpolating the experts' estimates. Simulating biased estimates reveals varying sensitivity of the considered test statistics to these biases. Expert weights derived using one measure of statistical accuracy were evaluated with other measures to assess their performance. The main conclusions are (1) (Formula presented.) overlooks important biases, while chi-square and (Formula presented.) behave similarly, as do (Formula presented.) and (Formula presented.). (2) All measures except (Formula presented.) agree that performance weighting is superior to equal weighting with respect to statistical accuracy. (3) Neither distributions can effectively predict the position of a removed quantile estimate. These insights show the behavior of different scoring rules for combining uncertainty estimates from expert or models, and extent the knowledge for best-practices.

Manufacturing imperfections are an inherent aspect of the production process, affecting the reliability and performance of all engineered components, including actuators. This work investigates the effects of manufacturing uncertainty in coil assembly. Given the manufacturing tolerances of geometrical and material properties of wires and coils, the coil’s electrical response is studied from a probabilistic perspective. While many researchers acknowledge these uncertainties, they often do not quantify their effects on the system’s response. We propose the use of copulas to model nonlinear relationships and quantify the probabilistic dependence between design variables (i.e., wire and coil’s properties) and their effects on electrical responses. Additionally, tolerances of the design variables are determined through unstructured expert elicitation. The results show the impact of the thickness of the wire’s insulation in the performance of the coil. Furthermore, we establish a design space that is studied probabilistically, allowing for probabilistic design optimization.

Laser powder bed fusion (LPBF), a prominent metal-based additive manufacturing (AM) technique, enables the production of complex, neat-net-shape components with minimal material waste and reduced lead times. However, achieving high final product quality is challenging due to numerous process variables and intricate, nonlinear interactions introduced by LPBF’s thermal-mechanical mechanisms. This study employs a copula-based analysis using multi-physics numerical simulations to probabilistically map relationships among process and part quality variables. Dependence and tail-dependence analyses are performed to provide deeper insights into variable interactions, enabling the identification of preferred operational windows with a balanced trade-off between product quality and productivity. The developed methodology advances the understanding of uncertainty propagation in LPBF, contributing toward improved process optimization, repeatability, and reliability.

In absence of sufficient data, structured expert judgment is a suitable method to estimate uncertain quantities. While such methods are well established for individual variables, eliciting their dependence in a structured manner is a less explored field of research. We tested the performance of experts in constructing and quantifying a nonparametric Bayesian network, describing the correlation between river tributary discharges. Specialized software was provided to assist the experts. Expert performance was investigated using the dependence calibration score (a correlation matrix distance metric) and the likelihood of the joint distribution. Desirable properties of the dependence calibration score were investigated theoretically. Individual expert judgments were combined based on performance into a group opinion aka decision maker. All experts were able to create and quantify a correlation matrix between 10 variables that resembled the correlations between observed discharges well. The decision makers performed similarly to the best expert. Based on the metrics investigated, it mattered little which expert opinions and with what weight were combined in a decision maker. This is partly because all experts performed well. Adding a bad performing expert increased the positive effect of performance-based weighting, underscoring the importance of developing scoring rules for dependence elicitation. The overall results are promising: Aided by specialized graphical software, the experts in this study were able to quickly create and quantify dependence structures.

For the last two decades, significant damage to groyne structures has been observed in the German Elbe estuary. The main reason is the generation of primary ship-induced wave loading. The stern wave of the primary wave system appears as an overflowing over groyne, leading to damage at the crest and lee side of the structure due to the presence of high overflowing flow velocities (Melling et al., 2020). Therefore, overflowing flow velocities and damage were quantified by means of two different experimental setups; the flow experiments and the damage experiment.

The flow experiment involved testing scaled physical models under continuous free-flow conditions. A Particle Image Velocimetry (PIV) setup was used to capture flow velocities at the crest and lee side slope. A dimensionless flow velocity equation is obtained for overflowing flow over groyne structures. The damage experiment assessed the impact of overflowing waves at the crest and lee side on one of the scaled physical models. Measurements were conducted via Structure from Motion principles (SfM) and the damage is expressed in damage parameters S for varying wave heights and freeboard levels. This parameter describes the damage by width-averaged eroded area made dimensionless by the squared nominal stone diameter. Furthermore, the assessment considered the determination of the damage limits (initiation, intermediate, and failure) of a groyne structure for these waves.

The results revealed the relation between the wave height and the freeboard and damage. Furthermore, by regarding the flow velocity explicitly a more fundamental understanding, and more generally applicable design approach might be obtained. The insights gained from this research contribute to an enhanced understanding of groyne behaviour under overflowing long- period ship-induced waves. By highlighting the significance of the flow velocities for waves and freeboard levels, this study provides valuable information for optimizing the design and maintenance of estuarine groynes that are prone to these types of wave-induced loads.

Accurate estimation of extreme discharges in rivers, such as the Meuse, is crucial for effective flood risk assessment. However, hydrological models that estimate such discharges often lack transparency regarding the uncertainty in their predictions. This was evidenced by the devastating flood that occurred in July 2021, which was not captured by the existing model for estimating design discharges. This article proposes an approach to obtain uncertainty estimates for extremes with structured expert judgment using the classical model (CM). A simple statistical model was developed for the river basin, consisting of correlated generalized extreme value (GEV) distributions for discharges from upstream tributaries. The model was fitted to seven experts' estimates and historical measurements using Bayesian inference. Results were fitted only to the measurements were solely informative for more frequent events, while fitting only to the expert estimates reduced uncertainty solely for extremes. Combining both historical observations and estimates of extremes provided the most plausible results. The classical model reduced the uncertainty by appointing the most weight to the two most accurate experts, based on their estimates of less extreme discharges. The study demonstrates that with the presented Bayesian approach that combines historical data and expert-informed priors, a group of hydrological experts can provide plausible estimates for discharges and potentially also other (hydrological) extremes with relatively manageable effort.

The rapid changes in the shipping fleet during the last decades has increased the ship-induced loads and, thus, their impact on infrastructures, margin protections and ecosystems. Primary waves have been pointed out as the cause of those impacts, with heights that can exceed 2 m and periods around 2 minutes. Consequently, extensive literature can be found on their estimation mainly from a deterministic perspective with methods based on datasets limited to one location, making difficult their generalization. These studies propose either computationally expensive numerical models or empirical equations which often underestimate the extreme primary waves, hindering their use for design purposes. Moreover, a framework to allow the design of infrastructure under ship-wave attack based on probabilistic concepts such as return periods is still missing. In this study, a probabilistic model based on bivariate copulas is proposed to model the joint distribution of the primary wave height, the peak of the total energy flux, the ship length, the ship width, the relative velocity of the ship and the blockage factor. This model, a vine-copula, is developed and validated for four different deployments along the Savannah river (USA), with different locations and times. To do so, the model is quantified using part of the data in one deployment and validated using the rest of the data from this deployment and data of the other three. The vine-copula is validated from both a predictive performance point of view and with respect to the statistical properties. We prove that the probabilistic dependence of the data is preserved spatially and temporally in the Savannah river.

The magnitude of flood impacts is regulated not only by hydrometeorological hazard and exposure, but also flood protection levels (primarily from structural flood defenses) and vulnerability (relative loss at given intensity of hazard). Here, we infer the variation of protection levels and vulnerability from data on historical riverine, coastal, and compound floods and associated impacts obtained from the HANZE database, in 42 European countries over the period 1950–2020. We contrast actual damaging floods, which imply flood protection was locally inadequate, with modelled potential floods, i.e. events that were hydrologically extreme but did not lead to significant impacts, which imply that flood protection was sufficient to prevent losses. Further, we compare the reported magnitude of impacts (fatalities, population affected, and economic losses) with potential impacts computed with depth-damage functions. We finally derive the spatial and temporal drivers of both flood protection and vulnerability through a multivariate statistical analysis. We apply vine-copulas to derive the best predictors out of a set of candidate variables, including hydrological parameters of floods, exposure to floods, socioeconomic development, and governance indicators. Our results show that riverine flood protection levels are much lower than assumed in previous pan-European studies. North-western Europe is shown to have better riverine protection than the south and east, while the divide is not so clear for coastal protection. By contrast, many parts of western Europe have relatively high vulnerability, with lowest value observed in central and northern Europe. Still, a strong decline in flood vulnerability over time is also observed for all three indicators of relative losses, suggesting improved flood adaptation. Flood protection levels have also improved since 1950, particularly for coastal floods.

Constructing Bayesian networks (BN) for practical applications presents significant challenges, especially in domains with limited empirical data available. In such situations, field experts are often consulted to estimate the model’s parameters, for instance, rank correlations in Gaussian copula-based Bayesian networks (GCBN). Because there is no consensus on a ‘best’ approach for eliciting these correlations, this paper proposes a framework that uses probabilities of concordance for assessing dependence, and the dependence calibration score to aggregate experts’ judgments. To demonstrate the relevance of our approach, the latter is implemented to populate a GCBN intended to estimate the condition of air handling units’ components—a key challenge in building asset management. While the elicitation of concordance probabilities was well received by the questionnaire respondents, the analysis of the results reveals notable disparities in the experts’ ability to quantify uncertainty. Moreover, the application of the dependence calibration aggregation method was hindered by the absence of relevant seed variables, thus failing to evaluate the participants’ field expertise. All in all, while the authors do not recommend to use the current model in practice, this study suggests that concordance probabilities should be further explored as an alternative approach for the elicitation of dependence.

The severity of damages to riverine structures, such as groynes or revetments, across German estuaries has increased in the past years due to the increase in the ship-induced loads. However, few studies can be found in the literature focused on the damage of rock slopes under ship wave attack. In this study, the field data of a rock-armored groyne (lateral slope 1/4 and rocks with nominal diameter Dn50»12.6cm and high density ρs=3.7t/m3) tracked for a year by Melling et al. (2020) is analyzed; the field campaign began after the structure was rebuilt and finished when the structure already presented severe damage. During this field campaign, the incident ship-induced primary waves and water levels were recorded, and laser scans of the groyne armor were taken. Using those field laser scans, damage curves along the life of the structure were derived. Also, ship data from the AIS was retrieved. Then, each increment of the damage (increment of the dimensionless eroded area, ΔSe) was related to a ship-wave event and a passing ship. The most significant variable to describe ΔSe was found to be the primary wave height Hp, while the best explanatory variables for Hp were the partial blockage factor, the ship length and width and the relative velocity of the ship. The shape of the dependence between these variables is also analyzed by pairs using copula space. A clear tail dependence is observed between several pairs. For instance, the pair ΔSe and Hp presents upper tail dependence, meaning that the high values of ΔSe and Hp are more correlated than the smaller ones. This implies that models more complex than Gaussian copula, which is commonly used in Coastal Engineering applications, might be needed to model the probabilistic dependence between the variables.

The authors regret the acknowledgments in the published article are incomplete. The complete acknowledgments are as follows: “This research was supported by the Mexican National Council for Science and Technology (CONACYT) under project number 2019-000021-01EXTF-00564 CVU 784544. The authors would like to thank Dr. Andrés Antonio Torres Acosta, Research Professor at the Department of Sustainable and Civil Technologies, School of Engineering and Sciences, Tecnológico de Monterrey for his contributions to this research.” The authors would like to apologise for any inconvenience caused. ____________________________.

Around the world, an increasing amount of bridge infrastructure is ageing. The resources involved in the reassessment of existing assets often exceed available resources and many bridges lack a minimum structural assessment. Therefore, there is a need for comprehensive and quantitative approaches to assess all the assets in the bridge network to reduce the risk of collapsing, damage to infrastructure, and economic losses. This paper proposes a methodology to quantify the structural criticality of bridges at a network level. To accomplish this, long-run site-specific simulations are conducted using Bayesian Networks and bivariate copulas, utilizing recorded traffic data obtained from permanent counting stations. To enhance the dataset, information from Weigh-in-Motion systems from different regions was integrated through a matching process. Subsequently, the structural response resulting from the simulated traffic is assessed, and the extreme values of the traffic load effects are obtained for selected return periods. Site-specific bridge criticality as a performance indicator for traffic load effects is derived by comparing the extreme load effects with the design load effects. The outcomes are mapped to facilitate visualization employing an open-source geographic information system application. To illustrate the application of the methodology, a total of 576 bridges within a national highway network are investigated, and a comparison with a popular simplified method is shown. The methodology herein presented can be used to assist in assessing the condition of a bridge network and prioritizing maintenance and repair activities by identifying potential bridges subjected to major load stress.