The performance of conventional discrete torsional bracings in steel-concrete composite bridges: a survey of Swedish bridges

Transcription

1 The performance of conventional discrete torsional bracings in steel-concrete composite bridges: a survey of Swedish bridges Oscar Carlson and Lukasz Jaskiewicz Avdelningen för Konstruktionsteknik Lunds Tekniska Högskola Lunds Universitet, 2015 Rapport TVBK

2 Avdelningen för Konstruktionsteknik Lunds Tekniska Högskola Box LUND Division of Structural Engineering Faculty of Engineering, LTH P.O. Box 118 S LUND, Sweden The performance of conventional discrete torsional bracings in steel-concrete composite bridges: a survey of Swedish bridges Master's thesis by: Oscar Carlson and Lukasz Jaskiewicz Supervisor: Hassan Mehri, PhD candidate Div. of Structural Engineering Examiner: Roberto Crocetti, Prof. Div. of Structural Engineering Rapport: TVBK-5241 ISSN: ISRN: LUTVDG/TVBK-15/5241+(110p)

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4 Acknowledgements This Master s thesis was written at the Division of Structural Engineering at Lund Institute of Technology in corporation with Reinertsen Sweden AB. Roberto Crocetti (Prof.) from the above mentioned department was examiner of this thesis. We wish to express sincere appreciation to all individuals who have offered support, inspiration and encouragement during the course of this research. Special gratitude is extended to our supervisors: Hassan Mehri (PhD candidate) and Fredrik Carlsson (Reinertsen Sweden AB) for generously offering their time and good will throughout the preparation and evaluation of this document. Without their guidance and help this study would never have matured. We further wish to express gratitude to our families for their patience and motivational talks that kept us going throughout the course of this project. Oscar Carlson Lukasz Jaskiewicz Lund 2015 iii

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6 Abstract The torsional bracing system is a fundamental part of a bridge structure that provides torsional restraint to the steel girders and prevents lateral-torsional buckling of the main girders during construction when no lateral restraint, in form of the continuous concrete deck, is yet provided to the compressive flanges. This paper investigates the performance of conventional discrete torsional braces of seven randomly chosen Scandinavian steel-concrete composite bridges. Geometry of the bridges and type of torsional bracing systems utilized to control the twist of the cross section is first presented. Chosen calculation methods for lateral-torsional buckling of discretely braced beams are then comprehensively described. Obtained critical buckling moments are discussed in detail and the differences between the presented methods are explained and compared. The accuracy of the approaches is then compared with finite element method used to investigate the exact buckling behavior of the bridges. As a direct consequence of the obtained results, a separate analysis concerning the cross sections of multi-span bridges is done where the dimensions of the cross sections are reduced and buckling behavior of the beams studied. Finally, a comparative study of the exact solutions presented in this paper and numerical approach is done in order to find the source of error between the two methods. Suggestions concerning bridge geometry are presented by the authors to make the exact solutions even more reliable. Keywords: Conventional torsional bracing, composite bridges, Eurocode v

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8 Notation Abbreviations Eurocode 3 FE FEA FEM LTB SLS ULS UDL EN :2005 Finite element Finite element analysis Finite element method Lateral torsional buckling Serviceability limit state Ultimate limit state Uniformly distributed load Roman symbols A Area of compression flange Bottom flange area Top flange area Area of diagonal brace members Area of horizontal brace members Bottom flange width Total width of vertical web stiffener Top flange width Girder height c Distance from compression flange centroid to neutral axis Factor that allows for the shape of the bending moment diagram Parameter associated with the load level Moment diagram modification factor of unbraced beam vii

9 Moment diagram modification factor of braced beam Torsional warping constant d D Distance between top flanges in box-girders Destabilizing parameter Web depth e é Distance between shear center and bottom flange Distance between shear center and neutral axis Modulus of elasticity Yield strength Shear modulus Distance between flange centroids Distance between flanges in box-girders Height of cross frame Girder total height Moment of inertia of torsional brace Strong axis moment of inertia Weak axis moment of inertia Weak axis moment of inertia of compression section Weak axis moment of inertia of tension section Weak axis effective moment of inertia St. Venant s torsional constant k Effective length parameter Warping restraint parameter Span length Distance between torsional braces Length of diagonal brace members viii

10 Lateral-torsional buckling moment of single girder Lateral-torsional buckling moment of torsionally braced twin girder system Global buckling moment of twin girder system Design buckling resistance of beam Euler s critical buckling load S t Spacing of girders Distance from tension flange centroid to neutral axis Bottom flange thickness Vertical web stiffener thickness Top flange thickness Web thickness U V Parameter that depends on sections geometry Parameter related to slenderness Section modulus Plastic section modulus about strong axis Distance between center of gravity of box-girder cross section and uppermost point of the top flange Distance between center of gravity of box-girder cross section and bottom point of the bottom flange Distance between level of load application and shear center Greek symbols Imperfection factor with regard to lateral-torsional buckling Torsional brace stiffness In plane flexibility of girders Web distortional stiffness Torsional stiffness expressed for one brace ix

11 Torsional stiffness expressed for continuous bracing system Torsional brace stiffness expressed per meter Parameter that allows for the classification of the cross section Non-dimensional slenderness parameter Minor axis non-dimensional slenderness parameter Buckling factor with regard to lateral-torsional buckling x

12 Table of Contents 1. Introduction and background Lateral torsional buckling of beams Torsional bracing of beams Methods Numerical analysis method Modified Euler s column buckling formula Simplified method according to NCCI SN Case studies Single span bridges Bridge over Upperuds River, Götaland (Bridge 1530) Bridge over Ore River, Kopparberg (Bridge 1020) Bridge over Vanån River, Dalarna County (Bridge 983) Multi-span bridges Bridge over E6 highway, Götaland (Bridge 1385) Bridge over Sält River, Uddevalla-Svinesund (Bridge 1768) Bridge over Motala River, Östrergötaland county (Bridge 917) Bridge over Vallsund, Jämtland County (Bridge 1052) Results Single span bridges Multi-span bridges Comparison of results Discussion and conclusions Bridge design Calculation methods Conclusions Examination of multi-span bridges Introduction Methods Calculations Bridge Bridge Results Discussion and conclusions Discussion Conclusions xi

13 7. Parametric study Comparison of FEM and Equations (1.1) - (1.3) Introduction Methods Results Brace number and distance variation Brace stiffness variation Cross section variation Distance between girders Discussion and conclusions References Appendix A - Brace stiffness A.1. Bridge A.2. Bridge A.3. Bridge A.4. Bridge A.5. Bridge A.6. Bridge A.7. Bridge Appendix B - Bracing location B.1. Bridge B.2. Bridge Appendix C - Critical bending moment of Bridge C.1. Critical bending moment according to Eq. (1.1), (1.2) and (1.3) C.2. Modified Euler s column buckling formula C.3. Simplified method according to NCCI SN Appendix D - Critical bending moment of Bridge D.1. Critical bending moment according to Eq. (1.1) - (1.3) D.2. Modified Euler s column buckling approach D.3. Simplified method according to NCCI SN xii

14 1. Introduction and background Slender structural members subjected to bending loads about the strong axis of the section may deform laterally and twist, a phenomenon known as lateral-torsional buckling. As a result, the cross section capacity of a deformed member can be reached long before the full plastic resistant moment has developed. According to current design criteria used in Europe, the slenderness of a section should be examined by estimating the elastic critical moment of the loaded member, however; no direct method for the calculation of the critical bending moment is provided. This thesis presents different approaches for estimation of the critical bending moment of slender members braced by discrete torsional braces as well as accounts for differences between these approaches Lateral torsional buckling of beams Lateral torsional buckling (LTB) of an I-beam is a failure mode that takes place when compression flange becomes unstable. When a beam experiences LTB a lateral out-of-plane movement between beam flanges as well as its twist occurs generating a torque about the shear center of laterally deflected beam, as shown in Figure 1. δ A Center of twist SECTION A-A A Figure 1 Geometry of buckled beam (Yura, 2001) Timoshenko and Gere (1961) provided the following equation for the elastic critical buckling moment of unbraced doubly-symmetric beam subjected to uniform moment. The formula can also be used for calculation of the local buckling mode in which compression flange buckles between torsional braces: (1.1) where moment diagram modification factor; distance between points along the length where twist is prevented; modulus of elasticity; weak axis moment of inertia; shear modulus; torsional constant; and warping constant. 1

15 Taylor and Ojalvo (1966) presented a solution for doubly symmetric beam subjected to uniform moment braced continuously by intermediate torsional braces. The solution assumes that compression flange buckles over longer length than between bracing points but buckling magnitude is resisted and controlled by bracing. The proposed torsional resistance of the system,, is following: (1.2) where critical bending moment evaluated with Eq. (1.1); and continuous brace stiffness given by Eq. (1.4). Global lateral-torsional buckling of a simply supported double-girder system can be evaluated according to Eq. (1.3) developed by Yura et al., (2008). (1.3) where spacing of the girders; span length; and weak respective strong axis moment of inertia Torsional bracing of beams The torsional brace stiffness expressed for one brace ( ) depends on several factors and in can general be divided into three major components, as expressed in Eq. (1.4). The equation does not only take into account the stiffness of the bracing but also the distortional stiffness of the web as well as the effect of web distortions (Yura and Phillips, 1992). (1.4) where torsional brace stiffness; web distortional stiffness; and in plane flexibility of girders. Bracing stiffness is governed mainly by type and position of torsional braces utilized for stabilization of the load bearing members. Cross beam located in the centroid of the main girders causes the flanges of adjacent girders to maintain a constant distance and makes the girders sway in the same direction, as shown in Figure 2 (Yura, 2001). The corresponding stiffness formula is given by Eq. (1.5). 2

16 S (1.5) θ Mbr Figure 2 Diaphragm system (U.S. Department of Transportation, 2012) where modulus of elasticity; moment of inertia of torsional brace; and spacing of girders. If the cross beam is positioned at the level of tension flanges instead, so called floor beam, the adjacent compression flanges will move in opposite directions (Yura, 2001). The behavior of the girders braced by floor beam as well as the corresponding torsional brace stiffness equation is presented in Figure 3 and Eq. (1.6). Mbr θ (1.6) Figure 3 Floor beam system (U.S. Department of Transportation, 2012) The torsional stiffness of the frame systems which rely on truss actions can be estimated by using truss analogy. As shown in Figure 4-6 the contribution of top and bottom struts of the compression-tension diagonal system as well as of the top strut of a K-brace system are conservatively considered zero force members and ignored. In tension-only system, horizontal struts are required but the contribution of the compression diagonal is not taken into account (Yura, 2001). The torsional stiffness provided by respective bracing system can be approximated by Eq. (1.7) (1.9). 3

17 F F 0 0 S FLc/S -FLc/S F hb F (1.7) Figure 4 Compression-tension diagonal system (U.S. Department of Transportation, 2012) where area of diagonal brace members; height of cross frame; and length of diagonal members. F 0 F F -2FLc/S F 2FLc/S -F F (1.8) Figure 5 K-brace system (U.S. Department of Transportation, 2012) where F F 2Fhb S area of horizontal brace members. +2FLc/S -F S 0 2Fhb S F hb F (1.9) Figure 6 Tension-only diagonal system; referred to as Z-brace (U.S. Department of Transportation, 2012) According to J. Yura (2001), the effects of cross section distortion at the locations where full depth stiffeners are utilized (see Figure 7) can be calculated with Eq. (1.10). 4

