Probabilistic methodologies for the safety assessment of short span railway bridges for high-speed traffic

PROBABILISTIC METHODOLOGIES FOR THE SAFETY ASSESSMENT OF SHORT SPAN RAILWAY BRIDGES FOR HIGH-SPEED TRAFFIC João Miguel dos Santos Pereira da Rocha 2015 A dissertation presented to the Faculty of Engineering of the University of Porto for the degree of Doctor in Civil Engineering. Supervisor: Professor António Abel Henriques Co-Supervisor: Professor Rui Calçada “Light thinks it travels faster than anything but it is wrong. No matter how fast light travels, it finds the darkness has always got there first, and is waiting for it.” Terry Pratchett “Yehi 'Or! (Let there be light!)” The Book of Genesis 1:3 Aos meus Pais e à minha Avó Abstract Transportation systems are vital in modern societies as they represent one of the most important means to achieve both social and economic development on a more than ever global world. Presently a fast, safe, efficient and, if possible, environmentally friendly mobility of people and goods is paramount. It is in this context that the high-speed railway systems arises as one of the most interesting alternatives to other means of transportation, explaining its rapid expansion across the globe in the last decades. Bearing in mind the importance of a safe and reliable high-speed railway network it was decided that the focus of this thesis should be directed to the development of a probabilistic methodology that enables the accurate safety assessment of short span railway bridges. This would complement most of the works done on this field to this date (that have adopted a deterministic approach) and would help understanding the effects of the intrinsic variability of parameters related to the train, track and bridge parameters on the dynamic behaviour of the system. A review of the state of the art regarding the dynamics of railway bridges is presented, with particular attention given to the resonance phenomenon due to its significance to small and medium span bridges. Excessive accelerations have caused several problems to short span railway bridges within the European high-speed railway network and it is important to understand the advantages, drawbacks and limitations of the different methods used to assess the dynamic response of railway bridges. Additionally, it is important to review the limit states prescribed by the current European standards and understand their origin to evaluate the adequacy of these proposals. Different probabilistic approaches were analysed and, taking into account the problem being studied, it became evident that the use of simulation techniques would be the most adequate approach. Two different simulation methods, namely the Monte Carlo and the Latin Hypercube, are combined with two different procedures to enhance the efficiency of the assessment. One of the methods is a tail modelling approach based on the extreme value theory that uses adequate functions to model the tail of the obtained distribution. The other one is an Enhanced Simulation procedure which uses an approximation procedure based on the estimates of the failure probabilities at moderate levels for the prediction of the far tail failure probabilities by extrapolation. To test the efficiency and accuracy of the different methodologies that are proposed a case study bridge was selected based on carefully selected range of potential solutions and taking into consideration the existing bridge portfolio on the current high-speed railway network. A ballasted filler beam bridge composed by six simply supported spans of 12 m each was selected and its behaviour was analysed for the crossing of a TGV-Double train. Several modelling particularities, including the complexity of the model used for both the train and the track, the consideration of train-bridge interaction and the existence of track irregularities are discussed in detail to understand their impact on the accurate assessment of the dynamic response of short span railway bridges. Additionally, a sensitivity analysis was carried out to determine the influence of each of the basic random variables on the dynamic behaviour of the train-bridge system. The proposed probabilistic methodologies are used to assess the safety of the case study bridge for two different criteria: the running safety of trains due to loss of contact between the wheel and the rail and the track instability due to excessive deck vibrations. The use of these criteria provide examples of limit state functions with varying degrees of complexity that test and validate their efficiency, robustness and reliability. Globally, the Enhanced Simulation procedure proved to be significantly more efficient than the tail modelling approach. It was also observed that if the response is not monotonic the use of Latin Hypercube simulation may affect the efficiency of safety assessment. Furthermore, it was demonstrated that if the computational costs can be reduced through a refined simulation method then the use of the Enhanced Simulation approach will results in even further benefits. The obtained results are extremely promising and indicate the feasibility of the application of this type of methodology more frequently due to the reasonable computational costs that are required. Resumo Os sistemas de transporte são vitais para as sociedades modernas uma vez que são um dos mais importantes meios de desenvolvimento económico e social num Mundo cada vez mais globalizado. Atualmente, a deslocação rápida, segura, eficiente e, se possível sustentável, de pessoas e bens assume extrema importância. Por estes motivos o transporte ferroviário de alta velocidade emergiu como uma alternativa muito apelativa aos meios de transporte mais tradicionais, explicando-se assim o seu rápido desenvolvimento nas últimas décadas. Tendo em consideração a relevância de uma rede ferroviária de alta velocidade segura e eficiente, a presente dissertação dedica a sua atenção ao desenvolvimento de uma metodologia probabilística que permita a avaliação da segurança de pontes ferroviárias de pequeno vão. Esta opção permite complementar o trabalho anteriormente desenvolvido nesta área (de carácter predominantemente determinístico) ajudando ainda a compreender os efeitos da variabilidade intrínseca aos parâmetros que condicionam a resposta do sistema ponte-via-comboio. Foi realizada uma revisão do estado da arte relativa ao comportamento dinâmico de pontes ferroviárias, dando-se particular atenção aos fenómenos ressonantes que são especialmente importantes para pontes de pequeno e médio vão. Diversos problemas devido a acelerações excessivas neste tipo de estruturas inseridas na rede ferroviária de alta velocidade Europeia foram reportados no passado. Deste modo é imperativo compreender-se as vantagens, desvantagens e limitações das diferentes metodologias utilizadas para aferir a resposta dinâmica das pontes. Adicionalmente, é importante rever os estados limites definidos nas normas Europeias e compreender a sua génese de modo a avaliar se são adequados. Foram analisadas diferentes metodologias probabilísticas para abordar o problema a estudar tendo-se concluído que a utilização de métodos de simulação seria a solução mais adequada. Deste modo, dois métodos de simulação, o método de Monte Carlo e o método do Hipercubo Latino, foram utilizados. Estes métodos foram combinados com dois procedimentos distintos de modo a melhorar a eficiência da avaliação da segurança. A primeira técnica baseia-se na teoria dos valores extremos e consiste na modelação das caudas das distribuições através de 2.3.2.6 Time step selection ......................................................................................... 2.27 2.4 Limit states ................................................................................................................... 2.28 2.4.1 Structural safety ................................................................................................. 2.28 2.4.2 Traffic Safety Checks ......................................................................................... 2.30 2.4.2.1 Vertical acceleration of the deck .................................................................... 2.30 2.4.2.2 Vertical deformation of the deck .................................................................... 2.34 2.4.3 Serviceability Limit State – Passenger riding comfort ...................................... 2.34 2.4.4 Running safety ................................................................................................... 2.36 2.4.4.1 Wheel flange climbing ................................................................................... 2.36 2.4.4.2 Track panel shift ............................................................................................. 2.37 2.4.4.3 Wheel unloading............................................................................................. 2.38 Chapter 3 – Structural reliability assessment 3.1 Introduction ................................................................................................................. 3.1 3.2 Fundamentals............................................................................................................... 3.3 3.2.1 Basic concepts ...................................................................................................... 3.3 3.2.2 Semi-probabilistic approach ................................................................................ 3.5 3.2.3 Probabilistic approach .......................................................................................... 3.6 3.2.4 The Fundamental Reliability Case ....................................................................... 3.7 3.3 Methods for the evaluation of the probability of failure ........................................... 3.11 3.3.1 Level I Methods ................................................................................................. 3.11 3.3.2 Level II Methods ................................................................................................ 3.12 3.3.3 Level III Methods............................................................................................... 3.15 3.3.4 Level IV Methods .............................................................................................. 3.22 3.4 Enhanced methodologies for estimating the probability of failure ........................... 3.23 3.4.1 Tail modelling .................................................................................................... 3.23 3.4.2 Enhanced simulation method ............................................................................. 3.25 3.5 Safety assessment framework ................................................................................... 3.29 Chapter 4 – Modelling of the train-track-bridge system 4.1 Introduction ................................................................................................................. 4.1 4.2 Case study ................................................................................................................... 4.2 4.2.1 Description of Canelas Bridge ............................................................................. 4.2 4.2.2 Previous research studies on Canelas Bridge....................................................... 4.5 4.3 Numerical models ....................................................................................................... 4.6 4.3.1 Track-Bridge numerical models .......................................................................... 4.6 4.3.2 Track irregularities modelling ............................................................................ 4.12 4.3.3 Basic random variables ...................................................................................... 4.17 4.3.4 Dynamic properties of the track-bridge system ................................................. 4.21 4.3.5 Train numerical models ..................................................................................... 4.24 4.3.5.1 Moving loads models ..................................................................................... 4.24 4.3.5.2 Moving masses models .................................................................................. 4.25 4.3.5.3 Suspended masses models .............................................................................. 4.26 4.3.5.4 Complete models ............................................................................................ 4.26 4.3.5.5 Wheel-rail interaction ..................................................................................... 4.28 4.3.5.6 Train numerical model developed .................................................................. 4.31 4.4 Dynamic response ........................................................................................................ 4.35 4.4.1 Train-bridge interaction effects .......................................................................... 4.36 4.4.2 Track irregularities effects ................................................................................. 4.38 4.4.3 Modelling particularities .................................................................................... 4.44 4.4.3.1 Influence of the track length before the bridge .............................................. 4.44 4.4.3.2 Validation of the track numerical model ........................................................ 4.49 4.4.3.3 Validation of the train numerical model ........................................................ 4.49 4.5 Variable screening procedure ...................................................................................... 4.51 4.5.1 Description of the methodology ......................................................................... 4.51 4.5.2 Preliminary approach ......................................................................................... 4.52 4.5.3 Sensitivity analysis accounting for the train-bridge interaction ......................... 4.54 4.5.3.1 Bridge deck acceleration ................................................................................ 4.54 4.5.3.2 Wheel unloading rate...................................................................................... 4.56 4.6 Concluding remarks ..................................................................................................... 4.57 Chapter 5 – Track stability assessment 5.1 Introduction ................................................................................................................. 5.1 5.2 Safety criterion ............................................................................................................ 5.2 5.3 Safety assessment – results and discussion ................................................................. 5.3 5.3.1 Simplified approach – moving loads ................................................................... 5.3 5.3.1.1 Dynamic response for a deterministic scenario ................................................ 5.3 5.3.1.2 Preliminary simulation analysis ....................................................................... 5.8 5.3.1.3 Simulation refinement .................................................................................... 5.14 5.3.1.4 Preliminary conclusions ................................................................................. 5.22 5.3.2 Train-bridge interaction ..................................................................................... 5.22 5.3.2.1 Dynamic response accounting for the interaction effects .............................. 5.23 5.3.2.2 Tail modelling approach ................................................................................. 5.29 5.3.2.3 Enhanced simulation approach ....................................................................... 5.41 5.3.2.4 Efficiency comparison .................................................................................... 5.44 5.4 Concluding remarks ..................................................................................................... 5.45 Chapter 6 – Train running safety assessment 6.1 Introduction ................................................................................................................. 6.1 6.2 Safety criterion ............................................................................................................ 6.2 6.3 Safety assessment – results and discussion ................................................................. 6.3 6.3.1 Simulation results................................................................................................. 6.3 6.3.2 Tail modelling approach ...................................................................................... 6.9 6.3.3 Enhanced Simulation approach.......................................................................... 6.13 6.3.4 Efficiency comparison ....................................................................................... 6.15 6.4 Concluding remarks .................................................................................................. 6.15 Chapter 7 – Conclusions and future developments 7.1 Conclusions ................................................................................................................. 7.1 7.2 Future developments ................................................................................................. 