modal analysis of pedestrian- induced torsional vibrations based on validated fe...

IN DEGREE PROJECT CIVIL ENGINEERING AND URBAN MANAGEMENT,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2017

Modal analysis of pedestrian-induced torsional vibrations based on validated FE models

SIMON CHAMOUN

MARWAN TRABULSI

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT

Abstract

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 accuracyis of great importance. However, more often than not, there are differences between FE models andthe structures being modelled, which can depend on numerous factors. These factors can consist ofuncertainties in material behaviour, geometrical properties and boundary- and continuity conditions.Model validation is therefore an important aspect in obtaining FE models that represents reality tosome 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 Kallhall bridge and the Smista bridge, have been modelledin a FE software based on engineering drawings and validated against experimental results with regardto their natural frequencies. Furthermore, in this thesis, a model has been developed in MATLAB basedon modal analysis that accounts for pedestrian-induced torsional vibrations, the 3D SDOF model. Thismodel 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 modelsof 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 torsionalvibrations 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 FEmodels corresponded well to the measured ones with the largest difference of 5 % obtained for the naturalfrequency of the first bending mode for the Smista bridge. Furthermore, the 3D SDOF model was ableto capture the dynamic response of torsional vibrations with an overall difference of less than 2 % incomparison to the FE models. The model can be improved by further studying the pedestrian-structureinteraction as well as studying the effect of using approximative functions describing the mode shapes.

Sammanfattning

Finita elementmodeller (FE-modeller) utgor en bas for manga olika typer av analyser som exempelvis dy-namiska analyser. Darmed ar det av stor betydelse att FE-modeller representerar det faktiska beteendetav verkliga strukturer med stor noggrannhet. Ofta ar det emellertid skillnader mellan FE-modeller ochde verkliga strukturer man modellerar. Dessa skillnader kan bero pa en rad faktorer sasom exempelvisosakerheter i materialbeteende, geometriska egenskaper samt upplag- och randvillkor. Modellvalideringar darfor en viktig aspekt i att erhalla FE-modeller som representerar verkligheten i olika omfattningar.Utover modellvalidering ar aven modellverifiering viktigt, inte endast for verifiering av FE-modellerutan aven for verifiering av andra teoretiska modeller inom omraden sasom utmaning-, fraktur- ochdynamiska analyser.

I detta arbete har tva GC-broar, Kallhall- och Smistabron modellerats i ett FE-program baseratpa konstruktionsritningar och validerats mot experimentella resultat med avseende pa de naturligafrekvenserna. Vidare har det i detta arbete utvecklats en modell i MATLAB som tar hansyn tillmannisko-inducerade torsionsvibrationer baserat pa modalanalys, benamnd 3D SDOF modellen. Mod-ellen har aven verifierats mot de tidigare namnda FE-modellerna.

Malet med detta arbete ar saledes uppdelat i tva delar, dar den forsta delen bestar av att utvecklatredimensionella FE-modeller av tva GC-broar samt validera dessa mot matdata vad galler de naturligafrekvenserna. Den andra delen bestar av att utveckla en modell for att analysera effekten av mannisko-inducerade torsionsvibrationer och undersoka huruvida modellen fangar den dynamiska responsen.

Resultaten visade att de naturliga frekvenserna for den forsta boj- och vridmoden fran FE-modellernamotsvarade de uppmatta frekvenserna med en storsta relativ skillnad pa 5 % for den fosta bojmoden forSmistabron. Vidare visade resultaten att den utvecklade 3D SDOF modellen kunde fanga den dynamiskaresponsen av torsionsvibrationer med en skillnad pa mindre an 2 % i jamforelse med resultat fran deFE-modellerna. Modellen kan forbattras genom att vidare studera interaktionen mellan fotgangare ochgangbro samt studera effekten av att anvanda approximativa funktioner som beskriver modformen.

Preface

The work presented in this master thesis was initiated by the engineering consultancy Tyrens AB andthe department of Civil and Architectural Engineering at the Royal Institute of Technology, KTH.

First and foremost, we would like to express our gratitude to our supervisor, Ph.D. Mahir Ulker-Kaustellfor all his valuable advice and guidance throughout the work of this thesis but also for all the enthusiasticand enlightening conversations.

We are truly grateful to Ph.D. student Emma Zall for taking her time to provide us with assistance andfor letting us take part of her research.

Furthermore, we would like to thank everyone at the bridge department at Tyrens AB for allowing us tocarry out our work at a welcoming place with such a genuine hospitality and for making it an enjoyabletime. A special thank goes to Viktor Tell and Joakim Kylen for always finding the time to assist uswith modelling issues.

Last but not least, we would like to thank our families and friends for their support and patience duringthe work of this thesis and during our entire study at KTH.

Stockholm, June 2017Simon ChamounMarwan Trabulsi

List of notations

Notation Description Unitan Load factor -aj Coefficient in Fourier series -b Width of bridge mbj Coefficient in Fourier series -c Damping Ns/mc Damping matrix Ns/mC Square damping matrix Ns/mC Generalized damping Ns/mE Modulus of elasticity N/m2

F Force NF Force vector Nf Periodic function -

F (w) Fourier transform of f -fa Aliasing frequency Hzfn Natural frequency HzFn Generalized force Nfp Walking frequency of a pedestrian HzFp Pedestrian induced force Nfs Sampling rate Hzg Gravitational acceleration m/s2

k Stiffness N/mk Stiffness matrix N/mK Square stiffness matrix N/mK Generalized stiffness N/mL Length of bridge mLel Element length m

m Mass kgm Mass matrix kgM Square mass matrix kgM Generalized mass kgmp Pedestrian mass kgN Number of samples -p Excitation function -q Modal displacement mt Time sT Total sampling time sT0 Period of harmonic function sTn Natural period su Displacement mvp Walking velocity of a pedestrian m/sxp Distance from node mβ Coefficient in Newmarks method -γ Coefficient in Newmarks method -

∆t Sampling interval sν Poisson’s ratio -ξ Damping ratio %ρ Density kg/m3

Φ Modal matrix -φn Natural mode -ω0 Cyclic frequency rad/sωn Natural cyclic frequency rad/sωnD Damped natural cyclic frequency rad/s

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Detailed description of the Kallhall bridge . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Detailed description of the Smista bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theoretical background 7

2.1 Structural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 The equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Natural frequencies and modes of vibration . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.5 Numerical evaluation of dynamic response . . . . . . . . . . . . . . . . . . . . . . 14

2.2 The finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Main idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Fourier transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Method 23

3.1 Validation of FE models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Modelling procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Verification of pedestrian-induced vibration model . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Description of the 2D SDOF model . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.2 Verification of the 2D SDOF model . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.3 Development of the 3D SDOF model . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.4 Verification of the 3D SDOF model . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Results 44

4.1 Result from the validation of the FE models . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.1 Experimental natural frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.2 Theoretical natural frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.3 Comparison between experimental and theoretical results . . . . . . . . . . . . . . 46

4.2 Results from the verification of the 3D SDOF model . . . . . . . . . . . . . . . . . . . . . 47

4.2.1 Verification against the simple FE model . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.2 Verification against the FE model of the Kallhall bridge . . . . . . . . . . . . . . . 50

4.2.3 Verification against the FE model of the Smista bridge . . . . . . . . . . . . . . . 53

5 Discussion and conclusions 56

5.1 Validation of the FE models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Verification of the 3D SDOF model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Bibliography

A Measurement results

B Convergence analysis

Chapter 1

Introduction

1.1 Background

Finite element (FE) models that represent the actual behaviour of structures with great accuracy is ofgreat importance in the field of structural engineering. The reason being is that FE models serve as thebase in different types of dynamic analyses, structural health monitoring and in general for verificationof different structural designs [2].

However, even though finite element modelling (FEM) is widely used in today’s society as an effectiveanalytical tool, there are more often than not differences between analytically evaluated dynamic prop-erties from initial FE models and experimentally evaluated dynamic properties. The differences occurdue to various assumptions and uncertainties in e.g. boundary- and continuity condition and material-and geometric properties. Different structures, e.g. bridges are subjected to movement in e.g. bearingsand hinges which complicates the boundary- and continuity conditions. All material properties are notconstant parameters nor linear in their behaviour as they might change with time as a result of e.g.damage and deterioration. Furthermore, this affects the geometric properties which in their self aredifficult to evaluate, as the amount of detail in the geometry varies from member to member [1]. Inorder to ensure that FE models corresponds to measured responses, model validation and possibly alsocalibration is required.

Model validation is the process of confirming that a model is capable of representing the real behaviourof a studied system. Model calibration on the other hand is a procedure which consists of correcting aninitially created FE model by e.g. changing various parameters or assumptions, so that the differencesbetween experimental data and the simulated response from the FE model are reduced to an acceptabledegree [12].

Two pedestrian steel bridges situated in Kallhall and Satra, northwest respectively south of centralStockholm, Sweden has been modelled and validated against experimental results. The former is asimply supported, three span bridge where the respective spans are 40 m, 33.6 m and 40 m, as presentedin Figure 1.1. The second bridge, the Smista bridge, is a continuous bridge built up of three spans,stretching over the highly-trafficked highway E4/E20 between Segeltorp and Satra. The respectivespans are 19.6 m, 47.5 m and 16.9 m and are presented in Figure 1.2. Both bridges are described moredetailed in sections 1.4 and 1.5.

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Figure 1.1: Elevation drawing of the Kallhall bridge. FL = pinned bearing, RL = roller bearing.

Figure 1.2: Elevation drawing of the Smista bridge. FL = pinned bearing, RL = roller bearing.

Model verification is important in terms of verifying theoretical models, other than FE models, whereone ensures that the model behaves as intendent. Model verification is thus beneficial in a wide rangeof fields e.g. fatigue-, fracture- and dynamic analysis [9].

In this thesis a model accounting for pedestrian-induced vibrations in pedestrian bridges with respectto the first torsional mode has been verified against the previously mentioned FE models. The model isbased on the work of a current Ph.D. student Emma Zall, whose research involves the development ofimproved models for analysing pedestrian-induced vertical vibrations.

1.2 Aim

The aim of this thesis is two parted where the first part is to develop three-dimensional FE models oftwo pedestrian bridges and validate them against measured data regarding the natural frequencies. Thesecond part is to develop a model for analysing the effect of pedestrian-induced torsional vibrations andto investigate whether the model captures the actual dynamic response of such loading by validating itagainst the previously mentioned FE models.

1.3 Limitations

A main limitation of this thesis is that in the development of the model for analysing pedestrian-inducedtorsional vibrations, the pedestrian-structure interaction have not been taken into consideration. Hence,the properties of the bridges are constant, independently of the presence of pedestrians. Furthermore,the effect of other sources of excitations has not been taken into consideration either, only the pedestrian-induced vibrations is analysed. Lastly, in both parts of this thesis only linear systems are studied.

