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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.

A short review of the present status of computer packages for the high order analytical perturbative calculations is presented. The mathematical algorithm and the quantum field theory methods used are briefly discussed. The most recent computer package HEPLoops for analytical computations in high energy physics up to four-loops is also discussed.

This paper presents a non-sampling based method for the simultaneous evaluation of quality and performance reliability of engineering systems with multiple time-variant responses due to multiple degrading components. This work provides a platform for robust design of degrading systems. The system performance degradations are related to component degradations using mechanistic models. The system soft failure is defined as the non-conformance of any response with respect to critical levels and such relations are easily modeled as time dependent limit-state functions. Then, for discrete time it is shown that an incremental failure set that emerges from a safe region can be written using only a pair of successive system instantaneous failure sets. The cumulative distribution function of soft failure is built by summing the incremental failure probabilities. A practical implementation of the proposed method can be manifest by first-order reliability methods (FORM) and second-order bounds. The proposed method can be used to assess initial quality and performance reliability of systems with combinations of designated means and tolerances. Examples of electro mechanical systems show the details of the formulation and the potential of the approach. Error sources and their magnitudes are discussed.

The geometry of high-dimensional spaces is very different from low dimensional spaces and possesses some counter-intuitive features. It is shown that, for high dimensions, the sampling points fall far away from the origin and concentrate within an intersection between a very thin shell and a suitable equatorial slab. The well-known First-Order Reliability Method (FORM), originally formulated for low dimensions, may work well in many engineering problems of high dimension. But it is not able to reveal the level of achieved accuracy. Considering the features of high-dimensional geometry, a novel linear response surface based on Support Vector Method (SVM) is proposed for structural reliability problems of high dimension. The method is shown to outperform FORM for structural reliability problems of high dimension in terms of robustness and accuracy.

The core of the paper consists of the illustration of a method for seismic reliability analysis of non-linear structures. The method, which is a development over a recent previous proposal, is quite comprehensive in scope since it includes consideration of randomness in the input, in the mechanical properties of the structure and in the limit-state, or capacity, conditions. Essentially, the problem is formulated as the out-crossing of the response process out of a (scalar) safe threshold, and this problem is solved by time-invariant FORM methods. An application to an idealised five-storey building demonstrates the salient theoretical and computational features of the method. The new approach is presented in a broader framework, which involves a discussion on present trends and capabilities in the area of probabilistic seismic design. For the sake of this discussion, two approaches just appeared in the literature are outlined, which are less demanding from a theoretical standpoint, and hence closer to engineering practice, but also less general in scope. It is argued that their introduction on code-assisted design would be feasible immediately with obvious advantages, while approaches of more rigorous nature would be given some more time to mature and to become more accessible to professional use.