# ANALYSIS AND MITIGATION OF VIBRATIONS INDUCED BY THE PASSAGE OF HIGH-SPEED TRAINS IN NEARBY BUILDINGS

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1 ANALYSIS AND MITIGATION OF VIBRATIONS INDUCED BY THE PASSAGE OF HIGH-SPEED TRAINS IN NEARBY BUILDINGS JOÃO MANUEL DE OLIVEIRA BARBOSA 3 Supervisors: Prof. Álvaro Azevedo (FEUP) Prof. Rui Calçada (FEUP Prof. Eduardo Kausel (MIT) Thesis presented to the Facult of Engineering of the Universit of Porto for the Doctor Degree in Civil Engineering

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3 Ao titio e à titia Aos meus pais Aos meus irmãos

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5 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Abstract The present dissertation addresses the subect of vibrations induced b the passage of high-speed trains. The main obective of the stud is the development of numerical tools that allow investigating distinct geometries of train and track, considering buildings in the proimit of the track, and also assessing the behavior of mitigation solutions. The problem of vibrations induced b moving vehicles is divided into three stages: a generation stage, in which the vehicle interacts with the track; a propagation stage, in which the forces that the train transmits to the track originate waves that propagate through track and ground; and a reception stage, in which the waves reach a nearb building, causing it to respond dnamicall. Since the geometric specifications of the problem var within the three stages, different strategies are chosen for each stage:. The generation stage involves a discrete structure (the vehicle) moving on top of a structure whose longitudinal dimension is infinite (track-ground sstem). In this wa, the problem is formulated in a moving frame of reference, being the equations solved in the frequenc domain;. For the propagation stage, since it is assumed that track and soil are invariant in the longitudinal direction, then the problem is formulated in the wavenumber-frequenc domain (.5D). In this wa, the three-dimensional problem is reduced to a series of two-dimensional problems of smaller dimensions that are faster to solve. The track is simulated with finite elements while the surface of the soil interacting with the track is simulated with boundar elements. Mitigation measures in the soil must be included in this stage; 3. In the reception stage, the three-dimensional structure to be analzed is irregular in all directions and therefore the.5d procedure cannot be applied. For this reason, a 3D frequenc domain formulation is used, in which the structure is simulated with finite elements and the soil is simulated with boundar elements. The eterior loads considered in this stage are calculated based on the results of the propagation stage. The response of the soil, which is of great relevance for the problem, is accounted for through the boundar element method (BEM). The fundamental solutions used to nurture the BEM are obtained with the thin-laer method, being this the main difference between the strateg adopted herein and the procedures followed b other authors that also use the BEM. The numerical procedures mentioned above are described in chapters -4. Additionall, in chapter 4, the link between the three stages and the distinct procedures is established. In chapter 5, the methodolog is applied to the stud of trenches as mitigation solutions. i

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9 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Acknowledgments It took me five ears to conclude this Doctoral work, and such would not have been possible without the contribution and support of the people that one wa or the other accompanied me throughout this process. To them, I want to demonstrate m appreciation. To m supervisors, Professors Álvaro Azevedo, Rui Calçada and Eduardo Kausel, for having accepted the responsibilit of guiding. I would like to stress the role of Professor Eduardo Kausel, who received me in the United States and who was essential for the success of m sta at MIT. To Professor Pedro Costa, for his comments and opinions given during the development of the thesis, but mostl during the last ear of work, which relates to Chapter 5 herein presented. To the colleagues developers of FEMIX, Prof. Álvaro Azevedo and Sérgio Neves, for making the software available, and for their effort in making FEMIX an attractive choice for people who want or need to develop their own calculation routines. To colleague Ricardo Nobre, PhD student in the field of computer science at FEUP, to the research group SPECS, to Professor Pedro Sobral from Universit Fernando Pessoa, to Professors João Cardoso, Rui Rodrigues and Jorge Barbosa from the Department of Computer Science of FEUP, and to Nuno Subtil from NVIDIA, for their support regarding GPU programming. To all colleagues from the High-Speed research group, headed b Professor Rui Calçada, for all help provided during these five ears. To all colleagues and friends that have worked in office H3 and that made the working environment ver friendl Alés de Miguel, Cristina Ribeiro, Fernando Bastos, Luís Martins, Mário Marques, Miguel Araúo, Nuno Santos, Ricardo Monteiro, Sérgio Neves. To the MIT office mates, for the same reason Dilip Thk, Swapnil Raiwade, Rositta Jünemann, Anthoula Agn, Zeid Alghareeb, Inez Azaiez. To Fundação para a Ciência e Tecnologia (FCT Portuguese Foundation for Science and Technolog), that provided me financial support through PhD grant SFRH/BD/4774/8 and through the research proect PTDC/ECM/ 455/9 Ground Vibration and Noise Induced b High-Speed Trains: Prediction and Mitigation. To MIT-Portugal program, for facilitating m research periods at MIT. At last, the most special acknowledgment goes to those who have followed me on a dail basis, ever since m birth. An immeasurable thank ou to m parents, Jorge Barbosa and Maria José Barbosa, m brothers, Ricardo Jorge and Rui Vasco, to ma granduncle and grand aunt, Manuel Sala (titio) and Maria Elisa (titia). Without their absolute support, I would not have started the PhD. v

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11 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Contents ABSTRACT... I RESUMO... III ACKNOWLEDGMENTS... V CONTENTS... VII. INTRODUCTION.... MOTIVATION.... STATE OF THE ART..... Eperimental campaigns Prediction models Countermeasures....3 OBJECTIVES AND ORIGINAL CONTRIBUTIONS OF THE PRESENT WORK ORGANIZATION OF THE DOCUMENT WAVE PROPAGATION IN THE SOIL: FUNDAMENTAL SOLUTIONS INTRODUCTION THIN-LAYER METHOD IN CARTESIAN COORDINATES DISPLACEMENTS IN THE WAVENUMBER-FREQUENCY DOMAIN....4 HORIZONTAL DERIVATIVES AND TRACTIONS IN THE WAVENUMBER DOMAIN TRANSFORMATION OF FUNDAMENTAL SOLUTIONS TO THE.5D DOMAIN Fundamental displacements Horizontal derivatives Fundamental tractions Vertical derivatives and internal stresses Validation MODELING UNBOUNDED DOMAINS Coordinate stretching PMLs for the TLM Eample of a laered domain SOLUTION OF THE EIGENVALUE PROBLEMS Eigenvalue problem for SH waves Eigenvalue problem for SVP waves CONCLUSIONS NUMERICAL TOOLS FOR SOIL-STRUCTURE INTERACTION INTRODUCTION D BOUNDARY ELEMENT METHOD Introduction vii

12 3.. Integral representation Regularization of the integral equation Discretization of the boundar Coupling BEM-FEM Weak coupling response to incoming wave fields Final considerations D FINITE ELEMENT METHOD Introduction D Finite Element Method Eample - dispersion curves of a UCI86-3 rail D BOUNDARY ELEMENT METHOD Introduction Formulation Horizontal boundar elements Vertical boundar elements Outgoing stress fields Coupling.5D BEM and.5d FEM Conclusions D BEM-FEM VALIDATION EXAMPLES Eample square tunnel in a laered medium Eample slab free in space CONCLUSIONS INVARIANT STRUCTURES SUBJECTED TO MOVING LOADS AND MOVING VEHICLES INTRODUCTION MOVING LOADS Introduction Constant moving loads Oscillating moving loads Eamples Conclusions MOVING VEHICLES Introduction Vehicle structure interaction Point mass moving on top of a beam on a Kelvin foundation Multi-degree of freedom vehicle moving on top of a ballast track Conclusion VIBRATIONS INDUCED BY A MOVING VEHICLE IN A NEARBY STRUCTURE viii

13 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings 4.4. Introduction and general description of the eample Step : eigenpairs of the soil (TLM) Step : transfer functions Step 3: generation stage dnamic forces Step 4: propagation stage response of the supporting structure Step 5: reception stage response of the building CONCLUSIONS REDUCTION OF VIBRATIONS BY MEANS OF TRENCHES INTRODUCTION PARAMETERS INFLUENCING THE EFFICIENCY OF TRENCHES Introduction Validation eample Influence of the trench depth Influence of the trench width Influence of the trench position Influence of the stiffness of the in-fill material Influence of densit of in-fill material Influence of the Poisson s ratio of the soil Influence of the ground stratification Influence of the modeling strateg Conclusions TRENCHES FOR THE MITIGATION OF TRAIN INDUCED VIBRATIONS Introduction D analses influence of the track D analses vibrations induced b the Alfa Pendular train Conclusions EFFECT OF TRENCHES ON A NEARBY STRUCTURE General description of the building Natural frequencies of the building Building response for the non-mitigated case Reduction achieved b trenches CONCLUSIONS CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH CONCLUSIONS RECOMMENDATIONS FOR FURTHER RESEARCH... APPENDIX I... 5 i

14 MATRICES D αβ FOR CROSS-ANISOTROPIC MATERIALS... 5 THIN-LAYER MATRICES FOR CROSS-ANISOTROPIC MATERIALS... 5 Linear epansion... 5 Quadratic epansion... 5 APPENDIX II... 7 EVALUATION OF I 4 USING CONTOUR INTEGRATION... 7 APPENDIX III... pɶ I AS A FUNCTION OF B uɶ I AND pɶ II... APPENDIX IV... 3 DISPLACEMENTS INDUCED BY DISK LOADS... 3 APPENDIX V... 5 VIBRATION MODES OF THE BUILDING... 5 REFERENCES... 7

15 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings. Introduction. Motivation High-speed railwa networks have been under construction all over Europe and Asia during the last decades. Currentl, in Europe, high-speed lines are being eplored in Spain, France, German, Belgium, Netherlands, Luemburg, Switzerland, Ital and Britain. In Asia, China holds the record for the largest high-speed network, with an etension of 75 km. It is epected that b the Chinese network will be epanded to a total of 5 km. In other countries, for eample Sweden, even though there is no high-speed network, trains can travel at speeds up to around km/h. In Portugal, in some sections of the Northern Line, the Alfa Pendular train can travel at speeds up to km/h and, until a few ears ago, there were plans to build high-speed railwa lines that would connect Portugal with Spain. Problems related to the increase of the travel speed of trains have been reported in the last ears. Shortl after the opening of the Swedish line between Göteborg and Malmö in 997, ecessive vibration levels were detected both in the railwa embankment and surrounding soil when the X- train travelled at speeds around km/h (Madshus and Kania, ). The high vibration levels were eplained b the increase of the train speed, which approached the Raleigh wave velocit of the soil, causing the resonance of the track-soil sstem and resulting in the potential instabilit of the train. Another problem that ma arise from the circulation of trains is the ecessive vibration level on nearb buildings originated b waves that propagate through the ground. The induced vibrations ma cause annoance to the occupants of the building, malfunction of sensitive equipment and, eventuall, structural and non-structural damage. Since during the planning stage of a high-speed railwa network it is not possible to avoid sensitive areas in which vibrations ma be problematic, such as zones of poor soil characteristics (soft soils with low wave velocities) or zones more or less urbanized (trainstations, for eample), prediction tools for the assessment of vibrations induced b the passage of trains reveal themselves ver valuable. This is the motivation for the present work. The main obective of the work is to develop numerical tools capable of predicting the vibration levels induced b trains in buildings and to stud the efficienc of countermeasures to mitigate those vibrations. In the net section, the state of the art on vibrations induced b vehicles is presented.. State of the art Vibrations induced b moving sources are not a problem that emerged with high-speed trains. Long before the appearance of the first high-speed trains in Japan, in the mid-sities, reports had been made regarding the impact that the circulation of vehicles could have on surrounding buildings. For eample, in the XIX centur, people were concerned with the effects that the passage of freight trains in a line et to be constructed in the proimit of the Roal Observator, in Greenwich, England, would have on its equipment (South, 863). This concern led to eperimental investigations on the subect. In another eample, in the middle of the XX centur, the National Research Council of Canada promoted eperimental campaigns to evaluate the vibrations induced b the passage of trolle buses in nearb buildings (Sutherland, 95). The campaigns were organized in response to the increasing

16 Chapter Introduction number of complaints received from inhabitants who sensed the vibrations and feared damage on their properties. The construction of the subwa sstems inside the cities (ecessivel close to residences), together with the increase of train capacities and comfort standards, gave more importance to the problem of vibrations induced b moving vehicles and led to several eperimental campaigns (Wilson et al., 983; Dawn and Stanworth, 979; Melke and Kramer, 983). The eperimental campaigns were carried out to assess if the vibrations induced b the circulation of trains could damage buildings, cause discomfort to inhabitants or cause the malfunction of sensitive equipment and, at the same time, to understand the mechanisms of generation of vibrations and to stud possible mitigation measures. In more recent ears, in part due to the continuous increase of the weight and speed of trains and in part due to the higher standards for comfort, the problem became even more important and, in addition to the problem of vibrations induced on buildings, a new problem emerged: at some lines resting on soft soils the train speed approached the propagating velocit of the waves and, as a consequence, large displacements in the embankment were observed, causing the risk of derailment of the train. This happened, for eample, in the ver well documented case of Ledsgard, Sweden (Hall, ), and in a line of the Northwest of France (Picou and Le Houedec, 5). In those lines, the circulation speed was reduced and, in several other lines around the world, new eperimental campaigns were conducted to assess the vibration levels. In addition, studies were performed on countermeasures to mitigate the vibrations and prediction models were developed or improved. Two strategies for the stud of the phenomenon of vehicle induced vibrations can be adopted: field measurements and numerical/analtical predictions. Field measurements are performed under real conditions and provide an enriched set of results, which account for all the factors that influence the phenomenon. The results of the eperimental campaigns indicate which aspects most influence the level of induced vibrations, thus showing which factors must be taken into account when developing numerical models. When large databases collecting eperimental data are available, it is possible to perform etrapolations in order to predict the vibration levels for scenarios with similar conditions (soil, track, vehicle, structure), and, consequentl, it is possible to develop empirical models. The drawback of field eperiments is their high cost and so it is desirable that the are emploed as less as possible. Nevertheless, eperiments are alwas needed to obtain inputs for numerical models and also to validate these models. On the other hand, to use prediction models is not as epensive as performing in situ eperiments but, contraril to eperiments, these models cannot reproduce the whole realit since the are based on assumptions and simplifications. On one side, empirical models that are developed based on eperimental data provide good estimates but their range of applicabilit is limited to scenarios with conditions similar to the eperiments. On the other side, numerical models are versatile and allow studing the influence of certain parameters on the vibration levels, but their accurac depends on the simplifications made and on the assumptions on which the models lie. Numerical models can usuall be adapted in order to account for countermeasures and stud their performance. In the following sub-sections, a historical overview of eperimental campaigns, prediction models and countermeasures for vibrations is presented.

18 Chapter Introduction With respect to rail traffic induced vibrations, Wilson et al. (983) reported eperiments made during the seventies to stud the use of floating slabs as a countermeasure to mitigate groundborne vibrations and to stud the influence of the properties of the bogies on the induced vibrations. Measurements were made in tunnels, on the free surface, and inside buildings. Later, in the late seventies, Dawn and Stanworth (979) measured the vibration levels on a wall of a single store building situated about 4m awa from a track during the passage of trains at speeds up to km/h. The noticed that the vibrations increased with the speed of the train and observed that the frequenc content of the response presented a peak on the passage frequenc of the sleepers. Dawn (983) performed further eperimental studies and confirmed that the passage frequenc of the sleepers is indeed a mechanism of ecitation. He also recognized that the critical velocit (to which corresponds the maimum ground response) occurred when the sleeper passage frequenc coincided with the resonance frequenc of the vehicle-track sstem. Melke and Kramer (983) reached the same conclusions in their eperimental studies. In more recent ears, with the increase of the train speed, the problem of induced vibrations has been given even more importance and new eperimental studies have been performed. In addition, with the construction of new railwa lines, their homologation tests enabled the eecution of new field eperiments. In German, Auersch (994; 5) performed measurements at three different sites near Würzburg during test runs of the ICE train with different configurations and at speeds between and 3 km/h. During the tests, which considered three different track conditions (surface line, bridge and tunnel), the vibrations of the vehicle, track and soil were recorded. The results showed that the quasi-static component of the ale load was important for the response of the track and the surrounding soil, and that its importance vanished rapidl with the distance. The results also suggested that the sleepers act as harmonic forces, whose intensit increases with the train speed, but remains constant when the sleeper passage frequenc eceeds the vehicle-track resonance frequenc. In Sweden, as a consequence of the high vibration levels observed shortl after the opening of the line between Göteborg and Malmö, in 997, the train speed was reduced at some locations and investigations were conducted during the Autumn of 997 and Spring of 998 to diagnose the problem and to find solutions (Madshus and Kania, ; Hall, ). A X- passenger train was used in a total of runs at speeds ranging from to km/h and the responses of rail, sleepers, embankment and ground (at the surface and its interior) were measured. It was observed that for speeds below 7 km/h the displacements of the ground were similar to those obtained considering static loading and therefore were independent from the speed. At speeds around km/h, the amplitude of the displacements increased drasticall, causing the risk of derailment of the train. In Belgium, the epansion of the railwa network allowed to eperimentall investigate the phenomenon in newl built high-speed lines. In December of 997, si weeks before the inauguration of the high-speed line between Brussels and Paris, an etensive eperimental campaign was organized b the Belgian railwa compan during the homologation phase of the line. Track response and free field vibrations up to 7 meters awa from the track were measured during the passage of a Thals train at speeds varing between 3 and 34 km/h (Degrande and Schillemans, ). Five ears later, in August and September of, the high-speed line between Brussels and Köln was also submitted to homologation tests (Kogut et al., 3). The tests were performed at two different sites, Lincent and Waremme, and 4

19 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings included the measurement of vibration levels at the track, in the free field and in a single famil dwelling located 5m awa from the track during the passage of Thals trains and IC trains at variable speed. The tests were complemented with the eperimental determination of some dnamic properties of the track and soil. The two sets of eperiments showed that the passage frequencies of bogies and sleepers and their higher harmonics could be noticed on the spectrum of the responses, namel in the near field. Also, differences in the registered responses for the different trains suggested that the induced vibrations depend on the train properties. In both cases, attenuation of vibrations with the distance to the track was detected. In Ital, Lai et al. (5) measured the transfer functions in two sections of a tunnel in the cit of Rome. The aim of the stud was to assess if the level of vibrations would affect the surrounding buildings. Since the line was not et operational, it was not possible to perform a direct measurement of vibrations induced b the railwa traffic. Hence, the transfer functions from the tunnel to the free field and to the interior of buildings were determined eperimentall using a mechanical hammer. The transfer functions would serve as inputs in a simple numerical model to predict the level of vibrations induced b future passing trains. In the Northwest of France, after it was observed that the ground presented ecessive displacements, Picou and Le Houedec (5) measured the vibrations on rails, sleepers and free-field during the passage of different trains. The results showed the influence of the train speed and of the tpe of train on the induced vibrations. In England, within the framework of the CONVURT proect, vibrations were measured at a site in Regent s Park, London, during 35 passages of a test train in a tunnel at speeds between and 5 km/h (Degrande et al., 6). Accelerations of the ale boes, of the tunnel, of the free field (both at surface and inside the soil) and on several floors of two buildings situated 7m awa from the tunnel were measured. Rail and wheel roughness have also been measured and track characteristics were determined b receptance tests. Analsis of the measured fields allowed concluding that the peak velocities on the ale boes and track increased with the train speed, a tendenc that was less pronounced in the free-field and in the buildings. In Beiing, China, a subwa line was planned to pass close to the Phsics Laborator of the Beiing Universit and so there was concern about the vibrations that would be induced b the rail traffic (Gupta et al., 8). To stud if certain equipments would need to change place, measurements were performed in the free-field near the lab and inside the building to evaluate the eisting vibration levels (induced b road traffic and people). In addition, measurements were made in a different line of the Beiing subwa sstem with similar characteristics. The superposition of the eisting background and the predicted vibration levels would provide the vibration level epected in the labs. In Northeast China, Xia analzed eperimentall the problem of vibrations in buildings induced b trains running on bridges. Trains running at speeds varing between 6 and 8 km/h (Xia et al., 5a) and between 6 and 37 km/h (Xia et al., 5b) were considered. It was observed that the vibration levels would increase with the weight and speed of the train and would attenuate with the distance to the railwa line. It was also observed that the vibrations were stronger at higher floors, eceeding in some places the levels allowed b the Chinese code. To finalize the field measurements, some eperimental campaigns have also been organized at the Portuguese Northern line b researchers from FEUP, whose obective was to characterize the site conditions and measure the vibrations induced b real traffic (Alves Costa et al., 5

20 Chapter Introduction a; dos Santos, 3). At the same site, further investigations have been conducted in order to evaluate the dispersion of the responses along the longitudinal directions. The results obtained from these campaigns are epected to be published in the near future... Prediction models Two tpes of prediction models can be considered: empirical and analtical/numerical. Empirical models are based on results from eperimental campaigns and usuall provide good predictions, but their range of applicabilit is limited to scenarios that are similar to the conditions under which the eperiments were performed, thus lacking versatilit. Description of empirical models can be found in works b Kurzweil ( 979), Melke (988) and Madshus et al. (996). These models use chains of transmission losses for the source-path-receiver sstem and consider parameters such as train speed, ale loads, suspension sstems, weight of the train, wheels and rail conditions, rail fastening sstems, tpe of track, tpe of tunnel and tpe of buildings. The model described b Madshus et al. (996) was developed based on a large number of vibration measurements made in Norwa and Sweden and was used for the planning of a high-speed railwa line in Norwa. In another empirical work, to evaluate problems of ecessive vibrations in preliminar stages, Bahrekazemi (4) presented a model that is based on measurements performed in several sites of Sweden. On the other hand, numerical and analtical methods are more versatile and can be efficientl used to stud the effect of train speed or weight, track tpe, material resilienc, ground conditions, etc. The drawback is that these methods rel on idealizations and simplifications, then failing to reproduce realit as accuratel as it would be possible with field eperiments. Nonetheless, depending on the degree of detail of the model, the obtained prediction can be acceptable and useful. To correctl model vibrations induced b vehicles, three stages must be accounted for in a numerical/analtical model: the generation stage, the propagation stage and the reception stage (Figure.). In the generation stage, the vehicle interacts with the track and induces a moving stress field on it. The stresses are transmitted from the vehicle to the track through contact surfaces (wheels or tres) that move in space. Due to the dnamic behavior of the vehicle and its interaction with the track, the vehicle is subected to accelerations and so the contact stresses, besides moving with the vehicle, also change their value with time. The non varing component of the contact stresses is called quasi-static ecitation (forces per wheel or tre) while the component varing with time is termed dnamic ecitation. In the propagation stage, the stress fields (or the vibrations) propagate through the track and part of them is transmitted to the soil. These stresses continue to propagate in the soil, being reflected or refracted whenever a different material or a barrier is encountered, and finall reach the building. In the reception stage, the vibrations that reach the building induce a dnamic response on it. The problem of vibrations induced b moving vehicles is three dimensional: the vehicle moves in one direction while the waves propagate in the soil in three directions. Modeling a three dimensional problem can become ver complicated and time consuming, even for the current computers. For this reason, the first works assumed that the phenomenon could be described b D models. For eample, in their review paper, Gutowski and Dm (976) mentioned that the vibrations generated along a road or a railwa track could be modeled as a line source as long as the roadwa was relativel uniform and the receiver was in the far field, but close enough to the source (less than /π times the length of the roadwa or the train). The authors supplemented that if the ground motions were dominated b the surface (Raleigh) 6

21 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings waves, then there would be no geometric damping and so the vibrations would attenuate onl due to material damping. Later, Verhas (979) compared the results of line source models with the results of point source models and concluded that b neglecting the geometric damping of waves inaccurate predictions would be obtained. This author suggested that the combination of the two models would ield better predictions, but no guidelines on how to combine the results from each model were indicated. In another D work where a finite element (FE) model was used, in order to account for the geometric damping of surface waves, the accelerations were corrected b a factor / r, being r the distance to the source (Taniguchi and Okada, 98). This methodolog was used to stud the efficienc of soil improvement via the lime pile technique as a countermeasure. Also using a D FE procedure, Chua et al. (995) determined the vibration levels in a four-store podium block due to the passage of trains in a double-bo tunnel, accounting both for the quasi-static and for the dnamic ecitation. The authors used an iterative nodal condensation procedure to avoid etremel large meshes. Figure.: Generation, propagation and reception of vibrations (Hall, 3) With the improvement of computational performance, both in terms of memor and speed, the use of 3D models became possible. One of the first works that considered the threedimensionalit of the problem was performed b Krlov (Krlov and Ferguson, 994; Krlov, 994; Krlov, 995). In their work, Krlov and his collaborators developed a model for surface trains where the forces transmitted to the ground through each sleeper are calculated analticall, and then, considering the sleepers as point sources, the field induced b each sleeper is combined, thus obtaining the response of the soil due to the passage of the train. Onl the quasi-static component of the ecitation is considered. The method for the calculation of the forces transmitted to the soil has been used b other researchers. 7

22 Chapter Introduction Conventional methods used for the analsis of three-dimensional problems, such as the FE method and the boundar element (BE) method, have also been emploed to analze the problem of induced vibrations. The FE method requires the discretization of the domain, which for 3D problems results in a large number of degrees of freedom and in sparse smmetric matrices. This method can be used to model irregular domains and, when applied in the time domain, can account for the non-linear behavior of materials. B itself, the classical FE method cannot simulate infinite domains, so special procedures need to be considered at the boundaries of the truncated domains in order to avoid fictitious reflections. Contraril, the BE approach onl requires the discretization of the boundar of the domain, thus resulting in less degrees of freedom, but, unlike the FE method, leads to full nonsmmetric sstems of equations. The BE method takes into account the radiation of waves towards infinit, but cannot account for non-linearities and requires the knowledge of the so called Green s functions (GF) or fundamental solutions. The hbrid FE-BE method combines the advantages of both approaches, being its use ver attractive when the coupling between irregular domains and unbounded domains is required. Regarding the FE approach, Hall (3) used a time domain methodolog and treated the reflections at the boundaries using dashpots. The considered mesh led to reasonable results onl up to the frequenc Hz and in the calculations onl the quasi-static component of the ecitation was considered. The results of the model showed a transient phenomenon that was not observed in real measurements and that was originated b the entrance of the loads in the model. However, this numerical phenomenon would have dissipated due to damping b the time that the waves reached the other etreme of the model, and so the results at that etreme were better. Using a model with 65 meters in the longitudinal direction, good results were limited to the near field. To obtain better results farther from the track, longer models would be needed, which would render the mesh impractical for calculation. The same author compared the results obtained with 3D models with those obtained with simpler D models and concluded that the D models could be used to stud certain effects of traffic induced vibrations but not to obtain good predictions of the induced levels of vibrations (Hall, ). In another work using the FE approach, Ekevid and Wiberg () followed a similar approach but instead of treating the boundaries of the mesh with dashpots, these authors used the scaled boundar finite element method (Wolf, 3). Even though the proposed methodolog accounted for the radiation of waves to infinit, the fact of using a 3D mesh required a ver large computational effort. Also following the FE approach, Ju used a 3D formulation to simulate soil vibrations due to a high-speed train crossing a bridge and to stud the efficienc of trenches (Ju, ) and of soil improvement (Ju, 4) as countermeasures. The boundaries were treated with first-order absorbing boundaries and the sstems of equations were solved using the preconditioned conugate gradients method, i.e., an iterative method. The calculation time of the problem was over one week. As for the BE and hbrid FE-BE approaches, Bode et al. () used the BE method to model the soil and the FE method to model the sleepers and the rails (dos Santos, 3). The methodolog was formulated in the time domain and was used to determine the vibrations in the free-field and to stud the influence of the soil-sleeper coupling scheme. The GF considered for the soil were the half-space Green s functions, thus limiting its discretization to the regions interacting with the sleepers. A similar strateg was followed b O'Brien and Rizos (5), but instead of using half-space GF, the used full-space GF, which demanded 8

25 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings in order to account for the presence of water in the ground (poroelasticit) and to stud the effect of trenches in the isolation of vibrations (Cao et al., ). As final eample concerning surface trains, Alves Costa () used.5d finite/infinite elements to simulate the Leedsgard case taking into account the large deformations that ma eist due to the poor qualit of the soil. With that purpose, he simulated the non-linear behavior of the soil under the track b considering equivalent elastic parameters that depended on the deformation level of the finite elements (Alves Costa et al., ). Additionall, he also used a.5d coupled FE-BE model to simulate the vibrations induced b the passage of trains in the Portuguese railwa sstem (Alves Costa et al., a). This last model was also used to stud the strateg for modeling the train (Alves Costa et al., b), and to stud ballast mats as mitigation measures (Alves Costa et al., c). From these two works it is concluded that train models can be reduced to ales, bogies and primar suspension sstems, and that ballast mats perform better when placed beneath the subballast laer, and not as well when placed between ballast and subballast laers. Regarding tunnels, Forrest and Hunt (6b) developed a model that assumes a tunnel with a clindrical shape surrounded b a soil of infinite etent. Since the soil is treated b means of wave equations of an elastic continuum, no free surface is considered and consequentl no surface waves are ecited. In the far field, it is likel that buildings receive more energ from such waves than from bod waves. Nonetheless, the model can be ver effective in the evaluation of the response near the tunnel, where surface waves have much less influence. The model was also etended to account for tracks and then was used to assess the behavior of floating slab tracks (Forrest and Hunt, 6a). The authors concluded that floating slabs ielded modest insertion losses and that under certain conditions the could even increase the transmission of vibrations. In the work b Hussein and Hunt (7), the model was further etended to account for tangential forces at the tunnel walls, and in the work b Hussein et al. (8) it was improved in order to permit the analsis of tunnels embedded in laered halfspaces. For that purpose, the authors assume that at a first step, the near field displacements are controlled b the dnamics of the tunnel and of the surrounding laer, i.e., the neglect the contribution of the other laers. At a second step, the response of the far field is calculated using the tractions calculated during the first step and assuming the proper stratification of the soil. Yang et al. (3) approached the problem using finite/infinite elements formulated in the wavenumber-frequenc domain (Yang and Hung, ). When compared with the.5d models mentioned so far, this approach has the advantage of considering the transverse stiffness of the track and of allowing more comple geometries both for track and soil. However, it ma become less attractive because the number of dofs increases significantl. The authors used the developed methodolog to stud the stiffness, damping and stratification of the underling soil, and concluded that increasing the stiffness results in the decrease of vibration levels, that increasing the damping results in a decrease of vibration levels onl if the loads move faster than the Raleigh wave velocit of the soil, and that the soil stratification is etremel relevant owing to the fact that the cut-off frequencies depend on the laers depths and because no waves can propagate below the first cut-off frequenc. Also regarding tunnels, Rieckh et al. () developed an invariant model in which the anisotrop of the soil is considered. The model is based on a boundar element formulation, and uses the.5d fundamental solutions of laered and anisotropic media, which are calculated with the method of potentials.

26 Chapter Introduction To conclude the discussion on invariant models, it must be mentioned that hbrid FE-BE methods formulated in the.5d domain can also be used (Sheng et al., 6; Galvín et al., ; Galvín et al., ). These hbrid approaches are more efficient and more versatile than the predictions that simpl rel on one of these methods. In what concerns periodic models, the differ from the invariant models inasmuch as the can account for geometric features that repeat themselves through the longitudinal direction. For the case of railwa tracks, the periodicit can account for the discrete sleeper support or for discontinuous slab tracks (Clouteau et al., 5; Chebli et al., 6; Sheng et al., 5; Gupta et al., 7; Gupta et al., 8). Vostroukhov and Metrikine (3) compared the results from periodic models with results from equivalent invariant models and concluded that the results were ver similar. In this last stud, onl the quasi-static component of the ecitation was considered. To complete the discussion on prediction models, one must add that in recent ears new approaches denominated b meshless methods (or mesh-free methods) were proposed and shown to be a promising alternative to the boundar element and finite element methods. The name meshless is given because these methods do not require the discretization neither of the interior nor of the boundar of the domain of interest. Within this famil of methods, a popular approach is the Method of Fundamental Solutions (MFS), which resembles the BE method in the sense that it requires the availabilit of the fundamental solutions, but overcomes the need for evaluating the boundar integral, thus avoiding complications associated with the singularities of the fundamental solutions. Researchers from Universit of Coimbra have successfull applied the MFS in acoustic problems (Godinho et al., ; Soares et al., ) and in some elastodnamic problems (Godinho et al., 3; Godinho et al., 9), reporting encouraging results. The main drawbacks of these methods are their high computational cost (mostl when weak form formulations are considered) and, in some cases, their lack of stabilit (mostl when strong form formulations are considered) (Godinho and Soares Jr, 3)...3 Countermeasures The control of vibrations induced b moving vehicles can accompan all three stages of the phenomenon: the generation stage, the propagation stage and the reception stage. Hemsworth () considers that the most effective and economical measures are those performed on the track, i.e., during the generation stage. For the rail traffic case, in order to control vibrations at the source, the following techniques are available: precision straightened rail, rail grinding, wheel truing, continuous welded rail, soft direct fiation fasteners, resilient materials under rails and sleepers, ballast, mats and floating slabs (Nelson, 996; Hemsworth, ; Thompson, 8). Rail straightening, rail grinding and wheel truing are maintenance works that improve the qualit of circulation of the vehicle and consequentl reduce the dnamic component of the ecitation. It is evident that the weight, speed and damping sstems of the vehicle also influence its dnamic behavior, so controlling these parameters also controls the induced vibration levels. The use of continuous welded rails avoids the impact forces that occur when a wheel passes a oint, thus reducing the vibrations. Regarding the use of soft fasteners, resilient materials, ballast mats and floating slabs, the obective is to absorb and attenuate the vibrations within the track, before the are transmitted to the ground. The softer the material, the more efficient is the isolation. However, the resilienc of the materials is limited b restrictions related to the safe operation of the vehicle. For this reason, the materials must be stiff enough so that large deflections do not occur. Floating slab tracks are

27 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings widel used, namel in subwa lines that pass under urban areas. Several studies and measurements confirm that when compared with fied slab tracks, the use of floating slab tracks is successful in reducing the vibrations at frequencies above the resonance of the track sstem (around Hz) (Grootenhuis, 977; Wilson et al., 983; Cui and Chew, ). For the control of vibrations during the propagation stage, the most commonl used measures are soil improvement and wave barriers, such as trenches and wave impeding blocks (WIB). The obective of soil improvement is to make it stiffer and consequentl to reduce its level of vibration. This technique was studied both eperimentall and numericall b Taniguchi and Okada (98) and Al-Hunaidi and Rainer (99a; 99b). Hildebrand (4) confirms that this technique is successful for low frequencies, though the author warns that it produces amplifications in some audible bands and can give rise to perceptible sound in buildings near the track. On the other hand, the obective of wave barriers is to reflect part of the waves that impinge on them. As a result, the part of the wave that is transmitted through the barrier carries less energ and consequentl the vibration levels behind the barrier are reduced. Trenches are efficient in reducing the vibrations originated b surface waves. According to Hubert et al. (), trenches close to the track can provide a considerable reduction in the vibration levels, namel in urban areas, where distances between buildings and tracks are too short to provide sufficient radiation damping. Several authors compared the efficienc of open trenches and in-filled trenches and concluded that the former are more effective than the latter (Beskos et al., 986; Dasgupta et al., 99; Yang and Hung, 997). However, the eecution of open trenches is more complicated than the eecution of in-filled trenches. With respect to the dimension of the trenches, for the case of open trenches onl the depth is relevant, while for the case of in-filled trenches both the depth and width are relevant. Ahmad and Al-Hussaini (99) derived epressions that can be used as guide lines to design trenches. In order for the trench to be effective its dimensions must be in the order of the length of the Raleigh wave, and so trenches are onl efficient in the isolation of high frequenc vibrations (Hung et al., 4). WIBs are intended to simulate rigid frontiers, so that waves are reflected. WIBs can be constructed under the buildings, in order to shield them from vibrations coming from below, or under the railwa track, simulating the presence of a bedrock (Hung and Yang, ). Takemia (3) studied numericall the reduction of vibrations due to the use of WIBs and concluded that the stiffening effects obtained b the installation of WIBs in soft laers could lead to a shift of the response from large and dnamic to small and quasi-static. Finall, at the reception stage, countermeasures are applied in the building. The measures can be applied at the foundation level, isolating the whole building from the soil, or at certain parts of the building. Both approaches were studied b Fiala et al. (7), who considered three different options to mitigate the vibrations induced b the passage of high-speed trains in a room: base-isolation (at the foundation), floating-floor and room-in-room (compartiment measures). As last comment concerning countermeasures, at the moment there is an undergoing European Proect named RIVAS (Railwa Induced Vibration Abatement Solutions - whose obective is to reduce the environmental impact of ground-borne vibrations while safeguarding the commercial competitiveness of the railwa sector. The areas of research of this proect include the mitigation at the source, at the track and at the propagation path. 3

29 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings In chapter 3, the numerical tools to solve soil-structure interaction problems are discussed. The 3D BEM-FEM algorithm is used to simulate the reception stage, in which an incoming wave field reaches a nearb building causing it to respond dnamicall. The.5D BEM-FEM procedure is used to simulate the propagation stage (and part of the reception stage). The 3D formulation is described first because the common reader is more likel to be familiar with this tpe of problems. The less intuitive.5d formulation is presented afterwards. The fundamental solutions used to nurture the BEM in both 3D and.5d domains are obtained with the TLM, and therefore the connection between the BEM and TLM procedures is also established in this chapter. Chapter 4 deals with the response of structures subected to moving loads and moving vehicles. The solution of the train-track interaction problem is presented, which corresponds to the missing stage of the problem, i.e., the reception stage. B now, the reader must be wondering wh the propagation stage is being addressed before the generation stage, which is in opposition to the natural order of events. This switch in the eplanation of events is chosen because the train-track interaction solution procedure requires the knowledge of the.5d transfer functions of the track-ground sstem, which are calculated with the.5d BEM-FEM procedure (i.e., the propagation stage). An eample is shown in which the link between all stages of the problem is carefull eplained. As an application of the tools described in chapters -4, in chapter 5 trenches are studied as a mitigation solution. The chapter is subdivided into 4 sections: in the first section, a parametric stud of trenches is performed; in the second section, the case of stud is described; in the third section, the reduction of vibrations on the soil surface is investigated; and in the last section, the effect of trenches on nearb buildings is addressed. In chapter 6, conclusions and recommendations for further research are presented. 5

30

31 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings. Wave propagation in the soil: fundamental solutions. Introduction Soil is a discontinuous medium, whose porous ma be partiall filled with water. When loaded, it generall presents non-linear and anisotropic behavior. Nevertheless, for the case of traffic induced vibrations, the level of distortion induced in the ground is small and thus its behavior can be idealized as linear and elastic (Galvín, 7). Moreover, according to Biot (Biot, 956a; Biot, 956b), when the frequencies of ecitation are smaller than a characteristic frequenc (inversel proportional to the permeabilit of the poroelastic medium), the relative movements between the liquid phase and the solid phase of the soil can be neglected, and so the soil can be idealized as a single-phased medium. For that reason, in this work the soil is handled as a linear viscoelastic solid, assumption that is also followed in several other works (Galvín, 7; Lombaert, ). Soil connects the track or the road to the nearb buildings and consequentl its dnamic behavior plas an important role in the phenomenon of traffic induced vibrations. In that sense, to know the response of an point in the soil elicited b a dnamic source at some arbitrar location is a powerful tool. These epressions that relate the response of a receiver with a source located anwhere in a solid are called Fundamental Solutions (Kausel, ). These solutions have been subect of stud during the XXth centur and are the essential ke to the Boundar Element Method (BEM), which is described in chapter 3. Closed form epressions for the fundamental solutions of some particular cases, namel the homogeneous isotropic full-space and the homogeneous isotropic half-space, have been derived b some researchers. Concerning the full-space, epressions for both point (3D) and line (D) sources were derived in the mid XIXth centur b Sir William Thomson (Lord Kelvin) and b Sir George Gabriel Stokes for static and dnamic loads, respectivel (Kausel, ). More recentl, Tadeu and Kausel () derived the fundamental solutions for line sources with sinusoidal variation in space and time (i.e.,.5d solutions). Regarding the dnamic response of half-spaces, one of the leading works was performed b Lamb (94). In his work, Lamb fails to obtain closed form epressions for the displacements of the halfspace, but obtains an approimation of the displacements in the far field b considering onl the contribution of the Raleigh waves, whose discover is due to Lord Raleigh two decades earlier (Raleigh, 885). Closed form epressions for the surface displacements of half-spaces (limited to the Poisson s ratio ν =.5 ) were obtained a half centur after the work of Lamb b Pekeris (955) and Chao (96), for vertical and horizontal point loads, respectivel. Moone (974) etended the work of Pekeris to account for arbitrar Poisson s ratios, but for the horizontal displacements the author onl succeeded in obtaining closed form epressions for Poisson s ratios smaller than ν =.63. All the previousl referred to closed form epressions are compiled in the book b Kausel (6). More recentl, Kausel () continued the work of the previous authors and generalized the analtical solutions for arbitrar Poisson s ratios and for an direction of load and displacements, thus concluding the work started b Lamb more than one centur ago. 7

32 Chapter Wave propagation in the soil: fundamental solutions The four works referred to above were developed based on the Cargniard-de Hoop technique (De Hoop, 96), which allows the direct evaluation of the double integral needed to transform the displacements from the wavenumber-frequenc domain to the space-time domain. Interestingl, the direct evaluation of onl one of these integrals is not possible and so closed form epressions of the fundamental solutions of half-spaces eist onl in the spacetime domain. Such is so because when using contour integration to evaluate the improper integrals to transform the displacements from the wavenumber to the space domain (Erigen and Suhubi, 975) or from the frequenc to the time domain (Park and Kausel, 4a), some branch integrals are obtained and these cannot be solved analticall. For the case of laered domains or sources/receivers inside homogeneous half-spaces, no closed form epressions are available and therefore it is needed to resort to numerical tools to determine the corresponding fundamental solutions. The most commonl used tools are based on integral transformation techniques, in which the fields of displacements are transformed to the wavenumber-frequenc domain and consequentl the wave equations are solved in that transformed domain. When necessar, the displacements can subsequentl be transformed back to the space domain and/or time domain through the numerical evaluation of the integrals that result from the inverse transformations. In the transformed domain, the solutions can be found using the transfer matrices derived b Thomson (95) and corrected b Haskell (953), using the stiffness matrices derived b Kausel and Roesset (98), using the method of Potentials (Tadeu et al., ; Tadeu and Antonio, ; Tadeu and António, ) or using the Thin-Laer Method (TLM) (Kausel and Peek, 98). The first three approaches handle the propagation of waves within each laer without an approimation. In opposition, the TLM approach is based on discretizations of the domain in the vertical direction and in approimations of displacements within the laers b means of the interpolation functions. Its advantage over the other three approaches is that it enables the analtical evaluation of at least one inverse transformation. As a drawback, it requires the solution of two eigenvalue problems. In this work, the TLM is adopted for the calculation of the fundamental solutions of laered soils, being the procedure and its.5d formulation presented in this chapter.. Thin-Laer Method in Cartesian coordinates The TLM was introduced in the seventies (Lsmer, 97; Lsmer and Waas, 97; Waas, 97) and since then it has found use in several areas related to wave propagation in laered media and in soil-structure interaction problems. The TLM is a semi-discrete numerical technique used for the analsis of wave motion in laered media, and consists in a finite element discretization in the direction of laering (for the case of soil, the vertical direction) combined with analtical solutions for the remaining directions, along which the material properties are assumed to be constant. A brief historical description of the method can be found in Park (). With respect to wave fields induced b moving loads or vehicles, the TLM is used in the works b Hanazato et al. (99), Jones and Hunt (, ) and Celebi and Schmid (5). In the first three references, the TLM is used in the contet of transmitting boundaries (Kausel, 988) or super-elements (Tassoulas and Kausel, 98). For the presentation of the TLM, the wave equation is first epressed in matri notation and is then discretized in the vertical direction. These steps follow the works from Park () and Barbosa and Kausel (). 8

33 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings One note before starting: cross-anisotropic materials are more general than isotropic materials since the can reproduce different behaviors of the medium when loaded in different directions. For the case of the soil response, this is an important aspect because the laers are formed b vertical sedimentation of particles and therefore the present different mechanical characteristics in the vertical and horizontal directions. The drawback of considering crossanisotropic materials is that the require the quantification of more elastic constants, thus requiring more eperiments in order to obtain the corresponding parameters. The constitutive matri D of a cross-anisotropic material is the positive definite matri defined b λ + G λ λt G > λ λ G λ + t Gt > λt λt Dt λ + G > D = (.) Gt ( λ + G) Dt > λt Gt G where λ and G are the Lamé constants in the isotropic plane (horizontal planes) and λ t, G t, D t are the Lamé constants and the constrained modulus in the transverse direction (vertical direction). When λt = λ, Gt = G and Dt = λ + G, the material reduces to an isotropic one. With the intention of being more general, the following formulation considers crossanisotropic materials. Consider a horizontall homogeneous and verticall stratified cross-anisotropic elastic medium of infinite lateral etent and characterized b the depth dependent mass densit ρ and the depth dependent constitutive matri = { d } ( i, =,...,6 ) D as defined in equation (.). Assume that the medium is subected to an arbitrar dnamic load b placed at some location. With dots denoting partial derivatives with respect to time, the dnamic equilibrium equation at an point can be written compactl in matri format as T ρu = i ɺɺ Lσ b (.) where the displacement vector u, the stress vector σ and the differential operator L are defined as = u u u T u z (.3) = T σ σ σ σ zz σ z σ z σ (.4) z = z z T L (.5) Additionall, consider the stress-strain and strain-displacement relations σ = Dε (.6) 9

34 Chapter Wave propagation in the soil: fundamental solutions ε = Lu (.7) = ε ε ε ε ε ε T ε zz z z (.8) The substitution of equations (.6) and (.7) in equation (.) results in the elastic wave equation (in the 3-D space) uɺɺ L DLu b (.9) T ρ = The differential operator L can be epressed as where the matrices L, L and z L = L + L + L z (.) L z are L = L = L z = (.) Since the domain under stud consists of homogeneous horizontal laers, the material properties are piecewise constant with depth and invariant in the horizontal directions leading T D = α =,, z. Thus, the term L DL in equation (.9) can be epanded to to ( ) α = z T L DL D ( D D ) ( Dz Dz ) being the material matrices D αβ defined b + D + ( D + D ) + D z z z z zz (.) and given in Appendi I. D = L D L,, =,, z (.3) T αβ α β α β Now, consider the internal stresses in horizontal planes, which are calculated b T T T z z zz z z s = σ σ σ = Lσ = L DLu (.4) If one removes an horizontal slice of the medium and treats it as a free bod in space, the dnamic equilibrium dictates the need to balance the internal stresses at the now eposed upper and lower surfaces with the eternal tractions t, i.e., t t s u u = = tl sl (.5) where t u and t l are the eternal tractions applied at the upper and lower boundaries of the removed domain and s u and s l are the internal stresses at the same locations.

