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Load Effect Modelling in Fatigue Design ofComposite BridgesAn assessment of Fatigue Load Models 3, 4 and 5 according to

SS-EN-1991-2 Actions on Structures – Part 2: Traffic loads on Bridges

Mathias DahlvikJohan Eriksson

June 2014TRITA-BKN. Examensarbete 419, 2014ISSN 1103-4297ISRN KTH/BKN/B-419-SE

c©Mathias Dahlvik & Johan Eriksson 2014Royal Institute of Technology (KTH)Department of Civil and Architectural EngineeringDivision of Structural Engineering and BridgesStockholm, Sweden, 2014

Preface

To begin with, we would like to direct a big thank you to ELU Konsult AB for givingus the opportunity to write this thesis. Special thanks are directed to Frank Axhag forcontinuous support and supervision throughout the entire process. We would also like tothank Bert Norlin and John Leander from KTH for their helpful input regarding modelling,regulation interpretation, report writing, etc.

Since this is our final report as students at KTH we would also like to thank all of ourfellow students, teachers and assistants for making these five years very special. Also,we want to use this preface as a opportunity to show appreciation for the support andinspiration given by family and friends during our time at the university.

Finally, we hope that you as a reader enjoy reading the report. A lot of effort has beenput down in order to make the report interesting and easy to follow.

Stockholm, June 2014

Mathias Dahlvik

Johan Eriksson

i

Abstract

At the turn of 2010/2011, Sweden went from designing structures according to nationaldesign codes to the new European standards Eurocode. For bridge engineers, this implieda change from a combination of BRO 2004 and BSK 07 to the Eurocode as the maindocuments, complemented by national documents such as TRVK Bro 11. The normtransition did not only change the calculation methods, but also turned a phenomenonthat never was of great importance for road bridges before into something that could limitthe carrying capacity of the structure. This phenomenon is called fatigue, i.e. repeatedload cycles, where each load is much lower than the ultimate limit state capacity, thatfinally results in collapse.

This master thesis investigates why fatigue is significant in the design today. This is donethrough a comparison of how the new and old regulations assesses fatigue. A bridge builtin 2011, designed by ELU Konsult AB according to the old regulations, was modelledin the finite element program LUSAS. Several lorry crossings from different fatigue loadmodels were then simulated. The output from LUSAS was then used to calculate theutilization ratios for three critical points along the bridge.

The result indicates that both regulations give rise to similar stress ranges, i.e. thedifference between the maximum and minimum stress obtained during a crossing. Thedifferences between the regulations are instead within the fatigue calculations, where themajor difference is the number of lorries crossing the bridge during its lifetime. Theutilization ratio according to the old regulations for the worst exposed point is 27.0%,corresponding to 9.13 daily crossings by heavy lorries, which is the maximum numberof daily crossings provided by BRO 2004. The lowest utilization ratio according tothe Eurocode is 70.0%, calculated for 137 daily crossings which is the lowest amountof crossings allowed. An interpretation of the Eurocode, which allows usage of fatigue loadmodel 5 even for smaller bridges, results in a utilization ratio of 56.0% which correspondsto 90.0 daily crossings, i.e. lower than the other fatigue load models provided by theEurocode but clearly above the old regulations.

The conclusion is that an alternative way of deciding the number of crossings shouldbe provided by the Eurocode. Today, the classification consists of four steps, which arevery rough. Instead, a proposal is given in this thesis which advocates usage of a linearfunction for deciding the number of design crossings based on the number of daily crossingsby lorries. The proposed alternative design method is between the two regulations withrespect to daily crossings and utilization ratio.

Keywords: Fatigue, Composite Bridges, Eurocode, BRO 2004, Fatigue Load Model

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Sammanfattning

Vid årsskiftet 2010/2011 övergick Sverige från att dimensionera byggnadsverk enligt nationellastandarder till den nya europastandarden Eurokod. För brokonstruktörer innebar dettaen övergång från en kombination av BRO 2004 och BSK 07, till att Eurokod blev dethuvudsakligt styrande dokumentet, med bland annat TRVK Bro 11 som ett dokumentmed tillhörande nationella val. Övergången medförde inte bara att verksamma konstruktörertvingades lära sig förändrade beräkningsmetoder, utan också att ett fenomen som tidigaresällan var dimensionerande för vägbroar nu kunde vara det som ställde högst krav påbärförmågan. Detta fenomen kallas utmattning, dvs. upprepade av- och pålastningar, varoch en betydligt lägre än brons maximala bärförmåga, som i slutändan resulterar i brott.

I detta examensarbete utreds det varför utmattning numera är en betydande del avdimensioneringen. Detta sker genom en jämförelse av hur de gamla och nya normernautvärderar utmattning. Som modell har en befintlig bro invigd 2011, dimensioneradav ELU Konsult AB enligt de gamla normerna, använts. Denna bro har modellerats ifinita element programmet LUSAS, varpå en mängd olika lastbilsöverfarter simulerats ochutmattningsutnyttjandet för tre utvalda kritska punkter beräknats.

Resultatet indikerar att båda normerna har liknande storlekar på spänningsvidderna,dvs. skillnaden på största och minsta spänningen som uppstår vid en överfart. Däremotråder det skillnader vid utmattningsberäkningarna, där den stora skillnaden är antalettunga fordon som passerar bron under dess livslängd. Enligt de gamla normerna ärutnyttjandegraden för den värst utsatta studerade punkten 27.0%, vilket är beräknatpå det högsta antalet dagliga passager från tunga fordon som BRO 2004 tillåter, d.v.s.9.13 dagliga passager. Enligt Eurokod uppgår den lägsta utnyttjandegraden till 70.0%,vilket motsvarar 137 dagliga överfarter vilket är det lägsta Eurokod tillåter. Vid ettalternativt sätt att tolka Eurokod, som tillåter användandet av utmattningslastmodell5 även för mindre broar, fås en utnyttjandegrad på 56.0% vilket motsvarar 90.0 dagligaöverfarter. Detta är något lägre än de andra utmattningslastmodellerna enligt Eurokodmen fortfarande högre än det gamla regelverket.

Slutsatsen av uppsatsen är att ett alternativt sätt att bestämma antalet överfarter bordeerbjudas i Eurokod, då indelningen idag består av fyra stora trappsteg vilket ger en väldigtsnäv indelning. I detta examensarbete presenteras ett förslag som innebär att antaletdimensionerande överfarter istället bör bestämmas som en rätlinjig funktion av antaletdagliga överfarter från tung trafik. Det föreslagna sättet ligger mellan de båda normernamed hänsyn till passager och utnyttjandegrad.

Nyckelord: utmattning, kompositbroar, Eurokod, BRO 2004, utmattningslastmodell

v

Table of Contents

1 Introduction 1

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

1.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Facts Regarding the Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Summary of Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.6 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.7 Normative References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.8 Abbreviations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.9 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Methods 9

2.1 Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Finite Element Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Element Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Element Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Support Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.4 Stress Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.5 Fibre Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Fatigue Load Models according to the Eurocode . . . . . . . . . . . . . . . 14

2.3.1 Fatigue Load Model 3 (FLM3) . . . . . . . . . . . . . . . . . . . . . 14



2.4 Fatigue Assessment according to the Eurocode . . . . . . . . . . . . . . . . 17

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2.4.1 The Lambda Method . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 Palmgren-Miner Method . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Fatigue Load Models according to BRO 2004 . . . . . . . . . . . . . . . . . 22

2.6 Fatigue Assessment according to BSK 07 . . . . . . . . . . . . . . . . . . . . 22

2.7 Interpretations, Simplifications and Conditions . . . . . . . . . . . . . . . . 24

2.7.1 Assumptions and Simplifications . . . . . . . . . . . . . . . . . . . . 24

2.7.2 Fatigue Load Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7.3 Fatigue Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7.4 The Nobs and LCN selection . . . . . . . . . . . . . . . . . . . . . . . 27

2.7.5 Dynamic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Results and Analysis 29

3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Properties of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.2 Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Stress Ranges and Critical Load Positions . . . . . . . . . . . . . . . . . . . 32

3.3 Fatigue Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.1 Analysis of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.2 Sensitivity Analysis - FLM3 . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.3 Sensitivy Analysis - FLM4 and FLM5 . . . . . . . . . . . . . . . . . 38

3.4.4 Sensitivity Analysis - BRO 2004 . . . . . . . . . . . . . . . . . . . . 38

4 Discussion 39

4.1 General Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Similarities between the Old and the New Regulations . . . . . . . . . . . . 39

4.3 Differences between the Old and the New Regulations . . . . . . . . . . . . 40

4.4 How to pass the Fatigue Verification . . . . . . . . . . . . . . . . . . . . . . 41

4.5 Pros and Cons with the Fatigue Load Models . . . . . . . . . . . . . . . . . 41

4.6 Most favourable FLM in the Design . . . . . . . . . . . . . . . . . . . . . . 42

4.7 Weaknesses in the Eurocode’s Fatigue Assessment . . . . . . . . . . . . . . 42

4.8 The Importance of Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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5 Conclusions and Future Outlook 47

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2 Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Bibliography 49

Appendix A Properties of the Main Beams A-1

Appendix B Hand Calculations B-1

Appendix C Shear Stress Assessment C-1

Appendix D Complete Calculations - FLM3 D-1

Appendix E Complete Calculations - FLM4 and FLM5 E-1

Appendix F Complete Calculations - BRO 2004 F-1

Appendix G Stress Evaluation with Increased Bottom Flange Size G-1

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Chapter 1

Introduction

1.1 Background

Ever since the Eurocode was implemented in Sweden at the turn of 2010/2011 there hasbeen an intense discussion regarding the impact on designs [1]. Within bridge design,the rumour claims that the implementation of the Eurocode clearly increased the requiredamount of steel compared to the old regulations. According to the rumour, the largestdifference appears in the bottom flanges at midspans, when assessed with respect tofatigue. The underlying reason is although not completely clear.

Fatigue is when repeated external loading gives rise stress variation that results in a crackpropagation within a structure [2]. The crack propagation may appear in structural steelparts, reinforcing steel bars, the concrete slab etc. and can have catastrophic consequencesif dealt with incorrectly. A high stress variation combined with many load cycles andweak design may result in total collapse. In some structures extensive cracking may existwithout seriously affecting the load-carrying capacity, although fatigue failure is said tohave occurred.

The majority of all engineering failures are caused by fatigue. The repeatedly applied loadrequired to cause fatigue failure is much lower than the allowed static design load. If thedesign is weak, failure due to fatigue can occur after only a few hundred stress cycles. Themost difficult thing with assessing fatigue is to determine and identify all the variationsthat could give rise to the phenomenon. In reality, very few structures are subjected purelyto static loading; some variations in stress always occur. A few examples of fatigue loadingare wind, live loads, temperature changes, vibrations or traffic.

Since the crack propagation usually arises under elastic nominal stresses the fatiguedamage starts at details where the stress is concentrated, such as near holes or welds.The development of a crack starts with an initiation phase as illustrated in Figure 1.1,i.e. initially the crack is almost impossible to see with the bare eye. Once the growth hasstarted and the crack propagates into the material, the growth is exponential until totalfracture occurs.

1

CHAPTER 1. INTRODUCTION

Figure 1.1: Crack propagation [3].

The most common way of assessing fatigue today is by using the Palmgren-Miner rule,which also is suggested by the Eurocode. The basic idea is to calculate the cumulativedamage that occurs during a components lifetime. Each partial damage is decided as thenumber of load cycles that is applied to the construction during its lifetime at a given stressrange divided by the potential number of load cycles that the construction could carry forthe same stress range. A stress range is defined as the difference between the maximum andminimum stress level in one load cycle. For example, if a component is exposed to a stressrange of 10MPa 10,000 times, and could carry 10MPa 100,000 times, the partial damagefrom this stress range is 10%, calculated as 10,000/100,000. The cumulative damage isthen obtained by summarizing the partial damage from all potential stress ranges thataffects the construction. The mathematical operations are further explained in Section2.4.2.

The Palmgren-Miner rule is named after Arvid Palmgren and M. A. Miner. Palmgren,born in 1890 in Falun, Sweden, was a Swedish civil engineer that operated as a researchengineer at the company SKF [4]. In 1919 he patented the self-adjustable roller bearing in19 different countries. Amongst his hardware research he worked on theories for estimatingthe service life of roller bearings and 1924 he published a cumulative damage theory forfatigue calculations. The theory did not get that much attention until the American M.A. Miner made some adjustments and re-published it in 1945. Therefore, the theory ismore known as the Palmgren-Miner rule these days and despite its many years on themarket, it is still one of the best ways of assessing fatigue.

SS-EN 1991-2 provides five different fatigue load models (FLMs); all with different set-upsregarding traffic intensity, axial loads, axial spacing etc. The first two of these models areonly appropriate for pure steel bridges and used to determine whether the life length ofthe bridge with respect to fatigue is limited or not, whilst the last three are appropriatefor all types of bridges and used to assess the life length with respect to fatigue. Thefatigue calculations are performed according to SS-EN 1993-1-9. Different methods areused for different FLMs. Before the Eurocode was introduced in Sweden, a combinationof BRO 2004 (used to specify the FLM) and BSK 07 (explained the fatigue calculations)was used in the fatigue assessment. Further on, this combination is referred to as the oldregulations.

2

1.2. AIM

1.2 Aim

The aim of this study is to investigate the differences and similarities between the newEurocode and the old regulations used in Sweden with respect to fatigue design of steel-concretecomposite road bridges. This is performed using a built bridge as a model, assessing itin different ways according to the scope defined in Section 1.3 and analysing the results.Previous studies indicate, as further discussed in Section 1.5, that the Eurocode is moreconservative than the old regulations, but the underlying cause is not thoroughly analysedand discussed.

1.3 Scope

The following bullets defines the scope of the study:

• Since the study’s aim is to look at differences between the design codes and not toperform a complete fatigue design of a bridge, the finite element model’s ability toaccurately reflect the actual bridge’s properties is not crucial. The important issueis instead to set up a model that generates reliable results to enable comparison ofthe different FLMs.

• The fatigue damage will be assessed using three different FLMs from SS-EN 1991-2Action on Structures - Part 2: Traffic loads on Bridges, namely:

– Fatigue Load Model 3 (Section 2.3.1).– Fatigue Load Model 4 (Section 2.3.2).– Fatigue Load Model 5 (Section 2.3.3).

• The fatigue damage should also be assessed using the previous Swedish regulationBRO 2004.

• Due to time constrains, only one bridge is assessed.

• The points at which the fatigue damage is calculated are limited to three, illustratedin Figure 1.2.

– The welded joints of the main girders at midspan. Since there is no joint in theexact midspan, two points, denoted midspan1 and midspan2, are assessed.

– The welded stiffener of the main beams at the midsupport, denoted midsupport.

Figure 1.2: 3D view of the investigated points along the bridge.

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1.4 Facts Regarding the Bridge

The road bridge is located in Ljungsbro, 15.0 kilometers north of Linköping, in ÖstergötlandsLän, Sweden. The bridge was inaugurated the 21st of June 2011 and designed by ELUKonsult AB according to the old Swedish regulations [1]. It is a steel-concrete compositetwo-span bridge with span lengths of 50.0 and 58.0 meter designed for two lanes of traffic,one lane in each direction, and a walking and biking path. Two 2.40 meter high mainbeams consisting of varying welded steel I-sections, sectioned in parts of 6.00-12.0 meters,carries the bridge. The main beams are connected with stiffeners spaced 8.00-12.0 meters.On top of the steel structure a concrete deck with a total width of 11.0 meters is connected.For further details regarding the dimensions see Appendix A.

1.5 Summary of Previous Work

Previous studies indicate that the Eurocode is more conservative than the old regulations[5, 6]. This is also valid when assessing fatigue, where the recurring main reason is thatthe Eurocode exerts more traffic in terms of lorries crossing the bridge. This fact is alsohighlighted by Robert Hällmark in the seminar Lastseminaruim - Eurocode [7]. A moredetailed study indicates that the main problem is welded bottom flanges and that topflanges rarely suffers from fatigue in steel-concrete composite bridges [8].

However, there is no complete comparison showing the complete differences between theFLMs provided by the Eurocode and the old combination of design codes.

1.6 Structure of the Thesis

Basically, the thesis is structured according to the IMRAD-structure (Introduction, Method,Results, Analysis and Discussion), with Chapter 1 being the Introduction. In the upcomingChapter 2, Methods, the approach used for the thesis is firstly presented. After that, thebasis of finite element modelling that applies for the used model, including simplificationsthat were made are presented. Thereafter comes an interpretation of how to assess fatigue,both according to the Eurocode and BRO 2004.

The results are presented and analysed in Chapter 3. This includes further details regardingthe finite element modelling as well as the fatigue assessments. The results are thendiscussed in Chapter 4. In the final Chapter 5, the thesis is concluded together with afuture outlook.

4

1.7. NORMATIVE REFERENCES

1.7 Normative References

This thesis is based on different normative texts. These references, that are listed hereafter,will be cited and emphasized in the text when used.

