Load Effect Modelling in Fatigue Design of Composite Bridges: An assessment of Fatigue Load Models 3, 4 and 5 according to SS-EN-1991-2 Actions on Structures – Part 2: Traffic loads on Bridges

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Load Effect Modelling in Fatigue Design of Composite Bridges

An 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 Dahlvik Johan Eriksson

June 2014

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

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Department of Civil and Architectural Engineering

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Preface

To begin with, we would like to direct a big thank you to ELU Konsult AB for giving us the opportunity to write this thesis. Special thanks are directed to Frank Axhag for continuous support and supervision throughout the entire process. We would also like to thank 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 our fellow 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 and inspiration 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 been put down in order to make the report interesting and easy to follow.

Stockholm, June 2014

Mathias Dahlvik Johan Eriksson

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Abstract

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

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

The result indicates that both regulations give rise to similar stress ranges, i.e. the difference between the maximum and minimum stress obtained during a crossing. The differences between the regulations are instead within the fatigue calculations, where the major difference is the number of lorries crossing the bridge during its lifetime. The utilization 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 number of daily crossings provided by BRO 2004. The lowest utilization ratio according to the Eurocode is 70.0 %, calculated for 137 daily crossings which is the lowest amount of crossings allowed. An interpretation of the Eurocode, which allows usage of fatigue load model 5 even for smaller bridges, results in a utilization ratio of 56.0 % which corresponds to 90.0 daily crossings, i.e. lower than the other fatigue load models provided by the Eurocode but clearly above the old regulations.

The conclusion is that an alternative way of deciding the number of crossings should be provided by the Eurocode. Today, the classification consists of four steps, which are very rough. Instead, a proposal is given in this thesis which advocates usage of a linear function for deciding the number of design crossings based on the number of daily crossings by lorries. The proposed alternative design method is between the two regulations with respect 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 nationella standarder till den nya europastandarden Eurokod. För brokonstruktörer innebar detta en övergång från en kombination av BRO 2004 och BSK 07, till att Eurokod blev det huvudsakligt styrande dokumentet, med bland annat TRVK Bro 11 som ett dokument med tillhörande nationella val. Övergången medförde inte bara att verksamma konstruktörer tvingades lära sig förändrade beräkningsmetoder, utan också att ett fenomen som tidigare sällan var dimensionerande för vägbroar nu kunde vara det som ställde högst krav på bärförmågan. Detta fenomen kallas utmattning, dvs. upprepade av- och pålastningar, var och 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 av dimensioneringen. Detta sker genom en jämförelse av hur de gamla och nya normerna utvärderar utmattning. Som modell har en befintlig bro invigd 2011, dimensionerad av ELU Konsult AB enligt de gamla normerna, använts. Denna bro har modellerats i finita element programmet LUSAS, varpå en mängd olika lastbilsöverfarter simulerats och utmattningsutnyttjandet 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. Däremot råder det skillnader vid utmattningsberäkningarna, där den stora skillnaden är antalet tunga fordon som passerar bron under dess livslängd. Enligt de gamla normerna är utnyttjandegraden för den värst utsatta studerade punkten 27.0 %, vilket är beräknat på 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 ett alternativt sätt att tolka Eurokod, som tillåter användandet av utmattningslastmodell 5 även för mindre broar, fås en utnyttjandegrad på 56.0 % vilket motsvarar 90.0 dagliga överfarter. Detta är något lägre än de andra utmattningslastmodellerna enligt Eurokod men fortfarande högre än det gamla regelverket.

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

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

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Introduction

1.1 Background

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

Fatigue is when repeated external loading gives rise stress variation that results in a crack propagation within a structure [2]. The crack propagation may appear in structural steel parts, reinforcing steel bars, the concrete slab etc. and can have catastrophic consequences if dealt with incorrectly. A high stress variation combined with many load cycles and weak design may result in total collapse. In some structures extensive cracking may exist without seriously affecting the load-carrying capacity, although fatigue failure is said to have occurred.

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

Since the crack propagation usually arises under elastic nominal stresses the fatigue damage 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 has started and the crack propagates into the material, the growth is exponential until total fracture occurs.