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1.4 Detailed description of the Kallhall bridge

The Kallhall bridge is a pedestrian bridge located in Kallhall, northwest of Stockholm, Sweden and wasbuilt in 2015, as a part of the rebuild of the Kallhall commuter station during 2014-2016. The bridge isa simply supported, three span bridge where the respective lengths are 40 m, 33.6 m and 40 m. Eachspan is independently connected and is inclined along its longitudinal direction with approximately 2◦,as shown in Figure 1.1. The bridge is built up of a three-dimensional frame structure with stiffeners andvarious plates as reinforcement in the deck plate and inside the frame structure.

The cross-section of the superstructure between the supports in the longest span, is presented in Figure1.3. The cross-section is built up of 6095 mm wide three-dimensional frames, with heights of 4056 mm.The frames are built of rectangular hollow beam sections and are braced at the joints, as seen in Figure1.1. The center-to-center (c/c) distance between the frame structures is 3640 mm in the longitudinaldirection. Furthermore, four main beams run longitudinally along each side of the frame, two in theupper and lower part of the cross-section, as shown in Figure 1.3.

Figure 1.3: The cross-section between the supports at the Kallhall bridge.

The bridge deck consists of a 10 mm thick steel plate which is supported with eleven U-shaped longitu-dinal stiffeners, having a c/c distance of 730 mm, as shown in Figure 1.3. These run through T-shapedcrossbeams, where the two crossbeams at the middle of the bridge have twice the web thickness incomparison to the others.

The cross-section at both ends of the span is presented in Figure 1.4, which have the same dimensionsas the cross-section between the supports, except for the height of the frame structure. From Figure 1.4it can be seen that the bridge deck is supported by plates inside a box section. There are also stiffeningplates inside the four main beams with a c/c distance of 3640 mm in the longitudinal direction.

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Figure 1.4: The cross-section of the support at the Kallhall bridge.

Lastly, the substructure is constructed as such that the bridge is supported by two abutments at bothends and two concrete piers which the respective spans are supported on. The piers are in turn restingon bedrock. At both ends there are roller bearings while the intermediate supports consist of pinnedand roller bearings as shown in Figure 1.1.

1.5 Detailed description of the Smista bridge

The Smista bridge is a three-span continuous bridge resting on four supports, as shown in Figure 1.2.The bridge is built in three parts which are welded together, resulting in a continuous structure overthe supports. The parts are 29.8 m, 29.04 m and 25.3 m long. The first and last span are inclined by 5% along its longitudinal direction while the middle span is arched with a 400 m vertical radius, whichcan be seen in Figure 1.2.

The cross-section between the supports is presented in Figure 1.5. The bridge is 4050 mm wide and1535 mm high. The cross-section consist of 200x200 mm longitudinal hollow box shaped beams on eachside which spans over the entire length of the bridge. Furthermore, vertical hollow box shaped beamsare connected to the longitudinal beams and to the crossbeams. These span along the bridge with a c/cdistance of 2640 mm as presented in Figure 1.6, which corresponds to the first part of the bridge. Thebridge deck consists of a 10 mm thick steel plate with a 2 % inclination. The deck is is connected tothe vertical hollow box shaped beams and is also supported by 180x180x10 mm crossbeams, with a c/cdistance of 660 mm as seen in Figure 1.2.

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Figure 1.5: The cross-section of the field at the Smista bridge.

Figure 1.6: Plan view of the first part of the Smista bridge.

As presented in Figure 1.5, the cross-section consists of 1335 mm high and 10 mm thick plates whichare connected to the vertical and longitudinal hollow box shaped beams. The plates in the midspanhave holes of varying width, increasing towards the middle of the bridge, as shown in Figure 1.2. Thevariation of the width of the holes corresponds to an architectural expression of the varying shear forcein the midspan and changes with approximately 250 mm/hole. The positions of the holes are notsymmetrically placed along the left and right side of the bridge. It can also be seen in Figure 1.6 thatat each weld seam, connecting the three parts, longitudinal and horizontal stiffening plates are present.There are also stiffening plates present in the 200x200 mm longitudinal hollow box shaped beams ateach weld seam.

The cross-section at the end- and intermediate supports, shown in Figure 1.7 and Figure 1.8, are notsignificantly different from the cross-section between the supports. At the end-and intermediate support,vertical hollow box shaped beams spans from the deck plate down to the supports. Also, several platesin the vertical and horizontal direction exist to increase the stiffness around the supports.

Lastly, the bridge is supported by abutments at the ends, which the bridge is also anchored to by steelrods in order to avoid uplift as shown in Figure 1.7. The intermediate supports, shown in Figure 1.6rests on skewed concrete piers where the angle between the supports is 117◦. The piers are in turnresting on bedrock.

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Figure 1.7: The cross-section of the end support at the Smista bridge.

Figure 1.8: The cross-section of the intermediate support at the Smista bridge.

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Chapter 2

Theoretical background

This chapter contains the relevant theoretical background for this thesis. It starts out with an extensivesection that covers the most fundamental concepts of structural dynamics. Followed is a brief intro-ductory to the theory behind the finite element method that is widely used for modelling of structures.Finally, this chapter ends with an overview of signal processing to showcase how to process signals frommeasurements to obtain valid results.

2.1 Structural dynamics

Structural dynamics covers the analysis of behaviour of structures when subjected to dynamic loading.A dynamic load is a time-varying load and can be in the form of walking people, wind or some otherkind of excitation such as an earthquake.

The equation of motion is the most fundamental equation in dynamic analysis and an overview ofthe theory behind it is thus given in this section. Solving the equation of motion gives for instancedisplacements for all locations on a structure during any given time when subjected to a dynamicload. The equation of motion also give rise to an eigenvalue problem which solution results in thenatural frequencies and modes of a system. Natural frequencies are the frequencies at which a structuretends to oscillate during free vibration i.e. in the absence of external loading. The deflected shapes ofthe structure during these oscillations at these certain frequencies are referred to as natural modes ofvibration, or simply as modes. These frequencies play a fundamental part in the design and analysisof bridges. The reason being that it is essential to consider that the natural frequency of a bridgedo not match the frequency of expected loads such as the walking frequency of pedestrians since thismay lead to discomfort for pedestrians but also structural damage. If the frequencies do coincide, oneshould conduct time history analysis to evaluate the dynamic behaviour of the bridge. This may bedone using modal analysis which is a powerful method for obtaining displacements and accelerations forstructures when subjected to dynamic loading. All derivations and statements made in the followingsections concerning structural dynamics are based on Chopra [3] if not stated otherwise.

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2.1.1 The equation of motion

Each structural member in a structure contributes with some degree to the mass, stiffness and dampingof the system. The degrees of freedom in the system are the number of independent displacementsneeded to describe the relative change in position of the masses in the system. In reality, structuresare composed of an infinite number of degrees of freedom. In structural dynamic the most basic way todescribe a system is as a single degree of freedom (SDOF) system. A SDOF system can be representedby e.g. a oscillator, allowed to only displace in the lateral direction. Figure 2.1 illustrates the oscillatorconsisting of a mass, m, a massless frame with stiffness k and a viscous damper with damping coefficientc. It is excited by an externally applied dynamic force F (t) that varies with time, t. The resulting time-dependent displacement, velocity and acceleration that occur is denoted u(t), u(t) and u(t), respectively.

Figure 2.1: Single degree of freedom system composed of an oscillator.

The external- and inertia force resisting the acceleration of the mass give rise to a resulting force, P (t),in accordance with Newton’s second law of motion:

P (t) = F (t)−mu(t) (2.1)

Elastic- and damping forces are internal forces that arise resisting the deformation and velocity of themass, respectively. The resultant is hence gives as

P (t) = cu(t) + ku(t) (2.2)

The equation governing the physical behaviour of the bridge i.e. the equation of motion is given bysetting Eq. 2.1 equal to Eq. 2.2:

mu(t) + cu(t) + ku(t) = F (t) (2.3)

Before proceeding to solving the equation of motion, some central concepts regarding structural dynamicsshould be addressed. The natural cyclic frequency, ωn, is given in units of radians per second and is foran undamped SDOF free vibration system given as

ωn =√k

m(2.4)

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The natural period of vibration, Tn, is the time required for the undamped system to complete one cycleof free vibration according to Eq. 2.1.1.

Tn = 2πωn

(2.5)

The natural frequency represents the number of cycles that occur during one second and is given inhertz (Hz) as

fn = 1Tn

(2.6)

The natural cyclic frequency is hence proportional to the natural frequency according to Eq. 2.1.1. Theterm natural is used to emphasize the fact that these are natural properties of the structure in freevibration.

fn = ωn2π (2.7)

The SDOF system described above is not always applicable to real life structures. The reason beingthat structures are composed of an infinite number of degrees of freedom and not all structures canbe idealized as SDOF systems since such idealizations may yield inaccurate results in terms of e.g.frequencies and mode shapes. In such cases, the dynamic behaviour can be described more accuratelyby discretizing structures into systems of elements with a finite number of degrees of freedom. Thissystem is referred to as a multi degree of freedom system (MDOF). The basis of the theory describing aMDOF system is analogous to the theory described above for the SDOF system. It is a generalizationfrom one to N number of dimensions where N is the number of degrees of freedom in the system. Theequation of motion for a multi degree of freedom (MDOF) system is thus given as

mu(t) + cu(t) + ku(t) = F(t) (2.8)

where m is the mass matrix, c the damping matrix and k the stiffness matrix. They are all matrices oforder N x N . Displacements, velocities and accelerations for each degree of freedom are now given astime-dependent vectors of order N x 1 and are denoted as u(t), u(t) and u(t), respectively.

F(t) is referred to as a load vector of order N x 1 describing the external forces acting on each degreeof freedom.

The mass matrix is determined by assuming the mass of the system to be concentrated at the nodesi.e. where the degrees of freedoms are located. The stiffness matrix is constructed by assembling thelocal stiffness matrix of each element. There are several ways to construct the damping matrix which isfurther discussed in the following section.

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2.1.2 Damping

The process by which the free vibration of structure decay in amplitude is caused by damping of thesystem. Damping occurs when kinetic and strain energy of the vibrating system dissipates due tovarious mechanisms acting simultaneously. Common sources of energy dissipation in bridge structurescould for instance occur due to friction at steel connections, inelastic behaviour of structures at largedeformations and opening and closing of cracks in concrete. To identify and mathematically describeeach source of damping that exist in a complex structure such as a bridge is a nearly impossible task.However, conducting vibration experiments on existing structures enables evaluation of the damping.

An usual way to express damping is by assuming viscous damping. A viscous damping force is propor-tional to the velocity of a moving body but oppositely directed relative its motion. The proportionalityto the velocity makes it convenient to express in mathematical terms. However, common sources ofdamping in bridge structures such as those described above are not viscous but rather due to othermechanisms. These mechanisms are neither easy to represent mathematically nor easy to measure com-pared to viscous damping [5]. Comparisons of conducted experiment and theory show that assumingviscous damping is sufficiently accurate in most cases [4].

Theoretical damping

Damping influences the nth natural cyclic frequency of vibration, ωn, of an system according to Eq. 2.9,where ξn is the damping ratio, a measure used to express the level of damping in a system and ωnD isthe damped nth natural cyclic frequency.

ωnD = ωn√

1− ξ2n (2.9)

Most practical structures have a damping ratio below 20 % [3]. The Eurocodes recommends a dampingratio of 0.5 % for railway bridges with span lengths over 20 m [7]. Consequently, the effect of dampingon the natural frequencies for bridges is negligible since ωnD = ωn for small damping ratios. However,damping might have a significant influence when evaluating displacements and accelerations. Thereare several commonly used methods to account for viscous damping. For simple SDOF systems, thedamping coefficient, c, is defined as

c = 2ξnmωn (2.10)

For MDOF systems, damping matrices may be constructed using for instance Rayleigh damping whichinvolves forming the damping as a linear combination of the mass and stiffness matrix. For the useof modal analysis which is further described in section 2.1.4, one can account for damping by usingestimated damping ratios obtained from e.g. measurements directly. This requires linearly elasticanalysis and that the system is classically damped i.e. that the natural modes of the system areuncoupled.

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Experimental damping

As mentioned earlier, conducting vibrational experiments on structures enables evaluation of the damp-ing. Commonly used damping estimation methods are the logarithmic decrement method and thehalf-power bandwidth method. The logarithmic decrement method evaluates the damping ratio in thetime domain representation. It accounts for the damping contribution from all excited modes of vibra-tion. If the damping contribution from each mode is of interest, the half-power bandwidth method canbe used.

The half-power bandwidth makes use of the width of the peak value in the frequency domain represen-tation. Consider Figure 2.2, an illustration of how a frequency domain spectrum may look. If ωn is thenatural cyclic frequency and its corresponding amplitude is A, then ωa and ωb are the frequencies oneither side of it with an corresponding amplitude of 1/

√2 A.

Figure 2.2: Illustration of half-power bandwidth.

The damping ratio is then given as

ξ = ωb − ωaωb + ωa

(2.11)

2.1.3 Natural frequencies and modes of vibration

The natural frequencies and modes of a structure are determined by evaluating the undamped freevibration i.e. in the absence of damping and external loading. In each natural mode, the systemoscillates at its natural frequency with all degrees of freedoms of the system vibrating in the same phase,passing through their maximum, minimum and zero displacement positions at the same instant of time.Free vibration occurs when a structure is disturbed from its static equilibrium position and allowed tovibrate without an externally applied dynamic force acting on it. This is achieved by introducing aninitial displacement and velocity.

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u = u(0) u = u(0) (2.12)

The equation of motion of an undamped MDOF system undergoing free vibration is governed by Eq.2.13.

mu+ ku = 0 (2.13)

This equation is based on a system of N number of degrees of freedom and homogeneous differentialequations coupled through the mass and stiffness matrix. The displacement can mathematically bedescribed as

u(t) = φnqn(t) (2.14)

where φn is the deflected shape of the system i.e. the mode shape and qn(t) is the time variation of thedisplacement which can be described as a harmonic function:

qn(t) = An cosωnt+Bn sinωnt (2.15)

An and Bn are constants that are to be determined using the initial conditions. Combining Eq. 2.14and Eq. 2.15 yields

u(t) = φn(An cosωnt+Bn sinωnt) (2.16)

Differentiating the displacement twice with respect to time gives the time variation of the acceleration:

u(t) = −ω2nφn(An cosωnt+Bn sinωnt) (2.17)

Substituting Eq. 2.16 and 2.17 into the equation of motion yields

[−ω2nmφn + kφn]qn(t) = 0 (2.18)

There are two ways to satisfy this equation. The first solution qn(t) = 0 implies that there is no motion ofthe system since u(t) always will be equal to zero. In the second solution, the natural cyclic frequenciesωn and mode shapes φn fulfil the following equation

[k− ω2nm]φn = 0 (2.19)

The trivial solution φn = 0 implies no motion and is hence not of interest. Nontrivial solutions areobtained if

det[k− ω2nm] = 0 (2.20)

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Eq. 2.19 is knows as an eigenvalue problem and has N real and positive roots for ω2n. These roots are

known as eigenvalues and represents the N natural frequencies ωn of the system. When the naturalfrequencies are known, the corresponding natural mode φn can be determined using Eq. 2.19.

2.1.4 Modal analysis

The simultaneous solution of the coupled equations of motion derived for MDOF systems in Eq. 2.8 isnot efficient for systems with a large number of DOFs and can be very computationally demanding. Areduction of the system’s complexity can be achieved by transforming the equations in terms of modalcoordinates. Simply put, a MDOF system with N DOFs is transformed into N SDOF systems. Thismethod is referred to as modal analysis and leads to an uncoupled set of modal equations. As a result,each modal equation can be solved independently to determine the contribution to the response fromthat specific natural mode. The responses from each natural mode are then combined to obtain thetotal response of the system. The classical modal analysis procedure is applicable for linear systemswhich are classically damped in order to obtain modal equations that are uncoupled, a central featureof modal analysis.

The orthogonality of natural modes implies that the following square matrices are diagonal:

K = ΦTkΦ M = ΦTmΦ (2.21)

For classically damped systems, the square matrix C is also diagonal:

C = ΦTcΦ (2.22)

The matrix Φ is called the modal matrix and contains the N modes of vibration. The diagonal elementsare thus given by

Kn = φTnkφn Mn = φTnmφn Cn = φTncφn (2.23)

Due to the orthogonality relations, the set of N coupled differential equations are transformed to a setof N uncoupled equations in modal coordinates according to

Mnqn(t) + Cnqn(t) +Knqn(t) = Fn(t) (2.24)

where

Fn(t) = φTnF (t) (2.25)

and qn are the unknown modal coordinates to be solved. Mn, Cn, Kn and Fn are referred to as thegeneralized mass, damping, stiffness and force for the nth natural mode φn. Thus, the parametersdepend only on the nth mode and the modal coordinates may be solved in the absence of any otherinformation about the other natural modes. Eq. 2.24 is of the same form as a SDOF system with

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damping according to Eq. 2.3. Thus the generalized damping can for each mode be defined in a manneranalogous to Eq. 2.10.

Cn = 2ξnMnωn (2.26)

The modal coordinates are solved using the relation between the generalized mass and generalizedstiffness according to Eq. 2.27.

Kn = ω2nMn (2.27)

Substituting Eq. 2.26 and 2.27 into Eq. 2.24 yields the following equation to solve for the modalcoordinates.

qn(t) + 2ξnωnqn(t) + ω2nqn(t) = Fn(t)

Mn

(2.28)

Once the modal coordinates of the nth natural mode have been determined, the contribution of thatparticular mode to the displacements is

un(t) = φnqn(t) (2.29)

Superposition of these modal contributions according to Eq. 2.30 gives the total displacement.

u(t) =N∑n=1

un(t) =N∑n=1

φnqn(t) (2.30)

2.1.5 Numerical evaluation of dynamic response

An analytical solution of the dynamic response is usually not feasible for systems where the dynamicforce varies arbitrarily with time. This can however be tackled by the use of numerical time-steppingmethods for integration of ordinary differential equations e.g. the equation of motion.

In time-stepping methods, the response is evaluated for a discrete number of time steps, ti, withi = 0, 1, 2, ... Two widely used numerical methods to evaluate dynamic responses are the central dif-ference method and Newmark’s method. The central difference method is based on a finite differenceapproximation of displacements. In contrast, Newmark’s method is based on assumed variation ofacceleration.

Two special cases of Newmark’s method are well-known: the constant average acceleration method andthe linear acceleration method. The difference between them lies in the choice of the parameters β andγ. These parameters define how the acceleration varies over a time step and determine the stability andaccuracy characteristics of the method. The constant average acceleration method has been used in thisthesis which is associated with choosing β and γ to 1

4 and 12 , respectively. The method is stable for any

given time step ∆t. However, to obtain accurate results ∆t must be chosen small enough since a toolarge time step will yield meaningless results due to the presence of numerical round-off .

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The following equations summarizes the time-stepping solution for the constant average accelerationmethod. This procedure is applicable for linear SDOF systems using classical damping. However, it iseasily applicable to the uncoupled equation in modal coordinates presented in Eq. 2.24 by replacing themass, damping, stiffness, force and displacements with their corresponding generalized form.

Initial calculations are made according to Eq. 2.31 to Eq. 2.35 where u0, u0 and F0 define the initialdisplacement, initial velocity and initial force, respectively.

u0 = F0 − cu0 − ku0

m(2.31)

a1 = 1β∆t2m+ γ

β∆tc (2.32)

a2 = 1β∆tm+

(γ

β− 1

)c (2.33)

a3 =(

12β − 1

)m+ ∆t

(γ

2β − 1)c (2.34)

k = k + a1 (2.35)

The calculations to be made for each time step i = 0, 1, 2, ... are given in Eq. 2.36 to Eq. 2.39.

Fi+1 = Fi+1 + a1ui + a2ui + a3ui (2.36)

ui+1 = Fi+1

k(2.37)

ui+1 = γ

β∆t(ui+1 − ui) +(

1− γ

β

)ui + ∆t

(1− γ

2β

)ui (2.38)

ui+1 = 1β∆t2 (ui+1 − ui)−

1β∆t ui −

(1

2β − 1)ui (2.39)

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2.2 The finite element method

The approach of using closed form differential equations for modelling the behavior of structures is onlyapplicable in the simplest cases of structures. Finite element (FE) analysis, also referred to as the finiteelement method is a method that provides a way of obtaining numerical solutions to more complexproblems. An general introduction to the FE is given as well as an description of the widely used shellelements.

2.2.1 Main idea

The main idea of the finite element method involves dividing a structure into a number of elements as ifthey were pieces of the structure. These elements are connected at points referred to as nodes that canbe visualized as pieces of glue that hold the elements together. Each node hence serves as a connectorbetween two or more elements. All elements sharing a node has the same displacement components atthat particular node [5].

In all applications of finite element analysis, one seeks to calculate a field quantity such as displacementsor stresses. The field quantity within an element is interpolated from values of the field quantity at thenodes using shape functions. Shape functions are used to idealize the variation of the field quantitywithin each element using linear or quadratic polynomials. This variation only yields approximatesolutions since the actual variation within the elements may need far more complex functions thanlinear or quadratic ones to be described correctly [5].

The main advantage with the finite element method compared to other analysis method is thus thatan arbitrary structure, of any form or size, can be idealized as a discretized set of elements connectedtogether. Hence, the field quantity of a structure can be evaluated by interpolating the field quantityover the elements. Additionally, the strength of the finite element method is its versatility since it isapplicable to a countless of physical problems.