SS-EN 1991-2 Eurocode 1: Actions on structures – Part 2: Traffic loadson bridges

SS-EN 1993-1-1:2005 Eurocode 3: Design of steel structures – Part 1-1: Generalrules and rules for buildings

SS-EN 1993-1-9:2005 Eurocode 3: Design of steel structures – Part 1-9: FatigueSS-EN 1993-2:2006 Eurocode 3: Design of steel structures – Part 2: Steel

bridgesSS-EN 1994-1-1:2005 Eurocode 4: Design of composite steel and concrete

structures - Part 1-1: General rules and rules for buildingsTRVK Bro 11 Trafikverkets tekniska krav BroTRVFS 2011:12 Trafikverkets författningssamlingBRO 2004 Allmänna tekniska beskrivningar för Broar. Utgivare:

VägverketBSK 07 Boverkets handbok om stålkonstruktionerBV Bro, utgåva 9 Broregler för nybyggnad. Utgivare: Banverket

1.8 Abbreviations and Definitions

Following abbreviations are used in this thesis:

BF Bottom FlangeFEM Finite Element MethodFLM Fatigue Load ModelLCN Load Cycle NumberNA National AnnexTF Top FlangeTP Transverse Position of the loadUR Utilization RatioÅDTt Average annual daily traffic from heavy lorries. In English, ÅDTt is

usually denoted AADTT, Average Annual Daily Truck Traffic, but sincethis thesis is based on the Swedish regulations, ÅDTt is used

5


1.9 List of Symbols

Roman Upper Case

Ci The fatigue normal stress resistance according to BSK 07Dd The cumulative damageFl.Pos The fibre location in the flangeL The span lengthLspa Lorry spacingMy Moment about the y-axisMz Moment about the z-axisN0 The reference value of Nobs

Nlor Number of lorriesNobs Number of heavy lorries crossing per slow lane and yearNR The number of cycles the design can endure a given stress rangeNtot The total number of crossings during the bridge’s life timePx The normal force in the x-directionQ0 A nationally selected parameter reflecting the expected average weight of

the lorriesQm1 The average weight of the lorries in the slow laneUR The utilization ratio in %

Roman Lower Case

frd The design value of the fatigue resistancefrk The characteristic value of the fatigue resistancefvrd The design value of the fatigue resistancenE,i The number of cycles under given stress rangent The potential number of load cycles that the construction can carry

during its lifetimentot The total number of expected stress cycles during the construction’s

lifetimeks The size effect factortd The life length of the bridge in years

6

1.9. LIST OF SYMBOLS

Greek Upper Case

∆σC The fatigue resistance according to the detail category∆σE The damage equivalent normal stress range∆σx The nominal normal stress range∆τ The nominal shear stress range∆τC The fatigue shear stress resistance∆τE The damage equivalent shear stress range

Greek Lower Case

γF f The partial coefficient for equivalent stress ranges with constant amplitudeγMf The partial coefficient for fatigue resistanceγn The partial coefficient of the safety classλ1 The damage factor due to the span lengthλ2 The damage factor due to traffic volumeλ3 The damage factor due to the expected life lengthλ4 The damage factor due to additional lanesλf The damage equivalent factor for stresses in midspansλmax The maximum value of the equivalent damage factorλs The damage equivalent factor for stresses at midsupportσrd The nominal normal stress rangeσx The nominal normal stress in the x-directionσx,min Minimum nominal normal stress range in x-directionσx,max Maximum nominal normal stress range in x-directionτrd The nominal shear stress range

7

Chapter 2

Methods

This section starts out with describing the approach used for the thesis. After that, themain steps in the finite element modelling are described, followed by an introduction tothe fatigue load models and fatigue assessment methods provided by the Eurocode andthe old Swedish regulations. Finally, all the assumptions and simplifications made in thethesis are presented.

2.1 Our Approach

To investigate if there are any differences between the regulations, a steel-concrete compositebridge across Motala Ström is assessed using both regulations. Originally, the bridge wasdesigned according to BRO 2004 by ELU in 2011 [1]. Drawings were provided by ELUKonsult AB at the start-up of the thesis and the bridge was modelled using the finiteelement software LUSAS.

The approach used for this thesis is shown in Figure 2.1 below. Note that this is aschematic illustration of the process and that when executed, an iterative approach, i.e.moving back and forward in between the steps once dissatisfying results are revealed, wasapplied. Information needed in the different steps was collected along the way from theEurocodes, the National Annexes, Handbooks, Supervisors, other Master Theses, ScienceReports and other available literature.

Finite ElementModeling

Design CodeInterpretation

FatigueAssessment

Results andAnalysis

Discussion andConclusion

Figure 2.1: Visualisation of the process.

9

CHAPTER 2. METHODS

2.2 Finite Element Modelling

The modelling and analysis were carried out in the software LUSAS Bridge Plus which isa world-leading finite element analysis (FEA) software [9]. The software offers guidancefor the design, analysis and assessment of all types of bridge structures.

To increase the reliability of the results from LUSAS smaller models that could verifyintended outcome were made. For instance, a simply supported steel beam with a concretedeck was modelled with full composite action and assessed with regard to stresses thatcould be compared to known results from hand calculations. This process was appliedthroughout the modelling process to assure that the model behaved as intended.

LUSAS implements FEA to accurately solve all types of linear and nonlinear stress,dynamic and thermal/field problems. The software is structured as most types of FEAsoftware on the market and consists mainly of two systems:

• Modeller: A graphical user interface for modelling and viewing of the results.Involves processes such as geometry, material, support and load definitions.

• Solver: The engine of the program, performs the analysis which was defined in themodeller.

2.2.1 Element Selection

All elements in LUSAS are divided into different groups according to their geometricalspace [9]. The different groups are point, line, surface, and volume elements. Under eachsection a variety of elements are defined and presented. The basic approach for creatinggeometries is presented below:

Point → Line → Surface → Volume

It is of great importance to bear this in mind while creating the geometry for the model;to add another point or line to an existing element can be a very difficult task and timeconsuming. Note that the geometry definition is not the same thing as the elementdefinition. A geometry is usually divided into several elements depending on the meshsize, where the points defining the elements are referred to as nodes.

Beam Elements

Beam elements are defined by a line which is defined by at least two nodes. Each node hasa certain degree of freedom (DOF) depending on the modelling space defined [10]. A 2Delement allows 3 DOFs for each node, namely lateral and axial translation and rotationwithin the elements own plane. A 3D element allows 6 DOFs at each node, one moretranslation direction and two more rotations. This is illustrated in Figure 2.2. Note thatif a 2D beam element is used, only translation in x- and y-directions and rotation aboutthe z-axis remain for each node.

10

2.2. FINITE ELEMENT MODELLING

Figure 2.2: Visualisation of the DOFs for a 3D beam element [10].

Each element is defined with properties such as moment of inertia I, Young’s modulus E,and the cross sectional area A. There are two different beam theories; the elementary beamtheory, also known as Euler-Bernoulli, and the Timoshenko. The Euler-Bernoulli theorydo not account for shear deformations which in general are applicable for thin beams. TheTimoshenko theory on the other hand do account for the transverse shear deformationswhich could have a significant effect for thick beams. The exact limit between thin andthick beams is debated, but as the ratio of thickness/length decreases, shear deformationsbecome less important.

Shell Elements

Shell elements are defined by a surface i.e. nodes and lines. The number of nodes aredependent of which element that is being used [10]. LUSAS provides triangular elementswith 3 or 6 nodes and quadratic elements with 4 or 8 nodes, where each node contains 5or 6 DOFs depending on the element definition. Just as for the beam elements there isa thin and thick formulation. The thin shells are based on the Kirchoff shell theory anddo not account for transverse shear deformations and the thick shell elements are basedon the Mindlin shell theory, which account for the transverse shear deformations. As theformulation states, the latter is more suitable for thicker shells.

Element Selection in this Thesis

The model in this thesis is modelled with Thick 3D beam elements (BMS3) and QuadrilateralThick shell elements (QTS4). Both element types have a linear interpolation order whichmakes them suitable for each other and the elements are capable of modelling the transverseshear deformations which is desirable, due to the high thickness/length ratio in they-direction of the bridge. Also, the element selection can accurately calculate nominalstresses which are used in the fatigue assessment. The mesh sizes are being controlledthrough the element definitions and with the help of so called "None elements". Theseelements were assigned to the edges of the bridge deck to control how the mesh is behavingnear the edges. This enables a complete control over the mesh sizes and its uniformity,which is shown in Figure 2.3.

11

CHAPTER 2. METHODS

Figure 2.3: Visualisation of the uniformity of the mesh.

2.2.2 Element Connection

The geometry of the bridge deck and main beams were created out of the same lines andnodes in LUSAS. This generates full composite action between the elements, i.e. no slidingoccurs. The elements are connected to one another using the same mesh size i.e. samemesh size on the 3D beam and the shell elements.

2.2.3 Support Modelling

Since both the beam and shell elements were created from the same lines in LUSASthe defining line for the beam elements is located in the top of the beam. To get thebeams supported in the bottom flange rigid elements were therefore introduced. The rigidelements were defined between two nodes, one at the top flange and another in the bottom,to support the beam at its bottom flange rather than its top. One of the end supports wasmodelled using a pinned support and the other supports were modelled as roller bearings.

2.2.4 Stress Ranges

The stress ranges were calculated using the nominal stresses based on cross sectional forcesobtained from the finite element model. The stresses were calculated as "EngineeringStresses" in LUSAS, which is a nominal stress calculation based on the Navier’s formula,since SS-EN 1993-1-9 Section 6.1 advises use of nominal stresses instead of geometricalstresses when assessing simple details with respect to fatigue. Therefore, this was appliedin the assessment. In addition, it keeps the detail level of the model at a manageable levelwith respect to run-time etc. The stress range is the difference between the maximum andminimum stress obtained for a lorry crossing of the bridge, defined in Equation 2.1.

∆σ =| σmax − σmin | (2.1)

12

2.2. FINITE ELEMENT MODELLING

To determine the maximum and minimum stress levels at a point, an influence line overthe bending moment about the y-axis for observed point and FLM was made. The reasonfor using this moment was because it has the greatest impact on the stress level in thex-direction. With the influence line as basis the most critical load positions could bedetermined and stresses calculated. When assessing a joint, the most slender of theconnecting profiles was used.

Investigated points in the midspan have both a tensile and compressive stresses while theinvestigated point in the support only have tensile stresses and therefore σmin was set to0. The compressive stress can, according to SS-EN 1993-1-9, be reduced to 60% of itsvalue if it represents the majority of the stress range. However, that is not the case in thisthesis, why the reduction will not be applied in the assessment.

2.2.5 Fibre Locations

To calculate the stress at desired points, fibre locations were assigned in the cross sectionaldefinitions in LUSAS. The stress σx is a function of the normal force and the bendingmoments around both bending axes and can be calculated by LUSAS for a chosen fibrelocation.

The fibre locations used in this thesis are visualised in Figure 2.4, where the top flangerepresents the stress assessment for the midsupport and the bottom flange the midspans.Since the loads act eccentric, three fibre locations were defined to fully represent the stressdistribution across the flanges. The vertical position of the BF points were chosen at theabsolute bottom flange since the welds are through the entire cross section. At the TF,the points were set in the middle of the top flange despite that the stiffeners are weldedat the inside. This decision was made to keep the calculation on the safe side.

Figure 2.4: Fibre locations in LUSAS.

13

CHAPTER 2. METHODS

2.3 Fatigue Load Models according to the Eurocode

As announced earlier, the Eurocode provides five different FLMs in SS-EN 1991-2 Actionon Structures - Part 2: Traffic loads on Bridges. However, this thesis focuses on FLM 3,4 and 5, which are further described in the following subsections.

2.3.1 Fatigue Load Model 3 (FLM3)

FLM3 is described in SS-EN 1991-2 Section 4.6.4. It is based on the damage due to acrossing of a single lorry which is translated into a lifetime damage using damage equivalentλ-factors. For further details of this method, called the Lambda Method, see Section 2.4.1.FLM3 consists of four axles located according to Figure 2.5, all carrying a force of 120 kN,i.e. 60 kN/tire.

Figure 2.5: The load group used in FLM3, SS-EN 1991-2 Figure 4.8.


FLM4 is described in SS-EN 1991-2 Section 4.6.5. It is based on five different frequentlorries which are weighed together based on the traffic type to reflect the traffic acrossthe bridge. The lorries vary in number of axles, axial load and contact area of the tiresaccording to Table 2.1. Further on in this thesis, those lorries are named FLM4A toFLM4E, from top to bottom.

For each frequent lorry, the passage of the bridge was simulated and evaluated using theRainflow-method combined with the Palmgren-Miner method which is further explainedin Section 2.4.2. The Eurocode specifies that the calculations should be performed underthe condition that the lorries are at the bridge one at a time.

14

2.3. FATIGUE LOAD MODELS ACCORDING TO THE EUROCODE

Table 2.1: The frequent lorries defined in FLM4, SS-EN 1991-2 Table 4.7.


FLM5 is described in SS-EN 1991-2 Section 4.6.6, but also in SS-EN 1991-2 AppendixB. It is based on the same principles as FLM4, but instead of using a fixed value of thenumber of heavy lorries crossing the bridge, Nobs, provided by the Eurocode, a projectspecific value based on estimations is applied. Also, the traffic type i.e. lorry percentagemay be specified in the specific project.

TRVFS 2011:12 6 kap 8§ states that byggherren, the developer, specifies how and if FLM5can be used in the specific project. This is also supported by the Swedish National Annex(NA) to SS-EN 1991-2:2003 – Traffic loads on bridges. However, according to TRVK Bro11 Section B.3.2.1.3j, such a specification may only be used if more than 24,000 heavylorries cross the bridge daily, which is not the case for this bridge’s location. In this thesisit has although been decided to follow the NA to add more value to the comparison.

15

CHAPTER 2. METHODS

The traffic distribution was approximated based on data from Trafikverket combined withour own estimations regarding the ordinariness of the different lorries. The result is shownin Table 2.2.

Table 2.2: Own estimations of lorry percentage based on information from Trafikverket.

Lorry Type Lorry Percentage [%]FLM4A 70.0FLM4B 10.0FLM4C 5.00FLM4D 10.0FLM4E 5.00

According to Trafikverket, the number of heavy lorries at the Swedish roads has beennearly constant during the last 30 years [11]. In this thesis it has although been assumedthat the traffic from heavy lorries will be three times higher in 120 years, which can beseen as a safe-side assumption. There is no exact data regarding the number of lorriespassing the bridge today, but a nearby bigger road was crossed by 45.0 lorries daily intotal for both directions in 2003 [12]. To compensate for the potential increment in trafficduring the last decade, it is assumed that 45.0 lorries pass the bridge in each directiontoday. The exact position of the used measurement point for the data is illustrated by thecircle in Figure 2.6 and the bridge is marked by the rectangular box.

Figure 2.6: The location of the measurement point and the bridge [12].

16

2.4. FATIGUE ASSESSMENT ACCORDING TO THE EUROCODE

2.4 Fatigue Assessment according to the Eurocode

The fatigue analysis was performed using nominal stresses, as recommended in SS-EN1993-1-9 Section 6.1. However, as specified in SS-EN 1991-2 Section 4.6.1, differentmethods are used for different FLMs; for FLM3, a simplified method called the λ-methodcan be used while FLM4 and FLM5 demand a more sophisticated method, such asPalmgren-Miner’s cumulative damage analysis. Therefore, these two methods were usedin the analysis and are presented in the following subsections.

2.4.1 The Lambda Method

The Lambda Method is a simplified method based on the crossing of a single lorry. Thefatigue calculation includes both nominal normal stresses and nominal shear stresses. Thedesign value for the nominal stress ranges is obtained by SS-EN 1993-1-9 Equation 6.1as:

γF f ·∆σE,2 = λ ·∆σ(γF f , Qk) (2.2)

γF f ·∆τE,2 = λ ·∆τ(γF f , Qk) (2.3)

Where;

- γF f is the partial coefficient for equivalent stress ranges with constant amplitude- ∆σE,2 is the damage equivalent normal stress range- λ is the equivalent damage factor, further explained below- ∆σ(γF f , Qk) is the nominal normal stress range, as described in Section 2.2.4- ∆τE,2 is the damage equivalent shear stress range- ∆τ(γF f , Qk) is the nominal shear stress range, as described in Section 2.2.4

The λ-factor is the equivalent damage factor that translates the single lorry crossing intoa lifetime representing factor. It is presented in SS-EN 1993-2 Section 9.5.2 Equation 9.9as:

λ = λ1 · λ2 · λ3 · λ4 but λ ≤ λmax (2.4)

Where;

- λ1 is the damage factor due to the span length- λ2 is the damage factor due to traffic volume- λ3 is the damage factor due to the expected life length- λ4 is the damage factor due to additional lanes- λmax is the maximum value of the equivalent damage factor

17

CHAPTER 2. METHODS

λ1 is decided according to SS-EN 1993-2 Figure 9.5, also presented in Figure 2.7. Onthe y-axis, λ1 is given, and the x-axis represents the critical length L. The left diagramrepresents field areas, where L is taken as the span length, and the right diagram representssupport areas, where L is taken as the average of the span lengths at each side of thesupport.