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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 cumulative damage that occurs during a components lifetime. Each partial damage is decided as the number of load cycles that is applied to the construction during its lifetime at a given stress range divided by the potential number of load cycles that the construction could carry for the same stress range. A stress range is defined as the difference between the maximum and minimum stress level in one load cycle. For example, if a component is exposed to a stress range of 10 MPa 10,000 times, and could carry 10 MPa 100,000 times, the partial damage from this stress range is 10 %, calculated as 10,000/100,000. The cumulative damage is then obtained by summarizing the partial damage from all potential stress ranges that affects the construction. The mathematical operations are further explained in Section 2.4.2.

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 research engineer at the company SKF [4]. In 1919 he patented the self-adjustable roller bearing in 19 different countries. Amongst his hardware research he worked on theories for estimating the service life of roller bearings and 1924 he published a cumulative damage theory for fatigue 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 is more known as the Palmgren-Miner rule these days and despite its many years on the market, 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-ups regarding traffic intensity, axial loads, axial spacing etc. The first two of these models are only appropriate for pure steel bridges and used to determine whether the life length of the bridge with respect to fatigue is limited or not, whilst the last three are appropriate for all types of bridges and used to assess the life length with respect to fatigue. The fatigue calculations are performed according to SS-EN 1993-1-9. Different methods are used for different FLMs. Before the Eurocode was introduced in Sweden, a combination of 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 old regulations.

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1.2. AIM

1.2 Aim

The aim of this study is to investigate the differences and similarities between the new Eurocode and the old regulations used in Sweden with respect to fatigue design of steel-concrete composite road bridges. This is performed using a built bridge as a model, assessing it in 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 more conservative than the old regulations, but the underlying cause is not thoroughly analysed and 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 to perform a complete fatigue design of a bridge, the finite element model’s ability to accurately reflect the actual bridge’s properties is not crucial. The important issue is instead to set up a model that generates reliable results to enable comparison of the different FLMs.

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

– Fatigue Load Model 3 (Section 2.3.1).

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

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

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

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

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

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1.4 Facts Regarding the Bridge

The road bridge is located in Ljungsbro, 15.0 kilometers north of Linköping, in Östergötlands Län, Sweden. The bridge was inaugurated the 21st of June 2011 and designed by ELU Konsult AB according to the old Swedish regulations [1]. It is a steel-concrete composite two-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 main beams 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 that the Eurocode exerts more traffic in terms of lorries crossing the bridge. This fact is also highlighted by Robert Hällmark in the seminar Lastseminaruim - Eurocode [7]. A more detailed study indicates that the main problem is welded bottom flanges and that top flanges rarely suffers from fatigue in steel-concrete composite bridges [8].

However, there is no complete comparison showing the complete differences between the FLMs 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 upcoming Chapter 2, Methods, the approach used for the thesis is firstly presented. After that, the basis of finite element modelling that applies for the used model, including simplifications that 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 regarding the finite element modelling as well as the fatigue assessments. The results are then discussed in Chapter 4. In the final Chapter 5, the thesis is concluded together with a future outlook.

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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 loads on bridges

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

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

bridges

SS-EN 1994-1-1:2005 Eurocode 4: Design of composite steel and concrete structures - Part 1-1: General rules and rules for buildings TRVK Bro 11 Trafikverkets tekniska krav Bro

TRVFS 2011:12 Trafikverkets författningssamling

BRO 2004 Allmänna tekniska beskrivningar för Broar. Utgivare:

Vägverket

BSK 07 Boverkets handbok om stålkonstruktioner BV Bro, utgåva 9 Broregler för nybyggnad. Utgivare: Banverket

1.8 Abbreviations and Definitions

Following abbreviations are used in this thesis:

BF Bottom Flange

FEM Finite Element Method FLM Fatigue Load Model LCN Load Cycle Number NA National Annex

TF Top Flange

TP Transverse Position of the load UR Utilization Ratio

ÅDT_t Average annual daily traffic from heavy lorries. In English, ÅDT_t is usually denoted AADTT, Average Annual Daily Truck Traffic, but since this thesis is based on the Swedish regulations, ÅDT_t is used