7.10 1.1 Chapter 1 Introduction 1.1 Scope of the thesis Transportation systems play a key role in modern societies. Social and economic development depend on the fast and efficient mobility of both people and goods. In this context, the highspeed railway system emerges as a reliable and appealing alternative to road-based means of transportation. The popularity and success of the high-speed railway network relies on two main aspects: firstly it is a safe and low energy means of transportation and secondly because of its high on-schedule rate. For this reason the high-speed railway network has developed and expanded rapidly across the world in the last decades. The first high-speed railway line was the Tōkaidō Shinkansen in Japan which connected Tokyo to Osaka and started operating in 1964, with trains running at 210 km/h. In Europe the first high-speed railway line started operating nearly 20 years later in 1981 in France, connecting Paris to Lyon, with a maximum speed of 260 km/h. Since then the European high-speed railway network has been expanding continuously and the operation speed has also increased. Nowadays it is already well established in several countries such as France, Germany, Italy, Spain and the United Kingdom and is planned to keep expanding in the near future, as is shown in Figure 1.1 and Figure 1.2. Chapter 1 1.2 Figure 1.1 – European High-speed railway network [adapted from Institut d’aménagement et d’urbanisme - Île-de-France (2012)]. Driven by the socio-economic potential of a European high-speed railway network the European directive 69/48/EC was published in July 1996 and was the initial step to establish the basis of the operational standards within the network. This work finally culminated with the publication of the Technical Specifications for Interoperability [TSI (2002)]. This document aimed to harmonise the European standards regarding the several components of the network, the rolling stock, the infrastructure and operation, and defines the technical basis for operational safety within the European high-speed railway network. Amongst the several components of the railway network, railway bridges have been identified as some of the most sensitive elements. Despite this fact, due to the several constraints involved in the planning of high-speed railway networks, which can either be political, territorial or technical, it can be observed that most networks have a significant percentage of bridges and viaducts. In China and Japan, for example, the percentage of bridges and viaducts in several of the lines exceeds 75% of the total length [Ishibashi (2004); Dai et al (2010)]. The European reality is slightly different and the percentage of bridges or viaducts tends to be lower. However, if we analyse the case of HS2, which is the biggest planned high-speed development in Europe and previewed to begin in 2017 in the UK, in a line with a length of approximately 200 km there will be around 350 bridges, which is still quite significant. Introduction 1.3 Figure 1.2 –High-speed railway network in numbers [http://www.uic.org/highspeed]. Therefore, the dynamic behaviour of railway bridges has been the research subject for many engineers since the introduction of railway as a means of transportation due to the importance of the dynamic effects caused by trains crossing over them. This is one of the most relevant aspects to take into account for a safe design and operation. However, the generalisation of high-speed railway lines over the last decades, the continuous demand for higher operational speed and the introduction of high-speed freight traffic has introduced new engineering challenges. Within the high-speed railway network the dynamic amplifications caused by trains moving over the bridges have proven to be the governing factor for bridge design. Recent research has shown that excessive vibrations have a higher tendency to occur for speeds above 200 km/h as a consequence of the resonance phenomena [ERRI (1999)]. This can lead to several problems, namely the instability of the ballast layer, the loss of contact between the wheel and the rail, the increase of fatigue-related damage or even affect the comfort of the passengers. The current European standards reflect the concern for excessive vibrations and all the consequences that arise from inadequate/inefficient design by imposing the application of a dynamic analysis in almost every case where the maximum line speed exceeds 200 km/h [EN1991-2 (2003)]. Short span railway bridges have been particularly identified as most critical as they are more prone to experiencing resonant effects. This has become evident following the reports of problems due to excessive vibrations in several bridges in the first European high-speed railway Chapter 1 1.4 line in France [Hoorpah (2005); Zacher & Baeβler (2009)], which required immediate repairs to the bridges shortly after construction to ensure they had adequate dynamic behaviour. Besides being very sensitive to resonance the dynamic response of short span railway bridges is also strongly affected by the non-structural environment, namely the track and train properties and even by the structural boundary conditions. All these aspects are particularly difficult to adequately quantify during the design stage and this was identified as one of the reasons why the initial design proved ineffective. It became evident that a better understanding of the resonance mechanisms was required in order to try to establish which parameters govern the dynamic response of these structures. The works of the European Railway Research Institute ERRI (1999), Majka & Hartnett (2008) and more recently of Doménech et al (2014) have tried to provide a response to this problem. Despite being extremely useful and providing some excellent insight into this issue none of the works has simultaneously considered the variability of the parameters related to the bridge, the track and the trains nor have they taken into consideration the existence of track irregularities in the analysis. It is also clear that the majority of the research studies on the assessment of the dynamic behaviour of railway bridges are deterministic in nature. The analysis is often limited to a specific scenario even when the existence of track irregularities is accounted for. This conventional approach uses fixed values to define the parameters, it may use a parametric analysis to assess the dynamic response and evaluates the results and safety based on deterministic limits defined in the standards which are the result of semi-probabilistic methods. In the few works that took the variability of parameters into account it is noticed that generally the variability is defined for only one of the components of the system and usually limited to a specific train speed. Taking advantage of the increasing computational capacity, the use of reliability based methodologies to assess the safety of railway bridges can be a reality in the present. This type of approach enables a more realistic assessment of the dynamic behaviour of the train-bridge system as it reflects the real variability of the parameters that affect the dynamic response. Furthermore, the output of this approach is a probability of a defined limit being reached (usually understood as a probability of failure) which allows the adjustment of the risk level according to the limit state being analysed and the problem that is being studied. This versatility is also reflected in the type of problems that can be addressed which are not limited to the design of new structures but Introduction 1.5 can also be applied to the assessment of existing structures when the lines are upgraded for higher speeds or heavier traffic. In the absence of sufficient studies and due to the existing knowledge gap in this topic the present thesis intends to propose an efficient, robust and accurate probabilistic methodology to assess the safety of short span railway bridges. 1.2 Motivation and objectives The motivation for the research work presented in this thesis is the paucity of studies carried out to assess the safety of high-speed railway bridges using probabilistic approaches that enable the variability of the parameters that govern the dynamic behaviour of the train-bridge system to be taken into account. As previously highlighted, most of the work developed in this field is deterministic in nature and the analysis is often limited to a specific scenario even when the existence of track irregularities is accounted for. As a consequence, there has been some debate about the adequacy of some of the limits defined in the current European standards, particularly with respect to the bridge deck acceleration limits. Due to the importance of accurately assessing the response of the train-bridge system, both for a cost efficient and safe design and operation, the development of adequate and efficient probabilistic methodologies is of the utmost importance and is extremely useful for Engineers both at the design stage and for the assessment of existing structures. Therefore, the main objective of this thesis consists of developing a probabilistic methodology that enables an accurate and efficient assessment of the safety of short span high-speed railway bridges whilst accounting for the variability of parameters of the bridge, the track and the train as well as for the existence of track irregularities. The use of such an approach enables a more realistic analysis of the behaviour of the trainbridge system which is translated into an increase in the accuracy of the assessment. Furthermore, this type of approach also enables the definition of the safety/risk level by an adequate selection of the probability threshold which can be adjusted according to the problem being studied. For this reason, and despite the fact that in this thesis the design stage view is adopted, this type of methodology can bring benefits to the analysis of either new or existing structures. Chapter 1 1.6 Additionally, the research presented in this thesis focus its attention on the short span highspeed railway bridges due to the fact that these structures are particularly sensitive to resonant effects as well as to the influence of the non-structural elements (such as the track properties and the train-bridge interaction effects) on the response of the train-bridge system. It is also an objective of this study to identify the parameters that govern the dynamic behaviour of this type of bridges and how their intrinsic variability affects the safety of the train-bridge system. In order to achieve these objectives a case study bridge that is representative of the population of short span bridges in the European high-speed railway network is selected and different limit states are analysed using the proposed methodologies in order to check the accuracy, efficiency and robustness of the methodologies as well as analysing the feasibility of the application of these methods in the future. 1.3 Layout of the thesis Taking into account the motivation and objectives of this thesis that have just been presented in the previous section, the contents of this thesis have been divided into seven chapters. A schematic representation of the contents of the thesis can be seen in Figure 1.3. Figure 1.3 – Schematic representation of the contents of the thesis. Dynamic behaviour of railway bridges 2.5 Despite the several modelling possibilities the model should be selected carefully in order to guarantee the best possible balance between the accuracy of the results and the computational costs. Dieleman & Fournol (2003) refer that in the particular case of short span bridges a significant difference between numerical and experimental results is often observed. The authors identified the following aspects as the main reasons for the observed differences: i) inaccurate definition of the span length; ii) inaccurate definition of the boundary conditions; iii) neglecting the track-bridge composite effect; iv) the use of moving loads to model the train; and v) neglecting the train-bridge interaction effects. Amongst the several aspects these authors focused mostly on the influence of the bearings stiffness on the dynamic response and were able to conclude that the use of elastic supports generally leads to satisfactory results. However, they point out that the bearings stiffness is a very difficult parameter to be accurately measure and represents a very sensitive modelling parameter. With respect to the modelling of the track several different proposals can be found in the literature. However, there are some aspects to distinguish between these models. Primarily they can be split according to the number of dimensions that are considered, dividing them into 2D [Calçada (1995); Zhai et al (2004)], 3D [Chellini & Salvatore (2007); Zabel & Brehm (2009)] and even 2.5D models, for some specific problems, such as the analysis of ground vibrations, [Costa (2011)]. This choice is mostly related with the type of analysis that is intended to be carried out. Another aspect concerns the track elements that are included in the numerical model, with some including all the track elements such as the rail, rail pads, sleepers, ballast and the existence of track irregularities, as well as the track-bridge composite effect [Calçada (1995); Zhai et al (2004)], whereas other authors choose fewer elements [Lou (2005)]. The latter models often neglect this effect but a parametric analysis carried out for a simply supported bridge with a small span shows the impact of the track-bridge composite effect on the dynamic response. The analysis considered both the higher and lower limits proposed in the European Standard EN19912 (2003), which represent a loaded and an unloaded track, respectively, as well as a scenario where the composite effect is neglected. The results are presented in Figure 2.1 and clearly indicate that the variation from loaded to unloaded track has a small impact on the dynamic response. However, when the track-bridge composite effect is neglected significant changes can be observed and the dynamic amplification shows an important increase. Chapter 2 2.6 a) Maximum deck acceleration at ½ span. b) Maximum deck acceleration at ¾ span. Figure 2.1 – Example of the influence of the track-bridge composite effect on the dynamic response. Finally, regarding the train model a different range of options can also be found in the literature. There are generally two main types of formulation used to model trains: one is the multibody dynamics formulation [Pombo (2004); Kwark et al (2004)], very common in Mechanical Engineering, and the other is the Finite Elements Method, mostly used in Civil Engineering [Calçada (1995); Yang et al (2004)]. The numerical models can either be two dimensional [Zhai et al (2001); Goicolea et al (2004); Doménech & Museros (2011)] or three 0 1 2 3 4 5 6 7 8 200 250 300 350 400 450 Acceleration (m/s2) Speed (km/h) loaded track unloaded track composite effect neglected 0 1 2 3 4 5 6 7 8 200 250 300 350 400 450 Acceleration (m/s2) Speed (km/h) loaded track unloaded track composite effect neglected Dynamic behaviour of railway bridges 2.7 dimensional [Kwark et al (2004); Xia & Zhang (2005); Zhang et al (2008); Lee & Kim (2010)]. The 2D models are the most commonly found in the literature and this can be explained by the fact that these are simpler, require less information about the train dynamic properties (which is often difficult to find) resulting in a model with fewer degrees of freedom and, consequently, with higher computational efficiency. However, the main drawback of these models is that they limit the analysis to a single direction (generally, the vertical direction). The 3D models are more versatile as they allow the analysis of all directions. These models tend to be more realistic and for some particular cases, namely when assessing passenger riding comfort, are necessary to obtain accurate results. The consequence of using more complex models is the increase of the computational time as the number of degrees of freedom that are taken into account is considerably larger. Given the importance of numerical modelling on the accuracy and efficiency of the dynamic analysis this topic is discussed in greater detail on Chapter 4. Several types of models that can be found in the literature are presented and the models used in this work are introduced and discussed. It is therefore important that the selection of the numerical model is carefully made and the degree of complexity to employ should always be related with the results that are to be analysed. This is to ensure that the computational costs of the analysis are feasible and adequate to the aim of the study. Besides the modelling of each subsystem their interaction is accounted for in the analysis is another aspect of great importance. Generically, this can be divided into two distinct groups: analysis that do or do not account for the train-bridge interaction. Due to the relevance of this topic for the present dissertation this is discussed in more depth on Section 2.3 where the different alternatives are presented and their advantages and disadvantages are analysed. 2.2.2 Resonance A particular case of dynamic amplification, which is particularly relevant due to the consequences of its occurrence, is resonance. Bridge resonance occurs when the excitation frequency, λ, matches or is a multiple of a natural frequency, nj, of the bridge. The resonant speed, vres, can then be estimated by: Chapter 2 2.8 ...,,,, 321 i i d nnv jijres  (2.1) where d represents the spacing between regular groups of axles. Recent research has shown that excessive vibrations have a higher tendency to occur for speeds above 200 km/h as a consequence of the resonance phenomena [ERRI (1999)]. Therefore, this is a crucial aspect to take into consideration when designing bridges for high-speed railway lines. The existence of resonance can significantly increase the dynamic response, which can lead to several problems. Some of these problems include the instability of the ballast layer, loss of contact between the wheel and the rail, increase of fatigue-related damage, discomfort of the passengers and, in more extreme cases, it can even lead to the collapse of the bridge. In the first high-speed railway line in Europe, which connected Paris to Lyon, several problems were reported due to the existence of resonance, resulting in significant maintenance and repair costs [ERRI (1999); Hoorpah (2005); Zacher & Baeβler (2009)]. Museros (2002) summarised the main problems that were detected due to excessive vibrations. The problems reported included the flying ballast phenomenon, which reduced the support to the rails, the enhancement of the ballast layer deterioration rate that affected the ballast bonding and promoted the appearance of voids beneath some sleepers (originating hanging sleepers). Moreover, the cracking of concrete elements was also observed, leading to a global stiffness reduction of the bridge which changed the critical train speeds over the structure. In order to provide a better understanding of this dynamic phenomenon and to prevent the serious consequences associated to its occurrence several researches have been dedicated to this topic in recent years. The aim of these studies is mainly to ensure the adequate performance of the train-track-bridge system and to develop methodologies that enable guaranteeing the safe design and operation of railway bridges and also the minimisation of operational/maintenance costs. The European Rail Research Institute (ERRI) carried out extensive studies that aimed to expand the knowledge on the dynamic behaviour of bridges in high-speed lines and to serve as a guideline to verify the safety demands and guarantee an adequate performance in service [ERRI (1999)]. The scope of the study was quite broad and dedicated some attention to the resonance phenomena. The report points out that for simply supported structures the design is generally Dynamic behaviour of railway bridges 2.9 governed by resonant effects that originate excessive deck accelerations, affecting the stability of the ballast layer and, consequently, creating a running safety risk. Yang et al (1997) investigated the dynamic behaviour of simply supported beams subjected to the passage of high-speed trains. This study identified the parameters that govern the dynamic behaviour of the structure and proposed an optimal design criterion for high-speed railway bridges based on both the condition of resonance and suppression. Xia et al (2006) evaluated the resonance effects on a high-speed railway bridge using theoretical formulations, numerical simulation and experimental tests. The authors divide the resonant response into three types according to different resonance mechanisms: the first is due to the periodical effects of the loading, the second is due to the rate of the loading and the third is a result of the periodically loading of the swing caused by track irregularities and wheel hunting movements. The work of Rigueiro (2007) was focused on the study of the dynamic behaviour of short to medium span railway bridges. The study analysed both the resonance and the suppression phenomena, highlighting the governing parameters. Furthermore, the