Figure 2.7: Selection of λ1, SS-EN 1993-2 Figure 9.5.

λ2 is calculated according to SS-EN 1993-2 Section 9.5.2(3) Equation 9.10 as:

λ2 = Qm1Q0·(Nobs

N0

)1/5(2.5)

Where;

- Qm1 is the average weight of the lorries in the slow lane- Q0 is a nationally selected parameter reflecting the expected average weight of thelorries, 410 kN or 445 kN in Sweden according to Trafikverket BRO 2011 SectionE.3.1

- Nobs is the number of lorries expected to cross the bridge each year per slow lane,further discussed in Section 2.7.4

- N0 is the reference value of Nobs, which is set to 0.5 · 106

λ3 is calculated according to SS-EN 1993-2 Section 9.5.2(3) Equation 9.11 as:

λ3 =(td

100

)1/5(2.6)

Where;

- td is the design life length of the bridge, expressed in years

18


The final lambda factor, λ4, is calculated using SS-EN 1993-2 Section 9.5.2(6) Equation9.12, also specified in Equation 2.7. However, in Sweden λ4 is set to 1.0 according toTrafikverket TRVFS 2011:12 Chapter 19 §17.

λ4 =[1 + N2

N1·(η2 ·Qm2η1 ·Qm1

)5+ N3N1·(η3 ·Qm3η1 ·Qm1

)5+ ...+ Nk

N1·(ηk ·Qmk

η1 ·Qm1

)5]1/5

(2.7)

Where;

- k is the number of lanes with heavy traffic- Ni is the number of heavy lorries in lane i per year- ηi is the lane factor for the load- Qmi is the average gross weight of the lorries in lane i

λmax is specified in SS-EN 1993-2 Figure 9.6, also presented in Figure 2.8. The sameprinciples as for λ1 applies for the axes.

Figure 2.8: Selection of λmax, SS-EN 1993-2 Figure 9.6.

The utilization ratio is finally determined using SS-EN 1993-1-9 Equation 8.2 as:

γF t ·∆σE,2∆σC

γMf

= UR ≤ 1.0 (2.8)

γF t ·∆τE,2∆τC

γMf

= UR ≤ 1.0 (2.9)

Where;

- γF t is the partial coefficient for equivalent stress ranges with constant amplitude- ∆σE,2 is the damage equivalent normal stress range

19

CHAPTER 2. METHODS

- ∆σC is the fatigue normal stress resistance according to the detail category, definedin SS-EN 1993-1-9 Table 8.1-10

- γMf is the partial coefficient for fatigue resistance- UR is the utilization ratio- ∆τE,2 is the damage equivalent shear stress range- ∆τC is the fatigue shear stress resistance according to the detail category, defined inSS-EN 1993-1-9 Tables 8.1-10

2.4.2 Palmgren-Miner Method

The Palmgren-Miner rule is based on the hypothesis that the fatigue damage to the steelis cumulative and non-reversible [3]. This means that the damage from one stress rangecan be calculated individually and then summed up to a total damage. The mathematicalexpression is defined in SS-EN 1993-1-9 Equation A.1 as:

Dd =k∑

i=1

nE,i

NR,i(2.10)

Where;

- Dd is the cumulative damage- nE,i is the number of cycles with the stress range γF f ·∆σi

- NR,i is the number of cycles the design can endure, further explained below

In this thesis, the number of stress cycles that the details can resist, NR,i, is the onlyunknown. Therefore, the equations describing the S-N, or Wöhler Curves in SS-EN1993-1-9 Figure 7.1, also shown in Figure 2.9, were rearranged according to below inorder to solve for NR,i. Note that there is no fatigue if NR,i > 100 · 106, i.e. if the stressrange is lower than the cut-off limit ∆σL. Also, note that Figure 2.9 is valid for normalstresses and that the same method, using another diagram, applies for shear stress fatigueassessment.

20


Figure 2.9: Fatigue strength curve for direct stress ranges, SS-EN 1993-1-9 Figure 7.1.

NR,i =

( ∆σC

γF t · γMf ·∆σi

)3· 2 · 106 if NR,i ≤ 5 · 106

( ∆σD

γF t · γMf ·∆σi

)5· 5 · 106 if 5 · 106 < NR,i ≤ 100 · 106

(2.11)

Where;

- NR,i is the potential number of load cycles that the construction can carry duringits lifetime if only subjected to stress ranges of magnitude ∆σi

- ∆σC is the fatigue normal stress resistance according to the detail category, definedin SS-EN 1993-1-9 Table 8.1-10

- γF t is the partial coefficient for equivalent stress ranges with constant amplitude- γMf is the partial coefficient for fatigue resistance- ∆σi is one occurring stress range for which fatigue is assessed- ∆σD is fatigue limit at constant amplitude, calculated as 0.737 ·∆σC

The design process aims to get a cumulative damage of less than one. The stress sequencecan be assessed by using either the Rain-Flow Method or the Reservoir Method, bothof them aiming to determine stress ranges and the number of stress cycles in a givensequence of load appliance. If performed correctly, both methods will result in the exactsame results. The methods are described in SS-EN 1993-1-9 Annex A, but will not beused in this thesis due to the simple shape of the load cycle diagrams and are thereforenot further explained.

21

CHAPTER 2. METHODS

2.5 Fatigue Load Models according to BRO 2004

The FLM of the old Swedish design codes is described in BRO 2004 Section 21.2226. Thismodel is similar to FLM3, i.e. based on the crossing of a single lorry. The load groupconsists of four axles, two in the front and two in the rear as visualized in Figure 2.10. Thedynamic effect of the load is assumed to be included in the axle loads. The load groupshould be placed so that the largest stress range occurs in the investigated beam. Onlytraffic from one lane is considered.

Figure 2.10: The load group according to BRO 2004.

2.6 Fatigue Assessment according to BSK 07

The failure criterion for normal stresses and shear stresses are presented in Equation 6:512aand 6.512b in BSK 07 as:

σrd ≤ frd = frk

1.1 · γn(2.12)

τrd ≤ fvrd = 0.6 · frd = 0.6 · frk

1.1 · γn(2.13)

Where;

- σrd is the nominal normal stress range- frd is the design value of the fatigue resistancefrk is the characteristic value of the fatigue resistance, further explained below- γn is the partial coefficient of the safety class- τrd is the nominal shear stress range- fvrd is the design value of the fatigue resistance

The characteristic value of the fatigue resistance is determined based on the detail categoryand the number of stress cycles. According to BSK 07 Figure 6.523, frk is based on S-NCurves and the equation for it is also specified in Equation 2.14. Note that Equation 2.14is valid for nominal normal stress ranges. If shear is assessed, the values are multipliedwith 0.6. Also, note that there is no fatigue if nt > 100 · 106.

22

2.6. FATIGUE ASSESSMENT ACCORDING TO BSK 07

frk =

C ·

(2 · 106

nt

) 13

if nt ≤ 5 · 106

0.885 · C ·(2 · 106

nt

) 15

if 5 · 106 < nt ≤ 100 · 106

(2.14)

Where;

- frk is the characteristic value of the fatigue resistance- C is the fatigue normal stress resistance according to the detail category, defined inBSK 07 Table B3:1-2

- nt is the number of cycles the design can endure, further explained below

In this thesis nt remains as the only unknown parameter in Equation 2.14. This resultsin the following rearrangement of the variables for normal stresses:

nt =

2 · 106(1.1 · γn · σrd

C

)3 if nt ≤ 5 · 106

2 · 106(1.1 · γn · σrd

0.885 · C)5 if 5 · 106 < nt ≤ 100 · 106

(2.15)

Where;

- nt is the potential number of load cycles that the construction can carry during itslifetime if only subjected to stress ranges of magnitude σrd

- γn is the partial coefficient of the safety class- σrd is one occuring stress range for which fatigue is assessed- C is the fatigue normal stress resistance according to the detail category, defined inBSK 07 Table B3:1-2

The utilization ratio can then be determined as:

ntot

nt= UR ≤ 1.0 (2.16)

Where;

- ntot is the total number of expected stress cycles during the construction’s life time,for bridges equal to LCN

- nt is the potential number of load cycles that the construction can carry during itslifetime if only subjected to stress ranges of magnitude σrd

- UR is the utilization ratio

23

CHAPTER 2. METHODS

2.7 Interpretations, Simplifications and Conditions

This sections specifies all the interpretations, simplifications and conditions that appliesto the thesis.

2.7.1 Assumptions and Simplifications

The following applies in the design:

• The bridge consists of two primary steel beams with varying cross sections throughoutthe length of the bridge. The exact geometry of the joints between the beams wasneglected. Vertical stiffeners and transverse bonds in and between the main girderswere also neglected in the model.

• On top of the primary beams, a concrete deck was modelled. The shape wassimplified into a rectangular shape with a thickness of 250mm corresponding tothe average thickness. The edge beams were neglected.

• Full composite action between the steel and the concrete was assumed.

• Elevation differences and pre-cambering were not included in the model.

• The Young’s modulus of the concrete deck was reduced over the midsupport to10.0% of the un-cracked modulus due to cracking in accordance with SS-EN 1994-1-1Section 5.4.2.3 and supervisor Frank Axhag, to obtain the worst-case scenario forboth field and support points. The cracking will lead to a decreased stiffness whichattracts less moment over the support but the composite section itself will let thesteel beam carry more of the moment due to the lower modulus. The reductionwas applied to a distance of six metres on each side of the midsupport, even thoughthe Eurocode allows a wider reduction length. The reason for not using the entirereduction was the sectioning of the primary beams.

24

2.7. INTERPRETATIONS, SIMPLIFICATIONS AND CONDITIONS

2.7.2 Fatigue Load Models

In the setup of the FLMs, both according to the Eurocode and BRO 2004, the followingwere considered:

• The contact area of the tires was neglected i.e. loads are modelled as point loadsinstead of distributed loads.

• The impact of paired tires in the axes types described in SS-EN 1991-2 Table 4.8was neglected. Instead, they are modelled as single point loads.

• Two slow lanes, denoted TPs, transverse positions, shown in Figure 2.11, was usedin the analysis of FLM4 and FLM5. The selections of the position of the slow laneswere made in accordance with SS-EN 1991-2 Section 4.6.1(4)-(5). The reason forusing two slow lanes in the analysis is because of the Nobs definition, i.e. per yearand slow lane. Since the bridge only has two lanes, both are considered as slow lanes.This is how the authors interprets the Eurocode.

• When executing FLM3, the following simplifications were made in order to reducethe workload:

– When assessing midspans, one single lorry was used since multiple lorries wouldcounter-act each other, resulting in lower stress ranges.

– When assessing midsupport, multiple lorries was used as the widest stress rangeis obtained when the lorries are placed in both spans. The spacing was set to44 meters, since that maximized the impact.

Figure 2.11: Location of the slow lanes used in the analysis.

25

CHAPTER 2. METHODS

2.7.3 Fatigue Assessment

In the fatigue assessment, according to the Eurocode and BRO 2004, the following applies:

• The bridge has a design life length of 120 years.

• Nobs was chosen based on the traffic category obtained according to TRVK Bro 11Section B.3.2.1.3h, further explained in Section 2.7.4.

• γMf was set to 1.35 to reflect "High Consequence of Failure" and "Safe Life Concept"according to SS-EN 1993-1-9 Table 3.1.

• γF f was set to 1.00 according to SS-EN 1993-2 Section 9.3.

• Since the focus of the thesis is regarding fatigue and not to make a complete designof the bridge, gravity loading and thermal expansion etc. were neglected in thecalculations.

• The detail categories were selected as SS-EN 1993-1-9 Table 8.3 Detail 9 for themidspan points and SS-EN 1993-1-9 Table 8.4 Detail 7 for the midsupport point.Their appearances are shown in Figure 2.12 and Figure 2.13 respectively. The sizeeffect was also included.

Figure 2.12: Detail category at midspans, SS-EN 1993-1-9 Table 8.3 Detail 9.

Figure 2.13: Detail category at midsupport, SS-EN 1993-1-9 Table 8.4 Detail 7.

26

2.7. INTERPRETATIONS, SIMPLIFICATIONS AND CONDITIONS

2.7.4 The Nobs and LCN selection

Nobs, the number of heavy lorries crossing the bridge per year and slow lane, is selected inaccordance with SS-EN 1991-2 Table 4.5, also presented in Table 2.3. Since Nobs is definedper slow lane and the assessed bridge has one lane in each direction, both were considered asslow lanes in FLM4 and FLM5. The fatigue damage was therefore cumulatively calculatedfrom both lanes. The basis for Nobs are big measurement campaigns on roads with heavycontinental traffic in several European countries in the 1970’s and 1980’s [13].

Table 2.3: Selection of Nobs, SS-EN 1991-2 Table 4.5.

Traffic categories Nobs per year and slow lane

1 Roads and motorways with 2 or more lanes perdirection with high flow rates of lorries 2.000 · 106

2 Roads and motorways with medium flow ratesof lorries 0.500 · 106

3 Main roads with low flow rates of lorries 0.125 · 106

4 Local roads with low flow rates of lorries 0.050 · 106

TRVK Bro 11 Section B.3.2.1.3h explains which traffic category to select. The criterionfrom TRVK Bro 11 is shown in Table 2.4. Note that if ÅDTt, the average annual dailytraffic from heavy lorries, is greater than 24,000 a special investigation regarding theconditions for the fatigue assessment must be performed.

Table 2.4: Selection of Traffic Category, TRVK Bro 11 Section B.3.2.1.3h.

Traffic Category ÅDTt Conditions1 6, 000 < ÅDTt ≤ 24, 0002 1, 500 < ÅDTt ≤ 6, 0003 600 < ÅDTt ≤ 1, 5004 ÅDTt ≤ 600

In BRO 2004 the amount of traffic is based on the Load Cycle Number (LCN). Twopossibilities apply; if the ÅDTt is below 10,000 lorries LCN is 100,000 and otherwise400,000. Note that LCN represents the entire life span of the bridge.

27

CHAPTER 2. METHODS

2.7.5 Dynamic Effects

The load models used did not generate any stress ranges large enough to cause fatigueonce the lorries left the bridge, as they were all less than 1.00MPa. An evaluation ofthe dynamic stress and the static is presented in Figure 2.14. The dynamic stress rangeswere calculated with LUSAS’s utility IMDPlus. The utility uses modal superpositiontechniques in the time domain to solve 2D and 3D moving load problems [9].

The evaluation was made for midspan1 and FLM4C, for eigenvalues within the range of0-30.0Hz and a damping factor of 0.500%. These parameters were selected in accordancewith SS-EN 1991-2 and BV Bro respectively. Note that these values are intended forrailway bridges but were used as a benchmark for this dynamic evaluation since theEurocode provides limited information regarding dynamic evaluations for road bridges.The eigenvectors were mass normalised to support the moving load utility within IMDPlus.The velocity of the lorry was set to 14.0m/s (50.0 km/h) to represent the reality.

0 10 20 30 40 50 60 70 80 90 100 110 120−20

−10

0

10

20

30

40

50

X-coordinate along the bridge [m]

Stress

atmidspan

1[M

Pa]

StaticDynamic

Figure 2.14: Comparison between Static and Dynamic stress influences for the midspanpoint.

The dynamic influence is limited since the graphs in Figure 2.14 correlate well. Thedifferences between the dynamic and static stress range is therefore not significant. Withthis in mind a static, rather than a dynamic, approach using the influence lines were usedwhen calculating the stress ranges for the different FLMs.

28

Chapter 3

Results and Analysis

3.1 The Model

This section presents the results of the modelling in LUSAS. A generalization of thereal bridge section is visualized in Figure 3.1 and the modelled bridge, according to thesimplifications in Section 2.7.1, is shown in Figure 3.2.

Figure 3.1: The general section of the bridge.

Figure 3.2: The modelled section of the bridge.

29

CHAPTER 3. RESULTS AND ANALYSIS

3.1.1 Properties of the Model

The two-span bridge was modelled in accordance with the drawings provided by ELUKonsult AB. The main steel beams are continuous with various cross sectional propertiesaccording to Table 3.1, with explanations of the dimensions in Figure 3.3. All numbersare presented in millimetres. For the plan view of the main beams see Figure 3.4. Notethat the widths are scaled up for visual reasons.

Table 3.1: Cross sectional properties [mm].Beam wbf tbf hw tw wtf ttf

A1 500 25 2, 356 15 400 20A2 600 35 2, 350 18 400 20B1 700 35 2, 353 17 400 20B2 700 40 2, 342 18 400 20C1 920 50 2, 313 19 640 40C2 1, 140 60 2, 302 20 1, 000 50D1 1, 080 55 2, 301 19 700 45D2 700 45 2, 327 20 700 30E1 860 45 2, 337 22 500 30E2 920 45 2, 345 19 920 45F1 800 40 2, 339 19 500 25F2 500 25 2, 357 17 400 20

Figure 3.3: Cross section with denotations.