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1.9 List of Symbols

Roman Upper Case

C_i The fatigue normal stress resistance according to BSK 07 D_d The cumulative damage

F l.P os The fibre location in the flange

L The span length

L_spa Lorry spacing

M_y Moment about the y-axis Mz Moment about the z-axis N₀ The reference value of N_obs Nlor Number of lorries

N_obs Number of heavy lorries crossing per slow lane and year

N_R The number of cycles the design can endure a given stress range Ntot The total number of crossings during the bridge’s life time P_x The normal force in the x-direction

Q0 A nationally selected parameter reflecting the expected average weight of the lorries

Q_m1 The average weight of the lorries in the slow lane U R The utilization ratio in %

Roman Lower Case

f_rd The design value of the fatigue resistance

f_rk The characteristic value of the fatigue resistance f_vrd The design value of the fatigue resistance

nE,i The number of cycles under given stress range

n_t The potential number of load cycles that the construction can carry during its lifetime

ntot The total number of expected stress cycles during the construction’s lifetime

k_s The size effect factor

t_d The life length of the bridge in years

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1.9. LIST OF SYMBOLS

Greek Upper Case

∆σ_C The fatigue resistance according to the detail category

∆σ_E The damage equivalent normal stress range

∆σ_x The nominal normal stress range

∆τ The nominal shear stress range

∆τ_C The fatigue shear stress resistance

∆τ_E The damage equivalent shear stress range

Greek Lower Case

γF f The partial coefficient for equivalent stress ranges with constant amplitude γ_{M f} The partial coefficient for fatigue resistance

γn The partial coefficient of the safety class λ1 The damage factor due to the span length λ₂ The damage factor due to traffic volume

λ3 The damage factor due to the expected life length λ₄ The damage factor due to additional lanes

λf The damage equivalent factor for stresses in midspans λmax The maximum value of the equivalent damage factor λ_s The damage equivalent factor for stresses at midsupport σrd The nominal normal stress range

σ_x The nominal normal stress in the x-direction

σx,min Minimum nominal normal stress range in x-direction σx,max Maximum nominal normal stress range in x-direction τ_rd The nominal shear stress range

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Chapter 2

Methods

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

2.1 Our Approach

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

The approach used for this thesis is shown in Figure 2.1 below. Note that this is a schematic 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, was applied. Information needed in the different steps was collected along the way from the Eurocodes, the National Annexes, Handbooks, Supervisors, other Master Theses, Science Reports and other available literature.

Finite Element Modeling

Design Code Interpretation

Fatigue Assessment

Results and Analysis

Discussion and Conclusion

Figure 2.1: Visualisation of the process.

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2.2 Finite Element Modelling

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

To increase the reliability of the results from LUSAS smaller models that could verify intended outcome were made. For instance, a simply supported steel beam with a concrete deck was modelled with full composite action and assessed with regard to stresses that could be compared to known results from hand calculations. This process was applied throughout 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 FEA software 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 the modeller.

2.2.1 Element Selection

All elements in LUSAS are divided into different groups according to their geometrical space [9]. The different groups are point, line, surface, and volume elements. Under each section a variety of elements are defined and presented. The basic approach for creating geometries 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 time consuming. Note that the geometry definition is not the same thing as the element definition. A geometry is usually divided into several elements depending on the mesh size, 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 has a certain degree of freedom (DOF) depending on the modelling space defined [10]. A 2D element allows 3 DOFs for each node, namely lateral and axial translation and rotation within the elements own plane. A 3D element allows 6 DOFs at each node, one more translation direction and two more rotations. This is illustrated in Figure 2.2. Note that if a 2D beam element is used, only translation in x- and y-directions and rotation about

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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 beam theory, also known as Euler-Bernoulli, and the Timoshenko. The Euler-Bernoulli theory do not account for shear deformations which in general are applicable for thin beams. The Timoshenko theory on the other hand do account for the transverse shear deformations which could have a significant effect for thick beams. The exact limit between thin and thick beams is debated, but as the ratio of thickness/length decreases, shear deformations become less important.

