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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
iv
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
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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
6
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).
12
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 and Figure 17.
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.
3.1.2.2. Technical aspects
The bridge is a box-girder bridge with open-topped trapezoidal cross section interconnected
by seven Z-type braces. The structure is fabricated of three steel subgrades: S420, S355
and S235. The first steel quality was used for constructing the box-girder while the other two
for fabrication of the bracing system as well as plate panels and end plates.
While the steel girder height is kept constant, the size of the flange and web varies along the
length of the bridge. The dimension of the top flanges located close to end supports is 48x600
mm and increases to 48x700 mm into the span to reach 50x800 mm in the mid-section. The
bottom flange and the webs change their dimensions in identical manner as shown
in Figure 18. The dimensions of the bottom flange vary from 30x2440 mm at bridge ends
to 50x2440 mm in its middle while the webs change their thickness from 19 to 17 mm. The
height of the box-girder has a constant value of 1930 mm.
Figure 18 – Structural steel distribution for the main girders of Bridge 1020
Bracing system comprises solid plates and braces of Z-type used at the support locations
respective as intermediate braces. As shown in Figure 19, the braces are spaced at equal
intervals of 8 m with the exception of two braces closest to the supports located at the distance
of 7,1 m from the bridge ends.
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
Shear force and bending moment diagrams during bridge construction phase are illustrated in
Figure 22 and Figure 23.
Figure 22 – Bending moment diagram for bridge 1020 in [MNm]
Figure 23 – Shear force diagram for bridge 1020 in [MN]
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.
3.1.3.2. Technical aspects
The bridge over Upperuds River is a box girder bride with open-topped trapezoidal cross
section. The bridge is made up of two types of steel subgrades: S355N and S275JR which
were used for fabrication of box-girder and end plates respective intermediate braces.
The box-girder changes in dimensions along the entire bridge length. The width and thickness
of the top flanges close to the supports are 25x560 mm and increase gradually to 40x770 mm
at 11,4 m into the span reaching 45x770 mm in the mid-section. Bottom flange as well as the
webs changes their dimensions in identical way as shown in Figure 24. The height of the
box-girder has a constant value of 2400 m.
Figure 24 – Structural steel distribution for the main girders of Bridge 983
Solid plates with thickness of 15 mm are used at the support locations while internal braces
of Z-type are used to prevent distortion of the cross section. Internal braces comprise HEA
120 and HEA 140 profiles and are spaced approximately every 8 m with exception of the
braces closest to the supports which are located 7.1 m from them. Plate panels at the brace
locations are used to strengthen the webs and the bottom flange. Additional vertical stiffeners
on the outside of the main girders are installed only above the supports. The brace positioning
along the bridge length is shown in Figure 25.
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
Effective torsional stiffness per unit length of girder obtained with help of Eq. (1.12)
is (see Appendix A A.3).
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
Shear force and bending moment diagrams during bridge construction phase are illustrated in
Figure 28 and Figure 29.
Figure 28 – Bending moment diagram for bridge 983 in [MNm]
Figure 29 – Shear force diagram for bridge 983 in [MN]
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.
3.2.1.2. Technical aspects
The bridge main load-bearing system consists of two I-shaped girders connected by
intermediate cross-frames. The bridge is fabricated of two types of steel: S460M and S355J2.
The first steel type was used for fabrication of the I-girders while the other type was used
to manufacture the braces, vertical stiffeners and end plates.
The total length of each of the two I-girders is 53,2 m whereas the length of the spans vary
due to skewed support installed in the middle. As a result, the free span length on respective
side is 25,6 m and 27,6 m. The main girders are equally spaced in the transversal direction
by a distance of 4 m and have a constant cross section along the entire length, see Figure 30.
The girders have a depth of 1300 mm with thickness of the web is 15 mm. Both top
and bottom flange are 600 mm wide and have a thickness of 30 respectively.