35 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings The first step for the formulation of the TLM is to discretize the domain in the vertical direction, i.e., to subdivide the medium into horizontal laers which are thin in the finite element sense, or in other words, which are small in comparison with the epected wavelengths and strain gradients. Thereafter, considering an arbitrar thin-laer as a free bod in space (Figure.), the displacements field inside the laer is approimated b means of interpolation functions, i.e. where = (, ) u = NU (.6) U U is a vector containing the nodal displacements (the nodes represent horizontal surfaces) and = ( z) U T T T T = u... u m, u = u u u z, =,,, m (.7) N N is an interpolation matri of the form [ N N ] N = I... mi (.8) with N being the interpolation functions, which depend on the vertical coordinate z, and with I being a 3 3 identit matri. (The subscript m is the number of nodal surfaces in each thin-laer, and m is the interpolation order. When m >, there eist inner surfaces that are equidistant from each other. For eample, m = 3 corresponds to a quadratic interpolation with one internal nodal surface, as shown in Figure., which depicts one thin-laer as a free bod in space, acted upon and dnamicall equilibrated b appropriate tractions applied onto the nodal surfaces.) z u, t h z um, t m Figure.: Discretization into thin-laers and thin-laer as a free bod in space ( m = 3 ). When substituting the interpolation (.6) into the wave equation (.9) and boundar conditions (.5), it can be verified that these equations are not satisfied eactl because the interpolation is onl an approimation of the actual field. As a result, one finds unbalanced bod forces r and boundar tractions q of the form ρɺɺ (.9) T b u + L DLu = r t s q = = q t s q m m m (.) The discrete wave equation is obtained b appling the method of the weighted residuals and b requiring the virtual work done b the unbalanced forces within the thin-laer and on its bounding surfaces to be zero. This results in the discrete thin-laer equation

36 Chapter Wave propagation in the soil: fundamental solutions U U U U U P = MUɺɺ A A A B B + GU (.) where the vector P contains the consistent eternal tractions at the interfaces of the thin-laer (which result from the eternal tractions t and the bod loads b ). The thin-laer matrices M, A αβ, B α and G are given b h T M N N dz (.) = ρ h T αα αα z α A = N D N d, =, (.3) h T = ( + ) dz A N D D N (.4) h h T T α α z zα α B = N D N dz N D N d z, =, (.5) h = T zz G N D N dz (.6) d in which h is the thickness of the thin-laer and N = N. Appendi I tabulates the above dz matrices for an individual thin-laer consisting of a cross-anisotropic material and considering both a linear and quadratic interpolation, i.e., m =,3, respectivel. After the individual matrices are overlapped in the usual finite element sense (i.e. laer b laer and in the natural top down order of the interfaces), one obtains a narrowl banded set of global sstem matrices and vectors which characterizes the complete stack of thin-laers. The resulting sstem of partial differential equations has the same form as equation (.), but its shape is now block-tridiagonal and has a correspondingl larger number of equations. In the remaining part of the present chapter, equation (.) refers to the complete assembl of thin-laers..3 Displacements in the wavenumber-frequenc domain To solve the sstem of linear partial differential equations (.), the displacements U and tractions P are transformed from the space-time domain to the wavenumber-frequenc domain b means of the triple Fourier transformations ( ω) = ( ) i( ωt k k ) U k, k, U,, t e d d dt (.7) ( ω) = ( ) i( ωt k k ) P k, k, P,, t e d d dt (.8) In the new domain, sstem (.) becomes ( ) ( ) P = k + kk + k + i k + k + ω A A A B B G M U (.9)

37 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings where i =. All matrices in this epression are smmetric, ecept for B and B which are skew-smmetric. Although this sstem could be easil solved for U, it is both possible and convenient to first change the sstem of equations into a full smmetric form b means of a similarit transformation. This is accomplished b multipling ever third row of the sstem (.9) b i and ever third column b i. This operation solel affects the vectors P and U and the matrices B and B, leaving the other matrices unchanged. As a result of this transformation, the sstem of equations is now ( ) p ɶ = k + kk + k + k + k + ω A A A B ɶ B ɶ G M u ɶ (.3) where pɶ and uɶ are obtained from P and U b multipling ever third row b i. Also, and B ɶ are obtained from B and B ɶ B b reversing the sign of ever third column [Note: in comparison with previous studies on the TLM (e.g. Kausel, 986), in this work a reversed sign for the i factor is used for reasons of convenience]. After solving the sstem of equations (.3), U is recovered b multipling ever third row of uɶ b i and the displacements in the space-time domain can be obtained at least formall from the triple inverse Fourier transform U(,, t ) U k, k, e dk dk dω (.3) ( π ) = 3 i( ωt k k ) ( ω) The integrals in equation (.3) can be evaluated numericall. However, b doing so the TLM loses its advantage over the integral transform techniques based on the transfer matrices or on the stiffness matrices, since these approaches, that also require the numerical evaluation of the inverse transformations, originate sstems of equations of smaller size (usuall, onl the interfaces between the laers need to be discretized). In this wa, a different procedure is followed, in which the displacement field is decomposed into a modal basis, similar to what is done in the modal superposition for linear dnamic analses. As a result of this decomposition, the sstem of equations (.3) can be diagonalized and that enables the evaluation in closed form epressions of at least one of the integrals of equation (.3). If the integral to be evaluated is the outer integral, the fundamental solutions are obtained in the wavenumber-time domain (Kausel, 994). If instead one changes the coordinates from Cartesian to clindrical and evaluates the integral in the radial wavenumber, then the fundamental solutions are obtained in the space-frequenc domain (Kausel and Peek, 98; Kausel, 98). Alternativel, if the inner integral of (.3) is evaluated, then the fundamental solutions are obtained in a mied space-wavenumber-frequenc domain (.5D domain), in which the plane-strain is the particular case k = (Barbosa and Kausel, ). In this work, in the propagation stage the geometr is assumed to be invariant and so the.5d fundamental solutions are of interest. On the other hand, in the reception stage the three-dimensionalit of the problem has to be considered and so the clindrical space-frequenc domain solutions must be used. In the ensuing, the sstem of equations (.3) is transformed in order to obtain the displacements U in the wavenumber-frequenc domain through modal superposition. As a first step in that direction, the order of the degrees of freedom is rearranged, grouping first all horizontal-, then all horizontal- and finall all vertical- z degrees of freedom. This rearrangement is suggested solel to reveal the special structure possessed b the matrices in the sstem of equations (.3) and the implications that the referred structures have on the 3

38 Chapter Wave propagation in the soil: fundamental solutions eigenvalue problems that are solved to find the modal basis. In practice, the degrees of freedom are ordered b interface and not b direction, which results in a reduction of the bandwidth of the matrices. Hence, after rearranging the degrees of freedom, the matrices ad vectors in sstem (.3) attain the following structures A O O A O O A = O A O A = O A O O O A z O O A z O A A O O O Bz A = A A O O Bɶ = O O O T O O O B z O O O O O G O O Bɶ = O O B z G = O G O T O B z O O O G z M O O u p M = O M O uɶ = u pɶ = p O O M z iu z ip z (.3) where O is the null matri, G G, M M M z, and Bz B z. In addition, and ecept for the matrices B z and B z, all sub-matrices are smmetric and block-tridiagonal. Having rearranged the order of the degrees of freedom, it is now convenient to define the radial wavenumber k, the propagation angle ϑ, the transformation matri T, its inverse T and the matrices A and C as k = k + k k = k cosϑ k = k sinϑ (.33) Icosϑ Isinϑ O T = sinϑ cosϑ I I O O O ki Icosϑ Isinϑ O = sin cos O O ki T I ϑ I ϑ O (.34) A O O A O A O = T B z O A z G ω M O B z C = O G ω M O z ω O O G M z (.35) in which I is the identit matri and O is a null matri, both with the dimensions compatible with the submatrices in equation (.3). B substituting each variable defined in equations (.33)-(.35) in the following equation, it can be shown that the sstem (.3) is the same as or equivalentl ( k ) ɶ ɶ (.36) - p = T A + C Tu ( k ) Tpɶ = A + C Tuɶ (.37) The modal basis needed to decompose the displacements in a summation corresponds to the solution of the right eigenvalue problem in k and r 4

39 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings ( ) k A + C r = (.38) Due to the structure of matrices A and C, the eigenvalue problem (.38) can be decoupled into two eigenvalue problems, one in the and z directions (generalized Raleigh problem) and the other in the direction (generalized Love problem): k R A O G ω M B z f T + = k Bz A z O G z ω M z Rfz { k ( ω )} A + G M f = L (.39) In this wa, the right eigenvalue problem (.38) has two sets of eigenpairs: one set associated T T T with the eigenvalues k R and right eigenvectors rr = f k Rf z and the other set associated with the eigenvalues k L and right eigenvectors Likewise, the left eigenvalue problem ( ) T r f. T L T = l k A + C = (.4) has two sets of eigenpairs: one set associated with the eigenvalues k R and left eigenvectors T T T lr = k Rf f z and the other set associated with the eigenvalues k L and left eigenvectors l T T L = f. The left and right eigenvectors satisf the orthogonal conditions (Barbosa and Kausel, ) l Ar = δ k, T R Rl l R l Cr = δ k T 3 R Rl l R l Ar = δ, T L Ll l T lrar Ll =, l Cr = δ k T L Ll l L T lrcr Ll = (.4) Having found the solutions of (.38), the rotated displacements Tuɶ are decomposed into a summation of the right eigenvectors, i.e. NR NL Tu ɶ = Γ r + Γ r (.4) R R L L = = where Γ R and Γ L are participation factors et to be determined and N R and N L are the number of degrees of freedom in the Raleigh and Love eigenvalue problems, respectivel. After replacing the identit (.4) in equation (.37) and after pre-multipling it b l latter becomes T Rl, the NR N l T Tp L T ɶ Rl = l Rl k A + C Γ r R R + Γ r L L (.43) = = Due to the orthogonal conditions epressed in (.4), equation (.43) is equivalent to T T 3 lrltpɶ lrltpɶ = k krl k Rl ΓRl Γ Rl = (.44) k k k Rl ( Rl ) 5

40 Chapter Wave propagation in the soil: fundamental solutions If (.37) is pre-multiplied instead b l T Ll, then T l Tpɶ llltpɶ = k k Ll ΓRl Γ Ll = (.45) k The combination of (.44), (.45) and (.4) ields T Ll kll or equivalentl N T T R N l L RTpɶ lltpɶ Tuɶ = r R + L = kr ( k kr ) r = k k (.46) L N T T R N l L RTpɶ l LTpɶ uɶ = T r R + L = kr ( k kr ) T r = k k (.47) L Equation (.47) can be further simplified into cos ϑ T sinϑ cosϑ T k cosϑ T u i z z f f p + f f p f f p k kr k kr kr ( k k R ) NR sinϑ cosϑ T sin ϑ T k sinϑ T u = i + z z f f p f f p f f p k kr k kr kr ( k k + = R ) kr cosϑ k sin T R u ϑ T T z i f i zf p + f zf p + f zfz p z k ( k kr ) k ( k kr ) k k R ϑ ϑ ϑ k k k k sin T sin cos T f fp f fp L L NL sinϑ cosϑ T cos ϑ T f fp + f fp = k kl k kl As a final step, taking into account the equalit (Barbosa and Kausel, ) ( R ) R ( R ) (.48) k T R T k fzf = fzf (.49) k k k k k k equation (.48) can be written in the more convenient form 6

41 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings cos ϑ T sinϑ cosϑ T k cosϑ T u i z z f f p + f f p f f p k kr k kr kr ( k k R ) N R sinϑ cosϑ T sin ϑ T k sinϑ T u = f i fp + f f p f fz p z + k kr k kr kr ( k kr ) = k cosϑ T k sinϑ u T T z i f i zfp + f zf p + f zfz p z kr ( k kr ) kr ( k kr ) k kr ϑ ϑ ϑ k k k k sin T sin cos T f fp f fp L L sin cos cos NL ϑ ϑ T ϑ T f fp + f fp = k kl k kl (.5) In the following, it will be implicitl understood that the eigenvalue problem for Raleigh (shear vertical pressure, SVP) waves will result in eigenvectors f, f z whose components at the th m elevation and while the eigenvectors th mode are written as ( m) ( m) φ, φ z and their eigenvalues are R k = k, f for Love (shear horizontal, SH) waves will have components ( ) written as φ m with eigenvalues k = kl. In the light of equation (.48), it is now convenient to define the set of kernels Kn given in Table.. K K K Table.: Kernels of fundamental solutions sinϑ cosϑ =, K = = k k k k k k k ( ) cos ϑ k sin ϑ 3 = =, K 4 = = k k k k k k k k k k k cosϑ ( ) ( ) k k sinϑ 5 = =, K6 = = k k k k k k k k k k k k ( ) ( ) ( ) ( ) k k k k From equation (.5) and in terms of the kernels in Table., the fundamental displacements ( mn) th th U k, k, ω at the m elevation in direction α due to a unit load applied at the n αβ ( ) elevation in direction β can be epressed as listed in Table.. 7

42 Chapter Wave propagation in the soil: fundamental solutions Table.: Fundamental solutions in the wavenumber-frequenc domain NR NL ( mn) ( m) ( n) ( m) ( n) = 3 φ φ + 4 φ φ U K K NR NL ( mn) ( m) ( n) ( m) ( n) = 4 φ φ + 3 φ φ U K K NR NL ( mn) ( m) ( n) ( m) ( n) ( mn) = φ φ φ φ = U K K G N R ( mn) ( m) ( n) z i 5 z U K φ φ =, N R ( mn) ( m) ( n) z i 6 z U K φ φ =, N R ( mn) ( m) ( n) ( nm) z = i 5 φz φ = z U K U N R ( mn) ( m) ( n) ( nm) z = i 6 φz φ = z = U K U.4 Horizontal derivatives and tractions in the wavenumber domain After the displacements are known, the horizontal derivatives in the wavenumber domain are easil obtained b multipling the displacements b ik, for the case of the -derivative, or b ik, for the case of the -derivative. Similarl, the second derivatives are obtained b multipling the first derivatives b ik or ik. In this wa, for the derivative displacements are multiplied b k, for the derivative multiplied b kk, and for the derivative the displacements are multiplied b the the displacements are k. th As for the consistent nodal tractions acting on one isolated thin-laer (sa the i thin-laer delimited b the global TLM interfaces l and m ), the can be calculated using eq. (.3) once the nodal displacements of that thin-laer are known. In this case, eq. (.3) refers to a single thin-laer and the vectors pɶ and uɶ contain the nodal values at the nn + nodes of the thinlaer ( nn + is used instead of m to avoid confusion between global TLM interface and number of nodal interfaces of the thin-laer): ( i) () ( i) p = ( i) tɶ ( nn+ ) pɶ ɶ tɶ ( i) () ( i) u = ( i) uɶ ( nn+ ) uɶ ɶ For a load in direction β, the modified nodal tractions are uɶ tɶ ( i ) ( i ) ( i ) ( ) ( ) ( ) ( ) i i k = tβ k tβ k t zβ ( k ) and ( ) ( ) ( ) ( ) the modified nodal displacements are uɶ i i i ( ) ( ) ( ) i i k = uβ k uβ k u zβ ( k ) (the word modified is used to take into account the multiplication b the factor i ; the first lower inde represents the direction of the traction or displacement, the second inde represents the direction of the source and the third inde, in parentheses, represents the nodal number). When there is no source acting in the interior of the considered thin-laer, the tractions ( i) () ( i) ( nn ) T T ( i) t αβ ( k ) for k =,..., nn are null and onl the tractions t αβ and t αβ + remain non-zero. These non- 8

43 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings zero values correspond to the tractions that the rest of the domain transmits to the thin-laer through the upper and lower interfaces. B replacing in eq. (.3) the displacements b their modal epansion as given in Table., the fundamental tractions are obtained also in terms of modal superposition. Hence, considering first a force applied at the global interface n and in the direction β =, the nodal tractions at the i th thin-laer are obtained b () ( l ) () ( l ) Γ Φ Γ Φ N R R R N L L L ( i) ( n) ( n) pɶ = A φ φ + + = () ( m) = () ( m) Γ Γ R ΦR L ΦL () ( l ) () ( l ) N R R R N L L L ( n) ( n) φ φ = () ( m) = () ( m) Γ Φ Γ Φ R R L L ( k A + B ) Γ ( k A k B G M) where Φ Γ Φ + + Γ Φ Γ Φ () ( l ) () ( l ) N R R R N L L n L ( n) + + ω φ + φ = () ( m) = () ( m) Γ Φ Γ Φ R R L L ( ) p k K3 k, k ( ) ( p) p Γ R = k K k, k p k K5 k, k ( ) p k K4 k, k ( p) p Γ L = k K ( k, k ) ( ) (.5) (.5) (.53) ( k ) ( k ) ( k ) ( k ) Φ R = φ φ φ z, T T ( k ) ( k ) ( k ) Φ L = φ φ, k = l,..., m (.54) Similarl, for a load in the direction the consistent nodal tractions are calculated b () ( l ) () ( l ) Γ Φ Γ Φ N R R L L R NL ( i) ( n) ( n) pɶ = A φ φ + + = () ( m) = () ( m) Γ Γ R Φ R L Φ L () ( l ) () ( l ) N R R R N L L L ( n) ( n) φ φ = () ( m) = () ( m) Γ Φ Γ Φ R R L L ( k A + B ) Γ ( k A k B G M) Φ Γ Φ + + Γ Φ Γ Φ () ( l ) () ( l ) N R R R N L L ( n) L ( n) + + ω φ + φ = () ( m) = () ( m) Γ Φ Γ Φ R R L L with (.55) 9

44 Chapter Wave propagation in the soil: fundamental solutions ( ) p k K k, k ( ) ( p) p Γ R = k K4 k, k p k K6 k, k ( ) p k K k, k ( ) ( p) p Γ L = k K3 k, k and for a load in the z direction, the tractions are calculated b ( ) (.56) (.57) with z() ( l ) Γ R Φ N R R ( i) ( n) pɶ = A φz + = z() ( m) Γ R ΦR z() ( l) N R R R ( n) φz = z() ( m) Γ R ΦR ( k A B ) Γ Φ + + z() ( l) N R R R ( n) φz = z() ( m) Γ R ΦR ( k A + k B + G ω M) ( ) Γ Φ p k K5 k, k ( ) z( p) p Γ R = i k K6 k, k p k K k, k ( ) (.58) (.59).5 Transformation of fundamental solutions to the.5d domain Having obtained the fundamental solutions eplicitl in terms of the two horizontal wavenumbers k and k, one can now proceed to carr out the inverse Fourier transform in k. In this work onl the case of space harmonic line loads is considered, being different load distributions treated in Barbosa and Kausel ()..5. Fundamental displacements Taking as eample the displacements in the direction induced b source in the same ( ) direction, it is seen in section.3 that the displacements U mn in the wavenumber-frequenc domain are given b (Table.) NR NL ( mn) ( m) ( n) ( m) ( n) = 3 φ φ + 4 φ φ (.6) U K K The corresponding fundamental displacement in the mied space-wavenumber-frequenc domain (.5D) is obtained with 3

45 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings u k U k K K k + + N N ( mn) ( mn) ( m) ( n) ( m) ( n) π π R L ik ik (,, ω) = e d = 3 φ φ + 4 φ φ e d (.6) Since the eigenvectors do not depend on the horizontal wavenumbers but instead on the frequenc ω, equation (.6) is equivalent to N + N + ( ) ( ) ( ) mn m n ( m) ( n) u k K k K k π π R L ik ik (,, ω) = φ φ 3 e d + φ φ 4 e d (.6) Similar conclusions are derived for the other components and thus, in order to obtain the fundamental displacements in the.5d domain, one simpl needs to evaluate the integrals ( ) + () i e k n = d,,...,6 π n = I K k n (.63) and then combine them in the same fashion as done with fundamental displacement u is obtained with ( mn ) NR NL ( mn) () ( m) ( n) () ( m) ( n) = 3 φ φ + 4 φ φ u I I K n in Table.. For eample, the (.64) () The integrals I n can be evaluated in closed form epressions b means of contour integration (Boas, 983). These epressions are summarized in Table.3, where k stands for () either the Raleigh or the Love poles, as appropriate. The evaluation of the integral I 4 with the contour integration technique is described in Appendi II. The evaluation of the remaining integrals follows similar steps. Table.3: Closed form epressions for I ( Im k k < ) () n + () i k i k k = e d e π = i k k I K k ( ) + () i k i k k k = e d e e π = k i I K k i k k k { } + () i k 3 = 3 e d e +i e π = i k + () i k 4 = 4 e d π sign I K k k k k I K k k ( ) + () i k i k k 5 = 5 e d e π = i k I K k + () i k i k k 6 = 6 e d e π = i k k k I K k k = i k k k e i e sign k i k k k k 3

46 Chapter Wave propagation in the soil: fundamental solutions.5. Horizontal derivatives One now proceeds to transform the horizontal derivatives to the.5d domain. Starting with direction, the first -derivative is obtained with u, = ik u while the second -derivative is obtained with u αβ, αβ = k u. Regarding the derivatives in the direction, taking as αβ eample the component and following the procedure eplained in subsection.5. for both first and second derivatives, the following two epressions are obtained (see section.4): N + N + mn m n m n u, k kk3 k kk4 k π π R L ik ik (,, ω) = iφ φ e d + iφ φ e d (.65) N + N + mn m n m n u, k k K3 k k K4 k π π R L ik ik (,, ω) = φ φ e d + φ φ e d (.66) Hence, to obtain the first and second -derivatives, the integrals and ( ) + () i e k n = d,,...,6 π n = I k K k n (.67) ( ) + () i e k n = d,,...,6 π n = I k K k n (.68) must be evaluated and multiplied b i and, respectivel, and then used in Table. in ( mn) ( mn) place of K. For instance, the epressions for u and u are n, i i u k I I, NR NL ( mn) () ( m) ( n) () ( m) ( n), (,, ω) = 3 φ φ 4 φ φ π π αβ (.69) u k I I NR NL ( mn) () ( m) ( n) () ( m) ( n), (,, ω) = 3 φ φ 4 φ φ π π (.7) For the remaining components similar epressions can be obtained. Closed form epressions for integrals I and I are given in Table.4 and Table.5, respectivel. () n () n After the first -derivatives are known, the cross derivatives the multiplication u, = ik u,. αβ αβ.5.3 Fundamental tractions u αβ, can be obtained through Such as described in the previous subsections, the fundamental tractions t αβ in horizontal planes result from the inverse Fourier transform in k of equations (.5), (.55) and (.58). Since the eigenvectors do not depend on the wavenumber k, the inverse Fourier transform is β ( p) accomplished b replacing in the equations (.5), (.55) and (.58) the matrices Γ and Γ b the matrices β ( p) L Λ and β ( p) R Λ β ( p) L, whose general epressions are R Λ = Γ + β ( p) β ( p) i k R R e dk π Λ = Γ (.7) + β ( p) β ( p) i k L L e dk π 3

47 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Table.4: Closed form epressions for I ( Im k k < ) () n ( ) + () i k sgn i k k I = k K e dk e π = i k I k K k k k k + () ik i k k k = e d e +i e π = ik ( ) { ( ) i k k k k } ( ) i k k k { } sgn I k K k k k k i I + () i k 3 = 3 e d e + e π = sgn + () ik 4 = k K4 e d e e π k = k k i k + () i k i k k 5 = 5 e d e π = i k I k K k sgn ( ) + () i k i k k 6 = 6 e d e π = i k I k K k k k k Table.5: Closed form epressions for I ( Im k k < ) () n Obtaining the matrices + k () i k k i k k I = k K e dk e π = i k sgn I k K k k k k ( ) { ( ) i k k k i } 3 i k 3 k k {( ) } i k 3 k k { } + () i k = e d e e π = + k I k K k k k k I + () ik 3 = 3 e d e i e π = i k + () i k 4 = k π K4 e dk = k e i e k k + k i k ( )( k k ) + () i k i k k 5 = 5 e d e π = i k I k K k + () i k i k k 6 = 6 e d e π = i k I k K k Λ and β ( p) R ( p ) n β ( p) L sgn k k k Λ is straightforward, as the results from the combination of the integrals I ( p =,, ). For instance, the matri Λ is I ( p) 5 ( k ), ( k ) z( p) ( p) Λ R = i I6, ( p) I, ( k ) z ( p ) R (.7) 33

48 Chapter Wave propagation in the soil: fundamental solutions.5.4 Vertical derivatives and internal stresses The z -derivatives can be obtained through the combination of the nodal displacements weighted b the derivatives of the associated shape functions. However, following that approach, the derivatives at the top and bottom interfaces of the thin-laers are not consistent with the tractions calculated in subsection.5.3, and so their accurac is inferior. To compensate for the lack of precision of the vertical derivative, Kausel (4) proposed an alternative strateg for their calculation. The procedure is based on the definition of secondar interpolation functions that are consistent with the tractions at the top and bottom interfaces of the thin-laer. In this work, that procedure is used to define the vertical derivatives and, subsequentl, the internal stresses at the internal nodal interfaces. Consider the th i thin-laer (of epansion nn ), from which the displacements nn + nodal interfaces, the horizontal derivatives ( ( i u ) αβ ( ), and ( ( i ) ( ) ( i) u αβ ( ), u αβ at the ( i) ( ) ), and the nodal tractions t αβ ) at the top and bottom interfaces are known for all directions α =,, z and for a source in the direction β. The tractions at the upper surface and the internal stresses at the same horizontal plane are related b T ( i) top top top t () = σ zβ σ zβ σ zzβ (.73) while the tractions at the lower surface and the internal stresses at the corresponding plane are related b T ( i) bottom bottom bottom t ( nn+ ) = σ zβ σ zβ σ zzβ (.74) In their turn, the internal stresses and the derivatives of displacements are related b σ σ (, u, ) (, u, ) ( ) = G u + zβ t β z zβ σ = G u + zβ t β z zβ = λ u + u + D u zzβ t β, β, t zβ, z The previous equation can be solved for the vertical derivatives, ielding u u u β, z β, z zβ, z = = ( σ zβ Guzβ, ) G ( σ zβ Guzβ, ) σ = t G t ( u, u, ) λ + zzβ t β β D t (.75) (.76) For each response direction α and for each source direction β, the values of the displacements at the nn + nodal interfaces and of the vertical derivatives at the upper and lower interfaces are now known. In this wa, it is possible to emplo the nn + 3 known quantities and use Hermitian interpolation to define a polnomial of degree nn + that approimates the vertical variation of the displacements. If the thin-laer is linear ( nn = ) and its thickness is h, then 34

49 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings ( ) 3 u z = A + B z + C z + D z (.77) αβ αβ αβ αβ αβ ( ) uαβ, z z = Bαβ + Cαβ z + 3Dαβ z (.78) ( ) ( ) i i αβ () αβ αβ () 3 ( i) ( i) αβ h h h αβ () αβ () = = ( ) ( ) i i αβ αβ (), z 3 h 3 h h h αβ (), z ( i) 3 3 ( i) αβ h 3h αβ (), z h h h h αβ (), z A u u B u u C u u D u u If instead the thin-laer is quadratic ( nn = ), then ( ) 3 4 (.79) u z = A + B z + C z + D z + E z (.8) αβ αβ αβ αβ αβ αβ ( ) 3 uαβ, z z = Bαβ + Cαβ z + 3Dαβ z + 4Eαβ z (.8) ( i) uαβ (3) 3 4 ( i) uαβ () 3 4 ( i) uαβ () ( i) uαβ (3), z 3 ( i) h h h uαβ (), z A αβ B αβ h ( h ) ( h ) ( h ) C αβ = h h h h = Dαβ E αβ 3 4 u ( i) αβ (3) ( i) uαβ () ( i) 5 h 6 h h h 4 h uαβ () ( i) 4 h 3 h 8 h 3 h 5 h uαβ (3), z ( i) 8 h 6 h 8 h h 4 h u αβ (), z (.8) After finding the left-hand side of equations (.79) and (.8), the vertical derivatives of the displacements at the nodal interfaces can be determined using eq. (.78) or eq. (.8). As for the vertical derivatives uαβ,z ( z) of points in the interior of the thin-laer, though the can be calculated using the same two equations, one chooses to use instead the original interpolation functions to determine these variables, i.e., nn+ ( i) αβ, z ( ) = ( ) αβ ( ), z (.83) = u z N z u With all the first derivatives known, eq. (.6) can be used to calculate the internal stresses. Regarding the calculation of the second derivatives involving the z direction ( z, z and z ), u αβ, z is obtained b multipling u αβ,z b ik, while u αβ,z is obtained b differentiating equation (.76) with respect to, i.e. 35

50 Chapter Wave propagation in the soil: fundamental solutions u u u β, z β, z zβ, z = = ( σ zβ, Guzβ, ) G ( σ zβ, Guzβ, ) σ = G t t ( u u ) λ + zzβ, t β, β, D t (.84) The traction derivatives σ α zβ, are calculated as eplained in subsection.5.3, with the eception that matrices Λ and β ( p) R Λ must be replaced with matrices β ( p) L respectivel. For the calculation of the matrices ( ) Λ and β (3) R Λ the integrals β (3) L + (3) 3 i e k n = d,,...,6 π n = β ( p+ ) R iλ and β ( p+ ) L iλ, (3) I n of the form I k K k n (.85) are needed. Closed form epressions for these integrals are given in Table.6. Table.6: Closed form epressions for I ( Im k k < ) (3) n + (3) k 3 i i k k k k I = k K e dk sgn( ) e π = i k I k K k k k k 3 i k 3 k k {( ) } + (3) 3 i k = e d e i e π = ik i k k k { } i k k k { ( ) } ( ) ( ) sgn I k K k k k k I + (3) 3 i k 4 3 = 3 e d e e π = i k (3) 4 = π sgn ( ) + 3 i k 4 k K4 e dk = k e e k k + k i k ( k k ) 3 + (3) 3 i k i k k 5 = 5 e d e π = i k I k K k ( ) ( ) k k k + (3) 3 ik i k k 6 = 6 e d sgn e π = i k I k K k [Note about Table.5 and Table.6: ( ) 3 ( ) ( k ) 3 k k k k k, with The remaining second derivative, (Achenbach, 973) which after being solved for k Im k k k must be calculated as < ; a direct use of the epression might possibl assign the wrong sign to the result.] u αβ,zz, can be calculated resorting to the Navier equation ( λ ) ρω G u + G + u + F = u (.86) i,, i i i u αβ ields (it is assumed that F =, i.e., no internal sources),zz i 36

51 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings u u u β, zz β, zz zβ, zz = = = ( λ )(,,, ) (,, ) ρω uβ + G uβ + uβ + uzβ z G uβ + uβ ( λ )(,,, ) (,, ) ρω uβ + G uβ + uβ + uzβ z G uβ + uβ ( λ )(,, ) (,, ) ρω uzβ + G uβ z + uβ z G uzβ + uzβ G λ + G G (.87).5.5 Validation To validate the equations derived in this and previous sections, the response of a homogeneous full-space obtained with the TLM is compared with the analtical solution derived b Tadeu and Kausel (). The isotropic full-space has mass densit ρ =, shear modulus G = and Poisson s ratio ν =.5, and is subected to a time harmonic line load of b,, z, t = δ δ z ep iωt ik, being the ecitation frequenc f = Hz the form ( ) ( ) ( ) ( ) ( ω = π ). The thin-laer model used to simulate the full-space consists of an elastic laer with thickness λ s divided into thin-laers of quadratic epansion a discussion on discretization errors can be found in Park and Kausel (4b) and supplemented at its upper and lower horizons with paraial boundaries (Seale and Kausel, 989), which are used to simulate the infinite domain in section.6 of this chapter, a more efficient strateg to model infinite domains is presented. The parameters λ s is the wavelength of the shear wave, which is defined b λ C f G ρ s s = Cs = (.88) The load is applied at the middle surface of the elastic laer. To avoid strong oscillations in the response, a small amount of damping is considered ( ξp = ξs =.5 ), which renders the wave velocities comple. In the first validation scenario, the displacements induced b a vertical load are computed as function of the wavenumber k at the horizontal plane z = and horizontal distances =.λ s, = λs and = 5λ s. Figure. depicts the comparison between the vertical displacements obtained with the TLM and those calculated with the analtical solution. As can be observed in Figure.b) and c), the match between the eact solution and the TLM solution is ver good. Nonetheless, ver close to the load (Figure.a), one can observe a rather small difference in the real part, which is due to discretization effects. This is because the thickness of the thin-laers is onl.λ s while the receiver is placed at one tenth of that distance from the source. Still, given the ecellent qualit of the comparison even at that short range, the results obtained clearl demonstrate the robustness of the TLM solution. Observe that at large distances the response decas ver fast with the wavenumber k beond the threshold ks = ω / Cs (the branch point), while below that value the response is highl wav. Hence, when computing the inverse transform from k space into space for remote points, one can truncate the integrals at the branch point, but then again because of the rapid oscillations one must consider a sufficientl dense number of points below that threshold. Conversel, for receivers at close range, the response functions are less wav, but the also 37

52 Chapter Wave propagation in the soil: fundamental solutions deca more slowl with k. Hence, their Fourier inversion must include points beond the branch point even if one can get awa with coarser spacing. a). b) uzz.5 uzz k c) k. uzz Figure.: Vertical displacements at a) =.λ s, b) = λs, and c) = 5λ s. Solid lines = TLM solution (real part blue; imaginar part red). Circles = analtical solution In the second validation eample, both displacements and derivatives are computed as function of the horizontal distance and considering all three directions for the load and for the response. The longitudinal wavenumber is k =.4 and the receivers are placed at the depth z = 4λ s and up to the horizontal distance ma = 4λ s. The stresses are not compared because the can be calculated based on the derivatives of displacements, and if the latter are correct, then the former are also correct. Likewise, the -derivatives are not represented because the result from the multiplication of other response fields (displacement or derivative) b ik. Figures.3 to.8 depict the comparison between the theoretical solution and the responses obtained with the TLM. All figures suggest that the two approaches ield results that are virtuall the same, thus confirming that the TLM is indeed capable of reproducing the wave motion with high accurac. This validates the epressions derived in the previous sections of this work. k 38

53 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings.5 u -3 u. u z u.5 u 5-3 u z u z 5-3 u z. u zz Figure.3: Displacements u αβ. Solid lines = TLM solution (real part blue; imaginar part red). Circles = analtical solution. u, 5-3 u,. u z, u,. u, -3 u z, u z, -3 u z,. u zz, Figure.4: Displacement derivatives u αβ,. Solid lines = TLM solution (real part blue; imaginar part red). Circles = analtical solution 39

54 Chapter Wave propagation in the soil: fundamental solutions.5 u,z -3 u,z. u z,z u,z.5 u,z. u z,z u z,z. u z,z. u zz,z Figure.5: Displacement derivatives u αβ,z. Solid lines = TLM solution (real part blue; imaginar part red). Circles = analtical solution.5 u,.5 u,.5 u z, u, u,.5 u z, u z,.5 u z,.5 u zz, Figure.6: Displacement derivatives u αβ,. Solid lines = TLM solution (real part blue; imaginar part red). Circles = analtical solution 4

55 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings.5 u,z.5 u,z.5 u z,z u,z u,z.5 u z,z u z,z.5 u z,z.5 u zz,z Figure.7: Displacement derivatives u αβ,z. Solid lines = TLM solution (real part blue; imaginar part red). Circles = analtical solution u,zz.5 u,zz.5 u z,zz u,zz u,zz. u z,zz u z,zz. u z,zz.5 u zz,zz Figure.8: Displacement derivatives u αβ,zz. Solid lines = TLM solution (real part blue; imaginar part red). Circles = analtical solution 4

56 Chapter Wave propagation in the soil: fundamental solutions.6 Modeling unbounded domains The TLM is a semi-discrete numerical technique for the analsis of wave motion in laered media and relies on a finite element discretization in the direction of laering. Due to the discrete character of the TLM, b itself the analses are limited to bounded domains. Nevertheless, in the mid eighties, the Paraial Boundaries (PB) were coupled to the TLM to allow the simulation of infinite domains (Seale and Kausel, 989). Ver briefl, the PB for the TLM can be obtained b epanding the stiffness matrices (Kausel and Roesset, 98) in a Talor series in the wavenumber and retaining onl the first three terms. Hence, this technique works ver well for small wavenumbers and not so well for higher wavenumbers. In other words, waves propagating verticall or almost verticall are mostl absorbed when the reach the paraial boundar while waves propagating with a considerable horizontal component are mostl reflected. For this reason, the PBs are usuall augmented with buffer (elastic) laers that are thick enough so that the component of waves that is reflected at the PB returns to the region of interest at a horizontal coordinate larger than the maimum distance of interest. Eplicit epressions for the PB matrices can be found in the thesis of Park (Park,, p. 84 for SH waves and p. 89 for the SVP waves) while comments and considerations about their stabilit can be found in the works b Kausel (988, 99). More recentl, Barbosa et al. () successfull coupled the Perfectl Matched Laer (PML) to the TLM, resulting in a more efficient technique to model unbounded domains. The PML is a numerical technique used for purposes similar to those of absorbing or transmitting boundaries, namel to suppress undesirable echoes and reflections of waves in infinite media modeled with discrete finite sstems. The technique was introduced in the nineties b Berenger (994), who developed and coupled it to the time-domain finite differences method for the analsis of electromagnetic fields. Initiall, the formulation of the PML followed the split-field approach, in which the sstems of equations are solved both for displacements and stresses, but later new formulations for the PML were derived, namel b considering the PML as an equivalent anisotropic material (Gedne, 996; Teieira and Chew, 998) or b stretching the coordinates to the comple space (Chew and Weedon, 994; Hugonin and Lalanne, 5). The PML has also been applied to elastodnamic problems, both in time and frequenc domains (Basu and Chopra, 3; Basu and Chopra, 4; Basu, 9; Harari and Albocher, 6). A good literature review on the subect can be found in (Kucukcoban and Kallivokas, ). In the work (Barbosa et al., ), the PML is formulated based on the coordinate stretching approach. This approach consists in stretching the real space to a comple space b means of position-dependent comple-valued scaling functions, which begin with unit values at the interface or horizon delimiting the elastic region and then attain progressivel larger comple values with the distance from this horizon, which causes the waves within the PML to attenuate eponentiall (Johnson, 8). Since there is no impedance contrast at the PML boundar, no reflections take place no matter what the angle of propagation of the waves entering the PML is. In the following subsections, it is shown how the coordinate stretching allows the simulation of infinite domains and then the PML is coupled to the TLM..6. Coordinate stretching Consider a plane wave travelling at an angle θ with the vertical direction (z) in a medium whose wave speed is C. This wave has the form 4