Shell Elements

Shell elements are defined by a surface i.e. nodes and lines. The number of nodes are dependent of which element that is being used [10]. LUSAS provides triangular elements with 3 or 6 nodes and quadratic elements with 4 or 8 nodes, where each node contains 5 or 6 DOFs depending on the element definition. Just as for the beam elements there is a thin and thick formulation. The thin shells are based on the Kirchoff shell theory and do not account for transverse shear deformations and the thick shell elements are based on the Mindlin shell theory, which account for the transverse shear deformations. As the formulation 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 Quadrilateral Thick shell elements (QTS4). Both element types have a linear interpolation order which makes them suitable for each other and the elements are capable of modelling the transverse shear deformations which is desirable, due to the high thickness/length ratio in the y-direction of the bridge. Also, the element selection can accurately calculate nominal stresses which are used in the fatigue assessment. The mesh sizes are being controlled through the element definitions and with the help of so called "None elements". These elements were assigned to the edges of the bridge deck to control how the mesh is behaving near the edges. This enables a complete control over the mesh sizes and its uniformity,

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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 and nodes in LUSAS. This generates full composite action between the elements, i.e. no sliding occurs. The elements are connected to one another using the same mesh size i.e. same mesh 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 LUSAS the defining line for the beam elements is located in the top of the beam. To get the beams supported in the bottom flange rigid elements were therefore introduced. The rigid elements 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 was modelled 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 forces obtained from the finite element model. The stresses were calculated as "Engineering Stresses" 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 geometrical stresses when assessing simple details with respect to fatigue. Therefore, this was applied in the assessment. In addition, it keeps the detail level of the model at a manageable level with respect to run-time etc. The stress range is the difference between the maximum and minimum stress obtained for a lorry crossing of the bridge, defined in Equation 2.1.

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2.2. FINITE ELEMENT MODELLING

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

Investigated points in the midspan have both a tensile and compressive stresses while the investigated point in the support only have tensile stresses and therefore σ_min was set to 0. The compressive stress can, according to SS-EN 1993-1-9, be reduced to 60% of its value if it represents the majority of the stress range. However, that is not the case in this thesis, 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 sectional definitions in LUSAS. The stress σ_x is a function of the normal force and the bending moments around both bending axes and can be calculated by LUSAS for a chosen fibre location.

The fibre locations used in this thesis are visualised in Figure 2.4, where the top flange represents 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 stress distribution across the flanges. The vertical position of the BF points were chosen at the absolute 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 welded at the inside. This decision was made to keep the calculation on the safe side.

Figure 2.4: Fibre locations in LUSAS.

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2.3 Fatigue Load Models according to the Eurocode

As announced earlier, the Eurocode provides five different FLMs in SS-EN 1991-2 Action on 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 a crossing of a single lorry which is translated into a lifetime damage using damage equivalent λ-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 frequent lorries which are weighed together based on the traffic type to reflect the traffic across the bridge. The lorries vary in number of axles, axial load and contact area of the tires according to Table 2.1. Further on in this thesis, those lorries are named FLM4A to FLM4E, from top to bottom.

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

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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 Appendix B. It is based on the same principles as FLM4, but instead of using a fixed value of the number of heavy lorries crossing the bridge, N_obs, provided by the Eurocode, a project specific value based on estimations is applied. Also, the traffic type i.e. lorry percentage may be specified in the specific project.

TRVFS 2011:12 6 kap 8§ states that byggherren, the developer, specifies how and if FLM5 can 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 Bro 11 Section B.3.2.1.3j, such a specification may only be used if more than 24,000 heavy lorries cross the bridge daily, which is not the case for this bridge’s location. In this thesis it has although been decided to follow the NA to add more value to the comparison.