influence of the track model in the dynamic response of the train-track-bridge system was also studied. The author observed that accounting for the track in the numerical model enabled the dissipation of the higher frequencies. It was also possible to conclude that neglecting the track in the model leads to an overestimation of the maximum bridge deck accelerations, particularly for resonant speeds. Rauert et al (2010) dedicated his attention to the influence of the continuous ballast layer over bridges with two structurally independent decks since generally this aspect is neglected in most research works. Through a monitoring campaign the authors observed that there is a considerable load transfer from the loaded to the unloaded deck (see Figure 2.2). This study concluded that the continuous ballast layer leads to a considerable increase of the overall stiffness of the structure, leading to a significantly lower dynamic response of the bridge, particularly for resonant speeds. This is explained by the fact that the resonant effects are generally associated with the structure’s first natural frequencies, which are the most affected by the additional stiffness provided by the continuous ballast. For research purposes the ballast interaction was simulated through spring elements that coupled the two separated bridge decks, requiring a complex 3D model. However, the authors recognised that in terms of common design practice a simplified approach would be of great advantage. For this reason, the introduction of a Chapter 2 2.10 component of additional bending stiffness which reproduces the effects of the continuous ballast over the independent decks was suggested. Figure 2.2 – Example for load transfer between loaded and unloaded bridge decks [Rauert et al (2010)]. Martínez-Rodrigo et al (2010) applied passive control techniques to mitigate the excessive vibrations due to resonance on short simply supported railway bridges. Two real bridges in Spain, one inserted in the high-speed railway network and the other inserted in the conventional railway network, were investigated in order to reduce their dynamic responses under the circulation of high-speed traffic. This work had two main objectives: to verify if the use of passive control techniques, through the application of fluid viscous dampers (FVDs), enabled the correction of the dynamic behaviour of the bridge, reducing the level of vertical acceleration to admissible values according to the European standards and to evaluate the technical feasibility of the proposed solution to real structures. The authors concluded that the introduction of the FVDs allowed an efficient control of the deck vibrations (see Figure 2.3) and that its application in a real bridge did not prove problematic from a technical point of view. Dynamic behaviour of railway bridges 2.11 Figure 2.3 – Influence of the fluid viscous dampers in the dynamic response [Martínez-Rodrigo et al (2010)]. Shin et al (2010) on the other hand focused their attention in the vehicles, namely in Korean Train eXpress (KTX) train. In order to prevent resonance due to the periodical effects of the loading the authors suggest the modification of the high-speed train configuration by inserting size-adjusted vehicle(s) into the existing train arrangement (see Figure 2.4). However, from a practical point of view this approach does not seem feasible due to the large variability of the dynamic properties of bridges that are inserted in the high-speed railway networks. Nevertheless, the work presents some very interesting observations regarding the potential of exploring resonance suppression. Figure 2.4 – Vibration reduction scheme based on insertion of size-adjusted vehicles [Shin et al (2010)]. Chapter 2 2.12 Romero et al (2011) studied the problem of confined structures, which represent a quite common case in the railway network for hydraulic passages or box culverts. The typical configuration of these structures is characterised by very short spans which are very prone to experience resonance phenomena. The main objective of this work was to understand the influence of soil-structure interaction in the dynamic response of the railway bridge. The results showed that this aspect is particularly important for the resonant speeds, as disregarding it may lead to a significant overestimation of the dynamic response (see Figure 2.5). Figure 2.5 – Influence of the soil-structure interaction in the dynamic behaviour of confined structures [Romero et al (2011)]. 2.3 Numerical evaluation of the dynamic response The methodologies to assess the dynamic behaviour of railway bridges have been evolving throughout the years. The Union International de Chemin de Fer (UIC) has been responsible for recommendations and codes of practice for the design and assessment of railway bridges in Europe. An extensive historical review of the UIC recommendations can be found in the works of James (2003) and Museros (2002). In the work of Museros (2002) an overview of the most important publications in the design and evaluation of the dynamic behaviour of railway bridges, highlighting their main contributions, is also included. Presently, the most recent recommendations of UIC have been incorporated in EN1991-2 (2003), which is the current guideline for the design of railway bridges. In order to numerically assess the dynamic response of a railway bridge EN1991-2 (2003) allows two different types of approach: quasi-static analysis or dynamic analysis. The selection of the most suitable method can be done by consulting the flowchart that is illustrated in Figure 2.6. Dynamic behaviour of railway bridges 2.13 Figure 2.6 – Flowchart for determining the need of performing a dynamic analysis [EN1991-2 (2003)]. It can be observed that for some very specific cases EN1991-2 (2003) allows using a quasistatic analysis, which results from the response obtained by a static analysis multiplied by a dynamic factor, Φ, that is introduced in order to account for dynamic nature of the loading. This Chapter 2 2.14 method is addressed in Section 2.3.1. For all the remaining cases EN1991-2 (2003) proposes the use of dynamic analysis, as described in Section 2.3.2. 2.3.1 Quasi-static calculation method If the requirements established by EN1991-2 (2003), shown in Figure 2.6, are verified a static analysis can be performed. The static response of the bridge must be evaluated under Load Models 71, SW/0 and SW/2. Furthermore, these results must be multiplied by a dynamic factor, Φ, which enhances the static effects of the load models. This dynamic factor is not to be perceived as a dynamic amplification factor since its application is limited to load models. The value this coefficient takes was defined according to a study that involved the analysis of several real bridges and intends to cover the dynamic envelope of all trains operating in the European highspeed railway network. The value of the dynamic factor depends on the quality of track maintenance and also on the “determinant” length, LΦ, which represents the length of the influence line for deflection of the element being analysed. Initially the application of this factor was limited to simply supported bridges, however, the introduction of LΦ enabled its application to any type of bridge. For standard maintenance tracks the dynamic factor, Φ3, can be determined by: 73.0 72.0 16.2 3    L (2.2) Additionally, the value of Φ3 is limited to the interval 1 ≤ Φ3 ≤ 2. In the case of carefully maintained tracks the dynamic factor, Φ2, is given by: 82.0 20.0 44.1 2    L (2.3) where Φ2 is limited to the interval 1 ≤ Φ2 ≤ 1.67. The values to be adopted for LΦ can be consulted in EN1991-2 (2003). Dynamic behaviour of railway bridges 2.21 This equation indicates that for any given time step t the external forces, F(t), and the internal forces, that can be divided into inertial forces, Fi(t), damping forces, Fa(t), and elastic forces, Fe(t), are in equilibrium. Scrutinising Eq. (2.14) and taking into account that the inertial forces can be obtained by multiplying the mass matrix, M, the viscous damping matrix, C, and the stiffness matrix, K, by the acceleration vector, u, the velocity vector, u, and the displacement vector, u, respectively, it is noticed that it can be re-written in the following form:   tuuu FKCM   (2.15) Due to its simplicity it has been reported that this method may not be sufficiently accurate for cases where the coupled behaviour of the train and the bridge significantly affect the dynamic response, which corresponds to cases where the mass of the vehicle is not negligible compared to that of the bridge. Short to medium span bridges are within the sort of structures where the effects of the mass of the vehicle are not small compared to that of the bridge and, in these cases, the moving loads method tends to overestimate the maximum accelerations of the bridge deck. Museros et al (2002) compared the dynamic behaviour of 25 simply supported bridges using both the moving loads method and the train-bridge interaction method. This study concluded that the interaction effects lead to a significant reduction (around 25%) of the maximum dynamic response. Similar conclusions were drawn in the research work of Goicolea et al (2002). In this work the two methods were also compared and scenarios with different bridge spans and damping were analysed. It was noticed that the differences were higher for structures with structural damping lower than 2%. For this reason EN1991-2 (2003) suggests the use of an additional damping, Δζ, which is a function of the span length, to reproduce the effects of the vehicle-bridge interaction and to overcome the referred limitations. Nonetheless, the additional damping method should be used with caution as Doménech et al (2014) concluded that in certain cases this approach overestimates the interaction benefits, which leads to a non-conservative prediction of the bridge dynamic response. Chapter 2 2.22 2.3.2.4 Train-bridge interaction method To perform a more realistic and accurate method to assess the dynamic behaviour the trainbridge interaction effects need to be taken into consideration. The train-bridge interaction method accounts not only to the dynamic properties of the bridge but also takes into account the dynamic properties of both the train and the track. In addition, and contrary to what is observed in the moving loads method, the load of each wheel varies in time due to the variation of the wheel-rail contact forces. This is a more advanced assessment method which requires specific software to be used. For this reason, this approach requires higher computational capacity and is more time consuming when compared to the moving loads approach. Due to the significance of the train-bridge interaction effects for an accurate assessment of the dynamic behaviour of railway bridges, particularly those with short to medium spans, the methodology is becoming more usual. The principles of the train-bridge interaction method are similar to those presented in the case of the moving loads method. Several approaches have been proposed to solve the train-bridge interaction problem. Regardless of the approach that is employed it is necessary to establish the equations of motion of the two systems, vehicle and structure, which can be achieved using either coupled or uncoupled sets of equations. A review of the advantages and disadvantages of each approach can be found in the works of Yang & Fonder (1996) and Lei & Noda (2002). As the name indicates, in the coupled approach the equations of motion of the vehicle and the structure are coupled into a single system of equations. Examples of application of this methodology can be found in Yang et al (1999), Yang et al (2004) and Neves et al (2012). Yang & Wu (2001) point out that this approach may demand a considerable computational effort since the position of each contact point changes over time, thus generally making the system matrix time–dependent and requiring it to be updated and factorised at each time step. The uncoupled approach formulates the interaction problem using two distinct sets of equations (one for each subsystem) which are solved separately. The uncoupled equations of motion are complemented with an additional equation in order to ensure the compatibility of the two subsystems. However, the compatibility of the two systems, train and bridge, must be verified at each instant in order to guarantee contact. This means that both subsystems are coupled by the compatibility of displacements and equilibrium of forces at the contact points. On the one hand, the train loading results in the deformation of the bridge. Simultaneously these displacements translate into actions on the train similar to an imposed settlement. The reactions Dynamic behaviour of railway bridges 2.23 on each of the train’s axles (which represent the contact points) represent the dynamic interaction forces between the two systems. Examples of application of this approach can be found in [Cruz (1994); Calçada (1995); Yang & Fonder (1996)]. To solve these two sets of equations iterative procedures are often used [Cruz (1994); Calçada (1995); Yang & Fonder (1996)]. Antolín (2013) points out that this approach has the advantage of reducing the size of the matrix used in the dynamic calculations. The equations of motion can be written as follows:                                                                 tF tF tu tu K K tu tu C C tu tu M M v s v s v s v s v s v s v s 0 0 0 0 0 0     (2.16) where the subscripts s and v represent the structure and the vehicle, respectively. However, Yang & Wu (2001) indicate that this procedure may exhibit a slow rate of convergence, particularly when a large number of contact points is considered. In order to overcome this drawback Neves et al (2012) developed a procedure in which the equations of motion are solved directly using an optimised block factorisation algorithm. Ribeiro (2012) proposed another development to the iterative procedure used by Cruz (1994) and Calçada (1995) by solving the equations of motion combining both the Mode Superposition method to solve the bridge subsystem and the Newmark method to solve the train subsystem. The use of the Mode Superposition method allows a more efficient analysis from a computational point of view, thus reducing the computational timings. However, Ribeiro (2012) points out that the existence of dampers on the trains does not enable the equations of motion of the train to be uncoupled, requiring therefore the use of the Newmark Method. In this work the approach introduced by Neves et al (2012) was used. Furthermore, it should be noted that most of the work done on this field, including the present dissertation, limits the analysis to the vertical interaction phenomenon. However, lateral interaction effects are extremely important in cases where lateral forces (such as wind or seismic actions) act on the trains. For this reason, this has been the subject of research work in recent years and several methodologies account for full interaction, i.e. account for both vertical and lateral interaction, have been proposed [Zhai et al (2009); Nguyen et al (2009); Antolín (2013); Montenegro (2015)]. Chapter 2 2.24 2.3.2.5 Methods to solve the equations of motion Besides the procedure selected to formulate the train-bridge interaction problem there are also several techniques to solve the system of equations of motion when using the Finite Element Method. The most common techniques are the Mode Superposition method and the direct integration methods [Clough & Penzien (1993); Chopra (1995)]. The Mode Superposition method consists on uncoupling the equations of motion by transforming the generalised coordinates into modal coordinates. Due to the orthogonality properties of the mode shapes the equations of motion from the set of N coupled differential equations can be transformed into a system of N independent linear equations. Therefore, it is possible to assess the contribution of each of the mode shapes to the global structural response.         Ni t ttt iii ,...,,, 212 2 i i iii M P YYY   (2.17) where i Y  , i Y  and i Y represent the generalised acceleration, speed and displacement, respectively, i  is viscous damping ratio, i  is the angular frequency, i P is the generalised load and i M is the generalised mass for mode i. To obtain the global response of the structure one simply needs to solve the N uncoupled modal equations and superposing the effects of each of the modal contributions, hence explaining the name of the methodology.         tYtYtYtu nn    2211 (2.18) where ϕn is the natural mode of order n. One of the features of this method is the fact that it enables the selection of the mode shapes that will be accounted for in the analysis of the dynamic response. This is particularly useful because it simplifies solving the system of equations of motion, making the method particularly efficient. Another advantage of the method is the possibility to define the damping ratio for each mode, which is more convenient and generally more reasonable since the modal damping ratios Dynamic behaviour of railway bridges 2.25 can normally be determined experimentally or estimated with adequate precision in many cases. One of the drawbacks of the method is the fact that it requires performing a modal analysis prior to the dynamic analysis in order to identify the structure mode shapes and the corresponding natural frequencies. Owed to the fact that superposition is applied this method is not suitable for the analysis of nonlinear problems. Another typical option to solve the equations of motion is based on direct integration methods, which adopt a step-by-step approach and makes use of integration techniques. Unlike the previous method, no transformation of the integration space is required. Among the several methods that use such an approach are the Newmark method, the Wilson-θ method and HilberHughes-Taylor (HHT) method [Cruz (1994); Calçada (1995); Neves (2008)]. In the present dissertation only the Newmark method is addressed. The Newmark method adopts a step-by-step approach and, as previously stated, makes use of integration techniques to solve the dynamic equilibrium problem. In this method the contribution of all natural frequencies is taken into account. Two parameters define the variation of acceleration over a time step and determine the stability and accuracy of the method: γ and β. The factor γ provides a liner varying weighting for the contribution of the initial and the final accelerations on the change of velocity, whereas the factor β provides a weighting between the initial and the final accelerations on the change of displacement. The Newmark method becomes unconditionally stable for: 2 1   (2.19) It is also possible to observe that the maximum efficiency in terms of numerical dissipation is registered for:   4 5.0 2     (2.20) Chapter 2 2.26 If 2/1  and 4/1  are adopted, the Newmark method is usually referred as the constant average acceleration method and satisfactory results are obtained from all points of view, including that of accuracy. Contrarily to the Mode Superposition method when using the Newmark method it is not possible to define a modal damping ratio. For this reason it is necessary to derive appropriate proportional damping matrices. Typically Rayleigh damping is used and is expressed as being proportional to both the mass and the stiffness matrices [Clough & Penzien (1993)]: KMC  21 aa (2.21) The two Rayleigh damping factors, a1 and a2, are obtained from the following equation:                            n m mn mn mn nm a a       11 222 2 1 (2.22) where ωm and ωn are the angular frequencies of the modes of order m and n, respectively, and ξm and ξn are the corresponding damping ratios. Figure 2.7 – Rayleigh damping [Clough & Penzien (1993)]. Dynamic behaviour of railway bridges 2.27 2.3.2.6 Time step selection A particularly important aspect to take into consideration when performing a dynamic analysis is the selection of the time step, Δt, to adopt in the analysis. The selection of an adequate time step is essential to accurately assess the dynamic response and to ensure that the most important modes of vibration are considered. There are several criteria in the literature regarding the selection of the most adequate time steps to adopt in dynamic analysis. One criterion is adopting a time step between Tn/10 and Tn/20, where Tn is the period of the highest mode that is to be considered in the dynamic analysis. ERRI (1999) suggests that the selected time step should be the minimum of the following two values: max 8 1 f t  (2.23) where fmax is the frequency of the highest mode of vibration considered in the analysis, and max 4vn L t  (2.24) where L is the span of the bridge, n is the number of modes considered in the analysis and vmax is the maximum train speed. The first criterion guarantees that the highest mode of vibration is represented by a minimum of eight points. This criterion is similar to those that had been referred previously, although it is slightly less conservative. The second criterion is defined in order to ensure that the excitation is accurately accounted in the dynamic analysis. It intends to guarantee that a given load moving over the bridge at a speed, vmax, is discretised into 4n intervals. Meixedo (2012) points out that the time step selection should not only be a function of the bridge natural frequencies but should also take into consideration the train frequencies as well as the frequencies related with the excitation promoted by track irregularities. Generally, the train frequencies are less limitative than those of the bridge. However, depending on the train speed to be analysed and on the irregularities wavelength range that is used in the analysis the latter criterion might Chapter 2 2.28 sometimes require a smaller time step than that predicted by the bridge natural frequencies to adequately model the excitation and obtain an accurate response. It is important to mention that when using the Newmark method and 2/1  and 4/1  is adopted the method does not introduce any dissipation to the higher frequencies. When proceeding to the integration of the equations of motion the contribution from very high frequencies related to higher modes can be observed. However, these effects are generally spurious due to the fact that these frequencies are not being adequately taken into account in the numerical integration [Rigueiro (2007)]. Since in this method no upper limit to the frequency of the modes considered is established the time step acts as a cut-off, solving this problem. 2.4 Limit states The European standards define several limit states that must be verified in order to guarantee the structural and traffic safety in the high-speed railway network. In order to perform according to the safety demands of the European standards a structure must verify Ultimate Limit States (ULS) as well as Serviceability Limit States (SLS), related to passenger comfort. Besides the traditional verifications, the European standard EN1990-A2 (2005) also define some specific verifications for bridges inserted in the high-speed railway network, which are labelled as Traffic Safety Checks and apply limits regarding deformations and vibrations of the bridge deck. In the following sections a review of the verifications required by the European standards to assess the safety of railway bridges is presented. 2.4.1 Structural safety Regarding structural safety, the design of bridges requires performing a dynamic analysis for the most unfavourable value of the two following scenarios:                  RT or HSLM 2 '' '1   (2.25) Dynamic behaviour of railway bridges 2.29 or   0/""71 SWLM  (2.26) where RT represents the real trains operating in the European high-speed railway network, HSLM represent the High Speed Load Models, LM 71 represents the static effects of vertical loading due to normal rail traffic and SW/0 which represents the static effects of vertical loading due to normal rail traffic on continuous beams. For the cases where no dynamic analysis is required the dynamic amplification factor, φ’, is given by:            76,0,325,1 76,0, 1 ' 4 K K KK K  (2.27) with 0 2nL v K   (2.28) The term φ’’ that has been previously introduced and represents an amplification factor to account for the existence of track irregularities and vehicle imperfections can be determined by:                                  2 20 0 2 10 1 80 5056 100 '' LL e nL e   (2.29) Chapter 2 2.30 where the parameter α is given by:           smv smv v /22,1 /22, 22  (2.30) For the cases of carefully maintained track EN1991-2 (2003) allows a 50% reduction of the parameter φ’’. 2.4.2 Traffic Safety Checks These verifications correspond to Serviceability Limit States from a structural point of view that must be verified in order to guarantee an adequate performance of the railway bridge. Despite corresponding to SLS from a structural perspective they represent an Ultimate Limit State for the running safety of the trains. The verifications regarding the running safety of trains over bridges include the assessment of the vertical acceleration of the deck, deck twist, vertical deformation of the deck, transverse deformation and vibration of deck and the longitudinal displacement of the deck. Furthermore, EN1990-A2 (2005) also includes a Serviceability Limit State that aims to guarantee passenger comfort. Since the focus of this dissertation is the vertical behaviour of the bridge only the verifications related to this behaviour will be discussed in the following sections. 2.4.2.1 Vertical acceleration of the deck Limiting the vertical acceleration of the bridge deck has two main objectives: the first is to avoid track instability due to the loss of interlock between ballast grains in ballasted tracks and the second is to prevent the loss of contact between the wheel and the rail due to excessive reduction of the wheel-rail contact forces. The ballast instability leads to the loss of the lateral resistance of the track which affects the running safety of the trains. The loss of contact between the wheel and the rail can originate the derailment of the train. Dynamic behaviour of railway bridges 2.37 each wheel and depends on the dynamic friction coefficient, μ, and the contact angle, γ. The Nadal factor, ηN, can be expressed as:    tan1 tan    Q Y N (2.31) Nadal’s criterion is of static nature and neglects the longitudinal creep forces. Furthermore, it was demonstrated to be conservative, particularly for small or negative values of angle of attack, since it does not account for the effects of friction in the non-flanging wheel. Therefore, a less conservative criterion was proposed by Weinstock (1984). This criterion presents a more realistic approach and is less sensitive to the variation of the friction coefficient. Weinstock’s criterion evaluates the flanging wheel using Nadal’s criterion but accounts for the non-flanging wheel by considering a Y/Q ratio equal to the friction coefficient. Therefore, the Weinstock factor, ηW, can be expressed as: B AA AA WQ Y         tan1 tan (2.32) where the subscript A represents the flanging wheel, whereas the subscript B represents the nonflanging wheel. According to the TSI (2002), the Y/Q ratio should not exceed 0.8 for either of the wheel flange climbing criteria. 2.4.4.2 Track panel shift Another derailment mechanism is the track panel shift, which corresponds to the lateral displacement of the track panel. This phenomenon is due to excessive lateral forces acting on the track. As the lateral displacement of the track panel builds up in some of the track elements, such as the rails and the sleepers, the loss of guidance of the wheels might be observed resulting in one of the wheels falling between the rails and the outer side of the track. The track panel shift Chapter 2 2.38 has become increasingly important due to generalised use of continuously welded rail and also to the higher operational train speed. Prud’homme (1967) proposed a criterion that limits the maximum lateral force applied on the track by a wheelset in order to prevent track panel shift: 3 210 stat wheelset Q Y  (2.33) where Qstat represents the static load per wheel. This criterion was adopted by TSI (2002). Recently some variations have been proposed to the original criterion in order to account for different track maintenance levels, ballast compaction level and sleeper type [Iwnicki (2006)]. 2.4.4.3 Wheel unloading The derailment by wheel unloading might occur when one or more wheels lose contact with the rails as a result of excessive vibrations. These excessive vibrations can be due to aspects such as track irregularities, crosswind or earthquakes. There are several methods to determine wheel unloading. A common method is the vector intercept, which is based on a geometrical analysis of the acting point of the overall resultant vertical forces [Andersson et al (2004)]. In its most usual form the wheel unloading factor, ηU, measures the reduction of the wheel load compared to its static value and is calculated by: sta dyn sta dynsta UQ Q Q QQ   1  (2.34) where Qdyn is the dynamic wheel load. When there is no unloading the parameter ηU takes the value of 0. On the contrary if a limit situation where the wheel loses contact with the rail is observed the dynamic wheel load is null and the parameter ηU is equal to 1. Carrarini (2006) refers that analysing a single wheel can be excessively conservative, as this criterion is not valid when complete unloading is reached, regardless of the severity of the wheel lift. A less conservative approach uses a similar principle but consists on analysing a complete Dynamic behaviour of railway bridges 2.39 bogie/wheelset. It should be noted that both approaches are allowed by the current European standard as indicated in EN14067-6 (2009), which limits the wheel unloading to 90% of the value of the static load, corresponding to a value of ηU equal to 0.9. A criterion that is strongly associated with wheel unloading is the train overturning criterion since the degree of unloading of the critical wheels is usually also used as a criterion for the assessment of the risk of overturning. The overturn factor, ηO, can be expressed as:      wheelset wheelset OQQ QQ 21 21  (2.35) where Q1 and Q2 are the vertical loads of each wheel of the same wheelset. Similarly to the wheel unloading TSI (2002) limit the overturn factor to 0.9. Chapter 2 2.40 3.1 Chapter 3 Structural reliability assessment 3.1 Introduction The safety of structures and their adequate performance while in service are the two key issues to take into account during the design stage. However, the complete safety of a structure is a concept that is not real as there is always some risk of failure. This risk is due to the uncertainties that characterise the parameters that influence the structural behaviour, which makes the structural reliability problem non-deterministic in nature. In order to classify the structural safety EN1990 (2002) defines that structures should be designed and maintained in order to display an adequate performance throughout their service life, with an appropriate balance between economy and safety level. For this reason structural reliability theory is based on mathematical statistics where the uncertainties are modelled by stochastic variables. Until the 19th century the design of Civil Engineering structures was carried out in an empirical way, mostly based on the experience of masons and builders. However, it was clear that uncertainty was a significant part of the structural reliability problem. The structural behaviour depends on several parameters that display, in most cases, a certain degree of uncertainty thus making it intrinsically a probabilistic problem. With the research carried out in this field a scientifically based method was developed for the assessment of structural safety: the admissible stress method. The fundamental idea of the method was to ensure that the maximum stress in the critical zones was lower than the material Chapter 3 3.2 capacity divided by a safety factor. The selection of the safety factor was somewhat empirical. The enhancements in Structural Engineering, namely a better understanding of the behaviour of materials and a more adequate evaluation of the loadings resulted in a reduction and a diversification of the safety factors. However, the method proved to have some imperfections, particularly by providing different safety levels for different structural elements and by making it difficult to assess the overall safety of the structure. For this reason the introduction of a probabilistically based approach to the safety problem was a natural development. This approach offers a more realistic model of the real phenomena. However, it requires statistical information for the description of the parameters involved to be used properly. Significant research has been dedicated to the topic of probabilistic structural reliability and it is possible to say that nowadays this is a field that is adequately documented and consolidated within the academic community [Jacinto (2011)]. Despite this this approach is not yet a common practice in Structural Engineering. This can be explained by the adoption of semiprobabilistic approaches by the design codes. In recent years the adoption of probabilistic approaches to assess structural safety has been increasing, indicating that these methods can now become more common and due to the advances in computational capacity their application to real structures is now becoming more feasible. In the particular field of railway bridges the use of probabilistic methods for safety assessment is very limited. From the literature review it can be seen that most of the work done in this research field was performed through deterministic analysis. This reveals that not much attention has been given to the variability of the parameters that are known to influence the dynamic response of the bridge or to the identification of the parameters that have a significant effect in the structural response. One of the few exceptions is the work of Cho et al (2010) which accounted for the variability of some parameters of both the bridge and the train and performed a reliability analysis of a box-girder railway bridge using an improved Response Surface Method. A prestressed concrete box girder bridge was used as a case study and the reliability analysis showed that bridge-related uncertainties have greater influence in the reliability indexes than train-related uncertainties. Nonetheless, the analysis was limited to a single train speed not enabling to conclude how the reliability of the train-bridge system is affected by this parameter. Even when the existence of track irregularities is accounted for, the analysis is often limited to a specific scenario. Au et al (2002) studied the behaviour of a cable stayed railway bridge for different track irregularity profiles and different track quality scenarios. It was found that the Structural reliability assessment 3.3 impact factor is not proportional to the magnitude of the roughness. However, it was noticed that this impact factor tends to increase for lower track quality. Lu et al (2009) proposed an extension of the pseudo excitation method for the analysis of the behaviour of vehicle-bridge coupled systems. Several examples are shown proving the efficiency of the proposed method against the Monte Carlo method. However, for both research works, the variability is limited to the track irregularity profiles. Johansson et al (2014) recently proposed a methodology for the preliminary assessment of existing railway bridges for high-speed traffic. The methodology divides the bridges into groups with similar characteristics. Afterwards, prediction bounds for the properties of the bridges within a group are determined through statistical analyses for each group of bridges and a probability distribution is generated for the entire bridge network. The dynamic response is assessed using a closed-form solution previously developed by the same authors [Johansson et al (2013)]. To take into account the uncertainties associated with the prediction bounds the Monte Carlo method is used to assess the dynamic behaviour of the bridges and based on the simulation results the probability of failure is determined. In this chapter the basis of structural reliability is presented. An overview of the fundamental concepts and the different approaches used to address this problem is provided along with the description of different methods used to assess structural safety. Furthermore, some enhancements of the most usual methods are presented in order to increase the efficiency of the safety assessment. The chapter ends with the presentation of the safety assessment framework that includes the basis of the methodology used to assess the safety of the train-bridge system. 3.2 Fundamentals In this section the fundamentals of structural reliability analysis are summarised. An overview of the basic concepts is presented along with the theoretical background to structural reliability. 