Figure 30 – Structural steel distribution for the main girders of Bridge 1385
Two different types of bracing systems are utilized in the construction, i.e. crossbeams over
the supports and K-type bracing in the spans. The crossbeams consist of standard HEB 800
steel profiles with different lengths depending on the location while the K-type bracing
system is made up of four HEA 100 beams. Vertical stiffeners are used on the inside of the
I-beams at the location of the torsional bracings and additional stiffeners placed on the outside
of the main girders are present only at the support locations. The center to center distance
between the braces is 6,65 m with the exception of the braces near the internal support where
the distance changes to either 5,65 or 7,65 m due to skewed support. Bracing placement
is presented in Figure 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
Effective torsional stiffness per unit length of girder obtained with help of Eq. (1.12)
is (see Appendix A A.4).
Shear force and bending moment diagrams during bridge construction phase are illustrated in
Figure 35 and Figure 36.
Figure 35 – Bending moment diagram for bridge 1385 in [MNm]
Figure 36 – Shear force diagram for bridge 1385 in [MN]
-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.
3.2.2.2. Technical aspects
The load-bearing superstructure consists of four similarly sized longitudinal I-shaped girders.
The total length of each of the girders is 60 m and the free span length between the supports
is 30 m on both sides. The structure is fabricated of two types of steel: S460 and S355. The
S460 steel type was used for fabrication of the bottom and top flanges in midsection while
S355 was used to manufacture other components of the I-girders as well as the braces.
The cross section of the girders is constant along the bridge length with exception of the part
over the internal support where upper and bottom flanges change their dimensions, as shown
in Figure 37.
Figure 37 – Structural steel distribution for the main girders of Bridge 1768
The girders are spaced 4,5 m and 5,5 m apart across the width of the bridge and are braced
together two-two, namely, no bracing between the inner girders is present. Support bracing
is provided by horizontal crossbeams while intermediate bracing is provided by K-type
bracing. The center to center distance between the braces as well as plane view of the bridge
is shown in Figure 38.
Figure 38 – Plan view of Bridge 1768 where S1, S2 and F1 symbolize bracing above the end supports, bracing
over internal support respective span bracing
Bracing type used at the support locations vary in size and dimensions. Crossbeams at the end
supports have a length of 4 m and a height of 850 mm while the height of the crossbeam used
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
Effective torsional stiffness per unit length of girder obtained with help of Eq. (1.12)
is (see Appendix A A.5).
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
Shear force and bending moment diagrams during bridge construction phase are illustrated in
Figure 42 and Figure 43.
Figure 42 – Bending moment diagram for bridge 1768 in [MNm]
Figure 43 – Shear force diagram for bridge 1768 in [MN]
-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.
3.2.3.2. Technical aspects
The load-bearing system of the bridge consists of four longitudinal I-girders with a total
length of 145,5 m. The length of the side spans is 41,5 m while the central span is 62,5 m
long. The structure is built of two types steel subgrades: S420 and S275JR. The S420 steel
quality was used for fabrication of the webs and flanges while S275JR was used to
manufacture other bridge components as braces and stiffeners. Bridge varying cross section is
shown in Figure 44.
Figure 44 – Structural steel distribution for the main girders of Bridge 917
The girders are spaced 6 m from each other and are braced two-two, that is, no braces are
present between inner girders. Entire bracing system is provided by bracing of K-type that
differs in dimensions and design depending on location along the span. The plane view of the
load-bearing system including different types of braces is shown in Figure 45. Precise center
to center distance between the braces is presented in Appendix B - Bracing location B.1.
Figure 45 – Plan view of Bridge 917 where S1, S2 and F1 symbolize bracing above the end supports, bracing
over internal support respective span bracing
Support bracing comprise four HEA profiles which differ in dimensions. Intermediate
bracing, on the other hand, is made of VKR and U-shaped profiles. Location of the vertical
stiffeners used on the inside of the I-girders follows exactly placement of the torsional braces.
