Report Grupp 11

8/2/2019 Report Grupp 11

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Royal Institute of TechnologyStockholm, 9th of December 2011

AF2203 Bridge Design, Advanced course

Technical ReportDesign of a pedestrian steel arch bridge

Professor: Raid Karoumi

Palmn, Anders820111-7150

[email protected]

Widn, [email protected]


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Table of contentsChapter 1 Introduction .............................................................. ................................................................. ............ 4

1.1 Aim and scope .............................................................................................................................................. 41.1.1 Obligatory requirements ....................................................................................................................... 41.1.2 Optional task ............................................................ ................................................................. ............ 41.1.3 Limitations ............................................................... ................................................................. ............ 4

1.3 The bridge .................................................................................................................................................... 51.3 Rules and regulations ................................................................................................................................... 51.4 Structure of the report................................................................................................................................... 51.6 Properties of the Bridge .............................................................. ................................................................. . 6

Chapter 2 Geometry ...................................................... ................................................................. ....................... 72.1Global geometry ............................................................................................................................................ 72.2 Local geometry ..................................................... ................................................................. ....................... 72.2 Cross sections ............................................................................................................................................... 82.3 Deck plate ............................................................. ................................................................. ....................... 92.4 Foundations .................................................................................................................................................. 92.5 Interactions ................................................................................................................................................... 92.6 The Model in LUSAS................................................................................................................................. 10

2.6.1 Mesh and elements ............................................................................................................................. 10

2.6.2 Local coordinate system ..................................................................................................................... 102.6.3 Material in LUSAS ............................................................. ................................................................ 11

Chapter 3 Loads and material .............................................................. ................................................................ 123.1 Material ...................................................................................................................................................... 123.2 Loads .......................................................................................................................................................... 123.3 Permanent loads ......................................................................................................................................... 12

3.3.1 Self weight ............................................................... ................................................................. .......... 123.3.2 Pavement ............................................................................................................................................ 123.3.3 Omitted permanent loads ............................................................... ..................................................... 13

3.4 Variable loads ....................................................... ................................................................. ..................... 133.4.1 Service vehicle ......................................................... ................................................................. .......... 133.4.2 Breaking force .................................................................................................................................... 143.4.3 Omitted variable loads ........................................................ .............................................................. .. 14

3.5 Load combinations ..................................................................................................................................... 14Chapter 4 Quality assurance........................................... ................................................................. ..................... 16

4.1 Model checking .......................................................................................................................................... 164.1.1 The mass of the model ........................................................ .............................................................. .. 164.1.2 Influence line ...................................................................................................................................... 16


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4.1.3 Load combination ............................................................................................................................... 174.2 Result line ............................................................. ................................................................. ..................... 184.3 Convergence analysis ................................................................................................................................. 18

Chapter 5 Section forces ........................................................... ................................................................. .......... 215.1 Result line ............................................................. ................................................................. ..................... 215.2 Denominations in LUSAS .......................................................... .............................................................. .. 215.3 Members of importance.............................................................................................................................. 215.4 von Mises stresses ...................................................................................................................................... 22

Chapter 6 Resistance verification in ULS ................................................................. ........................................... 236.1 General procedure ...................................................................................................................................... 236.2 Stiffening beam .......................................................................................................................................... 23

6.2.1 General ............................................................................................................................................... 236.2.2 Results and reflections ........................................................................................................................ 24

6.3 Arch ............................................................................................................................................................ 246.3.1 General ............................................................................................................................................... 246.3.2 Results and reflections ........................................................................................................................ 24

6.4 Hangers ...................................................................................................................................................... 256.4.1 General ............................................................................................................................................... 256.4.2 Results and reflections ........................................................................................................................ 25

6.5 Deck beams ................................................................................................................................................ 256.5.1 Generals .............................................................................................................................................. 25

6.5.2 Results and reflections ........................................................................................................................ 25

6.6 Deck plate ............................................................. ................................................................. ..................... 256.6.1 General ............................................................................................................................................... 256.6.2 Results and reflections ........................................................................................................................ 26

6.7 Fatigue resistance ....................................................................................................................................... 276.7.1 General ............................................................................................................................................... 276.7.2 Results and reflections ........................................................................................................................ 27

6.8 Conclusions resistance verification ............................................................................................................ 28Chapter 7 Serviceability ............................................................ ................................................................. .......... 29

7.1 General ....................................................................................................................................................... 297.2 Results and reflections................................................................................................................................ 29

Chapter 8 Optional tasks ........................................................... ................................................................. .......... 308.1 Refined quality assurance ........................................................... ................................................................ 30

8.1.1 General ............................................................................................................................................... 308.1.2 Results and reflections ........................................................................................................................ 30

8.2 LCC-analysis .............................................................................................................................................. 318.2.1 General ............................................................................................................................................... 318.2.2 Results and reflections ........................................................................................................................ 31


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8.3 Frequencies and mode shapes .................................................................................................................... 328.3.1 General ............................................................................................................................................... 328.3.2 Results and reflections ........................................................................................................................ 32

8.4 Global Buckling ......................................................................................................................................... 338.4.1 General ............................................................................................................................................... 338.4.2 Results and reflections ........................................................................................................................ 33

8.5 The influence of the pavement ................................................................................................................... 358.5.1 General ............................................................................................................................................... 358.5.2 Assumptions ....................................................................................................................................... 358.5.3 Results and reflections ........................................................................................................................ 35

Reference list ........................................................................................................................................................ 38Appendix ............................................................................................................................................................... 39

Appendix A: Load calculations ........................................................ .............................................................. .. 39Appendix B: Fatigue resistance ........................................................ .............................................................. .. 42Appendix C: Resistance verification for ULS ....................................................... ........................................... 43


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Chapter 1

Introduction

The objective of this report is to gather a greater understanding of the real procedure of designing and assessing abridge. This will be achieved through performing an assessment of an existing bridge for a new heavier servicevehicle. The geometry of this bridge is well suited for the three dimensional finite element tool LUSAS whichwill be the tool at hand for the solution process of its real behaviour. This introduction part will explain theprinciple and level of the full rapport.

1.1 Aim and scope

The model of the bridge to be studied is presented in chapter 3. The objective is to determine the maximumallowable axel load of a service vehicle passing the bridge. The calculations are obligatory to be performed all

the way just as in a real case. However the number of included loads and load cases are reduced. Further thefinite element model to be created needs to be sufficiently accurate for use in a real design or assessment project.

1.1.1 Obligatory requirements

There are some obligatory requirements of what needs to be included in this report.

A description of the bridge including figures of the main features A three dimensional finite element model of the bridge with a description of the model with figures and

relevant results. A technical report with appropriate assumptions and conditions outlined as well as the result of the design

calculations

Oral presentationAdditional requirements are that the technical report shall also treat the requirements specified in chapter 4 tochapter 7.

1.1.2 Optional task

The optional tasks presented and treated in this report are:

8.1 A refined quality assurance of the bridge model through a 2D model8.2 LCC-analysis8.3 A verification of frequencies, mode shapes and comfort criteria8.4 Global buckling

8.5 Influence of the pavement1.1.3 Limitations

The design is performed as close as possible to the actual constructed bridge as possible. However there aresome limitations introduced to the model and calculations. The limitations introduced are:

The number of included loads has been reduced, for example: no considerations have been given to windloads, uniformly distributed loads, concentrated loads and thermal variations.

The foundation of the bridge will not be designed or considered. Only the longitudinal result line is required, the transversal has been neglected. No verifications due to shear force and torsion are performed in the ULS calculations Only a linear-elastic response will be analysed

Some of these limitations might be considered in the optional tasks.


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1.3 The bridge

The pedestrian bridge to be examined in this project task is located in the Tantolunden in Stockholm, betweenthe commuter train station Stockholm Sdra and the rsta bridge, Figure 1.1.

Figure 1.1: Map displaying the bridge location (Eniro, 2011).

The simply supported steel bowstring arch bridge was constructed 2005 and is a connecting walk path over therailway track. On one side it is connected to the bedrock and on the other it is linked to an inclining walk path at

the position of the concrete foundation. A photo of the bridge is displayed in Figure 1.2 and 1.3.

Figure 1.2: The bridge to be studied, in North-East direction (Leander, 2011)

Figure 1.3: The bridge to be studied, in South-West direction (Eniro, 2011).

1.3 Rules and regulations

The design calculations have been carried out accordingly to certain guidelines given in: Eurocode - EN 1990, (CEN, 2002a) Eurocode 1 - EN 1991-1-1, (CEN, 2002b), EN 1991-2 (CEN, 2003) Eurocode 3 - EN 1993-1-1, (CEN, 2005a), EN 1993-1-7 (CEN, 2009), EN 1993-1-9 (CEN, 2005b), EN

1993-2 (CEN, 2006). Swedish standard TK Bro (Banverket and Vgverket, 2009). In order to understand and adopt different theories and methods for arch structures the book Arch

Structures (Sundquist, 2007) has been used.