18 bs stiffener h1 ts ( ) (1.10) tw Figure 7 Web stiffener geometry (U.S. Department of Transportation, 2012) where distance between flange centroids; web thickness; vertical web stiffener thickness; total width of vertical web stiffener. In torsional bracing systems the brace moments are reacted by vertical forces on the main girders reducing the torsional stiffness of the bracing system. The effect is most significant in twin girder systems where the relative displacement between the adjacent girders caused by the forces is the greatest. Yura (2001) gives the following formula for calculating the inplane stiffness of the girders: (1.11) where in plane flexibility of girders; strong axis moment of inertia. The brace stiffness in Eq. (1.2) is expressed for a continuous bracing system but it can also be adopted for multiple discrete torsional braces by summing the stiffness of all the braces and dividing it by the girder length according to expression below (Yura et al., 1992): (1.12) where torsional brace stiffness expressed per meter; number of braces. The equivalent continuous brace stiffness of a single brace located at mid-span is found by dividing the brace stiffness of the single brace by 75 percent of the beam length (Yura et al., 1992). 5

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20 2. Methods The analytical part of this research concerning calculation of the critical bending moment of the bridges consists of finite element analyses as well as hand calculations based on the approaches presented in this paper Numerical analysis method Finite element software, Abaqus, was used to numerically model the bridges and conduct linear eigenvalue buckling analyses. Four-node shell elements S4R were used to model the main beams and the cross frame members, where the dimensions and cross section schemes are comprehensively presented in Chapter 3. The mesh size was set to 50 mm for all the models in order to achieve god accuracy of the analyses. Figure 8 and 9 shows the typical finite element mesh of box-girder model as well as a connection detail. Figure 8 Box-girder system mesh Figure 9 I girder system mesh The structural steel was modeled as a linearly elastic isotropic material according to steel types used in respective bridge (see Chapter 3). Material parameters, that is, modulus of elasticity shear modulus as well as Poisson s ratio in elastic stage were kept constant with values of 210 GPa, 81 GPa and 0,30 respectively. For cases where the shear buckling occurred prior to the LTB, the elastic modulus of the girders webs was changed to 2100 GPa in order to avoid premature buckling of the structure. Table 1 shows the yield and ultimate tensile strength values of the construction steel types used in this study. Table 1 Nominal values of steel yield strength and ultimate tensile strength (Swedish Standard Institution, 2008 (1)) Standard and steel grade Nominal thickness of the element S S S S

21 Boundary conditions applied in Abaqus were modeled according to the theoretical assumptions that beams are restrained against lateral twist and displacement at the ends but free to warp. As a consequence, torsional rotations and lateral movements perpendicular to the web were restrained for the nodes at beam ends. Moreover, vertical displacements of the midpoints at both end sections were restrained while longitudinal displacement was restrained only at one section Modified Euler s column buckling formula Modified Euler s beam buckling formula is a method commonly used in Sweden to determine the capacity of a member with regard to buckling. The approach assumes that the lateraltorsional buckling behaviour of the beam can be represented by compression flange of the beam. The method is based on Eurocode 3 where reduction factor is utilized to account for instability phenomenon. Design normal force in the compression flange is obtained according to the following formula (Swedish Standard Institution, 2010): (2.1) where reduction factor; area of the compression flange; and = yield strength Reduction factor for buckling is defined as followed (Swedish Standard Institution, 2010): (2.2) where [ ) ; an imperfection factor for considering the effects of initial imperfections varying between 0,21 and 0,76 ; = factor recommended as 0,75 for I-sections; and = non-dimensional slenderness factor. The slenderness factor is calculated with help of Euler s critical buckling load according to Eq. (2.3) (Swedish Standard Institution, 2010): (2.3) where = critical buckling load of compression flange. The critical buckling load,, is calculated according to the Euler s buckling formula of a strut on an elastic spring foundation, Eq. (2.4) (Swedish Standard Institution, 2010). ) (2.4) where = Young s modulus; = moment of inertia of the compression flange about vertical axis; = buckling length factor; and = buckling length. 8

22 In theory, buckling length factor can vary depending on the stiffness of the torsional braces. However, this method always assumes that the theoretical brace stiffness is infinite and that the torsional braces possess enough strength and stiffness required for the compression flange to buckle between the braces. For this reason, the elastic springs are replaced by roller supports according to Figure 10. Lb Lb Lb Lb L Figure 10 Theoretical model of a simply supported beam braced by three torsional braces (Pettersson, 1971) Depending on the number of braces, the buckling length factor of the compression flange is chosen according to Table 2. For the integers not found in Table 2 the value of the buckling length factor is estimated using interpolation. Number of spans ( ) Table 2 Buckling length factors ,69 0,81 0,84 0,87 0,90 0,92 0,93 The modified Euler s beam buckling formula is used to evaluate the critical bending force in single and multi-span bridges with different types of cross section. When multi-span bridges are concerned, each span is calculated separately while in box-girder bridges only half of the cross section is studied. Choice of bracing type utilized to control the lateral displacement and rotation of the cross section is usually based on past experience and existing bridges of similar proportions. The distance between braces is usually set between 6 and 9 m for the same reason Simplified method according to NCCI SN002 The simplified method described in NCCI SN002 is based on Eurocode 3 where the reduction factor for lateral torsional buckling needs to be estimated in order to calculate the design buckling resistance moment, as shown in Eq. (2.5). where (2.5) (2.6) 9

