railway bridge response to passing trains
TRANSCRIPT
Railway bridge response to passing trains
Measurements and FE model updating
J o h a n W i b e r g
Doctoral Thesis in Civil architectural engineering
Stockholm, Sweden 2009
www.kth.se
TRITA-BKN. Bulletin 100, 2009ISSN 1103-4270
ISRN KTH/BKN/B--100--SE
Joh
an W
iberg railw
ay bridge response to passing trainsKTh
2009
Railway bridge response to passing trains
Measurements and FE model updating
JOHAN WIBERG
Doctoral ThesisStockholm, Sweden 2009
TRITA-BKN. Bulletin 100, 2009ISSN 1103-4270ISRN KTH/BKN/B--100--SE
KTH School of ABESE-100 44 Stockholm
SWEDEN
Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen i brobyggnadfredagen den 23 oktober 2009 klockan 10.00 i sal F3, Kungl Tekniska högskolan,Lindstedtsvägen 26, Stockholm.
© Johan Wiberg, september 2009
Tryck: Universitetsservice US-AB
Abstract
Today’s railway bridges are analysed in more detail for moving loads due tothe increase in speeds and axle loads. However, these numerical analyses arevery time consuming as they often involve many simulations using differenttrain configurations passing at different speeds and many considerations totake into account. Thus, simplified bridge, track and train models are oftenchosen for practical and time efficient simulations.
The New Årsta Railway Bridge in Stockholm was successfully instrumentedduring construction. A simplified 3D Bernoulli-Euler beam element FE modelof the bridge was prepared. The FE model was first manually tuned based onstatic load testing. The most extensive work was performed in a statisticalidentification of significantly influencing modelling parameters. Consequently,parameters to be included in an optimised FE model updating, with consid-eration also to synergy effects, could be identified. The amount of parametersincluded in the optimisation was in this way kept at an optimally low level.For verification, measurements from several static and dynamic field tests witha fully loaded macadam train and Swedish Rc6 locomotives were used. Theimplemented algorithms were shown to operate efficiently and the accuracyin static and dynamic load effect predictions was shown to be considerablyimproved.
It was concluded that the complex bridge can be simplified by means of beamtheory and an equivalent modulus of elasticity, and still produce reliable re-sults for simplified global analyses. The typical value of an equivalent modulusof elasticity was in this case approximately 25% larger than the specified meanvalue for the concrete grade in question.
The optimised FE model was used in moving load simulations with high speedtrain loads according to the design codes. Typically, the calculated verticalacceleration of the bridge deck was much lower than the specified allowablecode value. This indicates that multispan continuous concrete bridges are notso sensitive to train induced vibrations and therefore may be suitable for highspeed train traffic.
Finally, the relevant area of introducing the proposed FE model updatingprocedure in the early bridge design phase is outlined.
iii
Sammanfattning
Dagens järnvägsbroar analyseras mer i detalj för rörliga laster till följd avökade farter och axellaster. Dessa numeriska analyser är dock mycket tids-krävande och innebär ofta många simuleringar med olika tågkonfigurationervid olika farter och många andra omständigheter att ta hänsyn till. Därförväljs ofta förenklade bro-, spår- och tågmodeller för praktiska och tidseffektivasimuleringar.
Nya Årstabron i Stockholm instrumenterades framgångsrikt under dess upp-förande. En förenklad finit elementmodell beståendes av 3D Bernoulli-Eulerbalkelemet utarbetades. Modellen justerades först manuellt baserat på re-sultat från en statisk provbelastning. Det mest omfattande arbetet utfördesi en statistisk identifiering av signifikant inflytelserika modelleringsparamet-rar. Som en konsekvens av detta kunde parametrar som skulle inkluderas i enoptimerad uppdatering av finita elementmodellen, med hänsyn till synergief-fekter, identifieras. Antalet parametrar att inkludera i optimeringen hölls pådetta sett vid en optimalt låg nivå. För verifiering användes mätningar frånflera statiska och dynamiska fältförsök med ett fullastat makadamtåg ochsvenska Rc6 lok. De implementerade algoritmerna visade sig vara effektivaoch precisionen i predikterade statiska och dynamiska lasteffekter visade sigbli avsevärt förbättrad.
Det konstaterades att den konstruktivt komplicerade bron kan förenklas medhjälp av balkteori och en ekvivalent elasticitetsmodul och ändå ge pålitligaresultat för förenklade globala analyser. Det typiska värdet hos en ekvivalentelasticitetsmodul var i detta fall approximativt 25% större än det specificerademedelvärdet för betongklassen i fråga.
Den optimerade finita elementmodellen användes i simuleringar med hög-hastighetståg enligt normer. Beräknade vertikala accelerationer hos brobananvar klart lägre än det specificerat tillåtna normvärdet. Detta tyder på attkontinuerliga flerspannsbetongbroar inte är särskilt känsliga för tåginduceradevibrationer och därför kan vara lämpliga för höghastighetståg.
Slutligen presenteras ett tillämpningsområde för den föreslagna uppdaterings-proceduren av en finit elementmodell som verktyg redan vid framtagandet aven brodesign.
iv
Preface
The research work presented in this thesis was carried out at the Department ofCivil and Architectural Engineering, Royal Institute of Technology (KTH). It wasfinanced by the Railway Group at KTH, the Swedish National Railway Adminis-tration (Banverket) and the Division of Structural Design and Bridges at KTH.
I would like to express my greatest gratitude to my supervisor Prof. Raid Karoumiand assistant supervisor Prof. Håkan Sundquist for encouragement, support andexperienced guidance. Prof. Raid Karoumi especially, I very much appreciate yourcomments on the work and you letting me point out my own research direction andwork independently in good communication.
Adj. Prof. Costin Pacoste, thank you for your endless structural engineering knowl-edge, proofreading, valuable comments and always seeing things from another pointof view.
To Mr. Bo Eriksson-Vanke at the National Railway Administration (Banverket), youare the largest source of inspiration with your never-ending interest and experiencein civil engineering and for that I admire you.
I especially thank the PhD students, colleagues and friends at the Division of Struc-tural Design and Bridges and the Division of Concrete Structures – nobody men-tioned, nobody forgotten – for laughter, joy and fruitful discussions.
A very special thanks goes to the laboratory technicians Stefan Trillkott and ClaesKullberg for admirable installations of monitoring equipment and assistans in loadtesting.
Finally, without my family I would be nothing. Cecilia, Williott and Vidar, thankyou for patience, understanding and for always being there.
Stockholm, September 2009
Johan Wiberg
v
List of publications
This thesis consists of a summary and five appended papers.
Paper A Karoumi, R., Wiberg, J., and Liljencrantz, A. (2005). Monitoringtraffic loads and dynamic effects using an instrumented railwaybridge. Engineering Structures, 27(12):1813-1819.
Paper B Wiberg, J. and Karoumi, R. (2009). Monitoring dynamic be-haviour of a long-span railway bridge. Structure and Infrastruc-ture Engineering, 5(5):419-433.
Paper C Wiberg, J. (2009). An equivalent modulus of elasticity approachfor simplified modelling and analysis of a complex prestressed rail-way bridge. Submitted to Advances in Structural Engineering.
Paper D Wiberg, J., Karoumi, R., and Pacoste, C. (2009). Statisticalscreening of individual and joint effect of several modelling fac-tors on the dynamic FE response of a railway bridge. Submittedto Journal of Sound and Vibration.
Paper E Wiberg, J., Karoumi, R., and Pacoste, C. (2009). Optimizedmodel updating of a railway bridge for increased accuracy in mov-ing load simulations. To be submitted to Journal of Bridge Engi-neering.
Four of the papers were prepared in collaboration with co-authors. The author ofthis thesis took the following responsibility for the work in those papers:
Paper A Took part in the theoretical study and the algorithms. Pro-posed and made changes in writing the report. Presented thework in a pre-version article at The Second International Con-ference on Structural Engineering, Mechanics and Computation(SEMC 2004).
vii
Paper B Planned the work in collaboration. Performed the operationalmodal analysis. Wrote the report.
Paper D Planned the work in collaboration. Carried out all statistical andnumerical calculations. Wrote the report.
Paper E Planned the work in collaboration. Implemented the optimisationroutines and carried out all numerical calculations. Wrote thereport.
Additional relevant publications
Enckell, M., Karoumi, R and Wiberg, J. (2003). Structural health monitoring foran optimized pre-stressed concrete bridge. In The First International Confer-ence on Structural Health Monitoring and Intelligent Infrastructure (SHMII-1),Tokyo, Japan, November 13-15.
Karoumi, R., Wiberg, J. and Olofsson, P. (2004). Monitoring traffic loads and trafficload effects on the New Årstaberg Railway Bridge. In International Conferenceon Structural Engineering, Mechanics and Computation (SEMC 2004), CapeTown, South Africa, July 5-7.
Enckell, M. and Wiberg, J. (2005). Monitoring of the New Årsta Railway Bridge.Instrumentation and preliminary results from the construction phase. Technicalreport 2005:8. Royal Institute of Technology (KTH). Department of Civil andArchitectural Engineering.
Wiberg, J. (2006) Bridge Monitoring to Allow for Reliable Dynamic FE Modelling.A Case Study of the New Årsta Railway Bridge. Licentiate thesis, Royal Insti-tute of Technology (KTH).
Karoumi, R. and Wiberg, J. (2006). Kontroll av dynamiska effekter av passerandetåg på Botniabanans broar - Sammanfattning. TRITA-BKN. Report 97. RoyalInstitute of Technology (KTH). Department of Civil and Architectural Engi-neering. ISSN 1103-4289 (In Swedish).
Wiberg, J. (2007). Railway bridge dynamic characteristics from output only signalanalysis. In 2nd International Conference on Experimental Vibration Analysisfor Civil Engineering Structures (EVACES´07), Porto, Portugal, October 24-26.
Wiberg, J. and Enckell, M. (2008). Monitoring of the New Årsta Railway Bridge.Presentation of measured data and report on the monitoring system over theperiod 2003-2007. Technical report 2008:14. Royal Institute of Technology(KTH). Department of Civil and Architectural Engineering.
viii
Contents
I SUMMARY AND REVIEW OF THESIS 1
1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Aim and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Thesis contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Train-bridge modelling aspects 11
3 The research work 19
4 Conclusions 234.1 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Bibliography 27
A Yates’s algorithm 31
B The Nelder-Mead simplex algorithm 35
II APPENDED PAPERS 37
A Monitoring traffic loads and dynamic effects using an instru-mented railway bridge
B Monitoring dynamic behaviour of a long-span railway bridge
C An equivalent modulus of elasticity approach for simplified mod-elling and analysis of a complex prestressed railway bridge
D Statistical screening of individual and joint effect of several mod-elling factors on the dynamic FE response of a railway bridge
ix
E Optimized model updating of a railway bridge for increased ac-curacy in moving load simulations
x
Part I
SUMMARY AND REVIEW OF THESIS
1
Chapter 1
Introduction
1.1 Background
Today’s railway bridges are analysed in more detail for moving loads due to theincrease in speeds and axle loads. Moreover, modern railway bridges have moreslender designs and lower energy dissipation (damping), as a consequence of thedevelopment in construction and design methods. Consequently, these detailedanalyses of the dynamic response of bridges to passing trains are important inorder to guarantee the planned lifetime and economical assessment.
Fig. 1.1 illustrates the increase in train speeds for passenger traffic between 1820and 2040. Following the linear trend line from 1840 for maximum speeds worldwideresults in a predictable speed of 350 km/h in 2040. With the advancement inlocomotive and control technologies, railway trains that have a design speed of350 km/h or higher are however not uncommon already nowadays. Consideringonly the time after the Second World War, the development is much faster. In thiscase the trend points toward 450 km/h in 2040. Such high speeds call for morerealistic bridge, track and vehicle models in the simulations to accurately accountfor the dynamic load effects.
Willis and Stokes are considered the first to bring the problem of vehicle loadimpacts to the design desks of bridge engineers. Their historical research contri-butions on bridge vibrations caused by moving traffic, investigated the collapse ofthe Chaster Rail Bridge in England in 1847 (Willis, 1849; Stokes, 1849). Swedishpioneers were Lerfors and Hillerborg, but that was first in the 1940s. They investi-gated the most elementary cases of dynamic influences of moving loads on girdersin laboratory tests. Lerfors conducted the first part between 1943 and 1946. Fromthe end of 1946 Hillerborg worked on the problem, which resulted in his doctoralthesis “Dynamic Influences of Smoothly Running Loads on Simply Supported Gird-ers” (Hillerborg, 1951). Hillerborg also referred to Ödman (1948) as another early
3
CHAPTER 1. INTRODUCTION
Year
Max
imum
trai
nsp
eed/
kmh−
1World record
Scheduled rail traffic, worldwide
Scheduled rail traffic, Sweden
1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 2020 20400
50
100
150
200
250
300
350
400
450
500
550
600
Figure 1.1: Train speeds in passenger traffic. The dashed line for future scheduled railtraffic in Sweden indicates the soon opened Bothnia Line and the proposedhigh speed train at 320 km/h in 2025. Redrawn from Fröidh and Nelldal(2006).
Swedish contribution with his differential equation for calculation of vibrations pro-duced in load-bearing structures by moving loads.
A complete literature review of all the research conducted on the vibration of bridgesunder moving vehicles is to extensive to include here. However, Yang et al. (2004)present a comprehensive list of references, reviewed in short in the following. Theprevious investigations can be divided into two clearly different categories. Be-fore the advent of digital computers in the 1940s, research were concerned mainlywith the development of analytical or approximate solutions for simple, fundamen-tal problems. Typical researchers of this period were Timoshenko (1922), Jeffcott(1929) and Lowan (1935). Inglis (1934) conducted a general treatment on the dy-namics of railway bridges, constituting a ground for the development that followed.Digital computers, later followed by workstations, made it possible for researchersto adopt more realistic bridge and vehicle models in the analyses. Timoshenko andYoung (1955) and Biggs (1964) presented general structural dynamic works. Frýba(1972) included a detailed review of moving load problem investigations before the1970s, followed up with a 2nd and later 3rd edition (Frýba, 1999). Other literaturethat ought to be highlighted are Garg and Dukkipati (1984) and Frýba (1996).
After years of theory development the concept and performance of bridge moni-toring drastically increased in the last two decades, due to advanced technology
4
1.1. BACKGROUND
Figure 1.2: The author is installing of one of the strain transducers to be embedded inthe concrete section.
development within the fields of personal computers and monitoring techniques.This complemented the analytical and numerical studies with possibilities of sys-tem identification and model updating based on measurements and load testing.An important factor for the present interest in this area of research is also thesuccessful operation of high-speed railways in Japan and some European countries.
As an unique Swedish bridge monitoring project, the New Årsta Railway Bridge inStockholm was instrumented during construction with sensors placed on the soffitof the track slab and embedded in the bridge deck, see Fig. 1.2. The bridge design isconsidered complex and the main intention with the instrumentations was a deeperunderstanding of its structural behaviour. The bridge was not designed for highspeed traffic. Still, this instrumentation gave an opportunity to model and studymoving load simulations based on measurement validations. One of the typical loadtests with a fully loaded macadam train is demonstrated in Fig. 1.3. Fig. 1.4 givesemphasis to the unique but attractive geometry of the New Årsta Railway Bridge.Fig. 1.5 shows the great amount of reinforcement embedded in the concrete section.
During the process of this thesis work, different numerical models of the bridge wereconstructed and evaluated. These are, in chronological order, the early stage resultsof the 3D beam element FE model implemented by the author in Wiberg (2006),the finite difference method approach in Hannewald (2006), the 2D beam FE modelapproach in Gonzalez Silva (2008) and the volume element approach in Poisseroux(2009). In addition, there is also an ongoing structural health monitoring researchproject coupled to this bridge instrumentation, where modern fibre optic sensorsare implemented, see e.g. Enckell (2006).
5
CHAPTER 1. INTRODUCTION
Figure 1.3: A fully loaded macadam train positioned on the bridge during a static loadtest.
1.2 Outline
This thesis consists of a summary and five appended papers. All results are pre-sented in the papers.
The summary part has the intention of making the reader familiar with the sub-ject. It also explains the research structure, the relation between the papers andhighlights the role and objective of each paper in the overall research work. Asthe summary includes additional references, not included in the appended papers,a reference list is given. The summary also includes two algorithms as appendices.Appendix A presents the Yates’s algorithm, implemented in Paper D. Appendix Bexplains the optimisation solver based on the Nelder-Mead simplex algorithm, usedin Paper E.
The reader may read the papers independently but is encouraged to take part ofChapter 3 for an overview and then continue with the papers.
1.3 Aim and scope
Detailed numerical analyses of passing trains on bridges are very time consumingas they involve many simulations using different train configurations at differentspeeds and has lots of considerations to take into account. Thus, simplified bridgeand train models are often chosen for time efficient simulations.
6
1.3. AIM AND SCOPE
Figure 1.4: The smoothly shaped red concrete bridge.
Figure 1.5: A part of the bridge is being prepared with reinforcement and tendon tubesbefore casting.
7
CHAPTER 1. INTRODUCTION
The main objectives of the present work were:
• Instrument the New Årsta Railway Bridge for monitoring static and dynamicbridge behaviour.
• Implement a simplified FE model of the bridge for sufficiently accurate andtime efficient moving load simulations.
• Implement, test and demonstrate the potential in combining a simplifiedmodel with modern system identification techniques and FE model updat-ing routines.
• Verify the damping ratio in design codes for a prestressed continuous bridge.
• Study the possible influence of modelling parameters, separately and jointly,on load effects and dynamic characteristics of the bridge.
• Make dynamic simulations of high speed trains to investigate the sensitivityof the bridge to train induced vibrations.
Notice, it was not the intention to reflect the physical quantities of the bridge mostaccurately in the FE model, but to facilitate practical moving load simulations forlarge and complicated bridges. A more detailed and/or complex FE model systemwould probably render more accurate load effect predictions, but would at the sametime be impractical in time efficient moving load simulations.
The present work is based on the following main assumptions and limitations:
• Linear material properties.
• A bridge model with 3D Bernoulli-Euler beam elements.
• Accurately known positions of measuring sensors and vehicle loads in loadtesting.
• Vehicle axle loads represented as point forces.
1.4 Thesis contribution
The concept of simplified FE modelling, in combination with optimised FE modelupdating, based on measurements, is presented in this thesis.
From a scientific point of view, the main contributions consist of:
1. Instrumenting the bridge and performing measurements over a long period oftime.
8
1.4. THESIS CONTRIBUTION
2. Investigating the usefulness of measured data for bridge as well as vehicleidentification.
3. Introducing and testing the concept of an equivalent modulus of elasticityapproach, instead of in detail considering reinforcement, tendons, boundaryconditions and other contributors to the overall bridge stiffness.
4. Demonstrating the reliability of a simplified FE bridge model by using actualload effects and operational modal analysis.
5. Highlighting the potential and value of using statistical methods, i.e. hereinDesign and analysis Of Experiments (DOE), as a first step in FE modelupdating. To the author’s knowledge, the adopted method of factorial ex-perimentation has not been used before for discovering significant modellingfactors of the bridge system.
9
Chapter 2
Train-bridge modelling aspects
Due to the successful implementation and operation of high-speed railways world-wide, the dynamic response of railway bridges is receiving the most attention ever.These structural systems consist of the the vehicle, the track and the bridge. Inthe following, some modelling aspects are presented briefly to introduce the readerto the subject.
The vehicles are complex structures that can be modelled using masses, springsand dashpots, see Fig. 2.1. However, the most complex train models should only beused if e.g. acceleration levels in the train need to be checked and if the track is notwell maintained (i.e. the rail condition is poor and therefore rail irregularities needto be considered in the analyses), see e.g. Karoumi (1998) and Yang et al. (2004).For normal vehicle speeds, most of the excitation of the dynamic system is thencaused by the roughness of the rail surface and very little is caused by the elasticdisplacement of the bridge itself. Majka and Hartnett (2009) studied this effect ofrandom track irregularities, using the complex 3D vehicle model in Fig. 2.2.
v
(m1 +m2)g
(a) Moving force model
v
m1 +m2
(b) Moving mass model
vw
m1
m2
k c
(c) Sprung mass model
Figure 2.1: Vehicle model alternatives.
11
CHAPTER 2. TRAIN-BRIDGE MODELLING ASPECTS
Figure 2.2: Complex 3D model of a railway vehicle (Majka and Hartnett, 2009).
By neglecting the inertia effect of the vehicle, it only symbolises moving forcesaccording to Fig. 2.1(a). This kind of most simple vehicle model was for exampleadopted in the studies of simple beams by Timoshenko (1922) and extended tocontinuous beams by Chen (1978), or more recently by Dugush and Eisenberger(2002). In cases when the inertia effect of the vehicle is not regarded small, themoving mass model can be used, see Fig. 2.1(b). Such a vehicle model was imple-mented by Stanišič (1985), resulting in an exact closed form solution for a simplebeam carrying a single moving mass. A vehicle model that considers the elasticand damping effects of a suspension system is referred to as the sprung mass modelin Fig. 2.1(c). Vehicle models of this more sophisticated kind can make the simula-tions more realistic but also result in divergence or slow convergence in the iterationprocess of searching a large number of contact forces due to the interaction betweenvehicle and bridge (Yang et al., 1996). Instead, the use of simplified vehicle modelsmake the identification of key parameters, dominating the dynamic response of thebridge, much easier (Humar and Kashif, 1993).
Theoretically, the difference between the vehicle models in Fig. 2.1 is probably bestdemonstrated by reviewing the equation of motion for a simple beam under amoving train. This is presented here as in Yang et al. (2004) but can also befound elsewhere, see e.g. Karoumi (1998) or Olsson (1986).
Consider a simply-supported beam of length L travelled by a train at speed v,consisting of a number of identical cars of length d. The train is assumed to travelalong the centerline of the beam, i.e. no torsional action is present. Approximatingthe train as a series of lumped loads p, the corresponding load function can be givenas
F (t) =N∑i=1p · Ui(t, v, L) (2.1)
12
where
Ui(t, v, L) = δ[x− v(t− ti)] ·[H(t− ti)−H
(t− ti − L
v
)](2.2)
Here, δ denotes the Dirac delta function, x the coordinate of the beam, H(•) a unitstep function, ti the arrival time of the ith load at the beam, ti = (i− 1)d/v, andN the total number of moving loads. Consequently, the effect of the ith movingload is turned on by the term H(t− ti) when it enters the beam and is turned offby the term H(t− ti − L/v) when it leaves the beam.
Normally, each car is supported by two bogies, each of which consists of two wheelsets. In the following formulas however, each bogie is simplified and considered asone single component, a wheel assembly. By then letting Lc denote the distancebetween the two wheel assemblies of a car and Ld the distance between the rearwheel assembly of a car and the front wheel assembly of the following car, it followsthat the car length d is equal to the sum of Lc and Ld and that a train can berepresented as a sequence of wheel loads p with alternative intervals Lc and Ld.For a better representation of the load configuration, it is realised that each trainconsists of two wheel load sets, the first set representing the wheel loads of all thefront bogies and the second set the rear ones. Consequently, the distance betweenany two consecutive wheel loads in each set is then d. Due to the time lag, tc = Lc/v,between the two sets of moving loads the wheel load function in Eq. (2.1) is modifiedand represented as:
F (t) =N∑i=1p · [Ui(t, v, L) + Ui(t− tc, v, L)] (2.3)
The expressions given in Eq. (2.1) and Eq. (2.3) consider only the effect of movingforces according to the vehicle model in Fig. 2.1(a). They neglect the effect of inertiaof the moving masses and the interaction between the train cars and supportingbeam. If to consider those effects for the moving mass case, the load term p can bereplaced by the function (Bolotin, 1964)
F (p,M, v) = p−M(u+ 2vu′ + v2u′′) (2.4)
whereM denotes the vehicle mass (m1+m2) lumped at each load or wheel position,u is the vertical deflection of the beam and dots ( ˙ ) and primes (′) represent dif-ferentiation with respect to time t and coordinate x, respectively. Most importantis the physical meaning of the added terms in Eq. (2.4). The term Mu representsthe inertial force acting along the direction of deflection u of the beam, the term2Mu′ is the Coriolis force relating to the rate of inclination of the beam and theterm Mv2u′′ is the centrifugal force associated with the curvature of the beam,
13
CHAPTER 2. TRAIN-BRIDGE MODELLING ASPECTS
induced by the mass with speed v at the position of action. Consequently, basedon Eq. (2.4), the equation of motion for the beam can be written as
mu+ ceu+ ciIu′′′′ + EIu′′′′ =N∑i=1F (p,M, v) · [Ui(t, v, L) + Ui(t− tc, v, L)] (2.5)
where m denotes the bridge mass per unit length, ce the external damping coeffi-cient, ci the internal damping coefficient, E the modulus of elasticity of the beamand I the moment of inertia of the beam. Most commonly, the second and thirdterm on the left hand side is replaced with cu only.
Beam bridges for high speed trains are, however, often so stiff that the effects ofboth the Coriolis force and the centrifugal force are very much smaller than thatof the added masses. They can therefore generally be excluded without losingaccuracy in the solutions.
Olsson (1986) and Samani and Pellicano (2009) also showed that the behaviour ofbeams under moving forces or moving masses is very similar when the moving massis assumed to be small in comparison to the beam mass. Karoumi (1998) concludedthat too, assuming a rail surface with no roughness and a normal vehicle speed.Then, the moving force model is fully adequate and there is no need for usingcomplicated vehicle models. However, the moving force model has been foundnot suitable for the study of short span bridges (L � 20 − 25 m) since the resultthey produce (displacements and accelerations) are much larger than those obtainedfrom more sophisticated models with distribution of the loads due to the presence ofthe sleepers and a ballast layer, together with the train-bridge interaction (Museroset al., 2002). Thus, the track model in many cases also becomes important. How thebeneficial effect of the track (i.e. the rails, sleepers and ballast) should be consideredin an enhanced assessment is given emphasis to in SB-LRA (2007), which alsorefers to the investigations performed in ERRI D214 (1999). Thus, for enhancedassessments of short span bridges, the favourable effect of the track distributing theload should be considered in a track model, resulting in reduced acceleration levels.
Consideration of load distribution is not the only factor that complicates the movingload assessments on railway bridges. Fig. 2.3 includes potential factors to eventuallyconsider in more enhanced designs and assessments.
Powerful numerical methods, based on finite element methods, are most often em-ployed to analyse the dynamic behaviour of bridges and moving vehicles. Virtually,the limit of complexity in the subsystem models is then nonexisting. Obviously,the use of simplified vehicle and bridge models is helpful and requires less prepa-ration and computation efforts, still allowing the identification of key parametersdominating the dynamics in the structural system (Yang et al., 2004). However,the required time step Δt and the number of modes to include in a mode superpo-sition solution must be investigated in a study of convergence in results. Various
14
Bridge damping Irregularities (wheel, rail)
Cracked/uncracked concreteBridge/soil interaction
The mass of the train
Load distribution ballast/sleepers
Dynamic modulus of elasticity
The stiffness and damping of the track
The transition zone betweenembankment and bridge
Bridge/train interactionTrain damping
Boundary conditions
Figure 2.3: Example of modelling aspects that may be necessary to consider in moreenhanced moving load designs and assessments of railway bridges.
types of bridges has been studied in analyses of the moving load problem. Thesimply-supported beam is adopted most often but other more complex bridge mod-els also include the related effects of e.g. rail irregularities (Paultre et al., 1992) andtorsional vibration (Hsu, 1996). The objective with more realistic simulations withdesign loads or actual traffic loads, based on measurements, is to predict actualdynamic load effects instead of using the traditional dynamic amplification factors,described and evaluated in e.g. Chaallal and Shahawy (1998) and James (2003). Atypical example of this is the study of Karoumi and Wiberg (2006) where actualdynamic load effects on bridges along the Bothnia Line in Sweden were determinedbased on simulations.
Typical guidelines for the necessary kind of measurements and calculations hasalso been produced, see e.g. UIC (2006). In addressing dynamic effects and actualtraffic loads, an integrated international research project, Sustainable Bridges -Assessment for Future Traffic Demands and Longer Lives, recently considered thissubject in a background document on assessment of actual traffic loads (SB4.3,2007). This background document was followed by the guideline for assessing theactual load carrying capacity of existing railway bridges (SB-LRA, 2007).
The requirements in the Eurocodes, see CEN (2002), is reported on by Bucknall(2003), relating to high speed railway bridge design. The code includes requirementson design checks, acceptance criteria, structural analysis and structural propertiesto be adopted. Three principle behaviours that contribute to the total dynamicresponse of a railway bridge are identified: inertial response, resonance effects andadditional dynamic effects due to track irregularities, wheel defects and suspensiondefects. In addition, the importance of taking into account resonance effects andbridge deck acceleration, along with other deformations and load effects is focusedon. The code states explicitly that compliance with vertical deflection limits is notsufficient to guarantee satisfactory behaviour where resonance effects occur. Theserequirements were from 2005 also implemented in the Swedish railway bridge designcode, BV Bro (Banverket, 2008).
15
CHAPTER 2. TRAIN-BRIDGE MODELLING ASPECTS
In the following, some typical aspects to consider are presented, mostly based onthe Eurocode requirements for the design of high speed railway bridges. Whendetermining whether a structure will be prone to resonance effects, the relation-ship between frequency of loading (and associated traffic speed) and the naturalfrequencies of a structure is paramount. The natural frequency of a structure de-pends upon its element length or span, mass, stiffness, boundary conditions andvibration mode shape. Of these parameters, stiffness is the most difficult to predictaccurately.
Stiffness
The relationship between frequency of loading and the natural frequencies of thebridge is particularly important when any resonant peaks occur just above theloading frequency and hence the speed range being considered. It is thereforenecessary to make a lower bound assessment of the natural frequency (i.e. lowerbound value of stiffness) of the structure to obtain a lower bound (and safe) estimateof maximum permitted speed.
Damping and other dynamic characteristics
Damping in railway bridges is a complex phenomenon and its accurate descriptionand representation in a numerical model is difficult (Majka and Hartnett, 2008).Generally, it depends on the material, the state of the bridge (presence of cracks,ballast, support conditions) and the amplitude and frequency of vibrations (Frýba,1996). As the overall dynamic behaviour is more sensitive to the value of dampingrather than the mathematical model used to represent damping, viscous dampingassumptions are often made as they also result in easier computational methods.The assumption of viscous damping is sufficiently accurate for design, as bridgedamping values are highly variable. Highly importantly, the maximum accelerationor load effect is dependant upon the rate at which previous dynamic motions aredissipating through damping. The less the damping, the larger are of course themaximum dynamic effects. Therefore, lower bound values of damping should beused in design calculations to ensure that safe estimates of peak dynamic effects atresonance are obtained. However, as the dynamic vehicle/bridge interaction effectstend to reduce the peak response at resonance, especially for short span bridges,account of this may according to the codes be taken by increasing the value ofassumed damping with the term Δζ, see CEN (2002).
In assessment of existing bridges, a way to identify the dynamic characteristics ofa bridge (i.e. natural frequencies and mode shapes in addition to modal dampingratios) is to use powerful operational modal analysis, based on output-only analysesin ambient vibration, see e.g. Brincker et al. (2003). An assessment of dampingidentification methods is presented in Prandina et al. (2009).
Mass
The natural frequency of a structure decreases as the mass of the structure increases(providing other parameters such as stiffness do not change). Any underestimation
16
of mass will overestimate the natural frequency of the structure. As a result theloading frequency and hence speed at which resonance occurs will be overestimated.Therefore safe upper bound estimates of bridge mass are required to ensure a safeprediction of resonant speeds. In addition, the maximum acceleration of a structureat resonance is inversely proportional to the distributed mass of the structure.Therefore, it is important that a second case is also taken into account with lowerbound values of mass used in calculations to ensure that a safe estimate of peakdynamic acceleration effects at resonance is obtained.
17
Chapter 3
The research work
The requirement of more accurate dynamic analyses of railway bridges calls forreliable but rather simplified system models to make moving load simulations prac-tical for large and complex bridges. In the present research, the New Årsta RailwayBridge was instrumented and subjected to static and dynamic load testing for veri-fication and updating of a simplified Bernoulli-Euler beam FE model, intended formore accurate and time efficient analyses of passing trains on bridges.
Fig. 3.1 has the intention of clarifying the performed research work for the reader.The appended papers which are located in the middle suggest a methodology thatcan be adopted for analysis of the effect of passing trains on bridges. Two pos-sible directions of the developed and implemented analysis concept are indicated.The right hand side of Fig. 3.1, valid for assessment of existing bridges, briefly ex-plains the contribution in each paper and the total process work flow in this thesis,preparing the simplified model for dynamic load effect predictions based on systemidentification and measurements. The left hand side of Fig. 3.1, valid for design ofnew bridges, emphasises an alternative use in optimised bridge design, based onthe implemented statistical identification routine and the optimisation algorithm.The order of Paper C and Paper D is not strict in the context of the performedwork. Paper D could as well be placed before Paper C, which is indicated with thearrows in two directions between those two papers.