1.4 Structure of the report

Below is a brief description of the content of each chapter.

Chapter 2 Geometrydescription of the bridge. Chapter 3 Loads and materialdetailed properties of the bridge and the model.


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Chapter 4 Quality assurancea control of that the 3D model is accurate. Chapter 5 Section forcesoutlines the section forces at the points of interest. Chapter 6 Resistance verification in ULS. Chapter 7 Serviceabilityreviews the performance of the bridge. Chapter 8 Optional taskspreformed optional tasks accordingly to section 1.1.2 above. Reference listoutline of the used books, regulations, codes and other vital documents to this report.

1.6 Properties of the Bridge

A suspended arch bridge is described as a bridge where the bridge deck is connected to the arch at the supports,further the deck is carried by hangers connecting the arch and the stiffening beam in the deck. Also the idea withan arch bridge is that the arch itself should be constructed in such geometry resulting in basically no bendingmoment in the arch member due to the self-weight of the bridge.


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Chapter 2

Geometry

2.1Global geometry

The position of the bridge can be pinpointed through entering the coordinates in Table 2.1. The Swedishcoordinate system SWEREF 99 TM has been used and is based on the Meridian 15 degrees east of Greenwichwith a scale reduction factor of 0.9996. The N-coordinate commence from the Equator and the E-coordinatecommences from the earlier mentioned Meridian, but with an addition of 500 000m. The SWEREF 99 systemsare based on the metric system and conventionally the X-coordinate is positive in the North direction and the Y-coordinates positive in the East direction (Lantmteriet, 2011)

Table 2.1: Coordinates from SWEREF 99 TM (Eniro, 2011)

2.2 Local geometry

The main structural systems are made up by two arches both with curved centre lines, two stiffening beams and14 hangers. The bridge has a span width of 30 m. The global geometry of the bridge is presented in Figure 2.1-2.3.

Figure 2.1: Elevation of the bridge. Dimensions are in meter (Leander, 2011).

The secondary structural system of the bridge is made up of deck beams symbolised with F1- F3 in the figurebelow. On top of the deck beams the deck plate is found with a loading width of 3 meter see Figure 2.3.

X [m] Y [m]

6 578 755 673 640

6 578 722 673 652

SWEREF 99 TM


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Figure 2.2: A plan of the bridge deck with the notation F1- F3 for the deck beams. All dimensions are in meters (Leander,2011).

Figure 2.3: Section of the bridge in mid span. Dimensions are in meter (Leander, 2011).

2.2 Cross sections

As explained earlier the structural members are different depending on where in the structure they are located.Figure 2.4 displays the cross section of the circular cross sections of the arch and the hangers. Figure 2.5

demonstrates the cross section of the stiffening beam. As noted above, the deck beams are denoted F 1- F3 due tothe difference in design. The locations of these beams are presented in the Figure 2.2. The cross sections of thesebeams are presented in Figure 2.6. In order to perform the further calculations in this report the cross sectionproperties has been established, these values are presented in Table 2.1.

Figure 2.4: Cross-section of the arch (a) Figure 2.5: Cross section of the stiffening beamand of the hanger (b) (Leander, 2011) (Leander, 2011).


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Figure 2.6: Cross section of the deck beams F1, F2 and F3 (Leander, 2011).

Table 2.1: Cross-section properties

2.3 Deck plate

The deck plate is 16 mm thick and has the geometry as shown in Figure 2.7. Between the deck beams and thedeck plate there is a 2 mm thick rubber liner. The plate is connected to the beams with sparse bolts, for theconnection details read further under Section 2.5 Interactions. On top of the deck plate the pavement inconstructed out of a 50 mm thick Asphalt concrete layer.

Figure 2.7: Cross section of the deck plate (Leander, 2011).

2.4 Foundations

The bridge foundations will not be needed to be included in the FE-model. Realistic support conditions areneeded in order to get at scenario as close to the real case as possible. At the first support the bearings are

movable and at the seconded one the bearings are pinned. The support conditions are displayed in Figure 2.8.

Figure 2.8: Support conditions (Leander, 2011).

2.5 Interactions

The interactions to consider for this bridge are the connections between the deck plate and the deck beams. Dueto a weak connection between the deck beams and the deck plate, the plate should not be able to contribute to the

main bearing of the bridge. The connection shall be modelled to transmit the vertical forces but not thehorizontal ones. This will be achieved by using the joint elements in LUSAS.

Stiffening beam Arch Hanger Florbeam

Direction unit M1 A1 H6 F2

Cross section area [m2] - *10

-3m

213.05 6.85653 2.3889 10.5

Elastic Moment of inerita [m4] y-y *10

-6m

4320.3 72.63 4.36 244.3

z-z *10-6

m4

85.85 72.63 4.36 34.59Elastic section modulus [m

3] y-y *10

-5m

4177.9 48.66 6.87 135.7

z-z *10-5

m4

52.83 48.66 6.87 28.82


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2.6 The Model in LUSAS

2.6.1 Mesh and elements

This bridge analysis is completed in the finite element software LUSAS Bridge Plus. One of the key factors inderiving a trustworthy and accurate result in finite element software is to make sure the mesh of the model is

working correctly with appropriate element sizes (Pacoste, 2011). This model has been modelled with Thick 3DBeam Element and Thick Shell Elements assigned to the lines and surfaces respectively. It is highly important todivide the mesh for each structural geometry in equal divisions and shapes (Kringos, 2011). Kringos (2011)continues explaining that each element need to be able to correspond to the element it is connected to, in otherwords, there must be a node above another node if the connected elements shall be able to communicate witheach other.

Figure 1.4:To the left, the geometry of the bridge, to the right, the model in 3D views (LUSAS, 2010b).

The mesh used for the surfaces are a square quadrilateral mesh with the element QTS4, for all the Thick 3D

beam components the element BMS3 is used and for the joints between the deck beams and the deck plate JNT4is assigned. In Figure 1.5 to the left, the names of the elements can be reviewed.

2.6.2 Local coordinate system

It has also been of great importance to know how the local coordinate system is located in order to be able tocarry out the quality assurance and compare the forces and moments corresponding to each other. This isdisplayed to the right in Figure 1.5.


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Figure 1.5:To the left, the element types used for the 3D model, to the right, view of the local coordinates (LUSAS, 2010b).

2.6.3 Material in LUSAS

All the members in the 3D model have been assigned a linear-elastic isotropic material with the steel propertiesas described in Table 3.1. An Isotropic material behaves identical in every direction accordingly to Kringos(2011b). The only difference to the structural components is the modification of the deck plate with belongingpavement. The pavement and steel deck plate has been modelled as on solid material where combined densitiesof both have been assigned. This is explained later in section 3.3.2.


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Chapter 3

Loads and material

The design of the railway bridge has been carried out using the Eurocode 1: EN 1991-1-1 (CEN, 2002b) and EN1991-2 (CEN, 2003). The loads applied have been simplified as discussed in the section 1.2 Limitations.

3.1 Material

The material in the bridge is mainly steel with quality S355J2G3 which has the material properties as displayedin Table 3.1. In this table fyis the yield strengthfuis the ultimate limit strength, E.kthe characteristic modulus ofelasticity and is Poissons ratio.

Table 3.1: Material properties for steel grade S355J2G3 (Leander, 2011).

3.2 Loads

The design loads considered for this pedestrian bridge is General actions and Traffic loads. These load cases arespecified in the Eurocode 1: EN 1991-1-1 (CEN, 2002b) and EN 1991-2 (CEN, 2003). The coefficients forcalculating the load combination STR (structural) in ULS are displayed in Table 3.2.

Table 3.2: Loads and load coefficients (Leander, 2011).

3.3 Permanent loads

3.3.1 Self weight

The density for steel is 78kN/m

3

accordingly to table A.1 in EN 1991-1-1 (CEN, 2002b). The self-weight isapplied as a body force with a linear acceleration of 10 m/s2 in the vertical direction.

Figure 3.1: Self weight applied as a body force with the linear acceleration of 10m/s2 (LUSAS, 2010b).

3.3.2 Pavement

The pavement consists of gussasphalt (PGJA) which has a density of 24kN/m3 accordingly to Table A.6 in EN1991-1-1 (CEN, 2002b).


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Since a dynamic analysis of the bridge will be performed the mass of the pavement has been modelled as a co-vibrating mass. This is done through combining the density of the steel and the density of the asphalt andapplying that weight to the thickness of 16 mm of the deck plate see Equation 3.1. This weight is then applied asa body force with a linear acceleration of 10m/s2 Figure 3.2.

Equation 3.1: Weight combination of deck plate and asphalt.

Figure 3.2: Body force of the deck plate. Combination of steel and asphalt (LUSAS, 2010b).

3.3.3 Omitted permanent loads

The bridge is constructed as a simply supported bridge; therefore the loads support dislocation has no effect. Theloads that do not exist in this case are overburden, pre-stressing and pore pressure.

3.4 Variable loads

3.4.1 Service vehicle

The service vehicle to consider is presented in Figure 3.3 with the geometry of the wheel loads presented inFigure 3.4. The contact area of all four wheels are 0,2*0.2 m2 and the applied load is P at each front wheel and

P/2 at each of the rear wheels. A starting value of P has been set to 1 kN in order to calculate the maximumallowed load for the vehicle. Each wheel load has been applied as four discrete point loads with local coordinateswith at the centre of the vehicle

Figure 3.3: Service vehicle configuration (Leander, 2011). Figure 3.4: Wheel geometry and load (Leander, 2011).

In order to find the worst load case the service vehicle has been placed with a maximum eccentricity and appliedas a moving load function in LUSAS along a traffic line running along the x-axis of the bridge Figure 3.5.


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Figure 3.5: The moving load running with maximum eccentricity. The dotted line above the bridge is the traffic line(LUSAS, 2010b).

3.4.2 Breaking force

The load breaking force is set to 60%, Figure 3.6, of the vertical load and is applied in the same manner as thevertical load from the service vehicle, along the traffic line. After this a load combination is preformed to get themaximum values for the service vehicle.

Figure 3.6: Geometry of breaking force

3.4.3 Omitted variable loads

There are some actions that have been excluded in this report. These loads are: uniformly distributed load,concentrated load, wind load, creep and thermal variation. All these loads need to be considered in a real designprocedure. Other loads that are not accounted for are: ice and water flow, snow load and construction loads.

3.5 Load combinations

The governing load combination in the ULS is denoted STR in the Eurocode EN 1990 (CEN, 2002a). The effectof the loads should be combined as

or

The most unfavourable alternative of these two should be used.

Accordingly to Table A2.4(B) in EN 1990 (CEN, 2002a) the -factor shall be set to 0,85.

The different load combinations needed in order to receive a correctly combined result are presented in Table3.3. The setup is valid for the load combination STR and the following abbreviations are used: LTC load cases to


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consider, PLF: permanent load factor, VLF: variable load factor. For the fatigue evaluation, no safetycoefficients should be applied to the load. Use the characteristic values for the service vehicle.

Table 3.3: Load combinations to consider (Leander, 2011).


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Chapter 4

Quality assurance

4.1 Model checking

In order to verify that the created 3D model in LUSAS is correct, important output from the model needs to becompared with hand calculations.

4.1.1 The mass of the model

The hand calculations results in a total mass of the bridge of 40268 kg see Appendix A: Load calculations, whichare compared with the total mass of 40209 kg derived from LUSAS see Table 4.1. Comparing the results, onecan clearly see that this quality assurance successfully passes the check.

Table 4.1: Geometrical summary (LUSAS, 2010b).

4.1.2 Influence line

When comparing the generated influence lines displayed in Figure 4.2-4.3 with the correct influence line, Figure4.4, it clearly states that the theoretical model of the bridge behaves as it is supposed to. The resulting momentMy in Figure 4.2-4.3 is derived due to the fact that the bridge contains two stiffening beams and will therefore

half the correct values from Figure 4.4. This leads to a successful check of the influence lines.

Figure 4.1: Bending moment in the stiffening beam at the seconded hanger.This result is valid for a moving point load of 1kN over the midpoint of the deck plate (LUSAS, 2010b).


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Figure 4.2: Bending moment in the stiffening beam at the seconded hanger.This result is valid for a moving point load of 1kN over the stiffening beam (LUSAS, 2010b).

Figure 4.3: The correct theoretical influence line for the stiffening beam at the secondhanger with a unit point load of 1kN running over the stiffening beam (Leander, 2011).

4.1.3 Load combination

The load combination consists of comparing the maximum hand calculated load, at the specific point where theseconded hanger is attached to the stiffening beam, with the load derived from LUSAS for the same point. Thehand calculations (more extensively presented inAppendix A: Load calculations) are displayed in the equationsbelow where the designing load is the maximum out of the two equations 6.10a and 6.10b from EN 1990, A2(CEN, 2002a).

[ ] ()

The value of 2.352 kNm is compared with the value of 2.352 kNm from LUSAS using the load combinationSTR Fin max, see Table 4.2. Check passed!

Table 4.2: Load combination STR Fin max from LUSAS

Node Fx [N] Fy [N] Fz [N] Mx [Nm] My(*) [Nm] Mz [Nm]

Str max 1008 174673,00 -53,36 4700,03 -0,20 2351,95 2806,28


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4.2 Result line

Deriving results for shell elements are usually trickier than for a beam elements and the result volume is usuallyexcessively larger for shell structures. The Result line is created so that the results can easily be extracted fromexactly the same line with millimetre precision.

The result line for this analysis is placed so that the results are picked at the highest utilized side. In this case it isplaced in the middle of the wheel path for a case where the wheels are operating as close as possible to the sideof the deck plates, Figure 4.4

Figure 4.4: Position of the result line (Leander, 2011).

4.3 Convergence analysisThe theorem of finite element analysis states that a more exact result can be obtained through increasing thenumber of elements of which the model is constructed out of (Kringos, 2011a). In this section an evaluation ofthe element mesh sizes has been carried out in order to obtain the best element mesh size for this project. Theelement used for making the model are Thick 3D Beam Elements and Quadrilateral Thick Shell Elements asexplained in section 2.6.1. However this evaluation is carried out on the deck plate only which consisting of thelater.

In order to make sure that the mesh is correctly applied and that the correct element types and sizes are used aconvergence analysis needs to be performed in order to receive the wanted results. The finer the mesh the moreexact result will be obtained, but also the finer the mesh the more computing power and time spent (Pacoste,2011).

This convergence analysis is preformed of the deck plate where every surface is divided in equal divisions in x-direction and an equal divisions in y-direction with the aim to generate as close to a quadratic formed elementsas possible. Mesh size one: divisions in x=1, y=2. Mesh size two: divisions in x=2, y=4. Mesh size three:divisions in x=2, y=6. How the model is converging is presented in Figure 4.5 4.10. Where the graphs inFigure 4.84.10 are derived from the result line explained in section 4.2.

From the figures below it is evident that the result is not getting a lot more precise after the last simulation due toa finer mesh. Based on this analysis this report will be carried out with the mesh size of the second simulation,displayed in Figure 4.9, with a number of divisions x=2, y=4. This will save both time and computing power butwill not compromise the rehabilitee of the result.


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Figure 4.5: Number of divisions in: x=1, y=2 (LUSAS, 2010b).




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Figure 4.8: My as a graph through 2D through the middle of the deck plate.Number of divisions in: x=1, y=2 (LUSAS, 2010b).




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Chapter 5

Section forces

5.1 Result line

Because of the eccentric loading of the bridge by the service vehicle one side will carry more loads then theother. For the beam elements the results will be picked at the highest utilized side. The result line for the deckplate will be positioned in the middle of the wheel path, Figure 5.1.

Figure 5.1: Longitudinal result line in the deck plate (Leander, 2011).

5.2 Denominations in LUSAS

In LUSAS (2010b) the following notations are used for the sectional forces. In table 5.1 the sectional forces usedfor the design calculations in ultimate limit state are presented.

Fx axial forceFy shear force in the local y-axisF

z

shear force in the local z-axisMx torsion momentMy bending moment about the local y-axisMz bending moment about the local z-axis

5.3 Members of importance

Different cross-sections and members needs to be verified in the bridge construction, this verification is carriedout under Chapter 6. In order to perform this verification the section forces for each position of interest needs tobe extracted from LUSAS (2010b). In Figure 5.2 the positions of interest are displayed. These positions denotedin the figure below have been found to carry the maximum section forces which will be the designing forces andmoments later in Chapter 6. In Table 5.1 there is a notation F1 which indicates the deck beam in which thehighest section forces is occurring. This floor beam is located in the midspan of the bridge connecting to hanger

H4.

Figure 5.2: Notation of important members, the distance a = 8.6m (Leander, 2011).


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Table 5.1: Sectional forces used for the design calculations in ULS.

5.4 von Mises stresses

The verification of the deck plate will be carried out using von Mises stress. von Mises stress are a stressresultant with a direction that might not coincide with any of the axis x, y and z (LUSAS, 2010a). It is simply thehighest value of stress in any direction. With this knowledge von Mises has only been used for computing theresistance in the deck plate. In the other structural members the axial force and the bending moment has beenused through use of Naviers formula in order to find the highest stresses.

Section Fx [kN] My [kNm] Mz [kNm]

A1 495,4 69,2 3,4

M1 488,6 186,9 1

H6 85,9 2,6 15,5

F1 63,3 87,7 5,5


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Chapter 6

Resistance verification in ULS

The resistance for four members of the bridge is needed to be checked in order to verify that the resistance is notexceeding the yield point of the members. The four members are the stiffening beam, arch, one of the hangersand one of the floor beams. Also the stress in deck plate needs to be controlled and delimited.

These calculations will be carried out in the Ultimate limit state and the worst case out of the combinationspresented in CEN (2002a) Eq. 6.10a, 6.10b will be used. The result will indicate the maximum allowed weightof the service vehicle. In Figure 6.1 the notations and sections of the different members which resistance needsto be verified.

Figure 6.1: Member and section notations for the resistance verification (Leander, 2011).

6.1 General procedure

The initial step in the resistance verification procedure is to classify the cross-sections of the different members.

This is done through Table 5.2 in EN 1993-1-1 (CEN, 2005a). Further on the resistance will be calculatedthrough following Eurocode 3 EN1993-1-1 (CEN, 2005a) step by step. The reason for using this part is due to itsgeneral formulation and the fact that it easily can be simplified to fit various combinations of section forces.Finally the verification is made by using the equations 6.61 and 6.62 EN 1993-1-1 (CEN, 2005a), see below. Allthe ultimate limit state calculations are personated in Appendix C: Resistance verification for ULS

Eq. 6.61 (CEN, 2005a)and

Eq. 6.62 (CEN, 2005a)6.2 Stiffening beam

6.2.1 General

The resistance verification of the stiffening beam is carried out at the point M1 in Figure 6.1. The stiffening beamis exposed to both bending and axial tension. Lateral torsional buckling in the stiffening beam is prohibited dueto all the deck beams. The stiffening beam is always subjected to axial tensile forces mainly due to the deadweight. In order to perform this verification Eurocode 3 - EN 1993-1-1 (CEN, 2005a) has been studied andadopted. The criterion for the stiffening beam can be viewed below, where SF1 is the safety factor against failure

and are not allowed to exceed one.


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6.2.2 Results and reflections

When performing the cross section classification stiffening beam accordingly to Table 5.2 in EN 1993-1-1(CEN, 2005a) the assumption that only bending is present has been resulting in the worst case. Doing this itturns out that the web belongs to cross section class 1 and the flange to cross section class 2. Cross section class3 is dealing with both compression and bending moment and are therefore the one used for the resistanceverification of the stiffening. This is resulting in a usage of 44.6% of the total resistance for the stiffening beam,see Table 6.2.2. For the complete ULS calculation see Appendix C: Resistance verification for ULS.

Table 6.2.2: Design section forces and usage of total construction member resistance

6.3 Arch

6.3.1 General

The resistance verification of the arch is carried out at the point A 1 in Figure 6.1. The arch is always subjected toa compressive axial force due to the geometry and the dead weight of the structure. It will also be subjected tobending moments. Since this arch has a hollow circular cross-section it will not be susceptible to lateral-torsionalbuckling. It also can be assumed that the elastic resistance is in cross-sectional class 3. These last two

simplifications are quiet rough but will lead to a more manageable analyse process. There is though a must forchecking the in-plane and out of plane buckling accordingly to EN 1993-1-1 section 6.3.2.1 (CEN, 2005a) andthe critical buckling force is carried out through EN 1993-2 D.3, (CEN, 2006). These criterions are displayedbelow.

SF2y and SF2z cannot exceed 1.0. The reduction factors kyy, kyz, kzy, kzz has been introduced due to interactionbetween the axial force and the bending moment. The factors y, z are introduced due to flexural buckling andLT are dealing with lateral torsional buckling. This last factor is not accounted for in this design due to thelimitations stated in the beginning of the report.


The arch belongs to cross section class 2 but the brutal simplifications of that it is not prone to any torsionalbuckling and belongs to the elastic cross section class 3 is made. For the arch it is vital to check for both in andout of plane buckling which introduces some reduction factors as written about above. Performing thecomputation as presented in Appendix C results in a total usage of the arch resistance of 67.3% and 98.5% for inplane and out of plane buckling respectively. The highest value of these two becomes the designing one for the

arch, see Table 6.2.2.

Stiffening

Arch beam Hanger Desk beam

Fx [kN] 495.4 488.6 85.9 63.3

My [kNm] 69.2 186.9 2.6 87.7

Mz [kNm] 3.4 1,00 15.5 5.5

Usage of r tot esistance [%] 98.5 44.6 80.1 20.9


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6.4 Hangers

6.4.1 General

The resistance verification of the hangers is carried out at for one of the hangers H 1-H7 in Figure 6.1. Thehangers will be subjected to axial tensile force in combination with a bending moment. However, it will not be

susceptible to lateral torsional buckling. The resistance verification is carried out accordingly to Eurocode 3 -EN 1993-1-1 (CEN, 2005a). Where SF3 is the safety factor against failure and is not allowed exceeding the valueof one.


The hanger belongs to cross section class 1 and accordingly to the criteria above the usage of the resistance inthe most effected hanger comes to 80.1%, see Appendix C: Resistance verification for ULS.

6.5 Deck beams

6.5.1 Generals

The resistance verification of the deck beams is carried out at for the deck beam in the middle of the bridge.Since the deck plate will give sufficient restraint to the compression flange it will not be subjected to lateral-torsional buckling. The active forces will be a bending moment in combination with tensile axial forces. Theverification has been carried out through following Eurocode 3 - EN 1993-1-1 Section 6.3.3 (CEN, 2005a).Where SF4 is the safety factor against failure and not allowed to exceed one.


Since the deck beam also has an I-shape as the stiffening beam the same procedure is adopted resulting in a crosssection class 1 for the web and 3 for the flanges. Because the flanges belong to the higher cross section class, it isaccordingly to this class the resistance verification for the deck beam is completed resulting in a total usage of20.9% of its capacity.

6.6 Deck plate

6.6.1 General

The deck plate has been verified through following Eurocode 3 - EN 1993-1-7 (CEN, 2009). In the ultimate limitstate for the bridge the deck plate needs to fulfil the condition:

where and


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The extraction of values for this verification has been derived through using the load P equal to 1kN andmultiplying this with a factor of 40 in order to get an equivalent axial load of 40 ton. The measurements areextracted from a graph through 2D running under the wheel located closest to the centre line of the bridge Figure6.2. The resulting stresses at the bottom of the deck plate are around 47MPa and at the top 48MPa, see Figure

6.3 and 6.4 respectively.

Figure 6.2: Stress at the bottom of the deck plate between two hangers, to the left of the middle hanger (LUSAS, 2010b).

Figure 6.3: Graph displaying the stress at the bottom of the deck plate under the wheel closest to the centre line of the bridge(LUSAS, 2010b).

Figure 6.4: Graph displaying the stress at the top of the deck plate under the wheel closest to the centre line of the bridge(LUSAS, 2010b).


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Using the values derived from LUSAS in the graphs above it becomes evident that:

resulting in a safety factor of approximately 7.4 or a total usage of 13.5% of the deck plates capacity.

6.7 Fatigue resistance

6.7.1 General

A verification of the fatigue resistance of the connection between the sixth hanger, H6 in Figure 6.1, and thestiffening beam has been carried out in order to find the maximum allowed load cycles (load repetitions). Thegeometry of the connection is displayed in Figure 6.5.

Figure 6.5: The geometry of the butt welded hanger connected to the stiffening beam (Leander, 2011).


This verification follows the Eurocode 3 - EN 1993-1-9 (CEN, 2005b) with a selected welding geometryaccordingly to Table 8.6 figure 6 in the same Eurocode. The maximum stress range has been selected in the

element closest to the connection to the stiffening beam. In this element the highest stress value out of the fourfibres, Figure 6.6, has been selected in order to calculate for the worst case scenario.

Figure 6.6: The left displays the element (the black line) chosen displaying max stress. The right displays position of thehighest stress (LUSAS, 2010b).

With the stress range of 240MPa, see Figure 6.7, derived from LUSAS (2010b) the maximum number of yearlyload repetitions comes to 61. This number is derived using the maximum load of P = 60ton. For the fatiguecalculations see Appendix B: Fatigue resistance


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Figure 6.7: The spectrum displaying the stress range for one passage of the vertical load. The red plot is the fibre displayedin ir . an t rn ot is t ir from the chosen fibre (LUSAS, 2010b).

6.8 Conclusions resistance verification

In order to find the maximum allowed vehicle weight or P-load the calculation process needed to start off withan initial P-value of 1kN. Performing the computation of the resistance verification in ULS would indicate howmuch more the P-load could be increased. The P-load was set to 60kN and this value is the load behind theresults in Table 6.2.2 which clearly states that the out of plane buckling for the arch is the designing resistance.

Also a fatigue resistance verification was carried out for the connection between the hanger H6 and the stiffeningbeam with the consequence of a total number of annual passages for a service vehicle with P=60kN to 61 or7440 for the life time of the bridge. This would be more than enough for the maintenance of the bridge.


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Chapter 7

Serviceability

7.1 General

There are no specified limits for vertical deformations in the Eurocode concerning a pedestrian bridge.Nevertheless, there are some limitations in the Swedish publication TK Bro (Banverket and Trafikverket, 2009)stating that the deformation is not allowed to exceed 1/400 of the theoretical span length calculated on thecharacteristic values of the traffic load only.

7.2 Results and reflections

The values derived from the 3D model for the maximum displacement of the stiffening beam is 41.1mm atx=22.55m which is below the criteria stated in TK Bro (Banverket and Trafikverket, 2009), also displayedbelow. Figure 7.1 displays the deformation in the model.

There is however a comfort criteria in the Eurocode 3 EN 1993-2 (CEN, 2006) regarding vibrations. Thesecriteria state that the Eigen frequencies are not allowed to undermine 5 Hz for vertical vibrations and 2.5 Hz forhorizontal (transversal) and torsional vibrations.

Figure 7.1: Max displacement for only the vertical load. 41.1mm (LUSAS, 2010b).


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Chapter 8

Optional tasks

8.1 Refined quality assurance

8.1.1 General

In order complete a further check of the functionality of the 3D model, a refined quality assurance is carried out.This is achieved through creating a two dimensional beam model in LUSAS (2010b), see Figure 8.1.1, and thenapplying the same load as for the 3D model. The resulting section forces from the 2D model are then comparedto the section forces from the 3D model in order to verify the resemblances. Also any differences are discussedand taken into consideration.

8.1.2 Results and reflectionsSome simplifications have been made in order to generate the 2D model which reflects the 3D model and thereal case as good as possible. The same beam components (Thick 3D beams) have been used with the samestiffness as in the 3D model but in order to account for the fact that the 3D model distributes half the appliedload to each side (containing one stiffening beam and one arch) the applied load to the 2D model have been setto half value of the original load, see Figure 3.3. To avoid as many 3D effects as possible when modelling a 2Dcase with Thick 3D Bams Poissons ratio av n st to zro or t matria rortis. Since the P-load hadnot been derived jet upon completion of this task, P was set to 1000N. Furthermore a distributed load of3050N/m was applied across the length of the stiffening beam. This distributed load symbolise the weight of halfthe deck plate and half the deck beams.

Due to the fact that the breaking force applied a moment when using the moving load function in LUSAS thisbreaking force has been neglected in this analysis. Another simplification of the application of the load is that astationary load case has been used positioning the load at the midspan of the bridge, this is valid both for the 2Dand 3D model, see Figure 8.1.1 and 8.1.2.

Figure 8.1.1: Positioning of the load for the 2D simulation (LUSAS, 2010b).

Figure 8.1.2: Positioning of the load for the 3D case (LUSAS, 2010b).

Table 8.1 presents a comparison between the section forces in the 2D and 3D model. The notation A1, M1 andH6 is the positions where these values have been extracted, illustration of these positions can be found in Figure

6.1. It is evident after studying the section forces that all forces and moments are slightly higher in the 3D model.This can be explained through the fact that the vehicle load applied and the weight of the deck plate and deck


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beams, Figure 8.1.2, has some eccentricity resulting in a lever arm which increases the values. One factor toconsider when comparing these values is that the fairly low vehicle load of P=1kN has been used, this load mightnot have a such a big influence on the section forces because it is rather small in comparison with the self-weightof the bridge. Although, it could be reasoned the self-weight itself could result in a measure accurate enough forthe refined quality assurance.

Table 8.1: Comparison between the section forces for the 3D model and the 2D model.

8.2 LCC-analysis

8.2.1 General

The life cycle cost (LCC) is calculated out from an owner perspective with an annual interest rate of 4%. Whichgive the total cost of the bridge over its designed life of 120 year. The bridge is usually only trafficked bywalking people and people on bicycles. It seems unlikely that people crossing the bridge will get seriouslyinjured and that maintenance work will affect the train traffic driving under the bridge. Therefore the costs forthe society and for the users have not been taken into consideration. The initial cost of the bridge can be studiedin the upper part of Table 8.2.1.

( ) 8.2.2 Results and reflections

The expected life span of the bridge,T

, is set to 120 years.C

t is the sum of all costs incurred at timet

and thereal interest rate,p, is set to 4%. In Figure 8.2.2 a graph over the cost in relation to time is displayed and in Table8.2.1 the costs are outlined a bit more detailed.

Contractor and project leader costs are 25% of the total material cost also other unexpected costs are 30% basedon the total material cost. Figure 8.2.1 is displaying the costs as parts of the total life cost of the bridge.

Table 8.2.1: Total net present cost of the bridge

Figure 8.2.1 presents a pie-chart over the total costs at present value incurred over the bridges life time. Studying

this chart it becomes evident that the major costs, 68%, of the total cost for the bridge is incurred during theinitial construction phase. Moreover can be noted that the total cost of 2.08 million SKR seems like a rather lowvalue, but one needs to remember that the limitation of not considering the foundation work for the bridge.

3D 2D 3D 2D 3D 2D

A1 -124,1 -93,6 0,8 0,3 -6,3 -5,0

M1 192,3 143,4 4,1 4,1 -6,7 -6,2

H6 27,7 20,2 0,02 0,9 1,6 1,2

Fx [kN] [kN] My [kNm]

% M ass [kg] Cost [kr/unit ] Tot. Cost [kr] % of total

Steel 29773 30 893 190 kr

Pavement 10436 1.2 12 523 kr

Total Material cost 905 713 kr

Contractor and projec leader cost 25 226 428 kr

Unexpected costs 30 271 714 kr

Tot construction cost 1 403 855 kr 0.676

Intervals Unit Cost [kr/unit] Tot. Cost [kr] % of total

Major Inspection 3 years - 12000 95 128 kr 0.046

Painting 20 years m2

500 181 454 kr 0.087

Exchange of asphalt concrete 40 years m2

11000 235 315 kr 0.113

Rehabillitation of erosion protection 6 years m2

50 5 602 kr 0.003

Cleaning of the drainage system 1 years - 5000 123 870 kr 0.060

Cleaning from vegetation 5 years - 7000 31 955 kr 0.015

Total present value of maintenance 673 324 kr

Total present cost of bridge 2 077 179 kr 1.000

Initialcost

Maintenancecost


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Figure 8.2.1: Chart of total life cost. Figure 8.2.2: Total cost of the bridge compared with time.

Table 8.2.2: Area of the construction components needed to be painted

8.3 Frequencies and mode shapes

8.3.1 General

Accordingly to Section A.2.4.3.2 in CEN (2006), a verification of comfort criteria should be performed if thefundamental frequency of the bridge is less than 5Hz for vertical deformations and less than 2.5Hz for horizontaland torsional vibrations. Here the first ten frequency mode shapes will be evaluated for two different supportconditions, simply supported (pinnedfree) and pinned (pinnedpinned).


Extract the ten first frequency modes turned out that the first three mode shapes does not need to be considered.These mode shapes occurs with varying frequency due to the assumed stiffness in the horizontal directions of thesprings which is symbolizing the connection between the deck plate and the deck beams. This joint stiffness israther flexible and results in three different horizontal translations of the deck, one in x-direction, one in y-

direction and one rotational around the z-axis, displayed in Figure 8.3.1 to the left. Hence these mode shapesdoes not need to be considered.

Figure 8.3.1: To the left, one of the horizontal mode shapes of the deck due to high flexibility in the joints 1.05Hz, middle,first horizontal mode shape 2.73Hz, to the right, the first vertical mode shape 2.89Hz (LUSAS, 2010b).

68%

5%

9%

11%

0%

6%

1%

Life cost

Tot construction cost

Major Inspection

Painting

Exchange of asphalt c oncrete

Rehabillitation of erosion protection

Cleaning of the drainage system

Cleaning from vegetation

Length [m] Quantity Perimeter [m] Area [m2]

Arch 31.54 2 0.94 59.2

Stiffening beam 30.00 2 2.00 120.0Hangers 37.06 1 (tot length of all) 0.40 14.8

Floor beams, F1 4.10 2 0.60 4.1

Floor beams, F2 4.10 11 1.66 67.4

Floor beams, F3 4.10 28 0.93 94.3

Deckplate 30.00 1 3.41 81.4

Sum: 441.1

Constructionme

mbe


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Accordingly to Table 8.3.1 a comfort criteria needs to be performed for the two first vertical mode shapes whenthe bridge have the support condition pinned free in x-translation. All horizontal modes pass the criteriacontrol. When changing the support condition to pinnedpinned the comfort criteria needs to be performed onlyfor the first vertical mode, all others passes the comfort control. Further it is interesting to note that throughchanging the support condition to the later the second vertical mode shape changes from 7 th to 9th positionresulting in a higher frequency and passing the comfort criteria.