Web stiffeners utilized on the outside of the main girders are present only at the support
locations. Detailed design of braces is shown in Figure 46 and Figure 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
Shear force and bending moment diagrams during bridge construction phase are illustrated in
Figure 49 and Figure 50.
Figure 49 – Bending moment diagram for bridge 917 in [MNm]
Figure 50 – Shear force diagram for bridge 917 in [MN]
-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.
3.2.4.2. Technical aspects
The construction has an open-topped trapezoidal cross section and is manufactured of three
types of steel: S460M, S420M and S355JR. The first two types were used for fabrication
of the web and flanges while the third one for plates and bracing system.
The box girders vary in dimensions along the bridge length. The cross section is stiffest
in negative bending moment regions, that is, close to the internal supports where the
dimensions of the top and bottom flanges increase to 37x650 mm respectively 25x3300 mm.
The height of the box-girder has a constant value of 2,4 m. The manner in which the
box-girders change its dimensions is shown in Figure 51.
Figure 51 – Structural steel distribution for the main girders of Bridge 1052
Center to center distance between top flanges as well as the distance between the webs at the
level of the bottom flange are constant throughout the entire bridge length and have a value
of 4,5 m respective 3,1 m. A total number of twenty braces is utilized to prevent distortion
of the cross section. Center to center distance between the braces varies from 5,3 to 6,625 m
in side spans and from 5,3 to 8 m in the central span. A plane view of the bracing system
is shown in a figure below. Exact placement of the braces is shown in Appendix B - Bracing
location B.2.
Figure 52 – Plan view of Bridge 1052 where S1, S2 and F1 symbolize bracing above the end supports, bracing
over internal support respective span bracing
Bracings system comprises solid plates with thickness of 12 and 8 mm placed at the support
respective discrete locations. Unlike intermediate bracing plates, support bracing plates are
strengthened with vertical stiffeners. A detailed design of the bracing system is presented
in the Figure 53 and Figure 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).
Shear force and bending moment diagrams during bridge construction phase are illustrated in
Figure 56 and Figure 57.
Figure 56 – Bending moment diagram for bridge 1052 in [MNm]
Figure 57 – Shear force diagram for bridge 1052 in [MN]
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.
42
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 91 – First buckling mode of the girders with bottom flange thickness 100, 875 and 900 mm respectively
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 93 – First buckling mode of the girders with bottom flange thickness 23, 26 and 68 mm respectively
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
74
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:
Torsional brace stiffness:
Web distortional stiffness:
(
)
In plane flexibility of girders:
Effective torsional brace stiffness:
Effective torsional stiffness per
unit length of girder:
79
A.3. Bridge 983
Following data is used to calculate torsional stiffness of the system comprising seven Z-type
braces:
Torsional brace stiffness:
Web distortional stiffness:
(
)
In plane flexibility of girders:
Effective torsional brace stiffness:
Effective torsional stiffness per
unit length of girder:
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.
Torsional brace stiffness:
Web distortional stiffness:
(
)
In plane flexibility of girders:
Effective torsional brace stiffness:
Effective torsional stiffness per
unit length of girder:
81
A.5. Bridge 1768
Following data is used to calculate torsional stiffness of the system comprising four K-type
braces per span:
Torsional brace stiffness:
Web distortional stiffness:
(
)
In plane flexibility of girders:
Effective torsional brace stiffness:
Effective torsional stiffness per
unit length of girder:
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.
Torsional brace stiffness:
A.7. Bridge 1052
FEM is used to obtain torsional brace stiffness of plates.
Torsional brace stiffness:
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.
Table 11 – Cross section class control
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
√
(for details see Appendix C)
where
) ( (
)
)
(
)
√
where
Critical buckling moment becomes:
√ * √
+
Critical bending moment of the system:
where
(for details see Appendix C)
)
√
√ * √
+ (for details see Appendix C)
√
(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:
Table 12 – Cross section class control
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 <
The equivalent value for the critical bending moment is calculated below:
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):
The equivalent value for the critical bending moment is calculated below: