Q
d

Qd

Parametric design of class 4 steel cross
sections
A comparison between the reduced stress method and the
effective width method

Master’s thesis in Master Program Structural engineering and building technology

MALIN STENING
MAURICE KRONBERG

Department of Architecture and Civil Engineering
CHALMERS UNIVERSITY OF TECHNOLOGY
Master’s thesis ACEX30-18-84
Gothenburg, Sweden 2020





Master’s thesis ACEX30-20-xx

Parametric design of class 4 steel cross sections

A comparison between the reduced stress method and the effective
width method

MALIN STENING
MAURICE KRONBERG

Department of Architecture and Civil Engineering
Division of Structural Engineering
Lightweight Structures Group

Chalmers University of Technology
Gothenburg, Sweden 2020



MALIN STENING
MAURICE KRONBERG

© MALIN STENING, 2020.
© MAURICE KRONBERG, 2020

Supervisors: Gustav Good, Victor Andersson, Ramboll
Examiner: Professor Mohammad al-Emrani, Department of Architecture and Civil
Engineering

Department of Architecture and Civil Engineering
Division of Structural Engineering
Lightweight Structures Group
Chalmers University of Technology
SE-412 96 Gothenburg
Telephone +46 31 772 1000

Cover: An illustration of a steel bridge.

Department of Architecture and Civil Engineering
Gothenburg, Sweden 2020

iv



Abstract
Eurocode provides two different methods to design steel structures in cross section
class four which is the effective width method and the reduced stress method. The
first mentioned method is the most common method in Sweden and consists of a
rather time consuming calculation procedure. The second method results in a rather
conservative design. The aim with this thesis is to achieve a better understanding
of when and how to use either of the methods, and to achieve an effective design
process with the help of a developed design tool.

The thesis is conducted by a literature study about the two methods and the existing
studies of improved versions of the methods. The literature study is followed by a
parametric study where the methods are compared. The reduced stress method is
both calculated analytically and combined with the FE-programme Sofistik with its
geometry generated with Grasshopper, whereas the effective width method consists
of hand calculations.

Keywords: structural engineering, bridge engineering, steel cross section class 4,
parametric design, reduced stress method, effective width method.

v





Sammanfattning
Eurocode föreslår två olika beräkningsmetoder för att räkna på tvärsnitts klass fyra
stålkonstruktioner. Den första metoden är flitigt använd i Sverige och kallas för den
effektiva bredd metoden. Ett Problem med metoden är att den i vissa fall kan bli
tidskrävande. Den andra metoden kallas för den reducerade spännings metoden och
är känd för att vara den lite mer koncervativa utav de två. Målet med arbetet är att
identifiera var och när ena metoden kan vara mer fördelaktig än den andra samt att
uppnå en tidseffectiv design process med hjälp av ett genomarbetat design verktyg.

Examensarbetet består av en litteraturstudie för att först och främst erhålla en djup
förståelse kring de två undersökta metoderna, men även kunskap om studier som
har gjorts kring förbättringar av metoderna. Tre olika konstruktioner är undersökta
i det här examensarbetet där konstruktionernas kapacitet med de olika metoderna
är jämförda. Parametriska studier har genomförts på ett lådtvärsnitt för att under-
söka vilka parametrar som påverkar resultaten och hur metoderna bäst används.

vii





Acknowledgements
First of all we would like to thank our supervisors Victor Andersson and Gustav
Good at Ramboll as well as our examiner Mohammad Al-Emrani for their support
and guidance throughout this master’s thesis project. Their commitment not only
to the progress of our work but also our well being has been crucial for us. We would
also like to thank our opponents Amer Bitar and Pedro Cobucci for their thoughts
and questions as a part of the development of this thesis. Finally we would like to
thank the rest of Ramboll and in particular the bridge division at Ramboll for thier
support and interest in our work.

This report is performed as a final assignment written during the last six months of
our five years long education. We have gotten the possibility to apply our knowledge
from previous years at Chalmers to a specific but extensive project which has been
challenging but of course very educational. Moreover we have gotten the opportunity
to learn more about different programs and calculation methods which we’ve never
used before. We will bring this with us in our future work as engineers. We are
proud to get the opportunity to publish this thesis and we hope that the subject
will be further investigated in the future.

Malin Stening and Maurice Kronberg, Gothenburg, June 2020

ix





Contents

List of Figures xiii

1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 State of the art 3
2.1 Design models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Effective width method . . . . . . . . . . . . . . . . . . . . . . 3
2.1.2 Reduced stress method . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2.1 Von Mises yield criterion . . . . . . . . . . . . . . . . 6
2.1.3 Column- versus plate like behaviour . . . . . . . . . . . . . . . 6

2.2 Scientific studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Elaboration with load shedding . . . . . . . . . . . . . . . . . 8
2.2.2 Improvement of the interaction formula . . . . . . . . . . . . . 9

2.2.2.1 The V factor . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2.2 Addition to the V factor . . . . . . . . . . . . . . . . 11

2.2.3 RSM in structural design . . . . . . . . . . . . . . . . . . . . . 12
2.2.3.1 RSM on a high velocity railway station roof . . . . . 12
2.2.3.2 A combination formula for EWM . . . . . . . . . . . 13

2.2.4 Conclusions from the literature study . . . . . . . . . . . . . . 13

3 Single plate 15
3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Capacity comparison . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 I-Beam with two longitudinal stiffeners 19
4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.2 FE-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.3 Convergency study . . . . . . . . . . . . . . . . . . . . . . . . 22

xi



Contents

4.2 Result & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.1 Capacity comparison . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.2 Buckling analysis . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Box section 29
5.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1.2 FE-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.1.3 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1.4 Parametric study . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 Result & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2.1 Top plate thickness . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.1.1 Capacity comparison . . . . . . . . . . . . . . . . . . 36
5.2.1.2 Buckling analysis . . . . . . . . . . . . . . . . . . . . 39

5.2.2 Top plate stiffener thickness . . . . . . . . . . . . . . . . . . . 41
5.2.2.1 Capacity comparison . . . . . . . . . . . . . . . . . . 41
5.2.2.2 Buckling analysis . . . . . . . . . . . . . . . . . . . . 43

5.2.3 Side plate thickness . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.3.1 Capacity comparison . . . . . . . . . . . . . . . . . . 45
5.2.3.2 Buckling analysis . . . . . . . . . . . . . . . . . . . . 47