Notable is that all these frequencies are for the global structure of the bridge and that the decks frequencies arehighly dependent on the modelled spring stiffness between the deck beams and the deck plate. A furtherevaluation of the influence of the pavement stiffness to the global vibrational modes is performed in optionaltask 8.6.

Table 8.3.1: The ten first mode shapes of the bridge with the support condition pinnedfree

Table 8.3.2: The ten first mode shapes of the bridge with the support condition pinnedpinned

8.4 Global Buckling

8.4.1 General

In this part a global buckling analysis consisting of a verification of the load capacity needs to be performed fordifferent load positions. The influences of the different load positions needs to be compared, not only with eachother but also with the buckling analysis done in the ULS calculations in section 6.3. When exposing the modelof the positioned loads the buckling modes needs to be extracted for the arch. The first vertical mode for the archwill be the first in plane buckling mode and the first horizontal mode for the arch will be the first out of planebuckling mode. From these modes the load factor and the axial force will be extracted, multiplying the tworesults in the total buckling load for the arch.

8.4.2 Results and reflectionsThis analysis has been carried out with a Thick nonlinear 3D beam elements, and the maximum vehicle load ofP=60kN concluded under section 6.0. The positions of the load have been set to L/2, L/3 and L/4. For each of

Mode shape Frequency [Hz ] Kind of mode f1,vert > 5Hz f 1,horz > 2.5Hz

1 1.05 Deck

2 1.13 Deck

3 1.20 Deck

4 2.73 1st

Horizontal OK

5 2.891

stVertical

Not6 4.17 Arch Horizontal -

7 4.48 2nd

Vertical Not

8 4.55 Torsional OK

9 4.88 2nd

Horizontal OK

10 5.83 3rd

Horizontal OK

Not considered

Not considered

Not considered

Comfort criteria

Support:Pin

ned-Free

Mode shape Frequency [Hz ] Kind of mode f1,vert > 5Hz f 1,horz > 2.5Hz

1 1.05 Deck

2 1.13 Deck

3 1.20 Deck

4 2.77 1st

Horizontal OK

5 2.89 1st Vertical NOT

6 4.17 Arch Horizontal -

7 4.55 Torsional OK

8 4.92 2nd

Horizontal OK

9 5.10 2nd

Vertical OK

10 5.83 3rd

Horizontal OK

Comfort criteria

Support:Pinned-Pinned

Not considered

Not considered

Not considered


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these load positions the first 200 buckling modes have been extracted and examined in order to find the first in-and out of plane buckling. These modes have then been studied further in order to find the load factor and thehighest axial force, two of the first in- and out of plane buckling modes are displayed in Figure 8.4. Thecalculations for the buckling are presented in Table 8.4.1.

Table 8.4.1: Calculation of the buckling forces for each buckling mode

It is interesting that the buckling force for the out of plane is decreasing as the load is moved closer to either sideof the bridge, the reason for this might be due to the fact that a larger deflection is occurring in the middle of thebridge. The opposite is displayed for the in plane buckling force which is increasing as the load is movedtowards the either side of the bridge, a reason for this might be that the vehicle load is closing in on theconnection between the arch and the stiffening beam resulting in a higher transferred buckling load to the arch.The change in buckling load due to vehicle load position is displayed in Table 8.4.1.

Onwards a comparison is made with the values derived from the analysis of the buckling forces in ULS for thearch. Examining the Table 8.4.2 it becomes evident that the buckling loads from LUSAS are slightly higher thanthe values calculated accordingly to the Eurocode EN 1993-2 (CEN, 2006). A motivation for this might be thatin the ULS calculations the simplification of assuming the cross-section to be in cross-section class 3 is notxacty t cas. T arcs cross-section might belong to class 2 resulting in a higher resistance.

This buckling load control is highly interesting due to the fact that it verifies the statement in the ULS

calculations in section 6.0 which states that it is the out of plane buckling force that will result in the governingdesign values.

Table 8.4.2: comparison between buckling force derived from LUSAS and calculated accordingly to Eurocode

Position Mode Eigenvalue Load factor Axial force

[kN]

Buckling force

[kN]

Out of plane L/2 7 6.83 6.83 157.5 1 076

L/3 9 6.90 6.90 159.8 1 103

L/4 7 7.48 7.48 157.4 1 177

In plane L/2 72 30.03 30.03 106.0 3 183

L/3 84 30.30 30.30 160.1 4 851

L/4 77 32.48 32.48 288.3 9 363

Model Check Buckling force

[kN]

From Lusas Out of plane 1076.0

In plane 3183.5

Eurocode Out of plane 1045.3

In plane 3782.9


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Figure 8.4: Left displays the first in plane buckling mode. Right displays the first out of plane buckling mode (LUSAS,2010b).

8.5 The influence of the pavement

8.5.1 General

The task here is to investigate the influence of the stiffness of the pavement and determine the naturalfrequencies and mode shapes for the bridge. This is done through varying the stiffness of the pavementaccordingly to the variance of the temperature. The temperature ranges from -20C to 20C with steps of 10C.This analysis will examine the ten first mode shapes.

8.5.2 Assumptions

In orr to carry ot sc an anaysis t Yons mos or t ssasat a to stima ted for eachtemperature step. The reference value for this estimation is derived from Figure 8.5.1. The E-modulus isassumed to increase linearly with the value of 4GPa per every 10C, starting at 4GPa at 20C. The Poisons ratiohas been set to a value that also varies linearly between 0.35 and 0.2 for 20C and -20C respectively. A fewsimulations in LUSAS (2010b) quickly showed that Poisons ratio ad a minor influence on the naturalfrequencies, it will therefore not be investigated further due to the lack of time. Furthermore, the joint betweenthe deck beams and the deck plate has been modelled with a vertical stiffness of 100GPa and 10GPa in the twohorizontal directions.

Figure 8.5.1: Resilient modulus for asphalt at different temperatures (Erlingson, 2010)

8.5.3 Results and reflectionsIt becomes evident after studying the frequencies that the joint has a great impact on the influence of thepavement stiffness. A stiff joint results in a higher influence depending on the stiffness of the pavement and as


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expected if the joint is modelled with a very flexible behaviour the influence of the pavement stiffness convertsagainst having no influence at all, which of course is not the case in the real world. In Table 8.5 the differentnatural frequencies are presented and evaluated. The letters indicates the kind of mode shape, vertical,horizontal, torsional, arch horizontal or a combination of a few. Only the vertical and the ones containing ahorizontal component are analysed further, therefore the others have been crossed out. The comfort criterionstates that the vertical frequencies need to exceed 5Hz and the horizontal 2.5Hz (CEN, 2006). Comparing theresult in Table 8.5 results in the conclusion that the first vertical mode shape does not meet the criterion, but thefirst horizontal is conceded. This is displayed with the red and green colour in Table 8.5. Sinc t Yonsmodulus of the asphalt is assumed to vary linearly it seems like the natural frequencies follow the samerelationship with a linear increase as the temperature decreases to -20C.

Conirmin t assmtion o t varianc in Yons mos co on tro incrasin t natra

frequency of the simulation for 20GPa with f equal to 0.009Hz for every 4GPa up to 100GPa. This wouldresult in a natural frequency of 3.251Hz which is close to the simulated one of 3.214 for 100GPa. Therefor theassumption is valid and accurate enough for the real case situation.

Table 8.5: The variation of the natural frequencies with variation of the stiffness of the asphalt. The green indicates a passingthe comfort criterion and the red failing the comfort criterion.

The three columns to the right, 100GPa, Real model and Only steel deck plate is for comparison. The steel deckplate one is with just the deck plate without anything on it, the Real model has a combination of the density forsteel and asphalt combined and assigned to the thickness of the 16mm (as explained in section 3.3.2 of thisreport) and 100GPa is the true value for the pavement stiffness at -20C. Now in hindsight it is clear that the realmodel is a rather good approximation of the pavements influence on the vibration performance of the bridge, thisby studying Table 8.5. Even if it is slightly off, the difference is on the safe side and results in a worse case with

the vertical and horizontal frequencies lower than the real ones. This will of course make it harder to pass thecomfort criterion. In Figure 8.5.2 the variation of the influence of the first vertical and horizontal mode shapes isdisplayed.

