Bridge Monitoring to Allow for Reliable Dynamic Finite Element Modelling

8/13/2019 Bridge Monitoring to Allow for Reliable Dynamic Finite Element Modelling.

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c Johan Wiberg 2006Royal Institute of Technology (KTH)Department of Civil and Architectural Engineering

Division of Structural Design and BridgesStockholm, Sweden, 2006


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Abstract

Todays bridge design work in many cases demands a trustworthy dynamic analysisinstead of using the traditional dynamic amplication factors. In this thesis a reliable3D Bernoulli-Euler beam nite element model of the New rsta Railway Bridgewas prepared for thorough dynamic analysis using in situ bridge monitoring forcorrelation. The bridge is of the concrete box girder type with a heavily reinforcedand prestressed bridge deck. The monitoring system was designed for long termmonitoring with strain transducers embedded in the concrete and accelerometersmounted inside the edge beams and at the lower edge of the track slab.

The global nite element model used the exact bridge geometry but was simpliedregarding prestressing cables and the two railway tracks. The prestressing cables andthe tracks were consequently not included and an equivalent pure concrete modelwas identied.

A static macadam train load was eccentrically placed on one of the bridges twotracks. By using Vlasovs torsional theory and thereby including constrained warp-ing a realistic modulus of elasticity for the concrete without prestressing cablesand stiffness contribution from the railway tracks was found. This was allowed bycomparing measured strain from strain transducers with the linear elastic nite el-ement models axial stresses. Mainly three monitoring bridge sections were used,each of which was modelled with plane strain nite elements subjected to sectionalforces/moments from a static macadam train load and a separately calculated tor-sional curvature.

From the identied modulus of elasticity the global nite element model was updatedfor Poissons ratio and material density (mass) to correspond with natural frequen-cies from the performed signal analysis of accelerometer signals. The inuence of warping on the natural frequencies of the global nite element model was assumedsmall and the bridges torsional behaviour was modelled to follow Saint-Venantstorsional theory.

A rst preliminary estimation of modal damping ratios was included. The resultsindicated that natural frequencies were in accordance between modelling and signalanalysis results, especially concerning high energy modes. Estimated damping ratiosfor the rst vibration modes far exceeded the lower limit value specied in bridgedesign codes and railway bridge dynamic analysis recommendations.

Keywords: Bridge monitoring, FE modelling, signal analysis, torsion, warping,modulus of elasticity, natural frequency, damping, nite differences.

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Sammanfattning

I designen av dagens broar stlls ofta krav p trovrdiga dynamiska analyser fram-fr anvndandet av traditionella dynamiska frstoringsfaktorer. I denna avhandlingprepareras en plitlig 3D nit elementmodell av Nya rstabron med Bernoulli-Euler balkelement fr grundlig dynamisk analys genom att anvnda mtningar pbron som korrelationsmedel. Bron r av betong och typen ldbalkbro med extremtarmerat och efterspnt brodck. Mtsystemet var designat fr lngtidsmtningarmed trdtjningsgivare ingjutna i betongen och accelerometrar monterade i kant-balkarna och p sprplattans underkant.

Den globala nita elementmodellen anvnde den exakta brogeometrin men var fr-enklad stillvida att spnnkablar och de tv spren inte inkluderades. Istllet iden-tierades en ekvivalent modell bestendes av enbart betong.

En statisk last frn ett makadamtg placerades excentriskt p ett av brons spr.Genom att anvnda Vlasovs vridningsteori och drigenom beakta frhindrad vlv-ning hittades en realistisk elasticitetsmodul fr betongen utan spnnkablar ochstyvhetsbidrag frn spren. Detta var mjligt genom att jmfra uppmtta tjningarfrn trdtjningsgivarna med axialspnningar frn den linjrelastiska nita element-modellen. I huvudsak tre mtsektioner p bron anvndes och fr vilka var och enmodellerades med plana tjningelement utsatta fr tvrsektionskrafter och momentfrn den statiska makadamtgslasten och den separat berknade vridningskrknin-gen.

Frn den identierade elasticitetsmodulen uppdaterades den globala nita element-modellen betrffande tvrkontraktionstal och materialdensitet fr att verensstmmamed egenfrekvenser frn utfrd signalanalys p accelerometersignalerna. Inverkanav vlvning p den globala modellens egenfrekvenser antogs liten och brons vrid-ningsbeteende modellerades att flja Saint-Venants vridningsteori.En frsta preliminr uppskattning av modala dmpningskvoter utfrdes. Resultatenvisade p att egenfrekvenserna verensstmde vl mellan modell och signalanalysensresultat, srskilt betrffande moderna med hg energi. Uppskattade dmpnings-kvoter fr de frsta fria vibrationsmoderna verskred klart det lgre grnsvrdetspecicerat i bronormer och rekommendationer fr dynamisk analys av tgbroar.

Nyckelord: Mtning, nit elementmodellering, signalanalys, vridning, vlvning,elasticitetsmodul, egenfrekvenser, dmpning, nita differenser.

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Contents

Abstract i

Sammanfattning iii

Acknowledgements v

List of Symbols and Abbreviations xi

1 Introduction 1

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

1.1.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Aim and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Bridge Monitoring 7

2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 The Monitoring Concept . . . . . . . . . . . . . . . . . . . . . 7

2.2 Monitoring History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Dynamic Bridge Monitoring . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Monitoring Benets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Monitoring Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 The New rsta Railway Bridge 21

3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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3.3 Bridge Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Instrumentation and Data Acquisition . . . . . . . . . . . . . . . . . 25

4 Bridge Dynamics 35

4.1 Structural Dynamic Theory . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Analytical Natural Beam Frequencies . . . . . . . . . . . . . . 39

4.1.2 The FE Method . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Dynamic Amplication Factor . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Experimental Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.1 Time and Frequency Domain in Signal Analysis . . . . . . . . 47

4.3.2 Viscously Damped Free Vibration . . . . . . . . . . . . . . . . 544.3.3 Half-Power Bandwidth . . . . . . . . . . . . . . . . . . . . . . 56

4.3.4 Rayleigh Damping . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Design Code Values and Recommendations . . . . . . . . . . . . . . . 59

4.4.1 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.2 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.4 Fundamental Frequency . . . . . . . . . . . . . . . . . . . . . 61

5 Signal Analysis Fundamentals 63

5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.1 Anti-Alias Filtering . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.4 Signal Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.4.1 Savitzky-Golay smoothing . . . . . . . . . . . . . . . . . . . . 68

5.5 Digital Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.6 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.7 Signal Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Bridge FE modelling 75

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6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.1.1 Conditions and Assumptions . . . . . . . . . . . . . . . . . . . 75

6.1.2 Post-Tensioned Concrete Bridges . . . . . . . . . . . . . . . . 76

6.1.3 SOLVIAR FE Program Summary . . . . . . . . . . . . . . . . 766.2 Bridge Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2.1 Clothoid Shaped Transition Curve . . . . . . . . . . . . . . . . 78

6.2.2 Bridge Supports and Boundary Conditions . . . . . . . . . . . 79

6.2.3 Cross Section Parameters . . . . . . . . . . . . . . . . . . . . 80

6.2.4 Assembling of Beam Elements . . . . . . . . . . . . . . . . . . 82

6.3 FE Model Verication Methods . . . . . . . . . . . . . . . . . . . . . 84

6.3.1 Cross Section Stresses Comparison . . . . . . . . . . . . . . . 84

6.3.2 Frequency Analysis and the Subspace Iteration Method . . . . 92

7 Results 93

7.1 Cross Sectional Stresses . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.1.1 Torsional Curvature . . . . . . . . . . . . . . . . . . . . . . . 93

7.1.2 Static Load Testing Stresses from Macadam Train . . . . . . . 967.2 Natural Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2.1 Track Slab Frequencies . . . . . . . . . . . . . . . . . . . . . . 113

7.3 Damping Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8 Discussion and Conclusions 119

8.1 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8.1.1 Improved Structural Properties Prediction UsingARTeMIS Extractor . . . . . . . . . . . . . . . . . . . . . . . 121

8.1.2 Interaction Modelling . . . . . . . . . . . . . . . . . . . . . . . 121

8.1.3 Mode Superposition vs Direct Time Integration . . . . . . . . 121

Bibliography 123

A Section Generation and Property Listing in SOLVIA R 129

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List of Symbols and Abbreviations

Roman Upper Case

[C ] torsional rigidity matrix, see equation (6.24), page 90 Nm2

C torsional constant, see equation (4.29), page 42 m2

C n generalised damping for nth mode, see equation (4.14), page 37 Ns/m[C ] diagonal matrix of C n , see equation (4.12), page 37 Ns/m

C SC warping constant with respect to the shearcentre, see equation (6.9), page 87 m6

[C w ] warping stiffness matrix, see equation (6.24), page 90 Nm4

D d/dx , see equation (6.12), page 88

E modulus of elasticity, see equation (4.27), page 40 Pa

EC SC warping stiffness, see equation (6.9), page 87 Nm4

F dyn dynamic excitation force, see equation (4.32), page 43 N

F r axial/normal beam element force, see equation (6.8), page 85 N

F stat static external force, see equation (4.32), page 43 N

G shear modulus, see equation (4.31), page 42 Pa

GI r torsional rigidity, see equation (6.10), page 87 Nm2

H the Hanning window function, see equation (5.3), page 67 -

I moment of inertia, see equation (4.2), page 35 m4

I r Saint-Venant torsional inertia, see equation (6.10), page 87 m4

K n generalised stiffness, mode n, see equation (4.14), page 37 N/m

[K ] diagonal matrix of K n , see equation (4.12), page 37 N/m

L span length, page 36 m

L determinant length of the bridge, page 2 m

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M n generalised mass, mode n, see equation (4.14), page 37 kg

[M ] diagonal matrix of M n , see equation (4.12), page 37 kg

M s moment about the element s-axis, see equation (6.8), page 85 Nm

M t moment about the element t-axis, see equation (6.8), page 85 Nm

M x total torsional moment, see equation (6.27), page 90 Nm

P n generalised force, mode n, see equation (4.14), page 37 N

{P } vector of P n , see equation (4.12), page 37 NR radius of curvature, see equation (6.1), page 78 m

T n nth natural period, see equation (4.10), page 36 s

T D damped natural period, page 56 s

T t total duration of time for which the structural response is to bedetermined, see equation (4.54), page 54 s

X longitudinal cross section coordinate starting from a bridgepier, see equation (3.1), page 23 m

Z vertical cross section coordinate, see equation (3.1), page 23 m

Roman Lower Casea0 damping coefficient, see equation (4.24), page 38 1/s

a1 damping coefficient, see equation (4.24), page 38 s

ab acceleration from pure bending, see equation (5.12), page 72 m/s2

a t acceleration from pure torsion, see equation (5.11), page 72 rad/s2

c damping coefficient, see equation (4.55), page 54 Ns/m

ccr critical damping coefficient, see equation (4.57), page 55 Ns/m[c] damping matrix, see equation (4.1), page 35 Ns/m

c viscous torsional damping, see equation (4.2), page 35 Ns/m

f n nth natural frequency, see equation (4.10), page 36 Hz

f D damping force, see equation (4.56), page 54 N

f M Nyquist frequency, see equation (5.1), page 64 Hz

f s sampling frequency, see equation (5.1), page 64 Hz

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k stiffness or spring constant, see equation (4.55), page 54 N/m

[k ] stiffness matrix, see equation (4.1), page 35 N/m

k torsional spring constant, see equation (4.2), page 35 N/m

m mass, see equation (4.55), page 54 kg

[m ] mass matrix, see equation (4.1), page 35 kg

mx distributed moment per unit length about the longitudinalaxis of the beam, see equation (6.9), page 87 Nm/m

{p } external forces, see equation (4.1), page 35 N p(n ) polynomial of the same order as the number of DOFs of the

system, see equation (6.29), page 92

q modal coordinate, see equation (4.14), page 37 m

{q} modal coordinate vector, see equation (4.3), page 35 mr physical length coordinate along the the longitudinal axis of

the beam, i.e. the r-axis, see equation (6.8), page 85 m

s length of clothoid shaped curve from the initial point of theclothoid with R = , see equation (6.1), page 78 m

t time, page 35 s

u real coordinate, see equation (4.26), page 40 m

{u } real coordinate vector, see equation (4.1), page 35 mw(s, t ) warping function, see equation (6.8), page 85 -

x global length coordinate, page 36 m

Greek Upper Case

[ ] modal matrix, see equation (4.3), page 35 -

Greek Lower Case

normal strain, page 86 -

n nth eigenvalue, see equation (6.28), page 92

excitation circular frequency, see equation (4.32), page 43 rad/s

D damped natural circular frequency, see equation (4.58), page 55 rad/s

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ESD Energy Spectral Density, page 53

FE Finite Element, page 2

FFT Fast Fourier Transform, page 51

HBM Hottinger Baldwin Messtechnik, page 26

LVDT Linear Variable Differential Transformer, page 25

MDF Multi-Degree-of-Freedom, page 35

MEMS Microelectromechanical systems, page 32

P Pier, page 23

PSD Power Spectral Density, page 53

SDF Single-Degree-of-Freedom, page 35

SHM Structural Health Monitoring, page 4

U1 Western track (or outer track referring to the curve), page 23

UIC Union Internationale des Chemins de Fer (International Union of Rail-ways), page 2

VBI Vehicle Bridge Interaction, page 2

WIM Weigh-In-Motion, page 4

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Chapter 1

Introduction

1.1 BackgroundToday increasingly complex and unique bridges are being constructed. This is partlydue to the present design trend and the aesthetically demand from society on con-structional work in recent years. The architect, in turn, is given a freer hand in hisor her work. The result is often sophisticated and complicated production methodssince the material consumption are also optimised as a result of restrictive bud-gets. For bridges, and especially concrete bridges, this results in extremely slenderstructures with difficult and challenging reinforcement and casting work.

As a result, the trend in todays bridge engineering research includes monitoringof the actual behaviour of newly constructed bridges, but also in the process of updating older bridges for the actual dynamic load effects. Conference contribu-tions and journal papers in this eld of research clearly indicate this huge amountof monitoring activity, see for example (Zhao and DeWolf, 2002; Xu et al., 1997;Brownjohn et al., 2003; Xia and Zhang, 2005; Zhai et al., 2004; Lin and Yang,2005; Xia et al., 2005; Kim et al., 2005; Kumar and Rao, 2003; Frba and Pirner,2001). In addition, suitable monitoring equipment such as strain transducers, breoptic sensors and accelerometers has the possibility of following movements bothduring construction stages and for the bridge in operation. Therefore, the monitor-ing process is becoming more effective. Besides, not only due to the development of monitoring equipment technology but also in the computer capacity in the last fewyears the monitoring activity on bridges is drastically increased. Above all, moni-toring is however performed as an essential part in trying to fully understand theactual behaviour of complicated bridge structures.

In contrast, todays railway infrastructure is moving towards railway lines designedfor high-speed trains. This is to make the railway traffic competitive in the growingdemand for fast and safe means of transport. The development of high-speed linesinvolves the design of both new railway sections and updating of existing ones.At the same time allowable axle loads for particularly industry railway traffic areenlarged to make transports even more effective (James, 2001).

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CHAPTER 1. INTRODUCTION

of elasticity is used. This could be seen as the actual modulus of elasticity asthe concrete is heavily compressed due to prestressing and therefore regarded asuncracked. Reasonably, the concrete therefore on its own has a higher modulus of elasticity.

Two types of methods are included in the thesis and used to verify the developedFE model. Firstly, monitored strain results are compared with those achieved fromthe FE model when a macadam train is statically loading the bridge. Torsional andwarping displacements are then included in FE cross section analyses assuming planestrain. Consequently, both the Saint-Venant and Vlasov portion are included in thetotal torsional moment due to the eccentricity of the train load. This procedureidenties an equivalent modulus of elasticity to use for the simplied global bridgeFE model consisting of beam elements. Secondly, the traditional way of comparingactual natural frequencies, i.e. measured frequencies from signal analysis, with nu-merical results is used. The latter involves an updating of the remaining material

parameters of the FE model to achieve good correlation for natural frequencies.The thesis does not describe the monitoring equipment in detail. It describes wherethe strain transducers, accelerometers and data acquisition system are mounted/placedand how the data analysis is performed.

1.3 Thesis Layout

Chapter 2 includes a brief overview of bridge monitoring. Different types of mon-

itoring systems are presented. Examples are the Weigh-In-Motion (WIM) and theStructural Health Monitoring (SHM) systems. The possibility of using monitoringfor various purposes is presented. Above all the utility of using bridge monitoringas an instrument for reliable dynamic FE modelling is highlighted.

Chapter 3 is dedicated to the New rsta Railway Bridge. Here are the bridge andits history presented. This chapter presents the instrumentation and describes thedata acquisition system.

In chapter 4 the theory of structural dynamics is presented. Analytical naturalbeam frequencies are included and the concept of the dynamic amplication factor

is presented. The last part of this chapter deals with experimental dynamics forwhich design code values and recommendations are briey specied.

Chapter 5 is completely dedicated to the theory of signal analysis and its fundamen-tals.

In chapter 6 the bridge FE model is described and made assumptions are specied.At the end the FE model verication methods are presented.

Chapter 7 has the results. First torsional curvatures and cross sectional stresses fromthe static macadam load testing are given. Finally, global natural frequencies andthe fundamental track slab frequency are presented with the rst rough estimationof structural damping ratios. The natural frequencies achieved in the bridge design

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Chapter 2

Bridge Monitoring

2.1 GeneralThere has been a recent emergence of monitoring in civil engineering. This is pri-marily a result of complicated constructional work with innovative design which hasrequired long-term monitoring to follow the behaviour of structures. Monitoring istherefore nowadays used as an instrument to secure the condence of infrastructureowners. This is often referred to as SHM which for bridges could indicate an urgentneed of strengthening or even replacement of the bridge if it comes to the worst.Monitoring has also been shown to accurately assess the bridge behaviour from load-ing and is therefore used in the identication of structural properties. Thus, due to

the development within monitoring equipment and monitoring techniques, the useof bridge monitoring has increased. This has made the usage of bridge monitoring more appropriate than bridge health monitoring .

Traditional strain gauges have the possibility of monitoring both static and dynamicload effects on bridges. For frequency analysis purposes they are however oftencomplemented with accelerometers. In addition, also the most modern bre opticsensors have been and are being developed to perform monitoring of dynamic loadeffects besides only static measurements possibilities as previously has been the case.

2.1.1 The Monitoring Concept

In this section part of the content refer to a guideline for structural health monitoring(ISIS Canada, 2001).

Monitoring of bridges is a relatively new technique within civil engineering. It isthen often referred to as structural health monitoring (SHM). Monitoring is howevernot always related to the health of structures and should therefore be dened byits objectives and by the physical system and sensors required to achieve theseobjectives. However, in all types of bridge monitoring the in situ bridge behaviouris in some way monitored under various service loads.

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CHAPTER 2. BRIDGE MONITORING

Constructional work monitoring within civil engineering can be characterised hav-ing the basic concept of composing comprehensive integration of various sensinginstruments and the ancillary system. A complete monitoring system consists of

a sensory system composed of sensors, a data acquisition system, a data processing system, a communication system, and a evaluation and verication system.

A verication system can constitute a suitable developed FE model to assure the

reasonability of achieved monitoring results and to be used for further specic re-search.

Monitoring instruments have been available and used mostly in laboratory environ-ments for several years now. In addition, there exists several bridge design codes andguidelines that have strictly been used and followed in the past. Nowadays bridgemonitoring is carried out and its objectives are consistent with the objectives of theearlier normative praxis. Consequently, bridge monitoring is only a natural stepin the augmentation of current practice by integrating new technology into bridgemonitoring systems.

In achieving a comprehensive picture of a complete monitoring system it is necessaryto understand that it exists different kinds of structural monitoring, i.e. differenttypes of systems depending on the bridge characteristics to be measured. However, jointly for all monitoring systems are that they can be divided into two monitoringsubsystems: static and dynamic monitoring. Figure 2.1 illustrates the subdivisionof a typical monitoring system.

Each subsystem may involve different types of tests such as bridge behaviour testsincluding for example stress history monitoring. Monitoring can have the intentionof demonstrating a momentary bridge behaviour characteristic such as in the perfor-mance of temporary monitoring. Indeed, global and local changes in a structure can

be identied using long term monitoring. Dynamic monitoring makes it possible toidentify the real dynamic load effects and the typical structural parameters: naturalfrequencies, mode shapes and damping ratios. These are unique for each specicbridge.

Specic for every type of a bridge monitoring system is that it as the main taskrecords data concerning a certain type of bridge behaviour or bridge parameter.However, the process of constructing a suitable monitoring system for data as-sembling involves knowledge within many engineering areas: structural mechanics,structural dynamics, materials, sensors, data acquisition, signal analysis and process-ing, computers and data communication, etc. Consequently, a complete monitoringsystem is preferably divided into and includes the following parts:

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2.1. GENERAL

Monitoring system

Staticmeasurements

Dynamicmeasurements

Temporary/Periodic

monitoring

Long term/Continuousmonitoring

Temporary/Periodic

monitoring

Long term/Continuousmonitoring

Figure 2.1: Flowchart illustration of the subdivision of a typical monitoring system.The shaded surrounding box symbolises the complete monitoring system.Modied from (ISIS Canada, 2001).

Data acquisition Data communication Signal analysis and data processing Informative storage of processed data Intelligent selection of valuable/informative results

Data evaluation Presentation of results and documentation

An illustrative owchart of a preferably monitoring system is presented in Fig-ure 2.2. These monitoring subsets are most suitable for dynamic monitoring wherethe amount of collected data is voluminous.

Necessary in data acquisition is thorough planning of the instrumentation and se-lection of suitable sensors, i.e. equipment that can monitor the absolute values of,or changes in, the physical quantities of interest in the specic project. For bridge

monitoring these parameters mainly include:

strains/stresses, deformations/displacements, accelerations, structure temperatures, air temperatures, time, and

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Data acquisitionincluding sensor

installation and datacollection

Data

involving transmissionof data for processing

Data processingincluding signalanalysis filtering

Data storagecontaining enough

information forretrievability

Data evaluationinvolving interpretingof monitoring results

as structural responses

Data selectionincluding intelligent

assortments dependingon the considered

significance of the data

communication

Figure 2.2: Flowchart of an effective monitoring course of action in a complete mon-itoring system. Redrawn and modied from (ISIS Canada, 2001).

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Table 2.1: Historical development within the eld of structural monitoring activity,including the numerical structural dynamic techniques (Wenzel, 2003).

Period Developmental phase

19th century Development of relevant structural dynamics1920-1945 Execution of simple tests at clearly dened structures1965-1975 Development of the linear FE method1970-1980 Development of the forced vibration method1975-1990 Integration of the linear FE method1990-2000 Integration of the non-linear FE analysis1992-1995 Introduction of the ambient vibration method1993-1996 Introduction of computer measuring technology for dataSince 1994 Application of the ambient vibration method

Since 1995 Further developments of monitoring methods resultingin smart monitoringSince 1996 Commercial utilisation of monitoring equipment

important to choose data that is suitable for analysis and presentation. Thus, if the purpose of the analysis is known, and the evaluator has experience, the dataselection is probably performed before the evaluation. However, the data evaluationcould indeed be performed before the selection of valuable results. Therefore, theow in Figure 2.2 between the data selection and data evaluation is reversible.

2.2 Monitoring History

The history of dynamic monitoring is briey listed in Table 2.1. The structuraldynamic theory and numerical analysis facilities are included since they are closelyrelated to the monitoring process as in the updating of FE models. To make themonitoring activity history further illustrative it is graphically illustrated in Fig-ure 2.3.

Thus, the monitoring activity has drastically exploded in the last decade due to ad-vanced technology development within the elds of personal computers and smartmonitoring techniques. The term smart is then used to emphasise the meaning of in-telligent monitoring involving durable, reliable and economical monitoring systems.However, bridge testing is not a new phenomenon, exemplied in Figure 2.4.

Another example is Switzerland where dynamic testing of bridges has been per-formed since the 1920s (ISIS Canada, 2001). The fact is that all highway bridgeswith spans greater than 20 m have been and are being tested dynamically. Thesemonitoring tests use vehicles with prescribed axle distances crossing the bridges at

different speeds. In addition, bumps are introduced at different locations on the

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2.2. MONITORING HISTORY

1860 1900 10 20 30 40 50 60 70 80 90 2000Year

M o n i t o r i n g a c t i v i t y

T h e o r y D e v e l o p m e n t

1 s t W o r

l d W a r

2 n d W o r

l d W a r

F i r s t t e s t s

E u p

h o r i a

F o r c e

d v i

b r a t i o n t e s t s

F r u s t r a t i o n

P C i n p r a c t i c e

' S m a r t ' m o n i t o r i n g

Figure 2.3: Graphical illustration of the history of dynamic monitoring activity. Re-drawn from (Wenzel, 2003).