5.2.4 Side plate height . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2.4.1 Capacity comparison . . . . . . . . . . . . . . . . . . 49
5.2.4.2 Buckling analysis . . . . . . . . . . . . . . . . . . . . 51

5.2.5 Distance between vertical stiffeners . . . . . . . . . . . . . . . 53
5.2.5.1 Capacity comparison . . . . . . . . . . . . . . . . . . 54
5.2.5.2 Buckling analysis . . . . . . . . . . . . . . . . . . . . 56

6 Conclusion 59
6.1 Observed differences . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Pros and cons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.3 FEM-implementation and the parametric tool . . . . . . . . . . . . . 60
6.4 Recommendations for further studies . . . . . . . . . . . . . . . . . . 60

A Singel plate calculation I

B I-beam V
B.1 Changed stiffener width . . . . . . . . . . . . . . . . . . . . . . . . . V

C Box girder CI
C.1 Original cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . CI
C.2 Changed top plate thickness . . . . . . . . . . . . . . . . . . . . . . . CXII
C.3 Changed top plate stiffener thickness . . . . . . . . . . . . . . . . . . CXCIV
C.4 Changed side plate thickness . . . . . . . . . . . . . . . . . . . . . . . CCXXX
C.5 Changed side plate height . . . . . . . . . . . . . . . . . . . . . . . . CCLXVI
C.6 Changed vertical stiffener distance . . . . . . . . . . . . . . . . . . . . CCCVII

xii



List of Figures

2.1 Plate subjected to compression [ORI, 2019] . . . . . . . . . . . . . . . 3
2.2 Figure of von Mises yield criterion [Palaniswamy, 2012] . . . . . . . . 6
2.3 Plate like- vs column like behaviour of an stiffened rectangular plate

[Matsumoto, 2000] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Interaction column- and plate like behaviour [Johansson et al., 2007] . 7
2.5 Recistance levels according to [Johansson et al., 2007] . . . . . . . . . 8
2.6 Comparison different methods, α = 1 [Braun, 2010] . . . . . . . . . . 10
2.7 Comparison different methods, α = 3 [Braun, 2010] . . . . . . . . . . 11
2.8 Arch roof, high velocity railway station [Majowiecki and Pinardi, 2010] 12

3.1 Figure of the studied plate . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Flow shart of the calculation procedure with both methods . . . . . . 16

4.1 Cross section and section of the I-beam . . . . . . . . . . . . . . . . . 20
4.2 FE-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Maximum mid deflection of the beam versus mesh size . . . . . . . . 22
4.4 Load capacity of the I-beam with EWM and RSM . . . . . . . . . . . 23
4.5 Slenderness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.6 Two different buckling modes of the bridge with varying stiffener height. 26

5.1 Oceanpiren . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 Cross section of box girder . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3 Simplification of cross section . . . . . . . . . . . . . . . . . . . . . . 31
5.4 FE-Model of the bridge girder . . . . . . . . . . . . . . . . . . . . . . 32
5.5 Von mises convergency of the boxsection . . . . . . . . . . . . . . . . 33
5.6 Egenvalue convergency of the boxsection . . . . . . . . . . . . . . . . 34
5.7 Illustration of the studied parameters . . . . . . . . . . . . . . . . . . 35
5.8 Load capacity of the box girder bridge with the EWM and the RSM

with varying top plate thickness. . . . . . . . . . . . . . . . . . . . . 37
5.9 Slenderness with varying top plate thickness. Buckling mode a, b and c. 38
5.10 Slenderness with varying top plate thickness. Buckling mode a, b, c . 39
5.11 Three different buckling modes with varying top plate thickness. . . . 40
5.12 Load capacity with EWM and RSM with eigenvalues from Teddy for

different top plate stiffener thickness. . . . . . . . . . . . . . . . . . . 42
5.13 Slenderness with different top plate stiffener thicknesses. . . . . . . . 43
5.14 Three different buckling modes of the bridge with varying top plate

stiffener thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xiii



List of Figures

5.15 Load capacity of the box girder bridge with the EWM and the RSM
with varying side plate thickness. . . . . . . . . . . . . . . . . . . . . 46

5.16 Slenderness with varying side plate thickness. . . . . . . . . . . . . . 47
5.17 Three different buckling modes of the bridge with varying side plate

thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.18 Load capacity of the box girder bridge calculated with the EWM and

the RSM for different height of the side plates. . . . . . . . . . . . . . 50
5.19 Slenderness with different side plate heights. . . . . . . . . . . . . . . 51
5.20 Three different buckling modes of the bridge with varying side plate

height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.21 Load capacity with EWM and RSM with eigenvalues and stresses

taken from Teddy for different vertical stiffener distance. . . . . . . . 54
5.22 Slenderness with different vertical stiffener distance. . . . . . . . . . . 55
5.23 Three different buckling modes of the bridge with varying distance

between transverse stiffeners. . . . . . . . . . . . . . . . . . . . . . . . 56

xiv



1
Introduction

When designing a steel structure according to Eurocode, regards needs to be taken
for the risk of it’s compressed parts to buckle. This is accounted for in Eurocode by
a cross section classification with four different classes. Cross section class 4 means
that local buckling will occur before the yield stress is reached. The capacity of
class 4 cross sections shall consequently be limited with regard to this. [SIS, 2011a]
Modern steel bridges are often very slender in order to achieve a slim design and
save material. As a result their cross section tends to be in class 4 which reduces
their capacities.[Zizza, 2016]

1.1 Background
Eurocode provides two different methods to design steel structures in cross sec-
tion class 4, "Effective width method"(EWM) and "Reduced stress method"(RSM).
[SIS, 2011b]

EWM is a frequently used method to calculate the strength of steel plates in bridge
designs. Mostly the method corresponds quite well to the post buckling capacity
which is why the method often is preferred [Gren et al., 2019]. Plates subjected to
compression has capacity left after the plate starts to buckle. The EWM accounts
for this effect by calculating an effective width and introducing the effective cross
section.[Hansen et al., 2010] Some parts remains active whereas some parts of the
cross section are considered not to withstand load efficiently.[SIS, 2011b]

RSM, which is briefly described in EC3-1-5 Chapter 10, does not consider redistribu-
tion of stresses within the element which in some cases under predict the capacity of
the cross section which then gets over conservative. On the other hand, calculating
the effective cross sections is not needed which is a rather time consuming process.
Another benefit compared to EWM is that the method allows for calculating the
buckling capacity of multiaxially loaded structures which often is the case for mod-
ern bridges with a slender design.[Reis et al., 2019]

A problem is that the national application of Eurocode, [TSFS 2018:57, 2018] chap-
ter 19, does not recommend to use the RSM in Sweden. However, the method is
not used to the same extent whereas the EWM is well developed and used in design
of steel plates. Despite this, the RSM in combination with a FE-programme and
parametric design could lead to an optimized and more efficient design procedure.