57 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings ( ) ω ω i ω sinθ cosθ C C u, z, t = Ae t z (.89) No restriction is made regarding the vertical coordinate, and thus admit that this coordinate ma assume the comple values, and denote it b z. Equation (.89) is still valid but now z must be replaced b z. Assume also that the imaginar part of z depends on the depth (real part) and define z as ( ) z = z iψ z (.9) where Ψ ( z) is a function et to be determined. After replacing equation (.9) into (.89), the latter becomes ( ) ω ω i t sin z cos ω ω θ θ Ψ( z) cosθ C C C u, z, t = Ae e (.9) The aim is to attenuate the waves that enter a finite PML region defined b H > z > (Figure.9). For waves that propagate in the positive z direction ( cosθ pos > ), for the amplitude of the waves to deca as z increases, the eponential term ep( ω cosθ Ψ ( z) C) must decrease and consequentl Ψ ( z) must increase with z. Similarl, for waves that propagate in the negative z direction ( cosθ < ), for their amplitude to deca in the direction of propagation, neg Ψ ( z) must obe the same rule. A possible choice for ( z) rule is z ( z) ψ ( ζ ) where ψ ( z) is an alwas positive stretching function. Ψ that respects the established Ψ = dζ (.9) θ pos θ = π θ neg pos H z Figure.9: Propagation of a wave inside the PML region In theor, ψ ( z) might assume an shape as long as ( ) ( H ) Ψ < Ψ (Bienstman and Baets, ). However, once the domain is discretized so that the differential equations can be ψ z and therefore it is convenient that solved, spurious reflections occur due to changes in ( ) this function changes smoothl with z. A commonl used stretching function is (Basu and Chopra, 3) ψ ω z = ω H ( z) m (.93) 43

58 Chapter Wave propagation in the soil: fundamental solutions where ω controls the level of absorption of the wave and m > defines the rate of variation of the stretching function within the PML (Note: in section. the parameter m has a different meaning). This implies which can be written compactl as where ω Ω = ω m+ ωh z Ψ ( z) = ( m + ) ω ( m ) + H m ( z) Hζ + The stretched vertical coordinate then simplifies to (.94) Ψ = Ω (.95) m ( i ) which implies a total comple depth H H [ i ] z ζ = (.96) H z = z Ω ζ (.97) = Ω. Consider now a plane wave traveling at an angle θ with respect to the vertical direction z (Figure.9). In the stretched space, this wave can be epressed as ( ) u, z, t = A e = A e ω ω i ωt sinθ z cosθ C C ω ω i t sin z cos ω ω θ θ cosθ Ψ( z) C C C e (.98) When the wave reaches the bottom of the PML and is reflected, its amplitude is A ep ω cosθ Ψ H C = A ep ω cosθ Ω H C. When the wave reaches again the free ( ( ) ) ( ) surface of the PML, its amplitude is A ep( ω cosθ H C) of the wave is Ω. Hence, the total roundtrip deca H ω Ω cosθ C = e (.99) Accounting for the relation between the wave velocit, frequenc and wavelength ( λ = π C ω ) and defining the ratio η = H λ, then equation (.99) is equivalent to 4 cos e π Ωη θ = (.) Clearl, as long as the thickness of the laer is made proportional to the wavelength (i.e. η is chosen as a constant), the effectiveness of the PML for a given angle of incidence depends solel on the dimensionless parameter Ω. On the other hand, a ra entering the PML at with an inclination θ returns to the surface at a distance r = = H tanθ from the point of entrance, i.e. r η tanθ λ = (.) Equations (.) and (.) indicate that the higher the horizontal range of interest is, the higher the value needed for Ω, η, or both. 44

59 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings The suitabilit of PMLs for the simulation of a half-space is eamined net. For that purpose, consider an elastic half-space with shear modulus G = and shear wave velocit C s = ecited b an in-plane (SH) line source acting at the depth z s with frequenc ω = π. For an elastic half-space, the eact solution for the displacement observed at a receiver at range and depth z r is (Kausel, 6; p. 69) ( u = H ) ( k r ) H k r ig + ( ) ( ) s s k ω C s π λ = =, r ( z z ) = + (.), s r B contrast, the PML admits an eact solution based on an epansion in terms of the normal modes of the stratum, as given in Kausel (6; p. 3). Accounting for the fact that the vertical dimensions have been stretched, the solution is k s r ig = i k e u (, ω) = φ ( z ) φ ( z ) (.3) k H ( ) ω π = Cs H φ ( z ) ( ) π z = cos H Im k < (.4) (.5) where z s and z r are the stretched depths of the source and of the receiver, respectivel, and H is the total stretched depth. With λ s = Cs f =, and choosing for the PML the parameters η = H / λs =, a maimum range r ma = 5λ s = 5, a roundtrip deca of two orders of magnitude, i.e., =, and m =, the following can be inferred 5 tanθ = = = 5 ma ma ηλ s θ ma = 78.7 ln ln ma Ω = = = π η cosθ 4π.96 ( ) [ ] [ ] H = H i Ψ H = H iω = i =.5.87i cosθ ma =.96 This data is used to construct a PML that is subected to a SH line source at its surface z s = and then equations (.3) to (.5) are used to compute the displacements for receivers at variable positions on the surface z r =. Figure. compares the displacements thus computed against the displacements on the surface of the half-space predicted b equation (.). As can be seen, both the real and imaginar parts of the displacements agree perfectl until a range of about = 8λ s is eceeded. On the other hand, Figure. shows the ratio between the absolute values of the displacements obtained b the two approaches. This figure confirms that until the distance ma = 5λ s, the ratio of amplitudes is virtuall one, with an error less than %. Then again, at the distance = 8λ s the error has grown to approimatel 5%, and thereafter it increases substantiall. This shows that with the chosen parameters and as seen from its surface, the PML behaves essentiall as an elastic half-space, et is a perfect absorber of waves onl up to some maimum distance which depends on the parameters chosen. 45

60 Chapter Wave propagation in the soil: fundamental solutions.5 Re(u ) - PML Im(u ) - PML Re(u ) - Half-space Im(u ) - Half-space u λ Figure.: Displacements of the PML vs displacements of half-space s u u PML Half-space λ Figure.: Ratio of displacements at surface of the PML and those of the half-space.6. PMLs for the TLM In the contet of elastodnamics, two different waves with different wave speeds must be accounted for: the shear wave with speed C and the dilatational wave with speed C. If the s waves are considered separatel, the concept of the PML as eplained in the previous subsection is still valid, but now each wave has its own rate of attenuation inside the PML. In the following, a PML is constructed b means of a stack of thin-laers in the contet of the TLM. As shown in a previous work (Kausel and Barbosa, ), a ver simple wa to construct a PML with finite elements is to directl stretch the linear dimensions of the elements in accordance with their horizontal and vertical position within the PML. In the s p 46

61 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings TLM, this recipe translates into replacing the thicknesses of the thin-laers composing a PML b their comple counterparts, which depend on the location (i.e. depth) of the thin-laers within the PML. Thus, if it is assumed that the PML is divided into N equal thin-laers, then th the stretched thickness h of the thin-laer is m+ m+ h = z z = H iω N N N (.6) where N, with increasing downwards. On the other hand, in the TLM, each of the thin-laers is characterized b elementar laer matrices A, G, M, B see equations α α α z (.3), (.35) and (.39) two of which are proportional to the sub-laer thickness, while another one is inversel proportional to that thickness, i.e. A [ ] ; G = h [ ] ; [ ] α = h α M = h α (.7) Thus, in order to obtain the laer matrices for the PML, it suffices to substitute h in lieu of h. Thereafter, the laer matrices are overlapped as usual, which leads to the block-tridiagonal sstem matrices and the eigenvalue problems (.39). The parameters for the PML ( m, η and Ω ) can be selected following the strateg eplained in the previous subsection. However, since with the TLM the domain is discretized, the variation of the comple thicknesses of the sub-laers originates spurious reflections that influence the results. On one hand, these reflections become larger as Ω increases, and so thinner discretizations are needed for larger values of Ω. On the other hand, when Ω is not large enough, the waves are not attenuated sufficientl as the travel through the PML. This can be compensated with thicker PMLs, which also causes the increase of the number of degrees of freedom. In this wa, it is important to find a good compromise between the different parameters of the PML so that the minimize the number of degrees of freedom as much as possible. The following values are suggested as optimal parameters for the PMLs (Barbosa et al., ) H m = η = = 3 ma / λs n N = η Ω = 4η (.8) λ s In this equation, N is the number of thin-laers in the PML, n is the epansion order used in the discretization (the same as m in section.) and ma is the maimum horizontal distance of interest. Net, the PML technique is compared with Paraial Boundaries, which have alread been used to solve some eamples in the previous sections. For that purpose, consider a full-space ( ρ =, G =, ν =.5 and ξ = ξ = ) and submit it to a vertical D line load ( k = ) with s p frequenc ω = π. For the PML approach the full-space is modeled using two PMLs (one upper and one lower), while for the PB approach the full-space is modeled considering two buffer laers plus paraial boundaries, having the buffer laers the same depths and discretizations as the PMLs. The parameters chosen for the PML are m =, η =, Ω = 8 and N =, having the thin-laers quadratic epansion ( n = ). Figure. plots the relative errors of the two approaches, defined as the percent deviation of the absolute values of the vertical displacements at the depth of the source when compared to those of the eact solution. Clearl, the PML approach is substantiall more accurate than the PB approach, 47

62 Chapter Wave propagation in the soil: fundamental solutions with errors below % up to distances = λ s, while the errors in the second approach eceed % even at a distance lower than = 4λ s, and it becomes intolerable at larger distances. It is then clear that the PML approach outperforms the PB approach b a vast margin. The eample shown in this subsection considers a D model. For.5D models, the PML technique has been used to solve the eamples of the previous sections, having the PMLs onl one tenth of the thickness of the buffer laers. The results obtained are similar to the results shown in Figures.-.8, which confirms that the PML is also applicable in the.5d case..8 PML PB Relative Error Distance to the source Figure.: PML vs. Paraial boundaries.6.3 Eample of a laered domain Up to this point all eamples have been solved with the intent to validate the procedure and therefore onl models for homogeneous full-spaces are considered, since onl for these cases the fundamental solutions are available in closed form epressions. In this subsection, the eample of a laered half-space is considered. Laered half-spaces differ from homogeneous full-spaces due to the presence of surface waves, which result from the interaction between the bod waves at the surface between two different materials. For the case of elastic half-spaces, onl one surface wave eists, the Raleigh wave, and that wave is not dispersive, i.e., its velocit of propagation does not depend on the frequenc of ecitation. For the case of laered domains, depending on the frequenc considered, none or more than one surface waves ma eist. Because the are frequenc dependent, these waves are called dispersive waves. Theoretical works on propagation of waves in laered domains can be found in (Achenbach, 973; Erigen and Suhubi, 975; Ewing et al., 957). As for numerical works, Luco and Apsel (983) and Apsel and Luco (983) treat fied loads while De Barros and Luco (994) treat moving loads. The laered sstem considered in this subsection consists of a soft laer resting on a stiffer half-space. The material properties of the laer and half-space are Laer: C =.5, ρ =., ν =.3, ξ = ξ =.5, H = s s p 48

63 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Half-space: C =., ρ =., ν =.5, ξ = ξ =.5 s s p The laer is modeled b means of thin-laers of quadratic epansion, while the half-space is modeled using a PML whose parameters are η =, Ω = 6.67 and m =. The PML is divided into thin-laers of quadratic epansion. Eigenvalues of the laered sstem First, the solutions of the Raleigh eigenvalue problem (.39) for the frequencies f =. Hz, f =. Hz, f = Hz and f = Hz are shown. Figure.3 plots the eigenvalues obtained with the TLM and the eigenvalues of the eact formulation (stiffness matrices) obtained via search techniques. f =. Hz f =. Hz - - Im(k) -4-6 Im(k) Re(k) f = Hz Re(k) f = Hz Im(k) - - Im(k) Re(k) Re(k) Figure.3: Eigenvalues of the laered sstem obtain with the TLM (blue dots) and via search techniques (red circles) Even though the agreement between the eigenvalues obtained with the two distinct approaches does not look ver good, when the displacements are calculated b means of modal combination, the obtained results are quite good (as seen net). In Figure.3, it can be observed that the number of propagating modes (to which correspond real, or nearl real, eigenvalues) increases with the frequenc. Also, one can distinguish the eistence of two branches in the TLM modes that do not eist in the eact formulation. These modes are named Berenger modes and the are mathematical artifacts that are a consequence of the stretching of coordinates. The have been detected in some works related to electromagnetic waves (Bienstman and Baets, ; Derudder et al., ; Rogier and De Zutter, ; 49

64 Chapter Wave propagation in the soil: fundamental solutions Bienstman et al., ), but in these works, onl one branch eists. In the present work, the two branches are ustified b the eistence of two different bod waves. The branches start at the wavenumbers ks = ω Cs and k p = ω C p and the number of modes contained in each branch equals the number of degrees of freedom associated with the PML laer. Vertical displacements due to a vertical line load ( k = ) The modes obtained in subsection.6.3. are now combined in order to obtain the displacements in the interface between the laer and the half-space due to loads at the same elevation. Figures.4 and.5 plot the displacements for the frequencies f =. Hz and f = Hz obtained both with the TLM and with numerical integration on the wavenumbers of the displacements given b the stiffness matrices Re(u zz ) - TLM Im(u zz ) - TLM Re(u zz ) - Stiff. Mat. Im(u zz ) - Stiff. Mat. uzz Distance to the source Figure.4: Displacements of the laered half-space: TLM vs Stiffness matrices ( f =. Hz )..5. Re(u zz ) - TLM Im(u zz ) - TLM Re(u zz ) - Stiff. Mat. Im(u zz ) - Stiff. Mat..5 uzz Distance to the source Figure.5: Displacements of the laered half-space: TLM vs Stiffness matrices ( f = Hz ) 5

65 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings The comparison of the results obtained with the two approaches shows that ecept for ver small shifts, the displacements agree ver well. Hence, it is concluded that the PML is accurate in reproducing the behavior of infinitel deep stratified domains..7 Solution of the eigenvalue problems One of the more time consuming and comple steps in the TLM process is the solution of the eigenvalue problems (.39). Even though commercial software, like Matlab, contain functions capable of solving both the linear and the quadratic eigenvalue problems with comple matrices (for eample, eig and poleig), these functions do not take advantage of the banded structure of the matrices and therefore become inefficient for the solution of eigenproblems with dimension of ust a few hundred. Also, these routines, which are based on the QZ algorithm (Kressner, 5) and thus based on iterative rotations of the modal basis until convergence, ma ield eigenvectors that do not respect entirel the orthogonal conditions (.4) due to accumulation of round-of errors, hence introducing errors in the remaining steps of the modal combinations: though rare, this case has been observed when a large number of thin-laers and PMLs are used. Alternative approaches to the QZ algorithm are the methods based on the Power Method. This famil of methods is based on successive matri-vector multiplication until the direction of the resulting vector has converged to the eigen direction. Several variations eist, namel the Power Method itself, the Inverse Iteration method, in which instead of the matri-vector multiplication, a sstem of equations is solved, and the Inverse Iteration with Raleigh Shift method (Shit and Invert method), in which the eigenvalues are shifted in order to speed up convergence (Saad, 99). In this famil of methods, the eigenvectors are found one b one, being then deflated from the eigen base (remove them from the modal spectrum) in order to avoid convergence to the same pair. Proection methods can also be emploed: the most common ones are the Lanczos method (or Arnoldi method, for non-hermitian matrices) and the Locall Optimal Preconditionated Conugated Gradients method. These proection methods iterate with a group of vectors instead of a single vector and are used to etract a small spectrum range, not the full spectrum of eigenpairs (Saad, 99). Because the complete spectrum of the sstem of matrices is needed, the Inverse Iteration with Raleigh Shift is chosen for the solution of the eigenvalue problems (.39). In the net subsections, it is eplained how to appl this method taking into account the special structure of the TLM matrices..7. Eigenvalue problem for SH waves The linear eigenvalue problem to be solved has the form ( ) k A + C v = (.9) where matrices A and C are comple, smmetric and banded. The eigenvalues k and eigenvector v are also comple. The bandwidth of the matrices is if the thin-laers are of linear epansion and is 3 if the thin-laers are of quadratic epansion. The eigenvalue k, v problem admits N solutions, being N the dimension of the matrices, and if the pair ( ) is a solution of (.9), then the pair ( k, ) v is also a solution of (.9). In this wa, onl 5

66 Chapter Wave propagation in the soil: fundamental solutions half of the spectrum of the sstem needs to be computed. Taking into account the orthogonal conditions epressed in the second row of eq. (.4), the Inverse Iteration with Raleigh Shift method applied to the linear pencil (.9) results in the algorithm described in Table.7. Table.7: Algorithm for the solution of the linear eigenvalue problem v = rand( N ) Initial guess v = v v Av Normalize the initial guess against previousl T i w = Av Auiliar vector found eigenvectors ( i =,,... ) k = T T v Cv v w Raleigh quotient Iterate (until Raleigh quotient has converged) E = A + C From shifted matri k v = E w New approimation of eigenvector v = v v Av Normalize against previousl found T i eigenvectors (if k ki, i =,,... ) k T T = k + v w v Av Update the Raleigh quotient w = Av Auiliar vector T δ = v w v = v δ w = w δ Normalize the approimation The normalization step (against previousl found eigenpairs) is forced at the beginning of the procedure and during the iterative procedure in order to avoid finding the same solution twice. Notice however that at the iteration loop, the normalization is onl performed against the eigenvectors whose associated eigenvalue approimates the current Raleigh quotient. Another important aspect of the algorithm is the solution of the sstem of equations v = E w. Because matrices A and C are smmetric and banded, matri E contains similar structure. Hence, the application of Gaussian elimination to this sstem of equations is ver efficient. In opposition to the Power method or the Inverse Iteration method, the chosen approach requires the calculation and factorization of matri E at ever iteration. Nevertheless, because the cost of factorization of the matri is ver low (due to its slim banded structure), because the convergence is greatl improved (due to the application of the Raleigh shift), and 5

67 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings because the normalization (deflation) is imposed onl against the pairs that are in close proimit, then the Inverse Iteration with Raleigh Shift method turns out to be the most efficient tool among the three mentioned methods. One last comment: the speed of convergence depends on the qualit of the initial guess of v. In this work, the initial guess is a random value. However, because the eigenvalue problem needs to be solved for different values of ω ( C = G ω M ), if two successive frequencies are close enough to each other ( ω = ω + dω ), one can use the eigenvectors computed for ω as initial guesses for the eigenvectors of ω. This strateg shall improve the convergence of the procedure (eamples have shown that the average number of iteration per eigenpair reduces from 7-8 to -3; this strateg has not been used in this work due to complications that occur when two eigenvalues are too close from each other, and as a consequence the solutions ump from one branch to another)..7. Eigenvalue problem for SVP waves The first eigenvalue problem of eq. (.39) is equivalent to where ( k k ) A + B + C v = (.) A A Ο = Ο A z B Ο B z = T B z Ο G ω M Ο C = Ο G z ω M z v v f = f z Taking into consideration the original order of the degrees of freedom instead of the order organized b direction (see section.3), matrices A, B and C result smmetric and banded, being the bandwidth 4 if the thin-laers are of linear epansion or 6 if the thin-laers are of quadratic epansion. Sstem (.) can be further epanded and written as k R = k B C k v A Ο k v = k C Ο v Ο C v (.) and so it can be easil concluded that the eigenvalue problem has N solutions, being N the dimension of matrices A, B and C. Also, due to the smmetric properties of these matrices k, v is a solution of (.), then and due to the special structure of matri B, if the pair ( ) the pair ( k, ) v is also a solution. The modified eigenvector v coincides with for the components associated with the z direction, which have their sign reversed. v, ecept Because the quadratic eigenvalue problem is slightl more comple than the linear one, some eplanations are needed before presenting the solution procedure. Of equation (.), drop th the modal inde and consider a general iteration (sa, the l iteration) of the Inverse Iteration with Raleigh Shift: 53

68 Chapter Wave propagation in the soil: fundamental solutions kla + B C ul+ A Ο ul = C k C v Ο C v l l + l Handling simultaneousl both rows of the sstem (.), it is possible to obtain ul+ vl vl+ = kl ( k A + k B + C) u = Cv k Au l l l+ l l l (.) (.3) Notice that at each iteration, it is needed to solve a sstem of equations with dimension N and not N, as could be suggested b equation (.). As for the Raleigh quotient, its value is given b k l B C u T T l ul v l T T C O vl ul Bul + ul Cvl T T T T A O ul vl Cvl ul Aul ul vl O C vl = = (.4) However, in this work, instead of calculating the Raleigh quotient according to (.4), the quotient is updated in ever iteration according to the recursive formula (Waas, 97) u + Au v + Cv k k T T l l l l l+ = l + T T ul+ Aul+ vl+ Cvl + (.5) Equation (.5) works as well as eq. (.4), but is more convenient in terms of computational resources. The orthogonalit between a iteration of the converged (the i th eigenvector) is imposed through th eigenvector and an eigenvector alread u u T T kivi T T kivi = ( v Cvi kiu Avi ) ( v Cvi + kiu Avi ) v v vi vi (.6) As can be seen in eq. (.6), both the eigenvector associated with k i and the eigenvector associated with are being deflated. ki The algorithm used for the solution of the eigenvalue problem takes into account the maor steps described above and is listed in Table.8. 54

69 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Table.8: Algorithm for the solution of the quadratic eigenvalue problem v = rand( N ) Initial guess u = rand( N ) Initial guess (auiliar) Equation (.6) Normalize the initial guess against previousl found eigenvectors ( i =,,... ) w = Au Auiliar vector = Cv Auiliar vector k u Bu = + u T T T T v u wl Raleigh quotient Iterate (until Raleigh quotient has converged) E = Ak + Bk + C From shifted matri ( k ) u = E w ( ) v = u v k New approimation of eigenvector Equation (.6) Normalize against previousl found eigenvectors (if k ki or k ki, i =,,..., ) k u w = k + u Au v T T T T v Cv Update the Raleigh quotient w = Au Auiliar vector = Cv Auiliar vector u w δ = v = v δ w = w δ = δ T ( u Cu ) k T Normalize the approimations.8 Conclusions In this chapter, the Thin-Laer Method is introduced as a tool to calculate the response of laered domains (for eample, soil) to dnamic loads. The TLM is etended to the.5d domain and closed form epressions for the fundamental displacements, derivatives, tractions and stresses are given and validated. It is worth noting that the proposed methodolog relies on the solution of two eigenvalue problems that in no wa depend on the horizontal wavenumbers k and k, hence the need to be solved onl once for each frequenc. The 3D 55

70 Chapter Wave propagation in the soil: fundamental solutions fundamental solutions obtained with the TLM can be found in other works on the subect (Kausel, 98). A new and more efficient procedure, the Perfectl Matched Laer, is proposed for the simulation of infinite domains. The PML is used to solve eamples of full-spaces and laered half-spaces and is validated b means of the same eamples. This new procedure is much more efficient than the previous one (the paraial boundaries), and from now on it is recommended that the PB be replaced b the PML. In the last section of this chapter, efficient algorithms for the solution of the two eigenvalue problems are described. 56

71 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings 3. Introduction 3. Numerical tools for soilstructure interaction Soil-structure interaction received a lot of attention during the XX th centur. It is an interesting and comple problem, and man works covering a wide range of fields can be found in the literature. The interested reader is referred to the work (Kausel, ) for a historical overview on the subect. For the case of traffic induced vibrations, track and building are coupled through the ground, and so, both in the propagation stage and in the reception stage, the interaction between the soil and the track or the building must be considered. Due to the differences in the tpolog, the ideal numerical tool to model each of the sub-domains ma var. For eample, to model the soil, which is an infinite domain, the Boundar Element Method (BEM) is preferred because it can account for the radiation of waves to the infinit (Dominguez, 993). On the other hand, the use of the Finite Element Method (FEM) reveals to be more appropriate to model the behavior of buildings and tracks, since these structures are circumscribed and generall irregular. Additionall, in the propagation stage the geometr of the problem can be assumed invariant in the longitudinal direction, and thus a.5d formulation is advantageous, while in the reception stage the problem is limited to the structure under analsis and the surrounding environment, and therefore a 3D formulation is more appropriate. In this work, the propagation stage and the reception stage are decoupled. Hence, the wave field that the track induces in the soil is calculated disregarding the presence of the building in the far field. This simplification has been used b other authors (François, 8; Fiala et al., 7) and is valid when the distance between the building and the track is larger than the characteristic wavelengths of the soil, i.e., the simplification is valid for the medium and the high frequenc range. Nevertheless, in this work this assumption is also used for the low frequencies. In the present chapter, the 3D BEM, the.5d BEM and the.5 D FEM are described. 3. 3D Boundar Element Method 3.. Introduction The Boundar Element Method is a discrete numerical procedure that can be used to solve partial differential equations. This procedure relies on the discretization of the boundar of the domain, as opposed to the Finite Element Method, for which the whole domain must be modeled. For this reason, for problems involving ver large domains (such as the soil) the BEM becomes advantageous over the FEM as it avoids the discretization of the interior of the domain, which results in a reduced number of degrees of freedom. Another advantage of the BEM is that it deals with the radiation of waves to infinit eactl, contraril to the FEM, in which special procedures need to be applied at the boundaries of the truncated domain in order to model unbounded domains. As drawbacks, the BEM renders sstems of equations which involve full populated and non-smmetric matrices and requires the knowledge of the 57

72 Chapter 3 Numerical tools for soil-structure interaction Fundamental Solutions of auiliar domains, which can be homogeneous full-spaces, halfspaces or laered spaces. In this work the soil is assumed to be horizontall stratified in the work of Jones (), the importance of the inclination of laers is assessed. For this reason, the more appropriate auiliar domain is the laered half-space. Using the fundamental solutions of such auiliar domain, the BEM requires solel the discretization of the surfaces of the soil interacting with the structures. Unfortunatel, such fundamental solutions are not known in closed form epressions. An alternative would be to use the full-space fundamental solutions, for which analtical epressions are known. However, when considering such auiliar domain, the free-surface of the soil and of the interfaces between the different laers that characterize the soil must also be discretized, and that results in a substantial increase of the number of degrees of freedom, which renders the approach less attractive. Three-dimensional formulations of the BEM can be found in the works of Brebbia and Dominguez (99) for elastostatic problems and of Dominguez (993) for elastodnamic problems. For train induced vibration problems, the time domain 3D BEM has been used b Galvín (7) and O'Brien and Rizos (5), who considered the fundamental solutions of homogeneous full-spaces, b François (8), who considered the fundamental solutions of laered half-spaces, and b Bode et al. (), who considered the fundamental solutions of homogeneous half-spaces. The frequenc domain 3D BEM has been used b Hubert et al. (), who considered full-space fundamental solutions, and b Fiala et al. (7), who considered the laered half-space fundamental solutions. In this work, a frequenc domain 3D BEM procedure is coupled to the 3D FEM in order to obtain the response of a structure due to an incoming wave field. In the following section, the 3D BEM is presented. 3.. Integral representation Consider an elastic bod Ω with boundar Γ and two elastodnamic states described b the * * u, t u, t u, the displacement fields k ( ) and k ( ), the initial displacements uk ( ) and k ( ) * * initial velocities vk ( ) and vk ( ), the bod forces ρbk (, t) and bk (, t) * tractions pk (, t) and pk (, t), where,, 3 where is a point with coordinates = (,, z) ρ and the boundar k = corresponds to the, and z directions and. The two elastodnamic states are interrelated b the Reciprocal Theorem, which states (Achenbach, 973) Γ { } * * * * ( ) ( ) ρ ( ) ( ) ( ) ɺ ( ) ( ) ( ) p, t u, t d Γ + b, t u, t + u u, t + v u, t dω = k k k k k k k k Ω { } ( ) ( ) ρ ( ) ( ) ( ) ɺ ( ) ( ) ( ) = p, t u, t d Γ + b, t u, t + u u, t + v u, t dω Γ * * * * k k k k k k k k Ω (3.) In equation (3.), a dot over a variable represents the derivative with respect to time and the operator * represents the time convolution, defined b t ( ) ( ) = ( ) ( ) = ( ) ( ) f t * g t f t τ g τ dτ g t τ f τ dτ t (3.) Also, the Einstein notation is used, i.e., a repeated inde in a term implies the summation over that inde. For eample, 3 * * * * * k k = k k = k = p u p u p u p u p u (3.3) 58

73 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Now, assume that the second elastodnamic state (the one associated with the supperscript * ) is elicited b a bod force of the form ρb * k (, t) = δ δ (,) where l =,, 3 is the direction of the bod force, kl Dirac delta function and = ( ξ, ξ, zξ ) ξ (3.4) kl δ is the Kronecker delta, ( ) δ is the ξ is the point where the bod force is applied, which can be inside or outside the domain Ω. In this case, the displacements induced b such load correspond to the fundamental displacements in the time domain and, in the ensuing, are * denoted b ukl ( ξ,, t), in which the first inde represents the direction of the displacement and the second inde represents the direction of the bod load as given in equation (3.4). * Analogousl, the fundamental tractions at the boundar Γ are denoted b pkl ( ξ,, t) and are calculated with = 3 * * kl (,, ) σ kl (,, ) = p ξ t ξ t n (3.5) where n are the components of the vector n that is normal to the boundar Γ at the point * (pointing outwards) and where σ (,, t) kl ξ are the fundamental stresses in the time domain induced at the point b a point load at ξ. Assume also that the first elastodnamic state presents no bod forces and that it is initiall at rest, i.e., ρ bk (,, z, t) =, u k (,, z ) = and v k (,, z ) =. Under these two assumptions, equation (3.) becomes Γ * * (, ) (,, ) d Γ = (,, ) (, ) d Γ + κ (, ) p t u ξ t p ξ t u t u ξ t (3.6) k kl kl k kl k Γ where κ kl = δ kl if ξ Ω and κ kl = if ξ Ω. The integral equation (3.6) relates the displacements of a point ξ of the domain Ω with the displacements and tractions at the boundar Γ. In the frequenc domain, equation (3.6) becomes Γ * * (, ω) ( ξ,, ω) d Γ = ( ξ,, ω) (, ω) d Γ + κ ( ξ, ω) pɶ uɶ pɶ uɶ uɶ (3.7) k kl kl k kl k Γ where a tilde over the variables denotes their Fourier transform with respect to time. Hence, * * uɶ kl ( ξ,, ω) and pɶ kl ( ξ,, ω) represent the three-dimensional fundamental displacements and tractions in the frequenc domain, which can be calculated using the Thin Laer Method (TLM) (Kausel, 98) Regularization of the integral equation The fundamental displacements and fundamental traction are singular at the collocation point ξ and consequentl equations (3.6) and (3.7) are not valid if ξ belongs to the boundar Γ. Two approaches can be followed to deal with this problem: a limiting process in which a spherical portion of the domain with radius tending to zero is ecluded (or included) around the collocation ξ (Figure 3.) (Dominguez, 993); and a regularization procedure in which the singularities of the fundamental solutions are removed (François, 8). 59

74 Chapter 3 Numerical tools for soil-structure interaction Ω Γ ε n ξ S ε Γ Ω ξ n S ε Γ ε Γ Figure 3.: Eclusion (left) and inclusion (right) at point ξ ( Γ ε is the proection of S ε on Γ ) Following the first approach, equation (3.7) becomes Γ Γε * The integral u ( ω) * * ( ω) ( ξ ω) ( ξ ω) ( ξ ω) pɶ, uɶ,, d Γ + pɶ, uɶ,, ds = k kl k kl Sε = pɶ uɶ Γ + + pɶ S uɶ Γ Γε ( ξ,, ω) (, ω) d κ ( ξ,, ω) d ( ξ, ω) * * kl k kl kl k Sε (3.8) ɶ kl ξ,, ds vanishes for ε tending to zero (Dominguez, 993) and so Sε equation (3.8) is condensed to where ( ξ * * ω) ( ω) ( ξ ω) ( ξ ω) ( ω) c uɶ, = pɶ, uɶ,, d Γ pɶ,, uɶ, dγ kl k k kl kl k Γ Γε Γ Γε δ kl + * ckl = κkl + pɶ kl ( ξ,, ω) ds = * S p ε ɶ kl Sε Sε pɶ * kl ( ξ ω) ( ξ ω),, d S, if inclusion,, d S, if eclusion (3.9) (3.) When the boundar is smooth at ξ and the full-space fundamental solutions are being used, ckl =.5δ kl. For different geometries at the node ξ, Guiggiani and Gigante (99) and Mantic (993) indicate procedures to obtain the coefficient c kl, also considering the full-space fundamental solutions. For different fundamental solutions (such as the laered half-space fundamental solutions), no work addressing the calculation of the coefficients c kl was found in the literature. The remaining integrals in equation (3.9) must be evaluated in the Cauch principal value * sense. The fundamental displacements uɶ kl present a weak singularit at ξ and so the integral * * involving uɶ kl can be evaluated easil. The fundamental tractions pɶ kl present a strong * singularit at ξ and for that reason the evaluation of the integral involving pɶ kl is not straightforward. Cerrolaza and Alarcon (989) present a procedure based on a polnomial coordinate transformation that can be used to evaluate this tpe of integrals. Following the regularization procedure mentioned in the beginning of this subsection, the * strong singularit that is contained in the fundamental tractions pɶ kl is removed b emploing the rigid bod motion technique. The details of this technique can be found in (François, 8). For unbounded domains, the regularized integral equation that is obtained using this technique is 6

75 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings * ( ξ ω) ( ω) ( ξ ω) uɶ, = pɶ, uɶ,, dγ * in which p ( ξ,,) l k kl Γ kl Γ ( ξ ω) ( ω) ( ξ ) ( ξ ω) * * pɶ kl,, uɶ k, pɶ kl,, uɶ k, dγ ɶ are the fundamental static tractions. (3.) In this work, for the 3D BEM, equation (3.) is used. The reason wh equation (3.) is chosen instead of (3.9) is because with that equation the calculation of c kl is avoided. Furthermore, because the laered half-space fundamental solutions are used and since onl structures resting on the surface of half-spaces are considered, the calculation of the fundamental tractions is not necessar as the are zero (free traction boundar condition). [Note: For buried structures, a more convenient strateg can be used: if the collocation points are placed inside the part of domain to be ecluded instead of in the boundar, equation (3.7) can be used directl without the need for regularization because the collocation points are not contained in the integration path; hence, if one defines as man collocation points as nodes of the discretized boundar Γ, a sstem of equations that relates the boundar tractions and the boundar displacements is obtained, and from that sstem, a stiffness (or fleibilit matri) can be easil obtained. Nevertheless, the integration of the fundamental solutions over vertical boundaries is still needed and that can become troublesome.] 3..4 Discretization of the boundar To solve the integral equation (3.), the boundar Γ is divided into N e boundar elements and N n boundar nodes. Within each boundar element, the displacement and traction fields are approimated b means of interpolation functions and nodal values, i.e., ( e ) N n = ( e) ( e) (, ω) ( ω) ( ) uɶ uɶ S (3.) k k ( ) = ( e) N n = ( e) ( e) (, ω) ( ω) ( ) pɶ pɶ S (3.3) k k ( ) = in which uɶ k (, ω) and pk (, ω) th ( belongs to the e boundar element), e ) ( ) uɶ k ( ω) and p e k ( ω) th ( e) node of the e element, S ( ) ɶ are the interpolated displacements and tractions (in this case, ( ) ɶ are the nodal values of the are the associated interpolation (shape) functions evaluated at ( ) and e th N n is the number of nodes contained in the e element. The tpe of boundar elements and respective shape functions that are commonl used can be found in (Dominguez, 993). Now, assume that the collocation point ξ corresponds to one of the N n nodes of the boundar th (sa, the i node, with coordinates i ). In this case, after accounting for the approimations in equations (3.) and (3.3), equation (3.) becomes th 6

76 Chapter 3 Numerical tools for soil-structure interaction or in compact form, (, ω ) ( e ) Uɶ kl i e Ne N n ( i) ( e) * ( e) l ( ω) = k ( ω) kl (, i, ω) ( ) ( ) dγe e= = Γe uɶ pɶ uɶ S (, ω) ( e ) Pɶ kl i ( e ) Ne N n ( e) * ( e) uɶ k ( ω) pɶ kl (, i, ω) S( ) ( ) dγe e= = Γe + e ( i) * uɶ ( ω) pɶ ( ) k N e= Γe ( e,3dstat) Pɶ kl ( i ) kl, i, dγe ( e ) ( e) N N N N ( i) ( e ) ( e ) ( e ) ( e ) l k kl i k kl i e= = e= = e n e n ( ω) ( ω) ɶ (, ω) ( ω) ɶ (, ω) uɶ = pɶ U uɶ P + uɶ N ( i) (,3Dstat) k kl i e= e e ( ω) Pɶ ( ) B making l assume all the three directions, equation (3.5) assumes the vector form (3.4) (3.5) ( e ) ( e) N N N N ( i) ( e ) ( e ) ( e) ( e) i i e= = e= = e n e n ( ω) = ɶ (, ω) ɶ ( ω) ɶ (, ω) ɶ ( ω) uɶ U p P u Ne + Pɶ u e= i ( ) ɶ ( ω) ( e,3dstat) ( ) i where the bold variables have the following meanings (the arguments dropped) (3.6) i and ω were ( i) uɶ ( i) ( i) uɶ = uɶ ( i) u ɶz ( e ) ( e ) ( e ) Uɶ Uɶ Uɶ z ( e ) ( e ) ( e ) ( e ) Uɶ = Uɶ Uɶ Uɶ z ( e ) ( e ) ( e ) Uɶ z Uɶ z Uɶ zz ( e) uɶ ( e) ( e) uɶ = uɶ ( e) u ɶz ( e,3dstat) ( e,3dstat) ( e,3dstat) Pɶ Pɶ Pɶ z ( e,3dstat) ( e,3dstat) ( e,3dstat) ( e,3dstat) Pɶ = Pɶ Pɶ Pɶ z ( e,3dstat) ( e,3dstat) ( e,3dstat) Pɶ z Pɶ z Pɶ zz ( e) pɶ ( e ) ( e) pɶ = pɶ ( e) p ɶ z ( e ) ( e ) ( e) Pɶ Pɶ Pɶ z ( e ) ( e ) ( e ) ( e) Pɶ = Pɶ Pɶ Pɶ z ( e ) ( e ) ( e) Pɶ z Pɶ z Pɶ zz (3.7) After making the collocation point ξ correspond to all the N n nodes and after assembling the ( e) matrices U ɶ ( e ) into a square matri U and the matrices P ɶ ( e,3dstat) and P ɶ into a square matri P, the final sstem of equations is obtained { P( ω) + I} u( ω) = U( ω) pɶ ( ω) ɶ (3.8) 6

77 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings in which uɶ ( ω) is a vector that collects all the nodal displacements and ( ω) pɶ is a vector that collects all the nodal tractions. Equation (3.8) can be solved for uɶ, pɶ or a combination of the two vectors, depending on the unknowns of the problem Coupling BEM-FEM To model the behavior of structures composed of irregular and finite domains, the FEM is more adequate than the BEM. Hence, when such structures interact with the soil, the coupled BEM-FEM strateg ma be the most attractive option. In the frequenc domain, the FEM sub-domain is governed b the sstem of equations (neglecting the damping) in which M represents the mass matri and K represents the stiffness matri ( ω ) K M uɶ = Fɶ (3.9) Kɶ M ρn N T = dω FEM Ω K B DB T = dω FEM Ω (3.) (3.) and where the vectors uɶ and F ɶ represent the frequenc domain displacements and eternal forces, respectivel, and the variables ρ, D, N and B correspond to the densit, the constitutive matri, the shape function matri and the linear operator containing the derivatives of the shape functions. On the other hand, the behavior of the BEM sub-domain is described b equation (3.8). In the following, the superscripts F and B are used to distinguish the response fields of the FEM and BEM sub-domains, respectivel. FEM Consider a sub-domain Ω modeled b means of the FEM, which interacts along the FEM BEM surface Γ with a sub-domain modeled with the BEM and whose boundar is Γ (Figure 3.). To couple the two sub-domains, first it is convenient to separate the degrees of freedom of both sub-domains into two groups: one that contains the degrees of freedom belonging to FEM Γ (represented here b the inde I ); and another that contains the remaining degrees of freedom (represented b the inde II ). Taking into account the defined groups, sstems of equations (3.8) and (3.9) are written as (the argument ω is dropped) and B PI,I + I PI,II uɶ U I I,I UI,II pɶ I = B PII,I PII,II + I uɶ II UII,I UII,II pɶ II F Kɶ I,I Kɶ I,II uɶ I Fɶ I F = Kɶ II,I Kɶ II,II uɶ II F ɶ II (3.) (3.3) 63

78 Chapter 3 Numerical tools for soil-structure interaction FEM Ω FEM Γ BEM Γ BEM FEM Γ Γ Figure 3.: FEM sub-domain FEM Ω interacting with BEM sub-domain with boundar The two sub-domains are now coupled b enforcing the compatibilit of displacements and FEM the equilibrium of forces at the interface Γ. Since the interpolation functions ma differ from one sub-domain to the other, the compatibilit of displacements cannot be imposed at all FEM points of Γ. Hence, the compatibilit is enforced onl at the nodes of the BEM model interacting with the FEM domain, condition that is accomplished through the identit F B I I I BEM Γ N uɶ = uɶ (3.4) where the matri N I contains the FEM shape functions associated with the degrees of FEM freedom I and evaluated at the nodes of the boundar element model that belong to Γ. FEM Regarding the equilibrium of forces at Γ, the FEM formulation relates displacements with nodal forces while the BEM relates displacements with nodal tractions, and for this reason the equilibrium can onl be respected in an approimate manner. The equilibrium is therefore established b enforcing that the resultant of the boundar tractions equals the nodal forces, i.e., T T I ( ) I ( ) dγ I + I = N S pɶ Fɶ (3.5) FEM Γ Matri T, defined b the integral in (3.5), transforms the nodal tractions into equivalent N collects the FEM shape functions associated with the FEM nodal forces. Matri I ( ) nodes I and evaluated at while S ( ) BEM nodes I and evaluated at. I collects the BEM shape functions associated with the The sstem of equations (3.) can be solved for pɶ I and the following identit can be reached (see Appendi III) pɶ = A Buɶ + A Cpɶ (3.6) B I I II in which the matrices A, B and C are defined b ( ) A = U P P + I U (3.7) I,I I,II II,II II,I ( ) B = P + I P P + I P (3.8) I,I I,II II,II II,I ( ) C = P P + I U U (3.9) I,II II,II II,II I,II 64