The right hand side in Fig. 3.1 uses static and dynamic analyses to find an opti-mal FE model based on real measured load effects. This optimised FE model isthen adopted in new simulations for future trains having higher axle loads and/orspeeds. The left hand side in Fig. 3.1 instead uses simulations to find an optimalbridge design in the bridge design phase, based on design code requirements. As anexample, a typical problems that a bridge designer encounters is to find acceptablecombinations of span lengths, stiffness and mass distribution along the bridge, andso forth, resulting in a maximum vertical bridge deck acceleration lower than thecode limit of 3.5 m/s2. Using this maximum acceleration limit value as “measured”
19
CHAPTER 3. THE RESEARCH WORK
in the optimisation process, the optimal intended bridge characteristics are calcu-lated. This could be valuable and time-saving for bridge designers even though theoptimisation routines on their own can be time consuming in complicated cases.
The following are brief summaries of the appended papers:
In Paper A, Monitoring traffic loads and dynamic effects using an instrumentedrailway bridge, an example of vehicle and bridge system identification is presented.This paper is very much of a more general kind, demonstrating possibilities withrailway bridge measurements. The bridge in question is not the New Årsta Rail-way Bridge, analysed in the following papers, but an adjacent integral-type railwaybridge. A complete Bridge Weigh-in-Motion (B-WIM) system, with axle detectionand accurate axle-load evaluation was implemented, together with general evalua-tion of eigenfrequencies, prediction of possible wheel/rail defects and identificationof acceleration levels. Some very early but representative results are presented,and the efficiency of the algorithms and usefulness of the monitoring program high-lighted. This paper was included in a special issue of Engineering Structures, basedon a presentation of the author at the International Conference on Structural En-gineering, Mechanics and Computation (Karoumi et al., 2004). The B-WIM al-gorithm was later refined and presented as a Matlab® toolbox in Liljencrantz andKaroumi (2009).
In Paper B, Monitoring dynamic behaviour of a long-span railway bridge, thebridge in question, the New Årsta Railway Bridge, is introduced. The bridge andits instrumentation is described, together with the simplified FE model and its con-cept of an equivalent modulus of elasticity. The paper focuses on investigating thedynamic characteristics of the bridge using operational modal analysis, but withfixed monitoring sensors. In addition, extreme bridge acceleration values from dif-ferent train passages are collected and compared with the recommended limit valuein bridge design codes. However, the theory of operational modal analysis relieson movable accelerometers. Such a measurement was performed as a complementto this paper and presented by the author at the 2nd International Conference onExperimental Vibration Analysis for Civil Engineering Structures (Wiberg, 2007).Not much new information was received and results from these measurements areincluded in Paper E.
In Paper C, An equivalent modulus of elasticity approach for simplified modellingand analysis of a complex prestressed railway bridge, the possibility of using anequivalent modulus of elasticity in simplifying the modelling is investigated, basedon measured and predicted axial strain, including the Vlasov portion of the torsionalmoment due to constrained warping. The study involves several static load testswith a fully loaded macadam train and Swedish Rc6 locomotives. The definitionsof modulus of elasticity values are discussed, followed by a description of the fieldmeasurements and the analysis techniques.
In Paper D, Statistical screening of individual and joint effect of several modellingfactors on the dynamic FE response of a railway bridge, the potentials of statistical
20
ANALYSIS OF PASSING TRAINS ON BRIDGES
Paper A
Paper B
Paper C
Paper D
Paper E
Design based on optimisation Dynamic responses based on
system identification and
measurements
System identification
VehicleBridge
Bridge
Manual FE model tuning
Statistical identification of influencing modelling
parameters
Optimised FE model
Measurements
Measurements + FE modelling
Measurements + FE modelling
FE modelling
Simulations(predicted load effects )
Check code requirements
Statistical identification of influencing modelling
parameters
Optimised bridge design
\Measurements"+ FE modelling
FE modelling
Design code requirements=
\Measurements"
NEW BRIDGE TO BE DESIGNED EXISTING BRIDGE
Figure 3.1: Schematic thesis structure. The right hand side presents the subjects includedin the appended papers, intended for dynamic responses based on systemidentification and measurements. The left hand side emphasises the possibleimplementation in the design phase, resulting in an optimal bridge design.
21
CHAPTER 3. THE RESEARCH WORK
methods are highlighted. Factorial experimentation in simulating railway bridgedynamics is exemplified. Unlike the usual one factor at a time parameter studies,factorial experimentations also identify the effect of significant modelling parameterinteractions. The statistical effect estimations are based on main-effect and inter-action sum of squares in the theory of design and analysis of experiments. For largefactorial experiments these calculations become tedious and are simplified consider-ably by using Yates’s algorithm. The algorithm is fairly simple but its mathematicalproof is seldom given. It is therefore included here in Appendix A. Additionally, thepaper includes the necessary code for implementation of the statistical procedurein the mathematical software package Matlab®.
In Paper E, Optimized model updating of a railway bridge for increased accuracyin moving load simulations, an optimised updating method based on load testsand statistically identified influencing updating parameters is used for more timeefficient and accurate load effect predictions. A benchmark test is presented todemonstrate the high potential of the adopted Nelder-Mead simplex optimisationalgorithm. This algorithm is also described in detail here in Appendix B. Highspeed train model simulations are performed with the optimised FE model of theNew Årsta Railway Bridge and more accurately predicted load effects are exem-plified. In addition, the Matlab® syntax for implementation of the optimisationroutines is given. The high potential FE model updating procedure is used tradi-tionally, based on measurements, but the relevant area of introducing it in the earlybridge design phase is outlined.
22
Chapter 4
Conclusions
The New Årsta Railway Bridge in Stockholm was successfully instrumented duringconstruction. A simplified 3D Bernoulli-Euler beam element FE model of the bridgewas prepared. The FE model was first manually tuned based on static railway loadtesting. The most extensive work was performed in a statistical identification ofsignificantly influencing modelling parameters. Those were finally used in FE modeloptimisation based on both static and dynamic load testing, resulting in increasedaccuracy in predicted load effects from moving load simulations.
The following was concluded and noted:
• The study confirmed that, for a complex bridge structure, measurements arenecessary to obtain a reliable FE model to be used for dynamic analyses ofpassing trains.
• The complex bridge could be simplified by means of beam theory and anequivalent modulus of elasticity, still producing approximate but reliable re-sults for simplified global analyses. The typical value of an equivalent modulusof elasticity was in this case approximately 25% larger than the specified meanvalue for the concrete grade in question.
• With statistical identification of significantly effecting modelling parametersthe amount of parameters included in the optimisation were kept at an opti-mally low level.
• The implemented statistical and optimisation algorithms operated efficientlyand the accuracy in static and dynamic load effect predictions was consider-ably improved.
• The first pure bending and torsional frequencies for the New Årsta RailwayBridge were found at 1.30 Hz and 3.55 Hz, respectively.
23
CHAPTER 4. CONCLUSIONS
• The optimisation technique gave a modal damping ratio of between 0.92%and 2.10%. Results from peak picking methods and modern stochastic sub-space identification techniques in operational modal analysis varied consider-ably and generally gave lower damping ratios. This proved the difficulties indamping estimation. Also, the lower bound value of 1%, given in the designcodes for prestressed concrete bridges, is not as much on the safe side as wasearlier thought.
• The New Årsta Railway Bridge did not suffer from high bridge deck acceler-ations. This may indicate that this and other multispan continuous concretebridges are suitable for high speed train traffic.
4.1 Further research
Examples of possible subjects and recommendations for further research are:
• Detailed 3D modelling of the instrumented part of the bridge to include the in-plane stresses and a deformed cross section for comparison with the simplifiedmodel included here, especially in torsional vibration.
• Study the unique (in Sweden) slab track system on The New Årsta RailwayBridge in detail.
• Identification of changes in bridge behaviour, i.e. damage detection based onthe installed monitoring equipment as alarm system. With a more refinedbridge model it will be possible to detect local damages based on monitoringdata and the already implemented optimisation routine. In this way it shouldalso be possible to indicate eventual changes in track condition.