Since all horizontal mode shapes in this simulation is passing the comfort criteria the important once becomesthe vertical mode shapes. The first mode shape is always a vertical mode, but it is interesting to study the trendof the second vertical mode shape which is moving up in positions as the stiffness of the pavement is increasing.At the temperature of -20C and a stiffness of 100GPa the second vertical mode shape is still failing the comfortcriteria, but just failing so it could be assumed that if the stiffness was increased further to approximately 140GPa the second vertical mode shape would also pass the comfort criteria.

Mode

1 V 3.032 V 3.044 V 3.053 V 3.062 V 3.071 V 3.214 V 2.897 V 3.368

2 AH 3.170 AH 3.296 AH 3.380 AH 3.446 AH 3.499 AH 3.804 AH 3.960 AH 3.979

3 AH 4.174 AH 4.176 AH 4.176 AH 4.177 AH 4.178 AH 4.185 AH 4.176 AH 4.177

4 T 4.626 T 4.660 T 4.690 T 4.719 T 4.747 V 4.924 T 4.658 T 4.859

5 H+AH 4.721 V 4.757 V 4.771 V 4.783 V 4.793 T 5.255 V 4.686 V 5.485

6 V 4.738 H+AH 4.832 H+A.H 4.928 H+AH 5.020 H+AH 5.111 H+T+AH 6.290 V 6.949 AH 7.628

7 V 7.008 V 7.058 V 7.094 V 7.124 V 7.150 V 7.518 AH+T+H 7.542 AH+T 8.030

8 AH 7.625 AH 7.627 AH 7.629 AH 7.630 A .H 7.631 AH 7.641 AH 7.628 T 8.177

9 T+AH 7.692 T+AH 7.765 T+A.H 7.795 T+AH 7.814 T+AH 7.828 T+AH 7.941 T+AH 7.995 V 8.179

10 T 7.789 T 7.824 T 7.853 T 7.880 T 7.908 T 8.389 T 9.407 T+H 10.560

deck plate4 Gpa 8 Gpa 12 Gpa 100GPa AC+Steel 210GPa

Only steel

Pavement modeled Real model

V=Vertical mode

H=Horizontal mode

AH=Arch horizontal

T=Torsional

16 Gpa 20 Gpa

= passing the comfort criteria

= failing the comfort criteria


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Figure 8.5.2:To the left, the first vertical vibration mode for 10C. To the right, the first horizontal (antisymetrical) vibrationmode for 10C (LUSAS, 2010b).


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Reference list

Banverket, Vgverket, 2009. TK Bro. Banverket & Vgverket, ISSN: 1401-9612.

CEN, 2002a.Eurocode - Basis of structural design . CEN, EN 1990.

CEN, 2002b.Eurocode 1 Actions on structures Part 1-1: General actions. CEN, EN 1991-1-1

CEN, 2003.Eurocode 1 Actions on structures Part 2: Trafic loads on bridges. CEN, EN 1991-2

CEN, 2005a.Eurocode 3 - Design of steel structures - Part 1-1: General rules and rules for buildings. CEN, EN1993-1-1.

CEN, 2009.Eurocode 3 - Design of steel structures - Part 1-7: Strength and stability of planar plated structuressubject to out of plane loading. CEN, EN 1993-1-7.

CEN, 2005b.Eurocode 3 - Design of steel structures - Part 1-9: Fatigue. CEN, EN 1993-1-9.

CEN, 2006. Eurocode 3 - Design of steel structures - Part 2: Steel bridges. CEN, EN 1993-2.

Eniro. 2011. Kartor.http://kartor.eniro.seAccess 2011-12-05.

Erlingsson, S., 2010.Lecture: Material characterization AF2901. KTH. 2010-11-19.

Karoumi, R., 2011. Whole life costing of bridges. Stockholm: KTH.

Kringos, N., 2011a.Lecture: Modeling of pavement structures. AF2024. KTH. 2011-10-04.

Kringos, N., 2011b.Lecture: Intro to continuum mechanics. AF2024. KTH. 2011-10-04.

Lantmteriet. 2011. Tvdimensionella system SWEREF 99, projektioner.www.lantmateriet.se/templates/LMV_Page.aspx?id=4219Access 2011-12-05.

Leander, J., 2011. Project task: Assessment of a steel arch bridge . Stockholm: KTH.

LUSAS, 2010a.Application Examples Manual (Bridge, Civil and Structural) . Lusas version 14.5 Issue 1.

LUSAS, 2010b.LUSAS Bridge Plus software. Lusas version 14.5-.

Pacoste, C., 2011a.Lecture: Isoparametric formulation AF2024. KTH. 2011-09-08.

Sundquist, H. 2007.Arch structures. Technical Report 107. Stockholm: KTH.
http://kartor.eniro.se/http://kartor.eniro.se/http://kartor.eniro.se/http://www.lantmateriet.se/templates/LMV_Page.aspx?id=4219http://www.lantmateriet.se/templates/LMV_Page.aspx?id=4219http://www.lantmateriet.se/templates/LMV_Page.aspx?id=4219http://kartor.eniro.se/


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Hangers

Lh 37m This length is derived from LUSAS and is thtotal length of all the hangers.

Ah 0.002389m2 Cross sectional area

Vh Lh Ah 0.088m3

Volume for all the hangers

Deck beams F 1, F2 and F3

Ldb

4.1

4.1

4.1

m Length

Adb

0.0029

0.0105

0.003877

m2

Cross sectional area

Number of beams of each kindNo

2

11

28

Vdb No Ldb Adb

0.024

0.474

0.445

m3

Volume of all the deckbeams

Vdb 0.942m3Deck plate

td 0.016m thickness of the deck plate

Ad 0.05024m2


Ld 30m Length

Vd Ad Ld 1.507m3

Volume

Self weight of the bridge

Vself Vsb Va Vh Vd Vdb 3.764m3 The volume of all the steel in the bridgeWself Vself steel 293.559kN

Pavement

tp 0.05m thickness of the deck plate

Ap 0.1415m2


Lp 30m Length


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41

Vp Ap Lp 4.245m3

Volume

Wp Vp AC 101.88kN

Wbridge Wself Wp 395.439kN Weight of the total bridge

MassWbridge

g40269kg

Combination of the density of the pavement and deck plat

comb

steel td AC tp

td

153kN

m3

Check of Load combinations

g.j 1.35 0.85

Gk.j 1742.18N m self weigt including pavement, value taken from LUSAS

Qk.i 0.77N m traffic load, taken from LUSAS

M

g.j Gk.j

g. j Gk.j g. j Qk.i

2.352

2

kN m

max M( ) 2.352kN m


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42

Appendix B: Fatigue resistance

Appendix B: Fatigue res istance

In data

Steel quality S355

fy 355MPa Yield strength

r 0.0635m Radius

t 0.0063m Thickness of steel

A r2

r t( )2

2.389 103

m2


Partial coefficients

d 1.0 safety class 3, very sever damage

Mf 1.35 safe life

Ff 1.0 partial factor for fatigue

M0 1.0

d Mf Ff 1.35

n1 1 Number of cycles of stress range

Check over the butt weldStress range at the butt welded attachment, value derived

from LUSAS 240MPa

c 50MPa from Table 8.6, figure 6, if this is not the correct one, itis anyway the worst case for the weld.

D 0.737 c 36.85MPas

D= the endurance limit at constant stress range

L 0.549 D 20.231MPa s L = endurance limit at variable stress range

D OK! Criteria

i 324 MPa From the failure criterion one obtains

NR

c

i

3

2 106

7.4 103

Number

Dn1

NR

1.36 104

Accumelated damage

nevents1

D7.35 10

3 Number of events

nevents

12061.252 Number of events per year.


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43

Appendix C: Resistance verification for ULS

Appendix C: Resistance verification for ULS

General equation

Characteristic values M1 1.0fy 355 MPa

E 210 GPa

235 10

6 Pa

fy

0.814

Moment of inertia

Iy.el b h d t( )b h

3

12

b d( ) h 2 t( )3

12

Iz.el b h d t( ) 2t b

3

12

h 2 t( ) d3

12

Elastic section modulus for I-beams

Wy.el Iy.el h Iy.el

h

2

Wz.el Iz.el b Iz.el

b

2

NRkA( ) fy A [N]

Moment Resistance in section class 1 and 2

My.Rk.plWy.pl Wy.pl fy

Mz.Rk.plWz.pl Wz.pl fy

Moment Resistance in section class 3

My.Rk.elWy.el Wy.el fyMz.Rk.elWz.el Wz.el fy


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44

1. Stiffening beam

b1 0.325m With [m]

h1 0.360mHeigth [m]

d1 0.010m Web thickness [m]

t1 0.015m Flange thickness [m]

A1 0.01305m2

Cross section area [m2]

I1.y.el Iy.el b1 h1 d1 t1 3.203 104

m4

Moment of inertia, y-direction [m4]

I1.z.el Iz.el b1 h1 d1 t1 8.585 105

m4

Moment of inertia, z-direction [m4]

Only assuming bending when this results in the worst case.