There are a number of available FE software available for modelling of structures. Some of these aregeneral-purpose software while others are oriented towards more specific types of structures. The FEsoftware chosen for modelling in this thesis is BRIGADE/Plus. It is a FE software developed especiallyfor the use of structural analysis and design of bridges and other civil structures. In the reminder ofthis thesis, BRIGADE/Plus will be referred to as Brigade.

2.2.2 Elements

Elements are the basic building blocks in finite element analysis. An element is a mathematical relationthat defines how the degrees of freedom of a node relate to the other. A number of element types exist.The most commonly used element types in structural analysis are:

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• Truss elements

• Beam elements

• Solid elements

• Shell elements

What type of element type to use depends on the structure to be modelled and the aim of the analysis.Shell elements are used to model structural members in which the thickness is significantly smallerthan the other dimensions. Two common types of shell elements suitable for dynamic analysis exist inBrigade: continuum shell and conventional shell. For continuum shells, the thickness of the elementis defined by its nodal geometry and the entire three dimensional body is hence discretized. Theseelements have only translational degrees of freedom and use linear interpolation between the nodes. Incontrast, conventional shell elements define the geometry at a reference surface to discretize the body.They use linear or quadratic interpolation and their degrees of freedom account for both translation androtation. Figure 2.3 portrays the difference between continuum and conventional shell elements [6].

Figure 2.3: Conventional shell versus continuum shell [6].

Conventional shell elements in Brigade can be divided into three main subgroups: thin, thick andgeneral-purpose conventional shell elements. Thin conventional shell elements, such as STRI3 elementsare to be used when transverse shear flexibility is negligible and the Kirchhoff constrain must be satisfiedaccurately. The Kirchhoff constrain states that a line orthogonal to the shell reference surface shouldremain orthogonal to the shell reference surface after deformation. In contrast, thick conventionalshell elements, such as element type S8R are needed when transverse shear flexibility is important andsecond-order interpolation is desired. General-purpose conventional shell elements are used in mostcases since they usually provide accurate and robust solutions for most applications. Their advantagelies in their versatility to describe both thin and thick shells since they become discrete Kirchhoff thinshells as the thickness decreases and use thick shell theory as the thickness increases. One commontype of general-purpose conventional shell is the S4 element. It consists of 4 nodes and is stable againstphenomenons such as element distortion and parasitic locking. The corresponding element when usingreduced integration is the S4R. This type of element only has one integration location in contrast to theS4 which has four and the S4R is thus less computationally expensive [6].

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2.3 Signal processing

Signal processing involves the process of analysing signals from measurements. This involves trans-forming signals from time to frequency domain. In time domain, signals are evaluated according totheir response with respect to time. In contrast, frequency domain refers to the analysis of signals withrespect to frequency rather than time. Frequency domain representations are important in the field ofdynamics since they provide information about natural frequencies and damping ratios.

The Fourier transformation play an essential part in the process of transforming signals to the frequencydomain, where signals are decomposed into a sum of sine waves. An overview of the Fourier transforma-tion is hence given along with other fundamental concepts regarding signal processing. These conceptswill provide understanding of the basics in signal processing e.g how to address problems such as aliasingand leakage. The derivation are based on the derivation presented by Chopra [3] if not stated otherwise.

2.3.1 Fourier transformation

The Fourier expansions enables general functions of periodic and non-periodic character to be decom-posed into a series of trigonometric or exponential functions and continuous integrals of such. The twotypes of Fourier expansions are:

• Fourier series

• Fourier integral

A given periodic function can be expressed as a Fourier series, in terms of its harmonic componentsaccording to

f(t) = a0 +∞∑j=1

ajcos(jω0t) +∞∑j=1

bjsin(jω0t) (2.40)

whereω0 = 2π

T0(2.41)

The term jω0 corresponds to the jth harmonic cyclic frequency and T0 is the period of the function.The coefficients in Eq. 2.40 may be expressed as

a0 = 1T0

∫ T0

0f(t)dt (2.42)

aj = 2T0

∫ T0

0f(t)cos(jω0t)dt j = 1, 2, 3... (2.43)

bj = 2T0

∫ T0

0f(t)sin(jω0t)dt j = 1, 2, 3... (2.44)

Figure 2.4b) showcase how a signal can be represented by sinusoidal functions of different frequenciesin accordance with the Fourier series. Hence, based on the Fourier series the Fourier transformation

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enables a sampled signal in the time domain to be transformed into the different frequencies that thesignal is built up of which are represented in the frequency domain [10]. The procedure is schematicallypresented in Figure 2.4 where a sampled signal in the time domain, divided in its sinusoidal functions,is related to the same signals in terms of its frequencies in the frequency domain.

Figure 2.4: A signal shown in both time and frequency domain. a) The signal, from the upper part of the figure in b),shown as a set of sinusoidal functions, in both frequency (amplitude vs frequency) and time domain (amplitude vs time).b) The signal in the upper part of the figure decomposed into a set of sinusoidal functions in time domain, shown in thelower part of the figure. c) The same set of sinusoidal functions shown in a) but in the frequency domain [8].

The Fourier series is valid for a function which is periodic. For a given function that is non-periodic itcan according to the Fourier Theorem be represented by the complex Fourier integral:

f(t) = 12π

∫ ∞−∞

F (ω)eiωtdω (2.45)

F (ω) = 12π

∫ ∞−∞

f(t)e−iωtdt (2.46)

Eq. 2.45 is called the inverse Fourier transform while Eq. 2.46 represent the Fourier transform, alsoknown as the direct Fourier transformation of the function f(t).

The Fourier transform can be applied to both simple periodic excitation and non-periodic complicatedexcitation varying arbitrary in time. In the case of the latter, the transformation of data from the timedomain to the frequency domain does not occur in a continuous manner. Hence, a numerical evaluationof the Fourier integral, in form of digitalized samples of the time domain excitation must be performed.The process that transform the digitalized discrete samples from the time domain to the frequencydomain is called the Discrete Fourier Transformation (DFT) [8] and is schematically presented in Figure2.5

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Figure 2.5: a) A continuous input signal in time domain b) Discrete samples of the input signal c) Samples in the timedomain transformed to the frequency domain [8].

Eq. 2.47 and Eq. 2.48 defines a discrete Fourier transform pair, in analogy with Eq. 2.45 and Eq. 2.46which together define a continuous Fourier transform pair.

pn =N−1∑j=0

Pjei(jω0tn/N) (2.47)

wherePj = 1

T

N−1∑n=0

pne−i(jω0tn) ∆t = 1

N

N−1∑n=0

pne−i(2πj/N) (2.48)

In Eq. 2.47 and Eq. 2.48, pn is a set of N equally spaced discrete values of the excitation function p(t).T and ∆t represent the total sampling time and the sampling interval, respectively.

The DFT is a numerical evaluation and hence only an approximate representation while the Fouriertransform is a continuous, true representation of the excitation function. Hence, signal analysis andprocessing with DFT handle a large amount of discrete data points. In order to optimize the compu-tations, the DFT is today computed with a faster algorithm, called the Fast Fourier Transformation(FFT) based on the Cooley-Tukey algorithm.

The FFT transform the N time domain samples to N/2 equally spaced samples in the frequency domain,due to that the information about the phase is lost in the transformation procedure from time tofrequency domain. Choosing the sampling interval sufficiently small and the sampling time sufficientlylarge is necessary in order to accurately represent the input signal, as an insufficient number of samplesmay lead to distortions, e.g. aliasing and an insufficient sampling time may lead to leakage [8].

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2.3.2 Aliasing

Aliasing is a phenomenon which leads to erroneous result of the input signal being sampled. The reasonfor the phenomenon has to do with the sampling rate, used to sample the input signal. Consider Figure2.6, where an input signal is described as a sinusoidal function. If the input signal is sampled exactlyones every period, the result would be a constant line, which does not corresponds to the input signalthat was to be sampled.

Figure 2.6: The result of a bad sampling rate. a) Input signal b) Sampled signal c) The aliased signal [8].

In order to avoid problem with aliasing the sampling rate needs to be changed. The Nyquist Theoremstates that the sampling rate should be greater than twice the highest frequency of the input signalbeing sampled:

fs ≥ 2fmax (2.49)

where fmax is the highest frequency of the input signal, fs is the sampling rate defined as fs = 1∆t and

∆t is the sampling interval. Hence, suppose that a signal has the frequency f and that the samplingrate is fs, then:

If f < fs

2 - no aliasing occurs, according to the Nyquist Theorem.

If 12fs < f < fs - the signal undergoes aliasing, with an aliasing frequency of fa = fs − f .

If f > fs - the signal undergoes aliasing, with an alias frequency of fa = f − fs

In those cases where the sampling frequency is limited, the highest frequency contained in the signalneeds to be modified. This is done by applying a low-pass filter, also known as an anti-aliasing filter.Additionally, filters are also used to remove disturbing noise [8].

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2.3.3 Leakage

Leakage is a phenomenon which is caused by truncation of a signal in the time domain. The FFTcalculates the frequency spectrum based on a sample of the input from a complete period, which isassumed to be repeated at all time. Thus, the FFT assumes that the signals obtained from the samplingare periodic, repeated throughout the length of the total sampling time. If then the signal is truncatedat a non-integer number of cycles, leakage will occur [11]. The effect of leakage in the frequency domainis illustrated in Figure 2.7. This phenomenon can be reduced using windowing functions as e.g theHanning windowing function which is a non-negative bell-shaped curve.

Figure 2.7: Effect of leakage viewed in the frequency domain. a) Spectrum without leakage b) Spectrum with leakage[8].

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Chapter 3

Method

The following section is divided in two main parts. In the first part, a description of the process ofdeveloping the FE models is given along with a description of the validation process. The second partdescribes the process of developing and validating the model developed to account for pedestrian-inducedtorsional vibrations.

3.1 Validation of FE models

With the aim of obtaining FE models that accurately captures the dynamic properties of the Kallhalland Smista bridges, two FE models have been developed using the FE software Brigade. In order tovalidate the models, measurements have been conducted on each bridge whereby the natural frequencieshave been obtained along with the damping ratio for the first bending- and torsional mode of vibration.To assess the validity of the models, the natural frequencies obtained from the measurements have beencompared to the ones obtained from the FE analysis.

The following section will thus begin with a description of the modelling procedure along with theassumptions made in the process. Furthermore, a description is given about the set-ups used in mea-surements and the process of obtaining the natural frequencies from the measurement data.

3.1.1 Modelling procedure

The model of the Kallhall bridge was created using the graphical user interface and was based onengineering drawings. Only the largest span was modelled and the remaining spans were omittedfrom the analysis. The reason being that each span of the bridge function independent of the other.Furthermore, the Kallhall bridge was modelled without considering the piers. The basis behind thisdecision is that each pier is assumed to function as a massive rigid body. Hence, they are assumed tobe undeformable and the effect they may contribute with is thus assumed to be negligible.

For the Smista bridge, the modelling was performed by writing a script, also based on engineeringdrawings. In this case the entire bridge was considered including the piers due to its continuity overthe supports. Furthermore, the abutments in both models were not considered nor the soil-structureinteraction as it is out of the scope of this thesis.