Figure 2.4: Illustration of the testing of a steel truss in England to be used for arailway bridge in India in the 19th century. From (ISIS Canada, 2001).

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Figure 2.5: The Whipple Truss Bridge from 1867, still in service for buses and trucks,and with purely original members. From (DeLony, 1996).

deck of the bridges to further increase the dynamic load effects.

In the following examples are presented that can be found in (DeLony, 1996).

Furthermore, the prototype of the plate girder bridge, the Britannia Bridge (1850)across the Menai Straits, Wales (UK), designed by Robert Stephenson and WilliamFairbairn, was also subjected to testing. As a fact, Naviers theory of elasticity,published in 1826, did not convince Stephenson since, after all, very little was knownabout structural theory. Therefore, he relied mainly on empirical methods of testing,modied and retested a series of bridge model details.

Major bridge failures in the 19th century led to tremendous disasters in the USA,Great Britain and France. All disasters were based upon similar reasons: totalignorance of metallurgy resulting in uneven production methods and faulty cast-

ings. Furthermore, adequate inspections and maintenance were seldom correctlyperformed. Consequently, major bridge failures with human fatal outcomes resultedin development of standards, specications and regulations including bridge testingto protect the travelling public.

It however took another quarter century to develop better bridge design accordingto advanced theories of stress analysis, better material knowledge and simply dueto the respect of the forces of nature. Therefore, not until the middle of the 20thcentury, after the tragical Tacoma Bridge collapse in 1940 in the USA, the dynamicsof structures were fully understood and controlled.

The American Squire Whipple and other European engineers such as Collignon dida tremendous and pioneering work in the last quarter of the 19th century whenthe railroads inuenced the scientic design of bridges. This resulted n applica-tion of both analytical and graphical solutions of bridge design calculations andprobably the rst time ever full size member testing of bridges. Figure 2.5 showsthe Whipple Truss Bridge (1867), Normanskill Farm, Albany, New York (USA),which even today remains in service. Though, restricting only buses and trucks thebridge demonstrates the efficiency of Whipples design since all members are stillthe original.

Finally, Marc Sguin, who together with Guillaume-Henri Dufour designed the

worlds rst permanent wire cable suspension bridge, is mentioned. Sguin pro-

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Dynamic influences Structural properties

Dynamic stiffness,mass and damping

Natural frequencies(resonance)

Structure dynamiccharacteristics

Modes(mode shapes)

Cyclic loading

Impact loading

Forced vibration bymoving loads(eccentricity)

Vehicle/bridgeinteraction

Earthquakes

Wind and waveexcitations

(ambient vibration)

Structural dynamic response

NoiseVibrationsStress

Figure 2.6: A schematic illustration of the dynamic inuences and structural prop-erties that determine the structural vibration response. Redrawn andmodied from (Wenzel et al., 2003).

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2.3. DYNAMIC BRIDGE MONITORING

Structure(Bridge)

Structural model(Bridge FE model)

Dynamicmonitoring

Signalanalysis

Loading(e.g. traffic)

Actual bridgeresponse

Naturalfrequencies

Modaldamping

ratios

Frequencyanalysis

(mode shapes)

Rayleighdamping

Direct timeintegration

Modesuperposition

FEM responseresults

Linearanalysis

Updating bridgeproperties

Nonlinear/linearanalysis

Figure 2.7: Illustration of the interaction between the actual bridge structure to theleft and the bridge FE model to the right. The dynamic monitoring playsan important role and is integrated into the structural model.

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2.4 Monitoring Benets

The most obvious and important benets with bridge monitoring could be cate-gorised without order of precedence as follows:

To assess and understand actual bridge behaviour. To control and update numerical calculation methods and models. To verify calculation parameters used. To measure loads, dynamic load effects and load distributions. To upgrade existing bridges for higher axle loads and/or speed

This of course provides a properly installed and functioning monitoring system (sen-sors and data acquisition system).

2.5 Monitoring Possibilities

This section is used to further describe and exemplify the possibilities with bridgemonitoring.

For example the Canadian Network of Centres of Excellence on Intelligent Sens-ing for Innovative Structures (ISIS Canada) report on a demonstration project. Acamera has been installed and synchronised with a computer that records strainmeasurements, permanently stores photos, and records vehicles that exceed currentweight limits (ISIS Canada, 2001). In addition, a web page facilitates interactiveaccess to monitoring results and downloading of information from a database. Forintelligent processing of downloaded data a suitable software program is used.

ISIS Canada Network of Centres of Excellence among others has also developedadvanced technologies to facilitate remote monitoring. As a result a wireless trans-mission has been developed as well as an economical, portable microchip data acqui-

sition system. This intelligent data acquisition system accommodates transmissionby radio frequency, the Internet or satellite.

Research at the Division of Structural Design and Bridges at the Royal Institute of Technology (KTH) in Stockholm, Sweden, involves a cost-effective method for theassessment of actual traffic loads on bridges (Karoumi et al., 2005). The methoduses very few sensors to compose a complete Bridge Weigh-in-Motion (B-WIM)system. It includes axle detection and accurate axle load evaluations to increasethe knowledge of actual traffic loads and their effects on bridges. Consequently, theB-WIM monitoring together with developed algorithms converts the instrumentedbridges to scales for weighing of crossing vehicles.

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Chapter 3

The New rsta Railway Bridge

3.1 GeneralThe New rsta Railway Bridge in Stockholm, opened for traffic on 28 August 2005,is an optimised, slender and very complex eleven spans prestressed concrete bridgeconsisting of two unballasted tracks. The bridge has a unique design since it isshaped as curved surfaces in both the longitudinal and lateral directions. Therefore,the Swedish National Railway Administration (Banverket) initiated a monitoringprogram to study the static and dynamic behaviour of the bridge. As a whole themonitoring project includes static measurements conducted during the constructionphase, static and dynamic mmeasurements during a test period and for the bridge

in operation. A total of ve bridge sections have been instrumented which mainlyincludes the span second closest to the southern abutment and over the navigablechannel.

Totally, the monitoring system consists of as much as 40 bre optic sensors, 21strain transducers, 9 thermocouples and 6 accelerometers. These monitoring instru-ments are measuring the bridge behaviour through central data acquisition systemsmounted inside the bridge. The monitoring procedure has been simplied since thecentral data acquisition systems are connected to the outside world through Internetcable and therefore can be controlled and programmed from the office.

3.2 Background

The New rsta Railway Bridge is built as a compliment to the old rsta Bridgewith an upgrading from two tracks to four between Stockholm South and the newrstaberg further south of Stockholm. Thereby the track capacity is increased as afurther step in a positive development of rail traffic in the region.

In 1994 the Swedish National Railway Administration (Banverket) and the City of Stockholm announced for an international competition to come up with the designof the new railway bridge to be located west of the existing bridge crossing the so-

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CHAPTER 3. THE NEW RSTA RAILWAY BRIDGE

called rstaviken. Fifteen Swedish and nine foreign architectural and engineeringrms were invited for a pre-qualication of performing the design and constructionalwork. Finally ve of them were accepted and assigned to rene their proposals tocome up with a bridge complying with the prerequisites: an option of a supple-

menting pedestrian road and cycle way; consideration of the interaction with thealready existing bridge and the surroundings. An assessment group consisting of representatives from the City of Stockholm, the Swedish National Railway Admin-istration (Banverket) and technical specialists within different areas decided thata bridge proposed by the architectural rm Foster and Partners in collaborationwith the design rm Ove Aarup A/S., after some modications, had found the win-ning concept. They had proposed a prestressed bridge of Falun red concrete, witha double-curved underside and supported on piers with an elliptical cross-section.The assessment group declared the winning proposal as follows (Banverket, 2003):

A bridge that, despite its considerable width, is experienced as beinga slender and elegant structure. With its simplicity and rhythmicallyformed contours it interacts attractively and effectively with the existingbridge. The bridge has an advanced concrete structure and a relativelyhigh construction cost, but offers advantages from the point of view of maintenance and noise prevention.

However, due to the advanced design of the bridgeit could possibly be claimed torepresent what is practicable to achieve as a bridge structure within the frameworkof todays advanced calculation techniques and modern materialsa reference groupwas set up to investigate the possibilities of actually build the bridge and restrictionsto have in mind during the design work. They worked with experts from the leadingtechnical universities and institutes and their conclusion was that the design hadto be reworked. So, the rst design conducted by Foster/Aarup was reworked, notonly once but twice. As a result, the nal design was produced by the Danish rmCOWI A/S.

Then the Swedish National Rail Administration (Banverket), as a natural step in theprocess, conducted another international pre-qualication for contract procurement.Companies where assessed on the basis of resources, competence and quality systems.Nine companies were then shown compatible to implement the bridge structure.

Consequently, these companies were invited to tender for the building contract undercompetitive conditions. This led to six of them submitting tenders and those wereassessed on the basis of commercial and technical criteria. Finally, after evaluatingall six tenders Banverket decided that SKANSKA had the best tender and thereforegot the privilege of being chosen as the contractor.

3.3 Bridge Description

The total length of the New rsta Railway Bridge is approximately 833m. Onconstruction drawings however all dimensions are calculated for the centre line (CL)

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3.4. INSTRUMENTATION AND DATA ACQUISITION

Fibre optic sensors

Thermocouples

Fibre optic cableFibre optic sensors

Data acquisition systemCentral unit

Accelerometer

Electric cable

Strain transducers

Figure 3.4: Illustration of how a typically installed monitoring cross section could beinstrumented. From (Dahlquist and Franzn, 2003).

1997b). The concrete grade class is therefore related to the cubic strength. The

complete superstructure then uses K60, except for abutments and retaining wallswhere K40 is used.

The slender design of the bridge has required a high amount of reinforcement tosatisfy the needs for bearing capacity. Therefore, much non-tensioned reinforcementand tendons are used. The superstructure is so heavily reinforced that it contains anaverage of as much as 220kg/m3 non-tensioned reinforcement. This is approximatelytwice as much as usually used for RC bridges. Besides the bridge have about 50tendons spread over the cross section and extended along the complete bridge length.These are used for post-tensioning and xed inside durable plastic pipes to separatethem from the superstructure and make replacement of damaged tendons possible.

3.4 Instrumentation and Data Acquisition

Strain transducers, accelerometers, a linear variable differential transformer (LVDT)and bre optic sensors are placed on the bridge. These are mainly embedded in theconcrete and connected to the central data acquisition units placed inside the bridge.The principle of the monitoring system is illustrated in Figure 3.4.

Since this thesis concentrate on the identication of dynamic bridge properties itonly concerns the equipment possible to use in dynamic monitoring, i.e. accelerom-

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Table 3.1: Typical specications for the strain transducers manufactured by KTHand including strain gauges 1-XY11-6/120 from HBM connected as a fullbridge.

Parameter Value UnitNominal resistance 120 Measuring length 300 mmSensitivity 260 s/kN Gage factor (gage sensitivity) approx. 2 -

eters and strain transducers. The bre optic sensors are part of another ongoingmonitoring project and interested readers are referred to (Enckell-El Jemli, 2003;Enckell-El Jemli et al., 2003; Enckell et al., 2003; Enckell and Wiberg, 2005). Forthe dynamic monitoring purpose four cross sections are instrumented, concentratedto the navigable channel part of the bridge, i.e. mainly span P8P9 in Figure 3.1,and denoted with A, B , C and D . A symbolises the pier section at P9; B is thequarter point section in span P8P9 closest to P9; C is the midspan section in spanP8P9; D constitutes the midspan section in span P7P8. Sensors in these sectionsare logically named A, B, C or D depending on their positions, and followed by a K ,symbolising KTH, and a number. The accelerometers have either the abbreviationAV or AH in the end, symbolising an accelerometer in the vertical and horisontal direction, respectively. Figure 3.5 to 3.8 illustrates the instrumentation. The straintransducers are pre-mounted onto the reinforcement and embedded in the concretewhile most accelerometers are placed inside special electrical boxes mounted in theedge beams. The LVDT (BK11) in Figure 3.6 is used to check the long-term stabilityof the embedded strain transducer at the same location (BK5). Cables between themonitoring instruments and the central data acquisition unit are carefully placedprotected and embedded in the concrete.

Each strain transducer consists of four resistance wire strain gauges of the type1-XY11-6/120 from Hottinger Baldwin Messtechnik (HBM) connected as a fullWheatstone bridge. The laboratory technician Kent Lindgren at the Royal Instituteof Technology (KTH) has developed and implemented the design and assembly of the strain transducers. The four active strain gauges in each transducer are gluedonto a 300mm long hollow steel bar with a diameter of 10 mm and inside which thecurrent supply and signal cable is routed and thereby protected from damage, seeFigure 3.9 and 3.10. In addition, the strain gauges and the steel bar are encapsulatedwith several layers acting as protection shields. At each end of the strain monitor-ing instrument an anchor plate with a diameter of 50 mm is located to ensure thatthe deformations are only introduced at these. The strain transducers use a HBMstandard 6 wire circuit which compensates for the resistance of the supply cables,thus ensuring full voltage over the bridge. Some of the strain transducers technicalspecications are presented in Table 3.1. An in situ mounted strain transducer isshown in Figure 3.9 and Figure 3.10 illustrates its schematic construction. Typicalspecications for the strain transducers are to be found in Table 3.1.

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L o n g i t u

d i n a l s t r a i n t r a n s d u c e r s

A c c e l e r o m e t e r

T r a n s v e r s a

l s t r a

i n t r a n s d u c e r s

B K 2

B K 3

B K 6

B K 7

B K 1 0

B K 8

B K 9

B K 4

B K 1

B K 5 B K

1 1

L V D T

C o n n e c t i o n

b o x

P e d e s t r

i a n a n

d c y c l e p a t

h

B K C B

B K 1 A V

Figure 3.6: The location of strain transducers, accelerometer and LVDT in Section B,the quarter point section closest to P9 in span P8P9. Both longitudinaland transversal strains are measured. The LVDT (BK11) is used tocheck the long-term stability of the embedded strain transducer BK5.

Also notice the circular holes illustrating the plastic pipes for tendons,evenly distributed over the cross section.

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C K 2

C K 1

C K 7

C K 4

C K 5

C K 6

C K 3

T r a n s v e r s a

l s t r a


L o n g i t u

d i n a l s t r a


C o n n e c t i o n

b o x

A c c e l e r o m e t e r s

C K C B

C K 3 A V

C K 4 A H

C K 1 A V

C K 2 A V

P e d e s t r

i a n a n

d

c y c l e p a t

h

Figure 3.7: The location of strain transducers and accelerometers in Section C,the midspan section in span P8P9. Both longitudinal and transversalstrains are measured. Three of the accelerometers measure accelerationin the vertical direction while the fourth measures in the horizontal di-rection.

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P e d e s t r i a n a n

d c y c l e p a t h

C o n n e c t i o n b o x

A c c e l e r o m e t e r

D K 1 A V

D K C B

Figure 3.8: Location of the accelerometer in Section D, the midspan section in spanP7P8.

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Figure 3.9: A strain transducer mounted onto the reinforcement before casting.

50 Full Wheatstone bridge

Protection cover 10

300

Figure 3.10: Illustration of the strain transducer construction. Dimensions are givenin millimeters.

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Table 3.2: Typical specications for the Si-Flex TM , SF1500S accelerometers fromthe supplier Colibrys SA (Colibrys, 2006).

Parameter Value Unit

Linear output range 3 g peakSensitivity 1.2 V/gFrequency response (full signal) DC to 5000 HzNoise (10 to 100 Hz) 300 to 500 ngrms / HzLinearity error 0.1 %

(a) Accelerometer mounted inside

special developed cylinder.

(b) Accelerometer mounted inside a special box em-

bedded in the edge beam concrete.

Figure 3.11: Accelerometer placed inside special cylinder and on the bridge.

The accelerometers are based on the micro electromechanical systems (MEMS) sen-sors. The MEMS are small integrated devices or systems that combine electricaland mechanical components (EMPA et al., 2004). The application of the MEMStechnology to accelerometers is a relatively new development and the accelerome-ters used here throughout are the Si-Flex TM , SF1500S accelerometers. Some of themanufacturers technical specications are presented in Table 3.2 (Colibrys, 2006).In addition, the Si-Flex TM accelerometer includes DC offset adjustment.