1



1. Introduction

1.2 Aim
The aim of this thesis is to investigate how the two calculation methods for calcu-
lating steel structures with cross section class 4, differ with regard to capacity and
calculation time. How the already existing improvements of the methods influences
this and if they are safe and reliable in design. To elaborate further with calculation
time and capacity a parametric design tool based on RSM will be created. The idea
of the tool is to quickly generate a geometry to implement into a FE-software. The
outcome of this thesis should be a better understanding of when and where to use
either of the methods and a quick and effective design process with the help of the
design tool.

1.3 Objectives
A literature study and a parametric study, both with the focus on comparing the
two methods, will be made to investigate differences in capacity and calculation
time. To further reduce the calculation time and to optimize the parametric study,
a parametric design tool made in grasshopper will be created. During this process
a couple of questions shall be answered:

• Is there any difference between the methods? And if so when and where?
• What are the pros and cons of the methods?
• Could capacity be gained by combining RSM with FE?
• Will the parametric tool save time for a designer?

1.4 Limitations
The thesis will focus on steel structures with cross-section class 4 only. The calcu-
lations will only consider buckling capacity. The aim of this thesis is to compare
but not further improve the methods in Eurocode, or any existing improvements of
them in relevant litterateurs. LT-buckling will not be within the scope of this thesis
and the FE-analyses will not cover any non-linear behavior.

1.5 Method
The thesis is conducted by a literature study to receive an overview of the current
state of the art and help identifying possible problem formulations within the in-
vestigated area. Moreover the literature study will be followed up by a parametric
study where the two methods will be compared to each other on real case structures
to investigate the capacity. The case study will be preformed by Mathcad calcula-
tions and FE-analyses which will result in a parametric design tool with the purpose
of optimizing the design process. The idea of the design tool is to create a program
in the design software Grasshopper that generates a geometry into the FE-software
Sofistik where design- and critical stresses are automatically calculated.

2



2
State of the art

Steel plates in cross section class four can be designed with two suggested design
methods, the effective width method and the reduced stress method. Research and
numerous experiments have been performed over years which resulted in updated
versions of the methods. This chapter will contain background information regard-
ing the two methods, different improvements of the methods and real examples of
structures calculated with the RSM.

2.1 Design models
The methods originates from investigations by different scientists, like von Karaman,
Winter and von Mises. The following chapter contains background information
regarding the two earlier mentioned design models.

2.1.1 Effective width method
The width of the cross section which can resist load efficiently is established accord-
ing to figure 2.1 in case of uniform compression and internal compression elements
[ORI, 2019].

Figure 2.1: Plate subjected to compression [ORI, 2019]

3



2. State of the art

The EWM is based on Von Karaman’s investigations in 1932. His findings lead to a
formula which enables to calculate an effective width, which does not depend on the
total width nor the magnitude of the compressive stress. Karaman concluded that
there are stiffer parts along the edges because of redistribution of stresses. The mid
part in between the stiff edges can not resist load efficiently in case of plate buckling
whereas the stiffer parts can.[ORI, 2019] Several experiments was performed and a
modified formula regarding the effective width taking imperfections into account was
developed by Winter.[Structures and Ii, 2008] The equation was further modified in
the 1968 for cold formed steel which ended up with the same equation as eq. 4.2 in
EC3-1-5.

Formula by Von Karaman[Hansen et al., 2010]:

be = π√
3(1− v2)

t

√
E

σe
(2.1)

Formula by Winter[Hansen et al., 2010]:

be = 1.9t
√
E

σe
(1− 0.574 t

b

√
E

σe
) (2.2)

The updated version according to EC3-1-5 equation 4.2 [SIS, 2011b] when the plate
is subjected to uniform compression is:

be
b

=
√
σcr
σe

(1− 0.22
√
σcr
σe

) (2.3)

The calculation process regarding stiffened plates in CSC four starts with the check
of the local buckling of the plate fields between stiffeners under the assumption that
the stiffeners are stiff enough. Parts of the steel plate are reduced due to buckling
which leads to an ineffective part in between.

However, checks needs to be performed regarding the stiffeners since the stiffeners
could buckle itself locally or globally. If the stiffeners are not stiff enough, a part
of the plate and the stiffener can behave like a column exposed to axial forces with
transverse stiffeners as supports. The transverse stiffeners need to be stiff enough
to ensure a rigid support to the longitudinal stiffeners. The behaviour can however
also be somewhere in between column like or plate like behaviour which is explained
in detail in chapter 2.1.3. [SIS, 2011b]

4



2. State of the art

2.1.2 Reduced stress method
The reduced stress method is inspired by the Von Mises yield criterion where the
properties from the gross cross section is calculated. A reduced stress limit which
is lower than the yield stress is calculated based on the stress distributions of the
gross cross-section. Both the properties and the stress limit are calculated sepa-
rately for every part of the cross section and the weakest part with regard to this
will govern the whole cross section. This means that a large difference in capacity
between elements will be disadvantageous since no load shedding is accounted for.
[Gren et al., 2019]

The method is considered as a conservative method since the weakest section is
decisive when calculating the resistance of the whole cross section. The allowable
stress in the plate is reduced and a combination formula established from von mises
yield criterion enables RSM to check multiaxially loaded structures. The resistance
is checked according to the following formulas. [Zizza, 2016]

Cross section class three can be assumed if the condition in EC3-1-5 eq.10.1 is
fullfilled regarding multiaxially loaded plates. Unlike the EWM, the slenderness in
the RSM is calculated with a global slenderness formula according to [SIS, 2011b]
Chapter 10(3);

λp =
√
αult,k
αcr

=

√√√√ fy
σeq,cr

(2.4)

To calculate the global slenderness, αult,k and αcr needs to be estimated with the
following formulas, taken from [SIS, 2011b] Chapter 10(4) and 10(6) respectively.
Alternatively using a FE-program to obtain eigenvalues and von mises stresses to
calculate αult,k and αcr.

1
α2
ult,k

= [σx,Ed
fy

]2 + [σz,Ed
fy

]2 − [σx,Ed
fy

][σz,Ed
fy

] + 3[τEd
fy

]2 (2.5)

1
αcr

= 1 + ψx
4αcr,x

+ 1 + ψz
4αcr,z

+ [[1 + ψx
4αcr,x

+ 1 + ψz
4αcr,z

]2 + 1− ψx
2α2

cr,x

+ 1− ψz
2α2

cr,z

+ 1
α2
cr,τ

] 1
2 (2.6)

The reduction factor is the smallest value of ρx(αp), ρz(αp) and χw(αp) calculated
with the slenderness, λp, according to EC3-1-5 chapter 4.5.4 (1) and chapter 5.2(1).
The interpolation between column- and plate buckling to calculate the final reduc-
tion factor, ρc, is calculated according to chapter 2.1.3. Equation 2.7 is used to
verify if the condition is met which leads to the final reduction factor [SIS, 2011b]:

[ σxEd
fy/γM1

]2 + [ σzEd
fy/γM1

]2 − [ σxEd
fy/γM1

][ σzEd
fy/γM1

] + 3[ τEd
fy/γM1

]2 ≤ ρ2 (2.7)

An alternative method is to use the following interaction equation (Equation 2.8)
instead of the smallest reduction factor. The reduction factor is calculated with the
interaction formula in chapter 2.1.3 for each direction.

[ σxEd
ρxfy/γM1

]2 + [ σzEd
ρzfy/γM1

]2 − [ σxEd
ρxfy/γM1

][ σzEd
ρzfy/γM1

] + 3[ τEd
χwfy/γM1

]2 ≤ 1 (2.8)

5



2. State of the art

2.1.2.1 Von Mises yield criterion

Reduced stress method is based on von mises yield criterion[Gren et al., 2019]. It
can be concluded that yielding in a material will occur in case von Mises stress ex-
ceeds or is the same as the yield limit of a material loaded in tension. The condition
can however be investigated experimentally to determine whether the material is
yielding.

Von Mises yield criterion [Palaniswamy, 2012]:

σ =
√

1
2[(σxx − σyy)2 + (σyy − σzz)2 + (σzz − σxx)2 + 6(τ 2

xy + τ 2
yz + τ 2

zx)] (2.9)

The condition regarding von Mises formula leads to a surface or a elliptical curve in
case of plane stress(figure 2.2 ).

Figure 2.2: Figure of von Mises yield criterion [Palaniswamy, 2012]

2.1.3 Column- versus plate like behaviour

Different behaviour occurs depending on the plates geometry and dimensions. Ac-
cording to [SIS, 2011b], two types of behaviours are the plate buckling and column
buckling with an interaction equation in between to achieve a behaviour closer to
reality. Sometimes the stiffened plates behaves in a manner in between column and
plate like buckling depending on how stiff the plate is(figure 2.3). A large plate with
several stiffeners can buckle like a orthotropic plate even though the stiffness in dif-
ferent directions will differ. The interaction equation is however used in the EWM
and the RSM in the same way to obtain a final reduction factor. [Matsumoto, 2000]

6



2. State of the art

Figure 2.3: Plate like- vs column like behaviour of an stiffened rectangular plate
[Matsumoto, 2000]

Regarding calculation of the reduction factor for a plate with plate like behaviour,
the critical buckling stress needs to be calculated according to EC3-1-5 annex A.1
which is dependent on the degree of stiffening. A stiffer orthotropic plate can how-
ever possibly lead to column like behaviour or affect the reduction factor in the
interaction formula in SS-EN 1993-1-5 equation 4.13 (figure 2.4).[Matsumoto, 2000]

Figure 2.4: Interaction column- and plate like behaviour [Johansson et al., 2007]

The factor ρ is the reduction factor in accordance with the plate like behaviour
whereas χc is the reduction factor in case of column buckling. The factor ξ depends
on the width and length of the plate since the dimensions affects the behaviour of
the plate. A length much shorter than the width will lead to a behaviour similar to
column buckling and if the width increases the plate will successively behave more
in accordance with the plate buckling theory.[SIS, 2011b]

2.2 Scientific studies
Numerous papers has been written with focus on comparing and/or improving the
two methods. The common opinion is that RSM is in most cases more conservative
and therefore not as material efficient. A general conclusion is also that the EWM is
more tedious and not as general as RSM. It’s for example not possible in Eurocode to
combine multiaxial stresses in one calculation with EWM.[Samvin and Skoglund, 2016]
This section will go through interesting findings in relevant literature to get a

7



2. State of the art

better picture of what has been investigated and what conclusions that has been
drawn from it. The literature are scientific papers from 2007 to as recent as 2019
which gives a hint on how relevant this subject is, for example scientific papers like
[Braun and Kuhlmann, 2012], [Zizza, 2016] and [Reis et al., 2019].

2.2.1 Elaboration with load shedding
EC 1993-1-5 Chapter 10 (1) states that the resistance for buckling of a cross section
in CSC 4, calculated with RSM, shall be governed by the weakest element within it.
This is conservative since load shedding between elements is disregarded. In 2007 a
formulation of different resistance levels where proposed in [Johansson et al., 2007]
to achieve a more realistic stress distribution between elements. The resistance level
that can be utilized depends on the amount of strain that can be allowed within the
weakest element. The resistance levels are listed below and illustrated in a stress
strain diagram in Figure 2.5

Level 1 is as the original method in Eurocode where no shedding between elements
is allowed so that the buckling resistance depends on the weakest element.

Level 2 allows a limited stress distribution where load shedding up to the capacity
of the strongest element is considered.

Level 3 allows to increase the resistance of the cross section up to the yield strain of
the weakest element.

Figure 2.5: Recistance levels according to [Johansson et al., 2007]

A master thesis from 2016 that was carried out at KTH compared EWM and RSM
to a non-linear FE analysis [Samvin and Skoglund, 2016]. The RSM calculations
where made with all three resistance levels as well as a proposed forth option. This
proposed option used interpolation between the stress limits of the weakest and the
strongest element to calculate the resistance of the cross section. The comparison
was performed by parametric studies of an I-girder and a box-shaped column where

8



2. State of the art

load cases and slenderness alternated.

They concluded that RSM with level 3 resistance as well as the interpolated resis-
tance both gave similar results to EWM for the cases studied in the report. Although
problems aroused when axial compression and bending moment where combined, the
level 3 resistance calculations for RSM requires an iterative and tedious procedure
to calculate the neutral axis for that case. The interpolation formula is based on
bending resistance for axial stresses and the cross sectional area for uniform com-
pressive stresses respectively. The interpolation option is therefore not possible to
use when bending and axial compression is combined.

Another conclusion, which is quite generally known but nonetheless interesting for
this thesis is that it is very suitable to calculate the resistance using reduction factors
obtained from a FE-analysis in stead of analytically. The capacities gets closer to
reality and the calculations can be preformed fast once the model is built up. The
understanding of the designer becomes important as the governing resistance must
be interpreted from the analysis which can be hard for an inexperienced designer.

For further work on the subject, the authors suggest a similar investigation but on
a whole bridge instead of isolated beams and columns. If a whole bridge is analysed
it will be possible to get a better understanding of how much this deviation in ca-
pacities between methods can affect for example material use.

2.2.2 Improvement of the interaction formula
A couple of papers has been written regarding the conservative capacity prediction
when multiaxial loads are combined according to the interaction formula in Eu-
rocode. Benjamin Brown wrote two articles [Braun, 2010] and [Braun and Kuhlmann, 2012]
where a modification of the formula was proposed and compared to existing stan-
dards. Antonio Zizza suggested an addition to Brown’s modification in [Zizza, 2016]
which later on where tested for high strength steel by [Reis et al., 2019]

2.2.2.1 The V factor

Benjamin Braun proposes an alternative way to the verification equation for mul-
tiaxial loading in Eurocode regarding the reduced stress method. The proposed
change is due to the fact that reduced stress method does not consider redistri-
bution of stresses within elements. The modification should however lead to less
conservative results, by adding a V according to the following formula:[Braun, 2010]

[ σxEd
ρxfy/γM1

]2 + [ σzEd
ρzfy/γM1

]2 − V [ σxEd
ρxfy/γM1

][ σzEd
ρzfy/γM1

] + 3[ τEd
χwfy/γM1

]2 ≤ 1 (2.10)

Where
V = ρxρz if the stresses, σx and σz, are in compression. [Braun, 2010]

9



2. State of the art

Another developed concept is the German standard DIN18800-3 (2) which was com-
pared to EN1993-1-5 and equation 2.10. In the German standard, the slenderness
and reduction factors are calculated discrete by the stresses from each direction,
and the reduction factor depends on the calculated slenderness κx(λx,p), κz(λz,p),
κτ,p(λτ,p). [Braun and Kuhlmann, 2012]

[ |σx,Ed|
σx, Rd

]e2 + [ |σz,Ed|
σz,Rd

]e2 − V [ |σx,Edσz,Ed|
σx,Rdσz,Rd

] + 3[ τEd
τRd

]e3 ≤ 1 (2.11)

Where
σx,Rd = κxfyd
σz,Rd = κzfyd
τRd = κτfyd
e1 = 1 + κ4

x

e2 = 1 + κ4
z

e3 = 1 + κxκzκ
2
τ

V = [κxκz]6 if σx,Ed and σz,Ed are in compression.
V = σx,Edσz,Ed

|σx,Edσz,Ed|
for other cases.

As can be seen in figure 2.6, the result from the modified formula (equation 2.10)
does not differ significantly from the German standard DIN 18800-3 except for
very slender plates. Hence, the proposed formula needs to be more conserva-
tive in case of slender square plates. If the geometry differ from a square, the
modified formula is beneficial since it can account for that effect whereas DIN
18800-3 and EN1993-1-5 can not which leads to conservative and favourable re-
sults respectively.[Braun and Kuhlmann, 2012]

Figure 2.6: Comparison different methods, α = 1 [Braun, 2010]

10



2. State of the art

Figure 2.7: Comparison different methods, α = 3 [Braun, 2010]

The article concludes that the improved interaction formula entails reasonable re-
sults which corresponds quite good to the experiments. Moreover, Benjamin Brown
points out that the tensile effect for plates or some part of the plates loaded in
tension is not enough studied.[Braun and Kuhlmann, 2012]

2.2.2.2 Addition to the V factor

Antonio Zizza came up with an improved application to Benjamin Brauns formula
by introducing a "c", a changed e3, a square root and another equation for "V" when
the plate is subjected to tension. The improved interaction formula was developed
from experiments, proposed changes from the past, numerical analysis and compar-
ison between the buckling rules in Germany and the current design rules EC3-1-5.
[Zizza, 2016] √√√√[ σx

ρc,xfy
]2 + [ σz

ρc,zfy
]2 − V [ σx

ρc,xfy
][ σz
ρc,zfy

] + 3[
√

3τ
χwfy

]e3 ≤ 1 (2.12)

Where
e3 = 1, 25 + 0, 75χ2

w

V = ρc,xρc,z if the stresses, σx and σz, are in compression. [Braun, 2010]

V = 1
ρc,xρ

2−ζx
c,z

if at least one of the stresses,σx and σz, are in tension. [Zizza, 2016]

ρx = c[ 1
λp
− 0,22

λ2
p

]

c = 1, 25− 0, 25ψ ≤ 1, 25

ρc,z = χc,z + [ρz − χc,z]f

f = λ0,5
p ln[ξz + 1]0,9

ξz = σcr,p,z
σcr,c,z

− 1

11



2. State of the art

A recently published paper that focuses on design with high strength steel points out
the lack of availability to account for the positive effects of tensile stresses in buckling
calculations of slender plates subjected to multiaxial stresses.[Reis et al., 2019] This
is an important aspect when it comes to material use especially when high strength
steel is being used. High strength steel is expensive so the design must be as slim
as possible to make it economically advantageous to use it. The paper investigates
how well the improved interaction formula from [Zizza, 2016] describes the positive
effects of tension stresses for high strength steel. The investigation was conducted
by a lab test and a numerical analysis to compare to RSM. It could be concluded
from the investigation that it is beneficial to consider the positive effects of tension
stresses in the reduction factor. The results from RSM and the numerical analysis
agreed well which proves that the improved reduction factor gives a precise and
material efficient prediction of the capacity for the studied example.