79 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings The combination of equations (3.4), (3.5) and (3.6) ields Fɶ = TA B N uɶ TA Cpɶ (3.3) F I I I II and the use of eq. (3.3) in (3.3) renders the final sstem of equations in which and Kɶ + K Kɶ uɶ F F I,I BEM I,II I BEM F = Kɶ II,I Kɶ II,II uɶ Fɶ II II The sstem of equations (3.3) is solved for FEM Ω are obtained. The displacements equation (3.4). The boundar tractions pɶ I and displacements sstem of equations (3.) for these variables. (3.3) K BEM = TA B N I (3.3) FBEM = TA C pɶ II (3.33) F F uɶ I and uɶ II and as a result the displacements in uɶ of the BEM model are then obtained b emploing B I B uɶ II are obtained b solving the Eample: massless rigid footing resting on a half-space For the case of a rigid footing, its behavior is described b the displacements and rotations at some reference point. Assuming that the reference point coincides with the origin of the Cartesian coordinate sstem, which in turn is placed at the center of the footing (Figure 3.3), the displacements at an point = { } that belongs to the footing can be calculated with ( ) u U Foot u N= NI u uz = θ θ θz (3.34) where u α is the translation of the footing in the α direction and θ α is the rotation of the footing about the α ais, with α =,, z. To account for the soil reaction, the soil-structure interface is divided into N e boundar elements and N n nodes, all belonging to the group I of degrees of freedom. In addition, since the half-space fundamental solutions are used, the fundamental tractions are null. Hence, in equation (3.) onl the sub matrices U and I are not null, and consequentl the matri K BEM simplifies to K BEM = TU I,I N I, while the vector BEM I,I F vanishes. 65

80 Chapter 3 Numerical tools for soil-structure interaction z b Figure 3.3: Rigid footing resting on a half-space: boundar element mesh Net, to validate the procedure, the horizontal compliance ( CHH = uga, considering a unit horizontal load) and the vertical compliance ( CVV = uzga, considering a unit vertical load) of a square footing ( a = b ) are compared with the compliances obtained b Wong and Luco (978), for the dimensionless frequenc a = ωa Cs ( Cs = G ρ ) varing from to 4. The 6 size of the square foundation is a = 6 and the soil properties are G = 3.35, 4 ρ =.8 and ν = 3 (consistent units). The soil-structure interface is divided into equall sized squares, and within each square the displacement and traction fields are assumed to be constant (constant boundar elements). The fundamental solutions of the halfspace are obtained via the TLM with a model consisting of a elastic laer with thickness h = πc s 5ω =.λ s (divided into 5 thin laers of quadratic epansion) that rests on a PML with parameters m =, η = 3, n =, N = 5 and Ω = (see Chapter, equation.8). The elastic laer is added to the TLM model to improve the qualit of the fundamental solutions near the source (recall that the TLM is a discrete method and so, near a point source, where the displacement and stress fields var rapidl, thinner meshes are required in order to obtain a good accurac). Figures 3.4 and 3.5 plot the results herein obtained and the results reported b Wong and Luco. Though not perfect, the agreement is good, which validates the procedure...5 a Re(C HH ) Im(-C HH ) Re(C HH ) : Wong and Luco (978) Im(-C HH ) : Wong and Luco (978) CHH Figure 3.4: Horizontal compliance a C HH 66

81 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings. Re(C VV ) Im(-C VV ) Re(C VV ) : Wong and Luco (978) Im(-C VV ) : Wong and Luco (978) CVV Figure 3.5: Vertical compliance Note: In this eample, because the boundar elements are constant, square shaped, and resting at the surface of the half-space, the calculation of the boundar element matrices U I( II ),I( II) of equation (3.) can be accomplished with a simplified procedure: instead of integrating the displacements induced b a point load at the i th boundar node on the surface of the th boundar element, the 33 sub-matri associated with the i th boundar node (rows) and th boundar element (columns) can be taken as the displacements (all nine components) at the i th boundar node induced b a disk load applied at the th boundar element. The radius of the disk load must be such that the area of the disk is the same as the area of the boundar element. This simplified procedure ields ver good approimations as long as the boundar elements are horizontal, fairl square and of constant epansion. If the elements are not at the surface, this procedure cannot be used because it is not valid for the traction integrals. The displacements induced b disk loads are given in the work (Kausel and Peek, 98b) and transcribed in Appendi IV Weak coupling response to incoming wave fields As mentioned in the beginning of this chapter, in this work the propagation and the reception stages are decoupled, i.e., the wave-field that the track transmits to the soil is calculated disregarding the presence of the building in the far field. Hence, there is a weak coupling between the two sub-structures: the connection between the track and the structure is kept, while the connection between the structure and the track is not (Figure 3.6). t So, consider that the tractions p at the boundar Γ t (that result from the interaction between the track and the soil, and that are calculated without accounting for the removal of the volume of soil Ω s ) induce at the boundar Γ s the displacement field u and the traction field p (Figure 3.6). The displacements u are calculated b placing collocation points on the boundar Γ s and then using equation (3.7) with the derivatives of C VV Γ = Γ t. The traction field u in the strain-stress relations (.6). The derivatives of deriving equation (3.7) with respect to ξ see (Dominguez, 993) for details. a p is obtained using u are obtained b 67

82 Chapter 3 Numerical tools for soil-structure interaction Ω s Γ t t p s p u, p Γ s Figure 3.6: Weak coupling between track and structure The first step to obtain the response of Γ s is to make it traction free, a condition that is needed to account for the volume Ω s of soil that is ecavated. Such is accomplished b appling the tractions p at Γ s (same magnitude but opposite direction). These tractions induce an etra p displacement field u at Γ s calculated with ( ) p u = P + I U p (3.35) s where P s and U s are the boundar element matrices of the discretized boundar Γ s. The inc t displacement field u that p induces at Γ s (and that accounts for the volume of ecavated soil Ω ) is then s (for Γ s at the surface, s ( ) u = u + u = u P + I U p (3.36) inc p s s p is null and so inc u = u ). The second step to obtain the response of Γ s is to establish the compatibilit of displacements and equilibrium of forces between the soil and the structure. The incident displacement field inc u needs to be taken into account in the compatibilit equation (3.4), which becomes N uɶ = uɶ + uɶ (3.37) F B inc I I I The equilibrium equation does not change. After combining the compatibilit equation (3.37) and the equilibrium equation (3.5), a sstem of equations with the same form as equation (3.3) is obtained, with F BEM being calculated with 68

83 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings F = TA C pɶ TA B u (3.38) BEM inc II and with the remaining matrices and vectors being calculated as eplained in section 3..5 for Γ = Γ. s Eample: two weakl coupled rigid footings resting on a half-space Two square rigid footings ( a = b = 6 ) are placed at a distance d = 4a = from each other (center to center). Both footings have concentrated masses placed at their center of gravit ( M = 5 ), which results in inertial forces in the translational degrees of freedom. The footings rest on a half-space whose properties are the same of those of the previous eample. One of the foundations (footing ) is loaded verticall and the vertical response of both footings ( CVV, = uz,ga and CVV, = uz,ga ) is calculated considering full coupling and weak coupling between the two foundations. Figures 3.7 and 3.8 compare the responses obtained considering the two coupling schemes for the dimensionless frequenc a varing between and. The presence of the concentrated masses in the footings significantl modifies the behavior of the footing-soil sstem, as can be concluded from the comparison between Figures 3.5 and 3.7: both the shape and the magnitude of the response are different. It is also observed that the two coupling schemes ield different results. Nevertheless, the response of the weak coupling scheme follows the main trends of the response of the full coupling scheme..8.6 Re(C VV, ) - Full coupling Im(-C VV, ) - Full coupling Re(C VV, ) - Weak coupling Im(-C VV, ) - Weak coupling CVV, Figure 3.7: Vertical displacement of loaded footing: full coupling versus weak coupling a 69

84 Chapter 3 Numerical tools for soil-structure interaction.4. Re(C VV, ) - Full coupling Im(-C VV, ) - Full coupling Re(C VV, ) - Weak coupling Im(-C VV, ) - Weak coupling CVV, Figure 3.8: Vertical displacement of free footing: full coupling versus weak coupling 3..7 Final considerations In this section of chapter 3, the 3D BEM was introduced and coupled to the 3D FEM in order to perform dnamic analses of structures interacting with the soil. To validate the implemented methodolog, the eample of a rigid footing resting on a half-space was solved and the results were compared with results available in the literature. Then, the definition of weak coupling between two structures was introduced and an eample was shown where the results obtained considering weak coupling and full coupling were compared. It was observed that the two coupling approaches ielded different results, but that the maor trends of the responses were kept. In this work, the 3D BEM-FEM methodolog is used in the contet of railwa induced vibrations to analze the response of structures to incoming wave-fields. The wave-fields are calculated using a.5d BEM-FEM procedure, which is eplained in the following subsections. In the.5d BEM-FEM procedure, the presence of the building in the far field is disregarded, i.e., it is considered that the track and the buildings are weakl coupled. The consideration of buried structures is not attempted in this work, but some comments are proffered net about this issue: a. The calculation of the boundar matrices for non-horizontal boundaries using the TLM becomes much more complicated than to integrate the fundamental displacements on a horizontal surface or to use the simplified procedure eplained at the end of section The difficulties arise because the boundar integrals cannot be calculated directl (unlike in the.5d case, eplained in a later section of this chapter), which leads to the necessit of using special integration schemes (Gaussian integration, for eample) and consequentl an increase in the time needed for the computation of these integrals. Nevertheless, in the work (Kausel and Peek, 98a) the BEM is formulated both in D and 3D spaces using the TLM, and these formulation can be used for the analses of buried structures.. Alternative approaches can also be used: in recent ears, some authors suggested the use of the Perfectl Matched Laer together with finite elements to calculate 7

85 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings the stiffness matri K Γ s ( = K BEM ) of the soil interface (Basu and Chopra, 3; Harari and Albocher, 6). However, there are reports of incorrect results of the FEM-PML when the domain is laered, and that occurs due to grazing incidence of waves (Komatitsch and Martin, 7). In this work, during some tests with FEM-PML to simulate a single stratum, for certain frequencies, the response of the stratum diverged from the epected. 3. Another alternative for the calculation of the stiffness matri of the soil is the SASSI approach (Lsmer et al., 999). In this approach, a fleibilit matri F Ω that relates forces and displacements of a grid of nodes that delimit the volume of soil to be ecavated is calculated using the TLM. The fleibilit matri is then inverted, being thus obtained a stiffness matri ( K Ω = F Ω ). The stiffness matri K Γs of the soil is finall obtained b subtracting from K Ω the stiffness K Ωs of the volume to be ecavated, which is calculated with the FEM ( K s = K K s ). Γ Ω Ω 4. As a final comment, independentl of the approach that is followed, there is alwas a huge computational cost that cannot be avoided because of the need for a 3D mesh D Finite Element Method 3.3. Introduction When the geometr of the structure is invariant in one of the directions, space-wavenumberfrequenc domain (.5D) analses are usuall more advantageous than space domain (3D) analses. In essence, a.5d analsis consists in performing a Fourier transform of the longitudinal coordinate, which results in the reduction of the dimensionalit of the problem b one, and consequentl in the reduction of the 3D analsis to the summation of a set of D analses. In the contet of railwa induced vibrations, since in most cases it can be assumed that the geometr of the track and the profile of the soil are invariant in the longitudinal direction,.5d analses can be used to calculate the wave-fields that propagate in the track and soil and reach the buildings. The.5D analses have been used b several authors in this contet, as mentioned in the literature review presented in chapter. In this work, the track is modeled using.5d finite elements while the soil is modeled using.5d boundar elements. The two procedures are coupled in order to solve the track-soil sstem and to calculate the vibration field that propagates in the track and soil. Coupled.5D BEM-FEM schemes can be found in the literature, for eample in Galvín et al. () and in Sheng et al. (5). The first work uses laered half-space fundamental solutions calculated with the stiffness matrices while the latter uses the analtical full-space fundamental solutions that are given in the same reference. The fundamental solutions used in the present work are the.5d fundamental solutions eplained in chapter D Finite Element Method Consider an elastic bod that etends to infinit in one direction (longitudinal direction ) and whose cross section ΩS is constant (Figure 3.9). It is assumed that the material properties are invariant in the direction while within the cross section the material properties ma var in a stepwise fashion. 7

86 Chapter 3 Numerical tools for soil-structure interaction z G, ρ, ν n θ θ n G, ρ, ν Figure 3.9: Elastic solid: definition of invariance and angles θ Under the mentioned conditions, and following the same steps as for the TLM see chapter, equations (.) to (.3) the wave equation in the elastic bod can be written as (the following variables have the same meaning as in chapter ) ρuɺɺ D + ( D + D ) + ( Dz + Dz ) z + D + ( D z + Dz ) + Dzz = u b z z (3.39) On the other hand, the internal stresses s at an plane parallel to can be calculated with T T ( cosθ z sinθ ) s = L + L DLu (3.4) where θ is the angle of the outwards normal to the plane with respect to the ais. At the interface between different materials, the internal stresses must be in equilibrium, i.e., s = s, θ = π + θ (3.4) while at the boundar of the cross section tractions t, i.e., Ω S the internal stresses must balance the eternal s = t (3.4) As in the TLM case, one proceeds to discretize the domain to solve the wave equation. However, instead of discretizing onl in the vertical direction z, in this case the domain is discretized in the z plane. Hence, after dividing the cross section Ω S into plane finite elements, the displacement field in the solid is approimated b (,, z) = (, z) ( ) u N U (3.43) where N is a matri containing the interpolation functions and U is a vector containing the displacements of the associated nodes. The interpolation functions N used in the.5d case are the same functions that are commonl used in plane strain or plane stress problems. After inserting the approimation (3.43) in the wave equation (3.39) and boundar conditions (3.4) and (3.4), it can be verified that these equations are not rigorousl satisfied, due to the 7

87 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings presence of unbalanced bod forces and tractions. The discrete wave equation is derived after the application of the method of the weighted residuals and b requiring the virtual work done b the unbalanced forces within each finite element to be null, which results in the single finite element equation U U U F = MUɺɺ + G U B + G zu A B z + G zzu (3.44) The vector F contains the consistent eternal forces at the nodes of the finite elements (which result from the eternal tractions t and the bod loads b ), while the finite element matrices M, A, B α, G αα and G z are given b ( α =, z ) = T M ρn N ΩS = A N D N ddz T ddz ΩS B = N D N ddz + N D N T T ΩS ΩS α α α α α = G N D N T αα α αα α ΩS ddz G = N D N ddz + N D N T T z z z z z ΩS ΩS ddz ddz (3.45) (3.46) (3.47) (3.48) (3.49) being N = N and N z = N. After the assembl of the finite element matrices, one is left z with a global sstem of differential equations with the same form as (3.44). To solve it, the displacements U and forces F are transformed to the wavenumber-frequenc domain b means of the double Fourier transformations + + ( ω) = ( ) i( ωt k ),, e U k U t d dt (3.5) + + ( ω) = ( ) i( ωt k ),, e F k F t d dt (3.5) In the transformed domain, the sstem (3.44) changes into ( ) ( ) F = k + ik + z + + z + zz ω A B B G G G M U (3.5) All matrices in (3.5) are smmetric, ecept for B and B z, which are skew-smmetric. However, for cross-anisotropic materials whose constitutive matri is given in equation (.) of chapter, it is possible to transform these matrices into smmetric matrices b means of a similarit transformation that consists in multipling the rows of (3.5) that are related to the direction b i and the columns related to the direction b i. The referred to rows and columns have indees l = + 3, with =,,,.... This transformation solel affects the matrices B and B z and the vectors F and U. After the transformation, the sstem (3.5) becomes 73

88 Chapter 3 Numerical tools for soil-structure interaction ( ) ( ) f = k + k + z + + z + zz ω A B B G G G M u (3.53) where f and u are obtained from F and U after multipling the rows l b and B z are obtained from i. Also, B and B z b reversing the sign of the columns with inde l. After solving the sstem of equations (3.53) for u, the displacements U in the wavenumberfrequenc domain are recovered b multipling ever row l of u b +i. [Note: the above description considered the cross-anisotropic material defined in eq. (.). In some cases, it is more convenient to define materials whose direction of anisotrop is the longitudinal direction instead of the vertical one. For eample, it is common to model the sleepers (which are discontinuous) as a continuous anisotropic slab, where the in-plane behavior differs from the out-of-plane behavior (Alves Costa, ). In that case, the constitutive matri is more appropriate. ν z ν z Ez Ez Ez ν z ν z Ez E Ez ν ν z z Ez Ez Ez D = ( + ν ) (3.54) E ( + ν z ) E z ( + ν ) E E z and E are the in-plane and out-of-plane elastic modulus, ν z and ν ν is the cross Poisson s ratio. This last are the in-plane and out-of-plane Poisson s ratio, and z variable can assume negative values and must be such that matri D is definite positive. The equations presented in this section are still valid for this constitutive matri.] [Note: the above formulation considers onl volume elements. In order to consider different elements, such as beams or shells, different differential equations and discretizations have to be used, resulting in a final sstem of equations that will contain terms in k and k (Galvín et al., ). For an Euler beam, the sstem of equations is ( 4 ω ρ ) ( ω ρ ) ( 4 ω ρ ) f = EI k A u f = EAk A u f = EI k A u z z z 3 4 B (3.55) where E is the Young s modulus of the beam, ρ is the mass densit, I, z are the moments of intertia, A is the cross section area, f,, z represent the eternal forces and u,, z represent the displacements of the ais of the beam.] Eample - dispersion curves of a UCI86-3 rail To validate the implementation of the procedure presented in this section, the dispersion curves of a UCI86-3 rail are calculated and compared with the curves obtained b Gavric (995), who also used a.5d FEM procedure. Dispersion curves correspond to the pairs 74

89 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings ( k, ω ) that result in singular sstems of equations (3.5) or (3.53). In other words, the dispersion curves plot the values k that are solutions of ( ) ( ) k k ω A B B G G G M (3.56) det + + z + + z + zz = as function of the frequenc ω. The solutions of (3.56) ma be real or comple: the real solutions correspond to waves that propagate while the comple solutions correspond to waves that evanesce (i.e., attenuate) with the distance to the source. The.5D FEM methodolog presented in this section is emploed to determine the dispersion curves of the propagating modes of the rail. With that intention, the rail section is divided into 66 quadrilateral elements and a total of 4 nodes, resulting in the mesh shown in Figure 3.. For that mesh, the global matrices A, B α, G αβ and M are calculated and then used in equation (3.56). Figure 3.: Used mesh for the rail section (dimensions in mm) In Figure 3., the results obtained in this work (black) are compared with the curves obtained b Gavric (red). The agreement of the curves is good namel for the lower frequencies. For the higher frequencies (above 3 Hz), even though the shapes of the curves are the same, there is a shift between the results obtained in this work and the results obtained b Gavric. The disagreement might be ustified b discrepancies in the material parameters: while in this work the material properties are Young s Modulus E = GPa, mass densit 3 ρ = 7859 kg / m, and Poisson s ratio ν =.8, in (Gavric, 995) the material properties are not specified. [Note: equation (3.56) is solved for k b determining the eigenvalues and eigenvectors of ( ) ( ) k + k + z + + z + zz ω A B B G G G M f = (3.57) If the rows l = + 3 ( =,,,... ) of (3.57) are multiplied b k and the columns l are divided b k, the smmetric eigenvalue problem (3.57) can be reduced to a non-smmetric general eigenvalue problem of the form ( ) Ak + C f = (3.58) 75

90 Chapter 3 Numerical tools for soil-structure interaction where A is obtained b adding to matri adding to matri G G G M the columns l of B + B z.] + z + zz ω A the rows l of B + B z and C is obtained b 35 3 Wavenumber (rad/m) Frequenc (Hz) Figure 3.: Dispersion curves for the UIC86-3 rail: present work (black); Gavric (995) (red) 3.4.5D Boundar Element Method 3.4. Introduction The formulation of the.5d BEM is ver similar to the formulation of the 3D BEM. However, because in the.5d BEM the mesh is D, and because the TLM is used to calculate the fundamental solution, it is possible to evaluate the boundar integrals analticall via modal summation. This feature is not possible with methods such as the stiffness matri method, and represents one of the advantages of the TLM. In this section, the.5d BEM is presented and it is eplained how use the TLM to calculate the boundar integrals Formulation Recall the boundar integral equation epressed in (3.7), which concerns a three-dimensional bod. Since in the present case the bod is assumed to be invariant in the longitudinal direction, the boundar integral (3.7) is equivalent to + + ( ξ * *, ) (, ) ( ξ,, ) ( ξ,, ) (, ) κ uɶ ω = pɶ ω uɶ ω d dγ pɶ ω uɶ ω d dγ (3.59) kl k k kl kl k ΓS ΓS where Γ S is the boundar of the cross section Ω S. The Fourier transform of equation (3.59) with respect to the longitudinal coordinate results in the boundar integral equation in the.5d domain, which is (Galvín et al., ) 76

91 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings * ( ξ,, ) (,, ) ( ξ,,, ) κ u k ω = p k ω u k ω dγ kl k k kl ΓS ΓS ( ξ,,, ω) (,, ω) p k u k dγ * kl k In equation (3.6), uk (, k, ω) and pk (, k, ω) * * wavenumber-space-frequenc domain, while ukl ( ξ,, k, ω) and pkl (,, k, ω) (3.6) are the displacement and traction fields in the ξ are the.5d fundamental displacements and tractions discussed in chapter (Note: in chapter, these variables are written without the over bar). Also, the longitudinal coordinate is dropped, i.e., = (, z) and ξ = ( ξ, ξz ). When ξ belongs to the cross section Ω S then κ kl = δ kl, and when ξ does not belong to the cross section Ω then κ =. S Similarl to the 3D case, the boundar equation (3.6) is not valid for collocation points ξ that belong to the boundar Γ S. Hence, (3.6) must be regularized, and for that purpose the two approaches discussed in section 3..3 can be followed. However, as apposed to the option taken for the 3D case, for the.5d case the first approach is followed (i.e., the limiting procedure ), which ields the regularized boundar integral equation * ( ξ,, ω) (,, ω) ( ξ,,, ω) c u k = p k u k dγ ξ kl k k kl ΓS ΓS kl ( ξ,,, ω) (,, ω) p k u k dγ * kl k (3.6) The boundar coefficient c ξ kl depends on the geometr of the boundar at the collocation point ξ. The calculation of this coefficient is addressed in a later section of this work. To solve equation (3.6), the boundar Γ S is divided into N e boundar elements and N n boundar nodes, similarl to the 3D case: the difference is that in the.5d case the boundar elements consist in one dimensional elements in the z plane while in the 3D case the boundar elements consist in surface elements that can have an orientation in the z space. Within each boundar element the displacement and traction fields are approimated b means of interpolation functions and nodal values, i.e., ( e ) ( e ) where uk ( k, ω ) and k (, ) S ( e) ( ) the ( ) ( e ) N n ( e ) ( e) (,, ω) = (, ω) ( ) ( ) u k u k S (3.6) k k = ( e ) N n ( e ) ( e) (,, ω) = (, ω) ( ) ( ) p k p k S (3.63) k k = p k ω are the nodal values of the th node of the th e element, is the associated interpolation (shape) function evaluated at point ( is contained in th e element) and ( e) functions S ( ) ( ) N is the number of nodes contained in the ( e ) n th e element. The shape used for the.5d case are the same shape functions used for the D boundar elements and can be found in Dominguez (993). 77

92 Chapter 3 Numerical tools for soil-structure interaction Now, assume that the collocation point ξ corresponds to one of the N n nodes of the boundar th (sa, the i node, with coordinates i ). In this case, after accounting for the approimations in equations (3.6) and (3.63), equation (3.6) becomes ( e ) Ukl ( i, k, ω) ( e) Ne N n ( i) ( i) ( e ) * ( e) ckl ul ( k, ω) = pk ( k, ω) ukl (, i, k, ω) S( ) ( ) dγe e= = Γe ( e ) Pkl ( i, k, ω) ( N e ) N e n ( e ) * ( e) uk ( k, ω) pkl (, i, k, ω) S( ) ( ) dγe e= = Γe or in compact form, (3.64) N ( e) N ( i) ( i) ( e) ( e ) kl l k kl i e= = e n (, ω) = (, ω) (,, ω) c u k p k U k ( e) Since the shape functions S ( ) ( ) ( e) Ne Nn e= = e (, ω) (,, ω) u k P k ( e) ( ) k kl i (3.65) are polnomial functions (usuall up to the second degree), for the case of horizontall or verticall oriented boundar elements, the terms ( e) ( e) U, k, ω P, k, ω can be calculated in closed form epressions, as eplained ( ) and kl ( i ) kl i in the following sections. Equation (3.65) is now epanded so that l assumes all the three different directions. Equation (3.65) can then be replaced with ( e ) N N ( i) ( i) ( e ) ( e) e n ( k, ω) = ( i, k, ω) ( k, ω) C u U p e= = e Ne Nn e= = e ( i, k, ω) ( k, ω) P u ( e) ( ) (3.66) where the bold variables have the following meanings (the arguments dropped): i, k and ω are u u = ( i) ( i) ( i) u ( i) uz u u = ( e ) ( e) ( e ) u ( e ) uz p p = ( e ) ( e) ( e ) p ( e ) pz U U U U = ( e ) ( e ) ( e ) z ( e ) ( e ) ( e ) ( e ) U U U z ( e ) ( e ) ( e ) U z U z U zz P P P P = ( e) ( e) ( e) z ( e ) ( e) ( e) ( e) P P Pz ( e) ( e) ( e) Pz Pz Pzz C c c c = ( i) ( i) ( i) z ( i) ( i) ( i) ( i) c c cz ( i) ( i) ( i) cz cz czz Finall, after forcing the collocation point ξ to assume all the N n nodes and assembling the ( e) ( e ) matrices U into a square matri U, the matrices P into a square matri P and the 78

93 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings ( i) matrices C into a square block diagonal matri C, one arrives at the final sstem of equations { ( k, ω) + } ( k, ω) = ( k, ω) ( k, ω) P C u U p (3.67) in which u ( k, ω) is a vector that collects all the nodal displacements and ( k, ω) p is a vector that collects all the nodal tractions. Equation (3.67) can be solved for u, p or a combination of the two, depending on the unknowns of the problem. When the boundar elements are horizontall or verticall oriented, it is possible to calculate the boundar integrals and e (,,, ω) ( ) ( ) U = u k S dγ ( e ) * ( ) kl kl i e Γe e (,,, ω) ( ) ( ) P = p k S dγ ( e ) * ( ) kl kl i e Γe in closed form epressions. The net sections address this issue. (3.68) (3.69) Horizontal boundar elements Horizontal boundaries are defined b a constant depth. If it is assumed that the collocation point ξ is placed at the depth z n ( n th interface of the TLM model) and that the boundar element Γ e is at the depth z ( th m m interface of the TLM model) (Figure 3.), then the integrals (3.68) and (3.69) can be replaced b integrals of the form (the variables k, ω are dropped) ( ξ ) ( ) U = u S d (3.7) ( e) ( mn) ( e) kl Γe kl ( ξ ) ( ) P p S (3.7) ( e) ( mn) ( e) kl = kl ( ) d Γe n m ξ Γ e l l Figure 3.: Horizontal boundar element ( ) The variables u mn kl correspond to the fundamental displacements in the.5d domain ( mn) ( ) described in chapter (and referred to as u αβ ), while p mn corresponds to the.5d fundamental stresses ±, also described in chapter (the positive sign is used when the ( mn) σ α z β outwards normal of the boundar faces the positive z direction, while the negative sign is used otherwise). In order to complete the analtical evaluations of the integrals, equation (3.7) is first changed into a more convenient form. As seen in the previous chapter, the fundamental displacements are obtained through the inversion of the solutions in the wavenumber domain, i.e., kl 79

94 Chapter 3 Numerical tools for soil-structure interaction ( In this equation, mn ) ( ) ( ) different meaning than U e. kl u = U k k (3.7) + ( mn) ( mn) ik kl ( ) kl ( ) e d π U k are the wavenumber displacements defined in Table., having a kl Assuming that the ais is centered at the midpoint of the boundar element (of total width l ), after substituting (3.7) into equation (3.7), the latter becomes l ( e) ( mn) ik ( ξ ) ( e) U kl = U kl ( k ) e dks( ) ( ) d π l = π + ( e) The Fourier transform of S ( ) ( ) + l ( mn) ( e) ik i k ξ kl ( ) ( ) ( ) e d e d l U k S k is defined b (3.73) + l ( e) ( e) ik ( e) ik ( ) ( ) = ( ) ( ) e d = ( ) ( ) e d l S ɶ k S S (3.74) and after introducing this in equation (3.73) one obtains U U k Sɶ k k (3.75) + ( e ) ( mn ) ( ) ( e ) ( ) i k ξ kl = kl ( ) e d π The application of the same procedure to equation (3.7) ields ( The variable mn ) ( ) P k Sɶ k k (3.76) + ( e) ( mn) ( e) ik ξ kl = σ kzl ( ) ( ) ( ) e d π ± U kl k is calculated b modal superposition. If the horizontal boundar is placed at the interface between two thin-laers (a condition that is assumed to be true ( ) throughout the remainder of the formulation), then σ mn kzl corresponds to the consistent nodal ( ) tractions mn ( ) t kl at that interface and thus σ mn kzl ( k ) can also be obtained b modal superposition ( e) Sɶ k can be epressed analticall, then eqs. (.5)-(.59). For these reasons, if ( ) and ( ) ( e ) P kl can also be obtained analticall b modal superposition (in the BEM, the shape ( e) S are usuall polnomial functions whose Fourier transforms can be easil functions ( ) ( ) ( e) calculated). This idea is eplored net for ( ) n ( ) ( even though mn ) ( mn) uαβ ( ) and α zβ ( ) ( mn) ( ) σ ( k ) are finite, and therefore the values of U e and α zβ ( e) U kl n S( ) = S =, n =,,, but first observe that σ become singular when kl (, the variables mn ) ( ) ( e ) kl Uαβ k and P calculated with equations (3.75) and (3.76) are also finite. Hence, when the collocation point belongs to the horizontal ( ) boundar element, P e kl alread includes the factor c kl. In other words, using the proposed procedure, eq. (3.6) can be used directl in place of the regularized equation (3.6), with the term c kl being automaticall accounted for. 8

95 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings ( e) Case : S ( ) S ( ) = = ( ) The Fourier transform of S ( ), according to (3.74), is The coefficients ( e ) kl l kl kl i i i i k Sɶ ( k ) = e d = e e (3.77) l k U are then obtained with and the coefficients + l l i i ( ) i k k e ( mn) ξ + ξ U kl = U kl ( k ) e e dk π k (3.78) P with ( e ) kl + l l i i ( ) i k k e ( mn) ξ + ξ Pkl = tkl ( k ) e e dk π k (3.79) From equations (3.78) and (3.79) it can be concluded that the calculation of the coefficients ( e ) ( ) U and e ( mn ) u (section.5.) and kl P kl follow eactl the same steps as the calculation of kl ( ) ( mn ) ( ) t ( ) (section.5.3), changing onl the integrals I p ( ) b the integrals kl n J of the form ( p ) n ( ) The integrals I ( ).. n ( p) ( p ) l ( p ) l Jn = i In ξ In ξ + required to evaluate the coefficients U and ( e ) kl ( e ) kl (3.8) P are given in Table e Case : ( ) ( ) S = S = The Fourier transform of S ( ), according to (3.74), is l kl kl kl kl i i i i ik i l Sɶ ( k ) = e d = e + e + e e (3.8) l k k ( ) The coefficients U e are then obtained with kl + i i i i ( ) l l l l ( ) i k k k k e mn l ξ + ξ ξ ξ + U kl = U kl ( k ) e + e e e dk π + k k (3.8) and the coefficients P with ( e ) kl + i i i i ( ) l l l l ( ) i k k k k e mn l ξ + ξ ξ ξ + Pkl = tkl ( k ) e + e e e dk π + k k (3.83) From equations (3.8) and (3.83) it can be concluded that the calculation of the coefficients ( e ) ( ) U and e ( mn ) u (section.5.) and kl P kl follow eactl the same steps as the calculation of kl ( ) ( mn ) ( ) t ( ) (section.5.3), changing onl the integrals I p ( ) b the integrals kl n J of the form ( p ) n 8

96 Chapter 3 Numerical tools for soil-structure interaction ( p) l ( p ) l ( p ) l Jn = i n n I ξ + I ξ + + (3.84) ( p ) l ( p ) l In ξ In ξ + ( ) The integrals I ( ) 3.. e Case 3: S ( ) = S ( ) = n The Fourier transform of S ( ) required to evaluate the coefficients, according to (3.74), is U and ( e ) kl P are given in Table l kl kl kl kl kl kl i i i i i i ik i i l l Sɶ ( k ) = e d = e e + e +e e e + 3 (3.85) l 4k k k ( ) The coefficients U e are then obtained with kl ( e ) kl + i i i i ( ) l l l l ( ) i k k k k e mn l ξ + ξ l ξ + ξ U kl = U kl ( k ) e e e +e π + + 4k k and the coefficients P with ( e ) kl l l i ik ξ + ik ξ e e dk 3 k (3.86) + i i i i ( ) l l l l i k k k k e mn l ξ + ξ l ξ + ξ Pkl = tkl ( k ) e e e +e π + + 4k k l l i ik ξ + ik ξ e e dk 3 k (3.87) From equations (3.86) and (3.87) it can be concluded that the calculation of the coefficients ( e ) ( ) U and e ( mn ) u (section.5.) and kl P kl follow eactl the same steps as the calculation of kl ( ) ( mn ) ( ) t ( ) (section.5.3), changing onl the integrals I p ( ) b the integrals kl n J of the form ( p) l ( p ) l ( p ) l Jn = i 4 In ξ In ξ + + ( p ) l ( p ) l l In ξ + In ξ + + (3.88) ( p 3) l ( p 3) l i I n ξ In ξ + ( 3) The integrals I ( ) 3.3. n required to evaluate the coefficients U and ( e ) kl ( e ) kl ( p ) n P are given in Table 8

97 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Table 3.: Closed form epressions for I ( Im k k < ) ( ) n sign ( ) + ( ) i k i k k = e d e π = I k K k i ( k k ) I k K k k ( ) ( ) + ( ) i k i k k k = e d sign e i e π = + k k k sign + ( ) i k i k k k 3 = 3 e d = e e π ik I k K k ( ) sign e k + ( ) i k i k k 4 = 4 e d e π = + i k k k k k k I k K k + ( ) i k i k k 5 = 5 e d e π = ik k k I k K k ( ) + ( ) sign k i k I6 = k K6 e dk π = ik k k ( ) e i k k k k ( ) Table 3.: Closed form epressions for I ( Im k k < ) ( ) n I k K k i k k + ( ) ik e = e d i π = ( k k ) k k ( ) k i k k + ( ) k sign i k e e = e d π = + i k k k k k k I k K k + ( ) i k 3 = 3 e d π I k K k e e = + i k k k k ( ) ( ) ( ) k k ( ) k i k k k i k k + ( ) e i e k k 4 = 4 e d i π = + + k k k k k k k k k I k K k sign + ( ) i k i k k 5 = 5 e d e π = ik k k I k K k I k K k i k k + k i k e e d i 6 6 = π = k ( k k ) k k ( ) 83

98 Chapter 3 Numerical tools for soil-structure interaction Table 3.3: Closed form epressions for I ( Im k k < ) ( 3) n ( ) i k k + ( 3) 3 ik sign e = e d π = + i k k k k I k K k ( k k ) k i k k + ( 3) e 3 i e k k = e d i π = + + k k k k k k k k k k I k K k I ( 3) 3 = π + k 3 K ( ) ( ) ( ) k i k k e i sign e k k 3 e dk = + ik k k k k k k ( ) k i k k + ( 3) e 3 i sign k e k k k 4 = 4 e d π = + + I k K k i I k K k i k k + ( 3) 3 i k e 5 = 5 e d i π = k ( k k ) k k ( ) ( ) ( k k ) k ( k k ) k k k ( k k ) i k k + ( 3) sign k 3 ik e 6 = 6 e d π = + ik k k k k k k I k K k Considerations concerning horizontal boundaries ( ) The calculation of the coefficients U e kl involves onl the components of the modal shapes at the elevation of the collocation point and at the elevation of the boundar element. B ( ) contrast, the calculation of the coefficients P e kl involves the components of all TLM nodes that compose the thin-laer delimiting the boundar element. Since the boundar elements are placed at the interface between two consecutive thin-laers, a decision is required as to whether to consider the upper or the lower thin-laer. When the collocation point is not contained in the boundar element, it is immaterial which thin-laer is used. On the other hand, when the collocation point is contained in the boundar ( ) element, the value of P e kl depends on the thin-laer selected for the evaluation. The rule used in this work is that if the outwards normal faces up, the thin-laer located below the boundar ( e ) is emploed in the calculation of P kl, otherwise the thin-laer above is used. B following this procedure, collocation points on horizontal boundaries are circumvented as depicted in Figure 3.3. It is important to note that according to this procedure, when the collocation point ξ is at an edge of a boundar element ( = ± ξ l ), then the coefficient ( e ) P kl is calculated considering that the boundar is distorted as shown in Figure 3.4. This aspect is important for the treatment of corners, i.e., points where horizontal boundaries meet vertical boundaries (Figure 3.3b-c). 84

99 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings a) ε b) c) Figure 3.3: Eclusions in the domain at the collocation points. a) smooth horizontal boundar; b) concave corner; c) conve corner upper thin-laer ξ lower thin-laer n boundar deflected boundar Figure 3.4: Deflection of the horizontal boundar when the collocation point is at one etreme Due to the homogeneit of laered domains in the horizontal direction, the fundamental solutions depend on the horizontal distance between the source and the receiver, and not on their absolute horizontal coordinates. Hence, if the horizontal boundaries are dicretized in such a wa that the boundar elements have the same length and epansion order, then the ( ) boundar coefficients e ( ) U kl and P e kl can be reused whenever the distance between the collocation points and the boundar elements is repeated. This fact can reduce significantl the cost of computation of matrices U and P. Validation Consider a homogeneous full-space with mass densit ρ =, shear modulus G =, Poisson s ratio ν =.5 and hsteretic damping ξ = ξ =.. The full-space is simulated with a TLM p s model consisting of an elastic laer with thickness H = that is divided into 4 thin-laers of quadratic epansion and that is supplemented with two PMLs, one at the top and the other at the bottom, with parameters m =, η =, Ω = 8 and N = (see section.6). Consider also a horizontal boundar element (of quadratic epansion) whose width is l BEM =. and place it at the depth z BEM = and horizontal coordinate BEM =. The shape functions associated with the nodes of the boundar elements are (from left to right) 85

100 Chapter 3 Numerical tools for soil-structure interaction S ( ) = + 4 S ( ) = lbem S3 ( ) = + lbem lbem lbem lbem (3.89) In the following, the coefficients U kl and P kl associated with the middle node (node ) are calculated with the procedure described in this section and compared with the results obtained through simple numerical integration ( integrating points) of the analtical solutions (Tadeu and Kausel, ). Three collocation points are considered: the first is placed at the position ξ = (.5,.5) (outside the boundar element), the second at ξ = (.5, ) (right edge of the boundar element) and the third at ξ 3 = (, ) (center of the boundar element). For the last collocation point and for the integration of the analtical solutions, the boundar contour is deflected as indicated in Figure 3.3a in order to avoid the singularities. The assumed frequenc is f = Hz ( ω = π rad/s ). Figure plot the comparison between the results obtained with the two approaches. As can be observed from Figure 3.5, the two approaches ield practicall the same results, thus validating the procedure. Figure 3.6 provides eactl the same conclusions: discrepancies can be noticed for some components of U kl and P kl but these differences eist merel due to some residual values of the TLM results. In Figure 3.7, the agreement is also ver good: also in this eample, the differences are caused b residual values. Notice that the last two collocation points are placed inside the boundar element and that no special treatment is given to the fundamental solutions of the TLM and that the results are still correct. This confirms that using this procedure the coefficients c kl are alread accounted for during the calculation of the boundar integrals P kl : for the collocation point ξ 3, the components P, P and P zz correspond to ckl =.5δ kl (smooth boundar). 86

101 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings U U U z k -5-3 k k U U 5 U z k k -5. k U z U z U zz - k -5 k -. k... P P P z -.. k -..4 k -.. k P. P P z -.. k -.. k -..5 k P z -. P z -. P zz -.4 k -.4 k -.5 k Figure 3.5: Boundar coefficients U kl and P kl for ξ. Solid lines TLM solution; circles analtical solutions. Blue real component; red imaginar component 87

102 Chapter 3 Numerical tools for soil-structure interaction U U - - U z k -3.4 k k U - - U. U z -3-3 k k -5.5 k U z U z U zz - k -5 k -.5 k P -5 P P z k k k P P -5 P z k -. k k P z P z P zz -. k -. k -5 k Figure 3.6: Boundar coefficients U kl and P kl for ξ. Solid lines TLM solution; circles analtical solutions. Blue real component; red imaginar component 88

103 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings. -9 U U U z k - k.4 - k -3 U U. Uz k k -.5 k U z U z.5 U zz - k k k P.5 P P z k - k -5 k P P.5 P z k -.5. k -.4 k P z 5 P z. P zz.5-5 k k -.5 k Figure 3.7: Boundar coefficients U kl and P kl for ξ 3. Solid lines TLM solution; circles analtical solutions. Blue real component; red imaginar component 89