3.2.1 Basic concepts Since the safety and reliability of structures assumes such an important role in Engineering, structural reliability is a field that has been extensively studied in order to create tools for the Chapter 3 3.4 economic design of structures as well as accurate assessment of the safety level. A fundamental aspect to understand is the concept of reliability. According to EN1990 (2002) the term ‘reliability’ should be considered as the ability of a structure to fulfil the specified requirements for which it has been designed during its service life. This code states that generally the ability to comply with the specified requirements is evaluated in terms of a probability, thus making reliability a probabilistic concept. Therefore the reliability is a quantity that can be calculated or estimated. For this reason, in order to be realistic, structural reliability methods have a probabilistic nature, to account for the several sources of uncertainty that characterise these problems. Both in Thoft-Christensen & Baker (1982) and in Melchers (1999) the sources of uncertainty are divided into four main groups: physical uncertainty, statistical uncertainty, modelling uncertainty and uncertainties due to human factors. Therefore, an important part of the method is the identification of the parameters that influence the structural behaviour, known as the basic variables, as well as the definition of an appropriate probability distribution family and the estimation of the parameters of this distribution. Usually the selection of the distribution family is based on subjective information whereas the parameters of the distribution are estimated on the basis of available data or experience [Faber (2012)]. While accounting for the several sources of uncertainty is important, the definition of boundaries that measure the performance level that a structure must display during its service life is another key aspect. Nowak & Collins (2000) define the limit state as the boundary between desired and undesired structural performance. Therefore, the limit state functions or performance functions can be perceived as the basic requirements for adequate structural behaviour. Generally, two types of limit states are defined: serviceability limit states and ultimate limit states. Ultimate limit states are associated with severe damage to the structure, that reduces the structural capacity and raises concerns regarding structural and/or people safety [EN1990 (2002)]. In bridges the most common causes for ultimate limit states are bending, shear and loss of stability [Wisniewski (2007)]. Serviceability limit states are associated with less severe damage to the structure and generally concern the performance of the structure under normal use, the comfort of people, the structural appearance and its durability [EN1990 (2002)]. In railway bridges the most common serviceability problems are related with excessive deformations or vibrations and cracks. It is worth mentioning that EN1990 (2002) also distinguishes between reversible and irreversible serviceability limit states. Wisniewski (2007) points out that some Structural reliability assessment 3.5 authors separate the Fatigue Limit state from the ULS and recently some authors also propose the use of another limit state with respect to structural durability. If a structure does not meet the requirements established in the limit states it is considered to have failed. Since structural reliability relies on a probabilistic approach the structural response can be assessed statistically and the probability of violating a limit state, which represents the probability of failure, can be estimated. Assuming that the structural response is determined by the parameter x, the probability of failure can be expressed as:      F dxxfp Xf (3.1) where ΩF represents the failure region, which corresponds to the region where a given limit condition is not satisfied. The severity of the consequences that result from failing to meet a specific limit state is used to define the target reliability. Limit states that have greater consequences when exceeded must have a very small probability of occurrence and, consequently, higher reliability. Contrary to the reliability, safety is a qualitative concept [Schneider (2006)]. It is not possible to quantify the safety of a structure as it can only be classified as safe or unsafe. Therefore, it is necessary to assess the structural reliability and a structure is considered to be unsafe when its reliability is lower than a target reliability, which represents the minimum acceptable structural performance. 3.2.2 Semi-probabilistic approach One of the most common methods to evaluate structural safety and reliability is based on a semi-probabilistic approach. It constitutes the basis of most of the current standards and guidelines for the design of structures. In this approach the actions, E, and the resistance, R, are represented by their design values, Ed and Rd, respectively. In order to ensure safety the following condition must be met: dd RE  (3.2) Chapter 3 3.6 The semi-probabilistic approach reflects the uncertainty in a simplified manner, through the introduction of partial safety factors. These safety factors are applied to the characteristic values of the actions, Ek, and resistance, Rk, in order to enhance the loading and reduce the resistance. Typically the characteristic value for the actions corresponds to the 95% quantile whereas in the case of the resistance the 5% quantile is used. However, other quantiles can be used provided that safety factors are adequately adapted. Therefore, Eq. (3.2) can be written as: M k kf R E    (3.3) where  f and  M are partial safety factors with values greater than one. This method is based on more complex probabilistic approaches. However, the main purpose of this approach is to define a simple method to assess structural safety while ensuring the same safety level as more complex methodologies. It is within the designated Level I methodologies, which is a topic that is addressed in more detail in Section 3.3. 3.2.3 Probabilistic approach Another possibility for the assessment of structural reliability is adopting a probabilistic approach. In this method both the structural response and the loading are characterised by random variables, offering a more realistic representation of the structural reliability problem. The probability of failure is evaluated by structural reliability techniques. The basic variables are defined according to real or theoretical distributions and comprise the mechanical properties of the materials, the geometric imperfections and the loading on the structure among other significant properties. The use of probabilistic approaches in structural reliability problems has become more widespread in recent decades mainly due to the evolution of computers and their increasing capacities. The application of these methodologies in Civil Engineering is not limited to structural design assessment but has been also frequently applied to the structural assessment of existing structures. Since most European standards do not differentiate the structural assessment of existing structures from the design of new ones this approach is very useful to provide an Structural reliability assessment 3.13 The approximation function can be defined by expressing the performance function, g(X), in a Taylor series:          *2*** * *| 2 1XXgXXXXgXgZ X T X (3.16) Where X* represent the design point and g k  corresponds to the partial derivatives in terms of k. Therefore, it can be said that the essence of the FORM or SORM methods is to find the point that leads to the minimum distance between the limit state function and the origin of the normalised space [Melchers (1999)]. As the name indicates the FORM only accounts for the first order terms of the Taylor series of the function g(X) relative to the design point X*:     ** * XX X g XgZ X     (3.17) The limit state function is therefore approximated by a hyper surface tangent to the design point X* as illustrated in Figure 3.3. Figure 3.3 – Approximation to the performance function. Chapter 3 3.14 Generally second order surfaces such as paraboloids or spheres are used for the approximation in order to reduce the errors that result from using an approximation to the limit state function. Cornell (1969) introduced the reliability index concept by calculating the mean and the variance using first order approximation functions.       n iXi n iii i a Xaa 1 22 1 0   (3.18) However, this formulation has an important drawback which is the dependency of the reliability index, β, on the design point. In order to overcome this problem Hasofer & Lind (1974) proposed a methodology that performs a transformation of the original space into the normalised space (zero mean and unit variance). The variables Xi are substituted by the reduced variables Yi. The limit state function is also reformulated in the new coordinate system. The subsequent procedure is similar to the procedure described for the fundamental reliability case, the design point Y* is determined and afterwards the reliability index, β, can be calculated. If the limit state function is non-linear the determination of the design point is carried out using an iterative procedure. An interesting concept that is associated to this methodology is related to the physical interpretation of the direction cosines. The obtained value represents a sensitivity measure of the performance function at the design point for each of the basic variables, Xi. In the case of correlated and/or nonnormal basic variables the problem is more complex and some specific methods need to be applied. The work of Carrarini (2006) discusses some of them, including the Rosenblatt transformation, Rackwitz-Fiessler formula and the Orthogonal transformation. One drawback of the FORM and SORM approach is that in its formulation the probability distributions of basic variables, Xi, are not considered but only the first and second moments. Structural reliability assessment 3.15 3.3.3 Level III Methods The Level III reliability methods try to determine the probability of failure by calculating the integral that defines it. Considering performance function   XgM  , where X is the vector of the basic random variables, the probability of failure is represented by the following integral:         F dxxfXgPp Xf 0 (3.19) This integral can be solved analytically (only possible for a very limited number of cases) or numerically. Jacinto (2011) notes that the numerical solution of the integral can be difficult for problems with a significant number of basic random variables (generally higher than 5) or when the failure regions displays a complex geometry. Furthermore, it is not always possible to explicitly obtain the failure region function. This is typically the case when a problem is solved by the Finite Element Method, where these functions can only be defined point by point. The main consequence is that the failure domain is unknown and, consequently, the probability of failure cannot be calculated from an integral such as expressed in Eq. (3.19) as the domain is not completely known. In these cases one usual option to assess structural safety is by response surface methods. Seeing that the limit state function is not explicit the response surface method provides an analytical performance function,   Xg ~ , generally of the polynomial type, that defines the limit state that is to be assessed from a few properly selected deterministic analyses. In order to obtain an accurate assessment of the structural reliability the approximation function must adequately represent the limit state function in the vicinity of the design point. The response surface method can be summarised in the following steps [Henriques (1998); Wisniewski (2007)]:  Define a set of values for the basic variables in order to obtain   Xg ~ and assess the quality of the approximation;  Assess the structural response for each set of values that were defined using the Finite Element Method;  Determine the function coefficients using regression techniques based on the FEM analysis results; Chapter 3 3.16  Based on the analytical performance function,   Xg ~ , determine structural reliability through FORM/SORM methods or using simulation techniques. A careful choice of the polynomial degree of the analytical performance function must be carried out. The degree of   Xg ~ must be smaller or equal to the degree of   Xg in order to obtain a system of well-conditioned linear equations during the determination of the coefficients [Henriques (1998)]. However, Deng (2006) points out that this method becomes computationally impractical for problems involving a large number of nonlinear random variables, particularly when dependent random variables are involved. An enhancement to the traditional response surface methods is Artificial Neural Networks (ANN). When studying complex limit state functions this method proves to be particularly useful as obtaining an adequate fit, using the response surface method, might be very difficult. ANN are numerical algorithms that attempt to replicate the behaviour of the biological neural network in a computational model. One of the most interesting features of this methodology is the possibility of learning. This means that given a task to solve, such as determining the limit state function, ANN is capable of defining a model of the limit state surface from training examples and finding meaningful solutions without the need to specify the relationship between variables [Deng (2006)]. The main advantage of this methodology is that obtaining the limit state surface represents a small fraction of the computational time required by the numerical model, thus making it possible to analyse complex structures that display a non-linear behaviour [Chojaczyk et al (2012)]. Usually, and similarly to what is done with the traditional response surface methods, after determining the limit state surface using ANN the method is combined with Monte Carlo simulation or FORM/SORM methods in order to evaluate structural reliability. An alternative methodology is based on simulation techniques. This approach does not require an explicit limit state function, thus making it suitable to be used when FEM is applied in the assessment of the structural response. Since the determination of the integral presented in Eq. (3.19) is extremely difficult for most structural reliability problems, the use of simulation techniques allows one to obtain unbiased estimates of the value of the integral while adequately accounting for the non-regular structural behaviour. This method has emerged as an interesting alternative due to its simplicity and also because it is almost unaffected by the complexity of the studied problem and the number of basic random variables involved [Rubinstein (1981)]. Structural reliability assessment 3.17 The Monte Carlo method is the basis of most simulation methods and is generally selected for the analysis of complex systems [Rubinstein, 1981; Shinozuka (1972)]. In this method values are generated for the random variables, X(i) = (X1(i), X2(i), … , Xn(i)), according to their distribution. Next the structural response, Yi (i), is assessed for each trial and the global structural response is analysed as a sample with distribution of Y. Monte Carlo simulation is based on the random sampling concept and aims to artificially simulate a large number of experiments. The probability of failure, pf, or the structural reliability, β, can then be assessed using two distinct approaches. The most usual is a simple process that consists on counting the number of trials where the safety limit is exceeded over the total number of simulations: N z pf0  (3.20) where z0 corresponds to the number (or realizations) where structural safety is not verified and N represents the total number of trials. The total number of trials depends on desired accuracy for pf and also on the order of magnitude of the target pf. The necessary number of trials is directly proportional to the desired accuracy and inversely proportional to the value of the target pf. The second approach consists of statistical analysis of the results of all the realisations using the limit state functions. This enables the determination of the probability density function of the performance function, g(X), as well as the mean value, μM, and the standard deviation, σM. Assuming that g(X) is normally distributed the reliability index, β, can be determined using Eq. (3.15). The use of Monte Carlo simulation for structural reliability problems can be divided into two different classes: pure simulation methods and semi-analytical methods. The first class corresponds to the original formulation whereas the second class corresponds to cases where simulation techniques are combined with other methods that enable a more efficient assessment of the structural reliability [Henriques (1998)]. In its most usual form the method is also known as Crude Monte Carlo (CMC), due to its somewhat brute force nature, and can be adequately described by the following steps [Haldar & Mahadevan (2000)]:  Definition of the structural reliability problem with all its basic variables; Chapter 3 3.18  Definition of an appropriate probability distribution family and the estimation of the parameters of this distribution for each of the basic variables;  Numerical sampling of the basic variables based on their distribution;  Assessment of the structural response for each trial;  Obtain all the necessary statistical information from the N trials;  Evaluation of the quality of the estimation based on the accuracy and efficiency of the structural reliability assessment. The methodology is simple and its application depends essentially on the necessary number of trials, which define the computational costs. One way to estimate the necessary number of trials for an accurate assessment of the structural reliability is by analysing the variance or the coefficient of variation (CV) of pf. The variance or CV can be estimated assuming that each realisation is a Bernoulli trial. This way the number of failures in N Bernoulli trials has a Bernoulli distribution. The variance of pf can be expressed as:   N pp ff pf  1 2  (3.21) Consequently, the CV can be written as:   f ff ppN pp CV f   1 (3.22) The statistical accuracy in the estimate of pf is inversely proportional to the value of CV. Brodig et al (1964) suggested the following formula to make an initial estimate of the necessary number of trials, N, based on the confidence level, c: Structural reliability assessment 3.19   f p c N 1ln (3.23) Bjerager (1990) on the other hand suggested that the number of trials only needed to be related with the value of pf and a value in the range of 1/pf to 10/pf should be selected. [Melchers 1999] points out that despite being useful these ‘rules’ do not provide information about the accuracy of the method. One useful tool to assess accuracy is to analyse the evolution of both the estimated pf and its estimated variance for increasing sample sizes. The main criticism that is appointed to this method is the typically large number of simulations required for accurately assessing the reliability of Civil Engineering structures. Due to the rather small probabilities that are typically used in structural safety engineering problems [EN1990 (2002)], the computational costs of this method can, in some cases, be prohibitive. As an alternative to the Monte Carlo approach, the use of variance reduction techniques enables the refinement of the sampling process and increases the efficiency of the simulation. Among them the importance sampling techniques, the directional simulation method and the stratified sampling method are worth mentioning. The works of Rubinstein (1981) and Melchers (1999) provide a good overview of the various strategies for variance reduction when using simulation methods to solve structural reliability problems. Schuëler (2009) carried out a study in order to compare the efficiency of the standard Monte Carlo against methods where variance reduction techniques are employed. Using a building as a case study it was possible to observe that the use of importance sampling and directional simulation results in significant efficiency gains. Hurtado (2007) proposed a methodology that combines pattern recognition techniques with importance sampling. This work made it possible to conclude that the use of pattern recognition techniques achieves a significant improvement in the efficiency in the selection of the sample to use in the structural reliability analysis, resulting in an important reduction in the required number of samples. Grooteman focused his studies in directional simulation methods. In Grooteman (2011) an adaptive directional importance sampling was presented whereas in Grooteman (2008) a radialbased importance sampling method is discussed. The author recognises that importance sampling can prove more efficient but in some cases it requires information about the limit state which can Chapter 3 3.20 prove hard to define. For this reason he suggests that directional simulation is a good option to assess structural reliability problems due to the efficiency displayed by the method. Another variance reduction technique is based on stratified sampling methods. These methods divide the entire sample space, S, of X into m strata of equal marginal probability, Ωi. The probability of failure associated with each strata, Ωi, is defined by:      i idXXhXfp Xf (3.24) Each interval is characterised by the following probability:     idXXhpi (3.25) 1 1    m ii p (3.26) Thus, the probability of failure is determined by:             m if m iXXf i ipdXXhXfdXXhXfp 11 (3.27) Using discrete Monte Carlo simulation an estimate of pf can be obtained by:         i i n k k i i X m ii i fXf n P p 11 ˆ ~ (3.28) Structural reliability assessment 3.21 with a variance of:           m ii ii m i i X i i pn P xf n P f1 2 1 2 ~   (3.29) where:            i i i f X i i Xi P p dXXhXf P xf 2 2 222 1  (3.30) ni being the number of samples within the sub-space Ωi. Eq. (3.30) demonstrates that if an adequate stratification is selected a significant reduction of the variance is obtained, thus significantly increasing efficiency. One of the most common methods that uses stratified sampling techniques is the Latin Hypercube sampling method [Mckay et al (1979); Florian (1992)]. In this method the range of each variable Xi is divided into n strata of equal marginal probability, Ωi, ensuring that each variable Xi has all portions of its distribution represented by input values, sampling once from each stratum. Each interval is represented in the sample by the representative parameter which can be taken randomly within the interval or may be taken as the centroid of the interval [Florian (1992)]. Stein (1987) shows that this method is superior to standard Monte Carlo simulation with respect to both efficiency and precision of estimators provided that the response is a monotonic function (a function that is either entirely increasing or entirely decreasing) of the basic variables. The sampling process allows several possible configurations of the sample space. Therefore, the proper selection of samples representing the stratified sampled space is decisive for the efficiency of the method. The works of Morris & Mitchell (1995), Vořechovský & Novák (2003), Stocki (2005) and Beachkofski & Grandhi (2002) provide examples of different approaches regarding the selection of the optimal configuration of the sample space. Chapter 3 3.22 3.3.4 Level IV Methods The Level IV methods combine structural reliability with risk concepts. There are several definitions for risk, but the most general in the context of structural reliability is to understand it as the product of probability of occurrence of a given event, which in this case can correspond to the probability of violating a given limit state, by the consequences of the occurrence of such event, CF, namely the damages that are caused, the loss of lives, etc. [Melchers (1999)]. This perspective can be similar to a cost-benefit analysis and, therefore, the risk can be interpreted as the cost of the occurrence of a given event. The whole-life total cost of the structure, CT, corresponds to the sum of this cost with the initial costs (both of project and construction), CI, and the maintenance costs expected during the service life, CM. Taken this into consideration, the structural reliability assessment can be interpreted as an optimisation of problem with regard to finding an optimal probability of failure in order to maximise the following:     FfMIT CpCCBCB  maxmax (3.31) where B represents the benefits and, as previously explained, the last term represents the risk. Another formulation that can be employed is estimating the risk and ensuring that it is lower than the maximum admissible risk for structure. This type of approach is generally limited to structures that are extremely important, which are characterised by severe consequences when failure occurs. The use of methodologies that take risk into consideration has been more frequent in structural reliability problems in recent years. However, since this method exceeds the scope of the present dissertation this serves only as a brief reference and overview, acknowledging the existence of this approach and the potential that it offers when addressing this type of problem. Structural reliability assessment 3.29 Figure 3.5 – Example of the empirically estimated probability of failure and corresponding confidence intervals. 3.5 Safety assessment framework One of the objectives of this work was creating a simple, efficient and automatic procedure, which requires as little intervention from the user as possible, that allows the identification of the critical train speeds over a bridge and the assessment of the safety of the train-bridge system. A schematic representation of the proposed probabilistic methodology can be observed in the flowchart presented in Figure 3.6. After defining the basic random variables, which should include all the parameters which variability affects the dynamic behaviour of the train-bridge system, one has to generate the values that each variable will take for each simulation. To do so a random number generator can be used. This type of tool is available in a wide range of computer software. Afterwards the data files can be created. In this particular case study, two different data files need to be created: one for the structure, which includes the bridge and the track, and another for the train. An automatic data file generation procedure was developed for efficiency reasons. Chapter 3 3.30 Figure 3.6 – Proposed methodology. Following the generation of the data files, the dynamic analysis can be performed. An adequate time step needs to be selected for the dynamic analysis and batch processing is used to perform the dynamic analyses for efficiency purposes and to avoid the need for manual intervention. After the dynamic analyses the results must be processed. Due to the large amount of information an automatic procedure is used. The results processing tool obtains automatically the time history of the dynamic response for a selected bridge section and the maximum dynamic response for that simulation. To assess the safety of the train-bridge system the probability of failure is estimated using two different techniques. A tail modelling approach based on the extreme value theory that uses the Structural reliability assessment 3.31 appropriate functions to model the upper tail of the obtained distribution, as detailed in Section 3.4.1. An Enhanced Simulation technique which uses an approximation procedure based on the estimates of the failure probabilities at moderate levels for the prediction of the far tail failure probabilities, as has been detailed in Section 3.4.2. In order to enhance efficiency and ensure the accuracy of the estimates some criteria have been defined for each methodology and will be discussed in Section 5 where practical examples will be used to demonstrate the purpose, application and advantages of each criterion. The train-bridge system is considered to be safe if the estimated probability of failure is lower than 10-4, which corresponds to a reference value in JCSS (2001) to assess ultimate limit states for systems where failure has severe consequences. Chapter 3 3.32 4.1 Chapter 4 Modelling of the train-track-bridge system 4.1 Introduction In the previous chapters the problem that is being studied in this dissertation, as well as the methodologies proposed to achieve the objectives of this thesis, were presented. The current chapter is dedicated to presenting the case study which will be used to test the adequacy, efficiency and accuracy of the proposed methodologies. A ballasted filler beam bridge was selected as case study as this structural solution is representative of a significant part of the short span bridges that compose the current European high-speed railway network. A similar criterion was used in the selection of the train to be used in the analyses. The TGV double train, which is currently in operation in the European highspeed network, was selected due to having a particularly aggressive configuration for the span length in study. A thorough description of the case study bridge is presented along with the description of the geometrical and mechanical properties of the train. Based on the properties of both the bridge and the train, a set of random variables is selected and their variability and type of distribution is presented and discussed. Afterwards, the numerical models used to represent both the train and the bridge are presented, an explanation for all the options made during the development of the numerical models is provided and the numerical models are validated. Furthermore, several Chapter 4 4.2 particular modelling aspects are discussed in detail in order to provide a better understanding of their influence on the dynamic response of the train-bridge system. Finally, and due to the few probabilistic studies carried out in this field, it is important to understand which variables have more influence on the dynamic response. Therefore, a sensitivity analysis is carried out in order to identify the variables that govern several aspects of the dynamic response of the train-bridge system. 4.2 Case study 4.2.1 Description of Canelas Bridge For the purpose of this dissertation Canelas Bridge is selected as case study. Canelas Bridge is located in the Northern line of the Portuguese railway, near Estarreja at the 282.944 km, and is composed of six simply supported spans of 12 m each, with a total length of 72 m. The bridge deck is a composite structure consisting of a concrete slab with embedded rolled steel profiles. This kind of structural system is known as filler beam and is a common structural solution for small span bridges on the European high-speed railway network, especially in France and Germany [Martínez-Rodrigo et al (2010); Hoorpah (2005)]. The Canelas Bridge’s concrete slab has a height of 0.70 m and has embedded nine HEB 500 profiles. A side view of the bridge used as case study is shown in Figure 4.1. The bridge supports two ballasted tracks, however, each track is supported by a single half deck due to the existence of a longitudinal expansion joint (see Figure 4.2). Nonetheless, the abutments and columns support both decks. It should also be noted that despite the decks are independent the ballast layer is continuous over both half decks, which can originate some connection between them. This interaction between independent decks due to the continuous ballast layer is a topic that has been studied in recent investigations [Rauert et al (2010); Carvalho (2011)] but that is not considered in this dissertation. Modelling of the train-track-bridge system 4.3 a) Schematic representation b) Longitudinal section Figure 4.1 – Side view of the Canelas railway Bridge. Figure 4.2 – Detail of the longitudinal expansion joint. Furthermore, Canelas Bridge is located in a curve with radii of 892 m and 896 m for the inner and outer tracks, respectively. The grade line of the bridge along its longitudinal profile is horizontal and the rail is located at a 5.75m level. The maximum cant of the track is 0.179 m. The cross section of each deck has a width of 6.20 m and comprises of a concrete slab with a Chapter 4 4.4 width of 4.50 m and a height of 0.70m. There is also a cantilever walkay with a width of 1.70 m and a height that varies from 0.30 m to 0.50 m. Nine rolled steel profiles HEB 500 are embedded in the concrete slab with a spacing of 0.475 m. Headed studs with a diameter of 25 mm were welded to the top of the steel profiles in order to improve the bonding with the concrete slab. Cement plates were placed underneath the concrete slab, between the steel profiles, to be used as formwork during the concreting of the slab. It should also be pointed out that each deck has a small ballast retaining wall which forms part of the cantilever with a height of 0.60 m and a width of 0.30 m. Laminated neoprene elastomeric bearings are placed underneath each steel profile, with a total of 9 bearings per half deck in each abutment. The cross section of Canelas Bridge is depicted in Figure 4.3. Figure 4.3 – Canelas Bridge cross section. The reason for selecting of this bridge is that this is a very common structural form used on the European railway network. According to the European Project Sustainable Bridges (2004) composite bridges (steel/concrete or filler beam) represent 14% of the bridges in operation in the European lines, equivalent to more than 30,000 bridges. Furthermore, since the spans of the bridge are within the range of spans for which the European standards indicate that the interaction effects are more significant, it offers the possibility to improve the understanding of such effects on the dynamic behaviour of railway bridges. Modelling of the train-track-bridge system 4.5 4.2.2 Previous research studies on Canelas Bridge This bridge has already been used as case study in several research investigations. Rodrigues (2004) studied Canelas Bridge at the request of the Portuguese railway network infrastructure management company (REFER) to evaluate the consequences of increasing the line speed over the bridge from 140 km/h to 220 km/h. This investigation involved in situ dynamic tests in order to characterise the bridge from a dynamic point of view and to assess its dynamic response under train loading. Structural reliability, train running safety and passenger riding comfort were analysed to assess the bridge response to the increased line speed. Based on the obtained results it was concluded that all the safety criteria were verified for a higher train speed, thus validating the increase of the line speed over the bridge. Pimentel (2009) also carried out in situ tests on Canelas Bridge to determine its dynamic properties. This study aimed to characterise the railway traffic over the bridge and to assess its influence on the dynamic response. Ambient vibration tests were performed in order to identify the modal parameters of the bridge, namely its natural frequencies, modal configurations and damping coefficients. The characterisation of the railway traffic was carried out using a B-WIM (Bridge Weigh-in-Motion) algorithm. This algorithm is based on the strain measurement at three points of the structure, two located on the rail and another on the bridge deck. Such a technique enabled combining the advantages related with the strain measurement at the rail, which allow the identification of the train geometry, with the advantages of the strain measurements at the bridge deck, which allow obtaining the train axle loads. Silva (2010) developed several three-dimensional finite element numerical models of Canelas Bridge, with varying degrees of complexity. The aim of this study was to assess the influence of the inclusion of the track in the numerical model and also the effect of considering the adjacent span in the dynamic response of the bridge. The numerical models were calibrated using results from experimental tests on the bridge in order to guarantee a realistic representation of the existing structure. Structural safety was assessed for both static, using LM 71 multiplied by an adequate dynamic factor, and dynamic analysis, using the real trains that operate on the European railway network, as prescribed by the current European standards [EN1990 (2002)]. With the increasing complexity of the numerical model the obtained results are less conservative but demanded higher computational capacity and required longer computational times. The author also concluded that accounting for the connection between the decks due to the continuous ballast layer resulted in a significant decrease of the dynamic response, particularly for resonant speeds. Chapter 4 4.6 Carvalho (2011) continued the research carried out by Silva (2010), focusing his study on the model updating of Canelas Bridge using optimisation techniques. A genetic algorithm based on the theory of evolution was used for the optimisation in combination with results obtained from ambient vibration tests. Another aspect introduced in this research was the inclusion of elements that reflected the deterioration of the ballast connection between the decks in the numerical model, thus allowing a more realistic representation of the global dynamic behaviour of the bridge. This study concluded that the transversal connection due to the continuous ballast layer is reduced as the ballast in the “connection zone” is highly degraded as a consequence of the successive crossing of trains. Bonifácio (2012) also studied Canelas Bridge following the aforementioned research by Carvalho (2011). The main focus of this research was centred on the train-bridge interaction effects on the dynamic response of Canelas Bridge. This required the development of numerical models for the trains of both the Alfa-Pendular train, which operates in the Portuguese railway network, and the TGV train, that operates in the European high-speed network. This work concluded that the train-bridge interaction effects are only relevant for resonant speeds, enabling a more realistic analysis. Furthermore, it was not possible to notice significant differences of the dynamic response of the bridge due to the continuous ballast layer when using the moving loads method or the train-bridge interaction approach. This work also analysed the passenger riding comfort and it concluded that the maximum acceleration inside the vehicle tends to increase along the length of the train. 