the performance of conventional discrete torsional ... · the performance of conventional discrete...

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 - 5241

Avdelningen för Konstruktionsteknik

Lunds Tekniska Högskola

Box 118

221 00 LUND

Division of Structural Engineering

Faculty of Engineering, LTH P.O. Box 118

S-221 00 LUND, Sweden

www.kstr.lth.se

The performance of conventional discrete torsional bracings

in steel-concrete composite bridges: a survey of Swedish bridges

2015-05-26

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: 0349-4969

ISRN: LUTVDG/TVBK-15/5241+(110p)

iii

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

v

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

vii

Notation

Abbreviations

Eurocode 3 EN-1993-1-1:2005

FE Finite element

FEA Finite element analysis

FEM Finite element method

LTB Lateral torsional buckling

SLS Serviceability limit state

ULS Ultimate limit state

UDL 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

viii

Moment diagram modification factor of braced beam

Torsional warping constant

d Distance between top flanges in box-girders

D Destabilizing parameter

Web depth

e Distance between shear center and bottom flange

é 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

ix

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 Spacing of girders

t Distance from tension flange centroid to neutral axis

Bottom flange thickness

Vertical web stiffener thickness

Top flange thickness

Web thickness

U Parameter that depends on sections geometry

V 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

x

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

xi

Table of Contents 1. Introduction and background ........................................................................................................... 1

1.1. Lateral torsional buckling of beams ........................................................................................ 1

1.2. Torsional bracing of beams ..................................................................................................... 2

2. Methods ........................................................................................................................................... 7

2.1. Numerical analysis method ..................................................................................................... 7

2.2. Modified Euler’s column buckling formula ............................................................................ 8

2.3. Simplified method according to NCCI SN002 ........................................................................ 9

3. Case studies ................................................................................................................................... 13

3.1. Single span bridges ................................................................................................................ 14

3.1.1. Bridge over Upperuds River, Götaland (Bridge 1530) .............................................. 14

3.1.2. Bridge over Ore River, Kopparberg (Bridge 1020) .................................................. 17

3.1.3. Bridge over Vanån River, Dalarna County (Bridge 983) .............................................. 20

3.2. Multi-span bridges ................................................................................................................. 23

3.2.1. Bridge over E6 highway, Götaland (Bridge 1385) ........................................................ 23

3.2.2. Bridge over Sält River , Uddevalla-Svinesund (Bridge 1768) ...................................... 26

3.2.3. Bridge over Motala River , Östrergötaland county (Bridge 917) .................................. 29

3.2.4. Bridge over Vallsund, Jämtland County (Bridge 1052) ................................................ 32

4. Results ........................................................................................................................................... 35

4.1. Single span bridges ................................................................................................................ 35

4.2. Multi-span bridges ................................................................................................................. 35

4.3. Comparison of results ............................................................................................................ 37

5. Discussion and conclusions ........................................................................................................... 39

5.1. Bridge design ......................................................................................................................... 39

5.2. Calculation methods .............................................................................................................. 39

5.3. Conclusions ........................................................................................................................... 40

6. Examination of multi-span bridges ............................................................................................... 43

6.1. Introduction ........................................................................................................................... 43

6.2. Methods ................................................................................................................................. 43

6.3. Calculations ........................................................................................................................... 43

6.3.1. Bridge 1385 ................................................................................................................... 43

6.3.2. Bridge 1768 ................................................................................................................... 53

6.4. Results ................................................................................................................................... 54

6.5. Discussion and conclusions ................................................................................................... 55

6.5.1. Discussion ..................................................................................................................... 55

6.5.2. Conclusions ................................................................................................................... 56

xii

7. Parametric study – Comparison of FEM and Equations (1.1) - (1.3) ............................................ 57

7.1. Introduction ........................................................................................................................... 57

7.2. Methods ................................................................................................................................. 57

7.3. Results ................................................................................................................................... 58

7.3.1. Brace number and distance variation............................................................................. 58

7.3.2. Brace stiffness variation ................................................................................................ 65

7.3.3. Cross section variation .................................................................................................. 66

7.3.4. Distance between girders ............................................................................................... 72

7.4. Discussion and conclusions ................................................................................................... 73

References ............................................................................................................................................. 75

Appendix A - Brace stiffness ............................................................................................................. 77

A.1. Bridge 1530 ........................................................................................................................... 77

A.2. Bridge 1020 ........................................................................................................................... 78

A.3. Bridge 983 ............................................................................................................................. 79

A.4. Bridge 1385 ........................................................................................................................... 80

A.5. Bridge 1768 ........................................................................................................................... 81

A.6. Bridge 917 ............................................................................................................................. 82

A.7. Bridge 1052 ........................................................................................................................... 82

Appendix B - Bracing location .......................................................................................................... 83

B.1. Bridge 917 ............................................................................................................................. 83

B.2. Bridge 1052 ........................................................................................................................... 84

Appendix C - Critical bending moment of Bridge 1530 .................................................................... 85

C.1. Critical bending moment according to Eq. (1.1), (1.2) and (1.3) .......................................... 85

C.2. Modified Euler’s column buckling formula .......................................................................... 89

C.3. Simplified method according to NCCI SN002 ...................................................................... 91

Appendix D - Critical bending moment of Bridge 1020 .................................................................... 93

D.1. Critical bending moment according to Eq. (1.1) - (1.3) ........................................................ 93

D.2. Modified Euler’s column buckling approach ........................................................................ 96

D.3. Simplified method according to NCCI SN002 ...................................................................... 99

1

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.

1.1. 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.

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.

A

A

δ

Center of twist

SECTION A-A

2

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.

1.2. 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).

3

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).

Figure 3 – Floor beam system (U.S. Department of Transportation, 2012)

(1.6)

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).

Mbr θ

Figure 2 – Diaphragm system (U.S. Department of Transportation, 2012)

(1.5)

Mbrθ

S

4

Figure 4 – Compression-tension diagonal system (U.S. Department of

Transportation, 2012)

(1.7)

where area of diagonal brace members; height of cross frame; and length of

diagonal members.

Figure 5 – K-brace system (U.S. Department of Transportation, 2012)

(1.8)

where area of horizontal brace members.

Figure 6 – Tension-only diagonal system; referred to as “Z-brace”

(U.S. Department of Transportation, 2012)

(1.9)

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).

S

F

FF

F

0

-FLc/S

0

FLc/S

hb

FF

F0F-2FLc/S 2FL

c/S

-FF

2Fhb

S2Fhb

S

F

SF

F

F

-F0+2FLc/S

hb

5

Figure 7 – Web stiffener geometry (U.S. Department of

Transportation, 2012)

(

)

(1.10)

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 in-

plane 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).

bs

h1 ts

tw

stiffener

7

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.

2.1. 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

S235 235 360 215 360

S355 355 510 335 470

S420 420 520 390 500

S460 460 540 430 530

8

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.

2.2. 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 lateral-

torsional 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.

9

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.

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.

Table 2 – Buckling length factors

Number of

spans ( )

1 2 3 4 6 8 10

0,69 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.

2.3. 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).

(2.5)

where

√

(2.6)

Lb Lb Lb Lb

L

10

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 non-

dimensional 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 while for

the Class 3 sections ).

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.

11

√ √

(2.10)

Table 3 – Effective length parameter k (Chanakya, 2009)

Conditions of restraint at supports k

Compression flange laterally

restrained

Both flanges fully restrained against

rotation on plan

0,7

Nominal torsional restraint

against rotation about

longitudinal axis

Both flanges partially restrained against

rotation on plan

0,8

Both flanges free to rotate on plan 1,0

Compression flange fully restrained

against rotation on plan

0,75

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

Values of k Values of factors

W

1,0

0,5

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).

13

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.

Figure 11 – Geographical location of the bridges

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]

1530 1 I-girder Diaphragm 4 39,2

1020 1 Trapezoidal Z-type 7 75,4

983 1 Trapezoidal Z-type 7 62,8

1385 2 I-girder K-type 3-3 66,2

1768 2 I-girder K-type 4-4 74,4

917 3 I-girder K-type 5-8-5 158,1

1052 3 Trapezoidal Plate with opening 5-6-5 134,5

14

3.1. Single span bridges

3.1.1. Bridge over Upperuds River, Götaland (Bridge 1530)

3.1.1.1. 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.