S ection class for the web. EN 1993-1-1 Table 5.272 58.58 Cross section class 1

83 67.53 Cross section class 2


S1

h1 2 t1

d1

33

S1 72 The web belongs to cross section class 1

Cross section class for the flange. EN 1993-1-1 Table 5.2




S1

b1

b1 d1

2

t1

11.167

10 S1 14

The flange belongs to cross section class 2 but almoust cross section

class 3. Cross section class 3 is the worst case and therefore the

calculations are done in cross section class 3.

Elastic section modulus

W1.y.el Wy.el I1.y.el h1 W1.y.el 1.779 106

mm3

W1.z.el Wz.el I1.z.el b1 W1.z.el 5.283 105

mm3


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45

Resistans for uniform compression

N1.Rk NRkA1 N1.Rk 4.633 103

kN

Resistance moment

M1.y.Rk My.Rk.el W1.y.el M1.y.Rk 631.612 kN m

M1.z.Rk Mz.Rk.el W1.z.el M1.z.Rk 187.544 kN m

Design values for compression force and maximum moments from LUSAS

N1.Ed1C:\...\Forces and moment s from Lusas.xls

N1.Ed N1.Ed1 N

M1.y.Ed1C:\...\Forces and moment s from Lusas.xls

M1.y.Ed M1.y.Ed1 N m

M1.z.Ed1C:\...\Forces and moment s from Lusas.xls

M1.z.Ed M1.z.Ed1 N m

Formula 6.2 EN 1993-1-1

SF1

N1.Ed

N1.Rk

M1

M1.y.Ed

M1.y.Rk

M1

M1.z.Ed

M1.z.Rk

M1

Maximum value from equation 6.2

SF1.max max SF1 0.446

4. Floor beam

b4 0.240 m With [m]

h4 0.360 m Heigth [m]

d4 0.010 m Web thickness [m]

t4 0.015 m Flange thickness [m]

A4 0.0105 m2

Cross section area [m 2]

I4.y.el Iy.el b4 h4 d4 t4 2.443 104

m4

Moment of inertia, y-direction [m4]

I4.z.el Iz.el b4 h4 d4 t4 3.459 105

m4

Moment of inertia, z-direction [m4]


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46

Section class for the web. EN 1993-1-1 Table 5.2




S1

h4 2 t4

d4

33

S1 72 The web belongs to section class 1

Cross section class for the flange. EN 1993-1-1 Table 5.2




S4

b4b4 d4

2

t4

8.333

10 S4 14 The flange belongs to section class 3

Because the flange belongs to section class 3 all calculations for the stiffening beam are made in

section class 2.

Elastic section modulus

W4.y.el Wy.el I4.y.el h4 W4.y.el 1.357 106

mm3

W4.z.el Wz.el I4.z.el b4 W4.z.el 2.882 105 mm3


N4.Rk NRkA4 N4.Rk 3.728 103

kN

Resistance moment

M4.y.Rk My.Rk.el W4.y.el M4.y.Rk 481.868 kN m

M4.z.Rk Mz.Rk.el W4.z.el M4.z.Rk 102.321 kN m



N4.Ed N4.Ed4 N



M4.z.Ed4

C:\...\Forces and moment s from Lusas.xls



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47

Formula 6.2 EN 1993-1-1

SF4

N4.Ed

N4.Rk

M1

M4.y.Ed

M4.y.Rk

M1

M4.z.Ed

M4.z.Rk

M1

Maximum value from equation 6.2


Circular sections

General equations

Moment of inertia for circular hollow sections

Ic d di 64d

4di

4

Elastic section modulus for circular hollow sections

Wc.el d di 32d

4di

4

d

dA 0.2985 mdAi 0.2835 m

tA 0.0075 m

AA 0.00685653 m2

IA Ic dA dAi 7.263 105

m4

LA 31.54 m

50 2 33.09970

2 46.338

902

59.577

S2

dA

tA

39.8

502

S2 702

2 Arch

Outer diameter [m]

Inner diameter [m]

Thickness of steel [m]

Cross section area [m 3]

Moment of inertia [m4]

Length of arch [m]

Section class for the arch

Cross section class 1




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48

W2.el Wc.el dA dAi W2.el 4.866 104

N2.Rk NRkAA N2.Rk 2.434 103

kN

M2.y.Rk My.Rk.el Wc.el dA dAi M2.y.Rk 172.744 kN m

M2.z.Rk Mz.Rk.el Wc.el dA dAi M2.z.Rk 172.744 kN m


N2.Ed N2.Ed2 N





y1C:\...\Forces and moment s from Lusas.xls

y y1 m

z1C:\...\Forces and moment s from Lusas.xls

z z1 m

MEd.max max M2.y.Ed M2.z.Ed MEd.max 69.21 kN m

max max y2

max z2

max 63.63239 mm

The arch belongs to section class 2. But after a brutal simplification

the arch is calculated in cross section class 3.


Resistance moment


The maximum displacements in y- and z direction in the arch

Maximumn moment that acting in the arch

Resultant of the the maximum displacements


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49

3 Hanger

dH 0.127 m Wc.el d di 32d

4di

4

d

dHi 0.1144 m

tH

0.0063 m

AH 0.0023889 m2

IH 0.00000436218 m4

M3.y.Rk My.Rk.el Wc.el dH dHi M3.y.Rk 24.387 kN m

M3.z.Rk Mz.Rk.el Wc.el dH dHi M3.z.Rk 24.387 kN m

N3.Rk NRkAH N3.Rk 848.059 kN


N3.Ed N3.Ed3 N





Section class for the hanger50


702

46.338 Cross section class 2

902

59.577 Cross section class 3

S3

dH

tH

20.159

S3 502

The Hanger belongs to section class 1

According to Eurocode CEN 6.3.2.1

SF3

N3.Ed

N3.Rk

M1

M3.y.Ed

M3.y.Rk

M1

M3.z.Ed

M3.z.Rk

M1



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50

Reduction factor due to flexural buckling

Formulas (6.49) EN 1993-1-1

( )1

2 2

Reduction factor due to flexural buckling

( ) 0.5 1 0.2( )2

A Ncr A fy

Ncr

Cold formed hollow section gives the worst case of the imperfection factor . Table 6.

in EN 1993-1-1

0.49 Table 6.1 in EN 1993-1-1

Formula for the uniform moment factor Cmi

Cm L NEd Ncr 12

E IA max

L2

MEd.max

1

NEd

Ncr

Table A.2 EN 1993-1-1

Interaction factor

ki Cm NEd Ncr Cm

1NEd

Ncr

Table A.1 EN 1993-1-1

In plane bucklingPICTURE 6.2

Critical buckling force. Formula D.3 EN 1993-2

Ncr.y s I( )s

2

E I

s2

LA

2 Half length of the arch s2 15.77m

n 7 Number of hangersf 4.2m Height of arch

L

2

30 m Projected length of the arch

f

L2

0.14 actor n or

0.4 Buckling length factor

Ncr.y2 Ncr.y s2 IA Ncr.y2 3.783 103

kN

y AA Ncr.y2 y 0.802

y y y 0.969

y y y y 0.661

Cmy Cm L2 N2.Ed Ncr.y2

kyy ki Cmy y N2.Ed Ncr.y2


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Out plane buckling

Critical buckling force. Formula D.4 EN 1993-2

Ncr.z L I( )L

2

E I

Ncr.z2 Ncr.z L2 IA Ncr.z2 1.045 103

k

z AA Ncr.z2 z 1.526

z z z 1.989

z z z z 0.306

Cmz Cm L2 N2.Ed Ncr.z2

kzz ki Cmz z N2.Ed Ncr.z2

Equation 6.61 En 1993-1-1 for in plane buckling

SF2y

N2.Ed

y

N2.Rk

M1

kyy

M2.y.Ed

M2.y.Rk

M1

kzz

M2.z.Ed

M2.z.Rk

M1

Equation 6.62 En 1993-1-1 for out of plane buckling

SF2z

N2.Ed

zN2.Rk

M1

kyy

M2.y.Ed

M2.y.Rk

M1

kzz

M2.z.Ed

M2.z.Rk

M1

SF2y.max max SF2y 0.673

SF2z.max max SF2z 0.985

SFmax max SF1.max SF2y.max SF2z.max SF3.max SF4.max 0.985

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Report Grupp 11