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

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

Lorry Type Lorry Percentage [%]

FLM4A 70.0

FLM4B 10.0

FLM4C 5.00

FLM4D 10.0

FLM4E 5.00

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

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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-EN 1993-1-9 Section 6.1. However, as specified in SS-EN 1991-2 Section 4.6.1, different methods are used for different FLMs; for FLM3, a simplified method called the λ-method can be used while FLM4 and FLM5 demand a more sophisticated method, such as Palmgren-Miner’s cumulative damage analysis. Therefore, these two methods were used in 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. The fatigue calculation includes both nominal normal stresses and nominal shear stresses. The design value for the nominal stress ranges is obtained by SS-EN 1993-1-9 Equation 6.1 as:

γ_{F f} · ∆σ_E,2= λ · ∆σ(γ_{F f}, Q_k) (2.2)

γ_{F f}· ∆τ_E,2= λ · ∆τ (γ_{F f}, Q_k) (2.3) Where;

- γ_{F f} is the partial coefficient for equivalent stress ranges with constant amplitude - ∆σ_E,2 is the damage equivalent normal stress range

- λ is the equivalent damage factor, further explained below

- ∆σ(γ_{F f}, Q_k) is the nominal normal stress range, as described in Section 2.2.4 - ∆τ_E,2 is the damage equivalent shear stress range

- ∆τ (γ_{F 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 into a lifetime representing factor. It is presented in SS-EN 1993-2 Section 9.5.2 Equation 9.9 as:

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

- λ_max is the maximum value of the equivalent damage factor

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

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 = Q_m1 Q0

·

N_obs N0

1/5

(2.5)

Where;

- Q_m1 is the average weight of the lorries in the slow lane

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

- N_obs 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 N_obs, 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=

t_d 100

1/5

(2.6)

Where;

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

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The final lambda factor, λ₄, is calculated using SS-EN 1993-2 Section 9.5.2(6) Equation 9.12, also specified in Equation 2.7. However, in Sweden λ₄ is set to 1.0 according to Trafikverket TRVFS 2011:12 Chapter 19 §17.

λ4=

"

1 +N₂ N1

·

η₂· Q_m2 η1· Q_m1

5

+N₃ N1

·

η₃· Q_m3 η1· Q_m1

5

+ ... + N_k N1

·

η_k· Q_mk η1· Q_m1

5#1/5

(2.7)

Where;

- k is the number of lanes with heavy traffic

- N_i is the number of heavy lorries in lane i per year - η_i is the lane factor for the load

- Q_mi is the average gross weight of the lorries in lane i

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

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

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

γ_{F t}· ∆σ_E,2

∆σ_C γ_{M f}

= U R ≤ 1.0 (2.8)

γ_{F t}· ∆τ_E,2

∆τ_C γ_{M f}

= U R ≤ 1.0 (2.9)

Where;

- γ_{F t} is the partial coefficient for equivalent stress ranges with constant amplitude

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

- γ_{M f} is the partial coefficient for fatigue resistance - U R is the utilization ratio

- ∆τ_E,2 is the damage equivalent shear stress range

- ∆τ_C is the fatigue shear stress resistance according to the detail category, defined in SS-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 steel is cumulative and non-reversible [3]. This means that the damage from one stress range can be calculated individually and then summed up to a total damage. The mathematical expression is defined in SS-EN 1993-1-9 Equation A.1 as:

D_d=

k

X

i=1

nE,i

N_R,i (2.10)

Where;

- D_d is the cumulative damage

- n_E,i is the number of cycles with the stress range γ_{F f} · ∆σ_i

- N_R,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, N_R,i, is the only unknown. Therefore, the equations describing the S-N, or Wöhler Curves in SS-EN 1993-1-9 Figure 7.1, also shown in Figure 2.9, were rearranged according to below in order to solve for N_R,i. Note that there is no fatigue if N_R,i > 100 · 10⁶, i.e. if the stress range is lower than the cut-off limit ∆σ_L. Also, note that Figure 2.9 is valid for normal stresses and that the same method, using another diagram, applies for shear stress fatigue assessment.