2.2.3 RSM in structural design

There are two interesting papers that present cases where RSM have been used
in design of steel structures. A high velocity railway station roof in Florence
[Majowiecki and Pinardi, 2010] and the Queensferry Crossing in Scotland [Gren et al., 2019].
Eurocode did not allow the use of EWM in any of the cases but in the second paper
a proposed approach where applied in order to compare the methods.

2.2.3.1 RSM on a high velocity railway station roof

There are cases when EWM is not applicable, for example when the geometry of the
structure is complex so that multi-axial stresses occurs. The paper [Majowiecki and Pinardi, 2010]
mentions a real case where RSM has been used to verify the design of a railway
roof. The roof is located in Florence and has the shape of a cylindrical vault which
is formed with a grid of romboids as in figure 2.8 below. The roof cantilevers at
the openings and the usage of slender plates imposes reduction of the capacity with
regard to buckling which made RSM suitable for the verification.

Figure 2.8: Arch roof, high velocity railway station [Majowiecki and Pinardi, 2010]

12



2. State of the art

2.2.3.2 A combination formula for EWM

A recently published paper from "The Nordic Steel Construction Conference" in
Copenhagen suggests a combination formula to combine transverse and longitudinal
stresses for EWM. [Gren et al., 2019] The formula is presented below in equation
(2.13). It is very similar to the one for RSM but with some minor changes.√

[ σx
ρxfy/γM1

]2 + [ σz
ρzfy/γM1

]2 + 3[ τ

χwfy/γM1
]2 ≤ 1 (2.13)

The paper is written by Ramboll employees in an attempt to optimize the design of
the Queensferry Crossing which is a stay cable bridge with an orthotropic compos-
ite deck over three internal supports. The location of the stay cable planes creates
transverse stresses in the steel part of the bridge. The transverse stresses in combi-
nation with the global longitudinal stresses "forced" the designers to use RSM as it’s
not possible to preform multiaxial calculations in eurocode with EWM. A bottom
plate from the used design was compared to the same calculated by EWM and a
non-linear FE-analysis. Their findings where that EWM with the combination for-
mula describes a correct physical behaviour and generally leads to a higher capacity
compared to RSM. This combination formula is not yet tested for other cases nor
approved in design but if that will be the case then the attraction for EWM will
increase. It should however be pointed out that the earlier mentioned improvements
on the combination formula from [Braun and Kuhlmann, 2012] and [Zizza, 2016] for
RSM is not used in this investigation since that’s not added into Eurocode either.

2.2.4 Conclusions from the literature study
As of now it is not possible to account for load shedding between plates, stress
distribution within plates and the positive effects of tensile stresses in multiaxialy
loaded plates with RSM in Eurocode. This makes the method over conservative
compared to EWM but as these papers has shown there are suggested improvements
to account for these effects which can make it more appealing and material effective
to use. The method is known to be less tedious, more general and suitable to use
in combination with FE-analyses so it would be advantageous if it’s precision could
match EWM. It would be interesting to compare the EWM with the combination
formula proposed in [Gren et al., 2019] to the RSM with the modification to the
interaction formula in [Zizza, 2016]. This should preferably be done on a whole
bridge structure where some plates are subjected to multi-axial stresses where one
is in tension.

13



2. State of the art

14



3
Single plate

3.1 Method
To justify the calculation procedures on the more complex geometries a single plate
is first studied. The plate is loaded in pure compression on the short edges so the
two methods should give the exact same result for this case. The measurements of
the plate and the loading is presented in figure 3.1 and table 3.1 below.

3.1.1 Description
The single plate is three metres long and two metres wide with a thickness of 20
millimetres. It is loaded with a uniformly distributed load(Qd) on the short edges on
both sides so that a continuous stress field along the length of the plate will occur.
Because of the dimensions and since there is no stiffeners there will only be plate
like buckling. A figure sketch of the plate along with it’s geometrical- and material
properties is presented below in figur 3.1 and table 3.1.

Q
d

Qd

Figure 3.1: Figure of the studied plate

L b t fy E ν γM 0 γM 1
3 m 2 m 20 mm 235 MPa 210 GPa 0.3 1 1

Table 3.1: Measurement of the plate

The calculation procedure for the two methods looks a little different but some parts
coincide. Critical stresses are used to some extent in both methods and the reduc-
tion factors uses the same equations but with different slenderness approaches. An
attempt to express the two procedures with a flow chart is presented in figure 3.2
below. Note that this is the procedure for this particular example, the general idea
is always the same but some parts gets a little more complicated with more complex

15



3. Single plate

structures. An example is that the first assumption in the reduced stress method
calculation is that the transverse stresses and the shear stresses are set to 0. If that’s
not the case, then these stresses need to be interpreted in the amplification factors
along with the longitudinal stress, a formula for this is presented in chapter 10 in
[SIS, 2011b]. The parts that consider column-like behaviour is unnecessary for this
plate since it won’t behave like that but the calculations are needed to be able to
conclude that that’s the case.

Effective width method

Plate like 

slenderness

Effective 

width

Effective 

width

Column like

slenderness

Capacity 

verification

Plate like

Reduction factors

Collumn like

Collumn like

Interaction 

factor

Governing

stresses

Amplification

factors

Slenderness

Capacity

verification

Interaction

Critical stresses

Plate like

Reduced stress method

Figure 3.2: Flow shart of the calculation procedure with both methods

The idea with the flow chart is to follow the blue segment for the Effective width
method and the green for the Reduced stress method. The yellow and red parts are
for critical stresses and reduction factors respectively, these parts are relevant for
both methods why they need to be considered separately for both.

16



3. Single plate

Calculating the effective width is a bit iterative when column like behaviour is
considered. An effective width considering plate like behaviour is needed to calculate
the slenderness for column like behaviour. Both the plate like- and column like
reduction factors are then used in the interaction formula where a new reduction
factor is calculated and used to calculate the final effective width. This is why the
effective width is calculated twice in the flow chart.

