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

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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 TECHNOLOGY

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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 accuracy is of great importance. However, more often than not, there are differences between FE models and the structures being modelled, which can depend on numerous factors. These factors can consist of uncertainties in material behaviour, geometrical properties and boundary- and continuity conditions.

Model validation is therefore an important aspect in obtaining FE models that represents reality to some degree. Furthermore, model verification is also important in terms of verifying theoretical models, other than FE models, in fields such as fatigue-, fracture- and dynamic analysis.

In this thesis, two pedestrian steel bridges, the Kallh¨ all bridge and the Smista bridge, have been modelled in a FE software based on engineering drawings and validated against experimental results with regard to their natural frequencies. Furthermore, in this thesis, a model has been developed in MATLAB based on modal analysis that accounts for pedestrian-induced torsional vibrations, the 3D SDOF model. This model has been verified against the previously mentioned FE models.

The aim of this thesis is hence two parted where the first part is to develop three-dimensional FE models of two pedestrian bridges and validate them against measured data regarding the natural frequencies.

The second part is to further develop a model for analysing the effect of pedestrian-induced torsional vibrations and to investigate whether the model captures the actual dynamic response of such loading.

The results showed that the natural frequencies for the first bending- and torsional mode from the FE

models corresponded well to the measured ones with the largest difference of 5 % obtained for the natural

frequency of the first bending mode for the Smista bridge. Furthermore, the 3D SDOF model was able

to capture the dynamic response of torsional vibrations with an overall difference of less than 2 % in

comparison to the FE models. The model can be improved by further studying the pedestrian-structure

interaction as well as studying the effect of using approximative functions describing the mode shapes.

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Sammanfattning

Finita elementmodeller (FE-modeller) utg¨ or en bas f¨ or m˚ anga olika typer av analyser som exempelvis dy- namiska analyser. D¨ armed ¨ ar det av stor betydelse att FE-modeller representerar det faktiska beteendet av verkliga strukturer med stor noggrannhet. Ofta ¨ ar det emellertid skillnader mellan FE-modeller och de verkliga strukturer man modellerar. Dessa skillnader kan bero p˚ a en rad faktorer s˚ asom exempelvis os¨ akerheter i materialbeteende, geometriska egenskaper samt upplag- och randvillkor. Modellvalidering

¨

ar d¨ arf¨ or en viktig aspekt i att erh˚ alla FE-modeller som representerar verkligheten i olika omfattningar.

Ut¨ over modellvalidering ¨ ar ¨ aven modellverifiering viktigt, inte endast f¨ or verifiering av FE-modeller utan ¨ aven f¨ or verifiering av andra teoretiska modeller inom omr˚ aden s˚ asom utmaning-, fraktur- och dynamiska analyser.

I detta arbete har tv˚ a GC-broar, Kallh¨ all- och Smistabron modellerats i ett FE-program baserat p˚ a konstruktionsritningar och validerats mot experimentella resultat med avseende p˚ a de naturliga frekvenserna. Vidare har det i detta arbete utvecklats en modell i MATLAB som tar h¨ ansyn till m¨ annisko-inducerade torsionsvibrationer baserat p˚ a modalanalys, ben¨ amnd 3D SDOF modellen. Mod- ellen har ¨ aven verifierats mot de tidigare n¨ amnda FE-modellerna.

M˚ alet med detta arbete ¨ ar s˚ aledes uppdelat i tv˚ a delar, d¨ ar den f¨ orsta delen best˚ ar av att utveckla tredimensionella FE-modeller av tv˚ a GC-broar samt validera dessa mot m¨ atdata vad g¨ aller de naturliga frekvenserna. Den andra delen best˚ ar av att utveckla en modell f¨ or att analysera effekten av m¨ annisko- inducerade torsionsvibrationer och unders¨ oka huruvida modellen f˚ angar den dynamiska responsen.

Resultaten visade att de naturliga frekvenserna f¨ or den f¨ orsta b¨ oj- och vridmoden fr˚ an FE-modellerna

motsvarade de uppm¨ atta frekvenserna med en st¨ orsta relativ skillnad p˚ a 5 % f¨ or den f¨ osta b¨ ojmoden f¨ or

Smistabron. Vidare visade resultaten att den utvecklade 3D SDOF modellen kunde f˚ anga den dynamiska

responsen av torsionsvibrationer med en skillnad p˚ a mindre ¨ an 2 % i j¨ amf¨ orelse med resultat fr˚ an de

FE-modellerna. Modellen kan f¨ orb¨ attras genom att vidare studera interaktionen mellan fotg¨ angare och

g˚ angbro samt studera effekten av att anv¨ anda approximativa funktioner som beskriver modformen.

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Preface

The work presented in this master thesis was initiated by the engineering consultancy Tyr´ ens AB and the 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-Kaustell for all his valuable advice and guidance throughout the work of this thesis but also for all the enthusiastic and enlightening conversations.

We are truly grateful to Ph.D. student Emma Z¨ all for taking her time to provide us with assistance and for letting us take part of her research.

Furthermore, we would like to thank everyone at the bridge department at Tyr´ ens AB for allowing us to carry out our work at a welcoming place with such a genuine hospitality and for making it an enjoyable time. A special thank goes to Viktor Tell and Joakim Kyl´ en for always finding the time to assist us with modelling issues.

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

Stockholm, June 2017

Simon Chamoun

Marwan Trabulsi

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List of notations

Notation Description Unit

a

_n

Load factor -

a

_j

Coefficient in Fourier series -

b Width of bridge m

b

_j

Coefficient in Fourier series -

c Damping Ns/m

c Damping matrix Ns/m

C Square damping matrix Ns/m

C Generalized damping Ns/m

E Modulus of elasticity N/m

²

F Force N

F Force vector N

f Periodic function -

F (w) Fourier transform of f -

f

_a

Aliasing frequency Hz

f

_n

Natural frequency Hz

F

_n

Generalized force N

f

_p

Walking frequency of a pedestrian Hz

F

_p

Pedestrian induced force N

f

_s

Sampling rate Hz

g Gravitational acceleration m/s

²

k Stiffness N/m

k Stiffness matrix N/m

K Square stiffness matrix N/m

K Generalized stiffness N/m

L Length of bridge m

L

_el

Element length m

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m Mass kg

m Mass matrix kg

M Square mass matrix kg

M Generalized mass kg

m

_p

Pedestrian mass kg

N Number of samples -

p Excitation function -

q Modal displacement m

t Time s

T Total sampling time s

T

₀

Period of harmonic function s

T

_n

Natural period s

u Displacement m

v

_p

Walking velocity of a pedestrian m/s

x

_p

Distance from node m

β Coefficient in Newmarks method -

γ Coefficient in Newmarks method -

∆t Sampling interval s

ν Poisson’s ratio -

ξ Damping ratio %

ρ Density kg/m

³

Φ Modal matrix -

φ

_n

Natural mode -

ω

₀

Cyclic frequency rad/s

ω

_n

Natural cyclic frequency rad/s

ω

_nD

Damped natural cyclic frequency rad/s

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Chapter 1 Introduction

1.1 Background

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

However, even though finite element modelling (FEM) is widely used in today’s society as an effective analytical 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 occur due 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. bearings and hinges which complicates the boundary- and continuity conditions. All material properties are not constant 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 are difficult to evaluate, as the amount of detail in the geometry varies from member to member [1]. In order to ensure that FE models corresponds to measured responses, model validation and possibly also calibration is required.

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

Two pedestrian steel bridges situated in Kallh¨ all and S¨ atra, northwest respectively south of central

Stockholm, Sweden has been modelled and validated against experimental results. The former is a

simply supported, three span bridge where the respective spans are 40 m, 33.6 m and 40 m, as presented

in 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 S¨ atra. The respective

spans are 19.6 m, 47.5 m and 16.9 m and are presented in Figure 1.2. Both bridges are described more

detailed in sections 1.4 and 1.5.

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Figure 1.1: Elevation drawing of the Kallh¨ all 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, where one ensures that the model behaves as intendent. Model verification is thus beneficial in a wide range of fields e.g. fatigue-, fracture- and dynamic analysis [9].

In this thesis a model accounting for pedestrian-induced vibrations in pedestrian bridges with respect to the first torsional mode has been verified against the previously mentioned FE models. The model is based on the work of a current Ph.D. student Emma Z¨ all, whose research involves the development of improved 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 of two pedestrian bridges and validate them against measured data regarding the natural frequencies. The second part is to develop a model for analysing the effect of pedestrian-induced torsional vibrations and to investigate whether the model captures the actual dynamic response of such loading by validating it against 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-induced

torsional 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 Kallh¨ all bridge

The Kallh¨ all bridge is a pedestrian bridge located in Kallh¨ all, northwest of Stockholm, Sweden and was built in 2015, as a part of the rebuild of the Kallh¨ all commuter station during 2014-2016. The bridge is a simply supported, three span bridge where the respective lengths are 40 m, 33.6 m and 40 m. Each span 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 and various 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 Figure 1.3. 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 Figure 1.1. The center-to-center (c/c) distance between the frame structures is 3640 mm in the longitudinal direction. Furthermore, four main beams run longitudinally along each side of the frame, two in the upper and lower part of the cross-section, as shown in Figure 1.3.

Figure 1.3: The cross-section between the supports at the Kallh¨ all 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-shaped crossbeams, where the two crossbeams at the middle of the bridge have twice the web thickness in comparison to the others.

The cross-section at both ends of the span is presented in Figure 1.4, which have the same dimensions

as the cross-section between the supports, except for the height of the frame structure. From Figure 1.4

it can be seen that the bridge deck is supported by plates inside a box section. There are also stiffening

plates 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 Kallh¨ all bridge.

Lastly, the substructure is constructed as such that the bridge is supported by two abutments at both ends and two concrete piers which the respective spans are supported on. The piers are in turn resting on bedrock. At both ends there are roller bearings while the intermediate supports consist of pinned and 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 over the 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, which can be seen in Figure 1.2.

The cross-section between the supports is presented in Figure 1.5. The bridge is 4050 mm wide and

1535 mm high. The cross-section consist of 200x200 mm longitudinal hollow box shaped beams on each

side which spans over the entire length of the bridge. Furthermore, vertical hollow box shaped beams

are connected to the longitudinal beams and to the crossbeams. These span along the bridge with a c/c

distance of 2640 mm as presented in Figure 1.6, which corresponds to the first part of the bridge. The

bridge deck consists of a 10 mm thick steel plate with a 2 % inclination. The deck is is connected to

the vertical hollow box shaped beams and is also supported by 180x180x10 mm crossbeams, with a c/c

distance 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 which are connected to the vertical and longitudinal hollow box shaped beams. The plates in the midspan have holes of varying width, increasing towards the middle of the bridge, as shown in Figure 1.2. The variation of the width of the holes corresponds to an architectural expression of the varying shear force in the midspan and changes with approximately 250 mm/hole. The positions of the holes are not symmetrically placed along the left and right side of the bridge. It can also be seen in Figure 1.6 that at 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 at each weld seam.

The cross-section at the end- and intermediate supports, shown in Figure 1.7 and Figure 1.8, are not significantly 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 plates in 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 steel

rods in order to avoid uplift as shown in Figure 1.7. The intermediate supports, shown in Figure 1.6

rests on skewed concrete piers where the angle between the supports is 117

^◦

. The piers are in turn

resting 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 extensive section 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 from measurements 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 other kind of excitation such as an earthquake.

The equation of motion is the most fundamental equation in dynamic analysis and an overview of

the theory behind it is thus given in this section. Solving the equation of motion gives for instance

displacements for all locations on a structure during any given time when subjected to a dynamic

load. The equation of motion also give rise to an eigenvalue problem which solution results in the

natural frequencies and modes of a system. Natural frequencies are the frequencies at which a structure

tends to oscillate during free vibration i.e. in the absence of external loading. The deflected shapes of

the structure during these oscillations at these certain frequencies are referred to as natural modes of

vibration, or simply as modes. These frequencies play a fundamental part in the design and analysis

of bridges. The reason being that it is essential to consider that the natural frequency of a bridge

do not match the frequency of expected loads such as the walking frequency of pedestrians since this

may lead to discomfort for pedestrians but also structural damage. If the frequencies do coincide, one

should conduct time history analysis to evaluate the dynamic behaviour of the bridge. This may be

done using modal analysis which is a powerful method for obtaining displacements and accelerations for

structures when subjected to dynamic loading. All derivations and statements made in the following

sections 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 damping of the system. The degrees of freedom in the system are the number of independent displacements needed to describe the relative change in position of the masses in the system. In reality, structures are composed of an infinite number of degrees of freedom. In structural dynamic the most basic way to describe a system is as a single degree of freedom (SDOF) system. A SDOF system can be represented by e.g. a oscillator, allowed to only displace in the lateral direction. Figure 2.1 illustrates the oscillator consisting of a mass, m, a massless frame with stiffness k and a viscous damper with damping coefficient c. 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) − m¨ u(t) (2.1)

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

P (t) = c ˙u(t) + ku(t) (2.2)

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

m¨ u(t) + c ˙u(t) + ku(t) = F (t) (2.3)

Before proceeding to solving the equation of motion, some central concepts regarding structural dynamics should be addressed. The natural cyclic frequency, ω

_n

, is given in units of radians per second and is for an undamped SDOF free vibration system given as

ω

n

=

s k

m (2.4)

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The natural period of vibration, T

_n

, is the time required for the undamped system to complete one cycle of free vibration according to Eq. 2.1.1.

T

_n

= 2π

ω

_n

(2.5)

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

f

_n

= 1 T

n

(2.6) The natural cyclic frequency is hence proportional to the natural frequency according to Eq. 2.1.1. The term natural is used to emphasize the fact that these are natural properties of the structure in free vibration.

f

_n

= ω

_n

2π (2.7)

The SDOF system described above is not always applicable to real life structures. The reason being that structures are composed of an infinite number of degrees of freedom and not all structures can be 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 accurately by discretizing structures into systems of elements with a finite number of degrees of freedom. This system is referred to as a multi degree of freedom system (MDOF). The basis of the theory describing a MDOF system is analogous to the theory described above for the SDOF system. It is a generalization from one to N number of dimensions where N is the number of degrees of freedom in the system. The equation of motion for a multi degree of freedom (MDOF) system is thus given as

m¨ u(t) + c ˙u(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 of order N x N . Displacements, velocities and accelerations for each degree of freedom are now given as time-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 degree of freedom.

The mass matrix is determined by assuming the mass of the system to be concentrated at the nodes

i.e. where the degrees of freedoms are located. The stiffness matrix is constructed by assembling the

local stiffness matrix of each element. There are several ways to construct the damping matrix which is

further 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 the system. Damping occurs when kinetic and strain energy of the vibrating system dissipates due to various mechanisms acting simultaneously. Common sources of energy dissipation in bridge structures could for instance occur due to friction at steel connections, inelastic behaviour of structures at large deformations and opening and closing of cracks in concrete. To identify and mathematically describe each 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 proportionality to the velocity makes it convenient to express in mathematical terms. However, common sources of damping in bridge structures such as those described above are not viscous but rather due to other mechanisms. 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 assuming viscous damping is sufficiently accurate in most cases [4].

Theoretical damping

Damping influences the nth natural cyclic frequency of vibration, ω

_n

, of an system according to Eq. 2.9, where ξ

_n

is the damping ratio, a measure used to express the level of damping in a system and ω

_nD

is the damped nth natural cyclic frequency.

ω

_nD

= ω

_n

^q 1 − ξ

_n²

(2.9)

Most practical structures have a damping ratio below 20 % [3]. The Eurocodes recommends a damping ratio of 0.5 % for railway bridges with span lengths over 20 m [7]. Consequently, the effect of damping on the natural frequencies for bridges is negligible since ω

_n

D = ω

_n

for small damping ratios. However, damping might have a significant influence when evaluating displacements and accelerations. There are several commonly used methods to account for viscous damping. For simple SDOF systems, the damping coefficient, c, is defined as

c = 2ξ

_n

mω

_n

(2.10)

For MDOF systems, damping matrices may be constructed using for instance Rayleigh damping which

involves forming the damping as a linear combination of the mass and stiffness matrix. For the use

of modal analysis which is further described in section 2.1.4, one can account for damping by using

estimated damping ratios obtained from e.g. measurements directly. This requires linearly elastic

analysis and that the system is classically damped i.e. that the natural modes of the system are

uncoupled.

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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 the half-power bandwidth method. The logarithmic decrement method evaluates the damping ratio in the time 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 can be 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 ω

_n

is the natural cyclic frequency and its corresponding amplitude is A, then ω

a

and ω

b

are the frequencies on either 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 = ω

b

− ω

a

ω

_b

+ ω

_a

(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 free vibration i.e. in the absence of damping and external loading. In each natural mode, the system oscillates 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 to

vibrate without an externally applied dynamic force acting on it. This is achieved by introducing an

initial 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. m¨ u + ku = 0 (2.13)

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

u(t) = φ

_n

q

n

(t) (2.14)

where φ

_n

is the deflected shape of the system i.e. the mode shape and q

_n

(t) is the time variation of the displacement which can be described as a harmonic function:

q

_n

(t) = A

_n

cos ω

_n

t + B

_n

sin ω

_n

t (2.15) A

_n

and B

_n

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

u(t) = φ

_n

(A

_n

cos ω

_n

t + B

_n

sin ω

_n

t) (2.16) Differentiating the displacement twice with respect to time gives the time variation of the acceleration:

¨

u(t) = −ω

²_n

φ

_n

(A

_n

cos ω

_n

t + B

_n

sin ω

_n

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

[−ω

²_n

mφ

_n

+ kφ

n

]q

n

(t) = 0 (2.18)

There are two ways to satisfy this equation. The first solution q

_n

(t) = 0 implies that there is no motion of the system since u(t) always will be equal to zero. In the second solution, the natural cyclic frequencies ω

_n

and mode shapes φ

_n

fulfil the following equation

[k − ω

²_n

m]φ

_n

= 0 (2.19)

The trivial solution φ

_n

= 0 implies no motion and is hence not of interest. Nontrivial solutions are obtained if

det[k − ω

_n²

m] = 0 (2.20)

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

²_n

. These roots are known as eigenvalues and represents the N natural frequencies ω

_n

of the system. When the natural frequencies are known, the corresponding natural mode φ

_n

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 is not efficient for systems with a large number of DOFs and can be very computationally demanding. A reduction of the system’s complexity can be achieved by transforming the equations in terms of modal coordinates. Simply put, a MDOF system with N DOFs is transformed into N SDOF systems. This method 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 from that specific natural mode. The responses from each natural mode are then combined to obtain the total response of the system. The classical modal analysis procedure is applicable for linear systems which are classically damped in order to obtain modal equations that are uncoupled, a central feature of modal analysis.

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

K = Φ

^T

kΦ M = Φ

^T

mΦ (2.21)

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

C = Φ

^T

cΦ (2.22)

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

K

_n

= φ

^T_n

kφ

_n

M

_n

= φ

^T_n

mφ

_n

C

_n

= φ

^T_n

cφ

_n

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

M

_n

q ¨

_n

(t) + C

_n

q ˙

_n

(t) + K

_n

q

_n

(t) = F

_n

(t) (2.24) where

F

_n

(t) = φ

^T_n

F (t) (2.25)

and q

_n

are the unknown modal coordinates to be solved. M

_n

, C

_n

, K

_n

and F

_n

are referred to as the

generalized mass, damping, stiffness and force for the nth natural mode φ

_n

. Thus, the parameters

depend only on the nth mode and the modal coordinates may be solved in the absence of any other

information 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 manner analogous to Eq. 2.10.

C

_n

= 2ξ

_n

M

_n

ω

_n

(2.26)

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

K

_n

= ω

_n²

M

_n

(2.27)

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

¨

q

_n

(t) + 2ξ

_n

ω

_n

q ˙

_n

(t) + ω

²_n

q

_n

(t) = F

_n

(t)

M

_n

(2.28)

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

u

_n

(t) = φ

_n

q

_n

(t) (2.29)

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

u(t) =

N

X

n=1

u

_n

(t) =

N

X

n=1

φ

_n

q

_n

(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 dynamic force varies arbitrarily with time. This can however be tackled by the use of numerical time-stepping methods 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, t

_i

, with i = 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 difference approximation of displacements. In contrast, Newmark’s method is based on assumed variation of acceleration.

Two special cases of Newmark’s method are well-known: the constant average acceleration method and

the linear acceleration method. The difference between them lies in the choice of the parameters β and

γ. These parameters define how the acceleration varies over a time step and determine the stability and

accuracy characteristics of the method. The constant average acceleration method has been used in this

thesis which is associated with choosing β and γ to

¹₄

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 too

large 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 acceleration method. This procedure is applicable for linear SDOF systems using classical damping. However, it is easily applicable to the uncoupled equation in modal coordinates presented in Eq. 2.24 by replacing the mass, damping, stiffness, force and displacements with their corresponding generalized form.

Initial calculations are made according to Eq. 2.31 to Eq. 2.35 where u

₀

, ˙u

₀

and F

₀

define the initial displacement, initial velocity and initial force, respectively.

¨

u

₀

= F

₀

− c ˙u

₀

− ku

₀

m (2.31)

a

₁

= 1

β∆t

²

m + γ

β∆t c (2.32)

a

₂

= 1

β∆t m + γ β − 1

!

c (2.33)

a

₃

= 1 2β − 1

!

m + ∆t γ 2β − 1

!

c (2.34)

k = k + a b

₁

(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.

F b

_i+1

= F

_i+1

+ a

₁

u

_i

+ a

₂

˙u

_i

+ a

₃

u ¨

_i

(2.36)

u

i+1

= F ^b

_i+1

b k (2.37)

˙u

_i+1

= γ

β∆t (u

_i+1

− u

_i

) + 1 − γ β

!

˙u

_i

+ ∆t 1 − γ 2β

!

¨

u

_i

(2.38)

¨

u

_i+1

= 1

β∆t

²

(u

_i+1

− u

_i

) − 1

β∆t ˙u

_i

− 1 2β − 1

!

¨

u

_i

(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 only applicable in the simplest cases of structures. Finite element (FE) analysis, also referred to as the finite element method is a method that provides a way of obtaining numerical solutions to more complex problems. An general introduction to the FE is given as well as an description of the widely used shell elements.

2.2.1 Main idea

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

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

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

There are a number of available FE software available for modelling of structures. Some of these are general-purpose software while others are oriented towards more specific types of structures. The FE software chosen for modelling in this thesis is BRIGADE/Plus. It is a FE software developed especially for the use of structural analysis and design of bridges and other civil structures. In the reminder of this 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 relation that 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 smaller than the other dimensions. Two common types of shell elements suitable for dynamic analysis exist in Brigade: continuum shell and conventional shell. For continuum shells, the thickness of the element is defined by its nodal geometry and the entire three dimensional body is hence discretized. These elements have only translational degrees of freedom and use linear interpolation between the nodes. In contrast, 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 and rotation. 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 and

general-purpose conventional shell elements. Thin conventional shell elements, such as STRI3 elements

are to be used when transverse shear flexibility is negligible and the Kirchhoff constrain must be satisfied

accurately. The Kirchhoff constrain states that a line orthogonal to the shell reference surface should

remain orthogonal to the shell reference surface after deformation. In contrast, thick conventional

shell elements, such as element type S8R are needed when transverse shear flexibility is important and

second-order interpolation is desired. General-purpose conventional shell elements are used in most

cases since they usually provide accurate and robust solutions for most applications. Their advantage

lies in their versatility to describe both thin and thick shells since they become discrete Kirchhoff thin

shells as the thickness decreases and use thick shell theory as the thickness increases. One common

type of general-purpose conventional shell is the S4 element. It consists of 4 nodes and is stable against

phenomenons such as element distortion and parasitic locking. The corresponding element when using

reduced integration is the S4R. This type of element only has one integration location in contrast to the

S4 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 to their response with respect to time. In contrast, frequency domain refers to the analysis of signals with respect to frequency rather than time. Frequency domain representations are important in the field of dynamics 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 frequency domain, 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 concepts will provide understanding of the basics in signal processing e.g how to address problems such as aliasing and 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 two types of Fourier expansions are:

• Fourier series

• Fourier integral

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

f (t) = a

₀

+

∞

X

j=1

a

_j

cos(jω

₀

t) +

∞

X

j=1

b

_j

sin(jω

₀

t) (2.40)

where

ω

₀

= 2π

T

₀

(2.41)

The term jω

₀

corresponds to the jth harmonic cyclic frequency and T

₀

is the period of the function.

The coefficients in Eq. 2.40 may be expressed as

a

₀

= 1 T

0

Z

T0

0

f (t)dt (2.42)

a

_j

= 2 T

₀

Z

T0

0

f (t)cos(jω

₀

t)dt j = 1, 2, 3... (2.43)

b

_j

= 2 T

₀

Z

T0

0

f (t)sin(jω

₀

t)dt j = 1, 2, 3... (2.44)

Figure 2.4b) showcase how a signal can be represented by sinusoidal functions of different frequencies

in 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 the signal is built up of which are represented in the frequency domain [10]. The procedure is schematically presented 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 the lower 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 it can according to the Fourier Theorem be represented by the complex Fourier integral:

f (t) = 1 2π

Z

∞

−∞

F (ω)e

^iωt

dω (2.45)

F (ω) = 1 2π

Z

∞

−∞

f (t)e

^−iωt

dt (2.46)

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

The Fourier transform can be applied to both simple periodic excitation and non-periodic complicated excitation varying arbitrary in time. In the case of the latter, the transformation of data from the time domain to the frequency domain does not occur in a continuous manner. Hence, a numerical evaluation of 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 frequency

domain is called the Discrete Fourier Transformation (DFT) [8] and is schematically presented in Figure

2.5

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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 time domain 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. 2.46 which together define a continuous Fourier transform pair.

p

_n

=

N −1

X

j=0

P

_j

e

^i(jω⁰^tⁿ^{/N )}

(2.47)

where

P

_j

= 1 T

N −1

X

n=0

p

_n

e

^−i(jω⁰^tⁿ⁾

∆t = 1 N

N −1

X

n=0

p

_n

e

^{−i(2πj/N )}

(2.48)

In Eq. 2.47 and Eq. 2.48, p

_n

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 Fourier transform is a continuous, true representation of the excitation function. Hence, signal analysis and processing 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 to

frequency domain. Choosing the sampling interval sufficiently small and the sampling time sufficiently

large is necessary in order to accurately represent the input signal, as an insufficient number of samples

may 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 reason for the phenomenon has to do with the sampling rate, used to sample the input signal. Consider Figure 2.6, where an input signal is described as a sinusoidal function. If the input signal is sampled exactly ones every period, the result would be a constant line, which does not corresponds to the input signal that 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 Theorem states that the sampling rate should be greater than twice the highest frequency of the input signal being sampled:

f

s

≥ 2f

max

(2.49)

where f

_max

is the highest frequency of the input signal, f

_s

is the sampling rate defined as f

_s

=

_∆t¹

and

∆t is the sampling interval. Hence, suppose that a signal has the frequency f and that the sampling rate is f

_s

, then:

If f <

^f₂^s

- no aliasing occurs, according to the Nyquist Theorem.

If

¹₂

f

_s

< f < f

_s

- the signal undergoes aliasing, with an aliasing frequency of f

_a

= f

_s

− f . If f > f

_s

- the signal undergoes aliasing, with an alias frequency of f

_a

= f − f

_s

In those cases where the sampling frequency is limited, the highest frequency contained in the signal needs 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 FFT calculates the frequency spectrum based on a sample of the input from a complete period, which is assumed to be repeated at all time. Thus, the FFT assumes that the signals obtained from the sampling are periodic, repeated throughout the length of the total sampling time. If then the signal is truncated at a non-integer number of cycles, leakage will occur [11]. The effect of leakage in the frequency domain is illustrated in Figure 2.7. This phenomenon can be reduced using windowing functions as e.g the Hanning 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 of developing the FE models is given along with a description of the validation process. The second part describes the process of developing and validating the model developed to account for pedestrian-induced torsional vibrations.

3.1 Validation of FE models

With the aim of obtaining FE models that accurately captures the dynamic properties of the Kallh¨ all and Smista bridges, two FE models have been developed using the FE software Brigade. In order to validate the models, measurements have been conducted on each bridge whereby the natural frequencies have 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 been compared to the ones obtained from the FE analysis.

The following section will thus begin with a description of the modelling procedure along with the assumptions 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 Kallh¨ all bridge was created using the graphical user interface and was based on engineering drawings. Only the largest span was modelled and the remaining spans were omitted from the analysis. The reason being that each span of the bridge function independent of the other.

Furthermore, the Kallh¨ all bridge was modelled without considering the piers. The basis behind this decision is that each pier is assumed to function as a massive rigid body. Hence, they are assumed to be 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 engineering

drawings. In this case the entire bridge was considered including the piers due to its continuity over

the supports. Furthermore, the abutments in both models were not considered nor the soil-structure

interaction 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 direction as well as the arch were neglected in the FE model of the Smista bridge. However, the bridge was modelled with respect to its longitudinal inclined plane where it was assumed that the entire bridge had a 5 % inclination throughout its length. Hence, the true length has been accounted for with respect to the assumption. Regarding the Kallh¨ all bridge, the inclination was neglected as well. Instead, the bridge was modelled as straight but with respect to its inclined plane and its true length has thus also been 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 structural member. The reason being that the element thickness’s were assigned with respect to their middle surface. Furthermore, each bridge was modelled as one single three-dimensional part. This modelling approach provides rigid connections between all structural members and no further constraint had to be assigned at these connections.

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

• Density, ρ = 7850 kg/m

³

• Modulus of elasticity, E = 210 GP a

• Poisson’s ratio, ν = 0.3

Figure 3.1: The FE model of the Kallh¨ all 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 a significant contribution to the mass of the model. These non-structural components were accounted for by assigning regions with an equivalent density. The non-structural components include the:

• Paving in form of asphalt on the Kalh¨ all bridge

• Railings on the Kallh¨ all bridge

• Railing on the Smista bridge

• Curbstone on the Kallh¨ all bridge

Regarding the boundary condition, the behaviour of the bearings was represented by creating coupling

interaction between the bearings and the structural members. Figure 3.3 showcases the set up used to

represent the bearings.

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

The surface that rest on the actual support was coupled to a reference point, corresponding to the center of rotation of the actual bearing. In this connection, all translational and rotational movements were constrained. 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 the reference point was connected to e.g. the supporting piers, as shown in Figure 3.3 a), all translational and rotational movement were constrained. In the case where the second reference point was not connected to 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 the actual bearing. Boundary condition were then applied at the bottom of the piers, which were treated as completely fixed due to that they are resting on bedrock. The reason for modelling the bearing using this 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 were

studied. As previously mentioned, both bridges are simply supported. There were however uncertainties

on how the bearings were allowed to rotate and an analysis of the movements in the bearings was

performed. As a starting point, the bearings were allowed to rotate around all axes. Table 3.1 and

Table 3.2 showcase how a change in the properties of the bearings would affect the natural frequencies

of the FE models. The relative differences presented in the tables are differences with respect to the

first case. Noteworthy, is that the rotation around the Y-axis in the bearings always was free to rotate

in 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 Kallh¨ all bridge for different bearing conditions.

Case 1 - Simply Supported (SS)

Mode Natural frequencies [Hz] Rel. diff [%]

1

^st

vertical 3.31

1

^st

torsional 4.52

Case 2 - All fixed bearings

Mode Natural frequencies [Hz] Rel. diff [%]

1

^st

vertical 3.48 5.33

1

^st

torsional 4.63 2.41

Case 3 - SS, rotation prevented around X

Mode Natural frequencies [Hz] Rel. diff [%]

1

^st

vertical 3.31 0.00

1

^st

torsional 4.52 0.00

Case 4 - SS, rotation prevented around Z

Mode Natural frequencies [Hz] Rel. diff [%]

1

^st

vertical 3.31 0.00

1

^st

torsional 4.52 0.00

Case 5 - SS, rotation prevented around X and Z

Mode Natural frequencies [Hz] Rel. diff [%]

1

^st

vertical 3.31 0.00

1

^st