A pedestrian crowd walking on a footbridge causes the footbridge to vibrate. These vibrations become an issue of serviceability and can give rise to discomfort for the pedestrians, whereby they should, to as large extent as possible, be prevented. Currently, there is a lack of reliable models to describe a dynamic load on a footbridge, due to a walking crowd. Therefore, there is a need for such models. Lately, a great amount of research has been carried out on the subject pedestrian-induced vibrations of footbridges, though most of it with focus on lateral vibrations. Conversely, this project has been performed aiming to accurately model pedestrian-induced vertical vibrations of a general footbridge. For that purpose, starting from an existing model, a somewhat improved model, comprising three sub-model, has been developed. The sub-models are: one model of the pedestrian crowd walking along the footbridge, one model describing the load from the pedestrian footstep and one model describing the interaction between the pedestrians and the footbridge. In order to get statistically reliable results, numerous simulations of the pedestrian-induced vertical vibrations of a specific footbridge have been performed, using the developed model. Averaging the results over the simulations, we could conclude that the model gives an average error of 7 %, compared to experimental data. The measured quantity giving these results was the absolute maximum value of the acceleration at the midpoint of the footbridge. The achieved dynamical response of the footbridge is qualitatively satisfying, while the quantitative error is larger than we hoped for, whereby we conclude that further improvement of the model is needed before we are able to accurately model pedestrian-induced vertical vibrations of footbridges. / När en folksamling går över en gångbro uppstår vibrationer i gångbron. Dessa vibrationer påverkar brons användbarhet och kan ge upphov till obehagskänsla hos fotgängarna, vilket gör att vibrationerna i största möjliga utsträckning bör motverkas. I nuläget saknas pålitliga modeller för att beskriva den dynamiska last en gångbro utsätts för när en folksamling går över den. Således föreligger ett behov att utveckla en sådan modell. Under de senaste decennierna har mycket forskning utförts inom området människoinducerade vibrationer i gångbroar. Dock har merparten av denna forskning berört endast laterala vibrationer. Detta projekt däremot, har genomförts med syftet att, med ett noggrant resultat, modellera människoinducerade vertikala vibrationer i en generell gångbro. För att uppnå detta har jag utgått från en befintlig modell och från den utvecklat en ny modell bestående av tre delmodeller. De tre delmodellerna är: en modell som beskriver hur folksamlingen rör sig över gångbron, en modell som beskriver den kraft det mänskliga fotsteget uträttar på gångbron och en modell som beskriver interaktionen mellan fotgängarna och gångbron. För att uppnå statistiskt tillförlitliga resultat har modellen som utvecklats i detta projekt använts för att utföra åtskilliga simuleringar av människoinducerade vertikala vibrationer i en specifik gångbro. Om vi medelvärdesbildar resultaten över simuleringarna framgår det att modellen som utvecklats ger ett resultat som avviker med 7 % från experimentellt data. Detta gäller för den maximala accelerationen vid gångbrons mittpunkt. Den resulterande dynamiska responsen ser kvalitativt sett bra ut, medan den kvantitativa avvikelsen är större än vi hoppats på. Därför drar vi slutsatsen att vidare förbättringar av modellen behövs för att den ska kunna användas till att på ett noggrant sätt modellera människoinducerade vertikala vibrationer i gångbroar.

A footbridge in Slussen is planned to be built and will connect the area of Gamla Stan with Sodermalm. As an increasing number of footbridges with large span tend to become more flexible and light these days, the corresponding dynamic problems due to decreased stiffness and mass draw much more attention. Specifically speaking, reduced stiffness and mass lead to smaller natural frequencies, which make the structure more sensitive to pedestrian-induced loading, especially in lateral direction. Fortunately, in this master thesis, only the vibration in vertical direction is focused due to that the footbridge in Slussen project uses enough lateral bracings to make sure that the safety of lateral vibration is kept at an acceptable level. In order to analyze dynamic response of the footbridge, the real footbridge structure is converted into a FE model by the commercial software LUSAS. In this thesis, four different kinds of critical standards are introduced, which are Sétra [8], Swedish standard Bro 2004 [9], ISO 10137 [5] and Eurocode respectively. By comparing these four criteria, Sétra and Eurocode are finally chosen to be the standard and guidelines for this project. They give the basic theories about how to model the pedestrian loading and provide critical values to check the accelerations in both vertical and lateral direction. By using FE software LUSAS, natural frequencies of the footbridge and the corresponding mode shapes can be calculated directly. Then, according to these results and relevant theories introduced by Sétra, the pedestrian loading can be modeled and the acceleration response of any specific mode can be obtained as well. Finally, based on the worst case with excessive acceleration, the methods to reduce dynamic response will be presented. Commonly, there are two ways to reduce acceleration response. One method is to increase the stiffness of the structure. However, the increased stiffness is always accompanied with increased mass of the structure. Because of this reason, the other way that installing dampers is widely used in recent years. In this thesis, the tuned mass dampers (TMDs) are introduced in detail as well as the information about the design principles of it. With important parameters known, TMDs can be added to the model to check how the accelerations can be reduced.