The accelerometers are mounted inside special cylinders developed by the laboratorytechnician Claes Kullberg at KTH. One of these cylinders is illustrated in Figure 3.11where it is also seen how the accelerometers are placed inside boxes embedded inthe edge beam concrete.

The data acquisition system consists of two basic MGCplus Digital Frontend mod-ules together with carrier frequency ampliers named ML55B, both from HBM. Thesystem is mounted in a special cabinet located close to the transversal wall at pier 9.For safety protection the cables between the strain transducers and the data acqui-sition system are protected from concrete vibrator contact by being tightly mounted

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together with the reinforcement. The accelerometers are installed after completingthe bridge. Therefore, the power supply and signal cables between the accelerom-eters and the data acquisition system are not literally embedded in concrete, butwired freely in special tubes inside the concrete and on the concrete surface of thebridge interior.

For the bridge in service it is very difficult to reach the data acquisition system andmanually download the monitoring data to a portable PC. Therefore, the monitoringsystem is necessarily connected to the outside world through Internet cable. Thisconstitute a modern monitoring solution where everything can be mastered fromthe computer at the office and no manual in situ readings are necessary.

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Chapter 4

Bridge Dynamics

4.1 Structural Dynamic TheoryThe equation of motion for an MDF system

[m ]{u }+ [c ]{u }+ [k ]{u }= {p (t)} (4.1)contributes a system of coupled equations. Torsional motion in the form of a tor-sional angle are included in the equation of motion as (Brandt, 2000)

I + c + k = M (t) (4.2)for each torsional SDF and where M (t) is the torsional moment.

The mass and stiffness matrices are then by (nite element) discretisation of thebridge structure. The damping matrix is however unpractical to create by dis-cretisation and another procedure therefore has to be adopted. Consequently, thedamping coefficient, , in Equation 4.20 is determined by monitoring experimentsor based on experience, which is described in Section 4.3.

Transformation using the modal coordinates according to

{u (t)}= [ ]{q (t)} (4.3)where

[ ] =11 n 1...

...1n nn

= {1} {n} (4.4)

includes the n natural vibration modes, i.e. the mode shapes, of the structure. For

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CHAPTER 4. BRIDGE DYNAMICS

a discrete system they are vectors according to Equation (4.4) but they can indeedconstitute functions, as for the distributed system of a simply supported beam. Forexample the rst natural vibration mode is then composed of

1(x) = a sin( xL ) (4.5)

where a is a constant.

Natural circular frequencies and natural vibration modes are found by solving thefree vibration form of Equation (4.1), i.e. without damping and external forces equalto zero according to

[m ]{u }+ [k ]{u }= {0} (4.6)Neglecting the damping is correct if the damping is small which is an acceptableassumption for most structures. Assuming a solution of the form

{u }= {n}sin(n t) (4.7)and substituting Equation (4.7) into Equation (4.6) an equation valid for every timet is achieved, implying the eigenvalue problem

([k ]

n 2[m ])

{n

}=

{0

} (4.8)

The eigenvalue problem has the nontrivial solutions corresponding to

det([k ]n 2[m ]) = 0 (4.9)Equation (4.9) is referred to as the characteristic equation of the system. The naturalfrequencies are related to the natural circular frequencies according to

f n = n

2 =

1

T n(4.10)

The number of natural frequencies are equal to the number of degrees of freedomfor discrete systems while distributed systems have an innite number of naturalfrequencies.

Furthermore, introducing Equation (4.3) in Equation (4.1) and premultiply with[ ]T results in

[ ]T [m ][ ]{q}+ [ ]T [c ][ ]{q}+ [ ]T [k ][ ]{q}= [ ]T {p (t)} (4.11)This equation is rewritten as

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4.1. STRUCTURAL DYNAMIC THEORY

[M ]{q}+ [C ]{q}+ [K ]{q}= {P (t)} (4.12)where [M ] and [K ] are diagonal matrices since the natural vibration modes cor-responding to different natural frequencies can be shown to satisfy the followingorthogonality conditions when n = s (Chopra, 2001)

{n}T [k ]{s}= 0 and {n}T [m ]{s}= 0 (4.13)The damping[C ] is however a nondiagonal matrix, which is the case for all realstructures while {P (t)} is a column vector. Therefore Equation (4.12) is still asystem of N coupled equations through the damping terms according to

M n q n +N

r =1

C nr q n + K n q n = P n (t) (4.14)

Only when assuming classical damping, i.e. C nr = 0 if n = r the modal equationswill be uncoupled and thus reduced to

M n q n + C n q + K n q n = P n (t) (4.15)

where the diagonal matrix elements in Equation (4.12) are

M n = {n}T [m ]{n} (4.16)

C n = {n}T [c]{n} (4.17)

K n = {n}T [k ]{n} (4.18)Dividing Equation (4.15) by M n gives

q n + 2 n n q + n 2q = P n (t)M n

(4.19)

where the equality

n = C n2M n n

= C n

2 K n M n (4.20)

is used.

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Each SDF equation in the uncoupled system is possible to solve and give the modalcoordinate vector {q (t)}. Finally, the displacements are achieved by transformingback to real coordinates according to Equation (4.3).The problem however is how to obtain [C ] or [c ]. This is mostly done by experimentsor based on experience. In addition, transformation is only useful if [C ] is diagonal.It may therefore sometimes be better to solve the coupled equations in Equation (4.1)directly. Consequently, the damping matrix, [c], is necessary to determine. This ispossible through the damping ratios, n , which are identied by monitoring or chosenfrom experience. The two evaluation methods available are:

1. Calculate [c] from the modal matrix [C ].

From Equation (4.20) the diagonal modal damping matrix is composed as

[C ] = C 1 .. .

C n

(4.21)

Since[C ] = [ ]T [c ][ ] (4.22)

it follows that[c] = ([ ]T ) 1[C ][ ] (4.23)

2. Use Rayleigh damping.A damping matrix is used according to

[c] = a0[m ] + a1[k ] (4.24)

Modal coordinate transformation forms a diagonal modal damping matrix and

n = a02

1n

+ a12

n (4.25)

The drawback with the former method is that all damping ratios have to be knownwhich usually is not the case. Rayleigh damping on the other hand uses only twomodal damping coefficients. This means that only damping ratios for two naturalvibration modes has to be achieved. The natural vibration modes chosen shouldthen be taken so that they ensure reliable damping ratios for all other vibrationmodes. Experience shows that it is appropriate to use the lowest natural vibrationmode and the third or fourth lowest to calculate a0 and a1 and thereby reect theactual structural damping more correctly (Chopra, 2001).

Most systems are reasonably modelled as linear systems, i.e. with small elastic de-formations, with classical damping. The dynamic response is then determined us-ing classical modal analysis since classical natural frequencies and vibration mode

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shapes exist. Consequently, the equation of motion in Equation 4.1 can be solvedusing transformation to modal coordinates and resulting in an uncoupled system.Thus, the response of each natural vibration mode is calculated independently of the others. The total response constitutes a combination of separate modal re-sponses, recognised as mode superposition. Characteristic for each mode is that itresponds with its own particular deformation pattern, the so-called mode shape . Ithas its own frequency, the natural frequency, with its own modal damping. Thesethree constitute the structural dynamic characteristics. A SDF system accordingto Equation 4.19, with its own vibration properties, is consequently analysed whencomputing the specic modal response as a function of time . The solutions to theseSDF equations could be expressed in closed form for simple excitations that can bedescribed analytically or numerically for complicated excitations.

Systems with damping such that [C ] in Equation 4.12 is nondiagonal are referredto as having nonclassical damping. These systems differ from those with classical

damping. Systems with classical damping possess the same natural modes as theundamped systems. This is not the case for systems with nonclassical damping.Consequently, the classical modal analysis method with mode superposition is notpossible to use for these systems. The fact is that classical vibration modes donot exist for systems with nonclassical damping. Therefore, the equation of motioncan not be uncoupled and numerical time integration methods are required. Con-sequently, the solutions to these systems are always numerical even if the dynamicexcitation is simple. Furthermore, modal analysis can not be used for inelastic,i.e. nonlinear systems, independently of classical or nonclassical damping systems.However, systems with classical damping are always possible to solve using directintegration. This approach solves the coupled equations directly in the nodal dis-placements by using numerical methods. Two types of numerical solution methodsare typically used; the Newmark or Hilber-Hughes method.