104 Chapter 3 Numerical tools for soil-structure interaction Vertical boundar elements Vertical boundaries are defined b a constant horizontal coordinate BEM (Figure 3.8). If it is assumed that the load is applied at the depth z n ( n th interface of the TLM model) and that the th th boundar element is placed between depths z m and z m ( m and m interfaces of the TLM model), then the integrals in equations (3.68) and (3.69) can be replaced b integrals of the form (for convenience, the variables k, ω are dropped) m ( e) ( mn) ( ) kl = kl ( BEM ξ ) ( m) ( ) ( e) ( ) d m= m U u N z S z z (3.9) m ( e) ( mn) ( e) kl = ± σ kl ( BEM ξ ) ( m) ( ) ( ) ( ) d m= m P N z S z z (3.9) ( mn) ( mn) In these equations, the factors u ( ) N ( z) and ( ) N ( z) kl BEM ξ ( m) σ represent kl BEM ξ ( m) the verticall interpolated displacements and tractions fields, with N ( ) ( m) z being the TLM th shape function associated with the m interface. In equation (3.9), the positive sign must be used if the outwards normal is in the positive direction, while the negative sign must be used otherwise. n m m ξ Γ e l BEM l BEM Figure 3.8: Vertical boundar element ( Since mn ) ( ukl ( BEM ξ ) and mn ) kl ( BEM ξ ) σ are nodal values and therefore do not depend on the depth z, the epressions (3.9) and (3.9) can be replaced b m ( e) ( mn) ( ) kl = kl BEM m e m= m ( ξ ) ( ) ( ) ( ) ( ) d (3.9) U u N z S z z m ( e) ( mn) ( e) kl = ± kl BEM m m= m σ ( ξ ) ( ) ( ) ( ) ( ) d (3.93) P N z S z z ( e) Thus, onl the integrals of the form N( m) ( z) S( ) ( z) dz need to be evaluated. Since N( ) ( z ) m ( e) and S ( z ) are both polnomial functions, these integrals can be evaluated in closed form. ( ) As final note, since the displacements are interpolated in the vertical direction using polnomial functions, the singular behavior of the fundamental solutions is not captured. Hence, when the collocation point lies within the vertical boundar element, in the calculation ( ) of P e kl the term c kl is not accounted for. Nonetheless, since the boundar elements are verticall oriented and the fundamental solutions are smmetric with respect to vertical planes, the resulting value for the missing term is ckl =.5δ kl. In this wa, for nodes that belong to vertical boundar elements and that do not correspond to corners, the term 9

105 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings ckl =.5δ kl must be added to the diagonal of P associated with the node. When the node corresponds to a corner, two situations occur:. Concave corner (Figure 3.3b): in this case, because the horizontal boundar element alread accounts for the quarter of circle of the deflected boundar (Figure 3.4), then the factor c kl must onl account for the remaining semi-circle, and so c =.5δ ; kl kl. Conve corner (Figure 3.3c): the horizontal boundar element alread accounts for the quarter circle of the deflected boundar (Figure 3.4), and so the factor c kl is null. Validation Recall the homogeneous full-space used to validate the horizontal boundar elements, which is simulated with the same TLM model, and consider a vertical boundar element with width l BEM =., centered at ( BEM, z BEM ) = (,) and of quadratic epansion (the boundar element is contained in two distinct thin-laers). The shape functions of the boundar element are the same as in equation (3.89), with the argument being replaced b z. In the following, the boundar coefficients U kl and P kl associated with the middle node (node ) are calculated with the procedure described in this section and compared with the results obtained through simple numerical integration ( integrating points) of the analtical fundamental solutions (Tadeu and Kausel, ). Three collocation points are considered: the first is placed at the position ξ = (.5,.5) (outside the boundar element), the second at ξ = (,.5) (upper edge of the boundar element) and the third at ξ 3 = (, ) (center of the boundar element). The assumed frequenc is again f = Hz ( ω = π rad/s ). Figure plot the comparison between the results obtained with the two approaches. Such as in the case of the horizontal boundar element, a ver good agreement is obtained between the two procedures. There is however a shift in the real components of the boundar coefficients P z and P z for ξ = ξ and this difference does not vanish with the refinement of the TLM model. Nevertheless, as it is seen b the eamples described in the net section, despite these differences, the results obtained with the TLM-BEM ehibit a good qualit. 9

106 Chapter 3 Numerical tools for soil-structure interaction U U U z k -5-3 k k U U 5 U z k k -5. k U z U z U zz - k -5 k -. k.5..4 P P -. P z k -.4. k -.. k P P -. P z -.. k -.4. k -.. k P z P z P zz -. k -. k -. k Figure 3.9: Boundar coefficients U kl and P kl for ξ. Solid lines TLM solution; circles analtical solutions. Blue real component; red imaginar component 9

107 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings.5 U U U z -.5 k -.4 k k U U. U z - k k.5 k U z U z U zz - k k -.5 k.. P P P z - k -. k -. k P -. P P z -.4. k - k - k P z P z P zz -. k - k - k Figure 3.: Boundar coefficients U kl and P kl for ξ. Solid lines TLM solution; circles analtical solutions. Blue real component; red imaginar component 93

108 Chapter 3 Numerical tools for soil-structure interaction. U U U z -. k -.5 k - -3 k U U U z - k k -. k U z U z U zz - k - k -. k. - P.5 P. P z -.5 k k - -8 k P -. P.5 P z k k - k P z P z P zz - k - k - k Figure 3.: Boundar coefficients U kl and P kl for ξ 3. Solid lines TLM solution; circles analtical solutions. Blue real component; red imaginar component 94

109 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Outgoing stress fields To calculate the response of a close b structure due to wave-fields that are generated at the boundar Γ S, both the outgoing displacement field and the outgoing stress field that reach the structure must be known. The outgoing displacement field is calculated using eq. (3.6), where ξ corresponds to a point of the laered domain but outside the boundar Γ S, and so κ kl = δ kl. Its calculation follows what is described in the two previous subsections. The calculation of the outgoing stress-field can be based on the derivatives of the displacements, which are obtained through the derivation of eq. (3.6) with respect to ξ. For the generic direction δ =,, z, the derivative of equation (3.6) is * ( ξ ω) ( ω) δ ( ξ ω) ξ u, k, = p, k, u,, k, dγ l, δ k kl, ΓS ΓS ( ξ ω) ( ω) p,, k, u, k, dγ * kl, δξ k If δ =, the derivative can be easil calculated as ul, ( ξ, k, ω) = i k ul ( ξ, k, ω). (3.94) For the horizontal derivative, since the domain is homogeneous in the horizontal direction, eq. (3.94) is equivalent to * ( ξ ω) ( ω) ( ξ ω) u, k, = p, k, u,, k, dγ + l, k kl, ΓS ΓS ( ξ ω) ( ω) p,, k, u, k, dγ * kl, k (3.95) Conversel, in the vertical direction the domain is not homogeneous, and so the previous equivalence is not valid. Hence, in order to calculate the vertical strains u l, z, a better approach is to compute the displacements at two consecutive nodal interfaces and then calculate the derivative with u l, z (, k, ω) ( ξ +,, ω) ( ξ,, ω) u z k u k l l ξ = (3.96) Returning to the horizontal direction, after discretization of the boundar (3.95) gives rise to the matri equation z ξ,,, S Γ S, equation u = U p + P u ξ Γ (3.97) where P, is a matri that collects the coefficients P of the form ( e ) kl, e ( ω) ( ) P = p,, k, S dγ ( e) * ( ) kl, kl, i ( ) e Γe and U, is a matri that collects the coefficients U of the form ( e) kl, e ( ω) ( ) U = u,, k, S dγ ( e) * ( ) kl, kl, i ( ) e Γe (3.98) (3.99) The vectors u and p collect the nodal displacements and tractions at the boundar Γ S. 95

110 Chapter 3 Numerical tools for soil-structure interaction For vertical boundaries Γ e, the quantities in equation (3.98) and (3.99) are calculated as described in section 3.4.4, i.e., after calculating the nodal values of the interested quantities in the space domain (see chapter ), the integrals are calculated through polnomial integration. ( e) ( e ) ( ) For horizontal boundaries, the quantities U and P are calculated as U e ( ) and P e in section 3.4.3, with the integrals of the form kl, ( p ) i kl, I replaced with i. ( p+ ) I i Coupling.5D BEM and.5d FEM The coupling between the BEM and FEM sub-domains in the wavenumber domain follows the steps that are eplained in section 3..5 for the case of BEM-FEM coupling in the space domain. The main differences consist in the matri K ɶ, that is now ( ) ( ) K = A k + k B + B + G + G + G ω M (3.) z z zz and in the shape functions that are used in the calculation of T equation (3.5) that in this case must be the shape functions of D finite and boundar elements. The application of the steps epressed in equations (3.) to (3.33) ield the final sstem of equations, which is of the form F K I,I + K BEM K I,II u I FBEM = F K II,I K II,II u II fii kl kl (3.) F F This sstem is solved for u I and u II in order to obtain the.5d displacements of the FEM B domain. The displacements u I of the BEM domain are obtained subsequentl b application of equation (3.4) the over tilde is replaced b a bar and the boundar tractions p I and B displacements u II are obtained after solving the sstem of equations (3.) for these variables again, the over tildes are replaced b bars. [Note: in order to add K BEM to the sub-matri K I,I, after its calculation according to (3.3), the columns related to the dofs must be multiplied b i while the rows related to the dofs must be multiplied b i. For the same reasons, before subtracting the vector F BEM at the right-hand side of equation (3.), after its calculation according to equation (3.33), the rows F related to the dofs must be multiplied b i. After solving the sstem (3.) for of BEM F u I and F u II, the rows related to the dofs of these vectors must also be multiplied b i.] Conclusions In this section, a.5d BEM procedure based on the TLM fundamental solutions is presented. For horizontal boundar elements, the BEM coefficients are calculated directl based on a modal superposition, rendering accurate results and accounting for the singularities of the fundamental solutions. For vertical boundar elements, the verticall interpolated fundamental solutions are integrated analticall but the singularities are not accounted for: to account for the singular behavior of the fundamental solutions, the term c kl must be added à posteriori, being its value.5δ kl in smooth vertical boundaries or concave corners and being null in conve corners. When compared with the BEM procedures based on the analtical fundamental solutions of full-spaces, the proposed procedure presents the advantage of simulating horizontall laered domains with the same ease as homogeneous domains, and of avoiding the discretization of 96

111 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings free surfaces and laer interfaces. When compared with the BEM procedures based on fundamental solutions obtained with wavenumber transfer or stiffness matrices, the proposed method has the advantage of evaluating the inverse Fourier transform in closed form epression, which ield more accurate results. In addition, the proposed methodolog turn out to be more user friendl than other procedures based on the stiffness/transfer matrices since the definition of a proper wavenumber sample k is replaced b the subdivision of the laered domain into small thin-laers, a task that is far simpler. As maor drawbacks, the proposed procedure requires the solution of two eigenvalue problems for each frequenc, and since it is based on modal combinations, it might be inefficient for ver deep structures that require soil models with a large number of thin-laers and interfaces. The proposed methodolog onl considers horizontal and vertical boundar elements. If the actual boundar presents inclined surfaces, such geometr can be achieved b filling the irregular volume with finite elements. As a final remark, the TLM model must be compatible with the BEM mesh in such a wa that:. The horizontal boundaries are placed at the interface between two thin-laers and not inside a thin-laer;. The etremities of vertical boundaries correspond to interfaces between thin-laers and not to intermediate elevations within the thin-laers; 3. If there are boundar nodes inside vertical boundar elements (in constant and quadratic boundar elements, for eample), these nodes must be located at the interface of thin-laers; 4. It is not recommended that the horizontal boundar elements be smaller than the thickness of the thin-laers. Likewise, it is not recommended that the distance between vertical boundar elements at the same elevation be smaller than the thickness of the thin-laers D BEM-FEM validation eamples In sections 3.3 and 3.4, the.5d FEM and the.5d BEM are presented and validated: the.5d FEM is validated through the determination of the dispersion curves of a UIC86-3 rail, ( ) while the.5d BEM is validated through the calculation of the BEM coefficients U e kl and ( e ) P kl for horizontal and vertical boundar elements. In the present section, the coupled.5d BEM-FEM procedure is validated b means of two eamples: the first eample consists in the calculation of the response of a square tunnel inside a laered domain, while the second eample consists in the calculation of the response of a homogeneous slab free in space. The results of the.5d BEM-FEM procedure are compared with the results of the.5d FEM procedure Eample square tunnel in a laered medium In the present subsection, the dnamic compliances of a tunnel are computed and compared with the corresponding values obtained with a.5d finite element approach. The tunnel is massless, has rigid cross section, and is placed inside a horizontall laered domain. The geometr and properties of the problem are illustrated in Figure

112 Chapter 3 Numerical tools for soil-structure interaction ρ, G, ν, ξ u u u u ρ, G, ν, ξ z H H ρ, G, ν, ξ H H L ρ, G, ν, ξ l l l l Figure 3.: Square tunnel inside a horizontall laered domain Since the cross section of the tunnel is rigid, the displacements of the walls of the tunnel can be described as function of the translation and rotation of the tunnel, i.e., Tunnel Tunnel u u (, z) z Tunnel Tunnel Tunnel u z u (, z) = = z = Tunnel θ uz (, z) Tunnel θ Tunnel θ z Nu N u (3.) On the other hand, the pressures that the laered domain transmits to the walls of the tunnel induce at the center of the tunnel forces and moments that are calculated with (, z) ( ) (, z) Tunnel T T f = f f fz m m m z = N p, z dγ (3.3) Γ p z where Γ represents the boundar of the tunnel. After discretizing the surface of the laered domain that is in contact with the tunnel into boundar elements and N boundar nodes, the nodal displacements u at the boundar are obtained with The forces p (, z ) u N Tunnel = NUu NU = (3.4) u N (, z ) N N N Tunnel f are obtained from the boundar pressures p through u 98

113 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings p Tunnel T T f = N P NP = N (, z) S (, z) d Γ N (, z) SN (, z) dγ (3.5) Γ Γ p N with S (, ) z being the shape function associated to the Replacing eqs. (3.4) and (3.5) in equation (3.67) ields ( P ( ) U ) th boundar node. N U P + C N u Tunnel = f Tunnel (3.6) and so the compliance matri corresponds to the 6 b 6 matri F obtained with ( ( ) ) P U F = N U P + C N (3.7) In the subsequent eamples, the components of the compliance matri F are evaluated using the.5d BEM methodolog eplained earlier. The tunnel is given the cross section H = L = [m] and each edge of the tunnel is divided into 5 boundar elements of quadratic epansion (3 nodes per boundar element). The total number of nodes is then N = 4. To validate the results, the compliance matrices are also calculated using a finite element model coupled with PMLs, which are obtained as eplained in (Kausel and Barbosa, ). The ecitation frequenc is ω = π [rad s] and the wavenumbers k range from to 6 π [rad m] (3 wavenumbers). Homogeneous full-space 3 The material properties of the full-space are: mass densit ρu = ρ = ρ = ρl = kg m ; shear modulus Gu = G = G = Gl = Pa ; Poison s ratio ν u = ν = ν = ν l =.5 ; hsteretic damping ξu = ξ = ξ = ξl =.. The TLM model consists of the 4 macro-laers identified in Figure 3., where the upper and the lower semi-infinite elements are modeled with PMLs (with parameters η =, Ω = 8, N =, m = ; see chapter for definition of variables), and the middle laer satisf H = H = m and are divided into 4 thin-laers of quadratic epansion. Due to smmetr conditions, onl the components f, f, f zz, f θ θ, f θ θ, f θ z θ, f z θ = f z θz and f zθ = f θ z are non-zero. Also, due to the geometr of the problem, f = fzz, fθ θ = f θzθz and f θ = f z zθ. Hence, considering onl the five compliance components f, f, f θ θ, f θ θ and, it is possible to describe the entire sstem. In Figure 3.3, the 5 components of the f θ z compliance matri obtained with the proposed procedure (solid lines) are compared with the results obtained with the FEM (black circles). Blue is used for the representation of the real part, while red is used for the imaginar part. Figure 3.3 shows that the two approaches ield virtuall identical results, leading to the conclusion that both procedures are correct. It can also be observed that the in-plane components ( f, f θ θ and f θ ) present singularities at k z = ks = ω CS = π. It should be noted that, because in this eample the soil is a homogeneous, infinite space, it follows that the classical BEM that uses the fundamental solutions of a full-space has a clear advantage over the use of BEM-TLM. However, this problem of ver simple geometr is used solel for validation purposes. In the net eamples, the use of full-space fundamental solutions requires the discretization not onl of the edges of the tunnel but also of the free- 99

114 Chapter 3 Numerical tools for soil-structure interaction surfaces and of the interface between different laers, and now the TLM offers clear advantages inasmuch as these interfaces need not to be discretized f f f θ θ k π k π f θ θ k π -. 3 k π f θ z k π Figure 3.3: Tunnel compliances for the full-space case. Solid lines =.5D BEM (real part blue; imaginar part red). Black circles = FEM Homogeneous laer free in space The free laer consists of the two intermediate macro laers depicted in Figure 3. ( H = H = m ). The material properties of the free laer are the same of the full- space considered in the previous eample. The TLM model is similar to the one used therein, but with the upper and lower PMLs ecluded. Again, due to smmetr conditions, onl the components f, f, f zz, f θ θ, f θ θ, f θ z θ, f z θ = f z θz and f zθ = f θ z do not vanish. However, the identities f = fzz, fθ θ = f θzθ and f z θ = f z zθ do not hold, and so a total of eight components of the compliance matri are needed to describe the sstem. Figure 3.4 shows the eight components obtained with the proposed methodolog and with the FEM. Once again, the results obtained with the two procedures match perfectl.

115 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings f f k π k π. f. f θ θ k π k π f θ θ f θ θ k π k π f θ z f θ z k π k π Figure 3.4: Tunnel compliances for the free laer in space. Solid lines =.5D BEM (real part blue; imaginar part red). Black circles = FEM

116 Chapter 3 Numerical tools for soil-structure interaction Homogeneous half-space The material properties of the homogeneous half-space are the same as in the previous case. The TLM model differs from the model in the first eample in that the upper PML is ecluded. In this case, distinct components are needed to define the compliance matri, whose structure is f f θ f θ z f f z f θ f z fzz f zθ F = f f z f (3.8) θ θ θθ f θ f f θ θ θ θz f θ f f z θ θz θzθ z The components of the compliance matri are plotted in Figure 3.5. A good agreement is once again reached. Laered half-space The case of a non-homogeneous half-space is considered net. The properties of the laers, based on Figure 3., are the following: ρ =, G = (the upper half-space does not eist) u u ρ =. kg m, G =.Pa, ν =.5, ξ =., H = m 3 ρ =.3kg m, G =.Pa, ν =.3, ξ =., H = m 3 ρ = ρ, G = G, ν = ν, ξ = ξ l l l l Each of the phsical laers are modeled with 4 thin-laers based on a quadratic epansion. The lower half-space is modeled with PMLs with the same parameters used in the first eample ( η =, Ω = 8, N =, m = ). As in the case of the half-space, the components given b eq. (3.8) are needed to define the compliance matri F. These compliance components, obtained with the proposed procedure and with the FEM, are plotted in Figure 3.6. Again, the agreement between the proposed method and the FEM is ver good. It can be concluded from this eample that the BEM based on the TLM fundamental solutions can correctl simulate horizontall laered domains without the need to discretize the interfaces between laers, as is necessar in the standard BEM.

117 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings f f f zz k π k π k π f θ θ f θ θ -. f θ θ.6.4. z z k π k π k π f θ z f θ f z k π k π k π f θ fz θ f θ θ z k π -. 3 k π k π Figure 3.5: Tunnel compliances for the half-space. Solid lines =.5D BEM (real part blue; imaginar part red). Black circles = FEM 3

118 Chapter 3 Numerical tools for soil-structure interaction f f f zz k π k π k π f θ θ f θ θ f θ θ z z k π k π k π f θ f θ z f z k π k π k π f θ k π.6.4. f θ z k π f θ θ.4. z k π Figure 3.6: Tunnel compliances for the laered half-space. Solid lines =.5D BEM (real part blue; imaginar part red). Black circles = FEM 4

119 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings 3.5. Eample slab free in space For the second eample an infinitel long slab with width L = m and thickness H =. m that is free in space is considered. The material properties of the slab are: mass densit 3 ρ = kg m ; shear modulus G = Pa ; Poison s ratio ν =.5 ; material damping ξ =.. The slab is submitted to a vertical line load at its top (at the middle alignment) and the displacements of the top right corner, center right point and bottom right corner are calculated. The ecitation frequenc is ω = π rad/s and the range of wavenumbers of the line load is k = [,6 π ] rad/m. The displacements of the points referred to above are calculated using a.5d FEM procedure and using a coupled.5d BEM-FEM procedure, and then compared. For the first approach the cross section of the slab is divided into a regular mesh of 8 solid elements of quadratic epansion (8 nodes per element), while for the second approach the cross section of the slab is divided into two sub-domains: a FEM sub-domain, with dimensions m.5m and divided into a regular mesh of 4 solid elements of quadratic epansion; and a BEM sub-domain with the same dimensions, whose lateral boundaries are divided into 4 boundar elements each and whose interface between the BEM and the FEM sub-domains is divided into boundar elements. The boundar elements are of quadratic epansion (3 nodes per element). The fundamental solutions of an elastic laer free in space, calculated with the TLM (the elastic laer is divided into 3 thin-laers of quadratic epansion), are used to nurture the boundar elements: this choice for the fundamental solutions avoids the discretization of the free lower surface of the slab, thus reducing the cost of computation of the BEM matrices. The results obtained with the two approaches are compared in Figure for the 3 points considered. As can be observed, the agreement is ver good even for the center right point, which belongs to the BEM-FEM interface, and for the bottom left corner, which belongs to the BEM domain. The good qualit of the results validates the two procedures. As a concluding remark regarding this eample, it is important to be aware that the use of the.5d BEM-FEM is inefficient when compared to the.5d FEM. The structure under analsis is finite and of relativel small dimensions, which means that the reduction in the number of degrees of freedom achieved with the BEM does not compensate the computational cost associated with the calculation of the BEM matrices. This eample is considered herein solel for validation purposes, and not to demonstrate the advantages of the.5d BEM. 5

120 Chapter 3 Numerical tools for soil-structure interaction 4 a) 4 b) u - u k k 5 c) u z k Figure 3.7: Displacements of the top right corner of the slab: a) u ; b) u ; c) u z. Solid lines =.5D BEM (real part blue; imaginar part red). Black circles = FEM.3 a) b)..5 u. -. u k k 5 c) u z k Figure 3.8: Displacements of the center right point of the slab: a) u ; b) u ; c) u z. Solid lines =.5D BEM (real part blue; imaginar part red). Black circles = FEM 6

121 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings u u k k 5 u z k Figure 3.9: Displacements of the bottom right corner of the slab: a) u ; b) u ; c) u z. Solid lines =.5D BEM (real part blue; imaginar part red). Black circles = FEM 3.6 Conclusions In this chapter, the numerical tools used for the solution of soil-structure interaction problems are presented and validated. To solve the interaction between the track and the soil, a coupled.5d BEM-FEM procedure is used, and for the case of the interaction between the building and the soil, a coupled 3D BEM-FEM procedure is used. Both procedures are based on the fundamental solutions obtained with the TLM, which is described in chapter. The.5D BEM-FEM procedures are developed in the space-wavenumber-frequenc ( k, ω ) domain. In order to use the responses obtained with this methodolog in a coupled 3D BEM- FEM procedure, the responses must first be transformed to the space-frequenc (, ω ) domain, which is accomplished b an inverse Fourier transform. After having calculated the responses in the ( k, ω ) domain for a discrete sample of the wavenumber k, the inverse transform can be calculated numericall b means of a summation. It is important to recall that when the proposed.5d BEM methodolog is used, the coefficients for the BEM matrices can be calculated in closed form epressions. This fact leads to fast and precise calculations of such coefficients. The drawback of this approach is the time required to calculate the eigenmodes of the soil, which can become large when the fundamental solutions are needed at deep positions. Nevertheless, for each soil profile, the eigenmodes onl have to be calculated once for each frequenc. Then, the can be stored and reused to analze different configurations of tracks, buildings and countermeasures. Concerning the BEM procedures, the step that consumes more time is the calculation of the BEM matrices P and U. The components of these matrices are obtained b appling a load 7

122 Chapter 3 Numerical tools for soil-structure interaction at one BEM node and b integrating the fundamental solutions over the surface of the BEM element, being this procedure repeated Nn Ne times ( N n is the number of boundar nodes and N e is the number of boundar elements). Since these Nn Ne calculations can be performed simultaneousl, parallel computing becomes of great advantage. Using CPU parallelization, the calculation of the matrices P and U can become up to N CPU times faster than without parallelization, since the Nn Ne calculation can be divided b N CPU (number of CPUs). During the last two decades, the GPU boards, commonl used for image processing, have been used for scientific calculation and in some cases a reduction of two orders of magnitude in the time needed for the calculations has been achieved (Hwu and Kirk, 9). The GPU processors allow a large number of processes to run simultaneousl, at the epense of slower clock speeds. Also, the memor access of GPUs is more complicated than the RAM access b the CPUs, and so the algorithms must be properl adapted so that time needed to access the memor does not reduce the efficienc. During the PhD works, a GPU implementation has been attempted for the calculation of matrices P and U. The resulting program did work correctl, but an improvement in the calculation time could not be achieved, since at the end, the computational time needed to run a calculation with GPU parallelization was roughl the same computational time that was needed to run the same calculation parallelized with the CPU. The author believes that the reason for not achieving better performances is related to the GPU memor access. The results contained in this chapter were obtained with MATLAB. No special attention was given in terms of the efficienc of the calculations or versatilit of the subroutines, since the main purpose of these calculations was the validation of the procedures described throughout this chapter. For chapters 4 and 5, in order to obtain better computational costs, the.5d BEM-FEM and the TLM were implemented in FEMIX, which is a finite element code written in C programming language ( 8

123 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings 4. Invariant structures subected to moving loads and moving vehicles 4. Introduction In problems of vibrations induced b moving vehicles, the vehicle transmits to its supporting structure a set of moving forces (as man as the number of contact points/surfaces). For the cases in which the dnamic behavior of the vehicle is considered, the interaction between the vehicle and the supporting structure causes the vehicle to respond dnamicall, and consequentl the contact forces between the two structures are not constant in time. Even though in general the most significant contribution to the magnitude of the contact forces is attributable to the weight of the vehicle (quasi-static component), the oscillating (dnamic) component, whose contribution to the total force is smaller, has an important role in the response of the supporting structure, speciall for points located at remote positions. Hence, it is of great importance to consider the dnamic component of the transmitted forces in order to obtain accurate predictions of the vibrations in the nearb buildings and in the far field. The dnamic response of the vehicle is caused, among other aspects, b longitudinal variations of the stiffness of the supporting structure and b geometric irregularities observed at the contact surface between the vehicle and the structure. In this work, because it is assumed that the track-soil sstem is invariant in the longitudinal direction, the variations of stiffness cannot be considered, and therefore the ecitation associated with the discrete sleeper support is not accounted for. Regarding the geometric irregularities at the contact surface between the vehicle and the supporting structure (in the railwa case, wheel-rail contact), the are considered b means of position dependent gap/irregularit profiles. These irregularit profiles ma account for both the unevenness of the track and the imperfections of the wheels. In the present chapter, the.5d BEM-FEM procedure presented in the previous chapter is emploed in the analsis of longitudinall invariant structures that are subected to moving loads. The results obtained with the.5d BEM-FEM are in the ( k, ω ) domain, and in order for them to provide meaningful information, the must be transformed to the space-frequenc domain, and in some cases also to the space-time domain. The transformation to these domains can be simplified due to the moving nature of the perturbation. After describing the solution procedure for the case of moving loads, the vehicle-structure interaction problem is addressed. The solution of this interaction problem returns the forces that the vehicle induces in the structure. In the contet of vibrations induced b railwa traffic, this corresponds to the generation stage, the last stage to be described in this work (the propagation and the reception stages were eplained in the previous chapter). Having characterized the forces that the train transmits to the track, the tools eplained in chapter 3 and the epressions for moving loads described in this chapter can be used to determine the response of the track, soil and nearb buildings. 9

124 Chapter 4 Invariant structures subected to moving loads and moving vehicles 4. Moving loads 4.. Introduction The results obtained with the.5d BEM-FEM procedure are in the wavenumber-frequenc domain ( k, ω ), and in order for them to be meaningful and be used as inputs in the 3D BEM- FEM procedure, the results have to be transformed to the space-frequenc domain (, ω ), and in some cases, to the space-time domain (, t ). The transformation to the space-time domain requires the evaluation of the double inverse Fourier transform in which H ( k, ω) h (, ω ) + + ik iωt,, e d e d π π ( ) = ɶ ( ω) h t H k k ω (4.) ɶ corresponds to some response field in the wavenumber-frequenc domain (displacement, derivative, traction, etc) and in which h(, t ) corresponds to the associated response field in the space-time domain. When the response is needed in the space-frequenc h, ω is required, onl the inner integral needs to be evaluated. domain, i.e., when ( ) In some practical cases, the longitudinal and temporal variations of the load are independent of the variation within the cross section of the domain in such a wa that at an given instant P, t can be factorized into the product t and an given cross section, the load vector ( ) (, t) = p(, t ) P p (4.) where p is a vector containing the distribution of the load within the two dimensional cross section (this vector is to be used as right-hand side of equation (3.)) and where p(, t ) is a function representing the evolution of the load in time and with the longitudinal coordinate. Hɶ k, ω can be written as Under this condition, the response fields ( ) being h( k, ω) (, ω) = ɶ (, ω) (, ω) H ɶ k h k p ɶ k (4.3) ɶ the response function (usuall called transfer function) obtained with the.5d BEM-FEM tool (and considering that the load vector corresponds to p ) and being pɶ ( k, ω) the wavenumber-frequenc content of the load, calculated with + + ( ) ( ) i k -i ω = ωt pɶ k, p, t e d e dt (4.4) For general structures in which the transfer functions h( k, ω) ɶ cannot be determined analticall and for general load variations p(, t ), the inverse Fourier transforms (4.) can onl be calculated numericall based on the FFT technique or based on some numerical h ɶ k, ω need to be evaluated for a large range of k integration scheme, and so the values of ( ) and ω. For instance, an impulsive point load applied at the instant t = and at the longitudinal coordinate p, t = δ δ t and its wavenumber-frequenc content is ɶ ( ) = is defined b ( ) ( ) ( ) p k, ω =. Hence, the integral (4.) becomes

125 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings + + h t h k k h k k π π 4π ik iωt (, ) = ɶ (, ω) e d e d ω ɶ ( i, ω ) i ω (4.5) For loads moving with constant speed, the frequenc-wavenumber content p( k, ω) i ɶ contains a special structure that enables the calculation of one of the integrals in equation (4.) directl. This issue is addressed in the following sub-sections, where constant loads and oscillating loads moving with constant speed are considered and eemplified for the case of a beam on a Kelvin foundation. 4.. Constant moving loads Consider that a load moving with constant speed V and constant amplitude A crosses the longitudinal section = at the instant t =. The function p(, t ) associated with such load is and its wavenumber-frequenc content is (, ) δ ( ) p t = A Vt (4.6) + + ( ω) δ ( ) ik -iωt pɶ k, = A Vt e d e dt = + ( ) ik -it ( ω kv ) ik = A e e dt = π A e δ k V ω The insertion of (4.7) and (4.3) in equation (4.) ields (4.7) h t A hɶ k ω δ k V ω k ω (4.8) + + k iωt,, e d e d π i ( ) ( ) = ( ) ( ) and after solving equation (4.8) for the inner integral, the following result is obtained h (, ω) + + A -i i i (, ) ω (, ) e e d t V t A ω ω V h t = hɶ ω V ω ω = hɶ ( ω V, ω) e dω π V πv The previous equation reveals that for the case of moving loads, the transformation of the response from the wavenumber-frequenc domain to the space-frequenc domain can be accomplished without solving an integral. Instead, such inverse transformation is obtained h ɶ k, ω at the wavenumber-frequenc pair ( ω V, ω ). b evaluating the transfer function ( ) In the following eample, h( k, ω) h(, t ). For more general structures in which the transfer function h( k, ω) (4.9) ɶ can be determined analticall and consequentl so does ɶ cannot be evaluated in closed-form epressions, the remaining integral in eq. (4.9) can onl be solved numericall using, for eample, a discrete inverse Fourier transform (as in this work), an adaptive Filon method (De Barros and Luco, 994), a classical FFT, or a logarithm FFT (Talman, 978). This last technique is used in the works developed at Leuven (e.g., François et al., ). Before the presentation of the eample, it is important to note that the space-time domain t V and not of and t separatel. This aspect response field is a function of ( )

126 Chapter 4 Invariant structures subected to moving loads and moving vehicles indicates that for moving loads with constant amplitude, the response field moves with the load and is constant in time. Beam on a Kelvin foundation subected to a constant moving load The eample of a beam on a Kelvin foundation (Andersen and Nielsen, 3) is used net to illustrate the procedure. This eample is chosen because the displacements in the wavenumber-frequenc domain of such structure are known in closed-form epressions and consequentl the integral (4.9) can be evaluated analticall. In this contet, consider an Euler beam with fleural stiffness EI and unit mass m resting on a Kelvin foundation with stiffness k and damping c, as represented in Figure 4.. f ( t ) V EI, m k, c Figure 4.: Beam on a Kelvin foundation subected to a moving load The wavenumber-frequenc domain displacement uɶ of the beam is uɶ ( k, ω) = 4 (4.) EI k + k + iωc ω m and, according to (4.9), the space-time domain displacement induced b a constant moving f t = A ) is load ( ( ) iω t 3 + V AV e, = d π EIω + kv + iωcv ω mv ( ) u t ω (4.) Since the integrand in (4.) is a regular epression, and since it is bounded in the comple plane, the integral can be evaluated b means of contour integration, which results in 3 4 i 4 AV ω (, ) i sign e t V u t = t EI V = k = ( ω ωk ) k imag( ω ) t > where the values ω represent the roots of the polnomial V (4.) EIω + kv + iω cv ω mv = (4.3) Furthermore, for cases in which the damping is null ( c = ), there is a critical speed V cr around which the response of the beam is greatl amplified. This critical speed equals the lowest bending velocit of the sstem (Hung and Yang, ) and corresponds to the lowest velocit that ields the roots of the polnomial (4.3) real, and so V cr is given b

127 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings V 4EI k = (4.4) m 4 cr Net, the beam displacements are calculated with eq. (4.) and with a time domain finite element procedure (FEMIX, and then compared. The 6 mechanical and dnamic properties of the beam are EI =. Pa and m =.89 kg/m, 6 3 the stiffness and damping of the foundation are k = N/m and c = 3 Ns/m, and 3 the magnitude of the force is A = 95 N. The critical load speed of this sstem, according to (4.4), is V cr = 58.6 m/s. The beam displacements are computed for the load speeds V = 5 m/s, V = 5 m/s, V = 55 m/s and V = 7 m/s and plotted in Figure a) V = 5 [m/s] 4-3 b) V = 5 [m/s] 3 u(,t) 5 u(,t) Vt Vt c) V = 55 [m/s] -3 d) V = 7 [m/s] u(,t) u(,t) Vt Vt-+ Figure 4.: Beam displacements induced b a constant moving load. Blue line = eq. (4.); black circles = time domain FEM (FEMIX) The displacements represented in Figure 4.a show that for speeds below V cr, the behavior ahead of the load ( Vt + < ) and behind the load position ( Vt + > ) are identical. This fact is observed mostl because the roots ω are comple and characterized b a significant imaginar component, which causes the response to evanesce awa from the load. As the load speed V increases, the roots ω come closer to the real ais and therefore the response decas slower with the distance to the position of the load. 3

129 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings (the forcing frequenc associated with such load is ω is added as an argument for convenience). The function p(, t, ω ) i (,, ω ) e ω ( ) t δ p t = A Vt (4.5) and its wavenumber-frequenc content p( k, ω, ω ) ɶ is + + ωt ( ω ω ) δ ( ) i ik -iωt pɶ k,, = A e Vt e d e dt = + ( ) ik -it( ω ω kv ) ik = A e e dt = π A e δ k V + ω ω The insertion of (4.6) and (4.3) in equation (4.) ields (4.6) h t ω A hɶ k ω δ k V ω ω k ω (4.7) + + k iωt,,, e d e d π i ( ) ( ) = ( ) ( + ) and after solving equation (4.7) for the inner integral, the following epression is obtained ( ) (, ω, ω ) h + A iω -iω V V i t,, e hɶ ω ω ω, e e d π V V h t ω = ω ω = iω t V + A iω V ω ω = e hɶ, ω e d ω πv V The force (, ) component, which results from the eponential factor ep( iω t) (4.8) f t ω described above has no phsical meaning since it contains an imaginar. However, this tpe of epression can be used to define real functions of the tpe cosine or sine, as stated in the net equations (, ω ) cos( ω ) f t = A t = A C C (, ω ) sin( ω ) f t = A t = A S S e e ω + e e i ω i t -i t ω ω i t -i t Hence, in order to obtain the response C (,, ) hs (, t, ω ) for sine tpe loads, equation (4.8) must first be used to calculate (,, ) afterwards, since h(, t, ω ) h(, t, ω ) (4.9) h t ω for cosine tpe loads or the response h t ω and =, the real or imaginar component must be retained according to the following (,, ω ) + h(, t, ω ) h t hc (, t, ω ) = = Re (,, ) h t ω h(, t, ω ) h(, t, ω ) hs (, t, ω ) = = Im h(, t, ω ) i (4.) Similarl to the case of constant moving loads, for oscillating moving loads no integral needs to be evaluated in order to obtain the space-frequenc domain response. Instead, the response h ɶ k, ω for the in that domain is obtained simpl b evaluating the transfer function ( ) 5

130 Chapter 4 Invariant structures subected to moving loads and moving vehicles wavenumber-frequenc pair ( ( ω ω ) V, ω ). In addition, the response of points that move with the same speed as the load is given b P iω P t + iω V ω ω ω V ω (, A h = Vt + P t) = e h πv ɶ V, e d (4.) in which P = Vt + is the distance between the moving point and the source. The previous equation clearl shows that the response fields move together with the load and oscillate with frequenc ω. Beam on a Kelvin foundation subected to a cosine tpe moving load For a load f ( t) A cos( ω t) in which =, the beam displacements are calculated with (,, ω ) Re (,, ω ) uc t = u t (4.) iω t 3 + V A i V ω e V = π EI ( ω ω ) + kv + iωcv ω mv u(, t, ω ) e dω (4.3) Since the integrand in (4.3) is a regular epression and since it is bounded in the comple plane, the integral can be evaluated b means of contour integration, resulting in 3 4 i i 4 A V ω ω (,, ) i e sign e t V V u t ω = t EI V = k = ( ω ωk ) k imag( ω ) t > In the previous equation, the values ω represent the roots of the polnomial V (4.4) ( ) ω ω ω ω EI + kv + i cv mv = (4.5) Similarl to the case of constant moving loads, when damping is null, there is a velocit above which the polnomial (4.5) presents real roots. However, in this case, for low frequencies ω, the minimum velocit that ields real roots onl ields two of the poles real, and so there is a second velocit above which all the roots are real. At these two critical velocities, two roots of (4.5) become real and with the same value (double roots), which results in the amplification of the displacements of the beam. In Figure 4.4, the (logarithm of) maimum of the displacements u C is plotted as a function of the ecitation frequenc ω and load speed V for a beam on a Kelvin foundation with the same properties of the sstem described in section 4.. (with c = Ns/m ). 6

132 Chapter 4 Invariant structures subected to moving loads and moving vehicles -4 a) V = 3 [m/s] 3-3 b) V = 5 [m/s] 5 uc(,t) 5 uc(,t) c) V = 7 [m/s] uc(,t) Figure 4.5: Beam displacements induced b harmonic moving loads In Figure 4.6, the maimum displacements observed at the beam are plotted as a function of the load speed for the conditions referred to above (blue line) and for null damping (red line). The two critical velocities can be identified for the case of no damping, while for the case of damping, the critical velocities appear to merge into one. Such feature is observed for the damped case because the critical velocities are near each other and because the large amount of damping used not onl reduces the maimum displacements as it also widens the bell shaped response associated with each velocit. If the damping is reduced or if the ecitation frequenc ω is increased (and consequentl the critical velocities are moved farther apart), then, even for the damped case, the two critical velocities can be noticed. The eample solved in this subsection considers cosine tpe loads. For sine tpe loads, the same conclusions can be drawn and the response of the structure is practicall the same, being it simpl shifted in time and in space. In the net subsection, structures for which the response fields cannot be determined analticall are studied. 8

133 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings..8 ma(uc) V [m/s] Figure 4.6: Beam maimum displacements as function of the load speed. Blue line with damping; Red line no damping 4..4 Eamples Net, equations (4.9) and (4.8)-(4.) are used to calculate the response of longitudinall invariant structures of which the wavenumber transfer functions (and therefore the integrands of the mentioned equations) cannot be determined analticall. The case stud considered in this subsection consists in a fleible slab resting on the surface of a half-space (homogeneous or laered) that is subected to a vertical load f ( t ) that moves with speed V along the middle top alignment (point A), as eemplified in Figure 4.7 (it is assumed that the load crosses the section = at t =, i.e., = ). The material properties 3 of the slab are: densit ρ = 45 kg/m ; Young s modulus E = 3 GPa ; and Poisson s ratio ν =.. The slab is modeled with 8 four-node volume elements with dimensions.5.3 m and the interface between the slab and the half-space is divided into 8 boundar elements of constant epansion. f ( t ).3m A V B C D z E m m m 4m 5m Figure 4.7: Slab resting on a half-space submitted to a moving load Eample Slab on a homogeneous half-space subected to a constant moving load In this first eample, the slab is subected to a constant moving load defined b f ( t ) = N and the foundation, which consists in a homogeneous half-space, is given the following 3 properties: densit ρ = 8 kg/m ; shear modulus G =.5 GPa ; Poisson s ratio ν =.5 9

134 Chapter 4 Invariant structures subected to moving loads and moving vehicles (the corresponding bod wave velocities are C s = 5 m/s and C p = 433 m/s ). A small amount of hsteretic damping ξp = ξs =. is considered, which renders comple wave velocities = + sign( ) i and ( ) C C ω ξ p p p C = C + sign ω iξ (Dominguez, 993). s s s For constant moving loads, the time domain response fields (, ) h t are obtained through the h ɶ k, ω cannot be evaluation of eq. (4.9), but since in this eample the transfer functions ( i ) determined analticall, then the integral in (4.9) has to be evaluated numericall, being approimated with iωi t V Nω A ω h(, t ) h ɶ ( ωi V, ωi ) e (4.7) πv i= Nω Due to the conugate propert h( k, ωi ) = h( k, ωi ) ɶ ɶ, eq. (4.7) can be further simplified to h t h ɶ V h ɶ V πv (4.8) N ( ) ( ) ( ) ( ) ( ) ω A ω, Re, cos ωi ωi ω i t Im ωi, ωi sin ω i t V V i= In the following, equation (4.8) is used to calculate the time domain displacements at the cross-section = and for V = m/s at point B, situated at the edge of the slab, and at points C, D and E, situated at the surface of the half-space. Two TLM models are tested: in the first model (TLM-), the half-space is modeled with ust one PML with parameters η =, Ω = 4, m = and N = (see chapter ); in the second model (TLM-), the half-space is modeled with an elastic laer of thickness H = m, divided into 4 quadratic thin-laers, and with a PML (same parameters as for model TLM-). The displacements obtained with these TLM models are compared with the displacements obtained using a time domain methodolog (TD) (dos Santos et al., a; dos Santos et al., b) and with the h ɶ k, ω obtained displacements obtained using eq. (4.8) together with the transfer functions ( i ) from a.5d BEM-FEM procedure based on the stiffness matrices of Kausel and Roesset (98) (SM). For the procedures based on eq. (4.8), 5 frequencies with a step of. Hz are used. For the time domain procedure, a m long 3D model divided into 4 longitudinal sections and a time step of. s are used. Figure 4.8 plots the transverse displacements of the points B, C, D and E obtained with the four mentioned approaches, Figure 4.9 plots the longitudinal displacements and Figure 4. plots the vertical displacements. From Figure 4.8, it can be observed that the four approaches ield significantl different transverse displacements: at the slab (Point B), the TLM-, the TLM- and the SM results tend approimatel to the same maimum value, but the shapes of the curves are different; still regarding point B, the TD solution differs both in shape and sign from the remaining solutions (a ustification for this could not be found); at the surface of the half-space (points C, D and E), the results of the SM model tend the follow the TD results while the TLM- and the TLM- ield smaller displacements. Despite this fact, the passage of the load at the crosssection is noticed in all four approaches. As for the longitudinal and vertical displacements represented in Figure 4.9 and Figure 4., respectivel, a better agreement is, in general, observed.