30

3.1. THE MODEL

Figure 3.4: Plan view of the main beams.

Material properties used in the model were taken from the built-in material library inLUSAS which is in accordance with the Eurocode. The steel was given a Young’s Modulusof 210GPa and a Poisson’s ratio of 0.300. The concrete was given a Young’s Modulus of33GPa and a Poisson’s Ratio of 0.200, with a reduction of the Young’s Modulus to 10.0%of its value over the midsupport due to cracking, i.e. along part C2 in Figure 3.4. Controlcalculations were made whether the concrete should be cracked or not over the supportfor the midspan calculations. The worst case scenario for the midspan stresses is whenthe concrete always is considered cracked over the support.

3.1.2 Convergence Study

To decide the optimal mesh size for the model a convergence study was carried out. Threedifferent mesh sizes were assessed based on computational effort and stress convergence.The study was made with FLM4A at midspan1 mid BF for TP1. Bear in mind thatsplitting the lines defining the shell elements in half increases the number of shell elementsin the model by a factor of four, resulting in a lot more equations to solve. The resultsare presented in Table 3.2.

Table 3.2: Convergence Study.

Mesh size ∆σ Time1.00m 22.71MPa Quick0.500m 22.79MPa Medium0.250m 22.81MPa Very Slow

The computational time between mesh size 1.00m and 0.500m was tolerable but theincrement in computational time between 0.500m and 0.250m was not. The differencebetween 0.250m and 0.500m element size is less than 0.1% and therefore mesh size 0.500mwas used. This led to a rather quick model which was preferable when a lot of calculationswere made.

31


3.2 Stress Ranges and Critical Load Positions

When making the assessment, only the stress ranges from the nominal normal stress werewide enough to give rise to fatigue. As shown in Appendix C, the shear stress rangeswere very small and are therefore not considered in the analysis, i.e. in the remainingof this thesis ”stress ranges” refer to nominal normal stress ranges only. For FLM4 andFLM5, the shape of the influence line at midspan points indicated that only one stresscycle occurred during one lorry crossing. At the midsupport, more than one stress cycleoccurred, but since the largest of the stress ranges did not give rise to fatigue no stresscycle counting was needed.

Figure 3.5 illustrates the obtained stress ranges for each FLM. Note that only the worstpoint at the cross section for each FLM-longitudinal point-combination is included. Thedashed line indicates roughly at what stress range level no fatigue occur for each assessment;21.0MPa for the Eurocode based FLMs and 36.0MPa for the BSK07 based FLMs. FLM3does not exclude fatigue at low stress ranges; therefore, the red line does not cross thosebars. Note that the difference between the design categories within each FLM is verysmall (less than 0.500MPa) and therefore neglected in this illustration.

FLM3 FLM4A FLM4B FLM4C FLM4D FLM4E BRO20040

20

40

60

80

100

Stress

Ran

ges[M

Pa]

MidsupportMidspan1Midspan2

Figure 3.5: Stress ranges from LUSAS.

The stress ranges were obtained by placing the lorries according to the principles shownin the following figures. Figure 3.6 illustrates, using the lorry FLM4E as an example, theload position that results in the largest tensile stress at the bottom flanges at the midspanpoints. Note that the lorry is slightly moved to obtain the absolute maximum for eachpoint and that this is just an illustration. The greatest compression forces at the bottomflanges of the midspan points are obtained by placing the lorry close to the centre of theother span, as illustrated in Figure 3.7.

32

3.2. STRESS RANGES AND CRITICAL LOAD POSITIONS

Figure 3.6: The worst load position for tensile stresses at midspan points, x=80.0m.

Figure 3.7: The worst load position for compression stresses at midspan points, x=37.0m.

33


Figure 3.8 shows the position that results in the highest tensile stress at the top flange atthe midsupport for FLM3, i.e. having one lorry at each side of the midsupport. Whenusing the other FLMs, the highest tensile stress were obtained when the lorry was placedclose to midspan. The lowest stress is zero, since no compression stress was observed atthe top flange over the midsupport.

Figure 3.8: The worst load position for compression stresses at midsupport point for FLM3,x1=34.0m and x2=78.0m. For the other FLMs, the left lorry in the figure is removed.

34

3.3. FATIGUE ASSESSMENT

3.3 Fatigue Assessment

The results of the fatigue assessment are shown in Figure 3.9 as utilization ratios, UR. Inthe figure, only the worst point at the cross section with respect to the utilization ratio foreach FLM and longitudinal point is shown. For complete results, including calculationsand details regarding input parameters, please see Appendix D for FLM3, Appendix E forFLM4 and FLM5, and Appendix F for BRO 2004.

FLM3 FLM4LocalTraffic

FLM4MediumDistance

FLM4Long

Distance

FLM5ActualLocation

BRO2004Low LCN

BRO2004HighLCN

0%

50%

100%

150%

200%

250%

300%

UtilizationRatios[%

]

MidsupportMidspan1Midspan2

Figure 3.9: Utilization ratios.

35


3.4 Analysis

In this section, the result is analysed and followed by a sensitivity analysis of the modelswith respect to their main input parameters.

3.4.1 Analysis of the Results

The main findings are presented below:

• The widest stress ranges for the midspan points are obtained in BRO 2004, due to aheavy and short load group. For the midsupport, the widest stress range is obtainedfor FLM3, since it allows usage of two lorries.

• In FLM3, the stress ranges at midsupport is about half of the value at midspans. Inthe other FLMs, the pattern is that the stress range at the midsupport is about afourth compared to the other points.

• It is observed that midspan1 is the most sensitive point with respect to fatigue ofthe points investigated. The difference compared to midspan2 is however small inmost FLMs.

• At the midsupport, fatigue only occurs when using FLM3. This is due to small stressranges for the other FLMs and that even small stress ranges cause fatigue damagein FLM3. Note though that the UR at the midsupport in FLM3 is 60.0%, so therewill be no failure due to fatigue.

• According to the Eurocode, the bridge only passes the fatigue verification when usingthe local traffic type in FLM4 and when applying FLM5.

• BRO 2004 indicates a very limited risk of fatigue for both possible LCN selections.

• The UR increases with the traffic type in FLM4 due to a higher percentage of heavierlorries.

36

3.4. ANALYSIS

3.4.2 Sensitivity Analysis - FLM3

The variables with the most effect at the results for FLM3 are the number of lorries crossingthe bridge, Nobs, and the average gross weight of the lorries in the slow lane, Qm1. Bothof these parameters are regulated by the norms. The Eurocode provides four differentpossible selections of Nobs, based on the traffic categories defined by SS-EN 1991-2 Table4.5 and TRVK Bro 11 Section B.3.2.1.3h. Basically, Nobs is dependent on the ÅDTt. Qm1is selected according to TRVK Bro 11 Section E.3.1f. Two possibilities are provided forthe Swedish conditions; 410 kN or 445 kN. For comparison, the nationally selected valuein the United Kingdom is 260 kN [14]. That value is based on measurements performedby the UK Department of Transport using weight-in-motion sites on roads across the UKin the year 2000. Also, in the UK, the value of γMf is equal to 1.15 instead of 1.35 as inSweden. This difference is however not included in this sensitivity analysis, but increasesthe differences between the national selections even more.

Table 3.3 below illustrates the FLM’s sensitivity for these parameters, using the worstobtained result as fixed input data, i.e. midspan1 inner flange position. As shown, inthe UK the bridge would just have a UR of 79.8% under current circumstances, whilstthe result in Sweden is 126%. It shall also be noticed that for most of the Nobs − Qm1combinations the bridge fails the verification and that 218% is the highest utilization ratiodue to λmax.

Table 3.3: Sensitivity analysis FLM3.

Nobs 50, 000 125, 000 500, 000 2, 000, 000

Qm1 [kN] Comment Min EC EC EC Max EC

200 - 61.4 % 73.7 % 97.3 % 128%

260 UK 79.8 % 95.8 % 126% 167%

300 - 92.1 % 111% 146% 193%

400 - 123% 147% 195% 218 %

410 SWE 126% 151% 199% 218 %

445 Alt. SWE 137% 164% 216% 218 %

500 - 153% 184% 218 % 218 %

600 - 184% 218 % 218 % 218 %

37


3.4.3 Sensitivy Analysis - FLM4 and FLM5

For FLM4, and thereby also FLM5 since it is assessed the same way, the Eurocode providestwo main parameters that may vary: Nobs and the traffic type. As shown in Table 3.4,where the input data is from midspan1’s inner flange, the model is sensitive to changes intraffic type. A change from local to long distance traffic increases the UR by 4.19 times.Nobs has an even larger impact at the UR. By only changing Nobs from the lowest valueof 50,000 to the highest of 2,000,000, the impact on the UR is of the same size as theincrement of Nobs, i.e. by a factor of 40.0.

Table 3.4: Sensitivity analysis FLM4 presented as normalized factors with respect to localtraffic type and a Nobs of 50,000.

Nobs

Traffic Type 50, 000 125, 000 500, 000 2, 000, 000

Local 1.00 2.50 10.0 40.0

Regional 2.88 7.21 28.8 115

Long Distance 4.19 10.5 41.9 168

3.4.4 Sensitivity Analysis - BRO 2004

In BRO 2004, the major effect on the final result is from the Load Cycle Number (LCN),selected according to BRO 2004 Section 21.2226. The regulations offer two possibilities;if ÅDTt < 10,000 LCN is set to 100,000 and otherwise 400,000. The impact on the finalresult is directly proportional to the change of LCN, i.e. with a factor of 4.00.

38

Chapter 4

Discussion

4.1 General Discussion

To orientate oneself in the jungle of texts that the new regulations provide is challenging.Information needs to be gathered from several different parts of the Eurocode, TRVK Bro11 and TRVFS just to make a simple fatigue calculation. A lot of time writing thisthesis has therefore been spent interpreting how to actually assess the fatigue capacityof the bridge. Guidelines, such as Setrá, has therefore been valuable [2], but a Swedishadaptation would be of great use.

The method used for the modelling and the fatigue calculation, as well as the interpretationof the Eurocode, have been verified in discussion with the supervisors Frank Axhag fromELU Konsult AB and Bert Norlin from KTH. Also, previously performed studies supportthe reasoning and logic behind this thesis [2, 15, 16]. The result from LUSAS wasalso verified using simple hand calculations, with satisfying results. For further detailsregarding these calculations, see Appendix B.

Due to time constraints only one bridge was assessed which makes it hard to statisticallyvalidate the results. However, the design codes are the same independently of the bridgeassessed, meaning that the general results should be valid for other bridges as well. Theresult is also supported by previously performed studies according to Section 1.5, whichof course strengthens the validity of the study.

4.2 Similarities between the Old and the New Regulations

The first similarity is that both the Eurocode and BRO 2004 use similar FLMs. Table 4.1below presents the FLMs total weight, number of axles and distance between the first andthe last axle. Note that the lorries used in FLM5 are the same as FLM4. The most usefulcomparison is between FLM3 and BRO 2004, since they both use one lorry to representthe entire traffic during the bridge’s lifetime. The lorry used in BRO 2004 is heavier, buthas more spacing between the axles.

39

CHAPTER 4. DISCUSSION

Table 4.1: Presentation of involved lorries.

FatigueLoad Model

Weight[tonnes] Axles [no.] Length [m]

FLM3 48.0 4.00 8.40

FLM4A 20.0 2.00 4.50

FLM4B 31.0 3.00 5.50

FLM4C 49.0 5.00 11.0

FLM4D 39.0 4.00 11.2

FLM4E 45.0 5.00 14.1

BRO 2004 66.0 4.00 9.50

The second major similarity is the fatigue assessment, which is more or less the same forboth codes. Both are based on S-N curves and the theory and concept of the Palmgren-MinerMethod.

4.3 Differences between the Old and the New Regulations

The biggest difference between the regulations is undebatable the number of lorries crossingthe bridge. As specified in Section 2.7.4, the old regulation gives two options for the LCNover the bridge’s lifetime; 100,000 or 400,000 heavy lorries, which corresponds to 2.28 or9.13 daily crossings respectively during a lifetime of 120 years. This is to be comparedwith the Eurocode’s Nobs, which has a lowest value of 50,000 annually, corresponding to137 daily crossings, i.e. more than 15 times more traffic than the maximum level in BRO2004.

It can be argued which model is the most accurate, but the authors agree upon that BRO2004 is favorable, whilst the Eurocode might be conservative, since the closest availabletraffic data to the bridge’s actual location indicates a traffic intensity of 45.0 lorries daily.Values from Appendix F reveal that the bridge could withstand 1.50 million load cyclesat the highest stress range obtained from BRO 2004. Given an annual traffic intensity of50,000 the bridge would reach fatigue failure in less than 30 years. On the other hand,bridge failures due to fatigue are not very common in Sweden [17], meaning that the oldregulations worked perfectly fine and that the new regulations might be too conservative.

Another difference is that the Eurocode provides several different FLMs instead of justone. This makes it more sophisticated and should enable a better reflection of the realconditions. The Eurocode also handles the influence from several lanes, through λ4 inFLM3 and the Nobs definition in FLM4 and FLM5.

40

4.4. HOW TO PASS THE FATIGUE VERIFICATION

4.4 How to pass the Fatigue Verification

As shown in the analysis, the major fatigue problem occurs at the bottom flanges atmidspan. The easiest and most effective way to improve the design in order to pass thefatigue verification is therefore to increase the size of the bottom flange, both in thicknessand width. This gives an increased moment of inertia as well as a smaller value of z,the distance from the neutral layer to the outer tension fibre. The impact on the finalstress range, and therefore fatigue capacity, by increasing the size of the bottom flange istherefore two-folded.

The drawback with increased bottom flange size is of course the use of more steel whichresults in higher costs. How much more steel that is needed is impossible to say, due tothe extreme variation between the FLMs and the properties of the bridge. However, ashort test was made in FLM3, where the width and the thickness of the bottom flange wasincreased by 30.0% first at only steel section D2 and then throughout the entire beam.This increased the amount of steel with a total of 1.85% for the first case and 24.7% for thelatter. At the same time, the UR for midspan1 changed from 126% to 91.6% and 97.9%respectively, i.e. it now passes the fatigue verification. The reason for a higher UR whenthe entire beam is enlarged is probably that it results in a different moment distributionalong the beam, and therefore it could be beneficial to experiment with various thicknessesat critical points along the beam. For further details regarding this test, see Appendix G.

4.5 Pros and Cons with the Fatigue Load Models

The biggest advantage with FLM3 is the short assessment time. It uses only one load groupand makes several simplifications within the λ-factors that makes it easy to calculate. InSweden, the contribution from several lanes is set to 1.00 through λ4, i.e. only one laneis considered. The fatigue calculations are easy to execute and follow. The drawback isthat the origin of FLM3 is unclear since there is no logical explanation to either the loadgroup definition nor the λ-factors. The bridge engineer cannot relate the values to realityor other theories which makes it hard to assess the results using engineering experience.Therefore, FLM3 might lack in ability to reflect the reality.

FLM4 is better than FLM3 at reflecting the reality since it uses different types of frequentlorries. This ability is though limited due to only four possible selections of the trafficintensity and three possibilities regarding the traffic type. On the other hand, FLM4leaves room for engineering judgement since it is a straight forward method followingwell-known theories such as the Palmgren-Miner rule. This allows the engineer to noticepotential errors more easily compared to FLM3.

FLM5 is the most sophisticated model that allows the best correlation and adaptation tothe reality. The project team and project manager can evaluate and specify exactly howmuch traffic the bridge needs to withstand. If performed accurately, this will be the bestapproximation of the actual circumstances which allows optimization of the bridge andtherefore lower costs. This FLM should be used with caution since several sectors needto be involved to achieve a correct estimation of the traffic during the bridge’s lifetime.Accurate forecasts regarding how the traffic will change when the bridge is built and howit will develop in the future are some aspects that need extra attention.

41


4.6 Most favourable FLM in the Design

Depending on the traffic across the bridge, the "most favourable" FLM varies. For bridgeswith a low Nobs, the most favourable FLM is FLM4, or FLM5 if that is a possibility. WhenNobs increases, it gets more favourable to use FLM3. This is mainly due to λmax whichlimits the impact of increasing Nobs. The exact limit between which FLM is favourablecannot be generally concluded, due to the wide variety of bridge properties. However, inthis study it is only favourable to use FLM4 if Nobs is 125,000 or below, given that thetraffic is of local type.