• Develop and implement general computer code for statistical experimentaldesign and FE model optimisation in the process of FE model updating.With such routines included in a toolbox the user will be able to choose fromdifferent types of statistical and optimisation algorithms to use in the specificproject.
• The implemented routines for statistical identification and FE model optimi-sation are possible to use also in the bridge design phase. If this is furtherdeveloped and made user-friendly it may be a timesaving action for bridgedesigners.
• Generally, perform more measurements on this and other railway bridges toverify the recommended lower bound damping ratios in design codes and totest the performance of the operational modal analysis concept on railwaybridges.
24
4.1. FURTHER RESEARCH
• Implement B-WIM for measurements and real-time analyses of train trafficwhich can be used for assessment of bridges and for calibration of load modelsin design codes.
• Study the effect or influence of additional factors such as for example brakingand acceleration of trains, the crossing of two vehicles moving in oppositedirections, the mass ratio of the vehicles to the bridge, elastic bearings andsupporting columns.
In addition, the author actually put a great deal of effort into developing a vehicle-bridge interaction model that, based on the bridge deck curvature in both thehorisontal and vertical direction, was shown to operate successfully. This model ispossible to utilise for more detailed studies in the future.
25
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Chaallal, O. and Shahawy, M. (1998). Experimental evaluation of dynamic ampli-fication for evaluation of bridge performance. Technical report, Department ofConstruction Engineering. University of Quebec.
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Dugush, Y. A. and Eisenberger, M. (2002). Vibrations of non-uniform continuousbeams under moving loads. Journal of Sound and Vibration, 254(5):911–926.
Enckell, M. (2006). Structural Health Monitoring using Modern Sensor Technology– Long-term Monitoring of the New Årsta Railway Bridge. Licentiate thesis,Royal Institute of Technology (KTH).
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Frýba, L. (1972). Vibration of Solid and Structures under Moving Loads. NoordhoffInternational Publishing.
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Gonzalez Silva, I. (2008). Dynamic Behaviour of the New Årsta Bridge to MovingTrains – Simplified FE-Analysis and Verifications. Master thesis, Royal Instituteof Technology (KTH).
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Hillerborg, A. (1951). Dynamic Influences of Smoothly Running Loads on SimplySupported Girders. Doctoral thesis, Royal Institute of Technology (KTH).
Hsu, L. C. (1996). Impact responses of bridges traveled by high speed trains. Masterthesis, National Taiwan University, Taipei, Taiwan, R.O.C. (In Chinese).
Humar, J. L. and Kashif, A. M. (1993). Dynamic response of bridges under trav-elling loads. Can. J. Civ. Eng., 20:287–298.
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Karoumi, R. and Wiberg, J. (2006). Kontroll av dynamiska effekter av passerandetåg på botniabanans broar – sammanfattning. Technical report, Royal Instituteof Technology (KTH). Structural Design and Bridges. (In Swedish).
Karoumi, R., Wiberg, J., and Olofsson, P. (2004). Monitoring traffic loads andtraffic load effects on the New Årstaberg railway bridge. In Proc. of the In-ternational Conference on Structural Engineering, Mechanics and Computation(SEMC 2004), Cape Town, South Africa.
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30
Appendix A
Yates’s algorithm
Yates’s algorithm calculates sums of squares of all the contrasts simultaneouslyfrom a 2k factorial design. It achieves a saving in multiplying one form of matrixby a vector. Usually, multiplying a 2p × 2p matrix by a 2p vector requires 22p
multiplications. A matrix of the form
[a bc d
]⊗p
can however be multiplied by a 2p vector in p2p+1 operations by taking advantageof its particular structure.
The algorithm is simple to implement and is mathematically represented in thefollowing:
1. Define the tensor product of the matrices A = (aij)m×m and B = (bkl)n×n:
(aij)m×m ⊗ (bkl)n×n = (aijbkl)mn×mn
where aijbkl is at position (i− 1)n+ k, (j − 1)n+ l.
31
APPENDIX A. YATES’S ALGORITHM
2. Let Amn×mn be the matrix A expanded as:
A =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
a11 · · · a1m 0 · · · 00 a11 · · · a1m · · · 0...
......
0 0 · · · a11 · · · a1ma21 · · · a2m 0 · · · 0
0 a21 · · · a2m · · · 0...
......
......
...0 0 · · · am1 · · · amm
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
3. Let A and B be the expansions of A and B according to (2). Then:
A⊗B = AB
4. Generalise the result to powers, i.e.
A⊗p = Ap
Example
With m = n = 2:
A =[a bc d
], B =
[e fg h
]
A⊗B =[a bc d
]⊗[e fg h
]
=
⎡⎢⎢⎣ae af be bfag ah bg bhce cf de dfcg ch dg dh
⎤⎥⎥⎦
=
⎡⎢⎢⎣a b 0 00 0 a bc d 0 00 0 c d
⎤⎥⎥⎦
⎡⎢⎢⎣e f 0 00 0 e fg h 0 00 0 g h
⎤⎥⎥⎦
Multiplying A ⊗ B by the vector v, takes the sparsity advantage of the matri-ces A and B into account. Each row of B has only two non-zero elements, so
32
Bv is computed in 2 · 4 multiplications. Also A(Bv) is computed in 2 · 4 opera-tions. Thus, the number of multiplications is in general mn2 +m2n = mn(m+ n).Therefore, this example does not save anything since the total number of multipli-cations (16) is equal to the ordinary matrix multiplication. However, for m,n > 2,mn(m+ n) < (mn)2.
Multiplying A⊗p by a mp vector v, the mp ×mp matrix A has only m non-zeroelements on each row. The product Av involves m · mp multiplications. To getApv, the product is repeated p times, resulting in a total of pmp+1 multiplications.Consequently, the saving is huge even for a moderately huge p, compared to them2p multiplications of an ordinary matrix by vector product. If for example m = 2and p = 20, m2p ≈ 1012 but pmp+1 ≈ 4 · 106.
33
Appendix B
The Nelder-Mead simplex algorithm
The Nelder-Mead simplex algorithm is implemented in the optimisation toolbox ofMatlab® as the fminsearch solver. The algorithm uses a simplex of n+1 points forn-dimensional vectors x. Firstly, the algorithm makes a simplex around the initialguess x0 by adding 5% of each component x0(i) to x0 and uses these n vectors aselements of the simplex in addition to x0. Secondly, the algorithm modifies thesimplex repeatedly according to the following procedure:
1. Let x(i) denote the list of points in the current simplex, i = 1, . . . , n+ 1.
2. Order the points in the simplex from smallest function value f(x(1)) to largestf(x(n+ 1)). At each step in the iteration, the current worst point x(n+ 1) isdiscarded and another point is accepted into the simplex (or, in the case ofstep 7 below, all n points with values above f(x(1)) are changed).
3. Generate the reflected point
r = 2m− x(n+ 1)
wherem =∑ x(i)
n, i = 1, . . . , n
and calculate f(r).
4. If f(x(1)) ≤ f(r) < f(x(n)), accept r and terminate this iteration.
5. If f(r) < f(x(1)), calculate the expansion point s
s = m+ 2(m− x(n+ 1))
and calculate f(s).
a) If f(s) < f(r), accept s and terminate the iteration.
35
APPENDIX B. THE NELDER-MEAD SIMPLEX ALGORITHM
b) Otherwise, accept r and terminate the iteration.
6. If f(r) ≥ f(x(n)), perform contraction between m and the better of x(n+ 1)and r.
a) If f(r) < f(x(n+ 1)), i.e. r is better than x(n+ 1), calculate
c = m+ r −m2and calculate f(c). If f(c) < f(r), accept c and terminate the iteration.Otherwise, continue with Step 7.
b) If f(r) ≥ f(x(n+ 1)), calculate
cc = m+ x(n+ 1)−m2
and calculate f(cc). If f(cc) < f(x(n+ 1)), accept cc and terminate theiteration. Otherwise, continue with Step 7.
7. Calculate the n points
v(i) = x(1) + x(i)− x(1)2
and calculate f(v(i)), i = 2, . . . , n + 1. The simplex at the next iteration isx(1), v(2), . . . , v(n+ 1).
The iterations proceed until a stopping criterion is met.