23 where [ ) ; an imperfection factor for considering the effects of initial imperfections varying between 0,21 and 0,76; ; = factor recommended as 0,75 for I-sections; and = non-dimensional slenderness factor. The method, however; provides a number of simplifications in order to estimate nondimensional beam slenderness without having to calculate beam critical bending moment. The approach assumes that the buckling behaviour of the beam can be represented by compression flange of the beam plus one third of the compressed portion of the web, analysed as a strut. The solution for is given by Eq. (2.7) (SCI 2011): (2.7) where factor that allows for the shape of the bending moment diagram; parameter that depends on section geometry; parameter related to slenderness; destabilizing parameter to allow for destabilizing loads (i.e. loads applied above the shear center of the beam, where the load can move with the beam as it buckles); the minor axis non-dimensional slenderness of the member, given by in which where k is an effective length parameter (Table 3), ; and parameter that allows for the classification of the cross section (for Class 1 and 2 sections the Class 3 sections ). while for Factors used in the method are defined as following (SCI 2011): (2.8) where g = factor that allows in-plane curvature of the beam prior to buckling and is defined as ; ( ) ( ) (2.9) where = a warping restraint parameter; where no warping restraint is provided, and as a conservative assumption when the degree of warping restraint is uncertain, should be taken as unity; = parameter associated with the load level and is dependent on the shape of the bending moment diagram; = distance between level of load application and shear center. 10

24 (2.10) Table 3 Effective length parameter k (Chanakya, 2009) Conditions of restraint at supports k Compression flange laterally Both flanges fully restrained against 0,7 restrained rotation on plan Both flanges partially restrained against 0,8 Nominal torsional restraint against rotation about rotation on plan Both flanges free to rotate on plan 1,0 longitudinal axis Compression flange fully restrained 0,75 against rotation on plan Compression flange partially restrained against rotation on plan 0,85 If the restraint conditions at beam ends differ, the mean value of k should be used (Chanakya, 2009). Table 4 Values of factors and for cases with transverse loading corresponding to values of parameter k (European Committee for Standardization, 2006) Loading and support conditions Bending moment diagram W 1,0 0,5 Values of k Values of factors 1,132 0,972 0,459 0,304 W 1,0 0,5 1,285 0,712 1,562 0,652 A conservative assumption of may be obtained when = 1,0, = 0,9, = 1,0, = 1,0 and = 1,0 (SCI, 2011). 11

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26 3. Case studies Within this chapter, seven bridges chosen for the analysis are presented. Bridge location, geometry and most relevant data concerning bracing systems is described in detail. General information about the bridges and their geographical position is shown in Table 5 and Figure 11. Table 5 Summary of case studies discussed in this chapter Bridge name Number of spans Type of cross section Type of torsional bracing Number of braces per span Total bridge length [m] I-girder Diaphragm 4 39, Trapezoidal Z-type 7 75, Trapezoidal Z-type 7 62, I-girder K-type , I-girder K-type , I-girder K-type , Trapezoidal Plate with opening ,5 Figure 11 Geographical location of the bridges 13

27 3.1. Single span bridges Bridge over Upperuds River, Götaland (Bridge 1530) Background Bridge over Upperuds River is a single span steel-concrete composite bridge located in the eastern part of Mustadfors, Götaland. The total length of the structure is 39,2 m and it has a free width of 7,0 m providing one traffic lane in each direction Technical aspects The bridge is constructed of two 30 m long I-shaped girders interconnected by four intermediate cross-frames. Flanges and webs of the girders are of steel grade S460 while vertical stiffeners and attached crossbeams are of grade S355. The main girders are equally spaced in transversal direction by a distance of 4 m and have a constant height and web thickness of 1089 mm and 14 mm respectively. The thickness and width of the flanges and therefore the height of the web vary along the length of the bridge. The dimensions of the upper flanges close to the supports are 24x385 mm and increase to 34x475 mm at the distance of approximately 7,9 m into the span. Bottom flanges vary in width and thickness in the same manner measuring 12x620 mm close to the supports and 34x720 mm in central part of the bridge. The cross-sectional variation along the length is shown in Figure 12. TFL 24x385 WEB 14x1041 TFL 34x475 WEB 14x1021 TFL 24x385 WEB 14x1041 BFL 24x620 BFL 34x720 BFL 24x Figure 12 Structural steel distribution for the main girders of Bridge 1530 The two I-girders are strengthened on the inside with vertical stiffeners also used for crossframes connections. Additional stiffeners placed on the outside of the main girders are present only at the end supports. Beams lateral displacement as well as their twist is controlled by two types of crossbeams: HEA 450 used at the support locations and UPE 270 installed at intermediate positions. Exact placement of the braces is shown in Figure

28 S1 S Figure 13 Plan view of Bridge 1530 where S1 and symbolize bracing above the supports respective span bracing The design and vertical placement of the support and torsional braces is presented in Figure 14 and Figure HEA Figure 14 - Support bracing (S1)in Bridge 1530 where 1 15x100x396 mm, 2 20x290x1041mm and 3 20x200x1041 mm 7800 UPE Figure 15 - Intermediate bracing () in Bridge 1530 where 1 15x200x1041 mm Effective torsional stiffness per unit length of girder obtained with help of Eq. (1.12) is (see Appendix A A.1). 15

29 Shear force and bending moment diagrams during bridge construction phase are illustrated in Figure 16 and Figure 17. L [m] M [MNm] Figure 16 Bending moment diagram for bridge 1530 in [MNm] V [MN] -1-0,5 0 0,5 L [m] Figure 17 Shear force diagram for bridge 1530 in [MN] 16

30 Bridge over Ore River, Kopparberg (Bridge 1020) Background The bridge over Ore River is a one span composite bridge located in Kopparberg, Sweden. Length of the span is 62,2 m while the total length of the bridge is 75,4 m. The construction has a free width of 7,0 m and is open to a total of two lanes of traffic Technical aspects The bridge is a box-girder bridge with open-topped trapezoidal cross section interconnected by seven Z-type braces. The structure is fabricated of three steel subgrades: S420, S355 and S235. The first steel quality was used for constructing the box-girder while the other two for fabrication of the bracing system as well as plate panels and end plates. While the steel girder height is kept constant, the size of the flange and web varies along the length of the bridge. The dimension of the top flanges located close to end supports is 48x600 mm and increases to 48x700 mm into the span to reach 50x800 mm in the mid-section. The bottom flange and the webs change their dimensions in identical manner as shown in Figure 18. The dimensions of the bottom flange vary from 30x2440 mm at bridge ends to 50x2440 mm in its middle while the webs change their thickness from 19 to 17 mm. The height of the box-girder has a constant value of 1930 mm. TFL 48x600 WEB 19x1970 BF 30x2440 TFL 48x700 TFL 50x800 TFL 48x700 TFL 48x600 WEB 19x1950 WEB 17x1946 WEB 19x1950 WEB 19x1970 BF 48x2440 BF 50x2440 BF 48x2440 BF 30x Figure 18 Structural steel distribution for the main girders of Bridge 1020 Bracing system comprises solid plates and braces of Z-type used at the support locations respective as intermediate braces. As shown in Figure 19, the braces are spaced at equal intervals of 8 m with the exception of two braces closest to the supports located at the distance of 7,1 m from the bridge ends. S1 S Figure 19 Plan view of Bridge 1020where S1 and symbolize bracing above the supports respective span bracing 17

31 Detailed design of the bracing system is presented in Figure 20 - Figure Figure 20 Support bracing (S1) in Bridge 1020 where 1 30x275x870 mm, 2 25x330x850 mm, 3 20x3800x330 mm, 4 20x500x240 mm, 6 15x175x1400 mm and 7 15x2220/3162x1730 mm VARIES HEA100 HEA Figure 21 Intermediate bracing () in Bridge 1020 where 1 12x200x2343 mm, 2 12x350x1797 mm and 3 12x200x1797 mm Effective torsional stiffness per unit length of girder obtained with help of Eq. (1.12) is (see Appendix A A.2)

32 Shear force and bending moment diagrams during bridge construction phase are illustrated in Figure 22 and Figure 23. M [MNm] L [m] Figure 22 Bending moment diagram for bridge 1020 in [MNm] V [MN] L [m] Figure 23 Shear force diagram for bridge 1020 in [MN] 19

33 Bridge over Vanån River, Dalarna County (Bridge 983) Background Bridge over Vanån River is a one span composite bridge located close to Brintbodarna village, Dalarna County. The total length of the bridge is 62,8 m and it has a free width of 7,8 m. The distance between the supports is 62,2 m Technical aspects The bridge over Upperuds River is a box girder bride with open-topped trapezoidal cross section. The bridge is made up of two types of steel subgrades: S355N and S275JR which were used for fabrication of box-girder and end plates respective intermediate braces. The box-girder changes in dimensions along the entire bridge length. The width and thickness of the top flanges close to the supports are 25x560 mm and increase gradually to 40x770 mm at 11,4 m into the span reaching 45x770 mm in the mid-section. Bottom flange as well as the webs changes their dimensions in identical way as shown in Figure 24. The height of the box-girder has a constant value of 2400 m. 25x560 TFL 40x770 TFL 45x770 TFL 40x770 TFL 25x560 WEB 22x2480 WEB 20x2480 WEB 18x2480 WEB 20x2480 WEB 22x2480 TFL 30x2500 TFL 45x2500 TFL 50x2550 TFL 45x2500 TFL 30x Figure 24 Structural steel distribution for the main girders of Bridge 983 Solid plates with thickness of 15 mm are used at the support locations while internal braces of Z-type are used to prevent distortion of the cross section. Internal braces comprise HEA 120 and HEA 140 profiles and are spaced approximately every 8 m with exception of the braces closest to the supports which are located 7.1 m from them. Plate panels at the brace locations are used to strengthen the webs and the bottom flange. Additional vertical stiffeners on the outside of the main girders are installed only above the supports. The brace positioning along the bridge length is shown in Figure 25. S1 S Figure 25 Plan view of Bridge 983 where S1 and symbolize bracing above the supports respective span bracing 20

34 Detailed design of the support and intermediate braces is presented in Figure 26 - Figure Figure 26 Support bracing (S1) in Bridge 983 where 1 30x935x275 mm, 2 25x850x330 mm, 3 25x675x425 mm,4 25x500x265 mm, 5 15x2650x250 mm, 6 15x2000x175 mm and 7 15x3397x2240 mm VARIES HEA120 1 HEA Figure 27 Intermediate bracing () in Bridge 983 where 1 15x260x2539 mm and 2 15x460x2539 mm Effective torsional stiffness per unit length of girder obtained with help of Eq. (1.12) is (see Appendix A A.3). 21

35 Shear force and bending moment diagrams during bridge construction phase are illustrated in Figure 28 and Figure 29. M [MNm] L [m] Figure 28 Bending moment diagram for bridge 983 in [MNm] V [MN] L [m] Figure 29 Shear force diagram for bridge 983 in [MN] 22

36 3.2. Multi-span bridges Bridge over E6 highway, Götaland (Bridge 1385) Background Bridge over E6 highway is a two span composite bridge located north of Flädie, Götaland. The total length of the bridge is 66,2 m and it has a free width of 6,850 m with two traffic lanes, each 2,75 m wide Technical aspects The bridge main load-bearing system consists of two I-shaped girders connected by intermediate cross-frames. The bridge is fabricated of two types of steel: S460M and S355J2. The first steel type was used for fabrication of the I-girders while the other type was used to manufacture the braces, vertical stiffeners and end plates. The total length of each of the two I-girders is 53,2 m whereas the length of the spans vary due to skewed support installed in the middle. As a result, the free span length on respective side is 25,6 m and 27,6 m. The main girders are equally spaced in the transversal direction by a distance of 4 m and have a constant cross section along the entire length, see Figure 30. The girders have a depth of 1300 mm with thickness of the web is 15 mm. Both top and bottom flange are 600 mm wide and have a thickness of 30 respectively. TFL 30x600 WEB 15x1235 BFL 35x Figure 30 Structural steel distribution for the main girders of Bridge 1385 Two different types of bracing systems are utilized in the construction, i.e. crossbeams over the supports and K-type bracing in the spans. The crossbeams consist of standard HEB 800 steel profiles with different lengths depending on the location while the K-type bracing system is made up of four HEA 100 beams. Vertical stiffeners are used on the inside of the I-beams at the location of the torsional bracings and additional stiffeners placed on the outside of the main girders are present only at the support locations. The center to center distance between the braces is 6,65 m with the exception of the braces near the internal support where the distance changes to either 5,65 or 7,65 m due to skewed support. Bracing placement is presented in Figure

37 S1 S2 S Figure 31 Plan view of Bridge 1385 where S1, S2 and symbolize bracing above the end supports, bracing over internal support respective span bracing Detailed design of the bracing system is presented in Figure 32 - Figure HEB Figure 32 Support bracing (S1) in Bridge 1385 where 1 20x670x1100 mm, 2-20x520x1235 mm and 3-20x260x1235 mm HEB Figure 33 Support bracing (S2) in Bridge 1385 where 1 25x880x1100 mm, 2-20x520x1235 m and 3-20x260x1235 mm HEA HEA 100 HEA Figure 34 Intermediate bracing () where 1 20x260x1235 mm and 2-15x220x750 mm 24

38 Effective torsional stiffness per unit length of girder obtained with help of Eq. (1.12) is (see Appendix A A.4). Shear force and bending moment diagrams during bridge construction phase are illustrated in Figure 35 and Figure L [m] M [MNm] Figure 35 Bending moment diagram for bridge 1385 in [MNm] V [MN] -1-0,5 0 0,5 1 L [m] Figure 36 Shear force diagram for bridge 1385 in [MN] 25

39 Bridge over Sält River, Uddevalla-Svinesund (Bridge 1768) Background Bridge over Sält River is a two span composite bridge located in Knäm-Lugnet near Uddevalla-Svinesund. The structure carries E6 motorway and has a total length of 74,4 m. It has a free width of 18,5 m with two traffic lanes in each direction; 3,5 and 3,25 m wide Technical aspects The load-bearing superstructure consists of four similarly sized longitudinal I-shaped girders. The total length of each of the girders is 60 m and the free span length between the supports is 30 m on both sides. The structure is fabricated of two types of steel: S460 and S355. The S460 steel type was used for fabrication of the bottom and top flanges in midsection while S355 was used to manufacture other components of the I-girders as well as the braces. The cross section of the girders is constant along the bridge length with exception of the part over the internal support where upper and bottom flanges change their dimensions, as shown in Figure 37. TFL 20x500 (S355) WEB 17x1550 (S355) BFL 30x600 (S460) TFL 33x500 (S460) WEB 17x1527 (S355) BFL 40x750 (S460) TFL 20x500 (S355) WEB 17x1550 (S355) BFL 30x600 (S460) Figure 37 Structural steel distribution for the main girders of Bridge 1768 The girders are spaced 4,5 m and 5,5 m apart across the width of the bridge and are braced together two-two, namely, no bracing between the inner girders is present. Support bracing is provided by horizontal crossbeams while intermediate bracing is provided by K-type bracing. The center to center distance between the braces as well as plane view of the bridge is shown in Figure 38. S1 S2 S S1 S2 S Figure 38 Plan view of Bridge 1768 where S1, S2 and symbolize bracing above the end supports, bracing over internal support respective span bracing Bracing type used at the support locations vary in size and dimensions. Crossbeams at the end supports have a length of 4 m and a height of 850 mm while the height of the crossbeam used 26

40 in the middle is 1000 mm. The intermediate braces of K-type are made up of four VKRprofiles; two horizontals which are spaced 1110 mm apart from each other and two diagonals. Exact dimension of the bracing system is shown in Figure 39 - Figure 41. Vertical stiffeners used on the inside of the I-beams are present at the location of the torsional braces. Additional stiffeners placed on the outside of the main girders are present only at the support locations. A TFL 16x A WEB 12x850 TFL 20x350 Figure 39 Support bracing (S1) in Bridge 1768 where 1 20x250x1550 mm, 2 12x120x850 mm 9650 A TFL 20x WEB 16x A TFL 20x400 Figure 40 Support bracing (S2) in Bridge 1768 where 1 25x250x1527 mm and 2 25x180x1000 mm VKR 120x120x5 310 VKR 100x100x VKR 120x120x5 Figure 41 Intermediate bracing () in Bridge 1768 where 1 15x230x mm Effective torsional stiffness per unit length of girder obtained with help of Eq. (1.12) is (see Appendix A A.5). 27

