As far as the authors are aware, the threshold for vertical acceleration of the bridge deck was chosen based on the assumption that the induced dynamic loads would overcome gravity at higher accelerations, resulting in loss of contact between wheels and rail; however, the previous studies do not support this hypothesis. Considering these inconsistencies, a better understanding of the simplified design criteria is essential before conducting further studies suchas the calibration of partial safety factors. Therefore, this study considers a set of representative design scenarios to statistically compare wheel-rail contact loss with other criteria that can bederived from moving load models, such as vertical accelerations and bridge deck deflections. Based on the analyzes performed, deflection seems to be a better criterion than acceleration to control the running safety limit-state; although the results presented do not necessarily show avery strong correlation between these two criteria. Therefore, the k-means clustering approach isused together with 5% lower quantiles of the collected data to propose potential new thresholds. It should be noted that due to the limited number of analyzes, the approach presented in this study can be considered as a possible framework for further updates of the current design method rather than drawing general conclusions.

Traditionally, regulations employ semi-probabilistic methods with partial safety factors to control design limits. Calibrating these partial safety factors involves estimating the target reliability level and optimizing the partial safety factor values in order to minimize the deviation of the safety index between the considered design scenarios and the target value. This procedure necessitates performing a demanding amount of reliability analyses and is often carried out for simplified design situations. Therefore, high computational costs must be accepted for design problems formulated with complex computational models. This study implements a meta-modeling approach based on active learning in the partial safety calibration procedure, enabling its application to computationally intensive problems. Subsequently, the approach is applied to the running safety of ballasted high-speed railway bridges. This limit state implicitly accounts for the phenomenon of ballast destabilization, the occurrence of which disturbs the load path from the rail level to the bridge structure. The dramatic increase in train operating speeds in recent decades has increased the possibility of this design limit state being violated due to resonance. Despite the evident safety concerns, the adopted safety factors appear to be solely based on engineering judgments rather than calibration through higher-level reliability analysis. Therefore, the proposed calibration method is employed to determine the corresponding partial safety factors for various maximum allowable operating train speeds. The newly calibrated partial safety factors allow for a permissible maximum vertical acceleration of the bridge deck approximately 25% higher than the conventional design approaches. Therefore, incorporating these factors into the design procedure may lead to the construction of lighter bridges.

Increasing the operating speed of trains in modern railway networks can induce greater actions on the infrastructure than was previously the case. This is due, in particular, to the occurrence of the resonance phenomenon in railway bridges, which is the focus of this paper and was not traditionally considered as a concern. In this context, the vibrations experienced by bridges, both vertical accelerations and displacements, are limited by design regulations to ensure that the safety of train passages over bridges and the comfort of passengers are guaranteed. However, previous studies have shown that the conventional dynamic design methods do not always result in conservative designs, nor is the achieved safety always consistent. Therefore, a probabilistic approach is adopted in this study to optimize the cross-section properties of various railway bridges in a wide design range including section types, span lengths, and number of spans. For this purpose, an iterative line search-based optimization problem is formulated to minimize the thickness of the cross-sections under consideration and consequently the linear mass of the bridges. Meanwhile, the associated failure probabilities of the above dynamic limit states are constrained to be less than the desired level of safety by incorporating them into the optimization constraint. In this regard, First-Order Reliability Method (FORM) is adopted to perform reliability analyses. Thus, the obtained results are presented in the form of design curves that may assist designers to select minimum cross-section dimensions satisfying the desired level of safety in terms of dynamic limit states. This objective can be achieved using the proposed design curves without the need to construct associated complex computational models and perform computationally expensive dynamic analyses.

The operational safety of high-speed trains traversing ballasted bridges is contingent upon the prevention of the ballast destabilization, which can interrupt load transfer from the rail to the bridge. Current design regulations indirectly address this limit-state by specifying a threshold value for the vertical acceleration of the superstructure. This value represents the condition at which the inertial forces induced by train passage exceed the resistive forces. However, this approach is based on limited experimental data and the influence of numerous parameters remains unexplored. As a result, reliability analyses pertaining to running safety are hampered by a lack of knowledge, leading to greater epistemic uncertainties. In this study, the impact of such uncertainties on this dynamic system is investigated using surrogate-based Imprecise Structural Reliability Analysis (ISRA). For this purpose, parametric probability boxes are used to represent lower and upper bounds of the cumulative distribution function for basic random variables with epistemic uncertainties and surrogate models are adaptively trained to reduce computational costs. The obtained results show that neglecting the influence of epistemic uncertainties can lead to permissible operating train speed higher than the speed corresponding to the desired reliability level. In this study, an overestimation of about 13% was observed on average. Furthermore, the rough analyses carried out show that taking epistemic uncertainties into account can lead to a reduction of the system characteristic safety factor by up to 30%. This significant reduction underlines the importance of expanding the available knowledge on the phenomenon of ballast instability.

The prediction of structural vibrations induced by high-speed trains is crucial for designing bridge structures, as well as for the assessment of the dynamic compatibility between bridges and rolling stock in large railway networks. The computational effort involved in such dynamic compatibility checks is also large. However, obtaining an accurate prediction of the dynamic response is challenging due to numerous uncertain factors still under research. Interaction phenomena like vehicle–bridge (VBI), soil–structure (SSI), and track–bridge (TBI) interactions are crucial but complex to model, given the multitude of input parameters and the computational cost required. Therefore, the choice of a modelling approach for simulating the dynamic response of a train crossing a bridge will depend on the focus of analysis. For the purpose of sensitivity analyses – typical in early design stages –, compatibility checks (screenings), and also in non-deterministic analyses, computational cost becomes crucial. Consequently, for practical purposes interaction mechanisms are often simulated by means of simplified and conservative approaches, usually aligned with some design code recommendations. Moreover, traditional physical models, where the problem is idealised as a beam traversed by constant moving loads (travelling load model, TLM), have been commonly employed for such purposes. However, they neglect the load distributive effect of the track on the vehicle axle loads, which has proven to be relevant and beneficial, especially for short bridge spans. The main purpose of this work is thus to investigate the influence of the load spreading effect exerted by the ballasted track on the displacement and acceleration response of single-track, simply-supported (S-S) railway bridges of short-to-medium span lengths under operating conditions, since these structures are prone to experience inadmissible vibration levels under more demanding traffic conditions. A comprehensive analysis of several simplified approaches proposed by regulations and researchers is performed, plus a subsequent comparison vs. a finite element (FE) strategy to consider TBI in a refined manner. Conclusions about the adequacy of the simplified approaches are provided, along with a new data-driven formula for the prediction of the vertical displacement response in resonant conditions, that can be exploited to reduce significantly the computational effort.

There is a growing intent among engineers, stakeholders, and decision makers to use probabilistic methods for infrastructure assessment or design objectives. However, the corresponding limit state for such problems usually requires the construction of complex computational models, usually using commercial software without parallelization capability. Such a requirement makes performing reliability analysis computationally prohibitive, which is even more challenging for dynamic problems, since a very short time step is required to obtain sufficiently accurate predictions. This concern has led to several methods being proposed to surrogate the limit state function with a generally black box called a meta-model. A variety of them, such as Kriging, Polynomial Chaos Expansion (PCE), Artificial Neural Networks (ANN), and response surfaces (e.g., polynomial, spline, or radial-base functions), have been adopted for this purpose. These meta-models are typically trained on a limited data set collected by computing the true responses of carefully selected input variables. Their applicability for assessing the probability of failure has been studied individually in the literature for both benchmark and practical problems. However, as far as the authors are aware, no comparison has been made between them for dynamic problems. This comparison needs to be made from the point of view of both accuracy and performance (number of calls to the limit state function). In this context, this paper takes a systematic approach to evaluate their performance under identical conditions, i.e., with similar training datasets. For this purpose, the dynamic response of railway bridges with different spans excited by the passage of trains with a wide range of speeds is used as a reference problem.

Operating high-speed trains imposes excessive vibrations to bridges raising concerns about their safety. In this context, it was shown that some conventional design methods such as those related to the running safety suffer from a vague scientific background questioning their reliability or optimality. Therefore, the current article is devoted to updating the conventional design methodology, using Reliability-Based Design Optimization (RBDO) to propose the minimum allowable mass and stiffness which assures satisfying the target reliability. These proposed minimum design values can conceptually replace the conventional partial safety factor-based design method for running safety without the need for dynamic analysis. If the mass and stiffness resulting from the control of other limit states meet the proposed minimum values, the desired target reliability for running safety will be assured. This is achieved by adaptively training Kriging meta-models as a surrogate for the computational models decoupling the RBDO problem. In this regard, a new stopping criteria is proposed using mis-classification ratio of the cross-validated model; which reduces the generalization error of the trained meta-model and consequently the estimated failure probability. Moreover, due to the dependence of the design variables, the Copula concept is used to refine the augmented space and reformulate the RBDO problem.

Increasing operating speeds and axle loads of trains may induce higher verticaldeck accelerations on bridges, which may subsequently lead to the occurrence of ballast instabilities.This phenomenon not only increases maintenance costs but also leads to speed restrictionsunder operating conditions. In severe cases, it can also cause train derailment. Thishas been shown to be the governing criterion for the design of short to medium span lengthhigh-speed railway bridges, especially from the dynamic behaviour point of view. Despiteits substantial importance in the design of railway bridges, the conventional deterministic approachescannot achieve the desired level of safety (Allahvirdizadeh et al. 2020). Therefore,this article is devoted to the evaluation of the probability of violating running safety usingsimulation-based methods. In this context, different variables, including those for the bridge(span length, flexural rigidity, and geometric properties), for the train (axle load, dominantfrequency and damping ratio) and for the boundary conditions (soil and foundation properties)are considered. Due to the high computational cost and complexity of the consideredperformance function, a low-cost meta-model is trained using stack modelling concept as acombination of support vector machines (SVM), k-nearest neighbours (k-NN) and decisiontrees. Then, the range of maximum and minimum probabilities of exceeding the verticalacceleration threshold are evaluated as a function of train speed and bridge span length. Comparingthese boundaries with corresponding results of simply-supported bridges showed thatneglecting soil-structure interaction effects in shorter span bridges may result an underestimatedsafety of the system; however, for longer span bridges it may result in overestimatedsafeties.

Excessive vertical acceleration of ballastless railway bridges subjected to vibrations induced by passing trains is one of the governing design criteria for bridges in high-speed lines. However, to the authors' knowledge, the corresponding design limit is not based on a solid theoretical or experimental background. Moreover, the traditionally applied safety factor also suffers from these concerns. Therefore, in the present study, a crude probabilistic approach is adopted to evaluate the consistency and reliability of this safety factor. For this purpose, deterministically designed bridges (using conventional methods) with short to medium spans are considered. Then, their reliability is evaluated using simulation-based techniques and extreme value theory, i.e., tail approximation. Then, the existing safety factor is calculated to evaluate the consistency of the current approaches, and possible new values are proposed based on the desired target reliabilities.

Limiting the maximum vertical acceleration and deflection of the deck are two principal design criteriaof high-speed railway bridges. The former prevents ballast instability to ensure running safety, andthe latter attempts to limit the acceleration of the car-body below the level at which passengercomfort is disturbed. The previous studies are mainly concerned with the destabilization of the ballast,nevertheless the possibility of the maximum deflection occurrence should not be underestimated.Moreover, the literature indicates the need to improve the current design requirements including theminimum allowable mass and frequency of bridges, which requires solving optimization problems basedon modern requirements. Therefore, a probabilistic framework with simulation-based techniques is usedto evaluate the violation probability of the above limit states and distinguish dominant criteria underdifferent conditions, i.e., bridge span length and operational train speed. First, the performance of thesubset simulation method is compared with the Latin Hypercube-sampling based Monte-Carlo approachsupported by surrogate models. Polynomial chaos expansion (PCE) surrogate models are trained for thisobjective. Then, the resulting violation probabilities are evaluated for the two considered limit statesusing the approach with better performance.