4.3 Numerical models 4.3.1 Track-Bridge numerical models The Finite Element Method (FEM) was used in the numerical modelling of Canelas Bridge. The bridge was discretised with 2D beam elements using the finite element software FEMIX [Azevedo (2012)]. The numerical model was defined taking into account the design drawings. A single span was modelled and, as the two half slab decks are independent, only a single track was analysed. Furthermore, the numerical model assumed the bridge was in a straight section of a high-speed railway line. Since the structural system is very simple (simply supported bridge) and the goal was to perform a safety assessment (which may require a large number of simulations), Modelling of the train-track-bridge system 4.13  Elevation irregularities – variation in longitudinal-vertical plane;  Alignment irregularities – variation in the lateral direction of the horizontal plane;  Cross level irregularities – variation of the rail elevation along the longitudinal direction against the adjacent rail level;  Gauge irregularities – variation in the track gauge. An illustration of these types of irregularities is depicted in Figure 4.7. b) Elevation and cross level irregularities (side view) a) Schematic representation of the track c) Alignment irregularities (plan view) Figure 4.7 – Types of distributed irregularities (adapted from Rigueiro (2007)). There are two different ways to define the track irregularities: either using values measured experimentally [Xia et al (2003); Antolín et al (2013)] or through random generation using power spectral density functions [Claus & Schiehlen (1998); Nguyen et al (2009)]. Generally the use of experimentally measured track irregularities is limited to the analysis of specific cases. Since the aim of this study is to carry out a safety assessment of the train-bridge system, which requires a significant number of simulations, and due to the limited number of measured track irregularities profiles, the latter option proved to be more suitable. The generation of track irregularities using this method is possible because numerous measurements have shown that track irregularities represent a stationary and ergodic Gaussian random process that may be adequately described by power spectral density functions (PSD) [Claus & Schiehlen (1998)]. Chapter 4 4.14 Several railway administrations have proposed their own analytical expressions for the PSD functions, based on measured data, for practical application. Thus, each PSD function proposed depends on the maintenance and track quality levels used in each country. In this thesis the power spectral density function adopted by the French administration, SNCF, was selected. This function is expressed by:   3 6 1 10           r Ω Ω A ΩG (4.9) where A is a parameter that depends on the track quality and varying between 160 or 550, for a good or bad quality tracks respectively, Ω is the distance frequency and Ωr is the reference frequency, which takes the value of 0.307 m-1 [Frýba (1996)]. Note that SNCF limits the use of Equation (4.9) to wavelengths between 2 m and 40 m. In this study the wavelengths were limited to an interval between 3 m and 25 m, which corresponds to the wavelength range D1 according to EN13484-5 (2008). The numerical process to generate track irregularities has been described in Hu & Schiehlen (1997). The irregularity profile, r(x), is given by:        N iiii xΩAxr 1 cos2  (4.10) where Ωi is the distance frequency within the wavelength range, Ai corresponds to the amplitude and θi is the independent random phase angle that is uniformly distributed in the range between 0 to 2π. The distance frequency, Ω, corresponds to:   2 Ω (4.11) Modelling of the train-track-bridge system 4.15 where λ is the wavelength of the irregularity. Whereas the amplitude, Ai, can be determined through the PSD function, G(Ω), as follows:   iii ΩΩGA   2 1 (4.12) Usually the frequency increment is defined by establishing the lower and upper limits of the distance frequency range, Ωmin and Ωmax, respectively, and selecting an adequate number of discrete frequencies: N ΩΩ Ωminmax   (4.13) The adequate number of frequencies should be carefully selected because if the range of frequencies is large an inadequate value could cause the generated profiles to be dominated by a few frequencies and display patterns, which would make them not random. Taking into account the distance frequency range considered in the current study a set of 1,000 frequencies proved to be sufficient. In addition to this, and in order to guarantee that the frequency range is within the defined limits, the generated profiles are filtered. Two Chebyshev type II filters have been applied: a low pass and a high pass. Two particular aspects have been considered in the design of the filters. The first filter was that the filters ensured that wavelengths smaller than 3 m were removed. The second filter took into account the difficulties of filtering low distance frequency values (has to guarantee that the contribution of wavelengths up to 25 m were taken into consideration) allowing in some cases the contribution of waves with a slightly higher length. Furthermore, there are several documents that define the quality expected for a railway track. A review of the most relevant documents, which include ETI (2002), UIC518 (2005) and EN13484-5 (2008), was carried out by Vale (2010). In order to guarantee that the generated track irregularity profiles are in agreement with the European standards the values established in Chapter 4 4.16 EN13484-5 (2008) must be respected. The values defined in this standard for the alert limit are indicated in Table 4.1 and Table 4.2 and are valid for a track length of 200 m. Table 4.1 – Standard deviation of the longitudinal level limit. Train speed (km/h) Limit (mm) V ≤ 80 2,3 – 3,0 80 < V ≤ 120 1,8 – 2,7 120 < V ≤ 160 1,4 – 2,4 160 < V ≤ 220 1,2 – 1,9 220 < V ≤ 300 1,0 – 1,5 Table 4.2 – Mean to peak value isolated defect limit for the longitudinal level. Train speed (km/h) Limit (mm) V ≤ 80 12 – 18 80 < V ≤ 120 10 – 16 120 < V ≤ 160 8 – 15 160 < V ≤ 220 7 – 12 220 < V ≤ 300 6 – 10 In order to validate the quality of the generated irregularities profile one simply needs to invert the generation process. Using the generated irregularities profile the corresponding PSD function can be obtained through the Fourier transformation of the autocorrelation function. Afterwards, the obtained function is compared with the theoretical power spectral density function. The irregularities profile is validated if a match between the two curves is obtained. An example of this comparison for a validated profile is shown in Figure 4.8. Modelling of the train-track-bridge system 4.17 Figure 4.8 – Example of the validation of a generated track irregularities profile. 4.3.3 Basic random variables As previously indicated two numerical models were developed for the track-bridge system. The initial model was simpler and only intended to analyse the behaviour of the train-bridge system under moving loads models. This represented a preliminary assessment in order to understand and assess the feasibility of a probabilistic approach for the safety assessment of a railway bridge, to understand which variables were more significant for the dynamic response and also to evaluate how the number of variables considered influenced the required computational timings. For this reason, at this preliminary stage the basic variables of the problem to be studied were limited to bridge parameters. Taking into account the properties of the case study bridge, several bridge parameters that might have relevant nondeterministic properties, of which a variation might lead to a relevant variability on the structural response, were selected at this initial stage of the work. These variables may be divided into three different groups: mass, stiffness and damping. However, at this stage, and due to the importance of damping in the dynamic response of short span railway bridges, particularly for resonant speeds, this parameter was taken as deterministic and a damping coefficient, ξ, of 2% (which was obtained experimentally in previous research works on the structure [Pimentel (2009)]) was adopted. The selected random variables, their simulation identification number as well as their corresponding distribution and variability are defined in 1.00E-10 1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E-01 1.00E+00 1.00E+01 PSD [m3/rad] Ω[rad/m] PSD irregularities PSD analytical Chapter 4 4.18 Table 4.3. As it can be noticed from Table 4.3, besides bridge parameters only the contribution of the ballast to the overall weight of the structure was taken into consideration. Table 4.3 – Random variables, distribution functions and variability for the preliminary assessment. Variable (simulation) Distribution Mean (gaussian) or Min. (uniform) Std. Deviation (gaussian) or Max. (uniform) Concrete density weight (1,2) Gaussian 2.5 t/m3 0.1 (CV = 4%) Ballast density weight (3,4) Uniform 17 kN/m3 21 kN/m3 Ballast area (5,6) Uniform 1.48659 m2 2.76081 m2 HEB 500 area (7,8) Gaussian Nominal area 0.04 x nominal area Concrete elasticity modulus (9,10) Gaussian 36.1 GPa 2.888 (CV = 8%) Concrete weight (geometrical variation) (11,12) Uniform Minimum area Maximum area Concrete height (13,14) Gaussian Nominal value 10 mm Concrete width (15,16) Gaussian Nominal value 5 mm Neoprene shear modulus (17,18) Uniform 0.75 MPa 1.18 MPa Neoprene elasticity modulus (19,20) Uniform 420 MPa 600 MPa Regarding the second model, and having already the experience from this preliminary assessment, the variables that were considered included both bridge parameters as well as track parameters. This was possible because the preliminary assessment showed that a large number of random variables did not affect the efficiency of the analysis in a significant manner and, therefore, a higher number of random variables could be used without compromising the feasibility of the probabilistic approach. It should also be noted that in this model the presence of track irregularities is also accounted for and, as previously indicated, the track irregularity profiles were generated according to the process described in Section 4.3.2. Taken into account the numerical model that was developed (shown in Figure 4.6), the selected random variables for the bridge and the track are presented in Table 4.4 and Table 4.5, respectively. It should be noted that all variables where assumed to be independent. Modelling of the train-track-bridge system 4.19 Table 4.4 – Bridge random variables. Variable Distribution Mean (gaussian) or Min. (uniform) Std. Deviation (gaussian) or Max. (uniform) Concrete density weight (c) Gaussian 2.5 t/m3 0.1 (CV = 4%) Concrete elasticity modulus (Ec) Gaussian 36.1 GPa 2.888 (CV = 8%) Concrete height (hc) Gaussian Nominal value 10 mm Concrete width (bc) Gaussian Nominal value 5 mm HEB 500 area (As) Gaussian Nominal value 0.04 x nominal area Neoprene shear modulus (G) Uniform 0.75 MPa 1.2 MPa Structural damping () Gaussian 2 % 0.3 % Table 4.5 – Track random variables. Variable Minimum Maximum Reference Ballast density weight (b) 1.5 t/m3 2.1 t/m3 Fortunato (2005) Ballast elasticity modulus (Eb) 80 MPa 160 MPa SUPERTRACK (2005); INNOTRACK (2008) Ballast height (hb) 0.30 m 0.60 m IAPF (2003); UIC (2008) Ballast load distribution angle () 15 º 35 º Zhai et al (2004) Sleeper weight (Ms) 220 kg 325 kg ETI (2002) Rail pad stiffness (kp) 100 kN/mm 600 kN/mm ETI (2002) Track shear resistance (sr) (per metre of track) 1.0 × 104 kN/m 3.0 × 104 kN/m UIC (2001) Irregularity amplitude (A) 160 275 With respect to the bridge related basic variables the variability of the parameters related to the concrete where based on the work of Wisniewski (2007) whereas the variability of the neoprene shear modulus was based on Manterola (2006). Regarding the structural damping its variation was defined in accordance with the findings in Rodrigues (2004) and the recommendations from Cantieni (2009). For the variability of the ballast density weight the results of experimental tests carried out by Fortunato (2005) in the Portuguese Northern Railway Line were taken into consideration. The experimental tests show that the density weight of clean granite ballast varies between 14.8 kN/m3 and 17.7 kN/m3. In the case of contaminated ballast an increase of the density weight is observed reaching a maximum value of 21 kN/m3. The elasticity modulus of the ballast layer the variation range was defined by taking into consideration the results obtained from two European research projects: INNOTRACK (2008) Chapter 4 4.20 and SUPERTRACK (2005). INNOTRACK (2008) observed that the elasticity modulus for ballast made of soils with poor quality is 80 MPa. UIC719 (2008) recommendation suggests using an elasticity modulus of 130 MPa for current tracks where tests are not carried out to determine the characteristics of the ballast layer. SUPERTRACK (2005) identified that some ballasts where the elasticity modulus reached 160 MPa. Given the conclusions obtained by the aforementioned studies, it is possible to notice that the elasticity modulus of the ballast layer displays significant variability. For this reason, in the current work it was assumed that this parameter follows a uniform distribution ranging between the two most extreme values identified by previous research works. Another parameter that was taken as random was the area of the ballast layer. The recommendation from the Spanish standard IAPF (2003), which suggests a variation of 30% from the values specified by the design drawings, was used. Taking into account the recommendations for the variation of the ballast height produced by UIC (2008), a width of 5 m was defined for the ballast layer and the area variation is introduced by the variation of the height of ballast. The importance of the track shear resistance has already been highlighted in Section 2. This parameter depends essentially of two main aspects: the fastening system between rail and sleeper and the resistance to move of the rail-sleeper system relative to the bridge deck, which is due to the resistance provided by the ballast layer. Several railway administration companies have defined a plastic resistance, k, of 20 kN/m for unloaded tracks. The German Railway Administration (DB) considers a resistance of up to 60 kN/m for loaded tracks. Currently these limits are already part of EN1991-2 (2003). Therefore, the stiffness of the springs used to simulate the ballast shear resistance take the value of 1.0 × 104 kN/m per metre of track for an unloaded track scenario and 3.0 × 104 kN/m per metre of track for a loaded track scenario. Throughout the simulations the values used for the track shear resistance were randomly generated between these two limits. Finally, some other track parameters have been taken as deterministic due to the lack of sufficient information about an adequate distribution and because their variability would not significantly affect the results that are being analysed in this study. For the damping of both the ballast layer and the rail pads the value suggested by Zhai et al (2004) was used. The ballast damping value suggested by Zhai et al (2004) was 5.88 x 10-4 N.s/m and is based on experimental tests using the wheelset-dropping test (a typical test used in the Chinese and Japanese railway Modelling of the train-track-bridge system 4.21 lines). With respect to the rail pad damping the works of Zhai et al (2004) and Rigueiro (2007) were taken as reference. It was possible to observe that the values used in both works are very similar. For the purpose of coherency, the value adopted was the value suggested by Zhai et al (2004) that adopts a rail pad damping of 7.5 x 10-4 N.s/m. 4.3.4 Dynamic properties of the track-bridge system In order to understand the dynamic behaviour of the complex coupled train-track-bridge system it is necessary to know the dynamic properties of each subsystem. Therefore, a modal analysis was carried out to identify the main natural frequencies and the corresponding mode shapes for the average scenario, where all the random variables take their mean value. The results for the model developed for the preliminary assessment are shown in Figure 4.9. a) First bending mode: f = 9.168 Hz. b) Second bending mode: f = 34.571 Hz. Figure 4.9 – Track-bridge natural frequencies and mode shapes for the preliminary model. Similarly, the results for the complete track-bridge numerical model are presented in Figure 4.10. From the comparison between the two models it can be noticed that the first natural frequency is practically unaffected by accounting for the vertical deformability of the track. The second bending mode, due to the mode shape configuration, is more affected and accounting for the Chapter 4 4.22 track flexibility results in a reduction of the natural frequency. Both results are consistent and aligned with expectations considering the model assumptions. a) First bending mode: f = 8.926 Hz. b) Second bending mode: f = 31.550 Hz. Figure 4.10 – Track-bridge natural frequencies and mode shapes. It is also interesting to see how the variability of several parameters affects the natural frequency of the track-bridge system. For this reason, a modal analysis was carried out for a sample of 5,000 Monte Carlo simulations, which is a sample size that enables an accurate assessment of the obtained distribution. The obtained results (with the mean and standard deviation values) are illustrated in Figure 4.11. Modelling of the train-track-bridge system 4.29 According to Hertz theory, the normal contact force, Fn, is given by [Shabana et al (2008)]:   23 21 23 3 4//        BAKK KF hn (4.14) where δ is the penetration, β is a constant that depends on the ratio A/B, which is related with the geometric properties of the bodies in contact, and Kh is a generalised stiffness coefficient that depends on the material properties of the bodies in contact and on the curvatures of the surfaces at the contact point: i i iE K     2 1 (4.15) 630,       B A m n (4.16) and   630 23 2 1, / / /           BA BA nm (4.17) where E and ν are the Young modulus and the Poisson ratio of the materials, respectively, and m and n are constants that take values which can be consulted in specific abacus [Iwnicki (2006)]. It can be noticed that the Hertz normal contact force equation is non-linear. However, when the penetration is small (general case) and assuming that the bodies in contact have the same material properties, it is possible to establish a linear relationship between the contact force and the penetration [Nguyen et al (2009)]: Chapter 4 4.30   32 41 2 31 32 31 1 3 2 2 3 2 3/ // / /        rwnhnwV rr E FKFK  (4.18) where KwV is the linearised stiffness coefficient, rw is the wheel rolling radius and rr is the head radius of rail cross section. Given the importance of the wheel-rail contact stiffness on the evaluation of the contact forces and, consequently, on the wheel unloading coefficients, a parametric study is conducted in order to provide a better understanding on how the contact stiffness is influenced by the real variability of the parameters that affect its value. The variability of the Young modulus of the wheels and the rails in addition to the variability of the radius of both elements, is taken into consideration to account for the wear. The Young modulus of both the rail and the wheel are assumed to follow a normal distribution, whereas the radius of both elements are assumed to follow a uniform distribution. The parameters used for each variable are indicated in Table 4.6. Table 4.6 – Random variables for the parametric analysis of the linearised wheel-rail stiffness coefficient. Variable Distribution Mean (gaussian) or Min. (uniform) Std. Deviation (gaussian) or Max. (uniform) Rail Young modulus Gaussian 205.0 GPa 5.0 GPa Wheel Young modulus Gaussian 192.5 GPa 7.5 GPa Rail head radius Uniform 200 mm 300 mm Wheel radius Uniform 850 mm 1020 mm The variation of the stiffness coefficient is analysed for a sample of 5,000 Monte Carlo simulations. The obtained probability density function (PDF) is plotted against the PDF of a normal distribution in Figure 4.19. Modelling of the train-track-bridge system 4.31 Figure 4.19 – Distribution of the wheel-rail stiffness coefficient. It can be observed that the linearised wheel-rail stiffness coefficient follows a normal distribution with a mean of 1.61x109 N/m and a standard deviation of 4.1x107 N/m. The kurtosis and the skewness coefficients tests confirm the assumption of the normal distribution. 