Variations in dynamic properties of a steel arch footbridge1224131/FULLTEXT01.pdf · av balk-teori och finita element-modellering härled att troligen komma från förändringar i

IN DEGREE PROJECT THE BUILT ENVIRONMENT,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2018

Variations in dynamic properties of a steel arch footbridgeAn experimental study

MARTIN FÖLDHAZY

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT

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TRITA ABE-MBT-18161

Royal institute of technology (KTH)

School of architecture and the built environment

Department of structural engineering and bridges

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Abstract

This study separately investigates how temperature as well as two real load-situations affects the modal damping ratio and natural frequencies of a 64.9m span steel arch footbridge. Measurements of acceleration have been completed which covers a temperature span of

to . The natural frequencies of the five investigated modes were observed to decrease 2-6% as the temperature increased. This effect was with the help of beam-theory and finite element modelling deduced to originate mostly from changes in Young’s modulus of the materials, but also geometrical changes in steel because of thermal expansion. Further investigation included a static mass in the form of packed snow that was estimated to weigh 14 tons. The natural frequencies were observed to remain unchanged while the modal damping ratios decreased. The second load-case was an uncontrolled mass-event where a large group of pedestrians travelled over the bridge as two cars stood stationary at the quarter-point of the span. A large increase (146%) of the damping ratio was observed while the natural frequency of the first mode decreased 4%. This change was suggested come from the human structure interaction (HSI) partially because the natural frequency of the human body is close to the first vertical frequency of the bridge thus making humans act like dampers on the bridge when close to resonance, and that the number of pedestrians contribute to the modal mass of the system, thus decreasing the natural frequency.

Sammanfattning

Denna studie undersöker separat hur temperaturen såväl som två verkliga belastningssituationer påverkar de modala dämpnings kvoterna och egenfrekvenserna hos en 64,9 meter lång stål-bågs gångbro. Mätningar av accelerationen i bron har genomförts som täcker en temperatur på -10°C till 10°C. De naturliga frekvenserna hos de fem undersökta moderna observerades minska 2–6% när temperaturen ökade. Denna minskning var med hjälp av balk-teori och finita element-modellering härled att troligen komma från förändringar i Youngs modul av materialen, men även geometriska förändringar i stålet på grund av termisk expansion. Vidare undersökning innefattade en statisk massa i form av packad snö som uppskattades att väga 14 ton. Egenfrekvenserna observerades förbli oförändrade medan de modala dämpnings kvoterna minskade. Det andra lastfallet var ett okontrollerat massevenemang där en stor grupp fotgängare gick över bron medan två bilar var stationära en fjärdedel in på brons längd. En stor ökning (146%) av dämpnings kvoten för den första vertikala moden observerades medan egenfrekvensen minskade 4%. Denna förändring föreslogs komma från interaktionen mellan människan och bron, delvis för att människokroppens egenfrekvens ligger nära brons första vertikala frekvens vilket gör att människan agerar som en dämpare när de är nära resonans med bron, och att antalet fotgängare bidrar till den modala massan av systemet vilket sänker frekvensen.

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Preface

Tack Aura för att du spenderade många timmar till att hoppa på en bro med mig vid flertalet väldigt kalla tillfällen.

Tack Emma och Raid för guidning och tolkning av knepiga mätresultat.

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Table of contents

ABSTRACT ...................................................................................................................... 3

SAMMANFATTNING ....................................................................................................... 3

PREFACE ........................................................................................................................ 5

1 INTRODUCTION ....................................................................................................... 9

1.1 BACKGROUND .................................................................................................................................. 9 1.2 AIM AND SCOPE ................................................................................................................................ 9 1.3 LIMITATIONS ................................................................................................................................. 10

2 LITERATURE REVIEW ............................................................................................ 11

2.1 THE TEMPERATURE EFFECT ON NATURAL FREQUENCIES AND MODAL DAMPING RATIOS .................... 11 2.2 HUMAN-STRUCTURE INTERACTION ................................................................................................. 13

3 METHOD .................................................................................................................. 15

3.1 A BRIEF INTRODUCTION TO THE MAIN EQUATIONS .......................................................................... 15 3.1.1 The logarithmic decrement method .................................................................................... 15 3.1.2 The half power bandwidth method ..................................................................................... 15

3.2 THE BRIDGE ................................................................................................................................... 16 3.3 EXPERIMENTAL SETUP AND DATA ACQUISITION ............................................................................... 17

3.3.1 Temperature measurements ............................................................................................... 18 3.3.2 Two load-cases .................................................................................................................... 19

3.4 TREATMENT OF DATA AND NUMERICAL CALCULATIONS. ................................................................... 21 3.4.1 Logarithmic decrement method .......................................................................................... 21 3.4.2 Half power bandwidth method ........................................................................................... 23

3.5 STATISTICS. .................................................................................................................................. 24 3.6 FINITE ELEMENT MODEL. ............................................................................................................... 25

3.6.1 In general and geometry .................................................................................................... 25 3.6.2 Interaction and constraints ................................................................................................ 27 3.6.3 Material data input ............................................................................................................ 29

4 RESULTS AND DISCUSSION ................................................................................... 30

4.1 DAMPING RATIOS AND NATURAL FREQUENCIES VS TEMPERATURE .................................................. 30 4.1.1 1st vertical mode ................................................................................................................... 31 4.1.2 3rd vertical mode .................................................................................................................. 31 4.1.3 5th vertical mode .................................................................................................................. 32 4.1.4 5th torsional mode ................................................................................................................ 32 4.1.5 Finite element results and interpretation ........................................................................... 33

4.2 A STATIC MASS ............................................................................................................................... 35 4.3 A DYNAMIC MASS ........................................................................................................................... 37

5 CONCLUSIONS AND FURTHER RESEARCH ........................................................... 42

5.1 CONCLUSIONS ............................................................................................................................... 42 5.2 FURTHER RESEARCH ...................................................................................................................... 43

BIBLIOGRAPHY ............................................................................................................ 44

APPENDIX A ................................................................................................................. 46

5.3 COLD MEASUREMENT .................................................................................................................... 46 5.4 A STATIC MASS ............................................................................................................................... 53 5.5 MEDIUM MEASUREMENT ONE ........................................................................................................ 57 5.6 MEDIUM MEASUREMENT TWO ....................................................................................................... 64 5.7 WARM MEASUREMENT .................................................................................................................. 69 5.8 UNCONTROLLED MASS EVENT ......................................................................................................... 75

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Chapter 1

1 Introduction

1.1 Background

Damping ratios and natural frequencies in the codes are in general considered as fixed values. It is in SS-EN 1990 (2002) for example stated that a comfort criterion of a pedestrian bridge must be verified if the natural frequencies of the bridge is less than 5Hz for the vertical modes and 2.5 for lateral or torsional modes. However, a pedestrian bridge is during its life exposed to a collection of different loads. Vehicles, snow, pedestrians, wind, thermal loads are some examples, and not seldom a combination of the latter.

Phenomenon as lock in and resonance loading have caused large problems for structures such as bridges and arenas true out the history. An area of interest recent years have been on the human structure interaction, showing variations in modal damping ratio and natural frequencies of structures when exposed to a pedestrian load with internal dynamic parameters and walking frequencies. Modal damping ratios have frequently been observed to greatly increase while the HSI seem to have a more subtle influence on natural frequencies. Needless to say, the question that pedestrians have a positive influence on the combined dynamic response for the structure arises. Furthermore, the temperature effect on modal damping ratios and natural frequencies of structures has not been given much attention. Increasing temperatures naturally leads to a reduced stiffness of structural components, why the hypothesis that a temperature increase will result in a natural frequency decrease, have a basis given that a worst case scenario is desired from a design point of view.

1.2 Aim and scope

This report aims at presenting variances in a 64.9 span, steel-arch pedestrian bridge with an orthotropic walkway, covered with mastic asphalt by measuring acceleration in the bridge while under the influence of three different situations.

(1) Temperature: natural frequencies and modal damping ratios are studied during a temperature span of .

(2) A static mass: The influence of a layer of heavily packed snow that was estimated to weigh tons.

(3) An uncontrolled mass event: The human structure interaction.

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MATLAB was used as a signal processing tool where natural frequencies was obtained by peak picking, and modal damping ratios approximated with the logarithmic decrement and half power bandwidth method. The finite element program Abaqus was used for Visualization of the mode-shapes and further analysis of the natural frequencies by comparing the change because of an increase in Young’s modulus of the materials because of a decrease from 10 to -10 in temperature.

1.3 Limitations

Two accelerometers was used to measure the signal for the first, third and fifth vertical mode and the fifth torsional mode, all visible at midspan of the bridge resulting in only one hardware set-up during the measurement occasions. Natural frequencies, and modal damping ratios was obtained experimentally, mode shapes was analyzed with finite element. The study is limited to only one bridge, reducing the external validity.

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Chapter 2

2 Literature review

2.1 The temperature effect on natural frequencies and modal damping ratios

Articles on how temperature affects the modal damping ratios of a bridge are few in the literature and those who exist are often inconclusive about a hypothesized correlation. However, changes in natural frequencies in structures was experimentally determined by Xia et al. (2005) when investigating temperature effects on a RC-slab. The reduction of natural frequencies with increasing temperatures was deduced to come from variations of Young’s modulus of the materials. The author, using the well-known equation for natural frequencies of a simply supported beam, and extrapolated material properties establish that the temperature effect on Young’s modulus is more significant for the natural frequencies than the geometry change due to thermal expansion.