3.2 Result

The results are for both methods obtained from analytical calculations of the original
plate according to the procedure presented in section 3.1. The resulting capacity
is the ultimate capacity, in other words, the load that corresponds to at least 99%
utilization of the plate.

3.2.1 Capacity comparison

The capacity comparison between the two methods resulted in almost exactly the
same capacity with a difference of about 0.04 % which is a neglectful difference.
Notable is that the slenderness’s also are close with a difference of 0.03 %. Yet
the slenderness in the EWM is based on geometrical- and material properties of
the plate while the RSM bases it’s slenderness on the two amplification factors
which in turn are based on the governing- and critical stresses. The slenderness is
used to calculate the reduction factor in both methods but the reduction factors
are interpreted diferently in the two methods. The reduction factor in the EWM
is multiplied with the width to obtain an "effective width" which then determines
the normal force capacity. The reduction factor in RSM on the other hand is used
directly into the final verification calculation along with the governing stress to
obtain a "reduced stress". The results from both methods are presented in table 3.2
below.

EWM
λp 1.76
ρp 0.497
σcr.c 8.4356 MPa
σcr.p 75.92 MPa
ξ >1

beff 0.994
Nb.Rd 4672 kN
qEd 2336 kN/m

RSM
σx.Ed 116.85 MPa
σy.Ed 0 MPa
τEd 0 MPa
αcr 0.6497
αult.k 2.011
λp 1.7594
ρp 0.4973
qEd 2337 kN/m

Table 3.2: Results from analysis of the single plate with the EWM and the RSM.

17



3. Single plate

3.3 Discussion
The methods uses different approaches to calculate slenderness and capacity, yet
they obtain almost the exact same result for both. There is a reason for this, the
methods are mathematically the same for this case. Some criterion’s needs to be
fulfilled for this to occur which is the case in the analyzed plate. The structure should
only be conducted by one plate, stresses should only occur in one direction and
normal steel material properties should be used. When this occurs the expressions
for calculating slenderness and capacity contains the same variables and constants
that adds up to the same value. This wont be the case in the structures analyzed
in the following sections. The reason for the difference, which is very small but yet
a difference, is most likely because of rounding within the constants in the methods
or in our calculations.

18



4
I-Beam with two longitudinal

stiffeners

4.1 Method

To investigate a more complex structure with the two methods, an I-beam is studied.
The studied I-beam is an existing bridge girder which has already been checked for its
buckling capacity by Ramboll by using the EWM. The capacities from the existing
calculations will be compared with the results calculated with the RSM. The purpose
is to investigate how the capacities differs with the EWM and the RSM. Since the
beam will not only contain stresses in one direction, the capacities with the two
methods are expected to not give exactly the same results.

4.1.1 Description
The thin-webbed beam has symmetrical flanges and two longitudinal stiffeners in
the compression zone on one side as shown in figure 4.1a. The beam is actually part
of a bridge girder with vertical stiffeners at a distance of 8.914 m but here simplified
as a simply supported beam with four vertical stiffeners at distance 8.914m with
stiff ends which can be seen in figure 4.1b and 4.1c. The comparison is conducted
by applying an evenly distributed load QD which corresponds to the utilization ratio
one, i.e. maximum capacity with regard to buckling. The material properties of the
different elements are presented in table 4.1 and the measurements of the beam in
table 4.2 along with figure 4.1a.

Part fy [MPa] E [GPa]
Web and Stiffeners 390 210

Flanges 380 210

Table 4.1: Material properties of the beam

Measurement wf tf hw tw ws hs ts d1 d2 ss
Length [mm] 560 25 2000 12 200 90 12 500 500 8914

Table 4.2: Measurement of the beam

19



4. I-Beam with two longitudinal stiffeners

hw

tf

wf

ws

hs

ts

ts

tw

d1

d2

(a) Cross section of I-beam
Ss

(b) Section of I-beam

QD

L=Ss
(c) Simplification of I-beam

Figure 4.1: Cross section and section of the I-beam

4.1.2 FE-Model
The FE-model is built up as a 3D structure with both triangular and quadrilateral
shell elements to be able to catch the whole stress flows of the different plates. The
FE-model is presented along with boundary conditions in figure 4.2. The beam is
fixed for translation in y- and z-directions along the line at the left support (Se
figure 4.2b) whereas the right side is fixed in all directions (Se figure 4.2c).

20



4. I-Beam with two longitudinal stiffeners

The top flange is fixed in y-direction along the whole beam length to prevent insta-
bility of the beam. The actual girder would be prevented to move in that direction
by the bridge deck which makes it a fixation close to reality. At the lower part of
the transverse stiffeners, the beam is fixed in y-direction as the actual bridge would
be stayed by bracing (Se figure 4.2d). The longitudinal stiffeners and the rest of
the transverse stiffeners are not fixed in any direction but merely prevented by their
own moment of inertia as well as connection to other elements.

The beam is subjected to an evenly distributed load along the length that is placed
at the web center on top of the top flange. The centred placement of the load is in
this case not really necessary because of the fixation of the top flange but would in
other cases prevent instability problems.

(a) FE model of I-beam

(b) Left BC (c) Right BC (d) Mid BC at transverse
stiffeners

Figure 4.2: FE-Model

21



4. I-Beam with two longitudinal stiffeners

4.1.3 Convergency study
In order to get satisfying accuracy in the calculated results, a mesh convergence
study is performed. The maximum deflection in the middle of the beam from an
arbitrary uniform load is measured for different mesh sizes and a plot of the variation
is presented in figure 4.3 below.

-4.5

-4.4

-4.3

-4.2

-4.1

-4

-3.9

-3.8

-3.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

M
ax

 m
id

 d
ef

le
ct

io
n 

[m
m

]

Mesh size [m]

Convergency

Figure 4.3: Maximum mid deflection of the beam versus mesh size

A maximum tolerance of 1 % difference in deflection between mesh sizes is set as
a limit for convergence. What can be seen is that the deflection does not vary as
much with meshes finer than 0.4m and the difference for finer meshes is below 1%
why a mesh of 0.4m is chosen for the analyses.

4.2 Result & Discussion
The results are based on entirely analytical calculations for the EWM where both
moment capacity as well ass shear capacity are accounted for. Interaction between
moment- and shear capacity is also calculated with EWM. All according to chap-
ter 4-7 in [SIS, 2011b]. The results for RSM is based on FE- calculations where a
linear stress analysis and an eigenvalue buckling analysis are preformed and then
accounted for in the hand calculations according to chapter 10 in [SIS, 2011b].