4.3.5.6 Train numerical model developed Bearing in mind the several types of train model referred in the literature and their advantages and drawbacks, at a subsequent stage a FEM model of the TGV train was also developed. A complete 2D numerical model of the TGV train was developed. The car bodies and bogies were simulated by rigid bodies with masses Mc and Mb and rotational inertias Ic and Ib, respectively. The primary and secondary suspensions were simulated by spring-dashpot sets with stiffness kp and ks and damping coefficient cp and cs, respectively. The wheel sets were simulated by lumped masses, Me, whereas the wheel-rail contact stiffness was simulated by a spring with stiffness kh. A schematic representation of this numerical model is shown in Figure 4.20. 1.4 1.5 1.6 1.7 1.8 1.9 2 x 109 0 50 100 150 200 250 300 Kh (N/m) Frequency Mean = 1.61 x 109 N/m Std. deviation = 4.10 x 107 N/m Chapter 4 4.32 Figure 4.20 – Complete train numerical model. According to Goicolea et al (2002) it is common that the main effects of vehicle interaction with railway bridges are adequately captured using simplified interaction models. Due to the large number of simulations that is expected to be performed, these simplifications can result in a significant reduction of the required computational time to assess the safety of the train-bridge system. Therefore, two simplified train models were developed. The first simplified model simulates the car bodies through lumped masses instead of rigid bodies. The second simplified model eliminates the secondary suspensions and concentrates the car body and bogie masses on lumped mass elements connected to the primary suspension system. The validation of the developed numerical models is analysed in detail on Section 4.4.3.3. Due to the difficulty of obtaining information from train manufacturers the dynamic properties of the train were defined mostly according to the values presented by ERRI (1999). The variability of suspension parameters is defined in order to guarantee that the frequency of the bogies is within the typical frequency range of 4 to 8 Hz [Museros et al (2013)]. Regarding the car body mass, its value is assumed to be constant and the values defined in ERRI (1999) are used. The car body mass used for the power cars, the power passenger and the passenger cars are 51.5 tons, 35.86 tons and 22.525 tons, respectively. Moreover, the variation of the number of passengers on the train is also taken into account. The random variable selected to account for this feature is the occupancy rate of the train. The random variables selected for the train model are indicated in Table 4.7. Modelling of the train-track-bridge system 4.33 Table 4.7 – Train random variables. Variable Distribution Mean (gaussian) or Min. (uniform) Std. Deviation (gaussian) or Max. (uniform) Occupancy rate Uniform 0 % 100 % Bogie mass Uniform 2.32 t 3.48 t Wheel set mass Uniform 1.6 t 2 t Primary suspension stiffness Uniform 1300 kN/m 3900 kN/m Primary suspension damping Uniform 6 kN.s/m 18 kN.s/m Secondary suspension stiffness Uniform 290 kN/m 870 kN/m Secondary suspension damping Uniform 10 kN.s/m 30 kN.s/m Wheel-rail contact stiffness Gaussian 1.61x106 kN/m 4.1x104 kN/m To validate the numerical model of the train, its main mode shapes were analysed for the average case scenario and the results are presented in Figure 4.21. a) b) c) d) Figure 4.21 – Train natural frequencies and mode shapes. Chapter 4 4.34 The obtained results are well within the typical limits referred in the literature which validated the developed model. Similarly to what has been done for the track-bridge system a modal analysis was also carried out for a sample of 5,000 Monte Carlo simulations to analyse the type of distribution obtained for the several train mode shapes. The obtained results are illustrated in Figure 4.22 and include the mean and standard deviation values. a) Car body vibration b) Bogie vibration Modelling of the train-track-bridge system 4.35 c) Wheel-rail contact vibration Figure 4.22 – Train natural frequencies distribution. Contrary to what had been observed for the track-bridge system for the train mode shapes, no clear distribution can be identified. However, it is important to point out that the obtained mode shapes are still within the typical range of frequencies documented in the literature. 4.4 Dynamic response To provide a better understanding of the dynamic behaviour of the train-bridge system the average case scenario is analysed for different train speeds and the dynamic response is assessed for different sections of the bridge. Several significant modelling aspects are evaluated in detail to understand their effects on the dynamic response of the train-bridge system. Amongst these modelling aspects are the train-bridge interaction effects, the influence of the track irregularities and some other more specific modelling features such as influence of the track and train numerical models and the track extension before the bridge which are discussed in the following sub-sections. It should be noted that the time step selected for all the analyses is 0.002 s and all the analyses comprise a total of 5,000 steps, for a total of 10 s. The choice for the number of steps was based on the speed range analysed and aimed to guarantee that it allowed sufficient time for the train to cross the bridge and also to capture all significant response in free vibration after the train Chapter 4 4.36 leaves the structure. Regarding the selection of the time step value to adopt in the dynamic analysis, the recommendations presented in Section 2.3.2.6 were taken into consideration and a sensitivity study was also carried out. Time steps of 0.02 s, 0.005 s, 0.002 s and 0.001 s were analysed and it was possible to observe that the dynamic response of the train-bridge system can be adequately evaluate with a time step of 0.002 s, which offers a good balance between the accuracy and the computational time. 4.4.1 Train-bridge interaction effects Due to its importance for the dynamic response of short to medium span railway bridges, the first modelling aspect to be analysed was the influence of the train-bridge interaction. The analysis was extended to the full range of speed analysed, which goes from 200 km/h to 450 km/h, in incremental steps of 5 km/h. The maximum acceleration obtained for each speed in different sections of the bridge is shown in Figure 4.23, along with the comparison with results obtained using the moving loads method, which does not take into consideration the interaction effects. The results illustrated in Figure 4.23 indicate that, in general, accounting for the train-bridge interaction effects tends to reduce the maximum bridge deck acceleration. This difference is mostly noticeable in the resonant peaks. For non-resonant speeds, the differences between the two methods are not significant. However, it should be pointed out is that to be possible to compare the results between the two different methods, the analysis was carried out for a perfect track scenario. Another aspect worth noting was the observation that accounting for the interaction effects seems to originate a slight shift on the obtained dynamic response for speeds greater than 300 km/h. This suggests the existence of some degree of coupling between the two subsystems, in which the train mass contributes to global mass of the system, increasing the modal mass and affecting (in a small scale) the natural frequency of the system, which ultimately results in the differences in the dynamic behaviour of the structure. Modelling of the train-track-bridge system 4.37 a) ¼ span section. b) ½ span section. c) ¾ span section. Figure 4.23 – Influence of the train-bridge interaction effects on the dynamic response. Chapter 4 4.38 4.4.2 Track irregularities effects The importance of track irregularities on the dynamic behaviour of the train-bridge system is another aspect that requires adequate understanding. For this reason, a comparison was made between the bridge response obtained for a perfect track and for a track with irregularities and the obtained results are illustrated in Figure 4.24. a) ¼ span section. b) ½ span section. 0 1 2 3 4 5 6 7 200 250 300 350 400 450 Acceleration (m/s2) Speed (km/h) Irregularities Perfect track 0 1 2 3 4 5 6 7 8 200 250 300 350 400 450 Acceleration (m/s2) Speed (km/h) Irregularities Perfect track Modelling of the train-track-bridge system 4.45 similar on the bridge for each simulation. The comparison is made in terms of wheel rail contact force over the structure, several train speeds are analysed and it should be noted that the results are presented in terms of distance, where the coordinate 0 m indicates the beginning of the bridge. The obtained results are shown in Figure 4.28. a) Irregularities profile 1. b) Wheel rail contact force for profile 1. Chapter 4 4.46 c) Irregularities profile 2. d) Wheel rail contact force for profile 2. e) Irregularities profile 3. Modelling of the train-track-bridge system 4.47 f) Wheel rail contact force for profile 3. Figure 4.28 – Influence of the track length on the wheel-rail contact forces. An almost perfect match can be observed for all the three track extensions 2.5 m after the train entered the bridge. This indicates that the 10.5 m length is not sufficient for the accuracy level that is intended. For this reason, a new analysis was carried out using a 20 m track extension. In this case, the results are only compared with the 110 m track extension scenario. The results are illustrated in Figure 4.29. a) Profile 1. Chapter 4 4.48 b) Profile 2. c) Profile 3. Figure 4.29 – Assessment of the accuracy using a 20 m track length. The results obtained using a 20 m track extension are very similar to those obtained using a 110 m extension. Therefore, this is the extension that was selected for the numerical model to be used when assessing the safety of the train-bridge system. Modelling of the train-track-bridge system 4.49 4.4.3.2 Validation of the track numerical model As previously discussed in Section 2 and also on Section 4.3.1 several track models, with different degrees of complexity, can be found in the literature. The model chosen in this study was the two layer model, used by Calçada (1995). The reason to choose this model is that it is as accurate as other more complex models despite being simpler, which enables it to be more computationally efficient. In order to confirm this a comparison was made with the three layer model proposed by Zhai et al (2004). The obtained results are shown in Figure 4.30. Figure 4.30 – Dynamic response: 2 layer models vs 3 layer model. The results obtained by both models show a very good agreement with a slightly larger difference being notice for train speeds over 350 km/h. However, the observed differences were minor which validates the results obtained by the two layer model. For this reason, and for the purposes of the present study, the two layer model is considered to be as accurate as the three layer model. 4.4.3.3 Validation of the train numerical model Another important aspect that can have a significant impact on both the accuracy of the obtained results and the computational time required for the analysis is the train model. As previously mentioned, the work carried out by Goicolea et al (2002) demonstrated that the main Chapter 4 4.50 effects of vehicle interaction with railway bridges can be adequately captured using simplified interaction models. The two simplifications mentioned in Section 4.3.5.6 were tested to maximise the efficiency of the dynamic analysis in terms of computational timing. In order to validate the simplified model the results obtained using these models were compared with those obtained with the complete model, which was taken as the reference value. The results showed that disregarding the effects of the secondary suspension system leads to some variation to the results obtained when using the complete model. However, the model which simulates the car bodies through lumped masses showed similar results to those obtained with the complete model, as shown in Figure 4.31. Figure 4.31 – Validation of the simplified train numerical model. The results depicted in Figure 4.31 validate the use of the simplified model. Furthermore, and since the simplified model is 30% the size of the complete model, this large reduction in the total number of degrees of freedom resulted in a significant reduction of the required computational times thus making this model the most adequate to use in the safety assessment of the train-bridge system. Modelling of the train-track-bridge system 4.51 4.5 Variable screening procedure In order to understand the influence that each of the basic random variables has on the dynamic behaviour of the train-bridge system, a variable screening procedure (based on a simplified sensitivity analysis) was performed. The goal of such procedure is to identify which variables are relevant to the dynamic response of the bridge and which ones are not, being therefore considered deterministic in the simulation analysis. It should be noted that in this work four distinct response parameters were analysed in the screening procedure: natural frequencies, displacements, accelerations and wheel-rail contact forces. 4.5.1 Description of the methodology The procedure used on the present work is similar to the procedure used in Henriques (1998). The first stage of the procedure consists of selecting the random variables that allow the definition of the problem: the basic variables. Afterwards, a structural analysis is performed adopting the mean values for all the variables. Next, the bridge dynamic response is computed keeping all the variables with their mean values except for the ‘tested’ variable. The ‘tested’ variable value is modified from its mean value by two times the standard deviation. This step is repeated for all variables allowing for the evaluation of the sensitivity coefficients and importance indicators for each variable. The sensitivity coefficients and the importance indicators are obtained by comparing the difference between the reference results (corresponding to the analysis where all the variables are represented by mean values) and the results obtained for each ‘tested’ variable, as expressed in Equations (4.19) and (4.20), where y denotes the results, while x denotes the basic random variables:         mxk m im m kxh y x y / /y /x /y bk k k        (4.19)   CVbk k  b (4.20) Chapter 4 4.52 where σxk is the standard deviation of each variable, CV is the coefficient of variation for each variable and h is a coefficient related with the variation of the mean value of each variable (in this dissertation h = constant = 2). ym and xim are the reference value of the structural response and the mean value of the ‘tested’ variable, respectively, Δyk is the difference between the reference results of structural response and the results obtained for each ‘tested’ variable and Δxik is the difference between the mean value of the ‘tested’ variable and the value used in the analysis. After analysing these coefficients for all variables, the maximum importance indicator is determined and the relative importance of each variable is established by comparison with the maximum value obtained. 4.5.2 Preliminary approach At an initial stage of this work the random variables analysed were only related to the bridge parameters, as indicated in Table 4.3. The analysis had two essential goals: the first (and most important) one was the identification of the parameters that displayed a higher influence on the dynamic response; the second was to understand how the number of variables influenced the safety assessment procedure. For this reason, at the preliminary analysis, variables with a relevance smaller than a pre-established value (a 10% value was defined) are considered irrelevant for the variability of the dynamic response of the bridge. In the safety assessment these are taken as deterministic and their mean value is used for the simulation. It should also be pointed out that four distinct response parameters were analysed in the screening procedure: natural frequencies, displacements, accelerations and reactions. Due to the importance of the train speed on the dynamic response of the bridge, the sensitivity analysis was performed for several train speed ranging from 80 km/h to 450 km/h (speed increased in steps of 5 km/h). Some of the results obtained from the variable screening procedure can be seen in Figure 4.32. Modelling of the train-track-bridge system 4.53 a) Mid-span bridge deck acceleration b) Mid-span bridge deck displacement Figure 4.32 – Results of the sensitivity analysis. For a better readability, only part of the results are presented, allowing for an understanding of the variable selection criteria. It should also be pointed out that despite considering only 10 variables, both the upper and lower bound were analysed, leading to the 20 data sets presented in Figure 4.32, and the variable associated to the reference number in the picture is indicated in Table 4.3. When analysing all four response parameters previously referred, some variables stand out in terms of their influence on the dynamic response of the bridge. Among these variables are the inertia variation due to the variation of the section height, the ballast area, the elasticity modulus of the concrete and the vertical stiffness of the bridge bearings. The variables with less influence on the response (and which will hereafter be considered as deterministic) were the Chapter 4 4.54 geometric variation of the rolled steel profiles, the horizontal stiffness of the bridge bearings and the variation of the self-weight of the concrete elements due to the geometrical variations. However, since the last variable cannot be dissociated from the inertia variation, which has a significant relevance on the dynamic response, it was kept as random. 4.5.3 Sensitivity analysis accounting for the train-bridge interaction At a subsequent stage of this work a similar procedure was carried out whilst accounting for the variability of parameters of the structure, the track and the train. In this stage the train-bridge interaction effects were taken into consideration and this represents a more realistic analysis. In this stage two particular aspects of the response were analysed: the bridge deck acceleration levels and the wheel unloading rate. These were selected because they are two of the most significant indicators of the running safety of high-speed trains. Unlike what was done in the previous analysis, the sensitivity analysis was limited to adequately selected speed ranges. In the case of the bridge deck acceleration the analysis was limited to the [270 – 300] km/h speed ranges, as this corresponds to the range where the resonant effects are most significant. As for the analysis of the wheel unloading rate, the sensitivity analysis was carried out for the [405 – 435] km/h speed range, as this is the range where the first cases of loss of contact between the wheel and the rail were observed. 4.5.3.1 Bridge deck acceleration Having already analysed the most significant variables that affect the bridge deck acceleration, it is important to check if adopting a different type of methodology (train bridge interaction vs moving loads) leads to different conclusions. The obtained results of the sensitivity analysis for the bridge deck acceleration when accounting for the train-bridge interaction effects are illustrated in Figure 4.33.