Schubert el al. (2010) showed that damping ratios and natural frequencies on timber footbridges are dependent on temperature by investigating the effect of asphalt pavement, since asphalt is a viscoelastic material with properties that are strongly temperature dependent.

The authors investigated a laboratory, simply supported bridge made of timber with a 9.6m span and measured the natural frequencies and damping ratios with and without asphalt on the bridge at a low temperature (3 on the asphalt) and at a high temperature (47 on the asphalt). The result showed that at low temperature both the frequency of the first mode and the damping ratio increased when asphalt was on the bridge in relation to when there was no asphalt on the bridge. On the other hand, at high temperatures only the damping ratio was reported increased while the frequency of the first mode decreased when asphalt was on the bridge in relation to when there was no asphalt on the bridge. Another important note from their result was that both the natural frequency and the damping ratio decreased when the temperature were higher, and asphalt was on the bridge. It should be noted that only temperature intervals above the freezing point were studied.

Further investigation in this report included a theoretical model on the laboratory bridge with a viscoelastic layer on top to represent the asphalt. It was observed that a partial shear transfer in form of friction between the wood and the asphalt likely was the cause of change in frequencies and damping. And if shear transfer was assumed, both damping and natural frequencies decreased when the temperature was high in relation to low.

The report also included a case study on two pedestrian bridges which included studied temperatures down to -3 , the result on natural frequencies were consistent in the sense that they increased when the temperature decreased and thus consistent with Xia et al. (2005), but

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conclusion on damping ratios were difficult because of the large scatter and small change in damping.

One study that have been able to show this correlation was done by Li et al. (2010) where a large cable-stayed bridge was investigated and a negative correlation between modal damping ratios and temperature was establish and reasoned to come from friction at bearing and viscouselastic materials such as cable protectors. Their investigation of natural frequencies did, in agreement to previous studies also show a negative correlation with temperature.

A strong linear correlation between temperature and natural frequencies was later, again establish by Xia et al. (2012) when a laboratory steel beam, aluminum beam and reinforced concrete slab were studied. All three constructions showed a decrease in natural frequencies when the temperature increased for all modes. The research also included case studies done on a large suspension bridge and a 600m tall building that confirmed the linear correlation, however, here to, only degrees above the freezing point was investigated.

What was inconclusive was however the modal damping ratios investigated in both the laboratory environment the case studies. A large scatter was again present and no correlation with temperature could be determined.

Gonzales et al. (2013) reports on dynamic and material property change due to temperature on a one-span, ballasted, steel-concrete railway bridge. The frequencies of the first vertical and torsional bending modes increase when temperatures decrease with a relative change of 15%-35% where most of the change happens around 0 (e.g. not a completely linear correlation) whilst the investigated temperature range from -30 to +30 . An interesting phenomenon reported here is that the change in frequency took different paths going from sub-zero temperatures to above-zero temperatures than going the other way. It should be noted that the change in frequency here was suggested to come from changes in stiffness in the ballast and surrounding soil due to freezing of the water the soil contained and thus fairly consistent with Xia et al. (2005) and Schubert el al. (2010) in the sense that the change in modulus of the materials because of temperature significantly affect the natural frequencies. It was also deduced that the change in modulus in the concrete because of temperature could be neglected in contrast to the ballast and surrounding soil.

In an experimental study Cheynet et al. (2017) reports on temperature effects of natural frequencies and modal damping ratios of Lysefjord suspension bridge in Norway. Here, four lateral and vertical, and two torsional modes were investigated. All modes showed a decrease in natural frequency with increasing temperatures where the most drastic change came from torsional mode 1 with a decrease from 1.25Hz to 1.23Hz (1.6%). The modal damping ratios however did not show a distinct change with temperature. Some of the modes did show a reduction of damping ratio when the temperature increased, and others remained constant, all characterized by a large scatter. It was furthermore reasoned that the change in modal damping ratios came from change in wind speed, given the fact that in this area lower temperatures are connected to higher wind speeds. It should also be mentioned that the investigated temperature interval stretched from 0 to 20 and thus the subzero degrees are missing.

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2.2 Human-structure interaction

Human presence on structures have, in contrast to ambient temperature effects on damping, been given more attention recent years. This is not strange given bridge failure due to human-structure interaction where a well-known, recent example is the Millenium bridge in London (Reiterer and Hochrainer, 2004) And the historical example of the Broughton suspension bridge collapse in England 1831 described by Stewart-Drewry (1832) that was said to have been brought down due to an army group of sixty men walking in resonance with the bridge.

An experimental study investigating a laboratory bridge made from prestressed concrete with a 10.8m long span and a width of 2.0m was done by Zivanovic et al. (2009). The investigation included measurements on the bridge with a standing crowd and a walking crowd and compared damping ratios, natural frequencies and maximum acceleration against the bridge modal parameters while un-loaded. Result showed that the natural frequency of the bridge decreased when increasing the number of still-standing pedestrians but increased when increasing the amount of walking pedestrians. The damping ratio on the other hand increased by a factor 2.9 for the walking pedestrians and by a factor of 4 for the still standing pedestrians. The maximum acceleration was, in both cases, observed to decrease with increased presence of pedestrians on the bridge.

In agreement with conclusions made by previously mentioned study, James and Brownjohn (2000) investigation of the human structure interaction (HSI) of a one-way, prestressed simply supported concrete slab where a human was positioned in different postures. It was concluded that the natural frequency decreased with still-standing human presence on the structure, and damping increased with the largest effect coming from a standing position of the pedestrian with very bent knees interestingly. An equivalent mass, e.g. static mass with the same weight as the human, positioned on the slab was also observed to have the same influence on the modal parameters on the bridge as the human, but with considerably less effect. A two degree of freedom system created by the authors increased confidence in their result when modelling the human as a spring-mass damper (SMD) with f=5.27Hz and ζ=36% as the dynamic properties representing the human.

Kasperski (2014) investigated a cable-stayed, pedestrian-bridge with a total span of 66m by measuring the effect of a passive person standing, a single person walking and an uncontrolled mass event where 0.2-0.6 persons per second were to arrive at the bridge. Conclusions include that all load configurations increase the damping, but only marginally for the passive person, and considerably more for the uncontrolled mass event. Observations made shows a trend between the modal damping ratio and the arrival rate at the bridge why it was inferred that step frequency influences the damping ratio. Further observations included that a step frequency of 1.9Hz and a step length of 0.82m was reasonable for a pedestrian.

A recent study done by Archbold and Mullarney (2017) reports on a laboratory composite bridge made of a plywood deck with glass fibre reinforced polyester beams. The bridge was studied under four different spans and a range of support conditions. A number of different pedestrians were allowed to walk over the bride one at a time in different speeds e.g. slow, normal and fast while step width and length was closely monitored. Observed step frequencies of the pedestrian varied from 1.8, 2.1 and 2.3Hz respectively and the step length 0.77, 0.84 and 0.91 respectively and thus correspond very well with conclusions made by previously mention author regarding a normal walking speed and step length.

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Furthermore, in was observed that the natural frequency of the first mode was decreased by 29% when one pedestrian was standing still at midspan. It was also observed, in contradictory to Zivanovic et al. (2009), that the natural frequency was decreased by 6.7%, 14.1% and 12.5% when pedestrians walked over the bridge thus concluding that it is not only the mass of the pedestrian that affects the human-structure interaction, but also the stiffness of the pedestrian. Another interesting phenomenon reported here was the pedestrians step length and walking frequency that seemed to increase when it was close to the bridge natural frequency. This effect was hypothesised to come from the pedestrian’s disinclination to walk in resonance with the bridge, thus damping the bridge displacement.

There is, to the authors best knowledge, no studies where the main focus have been on investigating the suggested impact of a static mass on the dynamic properties of a structure. However the extensive literature review by Sachse et al. (2003) summarize a number of studies where it is not always definitive of increased damping while investigating a human-equivalent mass on the structure. The review of Hothan (1999) experiment on two steel beam structures that alternated spans resulting in 13 different experimental set-up’s shows that the influence on damping ratios from the presence of an equivalent static-mass is small, and even marginally reduces the damping ratio in some cases, whilst the damping ratio is greatly increased by the presence of humans on the structure and the natural frequency decreased in both cases.

Moreover, the experiments made by Falati (1999) that includes a comprehensive investigation of modal parameters on floor-slabs, makes comparisons of damping ratios. Experimental set-up included: rigidly attached false-floor panels; non-rigidly attached false-floor panels; and no floor-panels attached to the slab. Interestingly did the damping ratio only marginally increase (6%) when the floor-panels were rigidly attached compared to a large increase (66%) when the floor panels were non-rigidly attached. The difference of dissipated energy was deduced to come from an increase in friction between the slab and the floor-panels in the non-rigidly attachment set-up.

In the same investigation, the same author showed in that the damping ratio is greatly increased when a human is placed in the centre of the slab, compared to when a static equivalent weight is used. It was observed that this human effect on modal damping ratios was completely lost when studying higher frequencies. This result was later confirmed by Van Nimmen et al. (2016) that concluded that the HSI significantly increases the modal damping ratios on footbridges for modes lower than 6Hz.