3.1.1.2. 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.

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 cross-

frames 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 13.

30000

TFL 24x385

WEB 14x1021 WEB 14x1041WEB 14x1041

TFL 34x475 TFL 24x385

BFL 24x620 BFL 34x720 BFL 24x620

157767886 7886

15

Figure 13 – Plan view of Bridge 1530 where S1 and F1 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 15.

Figure 14 - Support bracing (S1)in Bridge 1530 where 1 – 15x100x396 mm, 2 – 20x290x1041mm and 3 –

20x200x1041 mm

Figure 15 - Intermediate bracing (F1) 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).

6000

S1 F1 F1 F1 F1 S1

6000600060006000

40

00

10

89

4000

HEA 450

57

0

3960

1

3 2

30

0

7800

10

89

4000

UPE 270

59

0

3960

1

30

07800

16

Shear force and bending moment diagrams during bridge construction phase are illustrated in


Figure 16 – Bending moment diagram for bridge 1530 in [MNm]

Figure 17 – Shear force diagram for bridge 1530 in [MN]

0

2

4

6

8

0 5 10 15 20 25 30

M [MNm]

L [m]

-1

-0,5

0

0,5

1

0 5 10 15 20 25 30

V [MN]

L [m]

17

3.1.2. Bridge over Ore River, Kopparberg (Bridge 1020)

3.1.2.1. 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.


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.


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.

Figure 19 – Plan view of Bridge 1020where S1 and F1 symbolize bracing above the supports respective span

bracing

62200

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 30x2440

10475 10475 20950 10475 10475

7100

S1

8000 8000 8000 8000 8000 8000 7100

F1 F1 F1 F1 F1 F1 F1 S1

18

Detailed design of the bracing system is presented in Figure 20 - Figure 21.

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

Figure 21 – Intermediate bracing (F1) 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).

3300

2890

17

80

720

3192

400 400720100 100

21

534

6

25

0

7800

3300

250

HEA100

2240100

19

80

VARIES 350 - 450

HEA 12021

3

18

30

100

7800

25

0

19





0

5

10

15

20

25

30

0 10 20 30 40 50 60

M [MNm]

L [m]

-2

-1

0

1

2

0 10 20 30 40 50 60

V [MN]

L [m]

20

3.1.3. Bridge over Vanån River, Dalarna County (Bridge 983)

3.1.3.1. 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.


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.


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.

Figure 25 – Plan view of Bridge 983 where S1 and F1 symbolize bracing above the supports respective span

bracing

62200

WEB 22x2480

25x560 TFL 40x770 TFL 45x770 TFL 40x770 TFL 25x560

TFL 30x2500 TFL 45x2500 TFL 50x2550 TFL 45x2500 TFL 30x2500

11100 11000 18000 11000 11100

WEB 20x2480 WEB 18x2480 WEB 20x2480 WEB 22x2480

7100

S1

8000 8000 8000 8000 8000 8000 7100

F1 F1 F1 F1 F1 F1 F1 S1

21

Detailed design of the support and intermediate braces is presented in Figure 26 - Figure 27.

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

Figure 27 – Intermediate bracing (F1) in Bridge 983 where 1 – 15x260x2539 mm and 2 – 15x460x2539 mm



3420

3050

22

40

400400 750750

4500

5

24

0072

1

634

8650

25

0

3500

250

HEA120

2300100

24

00

VARIES 310-520

HEA1401

2

23

00

100

8650

25

0

22





0

5

10

15

20

25

30

0 10 20 30 40 50 60

M [MNm]

L [m]

-2

-1

0

1

2

0 10 20 30 40 50 60

V [MN]

L [m]

23

3.2. Multi-span bridges

3.2.1. Bridge over E6 highway, Götaland (Bridge 1385)

3.2.1.1. 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.


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.


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 31.

25600 27600

TFL 30x600

BFL 35x600

WEB 15x1235

53200

24

Figure 31 – Plan view of Bridge 1385 where S1, S2 and F1 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 34.

Figure 32 – Support bracing (S1) in Bridge 1385 where 1 – 20x670x1100 mm, 2 - 20x520x1235 mm and 3 -

20x260x1235 mm

Figure 33 – Support bracing (S2) in Bridge 1385 where 1 – 25x880x1100 mm, 2 - 20x520x1235 m and 3 -

20x260x1235 mm

Figure 34 – Intermediate bracing (F1) where 1 – 20x260x1235 mm and 2 - 15x220x750 mm

6650 6650 6650 5650 7650 6650 6650 6650

56507650S1 F1 F1 F1 S2 F1 F1 F1 S1

40

00

4000

HEB 800

29606

50 1

30

0

3

2865

2 1

7650

30

04472

3192

65

0 13

00

HEB 800

2 1

3

7650

30

0

17

5

4000

HEA 100

HEA 100

HEA 100

13

00

95

01

75

1 2

3500

7650

30

0

25







-6

-4

-2

0

2

4

0 10 20 30 40 50M [MNm]

L [m]

-1

-0,5

0

0,5

1

0 10 20 30 40 50

V [MN]

L [m]

26

3.2.2. Bridge over Sält River , Uddevalla-Svinesund (Bridge 1768)

3.2.2.1. 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.


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.


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.



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

25000

TFL 20x500 (S355)

BFL 30x600 (S460)

WEB 17x1550 (S355)

TFL 33x500 (S460)

10000

TFL 20x500 (S355)

WEB 17x1527 (S355)WEB 17x1550 (S355)

BFL 40x750 (S460)BFL 30x600 (S460)

60000

25000

45

00

55

00

45

00

S1 S1F1 F1 F1 F1 S2 F1 F1 F1 F1

6000 6000 6400 6400 5200 6400 6400 6000 60005200

S1 S1F1 F1 F1 F1 S2 F1 F1 F1 F1

27

in the middle is 1000 mm. The intermediate braces of K-type are made up of four VKR-

profiles; 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.

Figure 39 – Support bracing (S1) in Bridge 1768 where 1 – 20x250x1550 mm, 2 – 12x120x850 mm

Figure 40 – Support bracing (S2) in Bridge 1768 where 1 – 25x250x1527 mm and 2 – 25x180x1000 mm

Figure 41 – Intermediate bracing (F1) in Bridge 1768 where 1 – 15x230x1550-1527 mm



4500

TFL 16x200

WEB 12x850

TFL 20x350

4000

A

A

44

0

16

00

12

11

2

9650

31

0

4500

TFL 20x300

WEB 16x1000

TFL 20x400

4000

A

A

52

0

16

00

12

11

2

9650

31

0

4500

11

10

VKR 120x120x5VKR 100x100x5

33

0

4040

VKR 120x120x5

16

00

1

11

2

9650

31

0

28





-10

-5

0

5

0 10 20 30 40 50 60M [MNm]

L [m]

-1,5

-1

-0,5

0

0,5

1

1,5

0 10 20 30 40 50 60

V [MN]

L [m]

29

3.2.3. Bridge over Motala River , Östrergötaland county (Bridge 917)

3.2.3.1. 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.


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.


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.



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 48.

TFL 25x400

WEB 18x1915-2225

BFL 25x400

TFL 45x825

WEB 24x2190-2700

BFL 50x1025

TFL 25x400

WEB 18x1900-2210

BFL 50x800

TFL 45x825

WEB 24x2190-2700

BFL 50x1025

TFL 25x400

WEB 18x1915-2225

BFL 25x400

Section 1 - 29000 Section 3 - 37500

145500

Section 5 - 29000Section 2 - 25000 Section 4 - 25000

60

00

60

00

60

00

S1 F1 S2 S2 S1F1 F1 F1 F1F1 F1 F1 F1F1 F1 F1 F1 F1F1 F1 F1F1

S1 F1 S2 S2 S1F1 F1 F1 F1F1 F1 F1 F1F1 F1 F1 F1 F1F1 F1 F1F1

30

Figure 46 – Support bracing (S1) in Bridge 917 where 1 – 20x950x1715 mm, 2 – 20x150x1335 mm,

3 – 35x350x620 mm and 4 – 12x930/1100x200 mm

Figure 47 – Support bracing (S2) in Bridge 917 where 1 – 25x950x2700 mm, 2- 25x200x2150 mm,

3 – 50x450x460 mm and 4 – 12x930/1100x300 mm

Figure 48 – Intermediate bracing (F1) in Bridge 917 where 1 – 20x250 mm

Torsional brace stiffness obtained with help of FEM is (see Appendix A

A.6).

3000 3000

42

01

10

01

95

600

27

01

10

03

45

17

15

150

150

HEA 140

HEA 240

HEA 200

43

2 1

11825

30

0

3000 3000

11825

39

52

02

52

80

750

24

52

02

54

30

27

00

150

HEA 200

HEA 360

HEA 300

200

4

3

2 1

30

0

3000 3000

6000

35

01

20

0

VA

RIE

S

USP 200

50

01

20

01

SECTION A-A

TFL 16x140

WEB 10x300

BFL 16x140A

A

30

0

31





-40

-30

-20

-10

0

10

20

0 20 40 60 80 100 120 140M [MNm]

L [m]

-3

-2

-1

0

1

2

3

0 20 40 60 80 100 120 140

V [MN]

L [m]

32

3.2.4. Bridge over Vallsund, Jämtland County (Bridge 1052)

3.2.4.1. 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.


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.


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.



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 55.

118500

TFL 20x450

WEB 19x1927

BFL 13x3300

Section 1 - 31550 Section 2 - 10000 Section 4 - 1000Section 3 - 36000 Section 5 - 31550

TFL 37x650

WEB 20x1882 (S460)

BFL 25x3300

TFL 20x400

WEB 19x1937

BFL 13x3300

TFL 37x650

WEB 20x1882 (S460)

BFL 25x3300

TFL 20x450

WEB 19x1927

BFL 13x3300

S1 F1 S2 F1 F1 S2 F1 F1 S1F1 F1 F1 F1 F1 F1 F1 F1F1 F1 F1

45

00

33

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

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

1003100

100

4500200

250

1150

18

00

12

67

1150

2500

5

4

2

3

7 8

9

6

6,275

30

0

Ø800

1003100

100

4500200

450

18

00

15

50

80

0

190 190560

2200

5

6

7

2

13

4

6,275

30

0

34

Figure 55 – Intermediate bracing (F1) 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).





1003100

100

4500200

250

18

00

15

50

2

3

1

6,275

30

0

-20

-15

-10

-5

0

5

10

0 20 40 60 80 100 120M [MNm]

L [m]

-1,5

-1

-0,5

0

0,5

1

1,5

0 20 40 60 80 100 120

V [MN]

L [m]

35

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.

4.1. 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.

4.2. 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 59.

36

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.

37

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 1530 1020 983 1385 1768 917 1052

0,8 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

24,8 112,6 113,8 53,9 52,7 371,7 137,0

Simplified method

according to NCCI

SN002

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 1768. 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.

38

In case of Bridge 1052 the largest bending moment value was obtained by means of FEM

and the lowest with modified Euler’s approach. The relative error between the

approaches was around 40%.

39

5. Discussion and conclusions

5.1. Bridge design

The results show that analyzed single span bridges are designed in a correct way when LTB is

concerned under construction time. Bearing and intermediate transverse stiffeners possess

enough strength and stiffness required to prevent flanges and webs against out of plane

deformations. Also bracing systems utilized to control the buckling of the main beams seem

to be chosen adequately to the bridge types making beams work like a single unit effectively

increasing their bending moment capacity.

Multi-span bridges analyzed in this report, on the other hand, fail due to shear buckling of the

webs, thus their design concerning lateral-torsional buckling under construction time seems to

not be optimal. However, even though the shear buckling occurs prior to LTB the girders

must not necessarily fail in shear. This is due to the fact that normally the girders possess

quite significant post-critical strength reserve in shear what in theory might lead to LTB.

The shear forces which occur in the multi-span bridges due to construction loads are

considerably lower than the girders’ shear resistance. As a consequence, the bridges are not

likely to buckle under construction time and the authors believe that their big cross section

dimensions are strongly dependent on the serviceability loads. The exception is Bridge 917

which shear force resistance is marginal compared with the loads the structure is exposed to.

It is obvious to the authors that the bridge design lacks crucial detailing which could prevent

the premature buckling of bridge webs. Vulnerable to out of plane deformations webs should

be strengthen by jack or longitudinal stiffeners in the compression zones.

5.2. Calculation methods

The calculated critical bending moment values vary significantly depending on a chosen

approach and none of the presented analytical methods can be considered universal. The

approaches are built on different assumptions and are bound to give different results,

however; it is worrisome that the differences are so huge especially when bridges with

trapezoidal cross sections are considered. From the obtained results it is rather obvious that

Bridge 1020 and 983 will not fail due to buckling between the braces, but to global buckling

instead, which means that the Eq. (1.1), modified Euler’s formula and NCCI approach cannot

be trusted.

The results also revealed that both Euler’s modified buckling formula and NCCI approach

give rather high critical bending moment values compared with other solutions. This might

depend on the fact that both approaches study instability of the beams only between the

torsional braces. In other words, both methods assume buckling of the beams between discrete

bracing points which give relatively large cross section depth to length ratio and as a result

high critical moment values (Bridge 1020, 983, 917 and 1052). Despite the fact that the two

methods are based on very similar assumptions the results obtained with the methods differ

significantly. None of the methods consider the stiffness of the braces, however; the modified

Euler’s approach considers the number of braces used to control the buckling of the girders

40

which is completely neglected in NCCI approach and the authors believe that this is what

makes the results differ so much.

The authors are also convinced that the best solution to calculate the bending moment of

torsionally braced beams would be simultaneous usage of Eq. (1.1) – (1.3) where the lowest

obtained critical bending moment value is chosen. As the analyses showed, with this approach

not only the critical bending moment of single span bridges could be predicted quite

accurately but also their buckling behavior. On the contrary to other analytical methods

presented in this paper, the exact solutions take into account buckling of the girders between

the torsional braces as well as the brace number and their torsional stiffness. Moreover, global

response of the double-girder system is also considered which is completely neglected with

the other approaches making the exact formulas more trustworthy.

Results of the analyses showed that the elastic bending moment of the bridges is highly

overestimated when NCCI and modified Euler’s buckling approach are used and when the

Eq. (1.1) – (1.3) are used simultaneously. In most cases the bending moment values calculated

according to the two solutions are several times higher than the values obtained with other

methods. FEAs revealed that the system and global buckling are the primary buckling modes

of analyzed single span bridges, while cross section of multi-span bridges buckles locally due

to excessive shear and bending stresses. Thus, the assumption that the primary buckling mode

always takes place between the bracing points is wrong. What is more, assuming that the

torsional braces always possess infinite stiffness seems also to be false as internal braces

utilized to control lateral-torsional buckling of Bridge 1530 deform due to excessive internal

forces.

5.3. Conclusions

The major finding of this research can be summarized as following:

The analyzed single span bridges have correct cross sectional dimensions and are

designed correctly when LTB under construction time is considered

The analyzed multi-span bridges fail due to shear buckling of the webs

Steel beam girder sections of multi-span bridges lack some important detailing which

could prevent premature buckling of the slender sections.

Modified Euler’s column buckling formula as well as NCCI approach give highly

unconservative critical bending moment values compared with FEM.

Assumption which modified Euler’s buckling formula and NCCI approach are based

upon, namely, that the LTB always takes place between the discrete torsional braces is

false

Assumption that torsional bracing systems always possess infinite strength and

stiffness which modified Euler’s buckling formula is based upon is false

Simultaneous usage of Eq. (1.1) – (1.3) where the equation with the lowest obtained

critical bending moment value is chosen, correctly predicts buckling behavior of the

41

analyzed single-span bridges as well as gives better critical bending moment results in

comparison to FEM.

43

6. Examination of multi-span bridges

6.1. Introduction

The numerical analyses conducted in the previous chapter showed that none of the multi-span

bridges chosen for this study is prone to fail due to LTB under construction time. However, it

was found that for all the structures shear buckling of the webs is the primary buckling mode.

The objective of this chapter, therefore, is to examine the reason behind the premature

buckling of the webs.

6.2. Methods

A set of calculations in accordance with Eurocodes and Swedish national application rules is

performed in order to investigate the cross section the multi-span bridges. Firstly, governing

loads acting on the girders during the construction time are calculated. Secondly, cross section

capacity in critical sections of the bridge is determined with help of methods used at

Reinertsen.

6.3. Calculations

In this worked example, only permanent and variable loads are concerned. Variable actions

which are considered here are traffic loads; thermal actions, wind or snow actions are ignored.

Bridges 1385 and 1768 are examined.

6.3.1. Bridge 1385

Design in the ultimate limit state (ULS)

Permanent loads

Cross-sectional properties of the load-bearing girders are presented below:

The self-weight of the construction steel is set to 79 kN/m3 and additional 10% of the

self-weight of the primary steel girders is added to account for the weight of the bracing

system. Characteristic permanent loads acting on one girder are as followed:

44

Variable loads

Variable loads acting on the main girders during construction time are self-weight

of the concrete as well as construction loads. The total width of the slab is 7,65 m and its

thickness varies marginally across the width of the bridge. As a simplification, an equivalent

slab with uniform thickness of 0,3 m is chosen. The cross section and dimensions of the

bridge deck are shown in Figure 61.

Figure 61 – Dimensions of the concrete slab over one I-girder

The self-weight of the concrete is set to 24 kN/m3 while the construction loads are set

to 1kN/m2 (Swedish Standard Institution, 2008 (2)). The characteristic variable loads acting

on one girder are:

3,825 kN/m

Load combinations

Various actions described above are combined and design values of their effects are

determined according to Eurocode 1. Most unfavorable load combination in the ULS is

calculated with Eq. 6.10b, thus force acting on one girder becomes:

Most sever consequence class is chosen ( ).

Bending moment and shear force

As a simplification, the length of both spans is set to 26,6 m. Given the loads acting on the

main girders the following critical bending moments and shear forces are obtained:

3,825

0,3

45

Capacity of the load bearing beams

Bending moment and shear capacity of the load bearing beams prior to hardening of the

concrete is calculated. Firstly, a class of cross section is determined; where a cross section is

classified according to the highest class of its compression parts, see Table 7.

Table 7 – Cross section class control

Web Top Flange Bottom Flange

SC1

SC2

SC3

CLASS

Cross section of the load bearing girders is a Class 3 cross section which can only develop

elastic distribution of stresses. As a consequence, center of gravity is located at the height

of 0,619 m from the lower edge of the bottom flange. Moment of inertia about the strong axis

becomes:

∑ )

Bending moment resistance of the cross section is then obtained to:

where according to Eq. (2.2)

With help of the obtained values bending moment capacity of the beam in the span and over

the internal support is calculated to:

46

The design shear resistance is calculated in accordance with Eurocode 3:

√

Contribution from the web is given by:

√

where

Shear buckling factor is obtained from shear buckling factor curve for non-rigid end post

and given by:

√

where

√

and is the shear buckling coefficient for the web panel ):

)

where

(

)

√(

)

)

This gives the contribution from the web to shear resistance to:

Contribution from the flanges is taken into consideration when as follows:

( (

)

)

47

where and = flange which provides the least axial resistance and ,

is the moment of resistance of the cross section consisting of the effective area

of the flanges

)

(

)

This gives the total design shear resistance to:

Verification:

Design in the serviceability limit state (ULS)

This worked example deals only with verification related to the longitudinal load effects

in the bridge and bridge functionality under normal use. Deformations affecting bridge

appearance, vibrations and fatigue are not considered here.

Permanent loads

The road pavement thickness, which covers the concrete is assumed to have a value of 0,1 m

and weight of . Characteristic permanent loads acting on one girder are as

followed:

(see 6.3.1)

(see 6.3.1)

Variable loads

Variable loads acting on the load bearing girders considered in this example are traffic loads.

Calculations are based on load model 1 (LM1), where the carriageway is divided into national

lanes, each 3 m wide (Swedish Standard Institution, 2007). Load model LM1 consists of

double-axle concentrated loads (tandem system, TS) along with uniformly distributed loads

(UDL) associated with notional lanes, which magnitude varies between different lanes. Load

values are adjusted by factors and , which are specified nationally, where ) is lane

number. Load magnitudes as well as corresponding factors are presented in the table below:

48

Table 8 – Load magnitudes and corresponding factors

Notional lane number UDL, [kN/m2] Factor TS, [kN] Factor,

1 9 0,7 300 0,9

2 2,5 1,0 200 0,9

3 2,5 1,0 100 0,9

Estimation of how much of the uniformly distributed load goes to one beam is done according

to Figure 62. In order to obtain most adverse effects only two notional lanes with widths of 3

respective 2,325 m are considered.

Moment around the girder to the right gives following:

Thus, each one of the girders must be designed for a load of:

The effect of tandem loads is calculated in the same manner according to Figure 63.

Figure 62 – Load model 1 for UDL where 0,5 m is required at each end to accommodate a safety barrier

q1k q2k

0,5 1,325 1,675 2,325

49

)

Thus, each one of the girders must be designed for a load:

Bending moment and shear force

Decisive load positions of the traffic load are dependent on the expected load effect and can

be determined from influence diagrams presented in Figure 64 to Figure 66, where traffic

loads are located arbitrarily in the longitudinal direction. Influence diagrams are created for

the beam sections where the bending moment forces as well as shear forces are expected to be

highest, that is, at the distance of 0,2L from the end supports as well as over the internal

support.

Figure 64 – Influence line for moment force in beam at x =0,2L

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0,1

0,12

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

η

β=x/L

Figure 63 – Load model 1 for TS where 0,5 m is required at each end to accommodate a safety barrier

1,0 2,0 1,0 1,825

Q1k Q1k Q2k

50

Figure 65 – Influence line for moment force in beam at internal support

Figure 66 – Influence line for reaction force in beam at internal support

The characteristic bending moment in the beams at the distance of 0,2L is calculated in

accordance to the Figure 67 where the maximum effect is obtained when UDL is placed over

one span while TS is present at the distance of 0,2L.

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

η

β=x/L

-1,2

-1

-0,8

-0,6

-0,4

-0,2

0

0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00

η

β=x/L

Figure 67 – Load model for maximum bending moment in beam at x=0,2L

0,5L 0,5L

0,2L

UDL

Selfweight

TS

51

Magnitude of characteristic bending moment in the span caused by the different actions is as

followed:

)

)

Most unfavorable load combination in the SLS is given by the following formula:

With consequence class set to 1,0 the design bending moment is the span becomes:

( )

The maximum load effect when bending moment at the internal support is considered is

calculated when UDL is placed over entire bridge length while TS is located at the distance of

0,27L from the end support, as shown in Figure 68.

Magnitude of characteristic bending moment over the internal support caused by the different

actions is as followed:

)

( ) ))

The design bending moment is:

( )

In this worked example only shear forces which occur over the internal support are

considered. According to Figure 66, UDL placed along bridge entire length as well as TS

positioned slightly to the left from the intermediate support give largest shear force values.

Figure 68 – Load model for maximum bending moment over the internal support

0,5L 0,5L

0,27L

UDL

Selfweight

TS

52

Magnitude of shear force close to the internal support caused by different actions is presented

below:

)

( ) ))

The design shear force is calculated to:

( )

Bending moment and shear force capacity

In order to estimate the bending moment and shear force capacity, an effective width of the

concrete slab must be calculated. With distance between shear studs 0,3 m and span length

26,6 m the effective slab width, , in the span and over the intermediate support is 3,4

respective 3,1 m (Swedish Standard Institution, 2009). Concrete strength class used for the

bridge is C35/45 and the deck is doubly reinforced in the longitudinal direction. The

reinforcing steel comprises B500B high bond bars with 16 mm diameter and characteristic

yield strength of = 500 N/mm2. Distance between the reinforcement bars is 0,14 m. Figure

69 shows a reinforced cross section for a steel-concrete composite bridge.

Bending moment and shear force capacity values are determined with help of the Excel

documents provided by Fredrik Carlsson from Reinertsen and only final results are presented.

Calculated results are following:

Figure 69 – Bridge cross section where denotes the effective slab width over one load bearing girder

beff

30020

Ø16s140

53

Cross section reduction

The calculated values showed that the cross section of the bridges is over dimensioned and

that its reduction is possible. The spacing of the shear studs in the direction transverse to the

shear force should be 2,5 to 4 times the diameter of the studs for solid slabs and other cases

respectively (Swedish Standard Institution, 2009). Moreover, the distance between shear

connectors and the edge of the flange of the beam should be greater than 25 mm which means

that the minimum width of the flange with 2 rows of studs should be at least 100 mm.

Based on the obtained results the dimensions of the top flange could be reduced to 350x14

mm in the span and 350x23 mm at the support while bottom flange dimensions could be

decreased to 500x25 mm in the span and 625x34 mm at the support.

6.3.2. Bridge 1768

Forces acting on the main girders as well as cross section capacity of Bridge 1768 are

calculated in the same way as for Bridge 1385. The results at ULS are following:

Design in the ultimate limit state (ULS)

VEd,support = 1,26 MN

MEd,span = 4,23 MNm

MEd,support = 7,54 MNm

VRd,support = 4,67 MN

MRd,span = 9,46 MNm

MRd,support = 17,4 MNm

Design in the serviceability limit state (ULS)

VEd,support = 3,86 MN

MEd,span = 18,33 MNm

MEd,support = 16,57 MNm

VRd,support = 4,67 MN

MRd,span = 24,5 MNm

MRd,support = 23,1 MNm

Cross section reduction

Based on the obtained results the dimensions of the top flange in the span and over support

can be reduced to 350x14 and 350x23 mm respectively. Bottom flange dimensions in the span

and over support can be decreased to 500x25 mm and 625x34 mm.

54

6.4. Results

Table 9 shows summary of the results from the calculations performed on Bridge 1385 and

Bridge 1768.

Table 9 – Acting forces and cross section capacity of Bridge 1385 and 1768 at ULS and SLS

Bridge 1385 Bridge 1768

ULS SLS ULS SLS

VEd,support [MN] 0,89 2,84 1,26 3,86

MEd,span [MNm] 2,92 12,80 4,23 18,33

MEd,support [MNm] 4,67 10,95 7,54 16,57

Original dimensions

VRd,support [MN] 3,89 3,89 4,67 4,67

MRd,span [MNm] 10,79 21,9 9,95 24,20

MRd,support [MNm] 11,89 14,15 14,50 22,39

Reduced top flange

VRd,support [MN] 3,61 3,61 4,44 4,44

MRd,span [MNm] 5,63 21,6 7,89 23,8

MRd,support [MNm] 7,67 11,52 10,13 17,70

Reduced bottom flange

VRd,support [MN] 3,74 3,74 4,52 4,52

MRd,span [MNm] 8,82 18,1 7,94 24,30

MRd,support [MNm] 9,71 11,07 13,52 17,44

For original dimensions of Bridge 1385 the load capacity factors for the critical shear force,

bending moment in the span as well as over the internal support were 22, 24 and 43%

at ULS correspondingly 73, 58 and 77% at SLS. For reduced top flange dimensions

the factors increased to 24, 39 and 88% at ULS as well as 79, 58 and 95% at SLS

and led to lateral torsional buckling of the bridge girders. Reduction of the bottom flange

dimensions gave the capacities to 24, 33 and 50% at ULS as well as 76, 59 and 96%

at SLS, however; did not have affect the buckling behavior of the beams. As before

the flange reduction, girders failed due to shear buckling close to the internal

support. Buckling behavior of the girders in both cases is shown in Figure 70.

Figure 70 – Buckling of Bridge 1385 with reduced top flange (a) and bottom flange dimensions (b)

The load capacity factors for Bridge 1768 at ULS and SLS were 28, 45 and 41% and 83, 75

and 72% respectively for original bridge dimensions. The factors increased to 28, 54 and 68%

55

at ULS as well as 87, 77 and 95% at SLS for reduced top flange dimensions and led

to similar buckling response of the main girders as in case of Bridge 1385, namely, beam

buckling between the braces. Reduction of the bottom flange dimensions gave the capacities

to 28, 53 and 56% at ULS as well as 87, 75 and 95% at SLS and caused shear buckling of the

web in the mid-span of the girders. Buckling behavior of Bridge 1768 after cross section

reduction is shown in Figure 71.

Figure 71 – Buckling of Bridge 1768 with reduced top flange (a) and bottom flange dimensions (b)

6.5. Discussion and conclusions

6.5.1. Discussion

Conducted calculations showed that the cross section of the chosen multi-span bridges was

possible. However, due to the fact that accidental loads as well as some variable loads were

ignored in the calculation process, the real load capacity factors could be slightly higher.

Nonetheless, the authors are positive that the cross sections of the chosen multi-span bridges

were not properly designed in relation to the loads acting on the bridge.

The shear buckling of the webs could also be caused by relatively large number of braces

utilized to control the twist of the cross section taking into consideration cross section

dimensions which resulted in unrealistic high proportions between cross section capacities

and design values. As presented in Table 9 the load capacity of the beams was quite large

compared to the actions the bridges were exposed to. On average, the distance between the

braces was 6,9 m and 6 m in Bridge 1385 and 1768 respectively. A quick calculation showed

that if the number of braces was decreased by 1 for each bridge, the girders would still

withstand all the loads applied to them both at ULS and SLS.

The authors believe that the relatively large cross section dimensions and choice of the

bracing system could be a result of calculation methods used to design the bridges. Both

structures were designed according to Swedish structural design code BRO2004 which could

have had a significant influence on the final bridge dimensions and design of the bracing

system.

The analyses also showed that for the chosen bridges the loads at ULS did not have any

influence on the design of the bridges. In both cases, the critical loads which governed the

cross section dimensions and probably number of torsional braces used to stabilize the girders

occurred at SLS.

56

Another interesting finding was that for the chosen bridges the loads at ULS did not have any

influence on the design of the bridges. In both cases, the loads which governed the cross

section dimensions and probably number of torsional braces used to stabilize the beams

occurred at SLS.

6.5.2. Conclusions

The following conclusions may be drawn:

The analyzed multi-span bridges were overdesigned considered LTB both at ULS

and SLS and reduction of their cross section was possible

Only reduction of the top flange dimensions influenced the buckling behavior of

the bridges at ULS

Reduction of the torsional braces in both bridges was also a possibility due to

relatively high load capacity factors

The analyzed multi-span bridges were designed for the serviceability loads

57

7. Parametric study – Comparison of FEM and Equations (1.1) - (1.3)

7.1. Introduction

It was observed throughout this study that the results obtained with Eq. (1.1) - (1.3) differ to

some extent from the FE results. The aim of this appendix, therefore, is to investigate how

cross section geometry, number of torsional braces and distance between the main girders

influence the relative error between the methods.

7.2. Methods

In order to see how the specific parameters affect the critical bending moments a number of

parametric studies are performed where only one parameter is changed at a time. Because of

the geometric simplicity as well as relatively good agreement between the equations presented

in this paper and FEM, Bridge 1530 is chosen for the analyses. Firstly, seven different

analyses are done in which number of braces varies from 1 to 7. In each of the analyses the

length of the girders is increased by a distance of 2 m until it reaches 54 m. Secondly, a

comparison of the critical bending moments between the two methods is done for fourteen

different UPE steel channels. The profiles selected for the analyses are chosen from steel

profile tables with accordance to European Standards. Furthermore, analyses and comparison

are made for varying cross section dimensions as well as the distance between the adjacent

girders.

58

7.3. Results

7.3.1. Brace number and distance variation

As shown in Figure 72, the critical bending moment curves obtained with help of Eq. (1.1)

and FEM are almost identical when only one torsional brace is utilized to prevent the

distortion of the cross section. Both curves follow the same path; however, the theoretical

bending moment is for the most part 12-18% lower that the moment calculated by FEM. Both

approaches predict local buckling of the beams as the first buckling mode (see Figure 72).

Figure 73 – First buckling mode of beams braced by one torsional brace for beam lengths 12, 22 and 32 m

Figure 72 – Critical bending moments of beams braced by one torsional brace vs. beam length

0,00

0,50

1,00

1,50

2,00

10 20 30 40 50 60

Mcr

/Mp

l

Beam length [m]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

Local

buckling Shear

buckling

59

When torsional bracing system comprises two torsional braces, the results obtained with the

two methods show lots of similarities. Both curves follow almost exactly the same path

(see Figure 74) and the relative difference between the critical bending moments is around

8-15%. The methods predict accordingly local buckling between the braces as the primary

buckling mode.

Figure 75 – First buckling mode of beams braced by two torsional braces for beam lengths 20, 28 and 36 m

Figure 74 – Critical bending moments of beams braced by two torsional braces vs. beam length

0

0,5

1

1,5

2

20 25 30 35 40 45 50 55

Mcr

/Mp

l

Beam length [m]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

Local

buckling

60

As shown in Figure 76, the relative error between the two methods increases when three

torsional braces are utilized to control the twist of the cross section. Curves look very alike

but the difference between the results gets larger and oscillates between 14-19%. Despite the

differences, however, both methods predict the same buckling modes, that is, system buckling

for beam lengths up to 22 m and local buckling for larger beam lengths.

Figure 77 – First buckling mode of beams braced by three torsional braces for beam lengths 22, 30 and 40 m

Figure 76 – Critical bending moments of beams braced by three torsional braces vs. beam length

0,00

0,50

1,00

1,50

2,00

20 25 30 35 40 45 50 55

Mcr

/Mp

l

Beam length [m]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

Local

buckling

Shear

buckling

61

In case when four torsional braces are installed at intermediate positions both solutions show

a good agreement when critical buckling moment is considered (see Figure 78).

The relative error between the two methods reaches up to 18 % for beam lenghts up to 23 m

but it quickly decreases to 7% for greater lenghts. The methods, though, predict slightly

different buckling behaviour of the bridge model. According to the theoretical solutons given

by Eq. (1.1) - (1-3), system buckling takes place for beam lenghts between 20 and 45 m and

local buckling between the brace points for larger beam lenghts. Results obtained by FEM;

however, point to system buckling as the only buckling mode.

Figure 78 – Critical bending moments of beams braced by four torsional braces vs. beam length

Figure 79 – First buckling mode of beams braced by four torsional braces for beam lengths 23, 30 and 40 m

0,00

0,50

1,00

1,50

2,00

2,50

3,00

20 25 30 35 40 45 50 55

Mcr

/Mp

l

Beam length [m]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

System

buckling

Shear

buckling

62

As shown in Figure 80, the outcome of FEM and theoretical approach on calculation of the

critical bending moment of beams braced by five torsional braces give very similar results

when type of buckling is considered. Both solutions accordingly predict a single wave

buckling of the beam system. Nevertheless, the relative error between the two solutions is as

large as 20% for beam lengths 27 m and 15% when the beam length exceeds 45 m.

Figure 80 – Critical bending moments of beams braced by five torsional braces vs. beam length

Figure 81 – First buckling mode of beams braced by five torsional braces for beam lengths 25, 30 and 40 m

0,00

0,50

1,00

1,50

2,00

2,50

3,00

20 25 30 35 40 45 50 55

Mcr

/Mp

l

Beam length [m]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

Shear

buckling

System

buckling

63

When bracing system comprises six torsional braces bending moment curves obtained with

respective method show relatively large dissimilarities in terms of critical bending moment,

as shown in Figure 82. For beam lengths between 26 and 28 m the relative difference between

the results is approximately 10% but the relative error gets as large as 25% with increasing

beam length. Despite fairly big differences between the results, both methods point to system

buckling as the first buckling mode.

Figure 83 – First buckling mode of beams braced by six torsional braces for beam lengths 26, 32 and 40 m

Figure 82 – Critical bending moments of beams braced by six torsional braces vs. beam length

0,00

0,50

1,00

1,50

2,00

2,50

3,00

20 25 30 35 40 45 50 55

Mcr

/Mp

l

L [m]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

System

buckling

Shear

buckling

64

Figure 84 shows the critical buckling moment curves of girders braced by seven torsionall

braces. Both methods show quite good agreement, that is, the relative error between the two

methods is in range of 10% only for beam lenghts up to 32 m. When the beam length is

increased the differences between the results get as big as 29%.

Figure 85 – First buckling mode of beams braced by seven torsional braces for beam lengths 27, 35 and 45 m

Figure 84 – Critical bending moments of beams braced by seven torsional braces vs. beam length

0,00

0,50

1,00

1,50

2,00

2,50

3,00

20 25 30 35 40 45 50 55

Mcr

/Mp

l

L [m]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

System

buckling

Shear

buckling

65

7.3.2. Brace stiffness variation

As shown in Figure 86, theoretical solution and FEA show a good agreement for most of the

UPE steel channels chosen for the analysis. The difference between the results lies in range of

10% for brace stiffness 0,5 – 4 MN/rad (UPE 80 – UPE 240). However, the curves drift apart

with increasing brace stiffness resulting in errors as large as 27% for UPE 400. Despite quite

big differences between the bending moment values both methods indicate similar buckling

responses of the main beams, in this case buckling of the entire system.

Figure 87 – First buckling mode of beams braced by UPE 80, 140, 200, 270, 330 and 400 steel profiles

Figure 86 – Critical bending moment vs. brace stiffness variation

0,00

0,50

1,00

1,50

2,00

2,50

3,00

0,00 1,00 2,00 3,00 4,00 5,00 6,00

Mcr

/Mp

l

Brace stiffness [MN/rad]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

System

buckling

66

7.3.3. Cross section variation

When thickness of the bottom flange is considered both methods give similar results only

when the flange thickness remains between 20 and 29 mm, see Figure 88. For higher

thickness values the error between the two methods increases and reaches up to 22% before

shear buckling takes place. Nevertheless, both methods accordingly indicate a system

buckling as a first buckling mode.

Figure 88 – Critical bending moment vs. bottom flange thickness

Figure 89 – First buckling mode of the girders with bottom flange thickness 20, 30 and 47 mm respectively

0,00

0,50

1,00

1,50

2,00

2,50

3,00

15 25 35 45 55 65 75 85 95

Mcr

/Mp

l

Thickness of the bottom flange [mm]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

System

buckling

Shear

buckling

67

When bottom flange width is considered, both methods show quite considerable disagreement

for very low width values. As shown in Figure 90, the theoretical results for bottom flange

widths between 100 and 200 mm are considerably lower than the results obtained with FEM.

According to the theoretical solution, the main girders buckle between the bracing points for

bottom flange dimensions within the range given above. The numerical analysis; however,

points to buckling of the twin-girder system giving 30% higher critical bending moment.

Nevertheless, both curves look identical for bottom flange widths greater than 250 mm.


Figure 90 – Critical bending moment vs. bottom flange width

1,00

1,50

2,00

2,50

3,00

0 100 200 300 400 500 600 700 800 900

Mcr

/Mp

l

Bottom flange width [mm]

FEM

EQ 1.1

EQ 1.2

EQ 1.3System

buckling Shear

buckling

68

As shown in Figure 92, bending moment values obtained with help of theoretical equations

and FEM differ for fairly low and large flange thickness values. For the analyzed values the

relative error between the approaches is no largen than 15%; however, the thicker the top

flange becomes the more the two curves diverge. Both methods accordingly predict buckling

of both girders.


Figure 92 – Critical bending moment vs. top flange thickness

0,00

0,50

1,00

1,50

2,00

2,50

3,00

0 10 20 30 40 50 60 70 80

Mcr

/Mp

l

Top flange thickness [mm]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

Shear

buckling

System

buckling

69

According to Figure 94, the methods show quite a big disagreement when the width of the top

flange changes in dimenions. The relative error concerning critical bending moment gets as

big as 250% for widths lower than 250 mm. Nonetheless; buckling behaviour of the beams

obtained with the methods is almost identical, that is, local buckling for the flange widths up

to 350 mm and system buckling for larger flange widths.

Figure 95 – First buckling mode of the girders with top flange width 250, 32, 400, 425, 525 and 650 mm

respectively

Figure 94 – Critical bending moment vs. top flange width

0,00

0,50

1,00

1,50

2,00

2,50

3,00

50 150 250 350 450 550 650

Mcr

/Mp

l

Top flange width [mm]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

System

buckling

Local

buckling

70

When varying depth of the web is considered both methods show lots of similarities, see

Figure 96. Bending moment curves follow fairly similar paths; however, the bigger web depth

value, the more the curves diverge from each other. For the most part, the error between the

methods lays within 10%, but for depths larger than 1350 mm the error exceeds 20%. Despite

the differencs in the results both methods point to system buckling as the first buckling mode.

Figure 97 – First buckling mode of the girders with web depth 500, 1500 and 1550 mm respectively

Figure 96 – Critical bending moment vs. web depth

0,00

0,50

1,00

1,50

2,00

2,50

3,00

400 700 1000 1300 1600 1900

Mcr

/Mp

l

Web depth [mm]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

System

buckling

Shear

buckling

71

Also for changing web thickness both methods give comparatively similar results. As shown

in Figure 98, when thickness of the web is within 25 and 60 mm obtained curves run next to

each other. For larger web thickness values the curves diverge giving an error in range of

23%. Buckling behaviour obrained with help of the theory and FEM shows that system

buckling takes place for all the webb thickness values higher than 15 mm.

Figure 99 – First buckling mode of the girders with web thickness 11, 41 and 83 mm respectively

Figure 98 – Critical bending moment vs. web thickness

0,00

0,50

1,00

1,50

2,00

2,50

3,00

0 10 20 30 40 50 60 70 80 90 100

Mcr

/Mp

l

Web thickness [mm]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

Shear

buckling System

buckling

72

7.3.4. Distance between girders

As shown in Figure 100, critical bending moment values obtained with help of theoretical

equation as well as FEM differ with changing distance between the girders. Both methods

give fairly close reults when the load bearing girders are spaced 1,5 – 3 m from each other.

For larger ditances the relative error increases to 30 % and remains constant.

Figure 101 – First buckling mode of the girders spaced 2, 4 and 6 m from each other

Figure 100 – Critical bending moment vs. distance between girders

0,50

0,80

1,10

1,40

1,70

2,00

1,50 2,50 3,50 4,50 5,50 6,50

Mcr

/Mp

l

Distance between girders [m]

FEM

EQ 1.1

EQ 1.2

EQ 1.3

System

buckling

73

7.4. Discussion and conclusions

The results of conducted analyses show that the relative error between numerical method and

the theoretical equations presented in this paper depends on many factors and its magnitude

differs from case to case.

When number of braces considered, the results are satisfactory for beams braced with 1-5

diaphragms. The difference between calculated bending moments is at its maximum 20% but

generally lies in range of 10-15%. The error; however, increases with higher number of braces

and for 6 and 7 torsional braces utilized to control the twist of the cross section the differences

between the approaches can reach up 28 % regardless beam length.

The results are also reasonable for varying brace stiffness, distance between the girders and

cross section dimensions exclusive top flange width. Both solutions predict the same buckling

behavior of the double-girder system as well as the relative dissimilarities between the

bending moment curves are not greater than 30%.

Significant differences between the approaches are noticeable for top flange width less than

350 mm. Despite the fact that both methods predict the same buckling behavior of the main

girders, exceptionally big error, which the authors do not have any explanation for, occurs.

According to the results, a relative difference between the critical bending moments obtained

with the two methods is as big as 250% when girders buckle locally. For flange widths over

350 mm when system buckling takes place, the results are close to identical.

Case studies have shown that the critical buckling moment of torsionally braced beams

depends on many factors and that the theoretical solutions presented in this paper almost

exactly predict buckling behavior of the double-girder system braced by diaphragms, yet not

their critical buckling moment capacity. In order to decrease the final error between the two

approaches and make the theoretical formulas more universal, the authors suggest some

limitations concerning bridge geometry and distance between the intermediate braces

depending on number of braces to be considered (see Table 10).

Table 10 – Bridge geometry limitations

Number of braces

75

References

Gilchrist, C. I., Yura, J. A. and Frank, K. H., (1997), Buckling Behavior of U-Shaped Girders,

Report No. 1395-1, Center for Transportation Research, University of Texas at Austin,

July

Chanakya, A., (2009), Design of Structural Elements: Concrete, Steelwork, Masonry and

Timber Design to British Standards and Eurocodes, Taylor & Francis, Oxon

Crocetti, R. and Mehri H., (2012), Bracing of Steel-Concrete Composite Bridge During

Casting of the Deck. In: Nordic Steel 2012 Construction Conference. Oslo, Norway, 5-7

September 2012

European Committee for Standardization, 2006. ENV 1993-1-1:1992 Eurocode 3: Design

of Steel Structures Part 1.1: General rules and rules for buildings. CEN, Management

Centre; rue de Stassart 36, B-1050 Brussels

Helwig, T., Frank, K. and Yura, J. (1997), Lateral-Torsional Buckling of Singly

Symmetric I-Beams, J. Struct. Eng., 123(9), 1172–1179.

Pettersson. O., (1971), Knäckning, Bulletin 24, Division of Structural Mechanics and

Concrete Construction, Lund Institute of Technology

SCI (2011), Stability of Steel Beams and Columns, Silwood Park, Ascot, Berkshire

Swedish Standard Institute, 2007. SS-EN 1991-2 Eurocode 1: Actions on structures –

Part 2: Traffic loads on bridges. No Place of Publication: SSI

Swedish Standard Institute, 2008 (1). SS-EN 1993-1-1:2005 Eurocode 3 - Design of steel

structures – Part 1-1: General rules and rules for buildings. No Place of Publication: SSI

Swedish Standard Institute, 2008 (2). SS-EN 1991-1-1:2005 Eurocode 1 – Actions on

structures – Part 1-6: General actions – Actions during execution. No Place of

Publication: SSI

Swedish Standard Institute, 2008 (3). SS-EN 1993-1-5:2006 Eurocode 3: Design of steel

structures – Part 1-5: Plated structural elements. No Place of Publication: SSI

Swedish Standard Institute, 2009. SS-EN 1994-2:2005 Eurocode 4: Design of composite

steel and concrete structures – Part 2: General rules and rules for bridges.

No Place of Publication: SSI

Swedish Standard Institution, 2010. SS-EN 1990 Eurocode: Basis of structural design.

No Place of Publication: SSI

Taylor, A.C. and Ojalvo, M. (1966), Torsional Restraint of Lateral Buckling, Journal

of the Structural Division, ASCE, ST2, April, pp. 115-129

76

Timoshenko, S. and Gere, J. (1961), Theory of Elastic Stability, McGraw-Hill,

New York

U.S. Department of Transportation (2012), Steel Bridge Design Handbook: Bracing

System Design, Publication No. FHWA-IF-12-052 – Vol.13

Yura, J. (2001). Fundamentals of Beam Bracing, AISC Engineering Journal, First Quarter

Yura, J. A., Helwig T., Herman, R. and Zhou, C. (2008), Global Lateral Buckling of I-

shaped Girder Systems, Journal of Structural Engineering, Vol. 134, No. 9

Yura, J. A. and Phillips, B. (1992), Bracing Requirements for Elastic Steel Beams, Report

No. 1239-1, Center for Transportation Research, University of Texas at Austin, May

Yura, J. A., Phillips, B., Raju, S. and Webb, S. (1992), Bracing of Steel Beams in

Bridges, Report No. 1239-4F, Center for Transportation Research, University of Texas

at Austin, October

77

Appendix A - Brace stiffness

In this appendix effective torsional brace stiffness of all the bridges is calculated. Eq.s

(1.4), (1.5), (1.8), (1.9), (1.10), (1.11) and (1.12) are used.

A.1. Bridge 1530

Following data is used to determine torsional stiffness of the system comprising four

diaphragms:

Torsional brace stiffness:

Web distortional stiffness:

(

)

In plane flexibility of girders:

Effective torsional brace stiffness:

Effective torsional stiffness per

unit length of girder:

78

A.2. Bridge 1020

Following data is used to calculate torsional stiffness of the system comprising seven Z-type

braces:



(

)





79

A.3. Bridge 983

Following data is used to calculate torsional stiffness of the system comprising seven Z-type

braces:



(

)





80

A.4. Bridge 1385

Following data is used to calculate torsional stiffness of the system comprising three K-type

braces per span:

FEM is used to obtain a value of torsional brace stiffness due to reverse design of the

diagonals.



(

)





81

A.5. Bridge 1768

Following data is used to calculate torsional stiffness of the system comprising four K-type

braces per span:



(

)





82

A.6. Bridge 917

Because of changing distance between flange centroids, web thickness as well as moment of

inertia of the girders, the authors do not know how to calculate effective torsional brace

stiffness of the bridge. For this reason, only torsional brace stiffness is presented. Due to

untypical design of diagonals a value of torsional brace stiffness is obtained by means of

FEM.


A.7. Bridge 1052

FEM is used to obtain torsional brace stiffness of plates.


83

Appendix B - Bracing location

Within this appendix exact location of torsional brace of Bridge 917 and 1052 is presented.

B.1. Bridge 917

Figure 102 – Brace placement in bridge; sections 1 (top) to 5 (bottom)

F1 F1 F1S1

7375 7375 7375 6885

F1 F1 S2 F1 F1

6000 6000 6000 6000

510490

F1 F1 F1 F1

7490 7500 7500 7500 7510

F1 F1 S2 F1 F1

6000 6000 6000 6000

510490

6875 7375 7375 7375

F1 F1 F1 S1

84

B.2. Bridge 1052

Figure 103 – Brace placement in bridge sections 1 (top), 3 (middle) and 5 (bottom)

F1 F1 F1

S1

5500 6000 6625 6625 6200

F1 F1

F1

300

F1

6200

F1F1 F1 F1

8000 7000 8000 6200

55006000662566256200

85

Appendix C - Critical bending moment of Bridge 1530

Within this appendix step by step calculations of the critical bending moment of Bridge 1020

are presented.

C.1. Critical bending moment according to Eq. (1.1), (1.2) and (1.3)

Critical bending moment of the bridge is evaluated with help of Eq. (1.1), (1.2) and (1.3),

where the lowest of the obtained values indicates the critical buckling moment of the bridge

girders

As the cross section changes its dimensions along the bridge length, mean dimensions of the

load bearing girders are calculated (see Figure 104):

An approximate torsional bracing effect for singly-symmetric cross sections can be estimated

by replacing moment of inertia with (see Figure 105).

Figure 105 – c and t factors in singly symmetric I-shape

(Gilchrist, 1997)

Center of gravity for the cross section is calculated to 0,465 m, which gives c and t values to:

y

x

compression flange

c

t

tension flange

Figure 104 – I-shape dimensions

Material properties are as followed:

htot

Ml

bt

Mlbb

tw

tt

tb

86

Effective moment of inertia around weak axis is:

where

Critical bending moment of singly-symmetric cross section leading to buckling between the

braces is determined according to exact solution suggested by Helwig et al. (1997). Buckling

length which is a distance between the braces is set to 6m.

√ * √

+

Effect of load height and non-uniform shape of the moment diagram of unbraced beam

(see Figure 106) is obtained with help of formula below, proposed by Helwig (Helwig et al.,

1997):

) )

where

0,624

1,089 m

where

Figure 106 – Factor , , and (Yura, 2001)

L/4 L/4 L/4 L/4

MA MB MC

Mmax

87

Factor becomes:

√

where

) ( (

)

)

(

)

where

Critical buckling moment is as followed:

√ * √

+

FEA have revealed that when the buckling moment for twin girder systems is calculated,

Eq. (1.2) needs to be multiplied by factor 2:

Critical bending moment of simply supported unbraced beam with singly-symmetric cross

section is determined according to Eq. (1.1) where the beam length is set to entire span length:

Effect of load height and non-uniform shape of the moment diagram of braced beam follows

the same calculation steps as when calculating -factor. The factor ; however, is

√

)

√

√ ( √

)

88

calculated independently for each beam length between the braces, and the lowest of

obtained values is used in the calculations:

Critical bending moment corresponding to buckling of the entire system is:

√

Critical buckling moment causing global buckling of the bridge is obtained with Eq. (1.3) to:

√

) )

(see Appendix A)

89

C.2. Modified Euler’s column buckling formula

Critical axial load in the compression flange is obtained with Eq. (2.4):

)

where

(buckling between the braces is assumed)

(see Table 2)

(moment of inertia about weak axis of the top flange)

According to Eq. (2.3) the slenderness factor becomes:

√

where

Reduction factor for buckling, , is now determined:

√

where

( ( ) ) for

With given reduction factor, maximum normal force acting on the compression flange is

calculated:

Bending moment which occurs in one girder due to self-weight of steel and concrete is

determined based on data given below:

Weight of steel

Area of I-girder

90

Weight of concrete

Width of concrete slab

Height of concrete slab

The bending moment value is calculated according to STR Set B (Swedish Standard

Institution, 2010) load combination:

Capacity check is carried out where cross section class is controlled first , as shown in

Table 11.


Web Top Flange

SC1

SC2

SC3

CLASS

The cross section is in class 3, thus elastic section modulus is used.

<

In order to compare the results with other solutions an equivalent value for the critical

bending moment is calculated with help of the following relationship:

where denotes earlier determined slenderness factor for the compression flange.

91

C.3. Simplified method according to NCCI SN002

Non-dimensional slenderness is obtained with help of Eq. (2.7):

√

√

where

(Table 4) for (Table 3)

√

√

where

√

for

√(

)

( )

where

for

(Table 4) for ,132 (Table 3)

(for shear centrum located at the height of 0,22m)

92

√ √

For Class 3 cross section, which gives = 1,16, non-dimensional slenderness becomes:

√

√

Conservative assumption where = 1,0, = 0,9, = 1,0, = 1,0 and √ = 1,0 gives

The reduction factor for is:

√

where

( ( ) )

for

The design buckling resistance moment is calculated with help of Eq. (2.5):

The equivalent value for the critical bending moment is calculated below:

93

Appendix D - Critical bending moment of Bridge 1020

Within this appendix step by step calculations of the critical bending moment of Bridge 1020

are presented.

D.1. Critical bending moment according to Eq. (1.1) - (1.3)

Mean values of cross section dimensions are explained in Figure 107 and presented below:

Figure 107 – Trapezoidal shape dimensions

7,77 m

Material properties are as followed:

According to Helwig (Helwig et al., 1997), the critical buckling moment causing the beams to

buckle between the bracing points is:

√ * √

+

where

(for details see Appendix C)

y

bb

d

y2

y1

e e'

C

h2 htot

Shear center

s s

94

√


where

) ( (

)

)

(

)

√

where

Critical buckling moment becomes:

√ * √

+

Critical bending moment of the system:

where


)

√

√ * √

+ (for details see Appendix C)

√


95

(for details see Appendix A)

Global buckling is calculated for half of the cross section according to Figure 108.

Figure 108 – S1-factor used for calculation of critical global buckling moment (Crocetti and Mehri, 2012)

Critical buckling moment leading to global buckling is:

√

y

C

S1

y1

xC1

96

D.2. Modified Euler’s column buckling approach

)

where

(buckling between the braces is assumed)

(see Table 2)

(moment of inertia about weak axis of the top flange)

√

where

√

where

( ( ) )

Moment due to concrete and self-weight:

Weight of steel:

Area of I-Girder:

Weight of concrete:

Width of concrete:

Height of concrete:

97

According to STR Set B load combination (Swedish Standard Institution, 2010):

Compression members class classification in presented in table below:


Web Top Flange

SC1

SC2

SC3

CLASS

Reduction of the web is done in accordance with SS-EN 1993-1-5:2006 (Swedish Standard

Institution, 2009):

⁄

√

Since the reduction factor is calculated as followed:

)

)

98

Following values are obtained:

Center of gravity

Effective moment of inertia

Effective section modulus

Stress during concreting phase

Axial force in the compression flange <


99

D.3. Simplified method according to NCCI SN002

The slenderness value is estimated according to Eq. (2.7) for conservative assumptions:

= 1,0, = 0,9, = 1,0, = 1,0 and √ = 1,0:

√

√

where

The reduction factor becomes:

√

where

( ( ) ) for

The design buckling resistance moment is obtained with Eq. (2.5):


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