4.1.1 Analytical Natural Beam Frequencies

In an analytical (exact) frequency analysis for the transverse vibration of beams thefollowing assumptions are used:

The beam is straight and uniform along the spans, i.e. the beam mass is evenlydistributed throughout the beam. The beam is composed of a linear, homogeneous, isotropic elastic material. The beam is slender. Only exural deformation is considered. Rotary inertiaand shear deformation is not considered. Only deformations normal to the undeformed beam axis are considered. Planesections remain plane.

No axial loads are applied to the beam.

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The shear centre of the beams cross section coincides with the centre of gravity,i.e. the plane of vibration is also a plane of symmetry of the beam, so thatrotation and translation of the beam are uncoupled.

The total transverse free vibration deformation as a function of the location x andthe time t is then the sum of the modal deformations according to (Blevins, 1979)

u(x, t ) = N n =1 An n (x) sin(2f n t + n ) (4.26)

where n (x) is the mode shape displacement value associated with the n th vibrationmode and An is a constant with the units of length. An and n are determined bythe means used to set the beam in motion.

Generally, i.e. for both single-span beams and uniform multispan beams, the natural

frequency of transverse vibration can be expressed as (Blevins, 1979)

f n = 2n2L 2 EI m , n = 1, 2, 3, . . . (4.27)

where m is the mass per unit length of the span and n is a dimensionless parameterwhich is a function of the mode number, boundary conditions and the number of spans. Values of n for different cases are for example possible to nd in (Blevins,1979).

For a multispan beam with unequal spans the fundamental frequency is representedwith 2n in Figure 4.1. The ends are clamped and the outermost spans have a differentspan length than the innermost spans. This case approximately corresponded to thetrack slab of the New rsta Railway Bridge in the transverse direction.

Concerning beams supporting a uniform axial load (e.g. from prestressing) the pre-vious assumption of no axial loads applied to the beam is not valid. However, atensile load which is applied axially to a beam increases the natural frequencieswhile a compressive axial load decreases the natural frequencies. Exact solutionsare however only available for single-span uniform beams with special boundaryconditions. An approximate formula for the effect of the axial load P on the natural

frequency is then

f n |P =0f n |P =0

= 1 + P |P b |212n

, n = 1, 2, 3, . . . (4.28)

where P b is the axial load required to buckle the beam.

Similarly to Equation (4.27) longitudinal vibrations have natural frequencies accord-ing to (Blevins, 1979)

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Figure 4.1: Fundamental natural frequency of a N span beam with extreme endsclamped, pinned intermediate supports and variable spacing in outermostspans. From (Blevins, 1979).

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f n = n2L 2 E , n = 1, 2, 3, . . . (4.29)

where is the mass density of the beam material and values of n for different casesare for example possible to nd in (Blevins, 1979).The natural frequency of torsion is a function of the torsional constant, C , denedas the moment required to produce a torsional rotation of one radian on a unitspanwise length divided by the shear modulus. However, a noncircular cross sectionwill generally warp as it deforms in torsion, see Section 6.3.1. For simple closed crosssections however, where the cross sectional dimensions are small compared with thelength of the beam, the effect of warping on the deformation is very small. Theeffect of warping on the natural frequency may even be neglected for thin-walledopened cross sections if (Gere, 1954)

1n

twD

LD

> 10 (4.30)

where D is the diameter of the cross section, tw is the typical minimum wall thicknessof the beams cross section and n is again the mode number.

Then, for slender, uniform, isotropic, homogeneous beams with circular cross sectionthe exact torsional natural frequencies are (Blevins, 1979)

f n = n2L 2

CGI p , n = 1, 2, 3, . . . (4.31)

where I p is the polar area moment of inertia (m4) of the cross section about theaxis of torsion. Notice though that this equation is valid if, and only if, warping isneglected.

4.1.2 The FE Method

When the cross section of the beam becomes complicated, and the beam in additionis curved, the assumptions in Section 4.1.1 are not valid. In addition, when for exam-ple the centre of gravity of the section does not coincide with the centre of rotationthe torsional and exural natural frequencies will be coupled. The warping effectcan also be sufficient. Then a numerical solution is most conveniently implemented.That is where the nite element method is introduced.

The nite element method is the most powerful and versatile technique that can beused in modelling of virtually any bridge type. It results in a nite number of DOFsand sets of coupled or uncoupled equations.

The type of bridge model used in a dynamic analysis should be as complex asrequired to obtain accurate and reliable results, but also as simple as reasonably

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4.2. DYNAMIC AMPLIFICATION FACTOR

possible to minimise the computational effort (UIC, 2006).

4.2 Dynamic Amplication Factor

It became natural to mention the dynamic amplication factor (DAF) when dealingwith dynamic bridge monitoring. Not since its concept actually is necessary but it isstill used in bridge design codes. In addition, it is insightful to present the complexityinvolved in calculating and choosing an appropriate dynamic amplication factor.

In monitoring dynamic load effects one follows some type of oscillating structuralresponse. This response gives an idea of the dynamic amplication factor as themagnication of the structural static load response. Experience, calculations andold praxis have then resulted in empirical dynamic amplication factor formulas

to be found in bridge design codes. Relating the dynamic response to the staticresponse in this way is and has been a user friendly design tool easy to use. It ishowever more convenient, and correct to talk about actual dynamic bridge response.This is since the empirical design code formulas only take a determinant length of the bridge into account and therefore impossibly can consider the moving load effectscorrectly. It should for example be obvious that the train speed has an inuence onthe dynamic bridge response. James has indeed highlighted the complexity withinthe subject (James, 2003). He emphasises that the dynamic amplication factor inreality results from a complex interaction between bridge properties (included thetrack for railway bridges) and the vehicle.

Already in 1931 the dynamic amplication of the static load was dened as thefactor by which the dynamic load exceeds the static load (Fuller et al., 1931). It ishowever important to realise that the dynamic amplication of the force may notnecessarily be the same as for example the dynamic stress amplication factor (ISISCanada, 2001). In spite of that, the same DAF value is usually used to calculatedifferent dynamic load effects and the dynamic load on its own, which for a SDFsystem with harmonic excitation is dened as (Chopra, 2001)

DAF force = F dynF stat

= 1

[1

(/ n )2]2 + [2 (/ n )]2

(4.32)

Figure 4.2 tries to illustrate the dynamic force amplication. As a result of vehiclespeed and for example track irregularities the varying dynamic load has its totalmaximum value of F dyn = ( 16n =1 F n (t))max . The static load is F stat = 2

2n =1 (mg)n .

Unfortunately, there is a lack of uniformity in calculating the dynamic amplicationfactor. For example, the Swedish National Railway Administration specify theirdynamic amplication factor as (Banverket, 2004)

D = 1.0 + 4

8 + L(4.33)

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F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 9 F 10 F 11 F 12 F 13 F 14 F 15 F 16

v

(mg )1(mg )1 (mg )2(mg )2

Figure 4.2: Illustration used to elucidate the dynamic amplication of the static load.The dynamic force is oscillating, i.e. varying with time. The dynamicamplication factor is then achieved when the dynamic force reaches itsmaximum value and the inequality ( 16n =1 F n (t))max > 2 2n =1 (mg)n isvalid.

L /m

D A F

D23

10 20 30 40 50 60 70 801

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Figure 4.3: Dynamic amplication factors dened according to railway bridge designcodes. D is from the Swedish National Railway Administration and isfrom Eurocode.

while the Eurocode (CEN, 2002) uses

2 = 1.44 L 0.2

+ 0.82, 1.00 2 1.67 (4.34)

for carefully maintained track, and

3 = 2.16 L 0.2

+ 0.73, 1.00 3 2.00 (4.35)

for track with standard maintenance. These differences are graphically illustrated

in Figure 4.3.

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4.2. DYNAMIC AMPLIFICATION FACTOR

Reference point

Dynamic response

Static responseMedian response

M i d s p a n r e s p o n s e

Load position

s

1

23

min 1 2

max stat stat

Figure 4.4: Midspan response illustration of a simply supported bridge. Differentparameters used for denition of the dynamic amplication factors areincluded. Redrawn and modied from (Biggs and Suer, 1956).