4.7 Weaknesses in the Eurocode’s Fatigue Assessment

The biggest drawback of the Eurocode is, according to us, its inflexibility regarding Nobs.As discussed by Mattias Öst in the report Fatigue Study of the Vårby Bridge, wherefatigue assessment according to the Eurocode is compared with fatigue assessment basedon measured data, an alternative of choosing Nobs should be provided by the Eurocode[5]. Therefore, an alternative way of selecting Nobs is presented in Figure 4.1. The soliddark line illustrates how the selection of Nobs is made today via ÅDTt, and the dashedline is a suggestion of an alternative way of making the selection, represented by Equation4.1. The alternative way of selecting Nobs eliminates the extreme steps obtained whenusing the Eurocode. Note that this suggestion is not official, but just a suggestion of animprovement provided by the authors of this thesis. One might argue that it is risky touse this alternative formula since it suggests an Nobs far below what is recommended inthe Eurocode for most ÅDTt’s, but compared with BRO 2004 this alternative estimationis still conservative. If the old regulations would be included in Figure 4.1, it would berepresented by a line with an Nobs of less than 1,000 for ÅDTt < 10, 000 and less than4,000 for higher values of ÅDTt, i.e. barely visible along the x-axis in this figure.

42

4.7. WEAKNESSES IN THE EUROCODE’S FATIGUE ASSESSMENT

0 2 4 6 8 10 12 14 16 18 20 22 24·103

0

0.5

1

1.5

2·106

ÅDTt

Nobs

Figure 4.1: Alternative way of selecting Nobs.

Nobs = 20, 000 + 82.5 ·ÅDTt (4.1)

Another potential argument against this approach could be that as soon as ÅDTt growsabove 242 lorries, Nobs is "underestimated". For example, if ÅDTt is 1,000 it gives anNobs of 102,500, when in reality, 365,000 heavy lorries cross the bridge annually. This isdue to the slope of the curve; an increment of 1 in ÅDTt only affects Nobs by 82.5 eventhough the real impact should be 365. On the other hand, a higher increment than 82.5would soon make the suggested model more conservative than the Eurocode. Therefore, itis reasonable to keep the model as suggested. Usage of the model, for local traffic type asspecified by the Eurocode, results in a UR of 33.2% for the worst exposed point, comparedto 70.0% as in FLM4. The difference is directly proportional to the decrement in Nobs,which changes from 50,000 to approximately 23,700, using Equation 4.1 with ÅDTt of 45lorries. Therefore, no further details regarding this calculation are provided.

A second drawback is that the Eurocode ignores the possibility of several heavy lorries beingon the bridge at the same time. This is somehow handled in FLM3, but not consideredat all in FLM4. The risk with this condition, specified in SS-EN 1991-2 Section 4.6.5(3),is that the partial damage with respect to fatigue is highly dependent on the magnitudeof the stress ranges. One large stress range is much more crucial than two small, i.e. thecrossing of two lorries at the same time is worse from a fatigue perspective than if theycross the bridge one-by-one. For instance, if the bridge is located nearby a harbour, itis not unlikely that several heavy lorries follow each other across the bridge. It could beargued that this is currently neutralized by an overestimation in Nobs, but it would bebetter if both factors reflected reality instead of trying to neutralize each other’s errors.

43


It is also interesting to notice that the axle spacings defined by the fatigue load modelsare relatively short, and the gross weight relatively high. Actually, neither one of the usedlorries are allowed on the Swedish roads according to Transportstyrelsen, as shown in Table4.2. The length-weight requirement is published in a table at their homepage [18], as wellas the maximum pressure from axles, bogies and triple axles [19]. This fact can howeverbe tuned down since the different norms claim that the dynamic effects are "included" intheir fatigue load models. How precise these dynamic effects are implemented can also bediscussed. Once again, the idea of neutralizing each other’s errors might not be the bestapproach.

Table 4.2: Validation of the FLM’s with the Swedish Transportstyrelsen’s regulationsregarding the lorries properties [18, 19].

Transportstyrelesen’s Regulations

FatigueLoadModel

Weight[tonnes]

Axles[no.]

Length[m]

Length-WeightRatio

AxlePressure

BogiePressure

TripleAxle

Pressure

FLM3 48.0 4.00 8.40 Not OK Not OK Not OK -

FLM4A 20.0 2.00 4.50 OK Not OK - -

FLM4B 31.0 3.00 5.50 Not OK OK Not OK -

FLM4C 49.0 5.00 11.0 Not OK Not OK - Not OK

FLM4D 39.0 4.00 11.2 OK Not OK OK -

FLM4E 45.0 5.00 14.1 OK Not OK OK -

BRO 2004 66.0 4.00 9.50 Not OK - Not OK -

A fourth issue regarding the Eurocode is that it assumes that all lorries are fully loadedevery time they cross the bridge, which does not accurately reflect the reality. It would bepossible to find a correction factor compensating for this, since this was done by the UKDepartment of Transport when they picked the national value of Qm1 [14]. On the otherhand, it is a safe side assumption to calculate as if they were always fully loaded.

Finally, it can also be discussed whether it is realistic to calculate the fatigue damagecumulatively. It is hard to imagine that the damage from the first passage has the sameimpact as the last one, especially keeping Figure 1.1 describing the crack propagation inmind. However, usage of a cumulative fatigue calculation according to the Palmgren-Minerrule is still the recommended method to use by the design codes, despite nearly 100 yearsof potential improvements since it was first published.

4.8 The Importance of Fatigue

Imhof presents, in his dissertation Risk Assessment of existing bridge structures, one of thelargest databases for bridge failures [17]. The database covers bridge collapses worldwidebetween 1444 and 2004. The first bridge in the database is the Rialto bridge in Veniceand it is followed by a gap of nearly 400 years. Only bridges that were successfullybuilt are included, even though the majority of failures in bridge construction occurredduring the construction phase. Imhof classifies the collapses into eight categories; limited

44

4.8. THE IMPORTANCE OF FATIGUE

knowledge, natural hazard, design error, overloading, impact, human error, vandalism anddeterioration. Failure due to fatigue is included in the category limited knowledge, sinceImhof classifies knowledge of fatigue as a “recently identified limit in structural behavior”.In fact, only five bridges have collapsed due to fatigue; one in 1967 and the others before1920.

Out of all collapsed bridges, five were located in Sweden. These were Sandö Bridge acrossÅngermanälven that failed in 1939 due to design error, Göteborg Arch Bridge whichfailed in 1959 due to limited knowledge, Gothenburg Harbour Bridge which failed due toan impact in 1977, and Almö Sound Bridge and Tjörn Bridge which both failed due toimpacts in 1980.

Table 4.3 shows how many collapses, in %, that occurred during different time intervals[17]. As shown, collapse due to limited knowledge was more common before 1940; today,limited knowledge is very seldom the reason of failure. The point is that, during the latterperiod, it can be assumed that regulations similar to BRO 2004 were used for the designall over the world and the number of fatigue failures dropped dramatically. Now, withthe new Eurocode, the fatigue design is even more conservative than before, even thoughfatigue was not a major problem using the old regulations. Altogether, this supports thearguments in Section 4.7 which aim to make the design code more adoptable to the actualcircumstances and thereby, in some sense, less conservative.

Table 4.3: Cause of bridge collapse for in-service bridges by date of failure [%], [17]

Collapse causeAll Bridges Before 1900 1900-1940 1941-1990 1991-2004(237 bridges) (35 bridges) (27 bridges) (117 bridges) (58 bridges)

Limited knowledge 9 14 30 7 1Natural hazard 40 31 37 37 50Design error 5 9 0 4 5Overloading 14 26 4 14 14Impact 25 17 29 30 19Human error 3 0 0 2 7Vandalism 1 3 0 0 2Deterioration 3 0 0 6 2Total 100 100 100 100 100

Another fact regarding fatigue is that it is usually a harmless failure since the crackspropagate step by step which gives time to discover the upcoming failure. However, oncefailure due to fatigue occurs, it can happen just within a couple of load cycles [3]. Theresults from the report A review of metallic bridge failure statistics, containing data from164 bridges, indicates that fatigue is the most common failure for non-collapsed bridges[20]. The conclusion, regarding that fatigue usually is discovered before resulting in a fullcollapse, is that bridge engineers pay a lot of attention to the phenomenon. Fatigue iscarefully handled in the design and regular bridge inspections discover the cracks beforethey reach their critical size.

45

Chapter 5

Conclusions and Future Outlook

5.1 Conclusions

The Eurocode is definitely more conservative than BRO 2004 when assessing fatigue.This is mainly due to the number of lorries crossing the bridge during its lifetime. The oldregulations calculated for a maximum number of daily crossings of 9.13 lorries, whilst theminimum number in the Eurocode is 137. Since no bridge designed according to the oldregulations in Sweden suffered a total collapse due to fatigue, this large increment seemstoo conservative and unmotivated. An alternative way of selecting Nobs, the number ofheavy lorries in each slow lane per year during the bridge’s lifetime, would therefore bepreferable. This thesis suggests selecting Nobs as a linear function of ÅDTt instead ofusing the current intervals.

The new regulations provide alternative methods for assessing fatigue, compared withthe old one that only provided one method. FLM3 in the Eurocode is simple and highlyapproximate, but very similar to BRO 2004. However, the Eurocode also provides FLM4and FLM5, models that enable more exact estimations of the actual traffic on the bridge.This could be of great value, especially for larger bridges for which the old regulationsmight lack in precision.

There are some similarities between the regulations. Both use a similar approach ofassessing the fatigue, with similar fatigue load models and calculation methods. Theseload models are not representative for the reality, since they are too short and heavy toaccess Swedish roads. The regulations claim that dynamics effects are included in themodels and that it is the reason for their appearance. The authors would prefer a betterreflection of the reality for all included parameters, instead of just approximating them alland working under the assumption that they all hopefully neutralize each others errors atthe end.

47

CHAPTER 5. CONCLUSIONS AND FUTURE OUTLOOK

5.2 Future Outlook

The main area of improvement seems to be in developing a better model for predicting thetraffic intensity, or more specifically Nobs. However, a regular change on a higher level, i.e.directly into the Eurocode, seems hard to incorporate at this stage due to the bureaucracybehind this European standard. Instead, focus should be in developing a model that canbe included as an alternative in the NA for FLM5. Such a model could be provided inTRVK Bro 11, maybe as an additional paragraph in Section B.3.2.1.3. Altogether, theuse of FLM5 should be encouraged for all types of bridges.

FLM3 could be improved by adapting Qm1 to the Swedish conditions as done in the UK.This could be done through measurements of the gross weight of the lorries passing ameasurement point. These measurements could be performed at many different roads andtimes of the year in order to give a representative value of the average gross weight of thelorries in Sweden, i.e. a statistically proven value of Qm1. Also, the probability of severallorries being at the bridge at the same time should be investigated.

Finally, a complete handbook explaining how to assess fatigue for the Swedish conditionswould be of great use.

48

Bibliography

[1] Frank Axhag, 2014. Thesis Supervisor from ELU Konsult AB.

[2] Sétra. Eurocode 3 and 4: Application to steel-concrete composite bridges, 2007.

[3] Bert Norlin. Fatigue of steel structures. Lecture Note in Advanced Bridge DesignCourse AF2203, 2013. Royal Institute of Technology, division of Structural Designand Bridges.

[4] Allan Palmgren. N Arvid Palmgren. In Svenskt Biografiskt Lexikon. Riksarkivet,2014.

[5] Mattias Öst. Fatigue Study of the Vårby Bridge - A Fatigue Stress Comparisonwith Eurocode. Master’s thesis, Luleå University of Technology, Department of Civil,Environmental and Natural Resources Engineering, 2013.

[6] Mikael Andersson and Erik Karlsson. Brolaster enligt eurocode - en jämförelse avdimensionerande brolasters påverkan enligt BRO 2002 och Eurocode. Master’s thesis,Chalmers Tekniska Högskola, Institutionen för bygg- och miljöteknik, 2006.

[7] Robert Hällmark. Lastseminarium - eurocode. erfarenheter från svenska vägbroar.Seminar Notes, 2012. Ramböll Sverige AB.

[8] Carlos Eduardo da Mata Bilé. Fatigue Assessment in Steel-Concrete CompositeBridges. Case Studies. Instituto Superior Técnico Graduate Student.

[9] LUSAS. Modeller Reference Manual, 2014.

[10] Robert D. Cook, David S. Malkus, Michael E. Plesha, and Robert J. Witt. Conceptand Applications of Finite Element Analysis, 4th edition. John Wiley & Sons Inc.,2001.

[11] Trafikverket. Fordons vikt, last och motor, 2011. Available from: http://www.trafikverket.se/PageFiles/46572/2_fordons_last.pdf.

[12] Trafikverket. Trafikflödeskarta tfk, 2003. Vägnummer 1120, punkt 8540018.

[13] P. Croce. Background to fatigue load models for eurocode 1: Part 2 traffic loads.Technical report, University of Pisa, Italy, 2002.

[14] S. Chakrabarti, C. Hendy, N. Adamson, and D. Iles. The uk national annexes to bsen 1993-2, bs en 1993-1-11, and bs en 1993-1-12, 2010.

[15] Peter Lykvist and Mathias Blom. Utmattning av vägbroar i armerad betong enligteurokoder. Examensarbete 15 hp inom Byggteknik och Design, 2011. KTH, ABE,Byggvetenskap, avd för Byggteknik och Design.

49

http://www.trafikverket.se/PageFiles/46572/2_fordons_last.pdf

http://www.trafikverket.se/PageFiles/46572/2_fordons_last.pdf

BIBLIOGRAPHY

[16] Y. Bouassida, E. Bouchon, P. Crespo, P. Croce, L. Davaine, S. Denton, M. Feldmann,R. Frank, G. Hanswille, W. Hensen, B. Kolias, N. Malakatas, G. Mancini, M. Ortega,J. Raoul, G. Sedlacek, and G. Tsionis. Bridge design to eurocodes worked examples.Technical report, JRC Scientific, 2010.

[17] Daniel Imhof. Risk assessment of existing bridge structures. PhD thesis, Universityof Cambridge, 2004.

[18] Transportstyrelsen. Bruttoviktstabeller, 2012. Available from: https://www.transportstyrelsen.se/sv/Vag/Yrkestrafik/Gods-och-buss/Matt-och-vikt/Bruttoviktstabeller/.

[19] Transportstyrelsen. Tillåtet axeltryck, 2012. Available from: https://www.transportstyrelsen.se/sv/Vag/Yrkestrafik/Gods-och-buss/Matt-och-vikt/Tillatet-axeltryck/.

[20] B.M. Imam & M.K. Chryssanthopoulos. A review of metallic bridge failure statistics.Faculty of Engineering and Physical Sciences, University of Surrey, 2010.

50

https://www.transportstyrelsen.se/sv/Vag/Yrkestrafik/Gods-och-buss/Matt-och-vikt/Bruttoviktstabeller/



https://www.transportstyrelsen.se/sv/Vag/Yrkestrafik/Gods-och-buss/Matt-och-vikt/Tillatet-axeltryck/



Appendix A

Properties of the Main Beams

This appendix gives a detailed description of the main beams. Denotations for thecalculations are shown at the left beam in Figure A.1. The right beam is included toshow the origin of the coordinate system used in LUSAS. Equations presented in thisappendix are generalized and therefore contain general denotations. Calculations of crosssectional parameters were carried out in the software MathCad 15.0.

Figure A.1: Cross section with denotations.

A-1

APPENDIX A. PROPERTIES OF THE MAIN BEAMS

The dimensions, as well as some cross sectional parameters calculated according to theequations below, are presented in Table 3.1. Table A.1 gives further explanations of thecontent in Table 3.1.

zgc =wbf · tbf ·

tbf

2 + hw · tw ·(tbf + hw

2

)+ wtf · ttf ·

(tbf + hw + ttf

2

)wbf · tbf + hw · tw + wtf · ttf

ztop = tbf + hw + ttf − zgc

Iy =wbf · t3bf

12 +wbf · tbf ·(tbf

2 − zgc

)2+ tw · h3

w

12 + hw · tw ·(tbf + hw

2 − zgc

)2+wtf · t3tf

12 +

+wbf · t3tf

12 + wtf · ttf ·(tbf + hw + ttf

2 − zgc

)2

A-2

Table A.1: Description of columns in Table A.2.

Column DescriptionBeam Beam segmentwbf Width of the bottom flangetbf Thickness of the bottom flangehw Height of the webtw Thickness of the webwtf Width of the top flangettf Thickness of the top flangeL Length of the beam segmentzgc Distance to centre of gravity measured from the bottom, see Figure A.1ztop Distance to centre of gravity measured from the top, see Figure A.1.

Used for offsetting the beam element in LUSASIy Moment of Inertia about the y-axis

Table A.2: Cross sectional properties.