135 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings -8 Point B -8 5 Point C u [m] - u [m] t [s] t [s] -8 Point D -8 Point E u [m] - u [m] t [s] t [s] Figure 4.8: Transverse () displacements: blue = TLM-; red = TLM-; black = TD; green = SM -7 Point B -8 5 Point C.5 u [m] u [m] t [s] t [s] 4-8 Point D -8 4 Point E u [m] u [m] t [s] t [s] Figure 4.9: Longitudinal () displacements: blue = TLM-; red = TLM-; black = TD; green = SM

136 Chapter 4 Invariant structures subected to moving loads and moving vehicles 5-7 Point B -7 Point C uz [m] -5 uz [m] t [s] t [s] 5-8 Point D -8 5 Point E uz [m] -5 - uz [m] t [s] t [s] Figure 4.: Vertical (z) displacements: blue = TLM-; red = TLM-; black = TD; green = SM When comparing the TD results (black lines) with the remaining approaches, it is noticed that for the earlier moments ( t <. s) the responses present ver distinct behaviors. These differences are ustified b the finite length of the 3D model used in the TD approach, a characteristic that violates the assumption of invariant cross-section considered in the.5d models. In this wa, while in the TD approach the entrance of the load in the finite element model induces a transient phenomenon, in the.5d approaches, since it is assumed that the load travels from minus infinit to plus infinit, the phenomenon is not present. The transient phenomenon dissipates due to internal damping and therefore after some time its contribution is minimal. Apart from the transient phenomenon, the remaining differences ma be ustified b the longitudinal and temporal discretizations required b the TD approach: the TD and the SM approaches are based on the fundamental solutions obtained with the stiffness matrices and therefore, theoreticall, the should ield the same values. This hpothesis has not been tested because to make the 3D mesh longer in the longitudinal direction and/or with thinner elements and smaller time-steps renders the calculation unfeasible (for the current m long model, more than two das were required to calculate the passage of the load from one edge of the model to the other edge). The differences between the results of the TLM-, TLM-, and SM approach can onl be hɶ k, ω. Figure 4. compares the transfer ustified b differences in the transfer functions ( ) functions u ( ω/ V, ω) ɶ calculated with the three approaches at the four points and, as epected, the present discrepancies, which ustif the distinct curves in Figure 4.8. Furthermore, when comparing the blue and red lines (TLM- and TLM-) with the green line (SM), one realizes that the differences are concentrated in the lower frequenc range ( ω < rad/s f < 3. Hz ),

139 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Figure 4.3: Vertical displacements for V = m/s ( ω = ): the displacements are multiplied b the shear modulus of the half-space (values in N/m) Figure 4.4: Vertical displacements for V = 3 m/s ( ω = ): the displacements are multiplied b the shear modulus of the half-space (values in N/m) 5

141 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings which is equivalent to N i ω A i -i ω ω ω i i i ω ω t t V ω V i ω ω + V hs (, t, ω ) Im e h(, ωi ) e + h(, ωi ) e V V πv ɶ ɶ (4.3) i= Hence, for each sampled frequenc ω i, the transfer functions h( k, ω) ɶ must be calculated for the wavenumbers k = ( ωi ω )/ V and k = ( ωi + ω)/ V, which are then used in eq. (4.3). Net, equation (4.3) is used to calculate the time domain displacements of points B, C, D and E (Figure 4.7) at the cross-section = and for V = m/s and ω = 4π rad/s (once again, it is assumed that the load crosses the section = at t =, i.e., = ). The two TLM models described in Eample (TLM- and TLM-) are used to simulate the half-space, and the results thus obtained are once again compared with the displacements obtained using the TD and the SM approaches. For the procedures based on eq. (4.3), 5 frequencies with a step of. Hz are considered. For the time domain procedure, the same model and time steps considered in Eample are used. Figures plot the transverse, longitudinal and vertical components of the displacements obtained with the four approaches. 4-8 Point B -7 Point C u [m] u [m] t [s] t [s] -7 Point D -7 Point E u [m] u [m] t [s] t [s] Figure 4.6: Transverse () displacements: blue = TLM-; red = TLM-; black = TD; green = SM 7

142 Chapter 4 Invariant structures subected to moving loads and moving vehicles -7.5 Point B -7 Point C u [m].5 u [m] t [s] t [s] -7 Point D -7 Point E.5 u [m] u [m] t [s] t [s] Figure 4.7: Longitudinal () displacements: blue = TLM-; red = TLM-; black = TD; green = SM -6.5 Point B -7 6 Point C 4 uz [m].5 uz [m] t [s] t [s] -7 Point D -7 Point E uz [m] uz [m] t [s] t [s] Figure 4.8: Vertical (z) displacements: blue = TLM-; red = TLM-; black = TD; green = SM 8

143 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Ecept for the TD response of point B, a good agreement can be observed between the results of the four distinct approaches. The differences observed for the TD approach at point B can be ustified b the longitudinal discretization, which is not fine enough to reproduce accuratel the response near the loaded point (note that for large negative and positive times, i.e., when the load is far from the reference section =, the agreement is good). When comparing the results of eamples and, it can be observed that in general the agreement is better for the second eample than for the first. The reason for this is that the time domain response in the second eample is dominated b the radial frequencies ω = ωvr ( VR V ) and ω = ωvr ( VR + V ), and not b the low frequencies, as in the first eample. The mentioned radial frequencies correspond to the interceptions between the dispersion curves of the foundation (for a homogeneous half-space, the dispersion curve corresponds to ω = VR k, where V R is the Raleigh wave speed) and the integration paths of equation (4.8) ( ω = ω + Vk and ω = ω Vk ). In opposition to what happens in eample, for the oscillation frequenc ω = 4 π[rad/s] there is no critical load speed at which the displacements are amplified, and the reason for that is because the integrating paths ω = ω + Vk and ω = ω Vk, that intercept the dispersion curves at the frequencies ω = ωvr ( VR V ) and ω = ωvr ( VR + V ), are never tangent to the mentioned curves (for ω = and for V = VR, the integrating path coincides with the dispersion line, i.e., are tangent everwhere, and that is the reason for the displacements to be amplified). This conclusion is in accordance with the work of Dieterman and Metrikine (997) and is supported b Figure 4.9, which plots the maimum displacement observed at point B for different forcing frequencies ω as a function of V, and where it can be observed that the displacements tend to decrease as V increases. Taking into account the conclusions obtained in section 4..3 for a beam on a Kelvin foundation, it could be epected the eistence of at least on critical velocit. However, in that case, the energ dissipates onl along the longitudinal and vertical directions, while for the case of a slab on a half-space (a pure 3D case) the energ also dissipates along the transverse direction, thus changing completel the behavior of the sstem. This aspect supports the importance of 3D models in the simulation of vibration fields induced b moving loads. Before proceeding to eamples of laered domains, the influence of the load speed on the shape of the response of the sstem is evaluated with the aid of snapshots of the vertical displacements induced b loads moving at the speeds V = m/s, V = 3 m/s and V = 5 m/s. The snapshots are represented in Figures

144 Chapter 4 Invariant structures subected to moving loads and moving vehicles ma(uz) [m] V [m/s] Figure 4.9: Maimum vertical displacement of point B as a function of the load speed V for: ω = (gra); ω = π rad/s (blue); ω = π rad/s (red); ω = 3π rad/s (black); and ω = 4π rad/s (green) Figure 4.: Vertical displacements for V = m/s ( ω = 4π rad/s ): the displacements are multiplied b the shear modulus of the half-space (values in N/m) 3

145 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Figure 4.: Vertical displacements for V = 3 m/s ( ω = 4π rad/s ): the displacements are multiplied b the shear modulus of the half-space (values in N/m) Figure 4.: Vertical displacements for V = 5 m/s ( ω = 4π rad/s ): the displacements are multiplied b the shear modulus of the half-space (values in N/m) 3

148 Chapter 4 Invariant structures subected to moving loads and moving vehicles maimum speed for the vehicle to circulate without causing the resonance of the track-soil sstem. ω/π 6 4 Dispersion Curves 4 3 ω/π k Phase Velocit k Vph [m/s] ω/π Vph [m/s] ω/π 7 Group Velocit 5 Vgr [m/s] Vgr [m/s] ω/π ω/π Critical Load Speed 5 Vcr [m/s] Vcr [m/s] ω /π 5 5 ω /π Figure 4.3: Dispersion curves, phase velocities, group velocities and epected critical load speeds for the laered domain a) (left) and for the laered domain b) (right) 34

151 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Other aspects that influence the dnamic behavior of the vehicle-track sstem are the geometr and the dnamic properties of the track and vehicle, and the travel speed of the vehicle. All these aspects must be considered when developing a numerical model for the solution of this problem. In this work, since it is assumed that the track-soil sstem is invariant in the longitudinal direction, the variations of the stiffness cannot be considered and consequentl the ecitation due to the discrete sleeper support is not accounted for. It must be mentioned, however, that there are works in the literature in which the periodicit of the track-soil sstem is considered (Gupta et al., 7; Gupta et al., 8; Gupta et al., ). The geometric irregularities at the contact between the vehicle and the supporting structure (in the railwa case, wheel-rail contact) are considered b means of a position dependent gap/irregularit profile. The irregularit profile ma account for the unevenness of the track and for the imperfections of the wheels (Wu and Thompson, ). Besides the assumption of invariance in the longitudinal direction, this work also assumes that the vehicle and the supporting structure are linear and that the contact between the two structures is also linear. These two assumptions, which are commonl used b other authors (Metrikine et al., 5; Lombaert et al., 6), allow the analses to be performed in the frequenc domain, which is ver convenient, since the.5d BEM-FEM ields results in the wavenumber-frequenc domain and their transformation to the space-frequenc domain is straightforward. Before proceeding to the description of the solution method, it must be mentioned that if the non-linear behavior of the vehicle or track is to be considered, or if the loss of contact between the wheels and the rail is to be studied, then frequenc domain procedures lose their applicabilit and therefore time domain procedures must be used (Lane et al., 7; Katou et al., 8; Neves et al., ). Hbrid methods, in which the response of the track-soil sstem is first obtained in the.5d domain, then transformed to the space-time domain, and finall given as inputs to time domain procedures, can also be applied (Grundmann and Lenz, 3; Müller et al., 8). Nevertheless, the increase of compleit associated with these approaches is considerable and would force simplifications in other components of the sstem, namel in the boundar conditions of the track and length of the model. In the net sub-sections, the procedure used to calculate the dnamic forces that the vehicle transmits to the supporting structure is eplained and eemplified Vehicle structure interaction Consider a vehicle moving with constant speed V on top of an invariant structure, as represented in Figure 4.5a. The vehicle contacts with the structure through N CP contact points which ma or ma not belong to the same longitudinal alignment (in the eample shown in Figure 4.5, the contact points belong to two distinct horizontal alignments). 37

152 Chapter 4 Invariant structures subected to moving loads and moving vehicles z a) V z b) V fi ( t ) f ( t ) i Figure 4.5: Vehicle-structure interaction cross-section perspective (left) and longitudinal perspective (right): a) coupled sstem; b) substructuring method The solution method for the vehicle-structure interaction consists in a substructuring technique (Figure 4.5b), in which the vehicle and the supporting structure are modeled independentl and in which the equilibrium of forces and the compatibilit of displacements is enforced between the N CP contact points of the vehicle and the corresponding points of the f t ( i =... NCP ) that the f t are known, the response supporting structure. The obective is to find the moving forces i ( ) vehicle transmits to the supporting structure. Once the forces ( ) fields both of the vehicle and of the supporting structure can be calculated using the governing equations of each of the domains. The compatibilit of displacements is imposed through the condition v in which u ( ) s ( ) (, ) δ ( ) u t = u + Vt t + u + Vt (4.3) v i i i i i i t is the displacement of the th s ui i + Vt, t is the displacement of the corresponding moving contact point of the supporting structure and s δ ui ( i + Vt ) is the irregularit/gap eperienced b the contact point. The quantities u i and δ u i are defined in a fied frame of reference, and so = i + Vt represents the longitudinal th position of the i contact point at the generic instant t, while i represents the corresponding position at the instant t =. s The displacements of the structure u ( Vt t) i i contact point of the vehicle, ( ) i i +, result from the contribution of all N CP contact points of the vehicle, and so the are calculated b the summation 38

153 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings N s i i i i = u Vt t u Vt t (4.33) CP s ( +, ) = ( +, ) f ( t) δ ( + Vt) s where ui ( + i Vt, t) f ( t) δ ( + Vt) are the displacements that the force f ( ) t (force transmitted th b the vehicle through the contact point) induces at the moving contact point i. The negative sign is used to account for the equilibrium condition. Since linear behavior and linear contact is assumed, eq. (4.3) can be transformed to the frequenc domain, becoming with s ( ω ) = ( +, ω ) + δ ( ω ) uɶ uɶ Vt uɶ (4.34) v i i i i + = v iω ( ω ) ( ) t v uɶ i ui t e dt (4.35) + s i i i i s iω ( + ω ) = ( + ) t uɶ Vt, u Vt, t e dt (4.36) + iω δuɶ ( ) ( ) e t i ω = δui i + Vt dt (4.37) Some comments must be made about these three frequenc domain variables. The variable s uɶ i ( i + Vt, ω ) represents the frequenc domain displacements of a point with longitudinal coordinates = i + Vt, i.e., a point that moves with the same speed as the set of loads. As mentioned in section 4..3 (equation (4.)), the response field of points moving with the same speed as the load oscillate also with the same frequenc as the load. Hence, accounting for (4.33) and (4.), the integral (4.36) can be replaced with where f ( ω ) calculated with i iω N CP V + i i s e ω ω ω V i ( i + ) = ɶ ( ) i = πv V uɶ Vt, ω f ω uɶ, ω e dω hɶ i ɶ is the frequenc content of the load transmitted b the and where ui ( k, ω) + = iω ( ω ) ( ) t (4.38) th contact point and fɶ f t e dt (4.39) ɶ is the transfer function that relates the displacements of the alignment th th associated with the i contact point with the forces at the alignment associated with the contact point. In this work, the referred to transfer function is calculated with the.5d BEM- FEM procedure eplained in chapter 3. On the other hand, the irregularit profile δ ui is a position dependent function, and so it is more convenient to define its Fourier transform in terms of the wavenumber k rather than the radial frequenc ω. This leads to 39

154 Chapter 4 Invariant structures subected to moving loads and moving vehicles + k δu k = u d (4.4) ( ) δ ( ) e i i i Hence, the Fourier transform (4.37) can be replaced with the more convenient epression ω i i ( ) e V uɶ i = ui (4.4) δ ω δ V v Finall, the vehicle displacements uɶ i are obtained through the solution of the differential equations that governs its behavior. In this work, a FEM procedure is used to solve these equations (whether the elements are fleible or rigid), and so the displacements of the vehicle are obtained b solving the linear sstem Kɶ Kɶ uɶ fɶ = Kɶ Kɶ uɶ ω V (4.4) in which the inde refers to the degrees of freedom that contact with the structure, the inde refers to the remaining dofs, K ɶ mn are the frequenc domain matrices obtained with Kɶ mn = K mn + iωcmn ωmmn (being K mn, C mn and M mn the stiffness, damping and mass matrices of the vehicle model), and in which uɶ is a vector containing the vehicle displacements and fɶ m is a vector containing the applied forces. The vector uɶ collects the v vehicle displacements uɶ i and the vector f ɶ collects the interaction forces f ɶ ( ω ). It is assumed that the remaining dofs are free of forces and therefore fɶ =. Sstem (4.4) can be condensed into and solved for uɶ, being thus obtained with ( ) m K ɶ K ɶ K ɶ K ɶ uɶ = f ɶ (4.43) uɶ = Ff ɶɶ (4.44) ( ) B collecting all irregularities/gaps δ ( ω ) F ɶ = K ɶ K ɶ K ɶ K ɶ (4.45) ɶ in the vector δ uɶ and all the coefficients u i defined in equation (4.38) in matri H ɶ, the equalit (4.34) can then be written as Ff ɶɶ = Hf ɶɶ + u ɶ (4.46) δ This equation can be solved for the interaction forces f ɶ, ielding The irregularit/gap profiles u ( ) ( ) f ɶ = F ɶ + H ɶ u ɶ (4.47) δ δ i are usuall defined as combinations of trigonometric functions of the tpe sine and cosine, i.e., N k δu A k A k (4.48) ( ) = c cos( ) + s sin( ) i,l l,l l l= h ɶ i 4

155 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Each harmonic l k is associated with a wavenumber content ui, l ( k ) ( ) ( i ) ( ) ( i ) ( ) i, l c,l s,l, l c,l s,l, l δɶ calculated with δu k = π A + A δ k k + π A A δ k + k (4.49) and with a corresponding frequenc content δ ( ω ) ɶ, which according to (4.4) is u i, l ω ( Ac,l + ias,l ) i i ω π ( Ac,l ias,l ) ω π i i ω V V δu ɶ i, l ( ω ) = e δ k, l + e δ + k, l V V V V (4.5) The insertion of gap profiles of tpe (4.5) in equation (4.47) leads to interaction forces f ɶ ω of the form ( ), l f ɶ ω ω, l ( ω ) = ( Bc,l + ibs,l ) δ k, l + ( Bc,l ibs,l ) δ + k, l V V (4.5) whose corresponding time domain functions are + iωt V f ( t) = f ɶ ( ω ) e dω = B cos( k V t) + B sin ( k V t ) (4.5), l, l c,l, l s,l, l π π The interaction forces are now known in terms of sines and cosines and the can be combined with equation (4.4) to obtain the response of the vehicle, or with the equations developed in section 4. to obtain the response of the supporting structure. [The factors V and π appear alternatel in the denominator and numerator of equations (4.5) and (4.5), and therefore can be disregarded in the numerical implementation of the method.] Before proceeding to the validation eamples, notice that the equations derived above are not restricted to the vertical direction and therefore the can be used to stud the transverse interaction as well. Nevertheless, in the railwa case, under the assumptions of straight lines and constant movement, the transverse forces assume significant values onl if there is a considerable misalignment of the rails in the horizontal direction, which is not likel to happen in high-speed lines, and therefore the horizontal interaction is not considered in this work. As for the interaction in the longitudinal direction, it is assumed that the vehicle moves with constant speed, which implies the resultant of forces in this direction to be null. Apart from the reactions of the supporting structure, it is assumed that no other force is applied at the vehicle, and so the reactions in the longitudinal direction must be null. For this reason, the interaction in the longitudinal direction cannot be considered. Hence, in this work, onl the vertical interaction is considered, but since the contact points can belong to different alignments, three-dimensional modeling of the vehicle becomes possible, thus enabling the stud of the rolling motion of the vehicle. Net, two vehicle-structure interaction problems are solved with the equations derived in this section and the results are compared with the results obtained with a time domain 3D-FEM procedure. The first eample consists of a point mass moving on top of a Winkler foundation, while the second eample considers a multi-degree of freedom vehicle moving on top of a ballast track Point mass moving on top of a beam on a Kelvin foundation Consider a mass M moving with speed V on top of an Euler beam with, fleural stiffness EI and unit mass m, that rests on a Kelvin foundation with stiffness k and damping c, as represented in Figure

156 Chapter 4 Invariant structures subected to moving loads and moving vehicles EI, m M V K, C k, c Figure 4.6: Point mass moving on top of a beam on a Kelvin foundation The moving mass contacts with the supporting structure b means of a suspension sstem consisting of a spring with rigidit K and a dashpot with damping C. Consider also that the surface of the Euler beam is uneven and that the profile of the unevenness is described b the δ u = A cos k + A sin k. The corresponding unevenness, as felt b contact function ( ) c ( ) s ( ) point, has frequenc content solel at the frequenc ω = k V, and its value is ( ω i V uɶ ( Ac ias ) e (4.53) δ = is the position of the point mass at the instant t = ; the factors π and V are neglected.) The vehicle responds also with frequenc ω, and thus its dnamic stiffness matri is K + iωc K iωc K ɶ = K iωc K + iωc ω M (4.54) while its fleibilit matri F ɶ (in this case a scalar) is K + iω C ω M F ɶ = (4.55) ω M K ( + iω C ) On the other hand, based on equation (4.4), the epression for h ɶ i is hɶ i 3 i 4 i i 4 V ω i iω V V = i e sign e EI V ( ω ω ) being the poles ω k and point, matri H ɶ is also a scalar. [Note: when i l = k = l k i k l imag ω l > V (4.56) ω l the roots of the polnomial (4.5). Since there is onl one contact =, the value h ɶ i returned b eq. (4.56) is null, a consequence of the sign factor. The sign factor is associated with the comple half-plane used in contour integration, which in turn depends on the sign of the imaginar component of i the term ω. When the imaginar part of this term is null (i.e., when l V i = ), the integrating functions do not vanish neither in the upper nor in the lower comple half-planes, and consequentl contour integration can no longer be applied. Nevertheless, the function h ɶ i 4

157 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings must be continuous with the argument i, and so, when i =, h ɶ i can be evaluated with (4.56) b calculating its limit when i tends to zero, either from the positive side or from the negative side.] With the values of F ɶ, H ɶ, and δuɶ known, equation (4.47) can be emploed to obtain f ɶ. To eemplif the procedure, the values indicated in Table 4. are given to the beam and foundation, point mass, and irregularit profile δ u. Table 4.: Properties of track and Vehicle Beam and foundation Point mass δ u 6 EI =. Pa M = 3 kg A c = m =.89 kg/m 9 K =.94 N/m A s =. m 6 k = N/m C = (no damping) k = π /.4 rad/m 3 c = 3 Ns/m V = m/s (as in section 4.) = For the properties indicated above, the following values are obtained: ω = π rad/s ; 9 F ɶ = m/n ; H ɶ = i m/n ; δ uɶ =. i m ; 4 5 f ɶ =.978 i.45 N. The interaction force is then of the form ( ) cos( π ).45 sin( π ) f t = t t, where the static term corresponds to the weight W of the SDOF (W = Mg, being acceleration). g = 9.8 m/s the gravit To validate these results, the eample is solved with a time domain FEM approach in which the mass-beam interaction problem is solved according to the algorithm described in Neves et al. (). Figure 4.7 compares the obtained interaction forces f ( t ) and the displacements of the beam at the section =. As can be observed, the results agree perfectl, which validates the epressions developed in this section. Similarl to the case of moving loads, the speeds V that lead to resonance of the mass-beam sstem can also be determined for a given irregularit profile δ u. Figure 4.8 plots the maimum interaction force f ( t ) and the maimum displacement of the beam for a range of speeds varing from 5 to m/s, considering both damped foundation and non-damped foundation. The unique peak of the interaction force, which coincides with the first peak of the beam displacements, occurs at the speed V m/s and corresponds to the resonance of the moving mass-suspension sstem. For ver stiff foundations, this resonance occurs at the speed V = K M k = 376 m/s, i.e., at the speed for which the displacements prescribed at the contact point oscillate with the natural frequenc of the mass-spring sstem ( ω nat = K M ). However, in this case, due to the fleibilit of the foundation, the resonance of the coupled 43

158 Chapter 4 Invariant structures subected to moving loads and moving vehicles mass-beam structure is shifted to lower speeds. No other speed causes the resonance of the mass-spring sstem, as suggested in the left plot of Figure f(t) [N] u(t) [m] t [s] t [s] Figure 4.7: Interaction force f ( t ) (left) and beam displacements ( ) u t (right). Blue line current work; black circles: time domain FEM ma(f) [N] V [m/s] ma(u) [m] V [m/s] Figure 4.8: Interaction force f ( t ) (left) and maimum beam displacement u( t ) (right). Blue line 3 c = 3 Ns/m ; red line - c = Ns/m The second peak of the beam displacements occurs at the speed V 78 m/s. This speed is the first speed at which the line ω = k V intercepts the ellow region of Figure 4.4: the line ω = k V relates the frequenc of the interaction force with the speed of the point mass, while the ellow regions of Figure 4.4 correspond to the pairs ( V, ω ) that lead to amplification of the beam displacements; it is therefore epected that the response is amplified when the line crosses this region. Finall, a third peak can be distinguished in the right plot of Figure 4.8 at the speed V = 5 m/s. This speed corresponds to the critical load speed of the beam (as defined in section 4.), and the associated peak is observed because the interaction force f ( t ) contains a static component that is independent of the moving speed. For the viscous foundation 44

159 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings 3 ( c = 3 Ns/m, blue line), this peak is strongl attenuated to the point that it can hardl be distinguished Multi-degree of freedom vehicle moving on top of a ballast track The second eample considered herein corresponds to a multi-degree of freedom vehicle moving on top of a ballast track (Figure 4.9). The track is composed of two UIC6 rails on top of a slab that rests on a ballast foundation, which lies on a rigid base (the rigid base is considered instead of a half-space in order to avoid the use of boundar elements in the time domain approach; also, since this eample is used onl for validation purposes, the slab is used instead of sleepers in order to make the.5d model and the time domain model geometricall the same). The rails and the slab are connected through rail pads..75 [m] Rigid mass Suspension sstem Wheel 4 3 Rail. [m].35 [m].5 [m] Slab Ballast Subballast.6[m].5[m].6[m].5[m].6[m] z Rail pad Figure 4.9: Multi-degree of freedom vehicle moving on top of a ballast track The vehicle consists of a rigid mass with bouncing ( u z ), pitching ( θ ) and rolling ( θ ) inertias, suspension sstems, and wheels. Contact springs establish the connection between the vehicle and the rail. The properties of the components of the problem are given in Table 4. (the properties of the ballast and subballast are taken from the work (Ribeiro et al., 9), while the multi-degree of freedom vehicle corresponds to a bogie of an Alfa-Pendular train (Ribeiro et al., 3)). The main differences between this eample and the eample in subsection reside in the transfer functions h ɶ i, which in this case cannot be determined analticall, being instead approimated b the summation i iω N i V i e ω i i i i i ω ω V ωi + ω ω V i ω i, ωi e + i, ωi e πv i= V V hɶ uɶ uɶ (4.57) and in the matrices F ɶ and H ɶ and the vectors δuɶ and f ɶ, which have dimension 4 instead of being scalars. 45

160 Chapter 4 Invariant structures subected to moving loads and moving vehicles Table 4.: Properties of track and vehicle Elastic modulus Mass densit Poisson ratio E 3 ( Pa) ρ ( kg/m ) 6 Ballast Subballast Slab..6 Elastic modulus Mass densit Poisson ratio Moment inertia Area 3 4 ( Pa) ρ ( kg/m ) ν ( m ) ( m ) E I A 9 Rail 78 Stiffness k ( N/m ) c ( N.s/m ) 6 3 ν Damping Rail pad Mass Pitching inertia Rolling inertia Stiffness Damping ( kg) ( kg.m ) ( kg.m ) ( N/m) ( N.s/m) M M M K C Rigid mass Rigid wheel Suspension Contact spring To ecite the rolling motion of the vehicle, different irregularit profiles are given to each of the rails: the left rail is given an irregularit profile of the form δu left =.5cos π.sin π +.sin π, while the irregularit profile associated ( 3 ) ( 3 ) ( 5 ) with the right rail is of the form δu right.sin ( π ).cos( π ) = +. The profiles are defined 3 5 b the two wavelengths λ = 3 m and λ = 5 m, whose corresponding wavenumbers are k = π rad/m and k 3 = π rad/m (these profiles are not based on an real measurement). 5 Net, the interaction forces between the wheels and rails are calculated for a vehicle moving with speed V = 8 m/s. For the.5d FEM model, the cross-section is divided into 8 solid elements of 4 nodes (to simulate ballast, subballast and slab), two spring-dashpots couples (to simulate the rail pads) and two Euler beams (to simulate the rails). For the 3D time domain model, 4 slices with an equivalent cross section and thickness.5 m are used, satisfing a total length of m. The results are plotted in Figure

161 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings f [N] f3 [N] t [s] t [s] f [N] f4 [N] t [s] t [s] Figure 4.3: Interaction forces f i : blue line = time domain approach; red line =.5D approach It can be observed that the results obtained with the two approaches are identical, eisting small differences at some local maima or minima, but being the responses in perfect phase. Furthermore, Figure 4.3 confirms that the interaction forces are entirel defined b two 6 frequencies, which in this eample are ω = kv = π rad/s and ω 3 = k V = 3π rad/s. The differences reported above can be ustified b the longitudinal discretization of the domain, which is required for the 3D time-domain approach, and that violates the assumption of continuit and infinite length of the.5d model. While for the eample of the Kelvin foundation treated in the previous section this problem can be easil solved b decreasing the discretization size and increasing the total length of the model used in the time-domain approach, for the current eample doing so results in an ecessive number of linear equations to be solved, and therefore in impracticable computational times. Nevertheless, given the good match between the two approaches, it can be concluded that the equations derived in this section ield good results also for structures with comple geometries and for vehicles contacting with the supporting structure at more than one contact point Conclusion In this section, the equations needed to obtain the interaction forces between a supporting structure and a moving vehicle are derived and validated. The derivation of the equations is based on the following three assumptions: ) both the supporting structure and the vehicle 47

162 Chapter 4 Invariant structures subected to moving loads and moving vehicles behave linearl; ) there is never loss of contact between the vehicle and the supporting structure; 3) the vehicle moves from minus infinit to plus infinit with constant speed. The first two assumptions allow the calculation to be performed in the frequenc domain, as the are consistent with the principle of superposition of effects. The third assumption allows for the calculation to be cast in the wavenumber-frequenc domain. The equations here derived are used to determine the interaction forces between a moving mass and a beam on a Kelvin foundation, and the interaction forces between a multiple DOF vehicle and a ballast track that lies on a rigid base. In both cases, the forces obtained with the derived equations are compared with the forces obtained with a space-time domain procedure. A good agreement between the two approaches is observed, which validates the equations. The eample of the point mass moving on top of a beam on a Kelvin foundation is also used to calculate the critical velocities of the sstem. It is concluded that the critical speeds are related both to resonance of the mass-suspension sstem and to the critical load speeds of the supporting structure (discussed in section 4.). The calculation of the interaction forces (generation phase) is the last step described in this work, since the propagation stage (response of the supporting structure) and the reception stage (response of a nearb structure/building) have alread been described in section 4. and sub-section 3..6, respectivel. In the subsequent section, an illustrative eample is solved in which the three phases are accounted for and linked. 4.4 Vibrations induced b a moving vehicle in a nearb structure 4.4. Introduction and general description of the eample In the previous chapters and sections it is discussed how to deal separatel with the three stages of vibrations induced b moving vehicles: the generation stage is discussed in section 4.3; the propagation stage is discussed in section 4. (these first two stages rel on the transfer functions calculated with the.5d BEM-FEM procedure eplained in chapter 3); and the reception stage is discussed in sub-section In the present section, all the three stages are put together in an illustrative eample used to eplain how to link all the tools developed in this work. In the eample considered net, the response of a building due to a train passing in a nearb track is calculated. The soil corresponds to the laered domain b) described in Eample 3 of sub-section 4..4, the track corresponds to the supporting structure represented in Figure 4.9, and the vehicle corresponds to the X train, whose geometries and properties can be found in Alves Costa (, p. 8) and are transcribed in Figure 4.3 and Table 4.3. Together with the train geometr, Figure 4.3 also presents the theoretical model used for each vehicle of the train: the vehicle model consists of 7 rigid bodies that represent the car-bod, the bogies and the ales; the distinct bodies are connected through the primar and secondar suspension sstems. Contact springs establish the connection between the ales and the rails (stiffness: 9.4 N/m ). The train moves at the speed V = 6 m/s. 48

163 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Car-bod Secondar suspension sstems Bogie Bogie Primar suspension sstems Ales Contact springs Contact springs Figure 4.3: X train geometr and single vehicle model (Alves Costa, ) Table 4.3: Properties of the vehicles of the X train (Alves Costa, ) Leading vehicle Middle vehicles Rear vehicle Car-bod Mass (kg) Bouncing inertia (kg.m ) Secondar Stiffness (N/m) suspensions Damping (N.s/m) Bogie Mass (kg) Bouncing inertia (kg.m ) Primar Stiffness (N/m) suspensions Damping (N.s/m) Ales Mass (kg) The same irregularit profile is considered for the left and right rails. This feature, together with the smmetr of the track, prevents the rolling motion of the vehicle, and therefore it is sufficient to consider a D model for the vehicle. The irregularit profile considered in this eample is artificial, and is generated according to the function (Alves Costa, ) N k δu A k A k (4.58) ( ) = c cos( ) + s sin ( ) = ( θ ) ( θ ) ( ) A = A cos A = A sin A = S k k (4.59) c s r 49

164 Chapter 4 Invariant structures subected to moving loads and moving vehicles where θ is a random variable with uniform distribution in the interval ], π [, S r is the power spectral densit function of the rail, and k is the increment in the longitudinal wavenumber. The function S r used to define the irregularit profiles is also taken from Alves Costa () and assumes the form ( ) Sr k =.36 k (4.6) The wavenumber sample contains N k = 6 equall spaced values ( k =.3 rad/m ), being the first k =.5 rad/m (the corresponding maimum wavelength of the irregularit profile is approimatel 4 m). Figure 4.3 plots the coefficients irregularit profile δ u( ). A c and A s and the corresponding Ac, As [m] k [rad/m] 6 8 δu [m] [m] Figure 4.3: Left image = coefficients A c (blue) and A s (red); right image = irregularit profile δ u( ) The nearb structure consists of a two stor building resting on the surface of the soil. Its geometr is indicated in Figure The floors and roof are assumed rigid, and the footings are assumed rigid and massless. The total mass per unit surface (including self-weight and overweight loads) is m floor = 6 kg/m for the floors and m roof = 8 kg/m for the roof. The columns are made of concrete and their cross section is.4.4 m. The are modeled with Euler beams and discretized as indicated in Figure The dimensions of the footings are.5.5 m, and each is divided into 5 square boundar elements of constant epansion. The distance between the center of the track and the closest column alignment is. m. 5

165 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings 3m D 3 D m m m m 3m D m A B C 7m 7m 7m 7m m m.5m.5m Figure 4.33: Geometr of the building and discretization of the columns 4.4. Step : eigenpairs of the soil (TLM) The first step to solve this problem corresponds to the calculation of the vibration modes of the soil. With that obective, the TLM matrices indicated in Chapter must be assembled and then used as inputs in the algorithms described in Table.7 and Table.8. The obtained eigenpairs are then used to calculate the transfer functions and boundar element matrices required for the three stages of the problem. In this eample, all integrals in dω are calculated numericall using a frequenc sample consisting of 5 frequencies, with frequenc step ω =.π rad/s ( f =. hz) and initial frequenc ω = ω. The TLM modes must be calculated for this sample of frequencies Step : transfer functions In the second step, the.5d BEM-FEM procedure described throughout sections 3.3 and 3.4 is used to calculate the transfer functions between sources at the rails and receivers at the rails, soil surface, and soil-building interaction surface. The eigenpairs calculated in step are used here to calculate the.5d boundar element matrices of the soil. Since the track and the irregularit profiles are smmetric, the transfer functions are calculated for a pair of vertical loads applied at the rails and with amplitude.5 N. Figure 4.34 plots the induced vertical displacements (transfer functions) of the rails and of receivers at the positions of columns A, B and C, for the wavenumber-frequenc pairs ( k, ω) = ( ω V, ω), ( k, ω) = ( ω V + k, ω) and ( k, ω) = ( ω V k, ω) (the second and third pairs relate to the integration paths associated with the wavenumber k = k of the irregularit profile). The first feature to be observed in Figure 4.34 is that the transfer functions for the pairs ( k, ω) = ( ω V + k, ω) (red line) are negligible when compared with those of the other pairs represented in the figure. Thus, the contribution of that branch for the final response is negligible. 5

166 Chapter 4 Invariant structures subected to moving loads and moving vehicles -8 Rail 4-9 Column A.5 3 hz [m/n] hz [m/n] ω [rad/s] ω [rad/s] 4-9 Column B 4-9 Column C 3 3 hz [m/n] hz [m/n] ω [rad/s] ω [rad/s] Figure 4.34: Transfer function at the rail and column: blue line - k k = ω V + k ; green line - k = ω V k = ω V ; red line - Regarding the path ( k, ω) = ( ω V, ω) (blue line), it is observed that its contribution is limited to the lower frequencies, and that the contributing frequenc range is narrower for receivers at the columns than for receivers at the rail. In fact, the farther a receiver is from the track, the narrower is the contributing frequenc range. As a last observation, the contribution of the path ( k, ω) = ( ω V k, ω) (green line) is more epressive in the frequenc interval [3,6] rad/s. This range limits the frequencies at which the referred to path intercepts the dispersion curves of Figure 4.3. In general, the absolute value of the transfer functions decreases as the distance to the track increases Step 3: generation stage dnamic forces After calculating the transfer functions for receivers at the rails, these functions are used in conunction with the irregularit profile and the FEM matrices of the vehicle in order to calculate the dnamic forces that the vehicle transmits to the track (procedure eplained in section 4.3). Figure 4.35 plots the dnamic forces of the front and rear ales of the train. 5

167 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings As epected, since the front vehicle is heavier than the rear one, the interaction forces of the front ale are larger than the interaction forces of the rear ale. Also observed in the figure is that for both ales the interaction forces oscillate around % above and below the static force, as epected according to Kruse and Popp () and Katou et al. (8). 3 Front Ale 5 F [N] - - F [N] ω [rad/s] t [s] Rear Ale.45 5 F [N] F [N] ω [rad/s] t [s].5 Figure 4.35: Interaction forces of front and rear ales. Left figures: cosine term A c (blue), sine term A s (red), and amplitude (black) of forces as a function of the frequenc of oscillation. Right figures: time domain forces Step 4: propagation stage response of the supporting structure After quantifing the dnamic forces, these forces and the transfer functions calculated in step are used in the equations derived in section 4. in order to obtain the response of an point of the track-soil sstem. Hence, in this step, the frequenc domain displacements of the boundar nodes of the discretized soil-building interaction surface are calculated, being for now the presence of the building neglected (weak coupling). These displacements serve as input for the last step, the reception stage, in which the response of the building is finall calculated. In order to illustrate some results of this fourth step, the frequenc and time domain displacements of the rails and of column A and the frequenc and time domain velocities of column A are plotted in Figure The results that are obtained when considering onl the quasi-static component of the interaction forces are also plotted in the same figure. 53

168 Chapter 4 Invariant structures subected to moving loads and moving vehicles 6-4 Rail -4 uz [m] uz [m] ω [rad/s] -4 Column A - displacements t [s] -5 - uz [m] uz [m] ω [rad/s] -4 Column A - velocities t [s] vz [m/s].6.4 vz [m/s] ω [rad/s] t [s] Figure 4.36: Displacements of rail and column A and velocities of column A: blue = quasistatic and dnamic forces; red = quasi-static forces onl In the first two rows of Figure 4.36, it is observed that the red line coincides with the blue line. This observation suggests that, at least up to the distance considered (approimatel m), the quasi-static component of the interaction forces is the maor contributor to the displacements. Nonetheless, the rapid oscillations of the blue line (observed in the time domain displacements of column A) indicate that the dnamic component of the interaction forces has a greater contribution to the velocities than the quasi-static component. Such is 54