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Chapter 3

3 Method

3.1 A brief introduction to the main equations

This section presents the main equations as described by Chopra 2013. In this study and common for bridges in general, only viscously underdamped vibration takes place. The result is presented as damping ratios, or fraction of critical damping. The damping ratio is defined as

(1)

Which is dimensionless parameter that depends on the dissipation of energy c, the stiffness k, and mass m of the system. Two methods for the estimation of the damping ratio have been used as described below.

3.1.1 The logarithmic decrement method

This equation ties together the damping ratio with the decay of damped, free vibration of a system. The equation is also only valid for underdamped systems, where the damping ratio is less than one.

(2)

(3)

Where in Eq.2, is defined as the logarithmic decrement and are two succeeding peaks (maxima) of the signal in the time domain, separated by peaks.

3.1.2 The half power bandwidth method

This method operates, in contrast to Eq.4, in the frequency domain of the signal. What characterizes the half power bandwidth method is that allows evaluation of damping ratios from forced vibration tests without knowing the applied force or the need for free decay in the time domain. It is furthermore, more approximate since two simplifications follows from the derivation. Still Eq.4 works well for small enough damping ratios, which is the case for the bridge described in section 3.2 below where all damping ratios are well under 5%.

(4)

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3.2 The bridge

This bridge is chosen for its location in Stockholm, Sweden, close to a large football arena. It is expected that many times a year a very large group of people will simultaneously walk over the bridge as a group, making it suitable for analysing the effect of human structure interaction on the dynamic properties of the bridge. Another reason is that on the walkway on this pedestrian bridge there is a 75mm layer of PGJA or mastic asphalt (Figure 1), making the bridge suitable for testing the shear transfer effect hypothesises in Schubert et al. (2010) and how temperature affects it.

Figure 1 shows the bridge walkway in crossection, the orthotropic deck consists of a concrete slab with edge beams connected to the steel crossbeams with shear studs. The mastic asphalt is, furthermore, not mechanically connected to the concrete walkway.

Figure 1: Crossection of the orthotropic deck of the case study bridge (PEAB anläggning, 2018).

As can be seen in Figure 2 it is a steel arch bridge with one span of 64.9m. It’s support consists of two concrete walls each resting on a concrete slab as a foundation with approximately 30 steel core piles per slab driven down thru a layer of clay and friction soil to bedrock. The superstructure consists, in addition to the walkway, of 17 crossbeams welded to the two main beams, two arches with a small inclination inwards and 22 hangers per arch, welded to the arch and the main beams. The west support (left in the figure) is pinned and the east support (right in the figure) is a roller support. The east support is shared with a beam bridge, but the two bridges are, other than that, not connected.

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Figure 2: The case study bridge (PEAB anläggning, 2018).

3.3 Experimental setup and data acquisition

Since many factors can affect the damping ratio, a challenge in measuring on a real bridge were control of background variables, so the hypothesised effect of temperature and different loading situation on the dynamic properties can be deduced. This was controlled by carefully choosing the measurement occasion, so the background variables were constant or absent whilst the variable under investigation were varying. The background variables were windspeed, unwanted pedestrians or other mass on the bridge, and trains running under the bridge. A complete description of the background variables for every measurement occasion is shown in Appendix A.

To capture the acceleration of the bridge two accelerometers of the manufacturer SENSR and brand CX1 were used. Each accelerometer has one sensor in each axis (x, y and z) and each sensor can measure up to 1.5g. The chosen sampling frequency was = 2000samples/s and an inbuilt low-pass filter were enabled at 200Hz in the accelerometers.

Figure 3: The accelerometer of the manufacturer SENSR and brand CX1.

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3.3.1 Temperature measurements

At the early measurement occasions different configurations of the position of the accelerometers and type of excitation (impact or continuously) were tried to study which frequencies were best suited for this study. This concluded in that the studied modes were the first, third and fifth vertical, and the fifth torsional since these modes were visible all visible at midspan resulting in only one hardware set-up per occasion.

The accelerometers were then placed at mid-span of the bridge, one on the north side and one on the south (Figure 4). At each measuring occasion two persons excited the bridge with an impact by simultaneously jumping one time on the bridge. Note that earlier measurement occasion had also resulted in the conclusion that the first vertical mode needed approximately 60s to complete its free decay down to zero, why each jump was 60s apart. And to make sorting of the data easier, each measured signal included three jumps, making one signal

approximately 180s long (Figure 5) which resulted in a frequency resolution of

Hz.

Figure 4: Hardware setup for the temperature measurements. Both accelerometers placed at mid span, one on the south side and one on the north.

At each measurement occasion 15 jumps were made in the middle in between the accelerometers, and 15 jumps at the south side of the bridge, next to one of the accelerometers (Figure 4). This resulted in 30 jumps for the calculation of the damping ratio of the vertical modes, and 15 jumps for the torsional mode, using the logarithmic decrement. Since each measured signal included three jumps, five signals from jumping in between the accelerometers were created, and five from jumping next to the accelerometer were created, this led to 10 calculations of the damping ratio for the vertical modes and 5 for the torsional mode, using half power bandwidth method.

It was shown by Ulker-Kaustell et al. (2010) that modal damping ratios and natural frequencies are linearly dependent on the amplitude of the measured signal. Why each jump is made by the same two persons, making the amplitude of the acceleration approximately the same for all signals.

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Figure 5: Example of a typical measured signal filtered at the first vertical mode. It contains three jumps and each jump is given time to decay down to zero, the amplitude is furthermore, approximately the same for all three jumps.

To fully investigate the hypothesized influence of temperature on the dynamic properties of the bridge, a number of these measurement occasions took place between February and May 2018 resulting in a temperature difference . These temperatures were measured in different ways. Two regular household thermometers were used to measure the ambient temperature of the bridge; the mean value of these temperatures was then used. To measure the surface temperature of the mastic asphalt an infrared thermometer was used. A number of samples over the walkway were measured and the mean value was then noted.

3.3.2 Two load-cases

Two load-case comparisons were made. One in winter temperatures, and one in spring temperatures. Furthermore, two different types of loading on the bridge were investigated.

3.3.2.1 A static mass

The winter temperature load-case was a static mass consisting of heavily packed snow on the bridge (it was packed by pedestrians walking over). A semi regular surface was formed estimated to be 2-5cm thick and 90% of the walkways with, since very few pedestrians walk at the outmost edge of the bridge, no packed snow was formed there. Using as the density of ice, and as the density of snow (SMHI, 2017). The density of the packed snow on the bridge was estimated to be 85% ice and 15% snow, or

The mass on the bridge can then be approximated to.

.

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The hardware setup, and data collection was, furthermore, the same as for the temperature measurements described above in section 3.2.1.

3.3.2.2 An uncontrolled mass event

The spring temperature load-case consisted of an uncontrolled mass event where a large group of pedestrians traveled over the bridge while two police cars were parked on the bridge according to Figure 6.

Figure 6: Measuring situation showing the hardware set-up and the pedestrian flow.

This was more of an observational study since no control over exact mass and walking frequency was exerted. The walking frequency was estimated to be and the amount of people was over the gray area on Figure 6. Both the accelerometers were placed, in contrast to the other measurements, on the south side of the bridge in fear of damaging them.

Pedestrian flow

Car 1 Car 2

Hardware set-up

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3.4 Treatment of data and numerical calculations.

This section describes the program made in MATLAB R2017b and the work flow in that program for the treatment of the data acquired with the accelerometers and the calculations of the frequencies and their modal damping ratios.

Figure 7: The program work-flow. In the figure LD stands for logarithmic decrement and HPB for half power bandwidth.

The workflow described in Figure 7 starts with deciding which kind of mode is to be analyzed. For vertical modes, the mean value between the accelerometers was obtained and for the torsional the difference divided by the distance (or the width of the walkway) was obtained. Then deciding which frequency is to be isolated, and the interval around that frequency that is to be let thru. Some “wider” frequencies need a larger interval than the “thin” frequencies for enough energy to be let through.

3.4.1 Logarithmic decrement method

The time interval T is defined for one jump and should start where the free decay of the signal starts, and end when the signal is around 0 . The program then identifies the maximums and minimums of the signal in the given time interval and fits a curve on the form

[ ] by adapting B and C to the maxima and minima of the measured signal with a least square algorithm.

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Figure 8: Example of what was considered a good (above) and a bad (below) decay of the first vertical mode.

The least squared fitted curves are the red curves in Figure 8. The left figure is an example of what is considered a good decay since the fitted curve follows the measured signal (blue). The figure to the right is an example of what is considered a bad decay since they don’t follow each other. In the example described in Figure 8 both signals were used for the calculation of the damping ratio, but it should be mentioned that some signals, that the author deemed were to poor, or contained too much error, that it was obvious that it would not give an accurate approximation of the damping ratio, was excluded completely.

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The program then randomly selects two different peaks in the given time interval T and calculates the damping ratio with the logarithmic decrement (Eq.3 and 2) between those peaks for the positive and negative side of the measured signal, and for the positive and negative fitted signal. The program does this times, each time selecting two new peaks and then takes the arithmetic mean of the calculations according to Figure 9. In the last square in this figure, the mean between the positive and negative side is the damping ratio for that specific jump and as shown there are two different damping ratios, one for the measured signal, and one for the fitted curve.

Figure 9: Calculation of damping ratio with logarithmic decrement. How the mean value was obtained.