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

N_R,i =











∆σ_C

γ_{F t}· γ_{M f} · ∆σ_i

3

· 2 · 10⁶ if N_R,i ≤ 5 · 10⁶

∆σ_D

γ_{F t}· γ_{M f} · ∆σ_i

5

· 5 · 10⁶ if 5 · 10⁶ < N_R,i≤ 100 · 10⁶

(2.11)

Where;

- N_R,i is the potential number of load cycles that the construction can carry during its lifetime if only subjected to stress ranges of magnitude ∆σ_i

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

- γ_{F t} is the partial coefficient for equivalent stress ranges with constant amplitude - γ_{M f} is the partial coefficient for fatigue resistance

- ∆σ_i is one occurring stress range for which fatigue is assessed

- ∆σ_D is fatigue limit at constant amplitude, calculated as 0.737 · ∆σ_C

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

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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. This model is similar to FLM3, i.e. based on the crossing of a single lorry. The load group consists of four axles, two in the front and two in the rear as visualized in Figure 2.10. The dynamic effect of the load is assumed to be included in the axle loads. The load group should be placed so that the largest stress range occurs in the investigated beam. Only traffic 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 6:512a and 6.512b in BSK 07 as:

σ_rd≤ f_rd= f_rk

1.1 · γ_n (2.12)

τrd≤ f_vrd= 0.6 · f_rd= 0.6 · f_rk

1.1 · γ_n (2.13)

Where;

- σ_rd is the nominal normal stress range

- f_rd is the design value of the fatigue resistance

- f_rk is the characteristic value of the fatigue resistance, further explained below - γ_n is the partial coefficient of the safety class

- τ_rdis the nominal shear stress range

- f_vrd is the design value of the fatigue resistance

The characteristic value of the fatigue resistance is determined based on the detail category and the number of stress cycles. According to BSK 07 Figure 6.523, f_rk is based on S-N Curves and the equation for it is also specified in Equation 2.14. Note that Equation 2.14

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2.6. FATIGUE ASSESSMENT ACCORDING TO BSK 07

f_rk =









 C ·

2 · 10⁶ n_t

¹₃

if nt≤ 5 · 10⁶

0.885 · C ·

2 · 10⁶ n_t

¹₅

if 5 · 10⁶ < n_t≤ 100 · 10⁶

(2.14)

Where;

- f_rk is the characteristic value of the fatigue resistance

- C is the fatigue normal stress resistance according to the detail category, defined in BSK 07 Table B3:1-2

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

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

nt=











2 · 10⁶

1.1 · γ_n· σ_rd C

3 if nt≤ 5 · 10⁶ 2 · 10⁶

1.1 · γ_n· σ_rd 0.885 · C

5 if 5 · 10⁶ < nt≤ 100 · 10⁶

(2.15)

Where;

- n_t is the potential number of load cycles that the construction can carry during its lifetime if only subjected to stress ranges of magnitude σ_rd

- γ_n is the partial coefficient of the safety class

- σ_rd is one occuring stress range for which fatigue is assessed

- C is the fatigue normal stress resistance according to the detail category, defined in BSK 07 Table B3:1-2

The utilization ratio can then be determined as:

ntot

n_t = U R ≤ 1.0 (2.16)

Where;

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

- n_t is the potential number of load cycles that the construction can carry during its lifetime if only subjected to stress ranges of magnitude σ_rd

- U R is the utilization ratio

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2.7 Interpretations, Simplifications and Conditions

This sections specifies all the interpretations, simplifications and conditions that applies to 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 throughout the length of the bridge. The exact geometry of the joints between the beams was neglected. Vertical stiffeners and transverse bonds in and between the main girders were also neglected in the model.

• On top of the primary beams, a concrete deck was modelled. The shape was simplified into a rectangular shape with a thickness of 250 mm corresponding to the 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 to 10.0 % of the un-cracked modulus due to cracking in accordance with SS-EN 1994-1-1 Section 5.4.2.3 and supervisor Frank Axhag, to obtain the worst-case scenario for both field and support points. The cracking will lead to a decreased stiffness which attracts less moment over the support but the composite section itself will let the steel beam carry more of the moment due to the lower modulus. The reduction was applied to a distance of six metres on each side of the midsupport, even though the Eurocode allows a wider reduction length. The reason for not using the entire reduction was the sectioning of the primary beams.

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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 following were considered:

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

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

• Two slow lanes, denoted TPs, transverse positions, shown in Figure 2.11, was used in the analysis of FLM4 and FLM5. The selections of the position of the slow lanes were made in accordance with SS-EN 1991-2 Section 4.6.1(4)-(5). The reason for using two slow lanes in the analysis is because of the N_obs definition, i.e. per year and 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 reduce the workload:

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

– When assessing midsupport, multiple lorries was used as the widest stress range is obtained when the lorries are placed in both spans. The spacing was set to 44 meters, since that maximized the impact.

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

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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.

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

• γ_{M f} 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.

• γ_{F 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 design of the bridge, gravity loading and thermal expansion etc. were neglected in the calculations.

• The detail categories were selected as SS-EN 1993-1-9 Table 8.3 Detail 9 for the midspan 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 size effect 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.

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2.7. INTERPRETATIONS, SIMPLIFICATIONS AND CONDITIONS

2.7.4 The N_obs and LCN selection

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

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

Traffic categories N_obs per year and slow lane 1 Roads and motorways with 2 or more lanes per

direction with high flow rates of lorries 2.000 · 10⁶ 2 Roads and motorways with medium flow rates

of 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 criterion from TRVK Bro 11 is shown in Table 2.4. Note that if ÅDT_t, the average annual daily traffic from heavy lorries, is greater than 24,000 a special investigation regarding the conditions 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 ÅDT_t Conditions

1 6, 000 < ÅDT_t≤ 24, 000

2 1, 500 < ÅDT_t≤ 6, 000

3 600 < ÅDT_t≤ 1, 500

4 ÅDT_t≤ 600

In BRO 2004 the amount of traffic is based on the Load Cycle Number (LCN). Two possibilities apply; if the ÅDT_t is below 10,000 lorries LCN is 100,000 and otherwise 400,000. Note that LCN represents the entire life span of the bridge.

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2.7.5 Dynamic Effects

The load models used did not generate any stress ranges large enough to cause fatigue once the lorries left the bridge, as they were all less than 1.00 MPa. An evaluation of the dynamic stress and the static is presented in Figure 2.14. The dynamic stress ranges were calculated with LUSAS’s utility IMDPlus. The utility uses modal superposition techniques 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 of 0-30.0 Hz and a damping factor of 0.500 %. These parameters were selected in accordance with SS-EN 1991-2 and BV Bro respectively. Note that these values are intended for railway bridges but were used as a benchmark for this dynamic evaluation since the Eurocode 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 14.0 m/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]

Stressatmidspan1[MPa]

Static Dynamic

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

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

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Chapter 3

Results and Analysis

3.1 The Model

This section presents the results of the modelling in LUSAS. A generalization of the real bridge section is visualized in Figure 3.1 and the modelled bridge, according to the simplifications 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.

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3.1.1 Properties of the Model

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

Table 3.1: Cross sectional properties [mm].

Beam w_bf t_bf h_w t_w w_tf t_tf

A1 500 25 2, 356 15 400 20

A2 600 35 2, 350 18 400 20

B1 700 35 2, 353 17 400 20

B2 700 40 2, 342 18 400 20

C1 920 50 2, 313 19 640 40

C2 1, 140 60 2, 302 20 1, 000 50

D1 1, 080 55 2, 301 19 700 45

D2 700 45 2, 327 20 700 30

E1 860 45 2, 337 22 500 30

E2 920 45 2, 345 19 920 45

F1 800 40 2, 339 19 500 25

F2 500 25 2, 357 17 400 20

Figure 3.3: Cross section with denotations.

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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 in LUSAS which is in accordance with the Eurocode. The steel was given a Young’s Modulus of 210 GPa and a Poisson’s ratio of 0.300. The concrete was given a Young’s Modulus of 33 GPa 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. Control calculations were made whether the concrete should be cracked or not over the support for the midspan calculations. The worst case scenario for the midspan stresses is when the 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. Three different 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 that splitting the lines defining the shell elements in half increases the number of shell elements in the model by a factor of four, resulting in a lot more equations to solve. The results are presented in Table 3.2.

Table 3.2: Convergence Study.

Mesh size ∆σ Time

1.00 m 22.71 MPa Quick

0.500 m 22.79 MPa Medium

0.250 m 22.81 MPa Very Slow

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

Load Effect Modelling in Fatigue Design of Composite Bridges: An assessment of Fatigue Load Models 3, 4 and 5 according to SS-EN-1991-2 Actions on Structures – Part 2: Traffic loads on Bridges