169 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings confirmed in the last row of Figure 4.36, and ustifies wh the dnamic component of the interaction forces have an important role in the vibrations at the far-field Step 5: reception stage response of the building In this last step, the response of the building due to an incoming wave-field is calculated using the 3D BEM-FEM procedure discussed in section 3.. With that obective, the eigen modes calculated with the TLM in the first step are used in this step to calculate the boundar element matrices U and P of the soil-building interaction surface. The frequenc domain displacements calculated in step 4 for receivers at the same surface are assembled in vector inc uɶ and used as indicated in subsection 3..6 to form the eterior forces F BEM. Knowing the matrices U and P, the vector F BEM and the FEM matrices of the building, one is read to emplo equations (3.6)-(3.3) and calculate the frequenc domain displacements of the building. inc Notice that in this eample the building rests at the surface of the soil and therefore uɶ corresponds to the incident displacements as calculated in step 4. If the structure was buried inc or partiall buried, then the incident stresses would also contribute to uɶ, as suggested in equations (3.35)-(3.36). In the ensuing, the responses of the floors and roof are calculated. The responses are calculated at the intersections between the slabs and the mid column of the alignment closest to the track, whose points are denoted with D i in Figure Figure 4.37 plots the frequenc and time domains displacements while Figure 4.38 plots the frequenc and time domains velocities. The response of point D is represented in blue, of point D in red and of point D 3 in black. The responses of points D i are accompanied with the incident displacements and velocities at footing A, represented in green. The transverse and longitudinal displacements represented in Figure 4.37 are in phase and tend to increase as the receiver is placed higher in the structure, which leads to the conclusion that the horizontal response of the building is dominated b the first horizontal mode. The natural frequenc of this mode is ω = 5 rad/s and, around that frequenc, the horizontal responses present a peak that is not observed in the incident displacements field. [In the time domain plots, the oscillations at earlier and later times are consequence of the low damping of the building and of the large frequenc step (. Hz) used in the inverse Fourier transform. Such would not be observed if the frequenc step was made smaller or if damping was considered] The vertical displacements are independent of the floor level and match almost perfectl the incident vertical displacements. It can thus be concluded that the building responds verticall as a rigid bod. In general, the smoothness of the structure displacements, when compared with the incident displacements, suggest that the higher frequencies are filtered b the building. The frequenc content of the velocities represented in Figure 4.38 confirms so: approimatel at the frequenc ω = 65 rad/s, the building velocities reduce to less than % of the value of the incident velocities; at the frequenc ω = 3 rad/s, the building velocities almost vanish, while the incident velocities vanish onl at the frequenc ω = 57 rad/s. These features lead to lower peaks and smoother oscillations in the time histor of the velocities. 55

170 Chapter 4 Invariant structures subected to moving loads and moving vehicles u [m] u [m] ω [rad/s] t [s] u [m] ω [rad/s] -4 u [m] t [s] -5 uz [m] 5 5 ω [rad/s] uz [m] t [s] Figure 4.37: Transverse, longitudinal and vertical displacements of points D (blue), D (red), D 3 (black) and incident displacements at point A (green): left = frequenc domain; right = time domain 56

171 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings v [m/s] ω [rad/s] v [m/s] t [s] v [m/s] ω [rad/s] v [m/s] t [s] vz [m/s] ω [rad/s] vz [m/s] t [s] Figure 4.38: Transverse, longitudinal and vertical velocities of points D (blue), D (red), D 3 (black) and incident velocities at point A (green): left = frequenc domain; right = time domain 4.5 Conclusions In this chapter, the problems of moving loads and of moving vehicles are addressed, being the epressions for the former given in section 4. and the epression for the latter given in section 4.3. Furthermore, in section 4.4, all procedures addressed in this work are linked b means of an eample in which the generation, propagation and reception stages are considered. 57

172 Chapter 4 Invariant structures subected to moving loads and moving vehicles Concerning section 4., after the derivation of the equations for moving loads, these are validated through the comparison of the results thus obtained with the results obtained with a time domain procedure. The derived epressions are also used to investigate the critical load speeds of the different sstems, namel: i) beam on a Kelvin foundation; ii) slab on a laered foundation. It is seen that the two sstems present different features and therefore the former, which is a simpler two-dimensional model, cannot reproduce accuratel the behavior of the second, which is a more comple three-dimensional model. Also, it is concluded that the stratification of the laered foundation influences the response fields and the critical load speeds, and therefore to consider homogeneous foundations ma ield incoherencies in the results. Concerning section 4.3, in which the interaction between an invariant structure and a moving vehicle is addressed, after the derivation of the epressions, these are validated through the comparison of the dnamic forces thus obtained with the dnamic forces obtained with a time domain procedure. The critical speed of a point mass moving on top of a beam on a Kelvin foundation is also studied and it is concluded that the critical speed is related to the resonance of the mass-spring sstem and to the critical load speeds of the foundation. The eample in section 4.4 links all the procedures eplained in this work and therefore makes the connection between the three stages of the studied problem: the generation stage, in which the vehicle interacts with the track inducing on it a moving force field; the propagation stage, in which the vibrations propagate through the track and foundation; the reception stage, in which the vibrations reach the building and cause it to respond dnamicall. The following characteristics are observed: i) the dnamic forces correspond to approimatel % of the total interaction force; ii) iii) iv) the contribution of the dnamic forces can be neglected in the calculation of the track response; the displacements of the soil at remote positions are dominated b the static component of the interaction forces; however, the dnamic component dominates the velocities; the horizontal response of the building increases with the floor level and is dominated b the first mode of the building, which is activated long before the train reaches the proimit of the building; v) the considered building filters the incident vibrations above the frequenc 65 rad/s ω =. The mentioned observation are specific of the problem that was solved in this chapter, and should not be generalized. 58

173 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings 5. Reduction of vibrations b means of trenches 5. Introduction The prediction and mitigation of vibrations induced b moving vehicles has been a concern to engineers for the past decades, as can be confirmed b the publications on the subect, which go back to the 9 th centur (South, 863). In fact, vibrations induced b traffic, including railwa transportation, are responsible for the annoance of inhabitants surrounding the transport infrastructures and, in the most severe situations, ma induce damage, at least in an aesthetic point of view, in sensitive structures such as old heritage buildings. The relevance of the above mentioned concern ustified several studies on the topic in order to reach a deeper understanding of the problem and of its mitigation, but despite all the progress that has been made in recent ears, there are still some gaps that deserve deeper studies, mainl in what concerns the comprehension and efficienc of mitigation measures. Nowadas, this topic is being obect of attention b the political instances, as indicated b the currentl running proects RIVAS ( and CARGOVIBES ( both subsidized b the European Commission and whose obective is to propose mitigation measures to obtain more sustainable and environmental friendl railwa infrastructures. The countermeasures for vibrations induced b railwa traffic can be classified b the location where the are applied: i) at the source; ii) at the propagation path; iii) at the receiver. Mitigation at the source ma involve, among other options, changing the properties of the trains (suspension sstem, masses, etc.), changing the tpe of the track (ballast track versus slab-track) and its resilienc (using rail pads, under-sleeper pads and/or ballast mats), and improving the rolling conditions of the vehicle (i.e., to reduce the defects of the wheels and the vertical roughness and unevenness of the rail) (Hemsworth, ). This last option appears to be the most efficient strateg because the improvement of the rolling qualit of the vehicle results in the reduction of the dnamic forces transmitted b the vehicle to the track, and, consequentl, in the reduction of the vibrations felt at large distances from the track (Nelson, 996). Nevertheless, due to the high cost of the wheel truing/rail grinding, it becomes economicall unsustainable to rel onl on this approach. Regarding the modification of the resilienc of the track, the main purpose is to achieve a considerable attenuation of vibration levels at high frequencies (Alves Costa et al., b; Bongini et al., ). However, the introduction of these elements on the track is also accompanied b the amplification of energ transmitted to the ground at the low frequenc range, which ma amplif the response of nearb buildings at their lower natural frequencies (Alves Costa et al., b; Alves Costa et al., 3). On the other hand, mitigation at the receiver (a structure to be shielded from vibrations) involves the application of elastic materials at the foundations, so that the whole building is isolated, or at certain floors/compartments, in order to separate them from the rest of the building. Fiala proposes different approaches based on this idea (Fiala et al., 7; Fiala et al., 8). The drawback of this option is that it is onl applicable, at least at a reasonable economical cost, to new buildings, since it involves drastic changes on the structural behavior 59

176 Chapter 5 Reduction of vibrations b means of trenches track. As seen in chapter 4, this interaction has direct influence on the vibration field induced in the ground. For this reason, in this work, the circulation of the train and its interaction with the track are taken into account in the assessment and discussion of different trench solutions. This chapter is organized as follows: section contains a comprehensive stud on the parameters that most influence the reduction achieved b trenches; section 3 investigates the efficienc of different trench solutions for the mitigation of vibrations induced b the passage of trains in an eisting line (near Carregado, Portugal); section 4 evaluates the influence of such measure on a nearb building; finall, section 5 summarizes the main conclusions of the chapter. 5. Parameters influencing the efficienc of trenches 5.. Introduction In order to obtain a more comprehensive insight about the influence of the parameters with more relevance to the efficienc of trenches, a parametric stud is presented. The stud is similar to the studies presented in some of the works mentioned in section 5.. The problem to be considered is depicted in Figure 5.: a homogenous half-space is submitted to a harmonic vertical point source and the response of the surface of the half-space is computed before and after the ecavation of a trench with width w and depth d at the distance l from a point source. Plane strain conditions are assumed (unless otherwise stated), which means that in the parametric stud the longitudinal wavenumber k is assumed to be null. The material properties of the soil are defined b the variables G Soil (shear modulus), ρ Soil (material densit), ν Soil (Poisson s ratio) and ξ Soil (hsteretic damping), while the material properties of the in-fill material are defined b the variables G Mat, ρ Mat, ν Mat and ξ Mat. l i e ωt w d Figure 5.: Trench in a half-space To analze the problem with the.5d BEM-FEM procedure, the trench limits are simulated with boundar elements of linear epansion, while the in-fill material is simulated with finite elements. The dimensions of the boundar elements and of the finite elements are such that elements per Raleigh wavelength are used. The fundamental solutions to be used in the BEM are calculated with TLM models such that for each boundar element there are two thinlaers of quadratic epansion. Additionall, for the TLM models, an elastic laer with 6

177 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings thickness λ R is placed between the lower surface of the trench and the lower PML simulating the lower half-space. The parameters to be studied are the trench depth d, the trench width w, the trench position l, the stiffness and densit of the in-fill material, and the Poisson s coefficient of the soil. The influence of stratification and of modeling strateg (D versus 3D) is considered afterwards. In order to avoid dependenc on the ecitation frequenc ω, all dimensions are made proportional to the Raleigh wavelength λr = π CR ω, in which C R is the Raleigh wave speed of the half-space. The Raleigh wave speed can be approimated b the epression (.874 (.97 ( ) ) ) C = G ρ + + ν ν ν (5.) R Soil Soil Soil Soil Soil Unless otherwise stated, the following values are assumed: d = λr, w =.λr, l = 5λR, ρmat = ρsoil, ν Soil =.5, ν Mat =.3, ξsoil = ξmat =.3. For each parameter investigated, three scenarios are considered for the in-fill material: G Mat = (open trench), GMat =.GSoil (softer material) and GMat = GSoil (stiffer material). For each scenario, the efficienc of the trench is evaluated through the position dependent ratio A r A r ( ) Trench z No trench z ( ) ( ) = u u (5.) Trench in which u z represents the vertical displacement at the surface of the soil after the No trench construction of the trench, u z represents the vertical displacement at the surface of the soil before the construction of the trench, and in which is the transverse distance to the point source (the smaller the ratio Ar ( ), the better the isolation). The average ratio A r, calculated with L Ar = Ar ( + l) d L (5.3) ma also provide important information and therefore is also represented. L represents the maimum distance of interest after the trench. In the net sub-section, a validation eample is shown. The parametric stud starts in sub-section Validation eample The suitabilit of the.5d BEM-FEM procedure for the simulation of trenches is assessed prior to its utilization in the corresponding parametric stud. With this purpose, the reduction achieved b an open trench is computed and compared with the results published in the literature and with the results obtained b means of a FEM approach. The validation problem consists of a trench with depth d =.λr and width w =.λ R, placed at the distance l = 5λR from a massless rigid footing with width b =.5λR. The footing is perfectl bonded to the underling homogeneous half-space and is submitted to a prescribed uniform vertical displacement. Figure 5. shows the ratios A r obtained in this work (blue curve), the ratios A r obtained b Yang and Hung (997) (red curve), and the ratios A r obtained with a D FEM approach (black curve). For the D FEM approach, an elastic region with width λ and R 63

178 Chapter 5 Reduction of vibrations b means of trenches depth λ R is modeled with 4 noded quadrilateral elements and such that there are 4 finite elements per Raleigh wavelength. The elastic region is augmented with the PMLs defined in the work of Kausel and Barbosa () in order to absorb the outgoing waves..5.5 A r λ Figure 5.: Reduction ratios A r for an open trench: blue line =.5D-BEM FEM; red line = Yang and Hung (997) ; black line = D FEM-PML Some differences can be observed between the three curves, but the trends are the same and the deviations of the curves are relativel small. Thus, it can be concluded that the.5d BEM-FEM procedure is adequate for this parametric stud and that the discretization used for the boundar elements is fine enough. It must be added, however, that the differences observed between the distinct curves, and namel after the trench (for λ R > 5 ), can be ustified b the dispersion associated with the discretization. The recalculation of the blue and black curves assuming 8 boundar elements per Raleigh wavelength (for the blue curve) and assuming 8 finite elements per Raleigh wavelength (for the black curve) leads to practicall coincident results Influence of the trench depth According to the literature, one of the parameters that significantl influence the efficienc of trenches is the depth. The influence of this parameter for the three trench solutions (open trench, trench filled with soft material and trench filled with stiff material) is represented in Figure 5.3 through the representation of the ratios A r and their average A r ( L = 5λR ) as a function of the trench depth. A maimum depth of d = 3λ is assumed. For the case of an open trench (Figure 5.3a), the efficienc of the trench appears to increase with its depth. However, above d = λr, increasing the depth does not seem to result in a considerable change of the trench efficienc, namel for < 3λR. For > 3λR, the ratio A r oscillates around one, which means that there are zones where the response is amplified. These amplification zones are ustified b horizontal shifts of the local minima due to the presence of the trench. Nevertheless, at these distances the response is so small that these amplifications have no practical consequence. R R 64

179 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings a) b) d/λr d/λr /λ R c) /λ R d).8 d/λr A r.6.4. Open trench Soft material Stiff material /λ R d/λ R Figure 5.3: Ratio A r as a function of the trench depth: a) open trench; b) trench filled with soft material; c) trench filled with stiff material; d) average ratio A r For the case of a trench filled with a soft material (Figure 5.3b), the ratios A r tend to decrease with the trench depth, reaching the minimum average value A r at the depth d =.5λR. Above this depth, the ratios A r start to increase, and when the trench depth d = λr is reached, the ratios A r remain approimatel constant. Therefore, for this case the best efficienc seems to be achieved when the depth of the trench is one half of the Raleigh wavelength. A deeper trench ma result in a reduction of displacements of 3% instead of 5%. Finall, for the case of a trench filled with a stiff material (Figure 5.3c), the best efficienc is obtained for the trench depth d =.5λR, but the average reduction is of onl %. Immediatel after the trench ( 5λR < < 6.5λR ) the reduction can reach up to 5%, but it decas as the distance to the source and trench increases. In all these cases, the displacements before the trench ( < l ) are amplified Influence of the trench width The other dimension that also influences the trench performance is its width. The influence of this parameter is evaluated net through the calculation of the ratios A r and A r for a trench width varing from.λ R to λ R. Figure 5.4 shows the obtained results. [A real trench is not likel to be wider than.5m. In this wa, for a soil with Raleigh wave speed C R = m/s (for eample), the width w =.5λR is achieved onl for frequencies above 65

180 Chapter 5 Reduction of vibrations b means of trenches 4 Hz. Thus, for the purpose of train induced vibrations, onl the results considering small trench widths (sa w <.5λ ) are of interest.] R a) b) w/λr w/λr /λ R /λ R c).8 d) Open trench Soft material Stiff material w/λr A r.6.4. /λ R.5.5 w/λ R Figure 5.4: Ratio A r as a function of the trench width: a) open trench; b) trench filled with soft material; c) trench filled with stiff material; d) average ratio A r For the open trench, its width does not have much influence on the efficienc, at least up to the distance = 3λR. For greater distances, increasing the width seems to enlarge the shielded zone (the darker blue area in Figure 5.4a etends farther as w increases). This tendenc is not observed in the average ratio A r because L = 5λR. For the in-filled trenches, the increase of the trench width leads, in general, to an increase of the trench performance. The optimal trench width is w =.λr for the soft material and w = λr for the stiff material. However, these values are not attainable for the frequencies ecited b trains, and so it is concluded that for realistic values, to increase the trench width is alwas beneficial Influence of the trench position With the aim of studing the influence of the trench position, the distance l between the point source and the trench assumes values between l =.5λR and l = λr and the ratios A r and A r are accordingl calculated. The results are represented in Figure

181 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings a) b) l/λr l/λr /λ R c) /λ R d).8 l/λr A r.6.4 Open trench Soft material Stiff material. /λ R l/λ R Figure 5.5: Ratio A r as a function of the trench position: a) open trench; b) trench filled with soft material; c) trench filled with stiff material; d) average ratio A r The results in Figure 5.5a-c suggest that the attenuation of the response after the trench ( > l ) is almost independent of the trench position: vertical stripes can be identified in the colored plots, which sustain this conclusion. The average ratio A r shows a similar trend: for the open trench, A r oscillates around the value A r =.3, being the maimum deviation equal to.; for the trench filled with the soft material, the mean value is A r =.75 and the maimum deviations are around.5; for the trench filled with the stiff material, the mean value is A =.85, and the maimum deviations are also around.5. r 5..6 Influence of the stiffness of the in-fill material The present and the following sub-sections investigate the influence of the mechanical properties of the in-fill material: the stiffness is considered first, and the material densit is considered second. The Poisson s ratio is not studied in this work, but it has been shown that its influence on the behavior of in-filled trenches is negligible (Yang and Hung, 997). The influence of the stiffness of the in-fill material is analzed b varing the shear modulus from GMat =.G Soil to GMat = GSoil. Figure 5.6 shows the corresponding ratios A r and A r. 67

182 Chapter 5 Reduction of vibrations b means of trenches a) b). GTrench/GSoil A r /λ R log(g Trench /G Soil ) Figure 5.6: a) Ratio A r as a function of the stiffness of the in-fill material; b) average ratio A r It can be clearl observed from Figure 5.6b that softer in-fill materials perform better than stiffer in-fill materials. Furthermore, for softer in-fill materials, the isolation efficienc improves with the decrease of the stiffness, while for stiffer materials, the isolation efficienc improves with the increase of the stiffness Influence of densit of in-fill material The second propert of the in-fill material to be studied is the densit. With that aim, the ratios A r and A r are calculated for densities ρ Mat varing between ρ Mat = and ρmat = ρsoil. Again, the scenarios of stiff in-fill material ( GMat = GSoil ) and of soft in-fill material ( GMat =.GSoil ) are considered. The open trench scenario is not applicable in this stud. Figure 5.7 shows the corresponding results. The main conclusion that can be inferred from the results shown in Figure 5.7 is that the densit of the in-fill material considerabl influences the behavior of trenches. While for soft in-fill materials a lighter material is beneficial, for stiffer materials it is convenient to have high densities. For the two stiffness scenarios studied, the densit has more influence on the behavior of the stiff material than on the behavior of the soft material Influence of the Poisson s ratio of the soil Since the soil is assumed homogeneous and since the trench dimensions are made proportional to the Raleigh wavelength λ R, then the unique parameter of the soil that ma influence the behavior of the trench is the Poisson s ratio. The influence of this parameter is investigated net through the calculation of the ratios A r and A r, which are represented in Figure

183 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings a) b) ρtrench/ρsoil ρtrench/ρsoil /λ R.8 c) Soft material Stiff material /λ R A r.6.4 Figure 5.7: Ratio A r as a function of the densit of the in-fill material: a) trench filled with soft material; b) trench filled with stiff material; c) average ratio A r For all three cases, the trend is that the average ratio A r increases (slightl) with the increase of the Poisson s ratio. The eplanation for this can be found in the ratio λp λ R ( λ P represents the wavelength of the P wave). This ratio increases with the increase of the Poisson s ratio and thus, for high values of this parameter, the dimensions of the trench become small when compared to the P wave. As a consequence, the reflection of P waves becomes negligible. The decrease of efficienc of the trenches due to higher Poisson s ratios is more obvious awa from the trench than in its proimit Influence of the ground stratification In general, the ground conditions do not correspond to homogeneous half-spaces, and therefore soil stratification ma need to be considered. In order to understand if disregarding the soil stratification ma affect the predictions, in this section the homogeneous half-space is replaced b a laer on top of a half-space. The material properties of the laer and underling half-space are indicated in the general description (sub-section 5..), ecept for the shear modulus of the half-space, which assumes values between GHalf =.G Laer and GHalf = 5GLaer. The thickness of the laer is scenario log(ρ Trench /ρ Soil ) H = λ. Figure 5.9 shows the ratios A r and R A r for this new 69

184 Chapter 5 Reduction of vibrations b means of trenches a) b) νsoil νsoil /λ R c) /λ R d).8 νsoil A.6 r.4 Open trench Soft material Stiff material. /λ R ν Soil Figure 5.8: Ratio A r as a function of the Poisson s ratio of the soil: a) open trench; b) trench filled with soft material; c) trench filled with stiff material; d) average ratio A r It can be seen in Figure 5.9 that the stratification of the ground considerabl changes the behavior of the trenches, namel when the lower half-space is stiffer than the upper laer. The influence of stratification is more noticeable for the open trench than for the in-filled trenches. 5.. Influence of the modeling strateg The modeling strateg is studied net to assess whether the simulations under plane-strain conditions are valid to infer about the behavior of trenches when acted upon b standing point loads (three-dimensional ecitation). Figure 5. plots the ratio A as function of the longitudinal ( ) and transverse ( ) coordinates for a point source applied at the origin (, ) = (,). For the open trench and for the trench filled with a soft material, the ratio A r after the trench is approimatel smmetric with respect to the loaded point. In opposition, for the stiff trench, one can notice the eistence of a ellowish triangular region inside which the displacements are not attenuated. This shadow region is a consequence of the high bending stiffness of the trench, being the angle of these triangular shaped shadow zones dependent on the ecitation frequenc, on the Raleigh wave speed of the soil, and on the bending dispersion of the trench. Epressions for this angle can be found in Coulier et al. (3b). This feature indicates that for standing harmonic solicitations, the best orientation of a stiff trench is not perpendicularl to the line that connects source and receiver, but instead at a certain angle. r 7

185 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings a) b) GHalf / GLaer GHalf / GLaer /λ R c) /λ R d) GHalf / GLaer /λ R A r G Half / G Laer Open trench Soft material Stiff material Figure 5.9: Ratio A r as a function of the stiffness of the lower half-space: a) open trench; b) trench filled with soft material; c) trench filled with stiff material; d) average ratio A r The average ratio A r has been computed for the central alignment =, and it has been found that this ratio is ver similar to the D ratio (results not shown here). In this wa, for open trenches and trenches filled with soft materials, since the ratio A r is almost smmetric with respect to the loaded point, D simulation can be used to estimate the reduction maps plotted in Figure 5.a,b. For stiff in-fill materials, such is not possible due to the presence of the shadow zones. Thus, at least for the case of stiff in-fill materials, 3D simulations are necessar. 5.. Conclusions In this section, a parametric stud was performed to evaluate the influence of some variables on the reduction of vibrations achieved b trenches. The following conclusions are drawn:. Increasing the depth of the trench has in general a beneficial effect, but above the value d = λr, increasing the depth does not improve significantl the efficienc of the trench. In the case of the trench filled with a soft material, the optimal value for the trench depth is one half of the Raleigh wavelength.. For the case of open trenches, the trench width has no significance. For in-filled trenches, to enlarge the width reveals to be beneficial. 7

186 Chapter 5 Reduction of vibrations b means of trenches a) b) /λr /λr /λr /λ R c) /λ R /λ R Figure 5.: Ratio A r for a 3D loading scenario: a) open trench; b) trench filled with soft material; c) trench filled with stiff material 3. The trench position does not influence much its efficienc. 4. Open trenches perform better than in-filled trenches, and within in-filled trenches, materials softer than the soil tend to perform better than materials stiffer than the soil. Trenches filled with soft materials perform better if the in-fill material is made softer, while trenches filled with stiff materials perform better is the in-fill material is made stiffer. 5. The densit of the material also influences the trench performance. The trend is that as the in-fill material is made heavier, the reduction provided b the trench increases. This parameter has more influence in the case of a stiff material than in the case of a soft material. 6. The greater the Poisson s ratio of the soil, the worse the efficienc of the trench is. This parameter has more influence on open trenches than on in-filled trenches. 7. To consider the ground stratification changes considerabl the performance of the trench. 8. For open trenches and trenches filled with a soft material, D simulations ma provide good information about the reduction achieved with trenches when acted upon b point loads. The same is not valid for trenches filled with stiff materials. For this last situation, the best position of trenches is obliquel to the line that links source and receiver. 7

187 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings The conclusions obtained in this stud are the same as those of the studies referred to in 5.. However, it must be mentioned that each parameter is being studied independentl of the others, and so, if more than one parameter varies in each stud, then different conclusions ma be reached. As last comment, it must be restated that these investigations concern standing line loads. In the case of moving loads, the response of the ground is characterized b a wide frequenc band and so the dimensionless stud becomes impossible, turning the problem much more comple. This issue is addressed in the net sections, in which trenches are used as mitigation measures for the vibrations induced during the passage of a train. 5.3 Trenches for the mitigation of train induced vibrations 5.3. Introduction In this section, distinct trench solutions are analzed and compared in the contet of vibrations induced b railwa traffic. This problem differs from the problem of vibrations induced b fied stationar loads in the fact that the motion induced b moving loads is characterized b a wide range of frequencies, while the response induced b stationar loads has the same frequenc as the load. The problem to be analzed is schematized in Figure 5.: a train runs on a surface track, net to which a trench is constructed in order to scatter the impinging waves and thus reduce the vibrations after the trench. For the stud, the train is considered as a set of moving harmonic loads whose amplitudes result from the solution of the train-track interaction problem described in chapter 4, the track and the interior of the trenches are simulated with finite elements, and the ground is simulated with boundar elements. Train simulated with a multi-bod approach Soil-track surface discretized with BEM Track modeled with FEM Soil modeled with BEM Open or in-filled trench: interior modeled with FEM; soil-trench surface discretized Figure 5.: Schematization of the problem The main obective of this section is the comparison between different trench solutions in the reduction of vibrations induced b railwa traffic. For this purpose, the geometr and material properties of an eisting and operational line are considered, being the scenario chosen for the analses a stretch of the Portuguese railwa network, near the town of Carregado. The reason for this option is the fact that the local conditions at this line have been obtained after some eperimental campaigns that have been carried out with that intent. The reader is referred to Alves Costa et al. (a) for more details on these eperimental campaigns. The following paragraphs describe the soil conditions, track geometr and properties, rolling material properties and trench solutions assumed in the analses. 73

188 Chapter 5 Reduction of vibrations b means of trenches Local soil conditions and TLM model for the soil The local properties of the ground have been determined eperimentall through cross-hole tests, CPT tests, and SASW tests, having the eperiments revealed that it consists of several laers of cla with distinct wave velocities (Alves Costa et al., a). In the following analses, the soil model is assumed to be simpler than the conditions obtained eperimentall, being the model composed b two elastic laers on top of a half-space, whose properties are indicated in Table 5.. Laer Thickness H [m] Table 5.: Soil stratification and properties Densit ρ [kg/m3] P wave speed C P [m/s] S wave speed C S [m/s] Hsteretic damping ξ = ξ For the TLM model, the laers one and two are divided into quadratic thin-laers with. m of thickness, while the lower half-space is simulated through the use of PMLs (Barbosa et al., ). Figure 5. shows the dispersion curves and the corresponding phase velocities for the considered soil profile. The maimum phase velocit corresponds to the shear wave speed of the lower half-space, while the minimum phase velocit corresponds to the Raleigh wave speed of the top laer (approimatel 43 m/s). 8 Dispersion Curves 35 Phase Velocit P S f [Hz] k [rad/m] V [m/s] f [Hz] Figure 5.: Dispersion curves (left) and corresponding phase velocities (right) of the soil. Properties of the track and corresponding numerical model The line at Carregado corresponds to a double ballast track, with a straight alignment, and is composed of UCI6 rails, Vossloh rail pads, concrete sleepers (spaced.6 m), and ballast and subballast laers. Despite the fact of being a double track, the numerical model used in the following analses corresponds to a single track, whose geometr and FEM mesh are represented in Figure 5.3. As previousl shown in Alves Costa et al. (a), the use of a single line model to simulate a double track is accurate enough. The elastic and dnamic properties of each component of the track are indicated in Table

189 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings.55 m.5 m.668 m Rail Subballast Ballast Sleeper Rail pad. m.4 m.5 m 7. m Figure 5.3: FEM model of the track Young s Modulus E [Pa] Table 5.: Properties of the components of the track Rails UIC6 (modeled with Euler beams) Mass Densit ρ [kg/m 3 ] Section Area A [m ] Vertical Stiffness Rail Pads Section Inertia I V [m 4 ] K V [kn/mm] Vertical Damping C V [kns/mm] Real: 6 Model * : Real:.5 Model * : 37.5 Real Model ** Sleepers Mass [kg/m] E [Pa] Dimensions [m 3 ] ρ [kg/m 3 ] E z [Pa] ν z E [Pa] ν ξ = ξ ν z P S Ballast E [Pa] ρ [kg/m 3 ] ν ξp = ξs Subballast E [Pa] ρ [kg/m 3 ] ν ξp = ξs * The rail pads are modeled with distributed line springs along the longitudinal direction ** The sleepers are modeled with anisotropic materials, as eplained in chapter 3 75

190 Chapter 5 Reduction of vibrations b means of trenches Another important aspect related to the track is its unevenness, which is responsible for the generation of the dnamic train-track interaction forces. The unevenness at Carregado has been measured with a EMI recording car, capable of measuring the unevenness profiles of rails for the range of wavelengths between.4 and 5 m. With the power spectral densit (PSD) of the measured unevenness profile, artificial profiles with 4 wavelengths ranging from.75 m to 3 m have been generated. These profiles are based on the epressions suggested in the norm ISO868 (995) and are such that their PSD approimate these of the measured profile. Figure 5.4 represents the PSD of the measured and artificial profiles, and the longitudinal variation of one possible random profile generated with the epressions of ISO868. The same unevenness profile is assumed for both rails. PSD [m 3 /rad] Wavelength [m] Unevenness [m] Longitudinal position [m] Figure 5.4: PSD of the unevenness (left: red = measured; blue = approimated) and random unevenness profile (right) Rolling stock and numerical model The trains that circulate in the Northern line near Carregado (Portugal) comprise commuter trains, with speeds of approimatel 3 km/h, Inter Cities trains, which can travel up to km/h, Alfa Pendular trains, which are the fastest trains operating in Portugal, with speeds up to km/h, and in some cases, freight trains, which travel with speeds below km/h. The train considered in this work is the Alfa Pendular. The Alfa Pendular is a high-speed train composed of si vehicles, fulfilling a total length of 58.9 m. Each vehicle is composed of the car-bod, two bogies and four ales. The car-bod is linked to the bogies at its etremities through the secondar suspension sstem, and the bogies are connected to a pair of ales through the primar suspension sstem. In the following analses each vehicle of the train is modeled as a two dimensional multi rigid-bod sstem in which the ales, the bogies and the car-bod are considered as rigid masses, and in which the suspension sstems that connect these components are simulated b means of springs and dashpots. All displacements not contained in the vertical-longitudinal plane are neglected. The contact stiffness is accounted for b means of springs between the wheels and the rails, that simulate the Hertzian contact (Wu and Thompson, ). The geometr of the train is represented in Figure 5.5 and the D model of each vehicle is represented in Figure 4.3. The dnamic properties assumed for the different components of the train are given in Table

191 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Figure 5.5: Geometr of the Alfa Pendular train Vehicle M [kg] Table 5.3: Dnamic properties of the train Ales Car bod Front bogie Rear bogie FF FR RF RR I M I [kg.m ] [kg] [kg.m ] [kg] [kg.m ] M I M [kg] M [kg] M [kg] M [kg] Primar suspension sstems Secondar suspension sstems Contact springs K [N/m] C [Ns/m] K [N/m] C [Ns/m] K [N/m] M = mass; I = momentum of inertia; K = stiffness; C = damping Trench solutions and numerical models The mitigation solutions analzed in this work are materialized with trenches constructed 7.5m awa from the central line of the track. The trenches are.4 m wide and two depths are considered: 3 m and 6 m. The trenches are open or in-filled with concrete or geofoam. The elastic and dnamic properties of these in-fill materials are listed in Table 5.4. From a practical standpoint, the open trench is not a viable solution as it requires some tpe of supporting sstem in order to avoid its walls to collapse. Nevertheless, this solution is taken into consideration for comparison purposes. As for geofoam, it consists in an epandable polstrene material with low densit which provides fleibilit in the design and easiness in its practical implementation. It has been investigated as a possible in-fill material for trenches b Alzawi and Hesham El Naggar (9, ; ). The concrete filled trenches can be materialized through buried concrete walls and have also been investigated in several eperimental and numerical studies (e.g., Celebi et al., 9). 77

192 Chapter 5 Reduction of vibrations b means of trenches E [Pa] (Young s modulus) Table 5.4: Properties of the in-fill materials ρ [kg/m 3 ] (Mass densit) Concrete ν (Poisson s ratio) ξ = ξ P (hsteretic damping) Geofoam E [Pa] ρ [kg/m 3 ] ν ξp = ξs S The behavior of the trenches is simulated with quadrilateral finite elements with 4 nodes and with dimensions.. m. The interface between the soil and the trench border is simulated b boundar elements of constant epansion in such a wa that there is one boundar element per face of the finite element mesh in contact with the soil (the same rule applies for the interface between the track and the soil) D analses influence of the track Before moving into the case of vibrations induced b traffic, the case of line loads ( k = ) is addressed so that some features of the numerical modeling ma be inferred. The impact resulting from the eclusion of the track from the mathematical model, in what regards the predicted soil response and the predicted trench efficienc, is investigated. For that purpose, two scenarios are considered: a first scenario in which the track is included and loads of the ω F t =.5e t are applied at each rail; and a second scenario in which the track is tpe ( ) i i ecluded and therefore a load of the tpe F ( t) the position = (central alignment of the track). ω = e t is applied directl at the ground surface at The differences in the ground surface responses are investigated first, and to this end, the vertical displacements are calculated as functions of the ecitation frequenc ω and of the Track transverse distance. The results are depicted in Figure 5.6, where the variables u z and No track u z correspond to the displacements in the presence and in the absence of the track, respectivel. In order to facilitate the interpretation of the differences between the two No track Track scenarios, the ratios u u are also depicted. No trench is considered at this point. z z Figure 5.6 confirms that the eclusion of the track from the mathematical model considerabl alters the predicted ground response. As can be observed, the displacements are two orders of magnitude higher for frequencies above 4 Hz when the track is ecluded. There is no significant difference for frequencies below Hz. 78

193 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings a) b) Ecitation frequenc [Hz] Ecitation frequenc [Hz] Ecitation frequenc [Hz] [m] c) [m] Track No track Figure 5.6: Displacements at the ground: a) log ( u z ) ; b) log ( z ) No track Track c) log ( uz u z ) u ; The difference observed for frequencies above 4 Hz can be eplained b the forces that are transmitted to the ground: firstl, the total transmitted force is not the same for the two scenarios (Figure 5.7a); secondl, in the first scenario, the transmitted force is distributed along the track-soil interface (Figure 5.7b), while in the second the force is concentrated at a single point. Therefore, as the force transmitted b the track is greater than the applied force, it could be epected that the predicted displacements were greater when the track was considered. However, since the load is distributed along the train-track interface, the near field displacements are smaller. For remote positions, the effect of the load distribution disappears and the predicted ground displacements are greater when the track is taken into account. This can be observed in Figure 5.7c, where the ground displacements are depicted for > 3 m and for the ecitation frequenc 8 Hz, with and without the track. The discrepanc between the two scenarios reveals that when the track is disregarded the predicted response of the near field is over-estimated, which can lead to an incorrect conclusion about the need for mitigation measures. [m] 79

194 Chapter 5 Reduction of vibrations b means of trenches 5 a) Vertical traction [N/m ] Vertical force [N] f [Hz] b).5 c) - Vertical displacement [m].5 Track No track -4-4 [m] [m] Figure 5.7: a) Resultant of the vertical forces between track and soil; b) distribution of tractions at the track-soil interface for f = 8 Hz; c) displacements at the ground surface for f = 8 Hz. The importance of the track for the correct estimation of the efficienc of trenches is eamined net. With that aim, the ratio A r, defined in equation (5.), is calculated as a function of the ecitation frequenc ω and of the transverse distance for the two scenarios (with and without track). Figure 5.8 presents the results obtained for the trench solutions with 3 [m] of depth. The comparison between the left and right columns of Figure 5.8 reveals differences in the estimated reduction levels: firstl, higher reduction levels are obtained when the track is not accounted for; secondl, there is a band near the frequenc 6 Hz that causes the displacements after the trench to be amplified, if the track is included, which does not occur if the forces are applied directl to the ground. In general, the estimated reduction is higher in situations in which the track is disregarded, and lower when the track is considered. For this reason, the track eclusion ma lead to under designed protective measures. The differences reported in this sub-section point out the need for the inclusion of the track in the numerical model, as it is observed that considering simpler models in which the track is disregarded ma result in over-estimated attenuation levels and, consequentl, in inappropriate designs of trenches. On the other hand, neglecting the track results in overestimations of the induced level of vibrations, which ma lead to an incorrect udgment about the need for mitigation measures. 8

195 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings a) b) Ecitation frequenc [Hz] [m] c) [m] Ecitation frequenc [Hz] Ecitation frequenc [Hz] Ecitation frequenc [Hz] [m] d) Ecitation frequenc [Hz] e) [m] f) Ecitation frequenc [Hz] [m] [m] Figure 5.8: Ratios A r for: a) open trench with track; b) open trench without track; c) geofoam trench with track; d) geofoam trench without track; e) concrete trench with track; f) concrete trench without track A common feature of the two scenarios is the amplification of displacements before the trench, which is a consequence of the reflection of waves at the trench. It is therefore important to eamine if this amplification effect ma influence the train-track interaction, potentiall causing an amplification of the interaction forces and resulting in the cancellation of the beneficial effects associated with the construction of trenches. This issue is addressed in the net sub-section, together with the evaluation and comparison of the efficienc of the distinct trench solutions. 8

196 Chapter 5 Reduction of vibrations b means of trenches D analses vibrations induced b the Alfa Pendular train In this sub-section, the vibrations induced b an Alfa Pendular train travelling at the speed V = 6 m/s are analzed. Contraril to the analses reported in 5.3., the present stud demands for the development of a 3D solution with train-track interaction. The dnamic interaction between the train and the track is taken into account through the formulation presented in chapter 4, for which the irregularit profile shown in Figure 5.4 is used. This sub-section is divided into three topics: in the first topic, the influence of the trenches on the train-track interaction is discussed; in the second, the different trench solutions are compared; in the last topic, the behavior of the trenches along the longitudinal coordinate is assessed. Influence of the trenches in the train-track interaction As described in sub-section 5.3., the construction of trenches causes the amplification of displacements in the ground surface located between the track and the trench. At a first glance, this amplification ma seem inconsequent, as the main obective is the attenuation of vibrations after the trench and not before it. However, since the vibrations are reflected at the trench and then come back towards the track, a deeper thought leads to the question: Will the reflections interfere in the train-track interaction phenomenon, causing the increase of the dnamic forces and the cancellation of the beneficial effects of the trench? This question is investigated net, through the comparison of the dnamic forces of the first wheel set for the mitigated and non mitigated situations. These forces and the relative deviations with respect to the non-mitigated scenario are overlapped in Figure 5.9 for the different trench solutions under analsis. Dnamic force [N] 5 5 3m concrete trench 3m open trench 3m geofoam trench 6m concrete trench 6m open trench 6m geofoam trench No trench Relative deviation m concrete trench 3m open trench 3m geofoam trench 6m concrete trench 6m open trench 6m geofoam trench Frequenc [Hz] Frequenc [Hz] Figure 5.9: Dnamic forces (left) and relative deviations (right) of first wheel set The maimum deviation of the magnitude of the dnamic forces is %, so it is concluded that the trenches do not affect the train-track interaction mechanism, at least for the track-trench distance considered in the present analsis. Though it could be epected that the waves that are reflected and that return to the track had more influence on the train-track interaction, the opposite happens, and the reason for that can onl be another reflection of waves when the reach back the track-soil interface. This aspect can be used in the benefit of the simulations, since, according to the conclusions, the train-track interaction problem needs to be solved onl for the non-mitigated scenario, enabling the use of the obtained results in the remaining situations. 8

197 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings The efficienc of trenches The efficienc of distinct trench solutions is now compared. The trench solutions are compared based on the insertion loss (IL) and on the running root means square (RRMS) of the vertical velocit induced b the passage of the train. Three receivers are considered, being all of them placed after the trench, namel at the transverse positions = m, = 5 m, and 3 = m ( = corresponds to the central alignment of the track). The longitudinal position of all receivers is =. The insertion loss IL(,, ω ) for a point with coordinates (, ) is calculated with IL ( ω) NoTrench z log u Trench z and the RRMS for the same point is calculated with where v (,, ) RRMS(,, t) = ( ω) ( ω) u = (5.4) t + t vz (,, τ ) dτ (5.5) t t z t is the vertical velocit in the time domain. The insertion losses and RRMSs for the three receivers are represented in Figure 5. (in one-third-octave bands) and in Figure 5., respectivel. For the calculation of the RRMS, the time window t =.5 s is used, as indicated in the norm DIN45- (999). The IL curves depicted in Figure 5. show that above the frequenc Hz, the displacements are attenuated for all trench solutions and all receivers considered. For this frequenc, the phase velocit of the Raleigh wave is approimatel 8 m/s (Figure 5.) and the associated wavelength is 8 m, so even though the depth of the shallower trench is onl /6 of the Raleigh wavelength, it is concluded that the 3 m deep trenches are able to reduce the vibrations. This feature could not be inferred from the D analses (Figure 5.8), thus highlighting the relevance of 3D analses for the understanding of the problem. Also based on the IL plots, it can be observed that the 6 m trenches tend to perform better than the 3 m trenches, as epected, and that no general trend can be distinguished in what concerns the influence of the distance between receivers and trench. As for the ranking of the in-fill material in terms of the performance of the trench, it is concluded that concrete trenches come first, empt trenches come second, though ver close to the concrete trenches, and geofoam trenches come last, wa below the other two. This ranking differs from the results of the D analses (Figure 5.8), for which open trenches come first, geofoam trenches come second and concrete trenches get last place. The RRMS plots can also provide information about the efficienc of the trenches. As observed in Figure 5., all trenches indeed succeed in reducing the velocities at all receivers, but with different levels of attenuation: as concluded from the IL plots, the concrete and open trenches provide ver similar reduction levels (apart from the first receiver, for which the concrete trenches outperform the open trenches), while the geofoam trenches ield smaller attenuation levels. Additional information that can be perceived from Figure 5. is that geofoam trenches perform worse when the receiver is placed farther from the trench, while no significant difference is noticed regarding open and concrete trenches. t 83