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Regarding the geometrical properties, the inclinations in both the longitudinal and transversal directionas well as the arch were neglected in the FE model of the Smista bridge. However, the bridge wasmodelled with respect to its longitudinal inclined plane where it was assumed that the entire bridge hada 5 % inclination throughout its length. Hence, the true length has been accounted for with respectto the assumption. Regarding the Kallhall bridge, the inclination was neglected as well. Instead, thebridge was modelled as straight but with respect to its inclined plane and its true length has thus alsobeen accounted for. Both the FE models are presented in Figure 3.1 and Figure 3.2, respectively.

Both bridges were modelled based on dimensions with respect to the centerlines of each structuralmember. The reason being that the element thickness’s were assigned with respect to their middlesurface. Furthermore, each bridge was modelled as one single three-dimensional part. This modellingapproach provides rigid connections between all structural members and no further constraint had tobe assigned at these connections.

Regarding material and section properties, the different sections each bridge is composed of were assignedtheir respective thickness, density, modulus of elasticity and Poisson’s ratio. The following materialproperties were assigned for both bridges:

• Density, ρ = 7850 kg/m3

• Modulus of elasticity, E = 210 GPa

• Poisson’s ratio, ν = 0.3

Figure 3.1: The FE model of the Kallhall bridge.

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Figure 3.2: The FE model of the Smista bridge.

Some components of the bridges were assessed to have a negligible structural stiffness while having asignificant contribution to the mass of the model. These non-structural components were accounted forby assigning regions with an equivalent density. The non-structural components include the:

• Paving in form of asphalt on the Kalhall bridge

• Railings on the Kallhall bridge

• Railing on the Smista bridge

• Curbstone on the Kallhall bridge

Regarding the boundary condition, the behaviour of the bearings was represented by creating couplinginteraction between the bearings and the structural members. Figure 3.3 showcases the set up used torepresent the bearings.

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Figure 3.3: a) coupling constraint between two structural members and bearing. b) coupling constraint between onestructural member and bearing.

The surface that rest on the actual support was coupled to a reference point, corresponding to the centerof rotation of the actual bearing. In this connection, all translational and rotational movements wereconstrained. Another reference point was then created, positioned slightly below the first reference point,that was either coupled to the piers or directly assigned boundary conditions. In the case where thereference point was connected to e.g. the supporting piers, as shown in Figure 3.3 a), all translational androtational movement were constrained. In the case where the second reference point was not connectedto a structural element, as shown in Figure 3.3 b), the boundary conditions were applied directly to it.A connection between the two reference points was then created to account for the movement in theactual bearing. Boundary condition were then applied at the bottom of the piers, which were treatedas completely fixed due to that they are resting on bedrock. The reason for modelling the bearing usingthis approach is to allow for movements between the bridges and the bearings.

Additionally, the influence of the bearing conditions on the natural frequencies of each bridge werestudied. As previously mentioned, both bridges are simply supported. There were however uncertaintieson how the bearings were allowed to rotate and an analysis of the movements in the bearings wasperformed. As a starting point, the bearings were allowed to rotate around all axes. Table 3.1 andTable 3.2 showcase how a change in the properties of the bearings would affect the natural frequenciesof the FE models. The relative differences presented in the tables are differences with respect to thefirst case. Noteworthy, is that the rotation around the Y-axis in the bearings always was free to rotatein all cases which is the rotation around the axis normal to the deckplates.

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Table 3.1: The natural frequencies of the Kallhall bridge for different bearing conditions.

Case 1 - Simply Supported (SS)

Mode Natural frequencies [Hz] Rel. diff [%]1st vertical 3.31

1st torsional 4.52

Case 2 - All fixed bearings

Mode Natural frequencies [Hz] Rel. diff [%]1st vertical 3.48 5.33

1st torsional 4.63 2.41

Case 3 - SS, rotation prevented around X



Case 4 - SS, rotation prevented around Z



Case 5 - SS, rotation prevented around X and Z



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Table 3.2: The natural frequencies of the Smista bridge for different bearing conditions.

Case 1 - Simply Supported (SS)

Mode Natural frequencies [Hz] Rel. diff [%]1st vertical 2.56

1st torsional 2.77

Case 2 - All fixed bearings



Case 3 - SS, rotation prevented around X



Case 4 - SS, rotation prevented around Z



Case 5 - SS, rotation prevented around X and Z



From the study, it was concluded that the influence of the bearing conditions did not have a significanteffect on the natural frequencies of the FE model and thus the initially assumed boundary conditionswere applied to the models.

Regarding the element and mesh size, an element type of S4R was chosen based on its advantagesdescribed in section 2.2.2. A fine mesh usually results in more accurate results. On the other hand, onewants to avoid the waste of computational time due to an overly refined mesh. A convergence studywas hence carried out to ensure appropriate choice of mesh size where the convergence of the naturalfrequencies were studied for the first five modes. The result of the study is presented in Table 3.3 andTable 3.4 for both FE models. As seen in the tables the relative differences between the two mesh sizesthat were studied are small and a mesh size of 0.05 m for both the FE models was considered sufficient.

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Table 3.3: Mesh convergence for the FE model of the Kallhall bridge.

Mode 50 mm 25 mm Rel. Diff

[Hz] [Hz] [%]1st vertical 3.31 3.31 0.041st torsional 4.52 4.52 0.032nd vertical 6.96 6.96 0.041sttransversal 7.19 7.19 0.092nd transversal 7.94 7.94 0.05

Table 3.4: Mesh convergence for the FE model of the Smista bridge.

Mode 50 mm 25 mm Rel. Diff

[Hz] [Hz] [%]1st vertical 2.56 2.56 0.001st torsional 2.77 2.77 0.051st transversal 5.15 5.14 0.052nd vertical 6.34 6.34 0.022nd torsional 6.69 6.69 0.03

3.1.1.1 Assumptions and effects

Models are representations of physical structures and do not represent the structure itself. It is thuscommon to make simplifying assumptions when constructing finite element models. These assumptionsshould however be made in a manner that does not affect the validity of the results. Following are theassumptions made in the modelling procedure which may have a significant contributing effect on theresults.

Inclination of the bridges

Even though the actual length of the bridges was accounted for, neglecting the inclinations in the modelmay affect properties e.g. the distribution of the mass. The inclinations have however been assumed tohave a negligible effect on the results due to their small magnitudes.

Connection and interaction between structural members

In the actual physical structures, all connections are welded to each other. The welding has beenassumed to provide a rigid body connection between the structural members. Furthermore, all boltshave been omitted from the models and consequently the interaction between the bolts and structuralmembers has been neglected.

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3.1.2 Measurements

To validate the FE models, measurements were performed on the Kallhall and Smista bridges wherethe accelerations were measured with two sensors. The measurements were performed for a number ofdifferent load cases, hereafter referred to as tests. The measurements were performed in cooperationwith another thesis work and hence the set-ups for the measurements was decided together to mutuallybenefit both parties. In the following section a description of the set-ups is presented, followed by thetests that were used to excite the bridges. Finally, a description of how the results were evaluated ispresented.

3.1.2.1 Set-ups

The instrumentation used included two three-axial CX1 Structural Response Monitor sensors, whereeach has three accelerometers that measure the acceleration in their respective axis. A GoPro camerawas used to record the different test and various interferences e.g. train and pedestrian passages. Astopwatch and metronome was used to control the frequencies of the excitations. Finally, a dieselgenerator was used as a power source for the computer. In total, seven participants took part in themeasurements, including the authors of this thesis.

Three sensor position layouts were used as illustrated in Figure 3.4. In the first layout (A), two sensorswere placed on each side of the bridge. One sensor was placed at the midspan while the second sensorwas placed 2 m from the midspan. The second sensor was used as a reference sensor and its positionstayed the same for all layouts. In the second layout (B), the first sensor at the midspan was placed atthe first quarter-point of the span. In the third layout (C), the first sensor was placed at the secondquarter-point of the span. On the Kallhall bridge the measurements were performed on the span thatwas considered in the modelling procedure. For the Smista bridge, the measurements were performedon the midspan. Layout A was used for all the tests except for two, which will be described in furtherdetail.

Figure 3.4: Schematic illustration of the sensor layouts where the diamond shape corresponds to the reference sensor.

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3.1.2.2 Tests

The following descriptions of the tests were carried out on both bridges for layout A, if not statedotherwise. Furthermore, during each test there was always one person who took notes of e.g. trainpassages, pedestrian interference and when these occurred. Each test was conducted three times, inorder to capture the free decay of the bridges. Some of the tests were performed more than once dueto a modification in how the test was carried out. Each one of these will be refereed to as a case. Themeasurements were, as previously mentioned, conducted in cooperation with another thesis work werethe author was studying the influence of pedestrians on the dynamic responses. Some measurementshave therefore been performed with the presence of pedestrians. Their mass have however been assumedto be insignificant in comparison to the total mass of the bridge and thus assumed to have a negligibleeffect on the natural frequencies.

Test 1 - Random testing

The first test was performed with random excitations in order to assure that the sensors were workingproperly.

Test 2 - Single jump

The following test was performed for both bridges. In the first case of the test, one person stood at themiddle of the span and jumped in the absence of other participants. In the second case the same personjumped while the other participants stood symmetrically on each side of the bridge at the middle of thespan. The third and fourth case were performed in the same manner as the first but using layout B andC.

Test 3 - Running

Different cases of the running test were performed on the Kallhall bridge and on the Smista bridge. InKallhall, the first case consisted of one person running across the span with a frequency of 2.7 Hz in theabsence of other participants. The second case was carried out in the same manner as the first case, butwith running frequency of 3.3 Hz. One of the sensors did not register any signal and the test was hencecarried out once more and labelled as case three. The fourth and last case was performed when oneperson was running across the bridge with a frequency of 2.7 Hz, with all of the participants standingat the middle of the bridge.

In Smista, the first case consisted of one person running across the bridge with a frequency of 2.7 Hz, inthe absence of other participants. The second case was conducted in the same manner as the first one,but in this case the participants stood symmetrically on each side of the bridge at the middle of thebridge. In the third case one person ran across the bridge with a frequency of 2.6 Hz, in the absence ofother participants.

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Test 4 - Walking

In Kallhall, one person walked across the span with a frequency of 2 Hz, without any other participantstanding on the bridge. In Smista, the same test was performed but in two cases. The first and secondcase were performed when one participant was walking with a frequency of 2 Hz and 2.4 Hz, respectively.Both cases were performed in the absence of other participants.

Test 5 - Synchronized running

Test 5 consisted of synchronized running. In Kallhall all the participants except for one ran in place atthe midspan with a frequency of 3.3 Hz and 4.5 Hz. In Smista all the participants ran in a synchronizedfashion across the bridge in two lines with a frequency of 2.6 Hz.

Test 6 - Synchronized walking

Test 6 consisted of synchronized walking in which all participants walked across the span in two lineswith a frequency of 2 Hz in Kallhall and 2.4 Hz in Smista.

Test 7 - Synchronized bobbing

Test 7 was only performed in Smista and consisted of synchronized bobbing where all of participantsexcept one, was bobbing at the middle of the span with a frequency of 2.4 Hz.

3.1.2.3 Evaluation of results

The experimental natural frequencies were evaluated for all but the first test i.e. for 6 tests, resultingin 12 cases for each bridge. Hence, the results of the natural frequencies presented in section 4.1.1 areaverage values from these tests. The same applies for the damping ratios where the standard deviationalso has been presented due to the scatter of the results. The reader is referred to Appendix A for theresults obtained from each measurement.

For each case, the natural frequencies for the first bending- and torsional mode was evaluated basedon the free vibrations of the bridges. The reason for evaluating the measurements during the freevibration of the bridges was to avoid associating the natural frequencies with other excitations withsimilar frequencies. Both the natural- and excitation frequencies may give rise to peaks in the frequencydomain spectrum. The consequence of capturing excitation frequencies close to the natural frequenciesis that a complication arises in distinguishing them from each other in the frequency domain. Thiscomplication is on the other hand avoided when only evaluating the signals during the period where thebridges are undergoing free vibration. For the tests and cases where the free vibrational signals couldn’tbe detected, the natural frequencies and damping ratios were not evaluated.

The signals were evaluated using MATLAB. Once each free vibrational signal had been detected andextracted from the entire signal, a bandpass filter in the form of a Butterworth filter was designed. Thefilter was designed with a lower cut-off frequency of 2 Hz and a higher cut-off frequency of 5 Hz. The

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signals with frequencies that pass through are those that lie within the frequency-range of 2 to 5 Hz andthe signals with frequencies that lie outside this range are hence rejected. Furthermore, to reduce theeffect of leakage, the signal was processed with a windowing function in the form of a Hann window.The effect of the filter and the window function in the time domain is illustrated in Figure 3.5 for testnumber 7, when performed on Kallhall bridge.

Figure 3.5: Comparison between the unprocessed signal and the processed signals in time domain for test 7 on theKallhall bridge.

Once the signal was filtered and windowed, the signal was converted from its original time domainrepresentation to a representation in the frequency domain using a fast Fourier transform algorithm.Figure 3.6 portrays the effect of the filter and the window function in the frequency domain spectrumfor the same test as above.

Figure 3.6: Comparison between unprocessed signal and processed signal in frequency domain for test 7 on the Kallhallbridge.

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The figure clearly displays that a processed signal in this case yields a much more suitable frequencyspectrum to apply the half-power bandwidth method to, which was the method used to estimate thedamping ratios for the bridges. The reader is referred to section 2.1.2 for a brief description on how toimplement the half-power bandwidth method.

For both the Kallhall and Smista bridges, the natural frequency of the first torsional mode was largerthan the one for the first bending mode. If the excitation frequency is significantly lower than thenatural frequency for the torsional mode, there lies a risk of not obtaining a noticeable peak in thefrequency spectrum for it. The reason is that the torsional mode won’t be excited notably comparedto the bending mode. This was the case when evaluating the results for several of the cases. This wasdealt with by subtracting the signal from the two sensor from each other. The accelerations from theresulting signal will be smaller than the initial one in terms of accelerations with respect to the bendingmode. In contrast, it will be larger than the initial one in terms of accelerations with respect to thetorsional mode. This has to do with how the bridge deforms during the bending- and torsional mode ofvibration. This approach of manipulating the signals will give rise to a more distinct peak of the naturalfrequency of the torsional mode in the frequency spectrum.

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3.2 Verification of pedestrian-induced vibration model

Emma Zall, a Ph.D. student at the Royal Institute of Technology, Sweden, has developed a model thatcalculates pedestrian-induced vertical vibrations with respect to the first bending mode. The model,which is implemented in MATLAB is based on modal analysis and is composed of a single degree offreedom system. The model will in the remainder of this thesis be referred to as the 2D SDOF model.The reader is referred to section 2.1.4 for a description of the theory behind modal analysis.

Based on the 2D SDOF model, a new model was developed. Analogously, this model calculates thepedestrian-induced torsional vibrations with respect to the first torsional mode. The procedure ofdeveloping this model involved a transformation from a system in two-dimensions, as was the case forthe 2D SDOF model, to the three-dimensional space and the model has thus been named the 3D SDOFmodel.

The following section will include a description of the 2D SDOF model followed by how the model wasverified against a simple FE model before using it to develop the 3D SDOF model. The procedure fordoing so is presented along with the verification process of the 3D SDOF model against the FE modelsof the Kallhall and Smista bridges.

3.2.1 Description of the 2D SDOF model

The 2D SDOF model, as previously mentioned, calculates the vertical accelerations that arise when oneor more pedestrians walk along a line on a bridge as illustrated in Figure 3.7.

Figure 3.7: Pedestrians, marked with circles, walking over a bridge in a line.

The model can thus be idealized to calculate the vertical accelerations that arise when one or morepedestrians walk along a two-dimensional simply supported beam, as shown in Figure 3.8. The modelis thus limited to only determining the response along the line where the pedestrians are walking and ishence not able to account for eccentric effects.

Figure 3.8: Simply supported beam with pedestrians walking along it.

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The procedure for which the model calculates the accelerations is shown in Figure 3.9. The model hastwo main categories of input, the first being the input parameters of the bridge and the second beingthe parameters of the pedestrians loading the structure.

Figure 3.9: A overview of the structure of the bending mode model.

As shown in Figure 3.9 the first parameters of the bridge properties are the natural frequency and theanalytical expression of the bending mode of vibration. The first bending mode shape is for a simplysupported span with a uniform cross section and distributed mass described by a sinusoidal functionaccording to Eq. 3.1 [3]

φ(y) = sin(πy

L

)(3.1)

where y is the position along the length of the bridge and L is the length of the bridge. The naturalfrequency and modal mass can either be obtained using the procedures described in section 2.1 or bythe use of FE software which was the case in this thesis. Furthermore, the modal damping and modalstiffness are determined using Eq. 2.26 and 2.27, respectively.

The second main input category are the properties of the pedestrians. These are described by a mass

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and a walking frequency from which the force that each pedestrian induce, Fp, is calculated with aFourier series according to Eq. 3.2 [13].

Fp(t) = mpg +N∑n=1

mpgansin(2πnfpt) (3.2)

where mp is the pedestrian’s mass, g is the gravitational acceleration which was set to 9.81 m/s2, t is thetime and fp is the walking frequency of the pedestrians. Furthermore, n is the regarded harmonic andan is the load factor for the harmonic n. In this thesis four harmonics were used and the load factorsfor each harmonic were calculated according to [13] as:

a1 = 0.41(fp−0.95) a2 = 0.069+0.0056fp a3 = 0.033+0.0064fp a4 = 0.013+0.0065fp (3.3)

Based on the force each pedestrian induce, the initial acceleration is calculated from the equation ofmotion. The acceleration is then solved in each time instant with Newmark’s method with respect tothe position of the pedestrians in each time instant. In this model the pedestrians are modelled suchthat they walk in a straight line with a certain distance from one another, the inter-pedestrian distance.Furthermore, the position of each pedestrian is defined with respect to their walking velocity, inter-pedestrian distance and the total length of the bridge, all of which are constant in time. Noteworthy,is that in this thesis, the results presented are only for when one pedestrian is considered, with thefollowing properties:

fp = 2 Hz, mp = 70 kg, vp = 1.25 m/s2

where vp corresponds to the walking velocity of the pedestrian.

Even though only one pedestrian has been considered, both the 2D SDOF and 3D SDOF models doprovide the ability to consider several pedestrians. However, in this thesis it was found sufficient tostudy the effect of only one pedestrian.

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3.2.2 Verification of the 2D SDOF model

The verification process began by firstly verifying the 2D SDOF model by comparing the results interms of vertical acceleration, with the result from a simple FE model. The results were based on onepedestrian modelled to move across the middle of the bridge with the previously mentioned properties.Furthermore, an arbitrary modal damping ratio of 1% was used in both the 2D SDOF model and thesimple FE model.

3.2.2.1 Simple FE model

The model that was created to verify the 2D SDOF model is a simple rectangular shell model representinga bridge and is presented in Figure 3.10. The bridge is 2.5 m wide, 30 m long and 1 m thick. The materialproperties defined for the bridge are as followed:

• Density, ρ = 2500 kg/m3

• Modulus of elasticity, E = 33 GPa

• Poisson’s ratio, ν = 0.2

Figure 3.10: The simple shell model, where the orange figures represent the nodes to which boundaries were applied.

The FE model is simply supported and the boundaries were applied directly to the four corner nodes, asshown in Figure 3.10. The model was meshed with a size of 0.125 m, obtained from a mesh convergenceanalysis for which the natural frequencies for the first bending- and torsional mode were studied.

3.2.2.2 Implementing pedestrian loading in the FE model

As previously described for the 2D SDOF model, one pedestrian was modelled to move along a line atthe middle of the bridge. The implementation of a moving pedestrian load in Brigade was performedby modelling the load as several concentrated forces applied at a number of nodes along the line thepedestrian is moving. Thus, a line was created in the FE model to represent the path the pedestrian iswalking along. This line which is placed along the middle of the FE model was partitioned in order toobtain nodes along its length. A script was written in order to express the variation of the load accordingto Eq. 3.2 for each concentrated force as a pedestrian moves over the bridge. The value of the load as apedestrian lied between two nodes, as shown in Figure 3.11, was interpolated linearly according to Eq.3.4.

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Fi = FpLel − xpLel

Fi+1 = FpxpLe

(3.4)

Figure 3.11: The value of the force induced by a pedestrian Fi and Fi+1 in node i and node i + 1 as a pedestrian Fp

moves between node i and node i + 1. xp corresponds to the distance the pedestrian has moved from node i and Lel

corresponds to the entire length of the element.

The loads were applied in a modal dynamics step which followed a frequency extraction step. In thefrequency extraction step, a frequency range was defined to account for the corresponding mode shapeof interest. Hence, the effect from a particular mode could be isolated.

3.2.2.3 Comparison of results between the 2D SDOF model and the FE model

The accelerations from the 2D SDOF model and the FE model were evaluated at the center of thebridge. The result obtained from the models is presented in Figure 3.12.

Figure 3.12: Comparison of results in terms of vertical vibrations between the 2D SDOF model and the FE model.

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The maximum acceleration obtained at the evaluated point was for the 2D SDOF model and FE modelequal to 18.96 mm/s2 and 19.03 mm/s2, respectively. The relative difference between the maximumaccelerations is 0.37 % and the 2D SDOF model was hence determined to accurately represent thedynamic response of the first bending mode and the development of the 3D SDOF model could thusbegin.

3.2.3 Development of the 3D SDOF model

The implementation of the 3D SDOF model was, as for the 2D SDOF model, made using MATLAB.As previously mentioned, the 3D SDOF model accounts for torsional vibrations in contrast to the 2DSDOF model which accounts for vertical vibrations. A torsional vibration mode deforms in a way that,when the left side of a bridge rises, the right side moves in the opposite direction and vice versa. Thatis, the two halves of a bridge span are twisted in opposite directions with the center line of the bridgemostly remaining undeformed. This is illustrated in Figure 3.13 which represents the undeformed shapeof the simple FE model to the left and its deformed shape to the right when the first torsional mode isexcited.

Figure 3.13: Underformed and deformed shape of the simple FE model when the first torsional mode is excited.

The 3D SDOF model calculates the acceleration in the same manner as for the 2D SDOF model andthe general description provided for in Figure 3.9 is thus also valid for the 3D SDOF model. Themajor difference lies in that one now has to account for displacements along both the longitudinal andtransversal direction of the bridge. Hence, a polynomial of two variables is needed to describe thetorsional mode. The mode shape in this model is thus described as a surface unlike the 2D SDOF modelwhere the mode shape was described with a polynomial of one variable i.e. a line. Hereafter, the referredbridges concerns the simple FE model and the FE models of the Kallhall and Smista bridges.

The torsional mode shape was obtained assuming a linear displacement in the transverse direction, alongthe length of the bridges. Let x define the points along the transversal direction of the bridges accordingto

0 ≤ x ≤ b (3.5)

where b corresponds to the width of the bridges. Furthermore, let y define the points along the longitu-dinal direction of the bridges according to

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0 ≤ y ≤ L (3.6)

where L corresponds to the total length of the bridge. A two-dimensional approximation of the torsionalmode shape is obtained by assuming that Euler-Bernoulli beam theory is valid i.e. plane sections normalto the cross section remain plane and normal after deformation. Thus, the inclination between twoarbitrary points in the transversal direction of the bridge along a certain point in the longitudinaldirection is always constant. The displacement of the mode shape in the transversal direction can bedescribed by a linear function varying from the displacement at x = 0 to x = b. Furthermore, if thedisplacements along the longitudinal direction of the bridge at x = 0 and x = b are given by the functionsu0(y) and ub(y), respectively, then the inclination-coefficient in the transversal direction, k(y), is givenas

k(y) = u0(y)− ub(y)b

(3.7)

The vertical displacement of the torsional mode shape can thus be described as a function according to

y = u0(y)− k(y)x (3.8)

The functions u0(y) and ub(y) have been obtained by extracting the displacements of the torsional modeat x = 0 and at x = b in the respective FE model. By importing the data to MATLAB, an analyticalfunction could be fitted to the data.

The presented approach of evaluating the mode shape is based on, as previously mentioned, that theinclination in the transversal direction is constant. This is based on the assumption that the anglebetween the undeformed- and deformed shape of the respective deckplates are constant for all values ofx along each point y. The angles for the respective deckplates of the FE models are presented in Figure3.14.

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Figure 3.14: Rotation around the longitudinal direction, UR2, given in radians for the simple FE model (left) and theFE models of the Kallhall bridge (middle) and the Smista bridge (right).

As can be observed, the angle is approximately constant. Hence, the bridges satisfy the Euler-Bernoullitheorem reasonably.

With the torsional mode shape now being represented by a surface, the 3D SDOF model is able todescribe the moving pedestrians along all points on the surface. Analogously, the accelerations can beevaluated in every point on the bridge, independently of the position of the pedestrians.

3.2.4 Verification of the 3D SDOF model

The 3D SDOF model was firstly validated against the simple FE model, previously described andsecondly against the FE models of the Kallhall and Smista bridges, for which the results are presentedin section 4.2.

The implementation of the moving load was performed as previously described in section 3.2.2.2 withone exception. The pedestrian was now modelled to move across the bridge on the far right edge ofthe simple FE model and on the far right edge of the deckplates for the FE models of the Kallhall andSmista bridges. The reason being that the torsional mode is excited at most when a dynamic load isapplied at the edge.

As mentioned in previous sections, the input parameters needed for the 3D SDOF model are the naturalmodes of vibration, natural frequencies, modal mass and damping ratios. The mode shapes were asexplained in the previous section obtained from the FE models along with the natural frequency andmodal mass. The damping ratio for the simple FE model was set to 1 % and the damping ratios usedfor the Kallhall and Smista bridges were those obtained from the measurements.

The results, again in terms of acceleration, were evaluated at the midpoint, in the longitudinal direction.In the transverse direction, the results were evaluated at five points i.e. at the left-, left quarter- , mid-,right quarter- and right points. Figure 3.15 illustrates the points where the results were evaluated.

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Figure 3.15: The points where the accelerations were evaluated in the 3D SDOF model and the FE models.

Lastly, for the 2D SDOF- and 3D SDOF models, convergence studies of the time increment were per-formed in order to choice a suitable time increment in both the MATLAB and FE implementation.In the convergence study the maximum acceleration in the different points were studied for varioustime increments. The time increments for the models were chosen if the relative difference between thetime increments were equal to or less than 0.5 %. The results of the convergence study is presented inAppendix B.

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Chapter 4

Results

As in the method section, this section consist of two parts. In the first part the results regarding thefrequencies and damping ratios from the measurements and from the FE models are presented, hereafterreferred to as experimental natural frequencies and theoretical natural frequencies, respectively. In thesecond part the results from the verification process of the 3D SDOF model are presented.

4.1 Result from the validation of the FE models

4.1.1 Experimental natural frequencies

In the following, the experimentally evaluated natural frequencies and damping ratios for both theKallhall and Smista bridges are presented in Table 4.1 and Table 4.2.

Table 4.1: The experimentally evaluated natural frequencies for the Kallhall and Smista bridges.

Mode Kallhall [Hz] Smista [Hz]

1st vertical 3.37 2.441st torsional 4.48 2.71

Table 4.2: The experimentally evaluated damping ratios and the standard deviation for the Kallhall and Smista bridges.

Mode Kallhall [%] Smista [%]

1st vertical 0.73± 0.36 0.87± 0.311st torsional 0.56± 0.22 0.93± 0.29

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4.1.2 Theoretical natural frequencies

The theoretical natural frequencies of both bridges are in the following presented in Table 4.3.

Table 4.3: The theoretical natural frequencies for the Kallhall and Smista bridges.

Mode Kallhall [Hz] Smista [Hz]

1st vertical 3.31 2.561st torsional 4.52 2.77

The respective modes for both bridges are also presented in Figure 4.1 to 4.4.

Figure 4.1: First bending mode of the Kallhall bridge.

Figure 4.2: First torsional mode of the Kallhall bridge.

Figure 4.3: First bending mode of the Smista bridge.

Figure 4.4: First torsional mode of the Smista bridge.

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4.1.3 Comparison between experimental and theoretical results

In Table 4.4 the relative differences are presented between the experimental and theoretical naturalfrequencies for both mode shapes and both bridges. It can be seen that there are only small differencesbetween the experimental and theoretical results and moreover that the largest difference of 5 %, isobtained for the first vertical mode of the Smista bridge while the smallest difference of mere 0.8 %, isobtained for the first torsional mode of the Kallhall bridge.

Table 4.4: Comparison between the experimental and theoretical natural frequencies.

Mode Kallhall [%] Smista [%]

1st vertical 2.03 51st torsional 0.8 2.1

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4.2 Results from the verification of the 3D SDOF model

In the following section the results are presented for the validation of the 3D SDOF model. The resultsfrom the 3D SDOF model compared to the simple FE model are firstly presented following the resultsfrom the 3D SDOF model and the FE models of the Kallhall and Smista bridges, respectively. In eachcase the results consist of tables for which the largest absolute acceleration in the previously mentionedpoints are presented along with the relative differences in each points. Additionally, the complete time-history analysis of the accelerations at the respective points are presented.

4.2.1 Verification against the simple FE model

Table 4.5: The largest absolute accelerations in the respective models at the different points and the relative differences.

Position FE model [m/s2] 3D SDOF [m/s2] Diff [%]·10−4 ·10−4

Left point 1.69 1.7 0.25Left quarter-point 0.85 0.85 0.2Midpoint 1.87 · 10−4 0 -Right quarter-point 0.85 0.85 0.24Right point 1.69 1.7 0.27

Figure 4.5: Comparison of results obtained from the simple FE model and the 3D SDOF model, at the left point.

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Figure 4.6: Comparison of results obtained from the simple FE model and the 3D SDOF model, at the left quarter-point.

Figure 4.7: Comparison of results obtained from the simple FE model and the 3D SDOF model, at the midpoint.

48

Figure 4.8: Comparison of results obtained from the simple FE model and the 3D SDOF model, at the right quarter-point.

Figure 4.9: Comparison of results obtained from the simple FE model and the 3D SDOF model, at the right point.

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4.2.2 Verification against the FE model of the Kallhall bridge



Left point 4.64 4.71 1.44Left quarter-point 2.57 2.54 1.35Midpoint 1.59 · 10−2 1.22 · 10−2 23.25Right quarter-point 2.58 2.56 0.78Right point 4.75 4.73 1.44

Noteworthy is that the relative difference of 23.25 % in the midpoint is due to the small magnitude ofthe values. More importantly, is that one can observe that from both models the acceleration in themidpoint is almost zero, as expected.

Figure 4.10: Comparison of results obtained from the FE model of the Kallhall bridge and the 3D SDOF model, at theleft point.

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Figure 4.11: Comparison of results obtained from the FE model of the Kallhall bridge and the 3D SDOF model, at theleft quarter-point.

Figure 4.12: Comparison of results obtained from the FE model of the Kallhall bridge and the 3D SDOF model, at themidpoint.

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Figure 4.13: Comparison of results obtained from the FE model of the Kallhall bridge and the 3D SDOF model, at theright quarter-point.

Figure 4.14: Comparison of results obtained from the FE model of the Kallhall bridge and the 3D SDOF model, at theright point.

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4.2.3 Verification against the FE model of the Smista bridge



Left point 1.39 1.38 0.77Left quarter-point 0.77 0.77 0.57Midpoint 0.12 0.12 3.81Right quarter-point 1.01 1 1.22Right point 1.63 1.62 1

Figure 4.15: Comparison of results obtained from the FE model of the Smista bridge and the 3D SDOF model, at theleft point.

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Figure 4.16: Comparison of results obtained from the FE model of the Smista bridge and the 3D SDOF model, at theleft quarter-point.

Figure 4.17: Comparison of results obtained from the FE model of the Smista bridge and the 3D SDOF model, at themidpoint.

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Figure 4.18: Comparison of results obtained from the FE model of the Smista bridge and the 3D SDOF model, at theright quarter-point.

Figure 4.19: Comparison of results obtained from the FE model of the Smista bridge and the 3D SDOF model, at theright point.

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Chapter 5

Discussion and conclusions

In this section, the result presented in section 4 will be analysed and discussed following suggestions offurther studies.

5.1 Validation of the FE models

As can be seen from Table 4.1 and Table 4.3, the results obtained from the FE models corresponds to theresults obtained from the measurements with small relative differences. Hence, the dynamic propertiesof the bridges are validated in terms of their natural frequencies. Consequently, the FE models are alsovalidated in this sense. Noteworthy, is that because the models corresponded well to the measurementregarding the natural frequencies, the mode shapes obtained from the FE models are also assumed tobe representative of the real mode shapes. In this thesis, the mode shapes were not validated as theresources were limited to two sensors.

It is interesting to understand why the natural frequencies of the FE models corresponded well to themeasurements and what the differences are between the FE models and the real bridges, as there aremany properties of the bridges that needs to be accounted for. Regarding the geometrical properties,great effort has been devoted in modelling the different details that exist in the bridges as accuratelyas possible. All the stiffening plates in e.g. the frame structure of the Kallhall bridge and the stiffeningplates in the support of the Smista bridge has been accounted for. Though, as mentioned all theinclinations of both bridges were neglected as well as the arch for the Smista bridge. These, along withthe previously mentioned assumptions in section 3.1.1.1 did not result in large differences regarding thenatural frequencies. However, if the inclinations of both bridges and the arch of the Smista bridge wasto be included it would most likely result in even smaller differences as the inclinations affect how thetotal mass is distributed and consequently the natural frequencies of both bridges.

Furthermore, as previously mentioned, some non-structural members were considered to have a signif-icant contribution to the mass and was thus accounted for. However, for these members their stiffnesscontribution was neglected. In reality, all members have some contribution to the overall stiffness. Twoexamples are the curbstone on the Kallhall bridge which is of concrete and spans along both sides ofthe deckplate along its length. The second example is the asphalt pavement along the deckplate of theKallhall bridge. The stiffness of the asphalt was neglected but an equivalent density was accounted for.In reality, the asphalt does contribute to the stiffness which also varies with the season of the year,

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increasing during the winter season.

Regarding the material properties there are less uncertainties in the behaviour of steel compared to e.g.asphalt or concrete. Unlike asphalt, the properties of steel do not change considerably with time and incomparison, to concrete, deterioration is of less concern. Both the Kallhall and Smista bridges are madeof steel and thus the modelling procedure is simplified in terms of describing the material behaviour.

Lastly, the behaviour of the bearings and supports are also an important modelling aspect. As presentedin section 3.1.1 the influence of the bearings was studied. As shown in Table 3.1, a change in themovements of the bearings did not affect the natural frequencies for the Kallhall bridge at all. Thus,one can conclude that the differences obtained between the measurements and the FE model of theKallhall bridge do not depend on the behaviour of the bearings. Furthermore, as previously mentioned,the piers are resting on bedrock and assumed to be massive undeformable rigid bodies. The resultsshows that omitting the piers from the analysis led to reasonably accurate results and the assumptionregarding the piers has thus been determined to be valid. For the Smista bridge, where the results arepresented in Table 3.2, one can conclude that if the bearings where modelled with properties as thosedescribed in the table, smaller differences would have been obtained.

5.2 Verification of the 3D SDOF model

The verification of the 3D SDOF model was based on the validated FE models. Based on the validityof the FE models they were also assumed to be valid in terms of analysing the dynamic response ofdynamic loading. Hence, from the basis of this assumption and the results presented in section 4.2, onecan conclude that the 3D SDOF model does accurately capture the dynamic response of pedestrian-induced torsional vibrations. An important factor is that the results are also based on the assumptionthat the deckplate of each FE model satisfy the Euler-Bernoulli theorem, i.e. plane sections normal tothe cross section remain plane and normal after deformation. This implies that the angle between theundeformed- and deformed shape of the respective deckplates is constant along the transverse direction.However, as shown in Figure 3.14 for the Kallhall bridge, the angle was not completely constant in thetransverse direction of the deckplate in the model, which might be due to the deformation of the framestructure and the stiffening beams. In comparison to Figure 3.14 for the Smista bridge, where the angle isalmost constant in the transverse direction throughout the length, except at the skewed supports. Thus,the differences presented in section 4.2 might depend on the accuracy in the approximation of the modeshape. Hence, for cases where the deckplate of a given structure does not satisfy the Euler-Bernoullitheorem at all, an alternative approximation of the mode shape might be necessary to implement.

Lastly, one might argue why the results from the 3D SDOF model were not compared to the results fromthe measurements in order to have verified but also validated the 3D SDOF model. The reason for this isthat at the time of the planning of the measurements it was not known in beforehand that the second partof this thesis would consist of the analysis of pedestrian-induced torsional vibrations. Hence, the set-upsfor the measurements were not decided based on analysing pedestrian-induced torsional vibrations.

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5.3 Further research

The 3D SDOF model is verified against validated FE models but without consideration to pedestrian-structure interaction which exist in reality. Hence, further studies need to be performed in analysingthe effect of the interaction for torsional vibrations with the proposed model. Furthermore, the modalmasses and the approximation of the mode shapes are based on the deformations obtained from the FEmodels. To further improve the model, from a practical point of view, further studies on the utilizationof approximative mode shapes of torsional modes could be performed along with a method of obtainingthe modal mass based on the mode shape. The torsional mode shape could e.g. be based on tables orsimplified functions. From a practical point of view this would imply that the torsional vibrations for agiven structure due to pedestrian loading can be analysed independently of a FE model. Thus, the onlyinput parameters would be the natural frequency of the torsional mode of vibration and the propertiesof the bridge. The natural frequency could be determined based on existing formulas or by conductingmeasurements on existing bridges. In the case of the former, from a practical point of view, this wouldbe highly practical and time-efficient in analysing pedestrian-induced torsional vibrations.

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[13] Zall, E., Anderson, A., Ulker-Kaustell,M., Karoumi,R., 2017. An efficient approach for consideringthe effect of human-structure interaction on footbridges. X International Conference on StructuralDynamics, EURODYN 2017.

Appendix A

Measurement results

The natural frequencies and damping ratios obtained from the measurements for the first bending- andtorsional mode are presented here. The reader is referred to section 3.1.2.2 for the content of what eachtest and case represents. Noteworthy, is that test 1 was solely performed for assuring that the sensorsworked properly and the data provided from it was thus not evaluated. Furthermore, two sensors wereused for the measurements and the results obtained from both of them in terms of natural frequenciesand damping ratios were identical. Thus, only the results obtained from one sensor are presented.

Table A.1 presents the results obtained for the Kallhall bridge from all conducted measurements.

Table A.1: Results obtained for the Kallhall bridge in terms of natural frequencies and damping ratios for the 1st bendingand torsional mode.

Natural frequency [Hz] Damping ratio [%]1st 2nd 1st 2nd

Test 1 - Case 1 - - - -Test 2 - Case 1 3.40 4.50 0.81 0.48Test 2 - Case 2 - - - -Test 2 - Case 3 3.38 4.50 0.30 0.19Test 2 - Case 4 - - - -Test 3 - Case 1 3.40 4.52 0.66 0.88Test 3 - Case 2 3.37 - 0.40 -Test 3 - Case 3 3.38 4.46 1.68 0.75Test 3 - Case 4 3.37 4.44 0.73 -Test 4 - Case 1 3.36 4.48 0.74 0.55Test 5 - Case 1 3.35 4.45 0.67 0.66Test 5 - Case 2 3.36 4.50 0.83 0.68Test 6 - Case 1 3.38 4.50 0.50 0.27Average 3.37 4.48 0.73 0.56

In ”Test 2 - Case 2” and ”Test 2 - Case 4”, the free vibrational signal of the bridge could not be detectedand the results were thus not evaluated based on reasons given in section 3.1.2.3. In ”Test 3 - Case2”, the natural frequency of the first torsional mode could not be detected nor the damping ratio as aconsequence. In ”Test 3 - Case 4”, the shape of the spectral representation of the torsional mode did

not allow for a proper use of the half-power bandwidth method and the damping ratio was thus notassessed. The results obtained for the Smista bridge are presented in Table A.2.

Table A.2: Results obtained for the Smista bridge in terms of natural frequencies and damping ratios for the 1st bendingand torsional mode.

Natural frequency [Hz] Damping ratio [%]1st 2nd 1st 2nd

Test 1 - Case 1 - - - -Test 2 - Case 1 2.45 2.73 0.63 0.71Test 2 - Case 2 2.41 2.70 0.62 0.80Test 2 - Case 3 2.44 2.72 1.10 0.55Test 2 - Case 4 - - - -Test 3 - Case 1 - - - -Test 3 - Case 2 - - - -Test 3 - Case 3 2.47 2.73 0.99 0.88Test 4 - Case 1 - - - -Test 4 - Case 2 2.45 - 0.68 -Test 5 - Case 1 2.45 2.70 0.41 0.86Test 6 - Case 1 2.46 2.71 1.25 1.26Test 7 - Case 1 2.39 2.67 1.28 1.44Average 2.44 2.71 0.87 0.93

For ”Test 4 - Case 2”, the natural frequency of the first torsional mode could not be detected and thusneither the corresponding damping ratio. For all other tests where values were not obtained, the freevibrational signal could not be detected and the signals was thus not evaluated.

Appendix B

Convergence analysis

The time increment convergence in both the FE models, the 2D SDOF model and the 3D SDOFmodel are presented in this appendix, with respect to the largest absolute accelerations. The resultsconvergence with the same rate in all of the evaluated points. Hence, the results are only presented forone point. In each table the convergence for both the FE model and the respective MATLAB model ispresented simultaneously.

Table B.1: Result of the convergence study at the mid-point for the simple FE model and the 2D SDOF model.

Simple FE model

Time increment [s] 0.005 0.0025Largest absolute acceleration [m/s2] 0.018957 0.018961Relative difference [%] 0.020561

2D SDOF model

Time increment [s] 0.005 0.0025Largest absolute acceleration [m/s2] 1.90E-02 0.019031Relative difference [%] 0.296509

Table B.2: Result of the convergence study at the right-point for simple FE model and the 3D SDOF model.

Simple FE model

Time increment [s] 0.005 0.0025 0.00125 0.000625Largest absolute acceleration [m/s2] 0.000161 0.000167 0.000169 0.000169Relative difference [%] 3.639384 0.895044 0.230488

3D SDOF model

Time increment [s] 0.005 0.0025 0.00125 0.000625Largest absolute acceleration [m/s2] 1.71E-04 0.00017 1.70E-04 1.70E-04Relative difference [%] 0.487451 0.121107 0.03023

Table B.3: Result of the convergence study at the right-point for FE model of the Kallhall bridge and the 3D SDOFmodel.

FE model of the Kallhall bridge


3D SDOF model


Table B.4: Result of the convergence study at the right-point for FE model of the Smista bridge and the 3D SDOFmodel.

FE model of the Smista bridge


3D SDOF model


TRITA BKN Master Thesis 509, Division of Structural Engineering and Bridges 2017

ISSN 1103-4297

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