Fig. B.1 below illustrates the points that may be calculated in the algorithm pro-cedure, along with each possible new simplex. The original simplex is representedin the bold outline.
x(n+ 1)
cc
m
c
r
s
v(n+ 1)
x(1)
Figure B.1: Illustrated Nelder-Mead Simplex algorithm procedure. Reproduction fromColeman and Zhang (2009).
36
Part II
APPENDED PAPERS
37
Paper A
Monitoring traffic loads and dynamic effects
using an instrumented railway bridge
Karoumi, R., Wiberg, J., and Liljencrantz, A. (2005). Monitoring traffic loads anddynamic effects using an instrumented railway bridge. Engineering Structures,27(12):1813-1819.
Paper B
Monitoring dynamic behaviour of a
long-span railway bridge
Wiberg, J. and Karoumi, R. (2009). Monitoring dynamic behaviour of a long-spanrailway bridge. Structure and Infrastructure Engineering, 5(5):419-433.
Paper C
An equivalent modulus of elasticity
approach for simplified modelling and
analysis of a complex prestressed railway
bridge
Wiberg, J. (2009). An equivalent modulus of elasticity approach for simplifiedmodelling and analysis of a complex prestressed railway bridge. Submitted toAdvances in Structural Engineering.
Paper D
Statistical screening of individual and joint
effect of several modelling factors on the
dynamic FE response of a railway bridge
Wiberg, J., Karoumi, R., and Pacoste, C. (2009). Statistical screening of individualand joint effect of several modelling factors on the dynamic FE response of arailway bridge. Submitted to Journal of Sound and Vibration.
Paper E
Optimized model updating of a railway
bridge for increased accuracy in moving
load simulations
Wiberg, J., Karoumi, R., and Pacoste, C. (2009). Optimized model updating ofa railway bridge for increased accuracy in moving load simulations. To besubmitted to Journal of Bridge Engineering.
List of Bulletins from the Department of Structural Engineering, RoyalInstitute of Technology, Stockholm
TRITA-BKN. Bulletin
1. Pacoste, C., On the Application of Catastrophe Theory to Stability Analysesof Elastic Structures. Doctoral Thesis, 1993.
2. Stenmark, A-K., Dämpning av 13 m lång stålbalk – ”Ullevibalken”. Utprov-ning av dämpmassor och fastsättning av motbalk samt experimentell bestäm-ning av modformer och förlustfaktorer. Vibration tests of full-scale steel girderto determine optimum passive control. Licentiate thesis, 1993.
3. Silfwerbrand, J., Renovering av asfaltgolv med cementbundna plastmodifier-ade avjämningsmassor. 1993.
4. Norlin, B., Two-Layered Composite Beams with Nonlinear Connectors andGeometry Tests and Theory. Doctoral Thesis, 1993.
5. Habtezion, T., On the Behaviour of Equilibrium Near Critical States. Licen-tiate Thesis, 1993.
6. Krus, J., Hållfasthet hos frostnedbruten betong. Licentiate thesis, 1993.
7. Wiberg, U., Material Characterization and Defect Detection by QuantitativeUltrasonics. Doctoral Thesis, 1993.
8. Lidström, T., Finite Element Modelling Supported by Object Oriented Meth-ods. Licentiate Thesis, 1993.
9. Hallgren, M., Flexural and Shear Capacity of Reinforced High Strength Con-crete Beams without Stirrups. Licentiate Thesis, 1994.
10. Krus, J., Betongbalkars lastkapacitet efter miljöbelastning. 1994.
11. Sandahl, P., Analysis Sensitivity for Wind-related Fatigue in Lattice Struc-tures. Licentiate Thesis, 1994.
12. Sanne, L., Information Transfer Analysis and Modelling of the StructuralSteel Construction Process. Licentiate Thesis, 1994.
13. Zhitao, H., Influence of Web Buckling on Fatigue Life of Thin-Walled Columns.Doctoral Thesis, 1994.
14. Kjörling, M., Dynamic response of railway track components. Measurementsduring train passage and dynamic laboratory loading. Licentiate Thesis, 1995.
15. Yang, L., On Analysis Methods for Reinforced Concrete Structures. DoctoralThesis, 1995.
16. Petersson, Ö., Svensk metod för dimensionering av betongvägar. Licentiatethesis, 1996.
17. Lidström, T., Computational Methods for Finite Element Instability Analy-ses. Doctoral Thesis, 1996. Bulletin 17.
18. Krus, J., Environment- and Function-induced Degradation of Concrete Struc-tures. Doctoral Thesis, 1996.
19. Silfwerbrand, J., (editor), Structural Loadings in the 21st Century. SvenSahlin Workshop, June 1996. Proceedings.
20. Ansell, A., Frequency Dependent Matrices for Dynamic Analysis of FrameType Structures. Licentiate Thesis, 1996.
21. Troive, S., Optimering av åtgärder för ökad livslängd hos infrastruktur-konstruktioner. Licentiate thesis, 1996.
22. Karoumi, R., Dynamic Response of Cable-Stayed Bridges Subjected to Mov-ing Vehicles. Licentiate Thesis, 1996.
23. Hallgren, M., Punching Shear Capacity of Reinforced High Strength ConcreteSlabs. Doctoral Thesis, 1996.
24. Hellgren, M., Strength of Bolt-Channel and Screw-Groove Joints in Alu-minium Extrusions. Licentiate Thesis, 1996.
25. Yagi, T., Wind-induced Instabilities of Structures. Doctoral Thesis, 1997.
26. Eriksson, A., and Sandberg, G., (editors), Engineering Structures and Ex-treme Events – Proceedings from a Symposium, May 1997.
27. Paulsson, J., Effects of Repairs on the Remaining Life of Concrete BridgeDecks. Licentiate Thesis, 1997.
28. Olsson, A., Object-oriented finite element algorithms. Licentiate Thesis, 1997.
29. Yunhua, L., On Shear Locking in Finite Elements. Licentiate Thesis, 1997.
30. Ekman, M., Sprickor i betongkonstruktioner och dess inverkan på beständig-heten. Licentiate Thesis, 1997.
31. Karawajczyk, E., Finite Element Approach to the Mechanics of Track-DeckSystems. Licentiate Thesis, 1997.
32. Fransson, H., Rotation Capacity of Reinforced High Strength Concrete Beams.Licentiate Thesis, 1997.
33. Edlund, S., Arbitrary Thin-Walled Cross Sections. Theory and ComputerImplementation. Licentiate Thesis, 1997.
34. Forsell, K., Dynamic analyses of static instability phenomena. LicentiateThesis, 1997.
35. Ikäheimonen, J., Construction Loads on Shores and Stability of HorizontalFormworks. Doctoral Thesis, 1997.
36. Racutanu, G., Konstbyggnaders reella livslängd. Licentiate thesis, 1997.
37. Appelqvist, I., Sammanbyggnad. Datastrukturer och utveckling av ett IT-stöd för byggprocessen. Licentiate thesis, 1997.
38. Alavizadeh-Farhang, A., Plain and Steel Fibre Reinforced Concrete BeamsSubjected to Combined Mechanical and Thermal Loading. Licentiate Thesis,1998.
39. Eriksson, A. and Pacoste, C., (editors), Proceedings of the NSCM-11: NordicSeminar on Computational Mechanics, October 1998.
40. Luo, Y., On some Finite Element Formulations in Structural Mechanics. Doc-toral Thesis, 1998.
41. Troive, S., Structural LCC Design of Concrete Bridges. Doctoral Thesis,1998.
42. Tärno, I., Effects of Contour Ellipticity upon Structural Behaviour of Hypar-form Suspended Roofs. Licentiate Thesis, 1998.
43. Hassanzadeh, G., Betongplattor på pelare. Förstärkningsmetoder och dimen-sioneringsmetoder för plattor med icke vidhäftande spännarmering. Licenti-ate thesis, 1998.
44. Karoumi, R., Response of Cable-Stayed and Suspension Bridges to MovingVehicles. Analysis methods and practical modeling techniques. DoctoralThesis, 1998.
45. Johnson, R., Progression of the Dynamic Properties of Large SuspensionBridges during Construction – A Case Study of the Höga Kusten Bridge.Licentiate Thesis, 1999.
46. Tibert, G., Numerical Analyses of Cable Roof Structures. Licentiate Thesis,1999.
47. Ahlenius, E., Explosionslaster och infrastrukturkonstruktioner – Risker,värderingar och kostnader. Licentiate thesis, 1999.