198 Chapter 5 Reduction of vibrations b means of trenches IL [db] IL [db] IL [db] Frequenc [Hz] m concrete trench 3m open trench 3m geofoam trench 3m concrete trench 3m open trench 3m geofoam trench 3m concrete trench 3m open trench 3m geofoam trench Point - Frequenc [Hz] Point Point 3 - Frequenc [Hz] IL [db] IL [db] IL [db] Frequenc [Hz] Frequenc [Hz] m concrete trench 6m open trench 6m geofoam trench 6m concrete trench 6m open trench 6m geofoam trench 6m concrete trench 6m open trench 6m geofoam trench Point Point Point 3 - Frequenc [Hz] Figure 5.: Insertion losses for distinct trench solutions and receivers 84

199 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings RRMS(t) [m/s] 8 Point -5 3m concrete trench 3m open trench 6 3m geofoam trench No trench 4 RRMS(t) [m/s] -5 Point 8 6m concrete trench 6m open trench 6 6m geofoam trench No trench 4 RRMS(t) [m/s] RRMS(t) [m/s] -5 5 t [s] 6 Point -5 3m concrete trench 5 3m open trench 3m geofoam trench 4 No trench t [s] 4 Point 3-5 3m concrete trench 3m open trench 3 3m geofoam trench No trench RRMS(t) [m/s] RRMS(t) [m/s] -5 5 t [s] 6 Point -5 6m concrete trench 5 6m open trench 6m geofoam trench 4 No trench t [s] 4 Point 3-5 6m concete trench 6m open trench 3 6m geofoam trench No trench -5 5 t [s] -5 5 t [s] Figure 5.: RRMS for distinct trench solutions and receivers The reason wh the concrete trenches outperform the geofoam trenches when acted upon b moving loads (as opposed to what happens in the D simulations) is related to their high bending stiffness. As eplained in some works (Coulier et al., 3a; Coulier et al., 3b; Dickmans et al., 3), the transmission of plane waves in the soil with a longitudinal wavelength smaller than the longitudinal bending wavelength of the barrier is hindered. In other words, when the longitudinal wavelength λ of a propagating wave is smaller than the free bending wavelength in the barrier the soil λ B, but not shorter than the Raleigh wavelength of λ R, then the waves are reflected at the barrier due to its bending stiffness (waves 85

200 Chapter 5 Reduction of vibrations b means of trenches travelling with longitudinal wavelength smaller than the Raleigh wavelength are evanescent in the transverse direction and therefore the barrier is ineffective for those cases). In plane strain conditions, the wavelength is infinite and so the bending stiffness has no influence, but in 3D conditions, waves can propagate in all directions and so surface waves that propagate λr with an angle θ such that sinθ > are reflected b the trench. As a result, for standing point λ B sources, there is a triangular zone delimited b the critical angle θ λr cr = sin, inside which λb the reduction of vibrations is not as significant as outside the mentioned zone. For the case of krv moving harmonic loads, the generated waves propagate with angles θ = sin, and therefore the waves that impinge the barrier with an angle higher than θ cr are also reflected. The wavenumbers of the free longitudinal bending waves k π B λb ω ω = for the 3 m concrete and 3 m foam trenches (vertical bending, transverse bending and torsion), and the Raleigh ω ω wavenumbers of the soil are represented in Figure 5.. Superimposed is the line = k V for ω = 8π rad/s (this line corresponds to the integration path for a moving load with speed V and oscillation frequenc f = 4 Hz). As can be observed, the portion of transversel propagating waves ( k < k ) that reach the barrier with angles greater than the critical angle θ ( k cr B R > k ) is greater for the concrete trench than for the geofoam trench. Furthermore, for the geofoam trench, the transverse bending stiffness and the torsional stiffness of the barrier contribute for the reflection of waves onl above the frequenc f = 6 Hz. These aspects ustif the considerable increase of performance of the concrete trenches and eplain wh the outperform the geofoam trenches. In general, the reflecting propert epressed in the last paragraph also supports the idea that D simulations are not sufficient to predict with enough accurac the abatement of vibrations induced b moving loads, since the obtained results will tend to underestimate the actual reduction. Frequenc [Hz] Raleigh wave k = ω- ω /V vertical bending transverse bending torsion vertical bending transverse bending torsion k [rad/m] Figure 5.: Raleigh wavenumber (blue) and free bending wavenumber of 3 m concrete trench (red) and 3 m geofoam trench (green) 86

201 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Behavior of trenches for various longitudinal positions The insertion losses shown in Figure 5. concern receivers at the longitudinal position =. For other longitudinal positions, since the amplitudes of the moving forces are the same, with or without a trench, it could be epected that the ratios between the responses of the two scenarios were the same, and thus that the insertion losses for = were representative of the problem. However, a deeper eamination of the epression that ields the displacements induced b a set of moving forces with multiple frequencies reveals the opposite. This epression is derived from eq. (4.8) and assumes the form N N ω ωi ω ω ω i ( i ) V u(,, ω) = P ( ωi ) u,, ω e V ɶ i= = V (5.6) in which N ω is the number of oscillation frequencies, N is the number of moving forces, P ( ω i ) is the amplitude of the th force for the i th oscillation frequenc, ω i is the i th oscillation frequenc, is the longitudinal position of the th force at t =, (,) are the i coordinates of some point, and u(, ω ω, ω V ) defined in equation (5.) becomes ɶ is the.5d transfer function. Thus, the ratio A r A (,, ω) = Nω N i= = r Nω N i= = P ( ωi ) uɶ Trench P ( ωi ) uɶ NoTrench ωi ω ω ω i i V,, e V ωi ω ω ω i i V,, e V ω ω ( ) ( ) (5.7) and, in order for this ratio to be location independent, the following condition is necessar uɶ uɶ Trench NoTrench ω ωi,, ω V = constant, i =... N ω ωi,, ω V The previous statement can be fulfilled onl if N ω =, and that leads to the conclusion that within the same transverse alignment and for multiple ecitation frequencies, the ratio A r in equation (5.7) and the insertion losses defined in eq. (5.4) change with the longitudinal coordinate. ω (5.8) To assess the variation of the insertion losses, these values are calculated for receivers placed at the alignment = 5 m and spread between the longitudinal positions = 3 m and = 3 m. The results are plotted in Figure 5.3. It is demonstrated that within each frequenc band the maimum variation can reach db, which is a considerable value. Nevertheless, above the frequenc Hz, all trench solutions offer an attenuation of vibrations for the great maorit of longitudinal positions (with ver few eceptions). 87

202 Chapter 5 Reduction of vibrations b means of trenches IL IL IL m concrete trench - Frequenc [Hz] 3m open trench - Frequenc [Hz] 3m geofoam trench - Frequenc [Hz] IL IL IL Frequenc [Hz] Frequenc [Hz] m concrete trench 6m open trench 6m geofoam trench - Frequenc [Hz] Figure 5.3: Insertion losses for receivers placed at = 5 [m] and 3 < < 3 [m]. The black solid line represents the IL ' curves for the maimum responses (defined in eq. 5.9). The dashed black line illustrates the IL assuming plane strain conditions For a given longitudinal position, the insertion loss varies between the maimum and the minimum values of the insertion envelope. Thus, for design purposes, considering onl the lower limit of the envelope ields over protective measures, while considering merel the upper limit ields ver relaed conditions. Therefore, it is recommended to consider a curve that passes through an intermediate value of both limits. A possible option is the curve calculated with the maimum responses, i.e., 88

203 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings NoTrench z Trench uz ( ω) ma u,, IL' (, ω) = log (5.9) ma,, ( ω) For each frequenc band, this curve considers the ratio of the response of points placed at different longitudinal positions. In this wa, the value IL ' cannot be considered as a reduction measure for a given point, but, instead, should be perceived as a reduction measure for the maimum response along a given alignment. The curves IL ' for the = 5 m alignment are represented in Figure 5.3 through solid black lines. For comparison purposes, the IL lines for D line loads are also represented in Figure 5.3 (black dashed line). It is observed that for the concrete trenches, the D results underestimate their efficienc, even when compared with the lower limits of the envelope. As for the open and geofoam trenches, the D results run close to the lower limit of the envelope, but there are frequenc bands where the D results predict amplification of displacements, a scenario that does not correspond to realit. This comparison corroborates that D results underestimate the efficienc of trenches and, therefore, plane strain conditions are not applicable for cases that include moving vehicles. One interesting aspect that can be observed in Figures 5. and 5.3 is the amplification of the response below 5 Hz. The low frequenc response of the soil is due to the quasi-static component of the interaction forces, and its effect evanesces with the distance (i.e., mainl evanescent waves contribute to the response of the ground in the low frequenc range). Since the trench is placed at a small distance from the track, the evanescent waves still reach the trench with a considerable amount of energ, and so their interaction with the trench produces new waves that reach further than what the would if no heterogeneit was encountered. That is in fact the reason wh amplifications are greater for 6 m trenches than for 3 m trenches (the energ of evanescent waves is distributed with the depth and, therefore, the deeper the trench, the greater the energ that collides with it), and wh amplifications are larger for concrete and open trenches than for geofoam trenches (the material contrast in the first two is greater). In an case, these amplifications are inconsequent because the contribution of this low frequenc content for the response of the ground surface is negligible when compared to the contribution of the medium and high ranges. This situation is represented in Figure 5.4, which plots the frequenc content of the vertical velocities evaluated at the receivers mentioned in the first paragraph of this sub-section for the non-mitigated scenario and for the 3 m and 6 m concrete trenches Conclusions In section 5.3, the efficienc of distinct trench solutions is investigated numericall with the aid of the.5d BEM-FEM approach. The investigations comprise D simulations ( k = ), carried out to assess the need for the inclusion of the track in the numerical models, and 3D simulations of moving vehicles, performed to compare and rate the trench solutions and to investigate some features of the problem, namel the influence of the trenches in the traintrack interaction phenomenon and the efficienc of trenches along the longitudinal direction. 89

204 Chapter 5 Reduction of vibrations b means of trenches -4 Without trench Vz [m/s] Vz [m/s] Frequenc [Hz] 3m concrete trench -4 Vz [m/s] m concrete trench Frequenc [Hz] Frequenc [Hz] Figure 5.4: Frequenc content of the vertical velocities for receivers placed at transverse position = 5 m and longitudinal position 3 < < 3 m The following conclusions are achieved throughout this section:. Neglecting the track leads to higher vibration levels in the soil surface (at least for short and medium distances) and to lower efficienc of trenches. Thus, its inclusion in the numerical model is necessar to assess the need for mitigation measures and to estimate the efficienc of the abatement solutions more accuratel.. The presence of the trench does not influence the train-track interaction phenomenon. This conclusion is important because it permits to calculate the train-track interaction forces for the non-mitigated scenario and to use these forces in the evaluation of trench like measures. 3. The behavior of trenches when acted upon b moving loads is ver different from their behavior when acted upon b D line loads. In fact, D simulations tend to underestimate the efficienc of trenches, and that is more pronounced for concrete trenches than for geofoam trenches. 4. Ranking the in-fill materials according to the efficienc of the trench, from the D simulations open trenches are placed first, geofoam trenches second, and concrete trenches in last place, while from the 3D simulations concrete trenches and open trenches come together in the first place, and geofoam trenches are placed last, clearl below the other two. 9

205 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings 5. In the studied eample, within the same longitudinal alignment the insertion losses ma var up to db, which is a considerable value. Thus, for design purposes, to consider onl the lower limit of the envelope leads to over protective measures, while to consider merel the upper limit ields relaed conditions. Therefore, it is recommended to consider the insertion loss curve based on the maimum responses of the alignment. 5.4 Effect of trenches on a nearb structure 5.4. General description of the building The response of a structure due to the passage of the Alfa train is now analzed. The obective is to assess the efficienc of the trench solutions in reducing the vibration levels inside a building situated near a railwa track. The target structure consists in a two stor building whose geometr is represented in Figure It is composed of vertical columns with cross section.4.4 m, horizontal beams with.4 m of width and.5 m of height, and slabs with.3 m of thickness. The structure considered in this eample differs from the structure considered in chapter 4 in two aspects: firstl, the slabs are fleible; secondl, the slabs are supported on beams and not directl on the columns. The material considered for all components of the structure is concrete, being its properties given in Table 5.4. A distributed mass of kg/m is added to the floors and a mass of kg/m is added to the roof-top. The footings are assumed massless and rigid, with dimensions.5.5 m. The distance from the nearest side of the building to the central alignment of the track is m. In terms of the numerical model, columns and beams are simulated with noded Euler beams, and the slabs are simulated with the 4 noded shell elements (the finite element model is created in ANSYS being the corresponding matrices eported and used in the implemented procedure). The maimum size of the elements is.5 m. Each rigid footing is divided into 5 equall sized square boundar elements (.3 m of side). The FEM mesh used in the numerical model is represented in Figure 5.5. A3 A A Figure 5.5: Finite element mesh of the building (generated with ANSYS) 9

206 Chapter 5 Reduction of vibrations b means of trenches 5.4. Natural frequencies of the building When a structure is subected to incident wave-fields (seismic waves, or, as in the problem at hand, vehicle induced vibrations), it is epected that the structural response be amplified at its natural frequencies. The natural frequencies of a structure depend on its geometr, on the material properties (elastic constants and densit), and on the boundar conditions. For the structure represented in Figure 5.5, assuming that it rests on a stiff foundation, the natural frequencies ω correspond to the solutions of ( ω ) det K M = (5.) in which K and M are the stiffness and mass matrices of the structure. There are as man natural frequencies ω as the number of rows and columns in K and M (approimatel 6 for this eample), and all of them are real. The lowest natural frequencies are depicted in the histogram of Figure 5.6, and the mode shapes associated with the natural frequencies below Hz are represented in Appendi 5.. Since the structure is smmetric, some natural frequencies ω are repeated, namel those associated with floor drifting and those associated with the bending of the slabs. The repeated frequencies are indicated in Appendi 5. together with the mode shapes. 6 5 N. of natural frequencies 4 3 Figure 5.6: Number of natural frequencies per frequenc interval (intervals of Hz) If more realistic boundar conditions are assumed, i.e., if the structure is considered to be resting on the soil (and if this is modeled with boundar elements), then the natural frequencies ω correspond to the solutions of Frequenc [Hz] ( BEM ω ω ) det K + K ( ) M = (5.) in which K ( ω) BEM is the frequenc dependent stiffness matri of the soil, calculated as eplained in chapter 3. Contraril to the solutions of (5.), since the soil absorbs waves and therefore introduces damping to the sstem, the natural frequencies ω of (5.) are comple. 9

207 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Nevertheless, some frequencies have small imaginar components and thus the structure can still resonate if ecited near these frequencies. Onl the almost real solutions ω are of interest to the current analsis. These solutions can be found b eciting the structure in a wide range of frequencies and then observing at which frequencies the response is amplified. In this wa, to find the natural frequencies, the structure is loaded in points A, A and A3 (indicated in Figure 5.5) and the displacements of the same points are calculated. The loads are applied both in the horizontal ( ) and vertical ( z ) directions, and the displacements are calculated in the direction of the loads. The obtained responses are represented in Figure 5.7, together with analogous results for the case of the rigid foundation. The results presented in Figure 5.7 show that the presence of the soil (blue curves) cancels some of the natural frequencies of the rigidl supported structure (red curves). Furthermore, the amplification of the response near the natural frequencies appears to be smaller when the soil is considered, which can be ustified b the damping introduced in the sstem due to the radiation of waves into the soil. Figure 5.6 and Figure 5.7 show that there is a set of natural frequencies that lies in the frequenc range below Hz, which according to the conclusions of section 5.3, is the frequenc range at which the trench solutions are ineffective. It can therefore be concluded that amplification problems ma arise below the frequenc Hz. This issue is investigated in the following sub-sections Building response for the non-mitigated case Before evaluating the reduction achieved b the trench solutions, the building response is first investigated considering the non-mitigated scenario. Figure 5.8 shows the horizontal and vertical components of the displacements of points A, A and A3 that are induced b the passage of the Alfa Pendular train. The displacements are represented in one-third-octave bands, and for comparison purposes, the incident displacements at the central footing are also represented. Figure 5.8a-b reveals amplification of the horizontal displacements u and u in the frequenc intervals -3 Hz and 6-8 Hz. These frequencies correspond to the lower natural frequencies of the building, and are associated with translations of the slabs and rotation about their vertical ais (see Appendi 5.). Above the frequenc Hz, the horizontal vibrations are almost completel filtered b the building, namel for the points A and A3. Observe that for the frequenc intervals -3 Hz and 6-8 Hz, the values of the displacements are consistent with the modal shapes of the associated natural frequencies: at the frequencies -3 Hz the horizontal displacements are greater at A3 than at A, which in turn are greater than at A (first, second and third modal shapes); at the frequencies 6-8 Hz the displacements are greater at A than at the other two points (fourth, fifth and sith modal shapes). Observe also that there is an amplification around the frequenc Hz, which is felt onl at point A. This amplification is ustified b the natural frequenc f = 8.69 Hz, whose associated modal shape corresponds to horizontal translations of the first floor. In what concerns the displacements in the vertical direction (Figure 5.8c), amplifications occur in the frequenc interval 6-3 Hz. This interval is eplained b the bending motion of the slabs, whose corresponding modal shapes are associated with natural frequencies above 8 Hz. Moreover, in the frequenc interval between and 5 Hz the amplifications are 93

208 Chapter 5 Reduction of vibrations b means of trenches interrupted, coinciding this interval with the range where no natural frequenc is observed (Figure 5.6 and Appendi 5.: natural frequencies ump from Hz to 5.7 Hz). log(u) Point A, horizontal force Rigid foundation Soil log(uz) Point A, vertical force Rigid foundation Soil log(u) log(u) Frequenc [Hz] Rigid foundation Soil Frequenc [Hz] Point A, horizontal force Point A3, horizontal force Rigid foundation Soil Frequenc [Hz] log(uz) log(uz) Frequenc [Hz] Rigid foundation Soil Frequenc [Hz] Point A, vertical force Point A3, vertical force Rigid foundation Soil Frequenc [Hz] Figure 5.7: Displacements of points A, A and A3 due to horizontal and vertical loads applied at the same points 94

209 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings u [m] a) 4 b) -7 A A A3 3 Central footing u [m] A A A3 Central footing Frequenc [Hz] uz [m] c) Frequenc [Hz] A A A3 Central footing Frequenc [Hz] Figure 5.8: Displacements of points A, A and A3 and incident displacements at the central footing: a) horizontal displacements; b) horizontal displacements; c) vertical z displacements Reduction achieved b trenches The reduction of vibrations in the building due to the si trench solutions described in section 5.3 is now studied. For that, the insertion losses defined b equation (5.4) are calculated for point A and represented in Figure 5.9 (for the horizontal directions, the insertion losses are calculated using the horizontal displacement in place of the vertical displacement; the results for points A and A3 are ver similar and are not shown here). Starting with the analsis of the horizontal vibrations, it can be observed that trenches lead to moderate amplifications in the frequenc range below 4 Hz, have practicall no influence on the frequenc interval 4- Hz, and reduce the vibrations above Hz. It has been seen in sub-section that the horizontal response of the building is dominated b the lower natural frequencies, which are below Hz. It can therefore be concluded that if there are problems associated with horizontal vibrations, the trenches will be ineffective, potentiall causing amplifications instead of attenuation. Furthermore, for frequencies above Hz the building filters the horizontal vibrations, and therefore the effect of the trench is irrelevant. 95

210 Chapter 5 Reduction of vibrations b means of trenches IL [db] IL [db] IL [db] Frequenc [Hz] Horizontal displacements 3m concrete trench 3m open trench 3m geofoam trench Horizontal displacements - Frequenc [Hz] m concrete trench 3m open trench 3m geofoam trench Vertical z displacements 3m concrete trench 3m open trench 3m geofoam trench - Frequenc [Hz] IL [db] IL [db] IL [db] Horizontal displacements 6m concrete trench 6m open trench 6m geofoam trench - Frequenc [Hz] Horizontal displacements 6m concrete trench 6m open trench 6m geofoam trench - Frequenc [Hz] Vertical z displacements 6m concrete trench 6m open trench 6m geofoam trench - Frequenc [Hz] Figure 5.9: Insertion losses for point A Concerning the effect of trenches on the vertical vibrations, in Figure 5.9 it can be observed that below the frequenc f = Hz the reduction is negligible, thus failing to reduce the vibrations at the lowest natural frequencies associated with the bending motion of the slabs (at these frequencies, an amplification of the vertical vibrations in the building is observed). Moreover, above the frequenc f = Hz, the insertion losses are ver modest (below db, which is equivalent to a reduction to one third of the original value). It can also be observed 96

211 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings that the 6m deep trenches perform better than the 3m deep trenches. Overall, it is concluded that trenches ma have a beneficial effect. As a final note concerning this topic, the insertion losses calculated for the building are quite similar to the insertion losses calculated for the ground response at the position of the building (incident wavefield). For instance, if the two plots at the bottom of Figure 5.9 are compared with the insertion losses of Point 3 represented in Figure 5., it can be observed that the are quite similar. The insertion losses for the horizontal response of the ground are not shown in this work, but similar conclusions can be obtained for that direction. 5.5 Conclusions In this chapter, the methods described throughout chapters -4 are emploed in the analsis of vibrations induced b trains and in the assessment of trench-like mitigation solutions. Chapter 5 starts with a parametric stud of trenches, then the behavior of distinct trench solutions is investigated, and finall the effect of the trenches on the vibrations inside a building is assessed. Based on the parametric stud, it is concluded that the behavior of trenches depends on several parameters such as its dimensions, stiffness and densit of the in-fill material, and properties and stratification of the soil. Each parameter is investigated separatel, and so different conclusions can be obtained if different default scenarios are assumed. In an case, the efficienc of trenches largel depends on their depth and width, being the trenches ineffective when these dimensions are much smaller than the characteristic wavelength (Raleigh wave) of the soil. For this reason, trenches are effective onl in the medium and high frequenc ranges. Concerning the mitigation of the vibrations induced b the Alfa Pendular train, some aspects are studied, namel the need for the inclusion of the track in the numerical model, the impact of the trenches in the train-track interaction phenomenon, and the variabilit of the reduction of vibrations provided b trenches along the longitudinal position. It is concluded that the track must be included in the numerical model, that trenches do not change significantl the train-track interaction forces, and that the relative reduction ma var up to db within the same longitudinal alignment. It is also observed that concrete and open trenches present similar reduction levels, while geofoam trenches do not perform as well as the other two (this ranking order differs from the D ranking). In section 5.4 the response of a building induced b the passage of an Alfa train is evaluated. It is observed that some amplifications occur at the natural frequencies of the structure, being the amplification of the horizontal response associated with the floor-shifting modes and the amplification of the vertical response associated with the slab-bending modes. Since the lower natural frequencies of the building are in the low frequenc range (below Hz), the trench solutions are ineffective in neutralizing these amplifications, possibl aggravating the horizontal response. Nevertheless, if problems are detected above Hz, then the trench solutions ma provide some protection. The reduction achieved inside the building resembles the reduction obtained for the soil surface at the location of the building. For this reason, the response of the building needs to be calculated onl for the non-mitigated scenario. As a final comment, it must be stressed that the results reported in section 5.3 and 5.4 concern the line near Carregado. The reductions obtained in different scenarios or at different train speeds ma not necessaril be equivalent to the results reported here. Further studies are 97

212 Chapter 5 Reduction of vibrations b means of trenches required to contribute to the development of rules of thumb for the design of this tpe of mitigation measures. 98

213 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings 6. Conclusions and recommendations for further research 6. Conclusions The work described in this dissertation addresses the problem of vibrations induced b trains in the surrounding environment. All stages relevant to the problem are considered, namel the generation stage, in which the train interacts with the track-ground sstem, the propagation stage, in which the vibrations propagate through the track and ground, and the reception stage, in which the mentioned vibrations reach the building and induce its dnamic response. Mitigation at the propagation stage using trenches is also addressed. The maor focus of the work is on the development and implementation of numerical tools for the simulation of the three stages of the problem. The selected numerical approaches are described throughout chapters -4. In chapter 5, the derived tools are used to analze trench like mitigation solutions. Numerical approaches have to fulfill several requirements in order to provide reliable predictions, namel:. Moving nature of the source while standing harmonic loads induce harmonic responses, moving loads induce transient responses characterized b a wide range of frequencies. Therefore, simplistic models in which moving loads are replaced b standing loads are not sufficient;. Train-track interaction if the interaction is disregarded, then the moving forces remain constant in time (quasi-static components) and the induced response evanesces with the distance to the track. In order to obtain the response at remote positions, the dnamic components must also be accounted for. These components result from the solution of a train-track interaction problem; 3. Adequate soil model soil connects the track with nearb buildings, and therefore its consideration is of great importance. The propagation of waves in the soil is affected b heterogeneities found along the propagation path. If these heterogeneities are neglected, the obtained results ma be inaccurate; 4. Mitigation measures in order to assess the efficienc of mitigation measures, the must be considered in numerical models, whether these measures are applied at the source, at the propagation path, or at the receiver. The above mentioned requirements can be fulfilled through the use of three-dimensional finite element models, where the components of train, track, soil and building can be simulated with a suitable tpe of element and constitutive behavior (eventuall nonlinear). However, the use of 3D FEM to solve soil-structure interaction problems (such as track-ground and buildingground interaction) leads to etremel large sstems of equations whose solution is time consuming. For this reason, and in order to simplif the problem, some assumptions are made, namel the consideration of: 99

214 Chapter 6 Conclusions and recommendations for further research. Linear behavior of the material and linear contact between train and track this assumption enables the analsis to be performed in the frequenc domain;. Constant cross section of the track this assumption allows the problem to be analzed in the wavenumber-frequenc domain (.5D domain). In other words, the 3D problem is reduced to a series of smaller D problems, which are faster to solve; 3. Constant speed of the train the.5d models are eact in the simulation of loads (constant or oscillating) moving at constant speed; 4. Horizontall stratified soil this assumption is commonl used in the literature, and allows the soil to be simulated with the boundar element method and with the thinlaer method, a combination that results in a ver efficient strateg; 5. Weak coupling between the nearb building and the track or in other words, waves that propagate from the track to a nearb building are accounted for, but waves that are reflected b the building and return to the track are neglected. This assumption allows the solution of the problem in two distinct phases: a first phase in which the generation and propagation stages are accounted for and in which the presence of the building is disregarded; a second phase in which the response fields calculated in the first phase are prescribed at the building, thus obtaining its response. Based on the above mentioned requirements and assumptions, the following strategies are followed: Generation stage (train-track interaction): the train is simulated with a 3D multi-rigid-bod approach, while the stiffness matrices for the track-ground sstem are obtained with the.5d transfer functions of that sstem (see the net point). The equilibrium and compatibilit equations are formulated in a moving frame of reference and solved in the frequenc domain. This procedure ields the dnamic interaction forces between train and track; Propagation stage: a.5d BEM-FEM approach is used to solve the soil-track interaction problem. The track is modeled with finite elements while the track-ground interface is modeled with boundar elements. Heterogeneities in the soil, such as trenches, are modeled with the same strateg. The interaction forces calculated in the generation stage are used as inputs in the.5d BEM-FEM, and the response of the ground at an position, including the positions of the nearb building, is calculated. The presence of the building is neglected at this point; Reception stage: a 3D BEM-FEM approach is used to solve the soil-structure interaction problem. The response fields calculated in the previous stage are transformed to the 3D space and used as inputs in the 3D BEM-FEM model, whose solution provides the response of the building. The fundamental solutions used to nurture the BEM (both.5d BEM and 3D BEM) are calculated with the thin-laer method (TLM). This aspect represents the main difference between the methodolog used in this work and the strategies followed b other authors. The TLM is described in chapter, where epressions for the.5d fundamental solutions are derived, and where perfectl matched laers (PMLs) are coupled with the TLM in order to simulate half-spaces. All the developments are validated through the calculation of the fundamental solutions of full-spaces and a posteriori comparison with analtical solutions. It is concluded that the TLM is in fact an efficient option for the calculation of fundamental

215 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings solutions of laered half-spaces. It is also concluded that the coupled TLM+PML scheme is significantl more efficient than the paraial boundaries (PB) approach, which prior to this work was the favorite procedure to model half-spaces with the TLM. The coupled BEM-FEM strateg is used to solve soil-structure interaction problems. The soil, of unbounded geometr, is modeled with boundar elements, while the structure (track or nearb building), of bounded and comple geometr, is modeled with finite elements. The advantages of the coupled BEM-FEM approach are the reduction of the number of degrees of freedom needed to model the domain (and in particular, the soil), and the consideration of the radiation of waves to infinit. The main disadvantage resides in the need for the calculation of the BEM matrices, which is usuall time consuming. The coupled BEM-FEM approaches are described in chapter 3. There, for the case of the.5d BEM, it is eplained how to use the TLM to calculate the boundar element coefficients without resorting to spatial integrations, and thus avoiding the complications associated with the singularities of the fundamental solutions. When compared with similar approaches, the main drawback of the.5d BEM-TLM strateg is the time needed for the calculation of the eigenmodes, which becomes large when the fundamental solutions are needed at deep positions. Nevertheless, for each soil profile, the eigenmodes onl need to be calculated once for each frequenc. The can thereafter be used to analze different configurations of tracks, buildings and countermeasures. The train-track interaction problem involves the coupling between a moving discrete structure and an invariant structure. While the moving vehicle is simulated in the 3D domain, the underling structure (track-ground sstem) is formulated in the.5d domain, and so the coupling between the two structures is not straightforward. This issue is addressed in chapter 4, where the epressions for the dnamic interaction forces and the transformation of the.5d results to the 3D space domain are described. The link between the distinct procedures and stages is eemplified also in chapter 4. It is convenient to point out that the.5d BEM-FEM procedure, the TLM, and the train-track interaction solution procedure have been implemented in the finite element software FEMIX ( The software is structured in such a wa that the user onl needs to provide information about the soil profile and respective TLM model, finite element model of track and mitigation measures, boundar element model for soil-structure interfaces, vehicle model (multiple rigid bod model), irregularit profiles, and frequencies of interest. The program then proceeds to complete all necessar operations and returns the outputs requested b the user, both in the time domain and in the frequenc domain. The user does not need to provide a transverse wavenumber sample for the transformation of the fundamental solutions from the (k, k, ω) domain to the (, k, ω) domain, since that transformation is accomplished internall b the TLM. The program is generic enough to account for surface lines or tunnels, stratification of the soil and obstacles in the propagation path (e.g., trenches). The 3D BEM-FEM procedure has not been implemented in the FEMIX software, but a MATLAB module has been written with the purpose of calculating the response of a building due to an incoming wave field. The subroutine requires a binar file with the incident displacement field (this file is an output of FEMIX), and tet files with the stiffness and mass matrices of the building. These matrices have to be written in the Harwell-Boeing format, as provided b ANSYS through the command HBMAT. This subroutine onl handles structures resting on the surface of laered half-spaces.

216 Chapter 6 Conclusions and recommendations for further research In chapter 5, the above mentioned programs are used to stud the efficienc of trenches as mitigation solutions. First a parametric stud is performed and then the behavior of trenches under realistic conditions is investigated. The main conclusion of the parametric stud is that the efficienc of a trench depends mostl on its dimensions, being the trench ineffective when its depth is much smaller than the characteristic wavelength of the soil (Raleigh wave). For this reason, trenches are effective onl in the medium and high frequenc range. In what concerns the use of trenches for the reduction of vibrations induced b trains, it is observed that ecluding the track from the numerical model results in underestimations of the induced vibrations and overestimations of the efficienc of trenches. It is also observed that trenches do not affect significantl the train-track interaction forces (at least for the problem analzed) and that concrete trenches and open trenches present similar reduction levels, while geofoam trenches perform worse than the other two. An important observation regarding the modeling of mitigation solutions is that D models are not appropriate for the simulation of the trench behavior, namel when the sstem is acted upon b moving loads. The results in chapter 5 show that D simulations overestimated the efficienc of geofoam trenches and underestimated the efficienc of concrete trenches. Regarding the reduction observed in nearb buildings, it is seen that trenches fail to diminish the amplifications at the low natural frequencies caused b the passage of trains. On the contrar, trenches are effective in reducing the vibrations at high frequencies (frequencies above Hz for the case considered). 6. Recommendations for further research Taking the FEMIX program and the MATLAB modules developed in the course of this thesis as framework for future investigations, the following studies are of great interest. Vibrations induced b underground trains: the eamples presented in this thesis onl consider surface lines. Nevertheless, trains running on tunnels are also a realit inside cities, and due to their close proimit to buildings, the ma have a negative impact on inhabitants and equipments. Mitigation measures: in chapter 5 of this thesis, trenches are studied as measures for the abatement of vibrations induced b trains. Nevertheless, the conclusions obtained in that chapter refer to the particular problem there considered, and so the etension of the conclusions to different scenarios ma be inappropriate. Thus, further studies are needed, and if possible, rules of thumb for their design should be attempted. Additionall, different mitigation solutions should be investigated. The developed programs are prepared to incorporate several mitigation solutions, whether the are applied at the source (e.g., suspension sstems of the train, tpe of track, resilient materials in the track), at the propagation path (wave impeding blocks, for eample), or at the target building (compartment isolation, for eample). Studies on these measures can be found in the literature, but further studies are needed. Additionall, improvements are required so that different studies become possible. For eample, to solve each of the 6m deep trench problems described in chapter 5 takes a couple of das, which renders sensitive analses impractical. In order to reduce the computation times, the following strategies are recommended. Reduced modal superposition: in the.5d BEM-FEM procedure, the step that is more time consuming is the calculation of the boundar element matrices. Their calculation

217 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings relies on a modal superposition, which in this work is performed combining all the vibration modes. It is known that vibration modes whose eigenvalues contain a large imaginar component are evanescent with the distance to the source, and therefore their contribution can be neglected when a collocation point and a boundar element are far from each other. To account for this feature ma save some time in the computation of the boundar matrices. Parallel computing (GPU vs CPU): the obective is to reduce the computation time, which for more comple geometries (such as when discontinuities/heterogeneities are considered in the soil) ma become ecessivel time consuming. In this work, GPU computation of the BEM matrices has been attempted, but the time reduction was not significant (CPU and GPU calculation took roughl the same time). The author believes that his implementation does not access the GPU memor in the most efficient wa, and so he believes that better performances can be obtained. Parallel computing ( supercomputers ): the parallelization attempted in this work is performed in the calculation of the BEM matrices, but parallelization can be implemented at a higher level. Since the same tpe of problem needs to be solved for each wavenumber-frequenc pair (k, ω), and since each of these.5d problems can be solved independentl from the others, then the calculations can be distributed across different computers. If multiple computers or if supercomputers are available, then distributing the calculations across several processing units is epected to result in a considerable speed up. Lastl, apart from the improvements associated with computation time, some of the limitations of the programs can be overcome so that new scenarios can be analzed. Buried structures: the formulation of the 3D BEM-FEM procedure in chapter 3 is generic, but the method has been implemented onl for structures resting at the surface. The reasons for this limitation are complications associated with the evaluation of the boundar integrals, which cannot be performed analticall. Furthermore, since the mesh is three dimensional, the time needed for the calculation of the BEM matrices ma render the problem impracticable. A promising alternative for the calculation of the stiffness matrices of the soil is the use of finite elements coupled with perfectl matched laers (FEM+PML). Periodic geometries: in this thesis, it is assumed that the geometr is invariant in the longitudinal direction. There are, however, scenarios for which periodicit cannot be neglected, as for eample, isolation of vibrations b means of rows of piles. In that eample, the space between consecutive piles plas an important role in the efficienc of the measure, and therefore invariant models are revealed to be inadequate, being a more appropriate model one that accounts for the periodicit of the geometr. 3

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219 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Appendi I Matrices D αβ for cross-anisotropic materials Based on the constitutive matri D defined in equation (.), the matrices (.3) are D λ + G = G G t D λ = G D z D αβ in equation λt = G t D D z G = λ Gt = λt D D z G = λ G + G t = G t λt D D z zz = λ t G t Gt = Gt D t Thin-laer matrices for cross-anisotropic materials Linear epansion The shape functions for this case are N = ζ N = ζ ζ = z / h where z = at the bottom surface of the thin-laer and z = h at its top surface. The evaluation of equations (.) to (.6) results in the following thin-laer matrices ρh I I M = 6 I I A A B aα α h Dαα Dαα = = 6 αα D Dαα ( D + D ) ( D + D ) ( D + D ) ( D + D ) h = 6 ( α, ) Dα z Dα z Dz α Dz α = = Dα z Dα z Dzα Dz α Dzz Dzz G = h Dzz Dzz The elementar matri Quadratic epansion The shape functions are now ( α, ) B ɶ α is obtained b changing the sign of ever third column of B α. 5

220 Appendices = ( ) N = ζ ( ζ ) N ( ζ )( ζ ) N ζ ζ 4 3 = ζ = z / h where z = at the bottom surface of the thin-laer and z = h at its top surface. The evaluation of equations (.) to (.6) results in the following elementar matrices 4I I I ρh M = 6 3 I I I I I 4I 4Dαα Dαα Dαα h Aa α = αα 6 αα αα, 3 D D D = Dαα Dαα 4D αα ( α ) ( D + D ) ( D + D ) ( D + D ) ( ) ( ) ( ) ( D D ) ( D D ) 4( D D ) 4 h A = D + D 6 D + D D + D Dα z 4Dα z Dα z 3Dz α 4Dz α Dz α Bα = 4 α z 4 α z 4 zα 4 zα, 6 D D D D = α z 4 α z 3 α z zα 4 zα 3 D D D D D Dz α 7Dzz 8Dzz Dzz G = h D D D D 8D 7D The elementar matri zz zz zz zz zz zz ( α ) B ɶ α is obtained b changing the sign of ever third columns of B α. 6

221 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Appendi II Evaluation of I 4 using contour integration The Residue Theorem (Boas, 983) states that the value of the contour integral for an contour in the comple plane depends onl on the properties of a finite number of points contained inside the contour (the poles). Using mathematic representation, this theorem states that where C ( ξ ) ξ π i Res ( ξ ) f d = f (AII.) ξ represents the th pole (inside the contour) of the single valued function f ( ξ ), and ( ) Res f ξ represents the residue of the pole ξ. For a simple pole, the residue is calculated with Res f ( ξ ) lim ( ξ ξ ) f ( ξ ) = (AII.) ξ ξ Equation (AII.) is valid for contour integrals in the anti-clockwise direction. For the clockwise direction, the right hand side is multiplied b. The integral to be evaluated is + k i k e, (AII.3) k ( k k ) I = dk k = k + k π or equivalentl, I 4 + k i k 4 = e dk π (AII.4) ( k + k ) k ( k k ) For this case, the function f ( ξ ) is f ( ξ ) = π ξ k ( + k ) ξ ( k k ) e whose poles are ( k is assumed to be real) iξ ξ = i k ξ = i k + = Im < ξ3 = k k ξ4 = k k ( ) ( ) ( ) ξ k ξ k k k k The residues of the poles, as defined b equation (AII.), are (AII.5) (AII.6) 7

222 Appendices i k Res f sign e ( ξ ) = ( k ) 4π k i k Res f sign e Res f Res f ( ξ ) = ( k ) 4π k ( ξ ) 3 ( ξ ) 4 k = 4π k k k k = 4π k k k e k e k i k k i k k (AII.6) In order to evaluate the integral (AII.4) using contour integration, the contours shown in Figure AII. are used. Im( ξ ) < ξ 3 ξ R Re( ξ ) R ξ ξ 4 > Figure AII.: Contours for the evaluation of I 4 The red contour must be considered when is smaller than zero, while the blue contour must be considered when is greater than zero. These conditions guarantee that the function f ( ξ ) vanishes in the semi-circumferences of infinite radius, which means that onl the integrals on the real ais are non-zero. Equations (AII.) and (AII.) are used to evaluate the contour integral, ielding ( ( ) ( 4 )) ( ( ) ( 3 )) π i Res f ξ + Res f ξ, > f ( ξ ) dξ = π i Res f ( ξ ) = (AII.7) C π i Res f ξ + Res f ξ, < Replacing the residues with the epressions (AII.6) leads to 8

223 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings C f ( ξ ) i k k k i k k π i sign ( k ) e + e, > 4π k 4 π k k k dξ = (AII.8) i k k k i k k π i sign( k ) e e, < 4π k 4 π k k k After some simplifications, equation (AII.8) becomes k k i k i k k I4 = f ( ξ ) dξ sign ( k ) e e = (AII.9) k C k k k which is equivalent to the epression indicated in Table.3. 9

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225 Analsis and mitigation of vibrations induced b the passage of high-speed trains in nearb buildings Appendi III B pɶ I as a function of uɶ and II Consider the sstem of equations (3.) I B PI,I + I PI,II uɶ U I I,I UI,II pɶ I = B PII,I PII,II + I uɶ II UII,I UII,II pɶ II After solving the first row for pɶ I and the second row for Substituting B B ( ) ɶ ɶ ɶ B ( ) ( ɶ ɶ ɶ ) UI,Ipɶ I = PI,I + I ui + PI,IIu II UI,IIp II uɶ = P + I U p + U p P u B II II,II II,II II II,I I II,I I pɶ B uɶ II, one obtains B uɶ II, defined in the second equation, into the first, ields B B ( ) ɶ ( ) ( ɶ ) U pɶ = P + I u + P P + I U pɶ + U pɶ P u U pɶ I,I I I,I I I,II II,II II,II II II,I I II,I I I,II II and after some simplifications, pɶ I can be obtained with B ( ) ( ) ɶ ( ) ɶ ɶ I,I I I,I I,II II,II II,I I I,II II,II II,I I U p = P + I P P + I P u + P P + I U p + I,II ( II,II + ) P P I UII,II UI,II pɶ II ( ) ɶ ( ) ( ) B UI,I PI,II PII,II I U II,I p I PI,I I PI,II PII,II I PII,I uɶ I + = ( ) + P P I U U pɶ I,II II,II II,II I,II II ( ) ( ) ( ) B ɶ I I,I I,II II,II II,I I,I I,II II,II II,I I pɶ = U P P + I U P + I P P + I P u + ( ) ( ) UI,I PI,II PII,II + I U II,I PI,II PII,II + I UII,II UI,II pɶ II

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