41 Shear force and bending moment diagrams during bridge construction phase are illustrated in Figure 42 and Figure L [m] M [MNm] Figure 42 Bending moment diagram for bridge 1768 in [MNm] V [MN] -1,5-1 -0,5 0 0,5 1 1, Figure 43 Shear force diagram for bridge 1768 in [MN] L [m] 28

42 Bridge over Motala River, Östrergötaland county (Bridge 917) Background Bridge over Motala River is a three span composite bridge located near Fiskeby Gård in Norrköping. The structure carries E4 motorway and has a total length of 158,1 m and free width of 22,85 m Technical aspects The load-bearing system of the bridge consists of four longitudinal I-girders with a total length of 145,5 m. The length of the side spans is 41,5 m while the central span is 62,5 m long. The structure is built of two types steel subgrades: S420 and S275JR. The S420 steel quality was used for fabrication of the webs and flanges while S275JR was used to manufacture other bridge components as braces and stiffeners. Bridge varying cross section is shown in Figure 44. TFL 25x400 TFL 45x825 TFL 25x400 TFL 45x825 TFL 25x400 WEB 18x BFL 25x400 WEB 24x BFL 50x1025 WEB 18x BFL 50x800 WEB 24x BFL 50x1025 WEB 18x BFL 25x400 Section Section Section Section Section Figure 44 Structural steel distribution for the main girders of Bridge 917 The girders are spaced 6 m from each other and are braced two-two, that is, no braces are present between inner girders. Entire bracing system is provided by bracing of K-type that differs in dimensions and design depending on location along the span. The plane view of the load-bearing system including different types of braces is shown in Figure 45. Precise center to center distance between the braces is presented in Appendix B - Bracing location B.1. S1 S2 S2 S S1 S2 S2 S Figure 45 Plan view of Bridge 917 where S1, S2 and symbolize bracing above the end supports, bracing over internal support respective span bracing Support bracing comprise four HEA profiles which differ in dimensions. Intermediate bracing, on the other hand, is made of VKR and U-shaped profiles. Location of the vertical stiffeners used on the inside of the I-girders follows exactly placement of the torsional braces. Web stiffeners utilized on the outside of the main girders are present only at the support locations. Detailed design of braces is shown in Figure 46 and Figure

43 HEA 140 HEA HEA Figure 46 Support bracing (S1) in Bridge 917 where 1 20x950x1715 mm, 2 20x150x1335 mm, 3 35x350x620 mm and 4 12x930/1100x200 mm HEA 200 HEA HEA Figure 47 Support bracing (S2) in Bridge 917 where 1 25x950x2700 mm, 2-25x200x2150 mm, 3 50x450x460 mm and 4 12x930/1100x300 mm VARIES A USP TFL 16x140 WEB 10x300 BFL 16x140 A SECTION A-A Figure 48 Intermediate bracing () in Bridge 917 where 1 20x250 mm Torsional brace stiffness obtained with help of FEM is A.6). (see Appendix A 30

44 Shear force and bending moment diagrams during bridge construction phase are illustrated in Figure 49 and Figure 50. M [MNm] L [m] Figure 49 Bending moment diagram for bridge 917 in [MNm] V [MN] L [m] Figure 50 Shear force diagram for bridge 917 in [MN] 31

45 Bridge over Vallsund, Jämtland County (Bridge 1052) Background Bridge over Vallsund is a three span composite bridge located in Jämtland County. The total length of the bridge is 134,5 m while the spans are 36,25 m, 46,0 m and 36,25 m. Free width of the bridge is 11,75 m Technical aspects The construction has an open-topped trapezoidal cross section and is manufactured of three types of steel: S460M, S420M and S355JR. The first two types were used for fabrication of the web and flanges while the third one for plates and bracing system. The box girders vary in dimensions along the bridge length. The cross section is stiffest in negative bending moment regions, that is, close to the internal supports where the dimensions of the top and bottom flanges increase to 37x650 mm respectively 25x3300 mm. The height of the box-girder has a constant value of 2,4 m. The manner in which the box-girders change its dimensions is shown in Figure 51. TFL 20x450 TFL 37x650 TFL 20x400 TFL 37x650 TFL 20x450 WEB 19x1927 BFL 13x3300 WEB 20x1882 (S460) BFL 25x3300 WEB 19x1937 BFL 13x3300 Section Section Section Section Section WEB 20x1882 (S460) BFL 25x3300 WEB 19x1927 BFL 13x3300 Figure 51 Structural steel distribution for the main girders of Bridge 1052 Center to center distance between top flanges as well as the distance between the webs at the level of the bottom flange are constant throughout the entire bridge length and have a value of 4,5 m respective 3,1 m. A total number of twenty braces is utilized to prevent distortion of the cross section. Center to center distance between the braces varies from 5,3 to 6,625 m in side spans and from 5,3 to 8 m in the central span. A plane view of the bracing system is shown in a figure below. Exact placement of the braces is shown in Appendix B - Bracing location B.2. S1 S2 S2 S Figure 52 Plan view of Bridge 1052 where S1, S2 and symbolize bracing above the end supports, bracing over internal support respective span bracing Bracings system comprises solid plates with thickness of 12 and 8 mm placed at the support respective discrete locations. Unlike intermediate bracing plates, support bracing plates are strengthened with vertical stiffeners. A detailed design of the bracing system is presented in the Figure 53 and Figure

46 200 6, Figure 53 Support bracing (S1) in Bridge 1052 where 1 25x1037/109x895 mm, 2 12x300x1226 mm, 3 20x250/144x1044 mm, 4 12x144x864 mm, 5 25x800x1150 mm, 6 12x150/150x400 mm, 7 12x150x727 mm, 8 20x300x3710 mm and 9 12x4076/3090x1267 mm 6, Ø Figure 54 Support bracing (S2) in Bridge 1052 where 1 20x192x725 mm, 2 18x200x1644 mm, 3 20x600x4300 mm, 4 8x150x2488 mm, 5 18x4284/3090x1526 mm, 6 25x350x1526 mm and 7 20x250x600 mm 33

47 , Figure 55 Intermediate bracing () in Bridge 1052 where 1 20x350x484 mm, 2 PL 8x150x4284 mm and 3 8x4284/3090x1526 mm Torsional brace stiffness obtained with help of FEM is (see Appendix A A.7). Shear force and bending moment diagrams during bridge construction phase are illustrated in Figure 56 and Figure M [MNm] Figure 56 Bending moment diagram for bridge 1052 in [MNm] L [m] V [MN] -1,5-1 -0,5 0 0,5 1 1, Figure 57 Shear force diagram for bridge 1052 in [MN] L [m] 34

48 4. Results In this chapter, the buckling modes of the bridges obtained from numerical analyses are presented. Critical bending moment values obtained from the FE buckling analyses and the theoretical solutions are presented in a table and compared. Shear forces and bending moment acting on the beams as well as beam design buckling resistance are also presented. The critical shear force,, shown in Table 6 was calculated for Eigenvalues obtained by means of FEA Single span bridges The numerical analyses revealed that the system and global buckling were the primary buckling modes for Bridge 1530 and Bridges 1020 and 983 respectively, see Figure 58. Figure 58 LTB of: a) Bridge 1530; b) Bridge 1020; and c) Bridge 983 Torsionally braced beams of Bridge 1530 buckled in a single wave and largest lateral and torsional displacement was generated at mid-span section. Stresses which occurred in braces due to LTB were large enough to force two of the internal braces to buckle. Bridges 1020 and 983 showed very similar buckling behavior, namely, the box-girders buckled in a single wave. However, the girders failed due to global buckling which did not involve any bigger change in cross sectional shape. Biggest torsional and lateral displacements occurred also at mid-span sections; nevertheless, the braces preserved their original geometry along the entire bridge length Multi-span bridges Results of FEA show that for all multi-span bridges local plate buckling occurred before the first lateral-torsional buckling modes were found, as shown in Figure

49 Figure 59 Shear buckling of: a) Bridge 1385, b) Bridge 1768, c) Bridge 917 and d) Bridge 1052 Mode of failure which occurred in Bridge 1385 and 1768 was shear buckling of the web. The buckling took place close to the internal support where the shear forces were expected to be largest. Bridge 917, on the other hand, failed due to combination of a shear buckling of the web as well as buckling of the compression flange. The local buckling occurred close to internal support where the shear forces in the web and compression forces in the bottom flange were the highest. Bridge 1052 failed due to local buckling of the webs. The buckling took place between the first and the second intermediate brace, that is, approximately 10 m from the end support. Change of the webs Young modulus, however; led to LTB of all the multi-span bridges. The buckling behavior of the structures is shown in Figure 60. Figure 60 LTB of: a) Bridge 1385, b) Bridge 1768, c) Bridge 917 and d) Bridge 1052 for E = 2100 GPa Bridge 1385 failed due to system buckling of the girders in the shape of half sine wave. Lateral-torsional failure mode of the rest of the bridges on the other hand was a local buckling between the discrete braces. The buckling occurred at the sections with lowest cross sectional dimensions at positive bending moment regions. 36

50 4.3. Comparison of results Table 6 shows the forces acting on the bridges during the construction time as well as the critical bending moment values calculated with help of the presented approaches. Table 6 Shear force (MN) and bending moment values (MNm) for the analyzed bridges where * denotes the occurrence of shear buckling prior to LTB of the girders Bridge ,8 1,6 1,8 0,9 1,3 2,5 1,1 1,8 1,1 1,2 4,7 3,6 0,1 1,8 3,1 4,8 6,8 3,9 4,7 8,0 5,6 5,9 25,9 28,1 4,7 7,5 28,8 16,8 Critical bending moment, FEM 13,9 17,8 21,6 54,9* 40,0* 41,7* 222,5* Eq. (1.1) 15,4 469,3 598,5 27,9 45,6 - - Eq. (1.2) 12,1 57,9 96,9 20,8 23,6 - - Eq. (1.3) 25,3 26,7 29,7 42,0 60,3 - - Modified Euler s column buckling formula Simplified method according to NCCI SN002 24,8 112,6 113,8 53,9 52,7 371,7 137,0 16,8 203,4 185,8 63,7 64,5 613,4 214,0 According to the results the critical elastic buckling moment of the bridges varied substantially by the type of calculation method used. When Bridge 1530 is considered, the largest critical bending moment value was obtained with Eq. (1.3) while the lowest with Eq. (1.2) which gave almost identical result as FEM. The relative difference between the calculated values was approximately 109%. For single span bridges with trapezoidal sections, namely, Bridge 1020 and 983 the highest and lowest critical bending moments were obtained with Eq. (1.1) and FEM respectively. The relative difference between the results varied from 2536% for Bridge 1020 and 2670% for Bridge 983. Eq. (1.1) and NCCI approach gave the lowest and highest critical bending moments for both Bridge 1385 and The relative error between the results was 206 and 177% respectively. In case of Bridge 917 and 1052 the authors were unable to perform the calculations according to Equations (1.1) (1.3) due to bridge changing geometry. In order to estimate the critical bending moment by means of modified Euler s formula as well as NCCI approach, bridge sections closest to internal support were chosen. The lowest results for Bridge 917 were obtained with FEM and the highest using NCCI approach which gave approximately 1300% higher critical bending moment value. 37