Even though monitoring is performed there are different denition alternatives forthe dynamic amplication factor. Some possibilities are illustrated and identiedusing Figure 4.4. Notice that this example should be seen as a reconstruction fromdata of a vehicle travelling over a simply supported bridge. A ctitiously scaledresponse is used that necessarily not reect the actual bridge behaviour in a realload case. The variation of both dynamic and static responses at midspan areillustrated with respect to the load position. The median response is included as anaverage of consecutive dynamic response peaks.

Regarding Figure 4.4 the following alternatives for a dynamic amplication factorfrom monitoring can be found in the literature (Bakht and Pinjarkar, 1990):

s + 1 s

, stat + 2

stat,

stat + 3 stat

, 2 max max + min

, max

2,

max 1

and max

stat

It becomes even more complicated if a reference point close to a bearing or supportis used. This point has hardly any static response at all, and when its maximum dy-namic response is divided by nearly zero a very large DAF is achieved. Furthermore,multispan bridges probably have their maximum static and dynamic responses atdifferent locations. Should it then be possible to calculate the dynamic amplicationfactor by combining different reference points?

In addition, there hardly ever exists any discussion or justication in selecting a

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Dynamic response

Static response

one cycle

Y/Y static.max.DIF=1+ Y

Y static,max.

Figure 4.5: Denition of DAF. Notice that DIF is specied instead of DAF. From(Kim et al., 2005).

particular denition (ISIS Canada, 2001). In other words, citing 26 published refer-ences the dynamic amplication factor denition chosen is taken as being axiomaticand therefore requiring no justication (Bakht and Pinjarkar, 1990).

The authors opinion however is that if the dynamic amplication factor has to beused instead of the actual dynamic bridge response, it must be based on a reasonabledenition. A good rule of thumb could be to dene the DAF as the ratio of themaximum absolute value of the difference between the dynamic and static responseto the maximum static response during one major period of the dynamic response,including the maximum static response, and adding it to one (Kim et al., 2005).This denition is illustrated in Figure 4.5

4.3 Experimental Dynamics

Much of the information found in this section is possible to nd in structural dynamicliterature, e.g. (Chopra, 2001; Tedesco et al., 1999; Battini, 2005).

Experimental dynamics has the purpose of determining the dynamic characteristicsof structures by means of monitoring. Most often the excitation is unknown andthe structural response referred to as ambient vibrations (random noise). Trafficand wind load are two examples of random forces. The structural response is thesum of a free vibration solution and a particular solution which depends on therandom load. To determine the structural properties from an ambient vibrationthe random decrement technique is used to convert the ambient vibration data tofree vibration data. The technique is for example described in (Johnson, 1999)where the excitation mechanism is eliminated from the response signal and leftis a freely vibrating mechanical system signal. For railway bridges however themajority of the response consists of a free vibration; the bridge structure is disturbedfrom its equilibrium state due to a crossing train load and vibrates freely withoutapplied forces. These free vibration response parts are often considered enough indetermining the unknown dynamic characteristics. Those are the bridges natural

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4.3. EXPERIMENTAL DYNAMICS

u

t

0

Figure 4.6: Example of a typical acceleration signal in the time domain. Recon-structed from (Battini, 2005).

frequencies, mode shapes and modal damping ratios.

In most cases accelerations are easier than displacements to measure. Therefore,accelerations most commonly constitute the monitoring signal magnitude.

Damping is one of the most important structural characteristics that affects theresponse. The damping phenomenon is however not yet well known and thereforehas to be assumed in dynamic analysis. Consequently, damping ratio estimations inthe bridge design code are necessary as reference values in designing new structureswhere damping cannot be measured.

4.3.1 Time and Frequency Domain in Signal Analysis

A typical bridge acceleration signal, free from noise is exemplied in Figure 4.6.

The fundamental and the concept of experimental dynamics is to use the signal for

signal analysis. In signal analysis the signal is transformed from the time domain tothe frequency domain where the dynamic characteristics are possible to determine.The basic theory is to represent a given signal as a combination of a number of sine and cosine terms. This means that a signal is composed of components withdifferent frequency and that the coefficients for the sine and cosine terms reveals thestrength of each frequency. By representing the signal in the frequency domain onemakes signal analysis and signal manipulations easier.

Since the trigonometric sine and cosine functions are related through the complexexponential function in Eulers formula as

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ei = cos + i sin (4.36)

where i is the complex unit it follows that

e i = cos() + i sin() = cos i sin (4.37)Consequently, the following fundamental expressions in signal analysis for sine andcosine are achieved

sin = ei e i

2i cos =

ei + e i

2 (4.38)

For a given integer n the notation

n = cos(2/n ) i sin(2/n ) = e 2i/n (4.39)is used and where n has important symmetry characteristics.

Furthermore, signal analysis is based on the properties of the Fourier transform,which is the standard tool for detecting sinusoids in a signal and thereby measure thesignal content for each frequency. One can however not transform to the frequencydomain continuously and must instead sample and digitise the time domain input.Assuming a function, or signal, depending on the time t , i.e. y(t), its sampled valuesat n locations evenly distributed in time becomes

yl = y[l] = y[l t], l = 1, 2, . . . , n (4.40)

The efficient tool for performing Fourier analysis of discrete time sequences is a tech-nique called the Discrete Fourier Transform (DFT), for nite duration signals. TheDFT of (y1, y2, . . . , yn ) is then a vector (Y 1, Y 2, . . . , Y n ) with the elements (Jnsson,2004)

Y l =

n

k=1 yk

(k 1)( l 1)n , l = 1, 2, . . . , n (4.41)

In matrix form, i.e.

Y 1Y 2Y 3...

Y n

=

1 1 1 11 1n 2n n 1n1 2n 4n

2(n 1)n

...

1 n 1

n 2(n 1)

n (n 1)2

n

y1y2y3...

yn

(4.42)

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4.3. EXPERIMENTAL DYNAMICS

the coefficient matrix is the so called Fourier matrix, [F ], with the elements

F lk = (k 1)( l 1)n (4.43)

The inverse Fourier matrix, [F ] 1, has elements given by shifting the sign for theexponents for n in [F ] and by dividing with the number of samples, n , accordingto

F 1lk = 1n

(k 1)( l 1)n (4.44)

This is veried through the matrix multiplication which gives [F ] 1[F ] = [F ][F ] 1 =[I ], where [I ] is the identity matrix. Consequently, the inverse DFT is written

y1y2y3...

yn

= 1n

1 1 1 11 1n 2n n 1n1 2n 4n

2(n 1)n

...1 (n 1)n 2(n 1)n

(n 1)2n

y1y2y3...

yn

(4.45)

or in its compact form

yl = 1n

n

k=1Y k (k 1)( l 1)n , l = 1, 2, . . . , n (4.46)

For real function values this complex exponential function that interpolates thefunction values becomes

y[t] = 1n

n

k=1

ak cos2(k 1)(t 1)

n+ bk sin

2(k 1)(t 1)n

(4.47)

which corresponds to

y[t] = 1n

n

k=1

ck sin2(k 1)(t 1)

n + n (4.48)

where t is measured in units of the sampling interval t.

In Equation (4.47) the DFT results in the amplitudes according to

ak = Re( Y k) and bk = Im(Y k) (4.49)

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and in Equation 4.48

ck = a2k + b2k and k = arctan akbk (4.50)Consequently, the DFT decomposes the discrete function y[t] into a nite number of vibrations with frequencies f k = ( k 1)/n , where k = 1, 2, . . . , n . Those frequenciesare then measured in number of cycles per time unit t and the sampling frequencynaturally has 1 cycle per time unit t. The coefficients Y k indicate the strength of the different vibrations while they also represent the function y[t] in the frequencydomain.

The component Y n/ 2+1 is of special interest as it represents the Nyquist frequency,f n/ 2+1 = 1/ 2, which is the highest frequency possible to represent at a given samplingfrequency (n is supposed to be even so that n/ 2 is an integer). This phenomenon isfurther discussed in Section 5.2. However, for real valued signals Y n/ 2+1 is always areal number. Furthermore, the components placed symmetrically around Y n/ 2+1 arecomplex conjugated so that

Y k = Y n +2 k , k = 2, 3, . . . , n / 2 (4.51)

This means that th

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Bridge Monitoring to Allow for Reliable Dynamic Finite Element Modelling