Finite element (FE) models serve as the base of many different types of analysis as e.g. dynamic analysis. Hence, obtaining FE models that represent the actual behaviour of real structures with great accuracy is of great importance. However, more often than not, there are differences between FE models and the structures being modelled, which can depend on numerous factors. These factors can consist of uncertainties in material behaviour, geometrical properties and boundary- and continuity conditions. Model validation is therefore an important aspect in obtaining FE models that represents reality to some degree. Furthermore, model verification is also important in terms of verifying theoretical models, other than FE models, in fields such as fatigue-, fracture- and dynamic analysis. In this thesis, two pedestrian steel bridges, the Kallhäll bridge and the Smista bridge, have been modelled in a FE software based on engineering drawings and validated against experimental results with regard to their natural frequencies. Furthermore, in this thesis, a model has been developed in MATLAB based on modal analysis that accounts for pedestrian-induced torsional vibrations, the 3D SDOF model. This model has been verified against the previously mentioned FE models. The aim of this thesis is hence two parted where the first part is to develop three-dimensional FE models of two pedestrian bridges and validate them against measured data regarding the natural frequencies. The second part is to further develop a model for analysing the effect of pedestrian-induced torsional vibrations and to investigate whether the model captures the actual dynamic response of such loading. The results showed that the natural frequencies for the first bending- and torsional mode from the FE models corresponded well to the measured ones with the largest difference of 5 \% obtained for the natural frequency of the first bending mode for the Smista bridge. Furthermore, the 3D SDOF model was able to capture the dynamic response of torsional vibrations with an overall difference of less than 2 \% in comparison to the FE models. The model can be improved by further studying the pedestrian-structure interaction as well as studying the effect of using approximative functions describing the mode shapes. / Finita elementmodeller (FE-modeller) utgör en bas för många olika typer av analyser som exempelvis dynamiska analyser. Därmed är det av stor betydelse att FE-modeller representerar det faktiska beteendet av verkliga strukturer med stor noggrannhet. Ofta är det emellertid skillnader mellan FE-modeller och de verkliga strukturer man modellerar. Dessa skillnader kan bero på en rad faktorer såsom exempelvis osäkerheter i materialbeteende, geometriska egenskaper samt upplag- och randvillkor. Modellvalidering är därför en viktig aspekt i att erhålla FE-modeller som representerar verkligheten i olika omfattningar. Utöver modellvalidering är även modellverifiering viktigt, inte endast för verifiering av FE-modeller utan även för verifiering av andra teoretiska modeller inom områden såsom utmaning-, fraktur- och dynamiska analyser. I detta arbete har två GC-broar, Kallhäll- och Smistabron modellerats i ett FE-program baserat på konstruktionsritningar och validerats mot experimentella resultat med avseende på de naturliga frekvenserna. Vidare har det i detta arbete utvecklats en modell i MATLAB som tar hänsyn till människo-inducerade torsionsvibrationer baserat på modalanalys, benämnd 3D SDOF modellen. Modellen har även verifierats mot de tidigare nämnda FE-modellerna. Målet med detta arbete är således uppdelat i två delar, där den första delen består av att utveckla tredimensionella FE-modeller av två GC-broar samt validera dessa mot mätdata vad gäller de naturliga frekvenserna. Den andra delen består av att utveckla en modell för att analysera effekten av människo-inducerade torsionsvibrationer och undersöka huruvida modellen fångar den dynamiska responsen. Resultaten visade att de naturliga frekvenserna för den första böj- och vridmoden från FE-modellerna motsvarade de uppmätta frekvenserna med en största relativ skillnad på 5 \% för den fösta böjmoden för Smistabron. Vidare visade resultaten att den utvecklade 3D SDOF modellen kunde fånga den dynamiska responsen av torsionsvibrationer med en skillnad på mindre än 2 \% i jämförelse med resultat från de FE-modellerna. Modellen kan förbättras genom att vidare studera interaktionen mellan fotgängare och gångbro samt studera effekten av att använda approximativa funktioner som beskriver modformen.

No Brasil, o estudo a respeito das vibrações induzidas por pedestres em passarelas vem sendo realizado a partir do início dos anos 1990, para passarelas de concreto e de aço. No entanto, ainda não há registro de estudos desse tipo e a norma brasileira NBR 7190 (1997) de projeto de estruturas de madeira apresenta somente critérios de dimensionamento para carregamentos estáticos, no caso de passarelas de madeira. Nesse contexto, este trabalho tem como objetivo geral a análise teórica, numérica e experimental de passarelas de madeira simplesmente apoiadas, com vistas a propor recomendações de projeto relacionadas ao conforto humano nessas estruturas. Este estudo foi desenvolvido a partir da análise de diretrizes normativas e critérios sugeridos por diversos autores, da análise teórica de passarelas dimensionadas segundo os critérios da NBR 7190 (1997), de modelos numéricos construídos em elementos finitos e de ensaios experimentais em modelo reduzido. Entre as conclusões alcançadas, destaca-se que a formulação de Euler-Bernoulli é adequada para o cálculo da primeira frequência natural nas direções transversal vertical e transversal horizontal em passarelas de madeira simplesmente apoiadas; entretanto, especial atenção deve ser dada para a avaliação da rigidez à flexão na direção transversal horizontal. Finalmente, sugeriu-se uma proposta para verificação do estado limite de serviço devido às vibrações induzidas por pedestres em passarelas de madeira simplesmente apoiadas, no tocante ao cálculo de frequências e acelerações, a ser inserida na futura revisão da norma brasileira de projeto de estruturas de madeira. / In Brazil, studies about pedestrian-induced vibrations on footbridges have been carried out since the early 1990s, for concrete and steel footbridges. However, there is no record of such studies and the brazilian standard NBR 7190 (1997) for the design of timber structures only presents design requirements for static loads, in the case of timber footbridges. Therefore, this paper aims to general theoretical analysis, numerical and experimental simply-supported beams timber footbridges, with a view to proposing recommendations for projects related to human comfort in these structures. This study was developed from the study of standards guidelines and criteria suggested by several authors, the theoretical analysis of footbridges design according to the criteria of the NBR 7190 (1997), numerical models using finite element and experimental tests on reduced model. Among the conclusions reached, it is emphasized that the formulation of Euler-Bernoulli is suitable for calculating the first natural frequency in vertical and lateral directions simply-supported beams timber footbridges; however, special attention should be given to the evaluation of lateral bending stiffness. Finally, we have suggested a proposal serviceability limit state verification due to pedestrian-induced vibrations on simply-supported beams timber footbridges, concerning the calculation of frequencies and accelerations to be included in a future review of the brazilian standard for the design of timber structures.

No Brasil, o estudo a respeito das vibrações induzidas por pedestres em passarelas vem sendo realizado a partir do início dos anos 1990, para passarelas de concreto e de aço. No entanto, ainda não há registro de estudos desse tipo e a norma brasileira NBR 7190 (1997) de projeto de estruturas de madeira apresenta somente critérios de dimensionamento para carregamentos estáticos, no caso de passarelas de madeira. Nesse contexto, este trabalho tem como objetivo geral a análise teórica, numérica e experimental de passarelas de madeira simplesmente apoiadas, com vistas a propor recomendações de projeto relacionadas ao conforto humano nessas estruturas. Este estudo foi desenvolvido a partir da análise de diretrizes normativas e critérios sugeridos por diversos autores, da análise teórica de passarelas dimensionadas segundo os critérios da NBR 7190 (1997), de modelos numéricos construídos em elementos finitos e de ensaios experimentais em modelo reduzido. Entre as conclusões alcançadas, destaca-se que a formulação de Euler-Bernoulli é adequada para o cálculo da primeira frequência natural nas direções transversal vertical e transversal horizontal em passarelas de madeira simplesmente apoiadas; entretanto, especial atenção deve ser dada para a avaliação da rigidez à flexão na direção transversal horizontal. Finalmente, sugeriu-se uma proposta para verificação do estado limite de serviço devido às vibrações induzidas por pedestres em passarelas de madeira simplesmente apoiadas, no tocante ao cálculo de frequências e acelerações, a ser inserida na futura revisão da norma brasileira de projeto de estruturas de madeira. / In Brazil, studies about pedestrian-induced vibrations on footbridges have been carried out since the early 1990s, for concrete and steel footbridges. However, there is no record of such studies and the brazilian standard NBR 7190 (1997) for the design of timber structures only presents design requirements for static loads, in the case of timber footbridges. Therefore, this paper aims to general theoretical analysis, numerical and experimental simply-supported beams timber footbridges, with a view to proposing recommendations for projects related to human comfort in these structures. This study was developed from the study of standards guidelines and criteria suggested by several authors, the theoretical analysis of footbridges design according to the criteria of the NBR 7190 (1997), numerical models using finite element and experimental tests on reduced model. Among the conclusions reached, it is emphasized that the formulation of Euler-Bernoulli is suitable for calculating the first natural frequency in vertical and lateral directions simply-supported beams timber footbridges; however, special attention should be given to the evaluation of lateral bending stiffness. Finally, we have suggested a proposal serviceability limit state verification due to pedestrian-induced vibrations on simply-supported beams timber footbridges, concerning the calculation of frequencies and accelerations to be included in a future review of the brazilian standard for the design of timber structures.