This method, for the measured signal converged the damping ratio at the second or third decimal, depending on length and quality of the signal. And for the fitted curve, well below the third decimal. The mean of the damping ratio from all the jumps for the measured signal of one measurement occasion, did not differ considerably from the mean of the damping ratio from all the jumps for the fitted curve from the same measurement occasion. Therefore, the result is based on calculations from the measured signal, and not the fitted curve.

3.4.2 Half power bandwidth method

Described in the last part of Figure 7 is the damping ratio calculated with the half power bandwidth method (Eq. 4). The frequency peaks tended to be noisy, and since this method estimates the damping by investigating how much the frequency “spreads out” from its peak down to half its power, the calculated damping ratios was not a representable approximation.

Figure 10: Above peak, larger frequency resolution with less noise. Below peak, smaller frequency resolution with more noise.

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In Figure 10, the below frequency peak comes from the FFT of an s long signal. The upper frequency peak comes from the same signal but the first 60s. To reduce the length of the signal in the time domain and recreate it in the frequency domain, thus decreasing the frequency resolution, proved, in this case to give a more accurate result. The damping ratio calculated on the above frequency peak gave, as the result will show, a damping ratio around the one calculated with the logarithmic decrement, whilst the damping ratio calculated on the below frequency peak typically gave a damping ratio reduced with a factor 10 or more, and this was as described above, because the calculation was done on a noisy peak that is thinner that the actual peak of the frequency.

3.5 Statistics.

It became clear that modal damping ratios are prone to a large scatter, why the 95-confidence interval was used continuously in the result. Two groups of data, one on damping ratios, and one on frequencies were tested for the normal distribution.

Figure 11: Normal distribution probability plots. Damping ratio above and frequency below.

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In Figure 11 the probability plot to the right is for the damping ratio and frequencie to the left. The data is shown as the blue dots and the normal distribution as the red line. It was estimated that they follow eachother sufficiently enough for normal distribuion arguments to be implemented on the data.

3.6 Finite element model.

3.6.1 In general and geometry

A FE- model created in the commercial software Abaqus/CAE was used to compare theoretical frequencies to measured frequencies and to visualize the mode-shapes.

Figure 12: Finite element model, with some of the main crossections.

The finite element model was based on the contruction blueprints. The support conditions were modeled as pinned in both supports and the piles were constraind at all degrees of fredom at the bottom to simulate being driven in bedrock. The glas railing shown in Figure 2 was considered as a co-vibrating mass by increaseing the density of the concrete for the edge-beams of the walkway. Figure 12 shows the main parts of the model by color and the dimensions of some of the main cross-sections of the structure. The arch and it’s corresponding loft-beams were converted to have rectangular, equivalent cross-sections with the same heigth, cross-sectional area and second area moment of inertia around the main and minor axis as their original cross-sections. This was done to take regard to the small extension of 10mm of the top and bottom flange of the original cross-sections. Figure 13 shows the dimensions of the original and the equivalent cross-sections defined in Abaqus.

26

Figure 13: Original cross-sections and the equivalent cross-sections defined in Abaqus.

Four different elements was used and listed below. Two beam elements and two shell elements. Furthermore, two different element sizes was used and listed in Table 1.

B32: Quadratic, shear-flexible Timoshenko beam element.

B31: Linear, shear-flexible Timoshenko beam element.

S8R: Eigth noded, quadratic stress-displacement shell, reduced integration.

S4R: Four noded, quadratic stress-displacement shell, reduced integration.

27

Table 1: Element types and sizes used for the different structural parts in Abaqus.

Structrual part (Material) Element type Element size [m]

Arch (steel) B32 0.125

Hangers (steel) B32 0.125

Loft beams (steel) B32 0.125

Main beams (steel) B32 0.125

Cross beams (steel) B32 0.125

Edge beams (concrete) B32 0.125

Piles (steel) B31 0.125

Walkway (concrete) S8R 0.125

Walkway (mastic asphalt) S4R 0.25

Foundation wall (concrete) S8R 0.25

Foundation slab (concrete) S8R 0.25

3.6.2 Interaction and constraints

3.6.2.1 Welded beam connections

All the structural parts made of steel was connected to each other by welding. This was simulated by using a connection type Beam in Abaqus, tying translational and rotational degrees of freedom of one node, to another node.

Figure 14: Beam connections.

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3.6.2.2 Support connections

The pinned support conditions were modelled with a connection type Join+Cardan allowing rotational degrees of freedom but constraining translation between the support wall and the support node for the superstructure.

Figure 15: Support connections and position of walkway layers.

3.6.2.3 Tie constraints

The concrete walkway, in reality connected with shear-studs to the cross-beams was positions with an eccentricity of half the thickness of the concrete walkway and half the height of the cross-beams and connected with a Tie-constraint, the same way as the mastic asphalt in turn was connected to the concrete walkway.

The support wall was also connected to the support slab by a tie-constraint and an eccentricity; however the piles were connected directly to the mid-surface of the support slab.

Figure 16: Foundation connections.

29

3.6.3 Material data input

Two analyses were made to investigate the influence of changes in natural frequencies because of changes in Young’s modulus. Material data were based on the equations below.

Tan et al (2017) (5)

Youssef and Moftah (2007) (6)

Schaffer et al (1992) (7)

Where Eq.5 is for the asphalt pavement, Eq.6 for concrete and Eq.7 for steel. is Young’s modulus at temperature and is the referens modulus for each material at a specfic temperature. Furtermore, was Eq. 6 and 7 extrapolated since the analytically predicted frequencies were compared at and as the measured frequencies.

Table 2: Material data input for the FE-simulations. Based on Eq.5-7

Material Young’s modulus [Gpa]

Young’s modulus [Gpa]

Density [ ]

Poassion’s ratio [-]

Steel 210.70 211.10 7800 0.3

Concrete(reinforced) 34.00 34.86 2400 0.2

Concrete(reinforced)+glass railing 34.00 34.86 3610 0.2

Mastic asphalt 9.19 1.19 2349 0.15

30

Chapter 4

4 Results and discussion

4.1 Damping ratios and natural frequencies vs temperature

Table 3 shows a summation of the damping ratio and its natural frequency with their corresponding 95% confidence interval at four temperatures for the measurements described in section 3.3.1. Table 4 shows the theoretically predicted frequencies, and compares them to the measured.

Table 3: Summation. Measured natural frequencies and modal damping ratios at four different temperatures for the four different modes under study.

Temperature(air/surface)

-9.0/data missing

-2.0/-2.2 0.5/1.0 12.0/11.5

Vertical mode 1

[%] 0.46 0.016 0.41 0.013 0.52 0.025 0.49 0.017 [Hz] 3.35 0.008 3.33 3.29 0.006 3.29

Vertical mode

3 [%] 0.68 0.016 0.60 0.023 0.91 0.035 0.74 0.026 [Hz] 5.51 0.022 5.45 0.011 5.39 0.012 5.34 0.010

Vertical mode

5 [%] 1.46 0.065 1.14 0.050 1.49 0.062 1.32 0.043 [Hz] 8.88 0.026 8.66 0.038 8.56 0.024 8.33 0.016

Torsional mode

5 [%] 1.17 0.046 0.74 0.026 0.81 0.032 0.83 0.060 [Hz] 10.61 0.110 10.55 0.030 10.51 0.050 10.45 0.030

The result is also shown graphically in section 4.1.1-4.1.4 below where the red rings represent a damping calculation made with logarithmic decrement. The blue cross

represent a damping calculation made with Half Power Bandwidth whilst the grey area represents the 95% interval for the above graphs in Figure 17-Figure 20. The blue cross also represents one frequency peak in the below graphs for the same figures.

In general, due to the few datapoints, no correlation can be determined. But a decrease in natural frequency can be observed with increasing temperatures for all modes. The modal damping ratios have a more arbitrary behaviour, where the largest increase can mostly be observed when passing over from negative to positive degrees, with the exception for the torsional mode.

31

Figure 17: Damping ratio above, natural frequency below with corresponding mode-shape of the bridge deck.

4.1.1 1st vertical mode

The largest difference of damping ratio for this mode were observed when passing over to positive degrees with a relative increase of 27%. The overall difference between winter and spring temperatures have overlapping confidence intervals and thus no change can be deduced. For the natural frequency the largest difference is the overall difference between winter and spring temperatures and were observed to decrease 2%.

4.1.2 3rd vertical mode

The largest difference of damping ratio for this mode were observed when passing over to positive degrees with a relative increase of 52%. The overall difference between winter and spring temperatures were observed to increase 10%. For the natural frequency the largest difference is the overall difference between winter and spring temperatures and were observed to decrease 3%.

Figure 18: Damping ratio above, natural frequency below with corresponding mode-shape of the bridge deck.

32

4.1.3 5th vertical mode

The largest difference of damping ratio for this mode were observed when passing over to positive degrees with a relative increase of 31%. The overall difference between winter and spring temperatures were observed to decrease 10%. For the natural frequency the largest difference is the overall difference between winter and spring temperatures and were observed to decrease 6%.

Figure 19: Damping ratio above, natural frequency and corresponding mode-shape of the bridge deck below.

4.1.4 5th torsional mode

The largest difference here were observed, in contrast to the vertical modes, when going from to . But there is still a relative increase of 10% when passing over to positive

degrees. The overall difference between winter and spring temperatures showed a relative decrease in 29% for the damping ratio. For the natural frequency the largest difference is the overall difference between winter and spring temperatures and decreased 2%.

Figure 20: Damping ratio above, natural frequency and corresponding mode-shape of the bridge deck below.

33

In Figure 21 below is a comparison of the length of the vertical signals. The time for vertical mode one to decay down to zero is approximately 50s, while for vertical mode three and five; the time is approximately 20s and 7s respectively.

Figure 21: Decay comparison of the vertical modes.

4.1.5 Finite element results and interpretation

Shown in Table 4 is the comparison between the theoretical predicted frequencies, under the assumption that only change in Young’s modulus because of temperature is what causes the change in frequency. Also visible are the respective measured natural frequencies which correspond well to the theoretically obtained ones.

Table 4: Theoretical frequencies, and their respective measured frequencies

Measured ( )

Theoretical ( )

Theoretical ( )

Measured ( )

Vertical mode 1 [Hz] 3.35 3.35 3.33 3.29

Vertical mode 3 [Hz] 5.51 5.35 5.29 5.34

Vertical mode 5 [Hz] 8.88 8.55 8.36 8.33

Torsional mode 5 [Hz] 10.61 10.56 10.36 10.45

A decrease in frequency with an increase in temperature is also observed for the theoretical natural frequencies as for the measured natural frequencies. This is expected since Young’s modulus also decreases with an increase in temperature, especially for the asphalt pavement. However, the difference for the theoretical natural frequencies is more marginal than for the measured natural frequencies, indicating that geometrical changes because of thermal expansion also affects the frequencies.

34

A simply supported beam analogy, similar to the one described by Xia et al. (2005) can investigate if it is likely that a change in geometry also causes a decrease in frequency.

A steel and concrete beam is examined separately. The beams have a rectangular crossections with initial geometrical and material properties at the reference temperature . Eq.6 and 7 is used to describe the changes in Young’s modulus while Eq. 8 and 9 (Schaffer et al (1992) and Siddiqui (2015)) is used to describe thermal expansion for concrete and steel.

(8)

(9)

Figure 22: Relative change in frequency because of temperature. Steel beam (above), concrete beam (below). Dotted line represents changes in frequency because of thermal expansion only, the dached line becase of Young’s modulus only and the solid line represents changes because of both.

35

On the y-axis shown in Figure 22 above is the relative change in frequency against the reference frequency at , for the first mode of vibration. Visible on the x-axis is the temperature. The dotted line represent change because of thermal expansion only, the dashed line because of Young’s modulus only, and the solid line represent change because of both. In both cases the change because of Young’s modulus is more significant for the change in frequency than the thermal expansion. However, the thermal expansion for the steel beam cannot be neglected in comparison. That is not the case for the concrete beam where change because of thermal expansion is insignificant in relation to its change in Young’s modulus. But given that the case-study bridge is a steel-arch bridge, it is feasibly that geometrical changes also contribute to the decrease in frequency. However, in reality the change in frequency because of geometrical changes is collaboration from the combined response of the structure, for example change in the arch height, in combination with hanger length.

The modal damping ratios are more challenging to quantify. Due to that the most significant difference was observed around zero degrees an explanation could be melting of ice in micro cracks in the concrete walkway or perhaps more movement in the bearings thus causing friction and dissipation of energy when passing over to positive degrees.

4.2 A static mass

The result from section 3.3.2.1 in Method are shown in Figure 23 below where the blue bar is the measurement with the static mass in the form of a semi-regular surface of packed snow, with an estimated weight of 14 tons. The red bar is the control measurement done when there was no mass on the bridge. The air temperature during the static mass measurement were

and it should be noted that the air temperature during the control measurement was . i.e. a different.

Figure 23: The influence of the static mass on natural frequencies and modal damping ratios.

36

On the x-axis the frequency is represented, and each mode clearly visible at its corresponding natural frequency and damping ratio. The natural frequencies for the static mass measurement do not differ remarkably. The modal damping ratios do however, seem to decrease with the static mass on the bridge. A relative decrease of 17%, 32%, 18%, 22% for each mode respectively according to the figure, Table 5 below also show the result. Table 5: Comparison of the modal properties between the empty and the snow loaded bridge

Vertical mode 1

Vertical mode 3

Vertical mode 5

Torsional mode 5

Empty [%] 0.46 0.016 0.68 0.016 1.46 0.065 1.17 0.046

[Hz] 3.35 0.008 5.51 0.022 8.88 0.026 10.61 0.110

Static mass loading

[%] 0.38 0.01 0.46 0.024 1.20 0.100 0.92 0.071

[Hz] 3.35 0.006 5.54 0.033 8.90 0.023 10.61 0.040

This decrease was not expected. The few studies on the subject Static mass available in the literature reports on a small increase in modal damping ratio (Falati ,1999 and James and Brownjohn, 2000). The only one reporting on a decrease was Hothan (1999) and then only marginally. Since it is unlikely that adding a static mass to the vibration would decrease the dissipation of energy, the only explanation feasibly to the author, is that the ice on the walkway added to the modal mass of the system, and perhaps the stiffness, without increasing dissipations of energy, thus decreasing the modal damping ratio.

37

4.3 A dynamic mass

The result from section 3.3.2.2 in Method are shown in Figure 24 below where the red rings represent a damping calculation made with Logarithmic Decrement while the blue cross

represent a damping calculation made with Half Power Bandwidth. What is different with this figure in contrast to the others is that the width of the grey area represents the confidence interval for the natural frequency, while the height of the grey area still represents the confidence interval for the damping ratio.

Figure 24: How the uncontrolled mass event affected the natural frequencies and damping ratio for the first vertical mode of vibration.

The above measurement occasion in Figure 24 with the large confidence interval is the one done with the dynamic mass in form of walking pedestrians and stationary cars on the bridge. The below measurement occasion is the control measurement done at the same temperature on the empty bridge. Why the dynamic measurements confidence interval is larger depends on the lack of control described in section 3.3.2.2. There were not exactly the same number of pedestrians on the bridge during the length of the measurement which was minutes. The walking frequency of the pedestrians did furthermore, probably differ a bit between the pedestrians and there were probably a number of pedestrians that changed their walking frequency while walking over the bridge. There was furthermore, no free decay of the signal why the logarithmic decrement was useless which reduced the number of damping calculations for the dynamic measurement. Due to this lack of control only the damping ratio of the first vertical mode was investigated because that mode was more present than the others. However, the frequencies were clearly visible for all modes except the fifth torsional, due to the experimental set-up. The damping ratio did however increase 146% relative the control measurement and the frequency decreased 4%. The frequencies for the other two modes are shown in Table 6 below and did interestingly not change due to pedestrian loading.

38

Table 6: Comparison between the modal properties of the empty and pedestrian loaded bridge

Empty bridge Pedestrians walking

Vertical mode 1 [%] 0.41 0.017 1.21 0.255 [Hz] 3.29 0.003 3.14 0.073

Vertical mode 3 [%] 0.74 0.026 Data missing

[Hz] 5.34 0.010 5.33 0.056 Vertical mode 5 [%] 1.32 0.043 Data missing

[Hz] 8.33 0.016 8.36 0.040 Torsional mode 5 [%] 0.83 0.060 Data missing

[Hz] 10.45 0.030 Data missing

In Figure 25 below are the frequency plots for the vertical modes 1,3,5. One of the control measurements are shown to the left, and the uncontrolled mass event to the right. Furthermore, the time-domain plots are also shown in the bottom of same picture, note that the y-axis scale on the time-domain plots are different. Clearly visible for the first vertical mode is how the height of the frequency-peak is decreased, and the width increased, when the pedestrians were present on the bridge indicating a larger damping ratio. However, this decrease in the frequency-peak were only observed for the first vertical mode and did interestingly increase for vertical mode 3 and 5 when pedestrians were present on the bridge. The increase in width of the frequency-peaks for vertical mode 3 and 5 are also not as distinct as for the first vertical mode.

39

Figure 25: Frequency and time-domain plots for one of the control measurements and the uncontrolled mass event. Note the different scales on the y-axis on the two time domain plots.

Control measurement Uncontrolled mass-event measurement Control measurement

Vertical mode 1

Vertical mode 3

Vertical mode 5

40

The result agrees with above mentioned research. The natural frequencies for the human body, summarized by Sachse et al (2003) range from 4.1-5Hz, and damping ratios from 23-53%. The effect, as shown by Van Nimmen et al. (2016) from HSI have a larger response in bridges with natural frequencies of 2.5-5Hz, since resonance then occur with the human body and the inherit modal damping ratio of the pedestrian loaded bridge will be larger than that of the empty structure. Compared to the bridge in question where Hz which is close to the natural frequency of the human body an increase in modal damping ratio was to be expected. However, the amount of pedestrian on the bridge do also affect the HSI. Using the well-known equation for the frequency of a simply supported beam, the relative change in frequency because of the contribution of an added mass can be shown graphically.

Figure 26: Relative change in natural frequency because of an increase in mass. Here is the additional mass evenly distributed over the beam, and is the mass of the empty beam. Clearly visible on the y-axis is a decrease in frequency for the beam as the additional mass increases.

41

Furthermore, contribution from walking frequencies also influence the effect of the HSI as shown by Zivanvic et al. (2009) and Archbold and Mullarney (2017) where modal properties of the structure differ when a passive person load the bridge in comparison to pedestrians and crowds thus showing that the HSI problem is more complex. Figure 27 below sums up the different factors, which combined creates the human structure interaction.

Figure 27: Human structure interaction.

42

Chapter 5

5 Conclusions and further research

5.1 Conclusions

Variations of the dynamic properties have been studied by measuring the acceleration in a steel-arch bridge over a temperature span from to , under a static mass and under an uncontrolled mass event separately. Finite element modelling and beam-theory helped together with a literature review to quantify the result.

The natural frequencies were observed to decrease 2, 3, 6 and 2% for vertical mode 1, 3, 5 and torsional mode 5 respectively. The decrease of frequency with increasing temperature is likely to come mostly from changes in Young’s modulus of the mastic asphalt, concrete and steel. And that the change in frequency because of Young’s modulus was greater than the geometrical changes. However, the geometrical changes of the steel because of thermal expansion was estimated to also contribute to the decrease of the natural frequencies, because steel makes up the framework of the bridge.

The damping ratio was observed to increase 146% and natural frequency decrease 4% for the first vertical mode under the uncontrolled mass event. This HSI effect was discussed to come from a resonance response between the bridge first vertical frequency and the natural frequency of the human body. Making humans present on the bridge to act similar to a tuned mass damper (TMD) and increasing the inherited damping ration of the first mode for the pedestrian-structure system compared to that of the empty bridge. The decrease in natural frequency was discussed to come from an increase in modal mass of the vibration. However only the natural frequency of the first mode was observed to decrease, and walking frequency of de pedestrians together with the number of pedestrians are likely to also influence the human-structure-response.

43

5.2 Further research

If temperature effects on modal properties is of significance from a design point of view is still unclear, most of the literature that reports on the subject have observed a marginal decrease in natural frequency with increasing temperature. The hypothesised temperature effect on modal damping ratios is however still undecided, and more testing is needed to quantify the effect and determine its potential magnitude.

The presence of pedestrians does seem to have a beneficial contribution to modal damping ratios of bridges. However, the likelihood of a load-case consisting of both a static mass and pedestrian loading is credible why it is this author’s opinion that further research on the influence on modal damping ratio from a static mass is of interest.

Lastly, a study to quantify the effect on structural modal properties from humans walking

and running frequencies is needed, so in combination with the effect from their internal modal properties can be better understood.

44

Bibliography

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Archbold P and Mullarney B. (2017). The relationship between pedestrian loading and dynamic response of an FRP composite footbridge. Bridge structures 13(2017) 147-157

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Cheynet E, Snæbjörnsson J, Bogunovi´c Jakobsen J. (2017). Temperature effects on the modal properties of a suspension bridge. Department of Mechanical and Structural Engineering and Materials Science, University of Stavanger. School of Science and Engineering, Reykjavík University, Menntavegur 1.

Chopra A. (2013). Dynamics of structures, fourth edition (s52-53, 48, 82-84). PEARSON

Falati, S. (1999) The contribution of non-structural components to the overall dynamic behaviour of concrete floor slabs. Thesis (PhD). University of Oxford, Oxford, UK.

Gonzales I, Ülker-Kaustell M, Karoumi R. (2013). Seasonal effects on the stiffness properties of a ballasted railway bridge. Engineering structures.

James M, Brownjohn W. (2000). Energy dissipation in one-way slabs with human participation. Nanyang Technological University

Kasperski M. (2014). Damping induced by pedestrians. Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014.

Li H, Li S, Ou J, Li H. (2010). Modal identification of bridges under varying environmental conditions: Temperature and wind. Struct. Control health monit. 2010; 17:495-512

Reiterer M, Hochrainer M. (2004). Damping of footbridge vibrations by tuned liquid column dampers: A novel experimental model set-up. Conference: 21st Danubia-Adria Symposium on Experimental Methods in Solid Mechanics (DAS 2004)

SS-EN 1990. (2002). Grundläggande dimensioneringsregler för bärverk. European Committee for Standardization CEN, Brussel. Sachse R, Pavic A, Reynolds P. (2003). Human-Structure Dynamic Interaction in Civil Engineering Dynamics: A Literature Review. The Shock and Vibration Digest 35(1).

Where cross-references are made to

Hothan, S. (1999). Einfluß der Verkehrslast – Mensch – auf das Eigenschwingungsverhalten von Fußgängerbrücken und die Auslegung linearer Tilger. Thesis (Dipl.-Ing.). Universität Hannover, Hanover, Germany. (17-19)

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Schubert S, Gsell D, Steiger R, and Feltrin G. (2010). Influence of asphalt pavement on damping ratio and resonance frequencies of timber bridges. Engineering structures.

Siddiqui M S, Fowler D W. (2015). A systematic optimization technique for the coefficient of thermal expansion of Portland cement concrete. Construction and building materials.

E. L. Schaffer, Chairman F. S. Harvey D. B. Jeanes R.H. lding T. T. Lie R. W. Fitzgerald, Past Chairman K. H. Almand J. R. Barnett B. Bresler J. F. Fitzgerald R. P. Fleming W.L.Gamble R.G. Gewain T. D. Lin S. E. Magnusson J. R. Milke M.M. Rudick. (1992). Structural fire protection. American Society of Civil Engineers. Page 223-228

SMHI. (2016). Snöns densitet, vatteninnehåll och tyngd. https://www.smhi.se/kunskapsbanken/meteorologi/vikten-pa-sno-1.10378 (downloaded 2018-04-13)

Tan Y, Sun Z, Gong X, Xu H, Zhang L, Bi Y. (2017). Design parameter of low-temperature performance for asphalt mixtures in cold regions. Construction and Building Materials.

Ulker-Kaustell M, Karoumi R. (2010). Application of the continuous wavelet transform on the free vibrations of a steel–concrete composite railway bridge. Engineering structures.

Van Nimmen K, Lombaert G, Roeck DE, Van den Broeck P. (2016). The impact of vertical human-structure interaction on theresponse of footbridges to pedestrian excitation. Journal of Sound and Vibration 402.

Xia Y, Chen B, Weng S, Ni YQ, X YL. (2012). Temperature effect on vibration properties of civil structures: a literature review and case studies. Civil Health Monitor2012;2(1):29–46.

Xia Y, Hao H, Zanardo G, Deeks A. (2005). Long term vibration monitoring of an RC slab. Temperature and humidity effect. Engineering structures.

Youssef M.A, Moftah M. (2007). General stress–strain relationship for concrete at elevated temperatures. Engineering structures.

Zivanovic S, Diaz M, Pavic A. (2009). Influence of Walking and Standing Crowds on Structural Dynamic Properties. Proceedings of the IMAC-XXVII Zivanovic S. (2012). Benchmark footbridge for vibration serviceability assessement under vertical component of pedestrian load. Journal of Structural Engineering.

46

Appendix A

This appendix shows the individual damping calculation for different jumps calculated with logarithmic decrement method (LD) and half power bandwidth method (HPB). It also describes different background variables for each measurement occasion separately.

5.3 Cold measurement

Temperature: -9◦C Wind: ~2 m/s No snow on the bridge Date: 2018-02-14

Test 1: Device 01 placed at mid-span, device 04 placed at quarter-span. Two persons jump at mid-span with approximately 45 s in between jumps Test 2: Device 01 placed at mid-span, device 04 placed at quarter-span. Two persons continuously jump at mid-span for approximately 30 s Test 3: Device 01 placed at mid-span, device 04 placed at quarter-span. Two persons jump at quarter-span with approximately 45 s in between jumps Test 4: Device 01 placed at mid-span, device 04 placed at quarter-span. Two persons continuously jump at quarter-span for approximately 30 s Test 5: Device 01 and 04 both placed at mid-span, device 01 on the north side and device 04 on the south side. Two persons continuously jump at the north side for approximately 30 s Test 6: Device 01 and 04 both placed at quarter-span, device 01 on the north side and device 04 on the south side. Two persons continuously jump at the north side for approximately 30 s Test 7: Device 01 and 04 both placed at mid-span, device 01 on the north side and device 04 on the south side. Two persons jump at mid-span in the middle between the devices with approximately 45 s in between jumps. The mean value of the acceleration was then obtained

47

Table 7: Damping with LD vertical mode 1

Test Device Type of mode

Jump number, or

Jump frequency

Un-fitted damping ratio with

LD [%]

Fitted damping ratio with

LD [%]

1

01

3.348 Hz Vertical ( )

0.489 0.467 01 0.512 0.470

01 0.370 0.484

04 0.450 0.463

04 0.493 0.470

04 0.442 0.488

3

01 0.468 0.435 01 0.456 0.430

01 0.450 0.482

04 0.452 0.468 04 0.531 0.467

04 0.409 0.500

7 01/04 mean 0.443 0.453 01/04 mean 0.467 0.482

01/04 mean 0.483 0.490

2 01 3.8 Hz 0.446 0.441

04 3.8 Hz 0.448 0.441

4 01 3.0 Hz 0.437 0.461

04 3.0 Hz 0.468 0.506

Mean 0.459% 0.468% Deviation 0.000356 [-] 0.000215 [-]

48



Jump number, or

Jump frequency

Un-fitted damping ratio

with LD [%]

1

01


0.661 01 0.656 01 0.704 01 0.694

04 0.617 04 0.642 04 0.670 04 0.667

3

01 0.660 01 0.760 01 -

04 0.692 04 0.702 04 0.725

7 01/04 mean 0.660 01/04 mean 0.656 01/04 mean 0.670

6 01/04 mean 3.7 Hz 0.630

Mean 0.675% Deviation 0.000353[-]