Beam wbf

[mm]tbf

[mm]hw

[mm]tw

[mm]wtf

[mm]ttf

[mm]L[m]

zgc

[m]ztop

[m]Iy

[m4]A1 500 25.0 2, 356 15.0 400 20.0 5.00 1.11 1.30 0.045A2 600 35.0 2, 350 18.0 400 20.0 9.00 0.992 1.41 0.057B1 700 35.0 2, 350 17.0 400 20.0 12.0 0.939 1.47 0.059B2 700 40.0 2, 340 18.0 400 20.0 12.0 0.905 1.50 0.063C1 920 50.0 2, 310 19.0 640 40.0 6.00 0.997 1.41 0.114C2 1, 140 60.0 2, 300 20.0 1, 000 50.0 12.0 1.08 1.34 0.182D1 1, 080 55.0 2, 300 19.0 700 45.0 6.00 0.960 1.44 0.137D2 700 45.0 2, 330 20.0 700 30.0 9.20 1.08 1.32 0.093E1 860 45.0 2, 340 22.0 500 30.0 12.0 0.944 1.47 0.092E2 920 45.0 2, 350 19.0 920 45.0 12.0 1.22 1.22 0.139F1 800 40.0 2, 340 19.0 500 25.0 7.60 0.948 1.46 0.077F2 500 25.0 2, 360 17.0 400 20.0 5.20 1.12 1.29 0.047

A-3

Appendix B

Hand Calculations

This Appendix presents a rough estimation of the normal stress ranges over the midsupport’smid TF for FLM3 using a hand calculation. The effective width of the concrete deck carriedby one of the two main girders is assumed to be half of the total width of the deck. Thesteel properties presented below are taken from Appendix A. All the following calculationswere carried out in MathCad 15.0.

As = 0.164 m2 Steel area for C2

zy = 1.08 m Centre of gravity measured from the bottom

ztf = 2.41 m Distance to top flange measured from the bottom

Is = 0.182 m4 Moment of inertia for the steel about the y-axis

Below, needed parameters for the concrete part of the structure are presented.

Lc = 5.5 m hc = 0.25 m Length and height of the concrete deck

Ac = Lc · hc = 1.38 m2 Area of the concrete deck

zc = 2.54 m Centre of gravity for the concrete deck measured from themain beams bottom

Ic = Lc · h3c

12 = 7.16 · 10−3 m4 Moment of inertia for the concrete deck about the y-axis

In order to calculate stresses for the composite section, an adaption of the cross sectiondue to different Young’s Modulus is needed. Note that the Young’s Modulus is reducedby 10% over the midsupport according to Section 2.7.1. Therefore, the concrete area isreduced using η, calculated as:

η = 210 GPa10.0% · 33.0 GPa = 63.6

B-1

APPENDIX B. HAND CALCULATIONS

The gravity centre and the second moment of inertia for the composite cross section cannow be calculated as:

zcomp =As · zy + Ac

η·(tbf + hw + ttf + hc

2

)As + Ac

η

= 1.25 m

Icomp = Is +As · (zy − zcomp)2 +(Ic

η+ Ac

η· (zc − zcomp

)2= 0.223 m4

The bridge is now assumed to have two spans of 55.0m each and FLM3 is assumed toconsist of one single point load of 480 kN only. All this to allow usage of engineering tablesfor calculation of the moment as:

Ms = 2 · −0.094 · P · L = 2 · −0.094 · 480 kN · 55.0 m = −4.96 MNm

The stress in the top flange may now be determined as:

σtf = Ms

Icomp· (ztf − zcomp) = −4.96 MNm

0.223 m4 · (2.41 m− 1.25 m) = −26.0 MPa

This value is now to be compared with the value obtained through LUSAS, which is26.1MPa. As the values are nearly equal, it can be concluded that the model behaves asintended.

B-2

Appendix C

Shear Stress Assessment

The stress ranges from shear forces were very small and therefore no fatigue problemsoccurred due to shear stresses. Below, the shear stress range for the midsupport, wherethe shear force is the highest, is calculated as a demonstration of its limited impact.FLM3 is used, since it gives the highest stress as it allows multiple lorries. The crosssectional parameters below are taken from beam C2, the beam crossing the midsupport(see Appendix A for further details), and the shear stress is calculated at the gravity centreof the cross section.

ztop = 1.34 m Centre of gravity measured from the topIs = 0.182 m4 Moment of Inertia for the steel about the y-axis

The first moment of area can now be calculated as:

Sy = wtf · ttf ·(ztop −

ttf2

)+ ttw ·

(ztop − ttf )2

2 = 1 m · 0.05 m ·(

1.34 m− 0.05 m2

)+

+0.012 m · (1.34 m− 0.05 m)2

2 = 0.082 m3

The shear forces are taken from LUSAS, giving following shear force range:

∆V = (570 kN + 470 kN) = 1, 040 kN

The shear stress range can now be calculated as:

∆τ = ∆V · Sy

Is · tw= 1, 040 kN · 0.082 m3

0.182 m4 · 0.02 m = 23.4 MPa

The obtained stress range is to be compared with the cut-off limit, ∆τL, where no fatigueoccur. ∆τL is calculated according to SS-EN 1993-1-9 Section 7.1(2) as:

∆τL =( 2

100

)0.2·∆τc = 0.457∆τc

The lowest value of ∆τc is, according to SS-EN 1993-1-9 Figure 7.2, 80.0MPa (size effectsneglected). If 80.0MPa is used, stresses below 0.457 · 80 = 36.6MPa results in no fatigue,i.e. shear stress ranges of 23.4MPa are negligible. Even if more points sensitive to shearstress exists in other parts of the bridge, it is doubtable that they would ever reach ∆τL.Note also that this assessment uses FLM3 with multiple lorries which is the worst loadgroup with respect to shear stresses.

C-1

Appendix D

Complete Calculations - FLM3

FLM3 is defined in SS-EN 1991-2 Section 4.6.4. It is based on the Lambda Method, whichtranslate the damage caused by a single lorry crossing into a lifetime damage by the useof equivalent damage factors called λ-factors. These factors are calculated according toSS-EN 1993-2 Section 9.5.2, also presented below. For the purpose of this thesis, onelambda is needed for the spans and another for the supports.

Below, complete calculations including normative references for the λ-factors are presented.A table summarizing the results of the fatigue assessment, including a detailed descriptionof the columns included in the table, is presented.

λ1 - Due to the span Length

λ1 is calculated from graphs defined in the Eurocode. Different values apply for span andsupport points, due to different influence lengths. Here, the length for the span points isLf = 58.8m, which corresponds to the span length. For the support, the influence lengthcorresponds to the average length of the nearby spans, i.e. Ls = 0.5 · (50.8 + 58.8) m =54.8m. This gives the following values for λ1:

Field: λ1,f = 2.55− 0.700 · Lf − 10.0 m70.0 m = 2.55− 0.700 · 58.8 m− 10.0 m

70.0 m = 2.06

Support: λ1,s = 1.70 + 0.500 · Ls − 30.0 m50.0 m = 1.70 + 0.500 · 54.8 m− 30.0 m

50.0 m = 1.95

λ2 - Due to the Traffic Volume

λ2 is calculated from the average weight and the number of the trucks in the slow lane(Qm1 and Nobs). These are combined with reference values Q0 and N0 specified in theSS-EN 1993-2 Section 9.5.2. The values used in this thesis, and the λ2 that is calculatedfrom them, is presented below.

Qm1 = 410 kN According to Trafikverket BRO 2011 Section E.3.1

Q0 = 480 kN Specified in the SS-EN 1993-2 Section 9.5.2

D-1

APPENDIX D. COMPLETE CALCULATIONS - FLM3

Nobs = 0.050 · 106 Nobs per slow lane, selected according to Section 2.7.4

N0 = 0.500 · 106 Specified in the SS-EN 1993-2 Section 9.5.2

λ2 = Qm1Q0·(Nobs

N0

)0.2= 410

480 ·(0.050 · 106

0.500 · 106

)0.2= 0.539

λ3 - Due to Expected Life Length

λ3 is calculated from the expected life length of the bridge, tLd, which is set to 120 yearsin this thesis.

λ3 =(tLd

100

)0.2=(120

100

)0.2= 1.04

λ4 - Due to Additional Lanes

In this thesis, the recommended value of λ4 according to Trafikverket TRVFS 2011:12Chapter 19 §17 is used, i.e. λ4 = 1.00.

λmax - Maximum λ-Values

λmax are calculated according to SS-EN 1993-2 Figure 9.6. Note that the value variesfrom field to support.

Field: λf.max = 2

Support: λs.max = 1.80 + 0.900 · Ls − 10.0 m50.0 m = 1.80 + 0.900 · 54.8 m− 10.0 m

50.0 m = 2.25

Calculation of Final λ-values

λ is now calculated according to SS-EN 1993-2 Equation 9.9.

Field: λf = min(λ1.f · λ2 · λ3 · λ4; λf.max) =

= min(2.06 · 0.539 · 1.04 · 1.00; 2.00) = min(1.15; 2.00) = 1.15

Support: λs = min(λ1.s · λ2 · λ3 · λ4; λs.max) =

= min(1.95 · 0.539 · 1.04 · 1.00; 2.25) = min(1.09; 2.25) = 1.09

The complete results of the calculations for FLM3 is shown in Table D.2. In Table D.1short descriptions of the columns in Table D.2 are given. The used detail categories arepresented in Figure D.1 for the midspans and Figure D.2 for the midsupport.

D-2

Figure D.1: Detail category at midspans, SS-EN 1993-1-9 Table 8.3 Detail 9.

Figure D.2: Detail category at midsupport, SS-EN 1993-1-9 Table 8.4 Detail 7.

Note that the values of ∆σC,x in Table D.2 includes the size effect, presented for points 1,2 and 3 below.

∆σC,1 = 80.0 MPa·ks,1 = 80.0 MPa·(25.0 mm

tbf,D2

)0.2= 80.0 MPa·

(25.0 mm45.0 mm

)0.2= 71.1MPa

∆σC,2 = 80.0 MPa·ks,2 = 80.0 MPa·(25.0 mm

tbf,E1

)0.2= 80.0 MPa·

(25.0 mm45.0 mm

)0.2= 71.1MPa

∆σC,3 = 80.0 MPa·ks,3 = 80.0 MPa·(25.0 mm

tbf,C2

)0.2= 80.0 MPa·

(25.0 mm50.0 mm

)0.2= 69.6MPa

D-3

APPENDIX D. COMPLETE CALCULATIONS - FLM3

Table D.1: Description of the columns in Table D.2.

Column DescriptionPoint The point that is observed, described in Figure 1.2. 1-Midspan1 (Beam

D2), 2-Midspan2 (Beam E1), 3-Midsupport (Beam C2)Nlor Number of lorries (varies between one and two)Lspa Lorry spacing (centre-to-centre distance in case of two lorries)Fl.Pos. Flange Position, described in Figure 2.4xmin The x-coordinate for the front axis of the first lorry at which the

minimum stress is obtained, given in metresxmax The x-coordinate for the front axis of the first lorry at which the

maximum stress is obtained, given in metresσx,min Minimum nominal stress in the x-direction given in MPa, obtained

from LUSASσx,max Maximum nominal stress in the x-direction given in MPa, obtained

from LUSAS∆σ The nominal stress range, given in MPa, calculated as σx,max − σx,min

∆σE,2 The nominal stress range, given in MPa, calculated as λ ·∆σ. Notethat λ is different for the span points (1 and 2) and the midsupportpoint (3)

∆σC,x Fatigue resistance, at point x in MPa, according to the detail categories

UR The utilization ratio, calculated as UR = γF t · γMf ·∆σE.2∆σc

SÅ

D-4

Table D.2: FLM3 - Lambda Method.

FLM3 - Lambda Method

Point Nlor Lspa Fl.Pos xmin xmax σx,min σx,max ∆σ ∆σE,2 ∆σC,x UR

1 1.00 - OuterBF 33.0 78.0 -12.2 33.0 45.2 52.1 71.1 99.0%

1 1.00 - MidBF 33.0 78.0 -12.7 38.6 51.4 59.2 71.1 112%

1 1.00 - LeftBF 33.0 78.0 -13.3 44.3 57.5 66.3 71.1 126%

2 1.00 - OuterBF 33.0 85.0 -6.66 31.2 37.9 43.7 71.1 82.9%

2 1.00 - MidBF 33.0 85.0 -7.01 39.4 46.4 53.5 71.1 101%

2 1.00 - LeftBF 33.0 85.0 -7.37 47.5 54.9 63.3 71.1 120%

3 2.00 44.0 OuterBF - 78.0 0.000 28.5 28.5 31.0 69.6 60.2%

3 2.00 44.0 MidBF - 78.0 0.000 26.1 26.1 28.4 69.6 55.1%

3 2.00 44.0 LeftBF - 78.0 0.000 23.8 23.8 25.9 69.6 50.2%

SÅ

D-5

Appendix E

Complete Calculations - FLM4and FLM5

The complete results of the calculations of FLM4 and FLM5 are shown in Tables E2-E13.Short descriptions of the columns in these Tables are given in Table E.1. Nobs used inFLM5 is based on the information that 45.0 lorries passes a nearby bridge daily. To beon the safe side, it is assumed that this number is three times higher in 120 years, givingNobs,F LM5 as:

Nobs,today = 45.0 crossings/day · 365 days/year = 16, 425 crossings/year

Nobs,120 = 3.00 ·Nobs,today = 3.00 · 16, 425 crossings/year = 49, 275 crossings/year

Nobs,F LM5 = 0.500·(Nobs,today+Nobs,120) = 0.500·(16, 425+49, 275) = 32, 850 crossings/year

E-1

APPENDIX E. COMPLETE CALCULATIONS - FLM4 AND FLM5

Table E.1: Description of the columns in the results tables to FLM4 and FLM5.


D2), 2-Midspan2 (Beam E1), 3-Midsupport (Beam C2)Lor.Type Type of lorry, A to E from top to bottom in Table 2.1Fl.Pos. Flange Position, described in Figure 2.4TP Transverse Position of the load, described in Figure 2.11xmin The x-coordinate for the front axis of the first lorry at which the




from LUSAS∆σi The nominal stress range, given in MPa, calculated as σx,max − σx,min

∆σC,x Fatigue resistance, at point x in MPa, according to the detail categoriesNR,i Potential number of load cycles, in millions, that the detail can handle

from stress range ∆σi calculated according to Equation 2.11. "-" meansthat the fatigue resistance is unlimited

T% The amount of the traffic connected to the lorry and traffic type,according to Table 2.1 for FLM4 and Table 2.2 for FLM5

ni The number of lorries of type i crossing the bridge during its lifetime,calculated as Nobs · 120·T% for a lifetime of 120 years

ni/NR,i The partial damage from lorry iUR The utilization ratio, calculated as the sum of ni/NR,i

E-2

TableE.2:

FLM4Po

int1,

i.e.midspan

1,part

1of

2.FL

M4

LocalT

raffic

Medium

Dist

ance

Long

Dist

ance

Nobs

=50,0

00N

obs

=50,0

00N

obs

=50,0

00

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

1A

OuterBF

132.0

76.0

-5.16

13.3

18.4

71.1

-80%

4.8

0.000

30.0%

40%

2.4

0.000

110%̇

20%

1.2

0.000

170%̇

1A

OuterBF

231.0

75.0

-3.31

13.4

16.7

71.1

-80%

4.8

0.000

40%

2.4

0.000

20%

1.2

0.000

1B

OuterBF

133.0

76.0

-7.99

20.0

27.9

71.1

25.9

5%̇0.3

0.012

10%̇

0.6

0.023

5%0.3

0.012

1B

OuterBF

232.0

76.0

-5.13

21.8

26.9

71.1

31.2

5%0.3

0.010

10%

0.6

0.019

5%0.3

0.010

1C

OuterBF

135.0

80.0

-12.4

33.2

45.5

71.1

3.10

5%0.3

0.097

30%

1.8

0.581

50%

3.0

0.969

1C

OuterBF

235.0

80.0

-7.94

27.2

35.1

71.1

8.25

5%0.3

0.036

30%

1.8

0.218

50%

3.0

0.364

1D

OuterBF

135.0

80.0

-9.81

27.0

36.8

71.1

6.52

5%0.3

0.046

15%

0.9

0.138

15%

0.9

0.138

1D

OuterBF

234.0

81.0

-6.30

20.6

26.9

71.1

31.1

5%0.3

0.010

15%

0.9

0.029

15%

0.9

0.029

1E

OuterBF

137.0

80.0

-11.2

31.1

42.3

71.1

3.86

5%0.3

0.078

5%0.3

0.078

10%

0.6

0.155

1E

OuterBF

236.0

80.0

-7.18

21.3

28.5

71.1

23.7

5%0.3

0.013

5%0.3

0.013

10%

0.6

0.025

1A

MidBF

132.0

76.0

-5.39

17.4

22.8

71.1

71.7

80%

4.8

0.067

46.5%

40%

2.4

0.033

147%

20%

1.2

0.017

220%

1A

MidBF

231.0

75.0

-3.37

10.5

13.9

71.1

-80%

4.8

0.000

40%

2.4

0.000

20%

1.2

0.000

1B

MidBF

133.0

76.0

-8.35

27.2

35.6

71.1

7.74

5%0.3

0.039

10%

0.6

0.078

5%

0.3

0.039

1B

MidBF

232.0

76.0

-5.23

16.5

21.7

71.1

91.6

5%0.3

0.003

10%

0.6

0.007

5%

0.3

0.003

1C

MidBF

135.0

80.0

-12.9

39.4

52.3

71.1

2.04

5%0.3

0.147

30%

1.8

0.882

50%

3.0

1.470

1C

MidBF

235.0

80.0

-8.09

23.8

31.9

71.1

13.3

5%0.3

0.023

30%

1.8

0.135

50%

3.0

0.225

1D

MidBF

135.0

80.0

-10.3

30.5

40.7

71.1

4.33

5%0.3

0.069

15%

0.9

0.208

15%

0.9

0.208

1D

MidBF

234.0

81.0

-6.41

18.4

24.8

71.1

46.9

5%0.3

0.006

15%

0.9

0.019

15%

0.9

0.019

1E

MidBF

137.0

80.0

-11.7

34.3

46.0

71.1

3.01

5%0.3

0.100

5%

0.3

0.100

10%

0.6

0.199

1E

MidBF

236.0

80.0

-7.32

20.7

28.0

71.1

25.7

5%0.3

0.012

5%0.3

0.012

10%

0.6

0.023

E-3


TableE.3:

FLM4Po

int1,

i.e.midspan

1,part

2of

2.FL

M4

LocalT

raffic

Medium

Dist

ance

Long

Dist

ance

Nobs

=50,0

00N

obs

=50,0

00N

obs

=50,0

00

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

1A

InnerB

F1

32.0

76.0

-5.62

21.5

27.2

71.1

29.8

80%

4.8

0.161

70.0%

40%

2.4

0.080

202%

20%

1.2

0.040

293%

1A

InnerB

F2

31.0

75.0

-3.43

7.65

11.1

71.1

-80%

4.8

0.000

40%

2.4

0.000

20%

1.2

0.000

1B

InnerB

F1

33.0

76.0

-8.72

34.5

43.2

71.1

3.63

5%0.3

0.083

5%0.6

0.165

10%

0.3

0.083

1B

InnerB

F2

32.0

76.0

-5.32

11.2

16.5

71.1

-5%

0.3

0.000

10%

0.6

0.000

5%0.3

0.000

1C

InnerB

F1

35.0

80.0

-13.5

45.6

59.1

71.1

1.42

5%0.3

0.212

30%

1.8

1.272

50%

3.0

2.119

1C

InnerB

F2

35.0

80.0

-8.24

20.5

28.7

71.1

22.6

5%0.3

0.013

30%

1.8

0.080

50%

3.0

0.133

1D

InnerB

F1

35.0

80.0

-10.7

34.0

44.7

71.1

3.29

5%0.3

0.091

15%

0.9

0.274

15%

0.9

0.274

1D

InnerB

F2

34.0

81.0

-6.53

16.2

22.7

71.1

73.1

5%0.3

0.004

15%

0.9

0.012

15%

0.9

0.012

1E

InnerB

F1

37.0

80.0

-12.2

37.4

49.6

71.1

2.40

5%0.3

0.125

5%0.3

0.125

10%

0.6

0.250

1E

InnerB

F2

36.0

80.0

-7.45

20.1

27.5

71.1

28.0

5%0.3

0.011

5%0.3

0.011

10%

0.6

0.021

E-4

TableE.4:

FLM4Po

int2,

i.e.midspan

2,part

1of

2.FL

M4

LocalT

raffic

Medium

Dist

ance

Long

Dist

ance

Nobs

=50,0

00N

obs

=50,0

00N

obs

=50,0

00

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

2A

OuterBF

132.0

87.0

-4.22

10.7

14.9

71.1

-80%

4.8

0.000

16.7%

40%

2.4

0.000

62.7%

20%

1.2

0.000

98.3%

2A

OuterBF

231.0

87.0

-1.86

14.4

16.3

71.1

-80%

4.8

0.000

40%

2.4

0.000

20%

1.2

0.000

2B

OuterBF

133.0

88.0

-4.37

17.3

21.7

71.1

91.6

5%0.3

0.003

10%

0.6

0.007

5%0.3

0.003

2B

OuterBF

232.0

88.0

-2.88

23.5

26.4

71.1

34.5

5%0.3

0.009

10%

0.6

0.017

5%0.3

0.009

2C

OuterBF

135.0

91.0

-6.77

31.9

38.7

71.1

5.08

5%0.3

0.059

30%

1.8

0.354

50%

3.0

0.591

2C

OuterBF

235.0

91.0

-4.46

26.8

31.2

71.1

14.8

5%0.3

0.020

30%

1.8

0.121

50%

3.0

0.202

2D

OuterBF

135.0

87.0

-5.36

25.4

30.8

71.1

16.0

5%0.3

0.019

15%

0.9

0.056

15%

0.9

0.056

2D

OuterBF

234.0

87.0

-3.54

21.6

25.1

71.1

44.0

5%0.3

0.007

15%

0.9

0.020

15%

0.9

0.020

2E

OuterBF

137.0

91.0

-6.11

29.9

36.0

71.1

7.29

5%0.3

0.041

5%0.3

0.041

10%

0.6

0.082

2E

OuterBF

236.0

91.0

-4.04

22.8

26.9

71.1

31.5

5%0.3

0.010

5%0.3

0.010

10%

0.6

0.019

2A

MidBF

132.0

87.0

-4.46

15.0

19.5

71.1

-80%

4.8

0.000

28.1%

40%

2.4

0.00

101%

20%

1.2

0.00

155%

2A

MidBF

231.0

87.0

-1.90

10.7

12.6

71.1

-80%

4.8

0.000

40%

2.4

0.000

20%

1.2

0.000

2B

MidBF

133.0

88.0

-4.60

27.6

32.2

71.1

12.7

5%0.3

0.024

10%

0.6

0.047

5%0.3

0.024

2B

MidBF

232.0

88.0

-2.94

16.8

19.7

71.1

-5%

0.3

0.000

10%

0.6

0.000

5%0.3

0.000

2C

MidBF

135.0

91.0

-7.13

40.0

47.1

71.1

2.80

5%0.3

0.107

30%

1.8

0.642

50%

3.0

1.070

2C

MidBF

235.0

91.0

-4.55

24.2

28.7

71.1

22.5

5%0.3

0.013

30%

1.8

0.080

50%

3.0

0.133

2D

MidBF

135.0

87.0

-5.65

31.6

37.2

71.1

6.17

5%0.3

0.049

15%

0.9

0.146

15%

0.9

0.146

2D

MidBF

234.0

87.0

-3.61

19.1

22.7

71.1

72.7

5%0.3

0.004

15%

0.9

0.012

15%

0.9

0.012

2E

MidBF

137.0

91.0

-6.44

35.7

42.1

71.1

3.92

5%0.3

0.077

5%0.3

0.077

10%

0.6

0.153

2E

MidBF

236.0

91.0

-4.12

21.6

25.7

71.1

39.2

5%0.3

0.008

5%0.3

0.008

10%

0.6

0.015

E-5


TableE.5:

FLM4Po

int2,

i.e.midspan

2,part

2of

2.FL

M4

LocalT

raffic

Medium

Dist

ance

Long

Dist

ance

Nobs

=50,0

00N

obs

=50,0

00N

obs

=50,0

00

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

2A

InnerB

F1

32.0

87.0

-3.12

23.8

26.9

71.1

31.4

80%

4.8

0.153

62.3%

40%

2.4

0.076

172%

20%

1.2

0.038

245%

2A

InnerB

F2

31.0

87.0

-1.93

7.05

8.98

71.1

-80%

4.8

0.000

40%

2.4

0.000

20%

1.2

0.000

2B

InnerB

F1

33.0

88.0

-4.84

37.9

42.8

71.1

3.74

5%0.3

0.080

10%

0.6

0.160

5%0.3

0.080

2B

InnerB

F2

32.0

88.0

-3.00

10.0

13.0

71.1

-5%

0.3

0.000

10%

0.6

0.000

5%0.3

0.000

2C

InnerB

F1

35.0

91.0

-7.49

48.0

55.5

71.1

1.71

5%0.3

0.175

30%

1.8

1.050

50%

3.0

1.750

2C

InnerB

F2

35.0

91.0

-4.64

21.6

26.3

71.1

35.3

5%0.3

0.008

30%

1.8

0.051

50%

3.0

0.085

2D

InnerB

F1

35.0

87.0

-5.93

37.8

43.7

71.1

3.51

5%0.3

0.086

15%

0.9

0.257

15%

0.9

0.257

2D

InnerB

F2

34.0

87.0

-3.67

16.7

20.3

71.1

-5%

0.3

0.000

15%

0.9

0.000

15%

0.9

0.000

2E

InnerB

F1

37.0

91.0

-6.76

41.5

48.2

71.1

2.61

5%0.3

0.115

5%0.3

0.115

10%

0.6

0.230

2E

InnerB

F2

36.0

91.0

-4.20

20.4

24.6

71.1

49.2

5%0.3

0.006

5%0.3

0.012

10%

0.6

0.012

E-6

TableE.6:

FLM4Po

int3,

i.e.midsupport,part

1of

2.FL

M4

LocalT

raffic

Medium

Dist

ance

Long

Dist

ance

Nobs

=50,0

00N

obs

=50,0

00N

obs

=50,0

00

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

[%]

T%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

3A

OuterTF

1-

76.0

0.000

5.93

5.93

69.6

-80%

4.8

0.000

0.000%

40%

2.4

0.000

0.000%

20%

1.2

0.000

0.000%

3A

OuterTF

2-

77.0

0.000

3.40

3.40

69.6

-80%

4.8

0.000

40%

2.4

0.000

20%

1.2

0.000

3B

OuterTF

1-

77.0

0.000

9.19

9.19

69.6

-5%

0.3

0.000

10%

0.6

0.000

5%0.3

0.000

3B

OuterTF

2-

77.0

0.000

4.62

4.62

69.6

-5%

0.3

0.000

10%

0.6

0.000

5%0.3

0.000

3C

OuterTF

1-

80.0

0.000

14.4

14.4

69.6

-5%

0.3

0.000

30%

1.8

0.000

50%

3.0

0.000

3C

OuterTF

2-

80.0

0.000

8.10

8.10

69.6

-5%

0.3

0.000

30%

1.8

0.000

50%

3.0

0.000

3D

OuterTF

1-

80.0

0.000

11.4

11.4

69.6

-5%

0.3

0.000

15%

0.9

0.000

15%

0.9

0.000

3D

OuterTF

2-

80.0

0.000

6.40

6.40

69.6

-5%

0.3

0.000

15%

0.9

0.000

15%

0.9

0.000

3E

OuterTF

1-

82.0

0.000

13.1

13.1

69.6

-5%

0.3

0.000

5%0.3

0.000

10%

0.6

0.000

3E

OuterTF

2-

82.0

0.000

8.84

8.84

69.6

-5%

0.3

0.000

5%0.3

0.000

10%

0.6

0.000

3A

MidTF

1-

76.0

0.000

5.98

5.98

69.6

-80%

4.8

0.000

0.000%

40%

2.4

0.000

0.000%

20%

1.2

0.000

0.000%

3A

MidTF

2-

77.0

0.000

3.50

3.50

69.6

-80%

4.8

0.000

40%

2.4

0.000

20%

1.2

0.000

3B

MidTF

1-

77.0

0.000

9.28

9.28

69.6

-5%

0.3

0.000

10%

0.6

0.000

5%0.3

0.000

3B

MidTF

2-

77.0

0.000

4.35

4.35

69.6

-5%

0.3

0.000

10%

0.6

0.000

5%0.3

0.000

3C

MidTF

1-

80.0

0.000

14.5

14.5

69.6

-5%

0.3

0.000

30%

1.8

0.000

50%

3.0

0.000

3C

MidTF

2-

80.0

0.000

8.51

8.51

69.6

-5%

0.3

0.000

30%

1.8

0.000

50%

3.0

0.000

3D

MidTF

1-

80.0

0.000

11.5

11.5

69.6

-5%

0.3

0.000

15%

0.9

0.000

15%

0.9

0.000

3D

MidTF

2-

80.0

0.000

6.75

6.75

69.6

-5%

0.3

0.000

15%

0.9

0.000

15%

0.9

0.000

3E

MidTF

1-

82.0

0.000

13.1

13.1

69.6

-5%

0.3

0.000

5%0.3

0.000

10%

0.6

0.000

3E

MidTF

2-

82.0

0.000

8.36

8.36

69.6

-5%

0.3

0.000

5%0.3

0.000

10%

0.6

0.000

E-7


TableE.7:

FLM4Po

int3,

i.e.midsupport,part

2of

2.FL

M4

LocalT

raffic

Medium

Dist

ance

Long

Dist

ance

Nobs

=50,0

00N

obs

=50,0

00N

obs

=50,0

00

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

T%

ni

ni/N

R,i

UR

3A

InnerT

F1

-76.0

0.000

6.03

6.03

69.6

-80%

4.8

0.000

0.000%

40%

2.4

0.000

0.000%

20%

1.2

0.000

0.000%

3A

InnerT

F2

-77.0

0.000

3.70

3.70

69.6

-80%

4.8

0.000

40%

2.4

0.000

20%

1.2

0.000

3B

InnerT

F1

-77.0

0.000

9.37

9.37

69.6

-5%

0.3

0.000

10%

0.6

0.000

5%0.3

0.000

3B

InnerT

F2

-77.0

0.000

4.08

4.08

69.6

-5%

0.3

0.000

10%

0.6

0.000

5%0.3

0.000

3C

InnerT

F1

-80.0

0.000

14.5

14.5

69.6

-5%

0.3

0.000

30%

1.8

0.000

50%

3.0

0.000

3C

InnerT

F2

-80.0

0.000

8.94

8.94

69.6

-5%

0.3

0.000

30%

1.8

0.000

50%

3.0

0.000

3D

InnerT

F1

-80.0

0.000

11.5

11.5

69.6

-5%

0.3

0.000

15%

0.9

0.000

15%

0.9

0.000

3D

InnerT

F2

-80.0

0.000

7.10

7.10

69.6

-5%

0.3

0.000

15%

0.9

0.000

15%

0.9

0.000

3E

InnerT

F1

-82.0

0.000

13.2

13.2

69.6

-5%

0.3

0.000

5%0.3

0.000

10%

0.6

0.000

3E

InnerT

F2

-82.0

0.000

8.84

8.84

69.6

-5%

0.3

0.000

5%0.3

0.000

10%

0.6

0.000

E-8

TableE.8:

FLM5Po

int1,

i.e.midspan

1,part

1of

2.FL

M5

Actua

lLocation

Nobs

=32,8

50

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

1A

OuterBF

132

.076

.0-5.16

13.3

18.4

71.1

-70

%2.8

0.00

0

24.8%

1A

OuterBF

231

.075

.0-3.31

13.4

16.7

71.1

-70

%2.8

0.00

0

1B

OuterBF

133

.076

.0-7.99

20.0

27.9

71.1

25.9

10%

0.4

0.01

5

1B

OuterBF

232

.076

.0-5.13

21.8

26.9

71.1

31.2

10%

0.4

0.01

3

1C

OuterBF

135

.080

.0-12.4

33.2

45.5

71.1

3.10

5%0.2

0.06

4

1C

OuterBF

235

.080

.0-7.94

27.2

35.1

71.1

8.25

5%0.2

0.02

4

1D

OuterBF

135

.080

.0-9.81

27.0

36.8

71.1

6.52

10%

0.4

0.06

0

1D

OuterBF

234

.081

.0-6.30

20.6

26.9

71.1

31.1

10%

0.4

0.01

3

1E

OuterBF

137

.080

.0-11.2

31.1

42.3

71.1

3.86

5%0.2

0.05

1

1E

OuterBF

236

.081

.0-7.18

21.3

28.5

71.1

23.7

5%0.2

0.00

8

1A

MidBF

132

.076

.0-5.39

17.4

22.8

71.1

71.7

70%

2.8

0.03

8

37.8%

1A

MidBF

231

.075

.0-3.37

10.5

13.9

71.1

-70

%2.8

0.00

0

1B

MidBF

133

.076

.0-8.35

27.2

35.6

71.1

7.74

10%

0.4

0.05

1

1B

MidBF

232

.076

.0-5.23

16.5

21.7

71.1

91.6

10%

0.4

0.00

4

1C

MidBF

135

.080

.0-12.9

39.4

52.3

71.1

2.04

5%0.2

0.09

7

1C

MidBF

235

.080

.0-8.09

23.8

31.9

71.1

13.3

5%0.2

0.01

5

1D

MidBF

135

.080

.0-10.3

30.5

40.7

71.1

4.33

10%

0.4

0.09

1

1D

MidBF

234

.081

.0-6.41

18.4

24.8

71.1

46.9

10%

0.4

0.00

8

1E

MidBF

137

.080

.0-11.7

34.3

46.0

71.1

3.01

5%0.2

0.06

5

1E

MidBF

236

.082

.0-7.32

20.7

28.0

71.1

25.7

5%0.2

0.00

8

E-9


TableE.9:

FLM5Po

int1,

i.e.midspan

1,part

2of

2.FL

M5

Actua

lLocation

Nobs

=32,8

50

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

1A

Inne

rBF

132

.076

.0-5.62

21.5

27.2

71.1

29.8

70%

2.8

0.09

3

56.4%

1A

Inne

rBF

231

.075

.0-3.43

7.65

11.1

71.1

-70

%2.8

0.00

0

1B

Inne

rBF

133

.076

.0-8.72

34.5

43.2

71.1

3.63

10%

0.4

0.10

9

1B

Inne

rBF

232

.076

.0-5.32

11.2

16.5

71.1

-10

%0.4

0.00

0

1C

Inne

rBF

135

.080

.0-13.5

45.6

59.1

71.1

1.42

5%0.2

0.13

9

1C

Inne

rBF

235

.080

.0-8.24

20.5

28.7

71.1

22.6

5%0.2

0.00

9

1D

Inne

rBF

135

.080

.0-10.7

34.0

44.7

71.1

3.29

10%

0.4

0.12

0

1D

Inne

rBF

234

.081

.0-6.53

16.2

22.7

71.1

73.1

10%

0.4

0.00

5

1E

Inne

rBF

137

.080

.0-12.2

37.4

49.6

71.1

2.40

5%0.2

0.08

2

1E

Inne

rBF

236

.080

.0-7.45

20.1

27.5

71.1

28.0

5%0.2

0.00

7

E-10

TableE.10

:FL

M5Po

int2,

i.e.midspan

2,part

1of

2.FL

M5

Actua

lLocation

Nobs

=32,8

50

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

2A

OuterBF

132

.087

.0-4.22

10.7

14.9

71.1

-70

%2.8

0.00

0

13.5%

2A

OuterBF

231

.087

.0-1.86

14.4

16.3

71.1

-70

%2.8

0.00

0

2B

OuterBF

133

.088

.0-4.37

17.3

21.7

71.1

91.6

10%

0.4

0.00

4

2B

OuterBF

232

.088

.0-2.88

23.5

26.4

71.1

34.5

10%

0.4

0.01

1

2C

OuterBF

135

.091

.0-6.77

31.9

38.7

71.1

5.08

5%0.2

0.03

9

2C

OuterBF

235

.091

.0-4.46

26.8

31.2

71.1

14.8

5%0.2

0.01

3

2D

OuterBF

135

.087

.0-5.36

25.4

30.8

71.1

16.0

10%

0.4

0.02

5

2D

OuterBF

234

.087

.0-3.54

21.6

25.1

71.1

44.0

10%

0.4

0.00

9

2E

OuterBF

137

.091

.0-6.11

29.9

36.0

71.1

7.29

5%0.2

0.02

7

2E

OuterBF

236

.091

.0-4.04

22.8

26.9

71.1

31.5

5%0.2

0.00

6

2A

MidBF

132

.087

.0-4.46

15.0

19.5

71.1

-70

%2.8

0.00

0

23.5%

2A

MidBF

231

.087

.0-1.90

10.7

12.6

71.1

-70

%2.8

0.00

0

2B

MidBF

133

.088

.0-4.60

27.6

32.2

71.1

12.7

10%

0.4

0.03

1

2B

MidBF

232

.088

.0-2.94

16.8

19.7

71.1

-10

%0.4

0.00

0

2C

MidBF

135

.091

.0-7.13

40.0

47.1

71.1

2.80

5%0.2

0.07

0

2C

MidBF

235

.091

.0-4.55

24.2

28.7

71.1

22.5

5%0.2

0.00

9

2D

MidBF

135

.087

.0-5.65

31.6

37.2

71.1

6.17

10%

0.4

0.06

4

2D

MidBF

234

.087

.0-3.61

19.1

22.7

71.1

72.7

10%

0.4

0.00

5

2E

MidBF

137

.091

.0-6.44

35.7

42.1

71.1

3.92

5%0.2

0.05

0

2E

MidBF

236

.091

.0-4.12

21.6

25.7

71.1

39.2

5%0.2

0.00

5

E-11


TableE.11

:FL

M5Po

int2,

i.e.midspan

2,part

2of

2.FL

M5

Actua

lLocation

Nobs

=32,8

50

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

2A

Inne

rBF

132

.087

.0-3.12

23.8

26.9

71.1

31.4

70%

2.8

0.08

8

50.6%

2A

Inne

rBF

231

.087

.0-1.93

7.05

8.98

71.1

-70

%2.8

0.00

0

2B

Inne

rBF

133

.088

.0-4.84

37.9

42.8

71.1

3.74

10%

0.4

0.10

5

2B

Inne

rBF

232

.088

.0-3.00

10.0

13.0

71.1

-10

%0.4

0.00

0

2C

Inne

rBF

135

.091

.0-7.49

48.0

55.5

71.1

1.71

5%0.2

0.11

5

2C

Inne

rBF

235

.091

.0-4.64

21.6

26.3

71.1

35.3

5%0.2

0.00

6

2D

Inne

rBF

135

.087

.0-5.93

37.8

43.7

71.1

3.51

10%

0.4

0.11

2

2D

Inne

rBF

234

.087

.0-3.67

16.7

20.3

71.1

-10

%0.4

0.00

0

2E

Inne

rBF

137

.091

.0-6.76

41.5

48.2

71.1

2.61

5%0.2

0.07

6

2E

Inne

rBF

236

.091

.0-4.20

20.4

24.6

71.1

49.2

5%0.2

0.00

4

E-12

TableE.12

:FL

M5Po

int3,

i.e.midsupport,part

1of

2.FL

M5

Actua

lLocation

Nobs

=32,8

50

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

3A

OuterTF

1-

76.0

0.0

5.93

5.93

69.6

-70

%2.8

0.00

0

0.00

0%

3A

OuterTF

2-

77.0

0.00

03.40

3.40

69.6

-70

%2.8

0.00

0

3B

OuterTF

1-

77.0

0.00

09.19

9.19

69.6

-10

%0.4

0.00

0

3B

OuterTF

2-

77.0

0.00

04.62

4.62

69.6

-10

%0.4

0.00

0

3C

OuterTF

1-

80.0

0.00

014

.414

.469

.6-

5%0.2

0.00

0

3C

OuterTF

2-

80.0

0.00

08.10

8.10

69.6

-5%

0.2

0.00

0

3D

OuterTF

1-

80.0

0.00

011

.411

.469

.6-

10%

0.4

0.00

0

3D

OuterTF

2-

80.0

0.00

06.40

6.40

69.6

-10

%0.4

0.00

0

3E

OuterTF

1-

82.0

0.00

013

.113

.169

.6-

5%0.2

0.00

0

3E

OuterTF

2-

82.0

0.00

08.84

8.84

69.6

-5%

0.2

0.00

0

3A

MidTF

1-

76.0

0.00

05.98

5.98

69.6

-70

%2.8

0.00

0

0.00

0%

3A

MidTF

2-

77.0

0.00

03.50

3.50

69.6

-70

%2.8

0.00

0

3B

MidTF

1-

77.0

0.00

09.28

9.28

69.6

-10

%0.4

0.00

0

3B

MidTF

2-

77.0

0.00

04.35

4.35

69.6

-10

%0.4

0.00

0

3C

MidTF

1-

80.0

0.00

014

.514

.569

.6-

5%0.2

0.00

0

3C

MidTF

2-

80.0

0.00

08.51

8.51

69.6

-5%

0.2

0.00

0

3D

MidTF

1-

80.0

0.00

011

.511

.569

.6-

10%

0.4

0.00

0

3D

MidTF

2-

80.0

0.00

06.75

6.75

69.6

-10

%0.4

0.00

0

3E

MidTF

1-

82.0

0.00

013

.113

.169

.6-

5%0.2

0.00

0

3E

MidTF

2-

82.0

0.00

08.36

8.36

69.6

-5%

0.2

0.00

0

E-13


TableE.13

:FL

M5Po

int3,

i.e.midsupport,part

2of

2.FL

M5

Actua

lLocation

Nobs

=32,8

50

Point

Lor.T

ype

Fl.Pos

TP

xm

inx

ma

xσ

x,m

inσ

x,m

ax

∆σ

i∆σ

C,x

NR

,iT%

ni

ni/N

R,i

UR

3A

Inne

rTF

1-

76.0

0.00

06.03

6.03

69.6

-70

%2.8

0.00

0

0.00

0%

3A

Inne

rTF

2-

77.0

0.00

03.70

3.70

69.6

-70

%2.8

0.00

0

3B

Inne

rTF

1-

77.0

0.00

09.37

9.37

69.6

-10

%0.4

0.00

0

3B

Inne

rTF

2-

77.0

0.00

04.08

4.08

69.6

-10

%0.4

0.00

0

3C

Inne

rTF

1-

80.0

0.00

014

.514

.569

.6-

5%0.2

0.00

0

3C

Inne

rTF

2-

80.0

0.00

08.94

8.94

69.6

-5%

0.2

0.00

0

3D

Inne

rTF

1-

80.0

0.00

011

.511

.569

.6-

10%

0.4

0.00

0

3D

Inne

rTF

2-

80.0

0.00

07.10

7.10

69.6

-10

%0.4

0.00

0

3E

Inne

rTF

1-

82.0

0.00

013

.213

.269

.6-

5%0.2

0.00

0

3E

Inne

rTF

2-

82.0

0.00

08.84

8.84

69.6

-5%

0.2

0.00

0

E-14

Appendix F

Complete Calculations - BRO 2004

The complete results of the calculations of BRO 2004 is shown in Table F.2. Shortdescriptions of the columns in Table F.2 are given in Table F.1.

F-1

APPENDIX F. COMPLETE CALCULATIONS - BRO 2004

Table F.1: Description of the columns in Table F.2.


D2), 2-Midspan2 (Beam E1), 3-Midsupport (Beam C2)Fl.Pos. Flange Position, described in Figure 2.4xmin The x-coordinate for the front axis of the first lorry at which the




from LUSAS∆σi The nominal stress range, given in MPa, calculated as σx,max − σx,min

∆σC,x Fatigue resistance, at point x in MPa, according to the detail categoriesnt,i Potential number of load cycles, in millions, that the detail can handle

from stress range ∆σi calculated according to Equation 2.15. A "-"means that the fatigue resistance is unlimited

nct,1 The first option of LCN according to BRO 2004, i.e. 100,000nct,2 The second option of LCN according to BRO 2004, i.e. 400,000UR The utilization ratio, calculated as UR = nt,i

nct,x

F-2

Table F.2: Results according to BRO 2004.

BRO 2004nct,1 nct,2

Point Fl.pos TP xmin xmax σx,min σx,max ∆σi ∆σC,x nt,i UR UR

1 OuterBF 1 33.0 79.0 -16.7 45.3 62.0 76.5 2.82 3.54% 14.2%

1 OuterBF 2 33.0 79.0 -10.7 34.5 45.2 76.5 31.9 0.313% 1.25%

1 MidBF 1 33.0 79.0 -17.4 52.0 69.4 76.5 2.01 4.97% 19.9%

1 MidBF 2 33.0 79.0 -10.9 31.4 42.3 76.5 44.3 0.226% 0.903%

1 InnerBF 1 33.0 79.0 -18.2 58.6 76.8 76.5 1.48 6.74% 27.0%

1 InnerBF 2 33.0 79.0 -11.1 28.3 39.4 76.5 62.8 0.159% 0.636 %

2 OuterBF 1 33.0 85.0 -9.12 41.7 50.8 76.5 17.7 0.565% 2.26%

2 OuterBF 2 33.0 85.0 -6.01 38.3 44.3 76.5 35.1 0.285% 1.14%

2 MidBF 1 33.0 85.0 -9.60 53.8 63.4 76.5 2.64 3.79% 15.2%

2 MidBF 2 33.0 85.0 -6.14 32.6 38.7 76.5 68.6 0.146% 0.583%

2 InnerBF 1 33.0 85.0 -10.1 66.0 76.1 76.5 1.53 6.54% 26.2%

2 InnerBF 2 33.0 85.0 -6.26 26.9 33.2 76.5 - 0.000% 0.000 %

3 OuterTF 1 - 78.0 0.000 19.4 19.4 75.9 - 0.000% 0.000%

3 OuterTF 2 - 78.0 0.000 10.8 10.8 75.9 - 0.000% 0.000%

3 MidTF 1 - 78.0 0.000 19.5 19.5 75.9 - 0.000% 0.000%

3 MidTF 2 - 78.0 0.000 11.5 11.5 75.9 - 0.000% 0.000%

3 InnerTF 1 - 78.0 0.000 19.5 19.5 75.9 - 0.000% 0.000%

2 InnerTF 2 - 78.0 0.000 12.1 12.1 75.9 - 0.000% 0.000%

F-3

Appendix G

Stress Evaluation with IncreasedBottom Flange Size

This Appendix explains the test at which the size of the bottom flange was increased by30.0% in both thickness and width, i.e. the bottom flange area was increased by 69.0%.This was performed in two steps, first, only the most critical beam, D2, and then theentire beam was enlarged. The first case changed the utilization ratio to 91.6% and thesecond to 97.9%, compared to the initial 126%. As a reference case, FLM3 at midspan1was used. Figure G.1 below illustrates how the cross section size increased.

Figure G.1: Illustration of the difference between the original and the enlarged crosssection.

Based on the information regarding the properies of the main beams given in Section 3.1.1areas and volumes for the different beam sections were calculated. This is summarized inTable G.1. As shown, the total amount of steel increases with 24.7%, from 10.9 to 13.6m3

if the entire beam is enlarged. However, if only D2 is enlarged, the total amount of steelchanges by just 1.84%.

The result of the fatigue calculations is shown in Table G.3. In Table G.2, short descriptionsof the columns in Table G.3 are given.

G-1

APPENDIX G. STRESS EVALUATION WITH INCREASED BOTTOM FLANGE SIZE

Table G.1: Cross sectional properties.

BeamOriginal Dimensions Enlarged DimensionsA [m2] L [m] V [m3] A [m2] L [m] V [m3]

A1 0.056 5.00 0.279 0.064 5.00 0.322A2 0.071 9.00 0.642 0.086 9.00 0.772B1 0.073 12.0 0.870 0.089 12.0 1.07B2 0.078 12.0 0.938 0.097 12.0 1.16C1 0.116 6.00 0.693 0.147 6.00 0.884C2 0.164 12.0 1.97 0.212 12.0 2.54D1 0.135 6.00 0.808 0.176 6.00 1.06D2 0.099 9.20 0.911 0.121 9.20 1.11E1 0.105 12.0 1.26 0.132 12.0 1.58E2 0.127 12.0 1.53 0.156 12.0 1.87F1 0.089 7.60 0.676 0.111 7.60 0.844F2 0.061 5.20 0.315 0.069 5.20 0.360

Total Volume: 10.9 13.6

G-2

Table G.2: Description of the columns in Table G.3.

Column Description

Point The point that is observed, described in Figure 1.2. 1-Midspan1 (BeamD2), 2-Midspan2 (Beam E1), 3-Midsupport (Beam C2)

Nlor Number of Lorries (varies between one and two)

Case The case being investigated, either enlargement of only D2 (called "D2"in the table) or the entire beam (EB)

Fl.Pos. Flange Position, described in Figure 2.4

xminThe x-coordinate for the front axis of the first lorry at which theminimum stress is obtained, given in metres

xmaxThe x-coordinate for the front axis of the first lorry at which themaximum stress is obtained, given in metres

σx,minMinimum nominal stress in the x-direction given in MPa, obtainedfrom LUSAS

σx,maxMaximum nominal stress in the x-direction given in MPa, obtainedfrom LUSAS

∆σ The nominal stress range, given in MPa, calculated as σx,max − σx,min

∆σE,2

The nominal stress range, given in MPa, calculated as λ ·∆σ. Notethat λ is different for the span points (1 and 2) and the midsupportpoint (3)

∆σC,x Fatigue resistance, at point x in MPa, according to the detail categories

UR The utilization ratio, calculated as UR = γF t · γMf ·∆σE.2∆σc

G-3

APPENDIX G. STRESS EVALUATION WITH INCREASED BOTTOM FLANGE SIZE

Table G.3: FLM3 - Lambda Method with 30% increased Bottom-Flange.

FLM3 - Lambda Method with 30% increased Bottom-Flange

Point Nlor Case Fl.Pos xmin xmax σx,min σx,max ∆σ ∆σE,2 ∆σC,x UR

1 1 D2 OuterBF 33.0 78.0 -8.64 22.1 30.7 35.4 71.1 67.2%

1 1 D2 MidBF 33.0 78.0 -9.12 27.2 36.3 41.8 71.1 79.4%

1 1 D2 LeftBF 33.0 78.0 -9.60 32.3 41.9 48.3 71.1 91.6%

1 1 EB OuterBF 33.0 78.0 -8.07 20.4 32.8 32.8 71.1 62.2%

1 1 EB MidBF 33.0 78.0 -8.68 27.9 36.6 42.2 71.1 80.1%

1 1 EB LeftBF 33.0 78.0 -9.30 35.5 44.8 51.6 71.1 97.9%

G-4

TRITA-BKN. Examensarbete 419, 2014ISSN 1103-4297

ISRN KTH/BKN/B-419-SE

www.kth.se

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