48. Battini, J-M., Plastic instability of plane frames using a co-rotational ap-proach. Licentiate Thesis, 1999.
49. Ay, L., Using Steel Fiber Reinforced High Performance Concrete in the In-dustrialization of Bridge Structures. Licentiate Thesis, 1999.
50. Paulsson-Tralla, J., Service Life of Repaired Concrete Bridge Decks. DoctoralThesis, 1999.
51. Billberg, P., Some rheology aspects on fine mortar part of concrete. LicentiateThesis, 1999.
52. Ansell, A., Dynamically Loaded Rock Reinforcement. Doctoral Thesis, 1999.
53. Forsell, K., Instability analyses of structures under dynamic loads. DoctoralThesis, 2000.
54. Edlund, S., Buckling of T-Section Beam-Columns in Aluminium with or with-out Transverse Welds. Doctoral Thesis, 2000.
55. Löfsjögård, M., Functional Properties of Concrete Roads – General Interre-lationships and Studies on Pavement Brightness and Sawcutting Times forJoints. Licentiate Thesis, 2000.
56. Nilsson, U., Load bearing capacity of steel fibre reinforced shotcrete linings.Licentiate Thesis, 2000.
57. Silfwerbrand, J. and Hassanzadeh, G., (editors), International Workshop onPunching Shear Capacity of RC Slabs – Proceedings. Dedicated to ProfessorSven Kinnunen. Stockholm June 7-9, 2000.
58. Wiberg, A., Strengthening and repair of structural concrete with advanced,cementitious composites. Licentiate Thesis, 2000.
59. Racutanu, G., The Real Service Life of Swedish Road Bridges – A case study.Doctoral Thesis, 2000.
60. Alavizadeh-Farhang, A., Concrete Structures Subjected to Combined Me-chanical and Thermal Loading. Doctoral Thesis, 2000.
61. Wäppling, M., Behaviour of Concrete Block Pavements – Field Tests andSurveys. Licentiate Thesis, 2000.
62. Getachew, A., Trafiklaster på broar. Analys av insamlade och Monte Carlogenererade fordonsdata. Licentiate thesis, 2000.
63. James, G., Raising Allowable Axle Loads on Railway Bridges using Simulationand Field Data. Licentiate Thesis, 2001.
64. Karawajczyk, E., Finite Elements Simulations of Integral Bridge Behaviour.Doctoral Thesis, 2001.
65. Thöyrä, T., Strength of Slotted Steel Studs. Licentiate Thesis, 2001.
66. Tranvik, P., Dynamic Behaviour under Wind Loading of a 90 m Steel Chim-ney. Licentiate Thesis, 2001.
67. Ullman, R., Buckling of Aluminium Girders with Corrugated Webs. Licenti-ate Thesis, 2002.
68. Getachew, A., Traffic Load Effects on Bridges. Statistical Analysis of Col-lected and Monte Carlo Simulated Vehicle Data. Doctoral Thesis, 2003.
69. Quilligan, M., Bridge Weigh-in-Motion. Development of a 2-D Multi-VehicleAlgorithm. Licentiate Thesis, 2003.
70. James, G., Analysis of Traffic Load Effects on Railway Bridges. DoctoralThesis, 2003.
71. Nilsson, U., Structural behaviour of fibre reinforced sprayed concrete anchoredin rock. Doctoral Thesis, 2003.
72. Wiberg, A., Strengthening of Concrete Beams Using Cementitious CarbonFibre Composites. Doctoral Thesis, 2003.
73. Löfsjögård, M., Functional Properties of Concrete Roads – Development ofan Optimisation Model and Studies on Road Lighting Design and Joint Per-formance. Doctoral Thesis, 2003.
74. Bayoglu-Flener, E., Soil-Structure Interaction for Integral Bridges and Cul-verts. Licentiate Thesis, 2004.
75. Lutfi, A., Steel Fibrous Cement Based Composites. Part one: Material andmechanical properties. Part two: Behaviour in the anchorage zones of pre-stressed bridges. Doctoral Thesis, 2004.
76. Johansson, U., Fatigue Tests and Analysis of Reinforced Concrete BridgeDeck Models. Licentiate Thesis, 2004.
77. Roth, T., Langzeitverhalten von Spannstählen in Betonkonstruktionen. Li-centitate Thesis, 2004.
78. Hedebratt, J., Integrerad projektering och produktion av industrigolv – Metoderför att förbättra kvaliteten. Licentiate thesis, 2004.
79. Österberg, E., Revealing of age-related deterioration of prestressed reinforcedconcrete containments in nuclear power plants – Requirements and NDTmethods. Licentiate Thesis, 2004.
80. Broms, C.E., Concrete flat slabs and footings. New design method for punch-ing and detailing for ductility. Doctoral Thesis, 2005.
81. Wiberg, J., Bridge Monitoring to Allow for Reliable Dynamic FE Modelling– A Case Study of the New Årsta Railway Bridge. Licentiate Thesis 2006.
82. Mattsson, H-Å., Funktionsentreprenad Brounderhåll – En pilotstudie i Upp-sala län. Licentiate Thesis 2006.
83. Masanja, D. P, Foam concrete as a structural material. Doctoral Thesis 2006.
84. Johansson, A., Impregnation of Concrete Structures – Transportation andFixation of Moisture in Water Repellent Treated Concrete. Licentiate Thesis2006.
85. Billberg, P., Form Pressure Generated by Self-Compacting Concrete – Influ-ence of Thixotropy and Structural Behaviour at Rest. Doctoral Thesis 2006.
86. Enckell, M., Structural Health Monitoring using Modern Sensor Technology– Long-term Monitoring of the New Årsta Railway Bridge. Licentiate Thesis2006.
87. Söderqvist, J., Design of Concrete Pavements – Design Criteria for Plain andLean Concrete. Licentiate Thesis 2006.
88. Malm, R., Shear cracks in concrete structures subjected to in-plane stresses.Licentiate Thesis 2006.
89. Skoglund, P., Chloride Transport and Reinforcement Corrosion in the Vicinityof the Transition Zone between Substrate and Repair Concrete. LicentiateThesis 2006.
90. Liljencrantz, A., Monitoring railway traffic loads using Bridge Weight-in-Motion. Licentiate Thesis 2007.
91. Stenbeck, T., Promoting Innovation in Transportation Infrastructure Mainte-nance – Incentives, Contracting and Performance-Based Specifications. Doc-toral Thesis 2007.
92. Magnusson, J., Structural Concrete Elements Subjected to Air Blast Loading.Licentiate Thesis 2007.
93. Pettersson, L., G., Full Scale Tests and Structural Evaluation of Soil SteelFlexible Culverts with low Height of Cover. Doctoral Thesis 2007.
94. Westerberg, B., Time-dependent effects in the analysis and design of slenderconcrete compression members. Doctoral Thesis 2008.
95. Mattsson, H-Å, Integrated Bridge Maintenance. Evaluation of a pilot projectand future perspectives. Doctoral Thesis 2008.
96. Andersson, A, Utmattningsanalys av järnvägsbroar. En fallstudie av stål-broarna mellan Stockholm Central och Söder Mälarstrand, baserat på teo-retiska analyser och töjningsmätningar. Licentiate thesis 2009.
97. Malm, R, Predicting shear type crack initiation and growth in concrete withnon-linear finite element method. Doctoral Thesis 2009.
98. Bayoglu Flener, E, Static and dynamic behaviour of soil-steel composite bridgesobtained by field testing. Doctoral Thesis 2009.
99. Gram, A, Numerical Modelling of Self-Compacting Concrete Flow: Discreteand Continuous Approach. Licentiate Thesis 2009.
100. Wiberg, J., Railway bridge response to passing trains. Measurements and FEmodel updating. Doctoral Thesis 2009.
The bulletins enumerated above, with the exception for those which are out of print,may be purchased from the Department of Civil and Architectural Engineering,Royal Institute of Technology, SE-100 44 Stockholm, Sweden.
The department also publishes other series. For full information see our home pagehttp://www.byv.kth.se
Railway bridge response to passing trains
Measurements and FE model updating
J o h a n W i b e r g
Doctoral Thesis in Civil architectural engineering
Stockholm, Sweden 2009
www.kth.se
TRITA-BKN. Bulletin 100, 2009ISSN 1103-4270
ISRN KTH/BKN/B--100--SE
Joh
an W
iberg railw
ay bridge response to passing trainsKTh
2009