49



Jump number, or

Jump frequency

Un-fitted damping ratio with LD [%]

1

01


1.437 01 1.674 01 1.392

04 1.467 04 1.615 04 1.293

3

01 1.880 01 1.247 01 1.294 01 1.746 01 1.576 01 1.459

04 1.776 04 1.486 04 1.528 04 - 04 - 04 -

7 01/04 mean 1.177 01/04 mean 1.463 01/04 mean 1.412

2

01 3.8 Hz 1.660 01 3.0 Hz 1.088

04 3.8 Hz 1.659 04 3.0 Hz 0.982

4 01 3.0 Hz 1.486

04 3.0 Hz 1.443

5 01/04 mean 2.45 Hz 1.475

Mean 1.469% Deviation 0.00212[-]

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Table 10: Damping with LD torsional mode 5

Test Device Type of mode Jump number, or

Jump frequency


LD [%] 5 01/04 torsion

10.610 Hz

Torsional ( )

2.33 Hz 1.197

7 01/04 torsion 1.122

6 01/04 torsion 3.65 Hz 1.207 Mean 1.175 %

Deviation 0.000465 [-]

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Table 11: Damping with HPB vertical mode 1,3,5 & torsional mode 5


Un-fitted damping ratio with HPB [%]

1 01


0.461

1 01 0.465

1 01 0.563 1 04 0.453 1 04 0.473 2 01 0.504 2 01 0.389 3 01 0.418 3 04 0.464 7 01 0.535 7 04 0.400

Mean 0.466% Deviation 0.000533[-]

1 01


0.647 1 01 0.714 1 01 0724 1 04 0.637 1 04 0.699 1 04 0.689 3 01 0.653 4 01 0.575 4 04 0.674 5 01 0.652 7 01 0.785 7 04 0.666

Mean 0.676% Deviation 0.000523[-]

1 01


1.215 1 01 1.475 1 01 1.325 1 04 1.341 1 04 1.381 1 04 1.444 2 01 1.604 3 04 1.425 7 01 1.492 7 02 1.573

Mean 1.428% Deviation 0.00118[-]

5 01/04 Torsion 10.610 Hz

Torsional ( ) 1.220

5b 01/04 Torsion 1.147 6 01/04 Torsion 1.125

Mean 1.164% Deviation 0.000497[-]

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Table 12: Frequencies and where they are visible

Test Device Frequency [Hz] Mode Visible at: 1 01 3.340


Mid-span 2 01 3.349 3 01 3.359 4 01 3.330 5 01/04 mean 3.353 7 01/04 mean 3.367 1 04 3.340

Quarter-span

2 04 3.362 3 04 3.359 4 04 3.330 6 01/04 mean 3.343

Mean 3.348 Deviation 0.0127

1 01 5.529


Mid-span 2 01 5.496 3 01 5.524 4 01 5.481 5 01/04 mean 5.528 7 01/04 mean 5.493


1 01 8.856

8.880 Hz

Vertical ( )

Mid-span 2 01 8.906

3 01 8.860 7 01/04 mean 8.919 1 04 8.857

Quarter-span 2 04 8.900 3 04 8.860


5 01/04 torsion 10.62 10.610 Hz

Torsional ( ) Mid-span

5 01/04 torsion 10.56 7 01/04 torsion 10.65


53

5.4 A static mass

Temperature: -11.5◦C Wind: ~2 m/s ~2-5 cm of packed snow on the bridge Date: 2018-02-28

Test 1: Device 01 and device 04 placed at mid-span. Device 01 at the north side and device 04 placed at the south side. Two persons jump at the south side with approximately 60 s in between jumps. Test 2: Device 01 and device 04 placed at quarter-span. Device 01 at the north side and device 04 placed at the south side. Two persons jump at the south side with approximately 60 s in between jumps.



Jump number, or

Jump frequency


with LD [%]

Fitted damping ratio with LD

[%]

1

01/04 mean


0.354 0.381 01/04 mean 0.398 0.419 01/04 mean 0.398 0.370

2 01/04 mean 0.390 0.388 01/04 mean 0.371 0.381 01/04 mean 0.386 0.355 01/04 mean 0.390 0.441

Mean 0.384 % 0.391 % Deviation 0.000160 [-] 0.000295 [-]



Jump number, or

Jump frequency


with LD [%]

1

01/04 mean 5.536 Hz Vertical ( )

0.504 01/04 mean 0.452

Mean 0.478 % Deviation 0.000368 [-]

54



Jump number, or

Jump frequency


with LD [%]

1 01/04 mean


1.401 1 01/04 mean 0.983 1 01/04 mean 1.510

2 01/04 mean 1.299 2 01/04 mean 1.660 2 01/04 mean 1.285 2 01/04 mean 1.117 2 01/04 mean 1.423 2 01/04 mean 0.978 2 01/04 mean 1.017 2 01/04 mean 1.030 2 01/04 mean 0.819

Mean 1.210% Deviation 0.00257 [-]



Jump number, or

Jump frequency

Un-fitted damping

ratio with LD [%]

1

01/04 torsion 10.608 Hz

Torsional ( )

0.962 01/04 torsion 0.832

01/04 torsion 1.076

01/04 torsion 0.793

Mean 0.916 % Deviation 0.00129[-]

55




1 01


0.374 1 04 0.378 1 01 0.377 1 04 0.381 1 01 0.391 1 04 0.400 1 01 0.404 1 04 0.362 2 01 0.364 2 04 0.367 2 01 0.404 2 04 0.398 2 01 0.388 2 04 0.436 2 01 0.400 2 04 0.329

Mean 0.385% Deviation 0.000240[-]

1 01

5.536 Hz

Vertical ( )

0.412

1 04 0.476 1 01/04 mean 0.438 1 01/04 mean 0.482 1 01/04 mean 0.418 2 01/04 mean 0.463 2 01/04 mean 0.487

Mean 0.454% Deviation 0.000310[-]

1 01/04 mean

8.903 Hz

Vertical ( )

1.092

1 01/04 mean 1.211 1 01/04 mean 1.136 2 01/04 mean 1.306 2 01/04 mean 1.035 2 01/04 mean 1.237 2 01/04 mean 1.166

Mean 1.169% Deviation 0.000914[-]

1 01/04 torsion

10.608 Hz Torsional ( )

0.904 1 01/04 torsion 0.933 1 01/04 torsion 0.922 1 01/04 torsion 0.912

Mean 0.918% Deviation 0.000126[-]

56


Test Device Frequency [Hz] Mode Visible at; 1

01/04 mean

3.341

Vertical ( )

Mid-span 1 3.341

1 3.355 1 3.362 2 3.346

Quarter-span

2 3.346 2 3.356 2 3.350 2 3.363 2 3.349


1

01/04 mean 5.542

Vertical ( )

Mid-span 1 5.505 1 5.546 1 5.550


1

01/04 mean

8.892

Vertical ( )

Mid-span 1 8.874

1 8.854 1 8.928 2 8.924

Quarter-span

2 8.900 2 8.928 2 8.886 2 8.963 2 8.885


1

01/04 torsion 10.610

Torsional ( )

Mid-span 1 10.600 1 10.640 1 10.580


57

5.5 Medium measurement one

Air temperature: -2.0◦C Asphalt surface temperature: -2.2◦C Wind: ~2 m/s No snow on the bridge Date: 2018-03-22

Test 1: Device 01 and device 04 are both placed at mid-span. Device 01 on the north side and device 04 on the south. Two persons jump at midspan, in between the devices with approximately 60 s in between the jumps Test 2: Device 01 and device 04 are both placed at mid-span. Device 01 on the north side and device 04 on the south. Two persons jump at midspan, at the south side on the bridge with approximately 60 s in between the jumps Table 19: Damping with LD vertical mode 1


Jump number, or

Jump frequency


LD [%]

1

01/04 mean


0.503 0.441

0.381

0.467

0.410

0.376

0.414

0.416

0.412

0.461

0.376

0.386

2

01/04 mean

0.430

0.443

0.387

0.446

0.431

0.336

0.429

0.380

0.441

0.367

0.352

0.393


58



Jump number, or

Jump frequency


with LD [%]

1

01/04 mean


0.614 0.614

0.577

0.573

0.611 0.575 0.671

0.598

0.440

2

01/04 mean 0.656 0.644

0.489 Mean 0.589

Deviation 0.0666

59



Jump number, or

Jump frequency


1

01/04 mean


0.972 1.516

1.283

0.979 0.932 1.355

1.029

1.135

1.002

1.205

1.182

1.190

1.166

1.072

2

01/04 mean

1.238 0.980

0.920

1.036

1.575

1.310

1.325 1.234

1.218 1.143

1.140 Mean 1.165

Deviation 0.1703

60



Jump frequency


LD [%]

2

01/04 torsion


0.741

0.777

0.795 0.705 0.799 0.817 0.752 0.678 0.800 0.652 0.774


61




1

01/04 mean


0.390

1 0.352

1 0.361 1 0.460 1 0.357 1 0.401 1 0.353 2 0.419 2 0.414 2 0.416 2 0.393 2 0.420 2 0.421


