The operational speed of the trains is intended to be significantly increasedforthcoming; which consequently questions the safety level of the current design conceptsconcerning different performance limits. Thus, the reliability of the simply supportedsingle-span bridges is assessed in the current article adopting the first-order reliabilitymethod (FORM) approach. In this regard, the dynamic response of the aimed bridges isinvestigated under the passage of a series of moving loads using available closed-form solutions in the literature. Hereof, axle load, car body/train configuration, flexural rigidity,damping, mass and model uncertainties are considered as random variables; while traintrack-bridge interactions are neglected and the procedure is repeated for a wide range ofspan lengths and train velocities. Then, the safety index corresponding to each case isevaluated by considering running safety as the limit state function; where, the bridge deckvertical acceleration is taken as the capacity of the system. The outcomes are presentedas average probability of exceeding the limit state versus train speed and categorized basedon span lengths.

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 article 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 widedesign 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 depth 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 in to 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.

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.

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.

The increasing speeds of modern trains lead to excessive vibrations on the bridges, which have the potential to destabilize the ballast particles of the track. The occurrence of this phenomenon not only increases the track maintenance cost, but can also disrupt the load path from the rail level to the bridge deck, posing a risk to the train running safety. The design regulations indirectly control this limit-state by limiting the vertical acceleration of the bridge deck to a threshold value. The assessments pertaining to this purpose often neglect the soil-structure interaction (SSI) effects considering that as a conservative assumption. Such effects can positively contribute by increasing the system damping, but they can also increase the bridge flexibility making it more susceptible to vibrations. Therefore, this study investigates the influence of considering/disregarding SSI effects on the running safety criterion using a probabilistic methodology. The results are classified based on the maximum permissible train speeds and the bridge span length. Due to the high computational costs of the reliability analyses, the limit-state is approximated by an ensemble of classification-based surrogate models using the stack-generalization concept. Subsequently, the upper/lower bounds of the failure probability in the presenceof SSI effects are compared with those obtained for simply-supported bridges. It is pointed out that neglecting SSI effects for shorter span bridges may lead to an underestimation of system safety. For longer span bridges, however, this may lead to an overestimation of safety, which means that a non-conservative system can be designed.

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.