1

01/04 mean


0.620 1 0.631 1 0.573 1 0.644 1 0.598 2 0.607 2 0.590 2 0.589 2 0.616 2 0.611


1

01/04 mean


0.939 1 1.081 1 1.060 1 1.090 1 1.381 1 1.098 1 1.075 2 1.050

62


Test Device Frequency [Hz] Mode 1 01/04 mean 3.328

Vertical ( )

1 01/04 mean 3.328 1 01/04 mean 3.330 1 01/04 mean 3.329 1 01/04 mean 3.326 1 01/04 mean 3.335 1 01/04 mean 3.325 2 01/04 mean 3.332 2 01/04 mean 3.332 2 01/04 mean 3.336 2 01/04 mean 3.332 2 01/04 mean 3.328 2 01/04 mean 3.339


1 01/04 mean 5.426

Vertical ( )



2 0.952 2 1.138 2 1.152 2 1.168 2 1.093


2

01/04 torsion


0.714 2 0.725 2 0.710 2 0.650 2 0.742 2 0.732


63

1 01/04 mean 8.606

Vertical ( )



2 01/04 torsion 10.50 2 01/04 torsion 10.56

Torsional ( )

2 01/04 torsion 10.55 2 01/04 torsion 10.54 2 01/04 torsion 10.57 2 01/04 torsion 10.56


64

5.6 Medium measurement two

Air temperature: 0.5.0◦C Asphalt surface temperature: 1◦C Wind: ~5 m/s No snow on the bridge Date: 2018-03-22

Test 1: Device 01 and device 04 are both placed at mid-span. Device 01 on the north side and device 04 on the south. Two persons jump at midspan, in between the devices with approximately 60 s in between the jumps Test 2: Device 01 and device 04 are both placed at mid-span. Device 01 on the north side and device 04 on the south. Two persons jump at midspan, at the south side on the bridge with approximately 60 s in between the jumps Table 25: Damping with LD vertical mode 1


Jump number, or

Jump frequency


LD [%]

1

01/04 mean

Vertical ( )

0.478 0.553

0.500

0.528

0.496

0.523

0.418

0.571

0.552

2

01/04 mean

0.459

0.443

0.586

0.562

0.581

0.580

0.512

0.436

0.621

0.497

65



Jump number, or

Jump frequency


with LD [%]

1

01/04 mean

Vertical ( )

0.935 0.889

0.798

0.969

0.906 1.058 0.999

0.831

0.836

0.714

0.788

2

01/04 mean 0.975 0.846

1.002

Mean Deviation

66

0.954

0.858

0.983

0.867

0.984 Mean

Deviation



Jump number, or

Jump frequency


1

01/04 mean

Vertical ( )

1.360 1.454

1.740

1.691 1.602 1.411

1.492

1.479

1.475

1.826

1.548

1.301 1.531

1.334

1.399

67

2

01/04 mean

1.509

1.252

1.499 1.527

1.827 Mean

Deviation Table 28: Damping with LD torsional mode 5


Jump frequency


LD [%]

2

01/04 torsion

Torsional ( )

0.737

0.777

0.914 0.813 0.740 0.850 0.758 0.847 0.838

Mean Deviation




1

01/04 mean

Vertical ( )

0.420

1 0.448

1 0.397 2 0.497 2 0.600 2 0.533 2 0.554 2 0.600

Mean

68

Table 30: Frequencies


Vertical ( )

1 01/04 mean 3.289 1 01/04 mean 3.298 1 01/04 mean 3.290 1 01/04 mean 3.288 2 01/04 mean 3.293 2 01/04 mean 3.276 2 01/04 mean 3.288 2 01/04 mean 3.290

Mean Deviation

1 01/04 mean 5.368

Deviation 1

01/04 mean

Vertical ( )

0.848 1 0.880 1 0.928 1 0.854 2 0.983 2 1.093 2 0.816 2 0.931

Mean Deviation

1

01/04 mean

Vertical ( )

1.298 1 1.554 1 1.576 2 1.367 2 1.374 2 1.411

Mean Deviation

2

01/04 torsion

Torsional ( )

0.833 2 0.873 2 0.791 2 0.812

Mean Deviation

69

1 01/04 mean 5.391

Vertical ( )

1 01/04 mean 5.411 1 01/04 mean 5.383 1 01/04 mean 5.381 2 01/04 mean 5.404 2 01/04 mean 5.404 2 01/04 mean 5.372 2 01/04 mean 5.383

Mean Deviation

1 01/04 mean 8.554

Vertical ( )

1 01/04 mean 8.558 1 01/04 mean 8.547 1 01/04 mean 8.590 1 01/04 mean 8.497 2 01/04 mean 8.544 2 01/04 mean 8.551 2 01/04 mean 8.601 2 01/04 mean 8.590

Mean Deviation

2 01/04 torsion 10.540 2 01/04 torsion 10.550

Torsional ( )

2 01/04 torsion 10.500 2 01/04 torsion 10.500 2 01/04 torsion 10.450

Mean Deviation

5.7 Warm measurement

Air temperature: 12.0◦C Asphalt surface temperature: 11.5◦C Wind: ~2 m/s No snow on the bridge Date: 2018-04-19

Test 1: Device 01 and device 04 are both placed at mid-span. Device 01 on the north side and device 04 on the south. Two persons jump at midspan, in between the devices with approximately 60 s in between the jumps Test 2: Device 01 and device 04 are both placed at mid-span. Device 01 on the north side and device 04 on the south. Two persons jump at midspan, at the south side on the bridge with approximately 60 s in between the jumps

70


Table 32: Damping with LD vertical mode 3 Test Device Type of

mode Jump

number, or Jump

frequency


with LD [%]

1

01/04 mean


0.781 0.783

0.615

0.744

0.841

2

01/04 mean 0.777 0.740

Mean


Jump number, or

Jump frequency


LD [%]

1

01/04 mean


0.480 0.485

0.482

0.470

0.474

0.431

0.642

0.514

0.471

0.524

0.476

0.523

0.563

2

01/04 mean

0.56

0.553

0.465

0.529

0.525

0.448

0.495

0.522

0.509

0.486

Mean Deviation

71

Deviation



Jump number, or

Jump frequency


1

01/04 mean


1.260 1.107

1.505

1.310 1.304 1.194

1.339

1.303

1.562

1.598

1.241

2

01/04 mean

1.381 1.345

1.247

1.262

1.253

1.364

1.579 1.372

Mean Deviation



Jump frequency


LD [%]

2

01/04 torsion


0.783

0.856

0.955 0.956 0.888 0.652 0.873 0.694 0.743 1.085 0.931 0.735

72

Mean Deviation




1

01/04 mean


0.437

1 0.430

1 0.438 1 0.487 1 0.505 2 0.430 2 0.434 2 0.436 2 0.439 2 0.510

Mean Deviation

1

0.806 1 0.712 1 0.702

73






1

01/04 mean

5.342 Hz

Vertical ( )

0.773 1 0.716 1 0.744 1 0.781 1 0.763 2 0.660 2 0.719 2 0.731 2 0.793 2 0.662

Mean Deviation

1

01/04 mean


1.264 1 1.106 1 1.281 1 1.398 1 1.218 1 1.370 1 1.297 1 1.264 1 1.373 1 1.203 2 1.101 2 1.353 2 1.289 2 1.266 2 1.419

Mean Deviation

2

01/04 torsion


0.770 2 0.824 2 0.653 2 0.840 2 0.893

Mean Deviation

74

1 01/04 mean 5.367




1 01/04 mean 8.323




2 01/04 torsion 10.470 2 01/04 torsion 10.410


2 01/04 torsion 10.450 2 01/04 torsion 10.470 2 01/04 torsion 10.470


75

5.8 Uncontrolled mass event

Air temperature: 12.5◦C Asphalt surface temperature: 13.5◦C Wind: ~2 m/s Walking pace of people: ~1 steps/s Less than 1 walkway surface per person Two vans positions at ca quarter-span of the bridge Date: 2018-04-15

Test 1: Device 01 and device 04 are both placed at mid-span. Both devices placed at the south side of the bridge Table 37: Damping with HPB vertical mode 1,3,5 & torsional mode 5

Test Device Type of mode Un-fitted damping ratio with HPB [%]

1

1.597

1.061

1.592 1.061 1.104 0.844

76


Test Device Frequency [Hz] Mode 1

01/04 mean 3.017

Vertical ( )

01/04 mean 3.05 01/04 mean 3.083 01/04 mean 3.167 01/04 mean 3.183 01/04 mean 3.200 01/04 mean 3.283 01/04 mean 3.133 01/04 mean

Mean Deviation

01/04 mean 5.317

01/04 mean Vertical ( ) 1.485 0.905

Mean Deviation

77

1

01/04 mean 5.333

Vertical ( )

01/04 mean 5.183 01/04 mean 5.333 01/04 mean 5.333 01/04 mean 5.367 01/04 mean 5.417 01/04 mean 5.350 01/04 mean

Mean Deviation

1

01/04 mean 8.333

Vertical ( )

01/04 mean 8.317 01/04 mean 8.35 01/04 mean 8.283 01/04 mean 8.433 01/04 mean 8.367 01/04 mean 8.383 01/04 mean 8.400 01/04 mean

Mean Deviation

01/04 torsion

Data missing

01/04 torsion

Torsional ( )

01/04 torsion 01/04 torsion 01/04 torsion

Mean Deviation

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