To investigate the I-beam further, it is of interest to study how the capacity is
influenced by the type of behaviour of the I-beam. If there is any relationship
between the capacity of the beam and the type of behaviour, i.e column like versus
plate like behaviour. For this purpose the two methods are calculated and compared
with different widths on the longitudinal stiffeners. The idea is to achieve more or

22



4. I-Beam with two longitudinal stiffeners

less support for the web to make it behave more or less as a plate. The results are
presented in section 4.2.1 and 4.2.2

4.2.1 Capacity comparison
The two methods are quite close to each other with the biggest difference at about
8% if interaction between moment- and shear capacity is considered in EWM. If
instead moment- and shear capacity are to be calculated separately in EWM the
results will differ more for some stiffener widths. The moment capacity is higher by
a lot in all cases so shear capacity will be governing when separated from moment
capacity. The results are ploted in figure 4.4 below.

Figure 4.4: Load capacity of the I-beam with EWM and RSM

The capacity increases with wider stiffeners regardless of method which is reason-
able since the beam, but mainly the web gains stiffness and support. What is more
interesting is that the moment capacity in EWM is barley changed while the shear
capacity increases a lot. The initial guess would be the other way around since the
stiffeners should increase the webs resistance to normal stress buckling rater than

23



4. I-Beam with two longitudinal stiffeners

shear buckling. The answer to this irregular behaviour is that the beam is, even for
the most narrow stiffeners, sufficiently supported in terms of pure moment capacity.
So an increase in stiffness from the stiffeners will not increase the moment capacity
substantially. The behaviour that is visualized in the figure is that the stiffeners
becomes more and more rigid so that most of the beam action occurs below the
lower stiffener. In a way the beam behaves more and more as an unstiffened, half
as high beam. The bottom stiffener, and possibly some part of the web above it,
acts as the top flange. A verry stiff top flange indeed, as all the part above it is
practically unbendable. The beam stays more or less the same in the context of mo-
ment capacity but is now increasingly robust against shear as the beam approaches
the smaller beam behaviour. The smaller beam doubles in height to width ratio
which explains this high increase of shear capacity. At a stiffener width of 400 mm,
it seems that the behaviour is totally in accordance with the small beam and not
much more shear capacity is left to be gained.

When the RSM curve is compared to the Interaction curve of EWM they are fur-
thest away from each other in the middle of the section where the beam changes
behaviour as previously explained. Almost as if one of the methods fail to capture
the transition between the original beam and the half as high one properly. If so,
could it be EWM that is unnecessarily conservative or RSM that is over predicting?
That would indeed be interesting to test in a lab on actual stiffened beams. The
answer might as well lie in the fact that the interaction calculation is a little bit over
conservative since the maximum shear and moment wont occur at the same section
of the beam.

Another finding is that the RSM curve seems to follow the inclination of the shear
curve at the lower stiffener widths and the moment curve with the wider stiffeners.
which is further investigated by comparing slenderness in figure 4.5 below. The RSM
uses a global slenderness for all elements regardless of buckling behaviour wheras the
EWM uses a different slenderness equation depending on the behaviour and element.

24



4. I-Beam with two longitudinal stiffeners

Figure 4.5: Slenderness

A lot of slenderness calculations are made in EWM but two of them are governing
depending on the stiffener width. Its the slenderness for column buckling of the
stiffeners and normal stress buckling of the top flange that are governing at different
occasions. What is interesting about this is that the slenderness curve for RSM
seems to change inclination at the point where the two EWM slenderness curves
overlap. As if the global slenderness for RSM is influenced by the change in be-
haviour of the beam. Even though the inclination of the column buckling curve in
EWM is a lot steeper than the part of the RSM curve in the same section.

Even more interesting is that this point is close to the point where the inclination
of the RSM curve in Figure 4.4 changes from one that follows the shear capacity
curve to the moment capacity curve in EWM. So it seams that there is a relation
between slenderness and buckling behaviour in both method as well as a relation
between buckling behaviour and capacity growth in RSM. The FE buckling analysis
is presented in the next section to see if the slenderness is describing the buckling
behaviour correctly.

25



4. I-Beam with two longitudinal stiffeners

4.2.2 Buckling analysis
The width of the stiffeners are changed in order to achieve different buckling modes,
which resulted in two governing buckling modes as presented in figure 4.6 below. The
first buckling mode, which occurs for smaller stiffener widths, is however buckling of
longitudinal stiffeners with the traversal stiffeners as supports. The next buckling
mode is buckling of the top flange in compression and it occurs when the longitudinal
stiffeners are wider. This is in agreement with the two governing stiffness parameters
in EFW.

(a) Buckling mode a

(b) Buckling mode b
Figure 4.6: Two different buckling modes of the bridge with varying stiffener
height.

Buckling mode Buckling behaviour Long. stiffener width [mm]
(a) Column buckling 50, 100, 150, 200, 250

of stiffeners.

(b) Normal stress buckling 300, 350, 400, 500
of compression flange.

Table 4.3: Buckling modes of the I-beam with varying stiffener heights.

26



4. I-Beam with two longitudinal stiffeners

The change in buckling behaviour seems to occur between the 250 and 300 mm
stiffeners which coincide quite well with the stiffness curves in Figure 4.5. Note that
the calculated slenderness values are in no way interpreted in the FE analysis which
makes it even more interesting that they are so closely related. The visualized
buckling behaviours in 4.6 makes it easier to understand why the capacity curve
for RSM follows the EWM shear capacity curve for smaller whidts and the EWM
moment capacity curve for wider stiffeners. Not only beacuse buckling mode a
occurs in the web and buckling mode b in the top flange but mode a looks a lot like
shear buckling as well. It might even be that the mode contains some shear buckling
behaviour since combined buckling modes are possible in the analysis. This will be
the case in the next analysed structure where a lot of sub plates and behaviours are
close to each other.

27



4. I-Beam with two longitudinal stiffeners

28



5
Box section

This chapter covers the method comparison on the third and final structure. It is
a section of an actual walking bridge which is under development at Ramboll as
this report is being written. The bridge is called Oceanpiren and will be located in
Helsingborg, Sweden. It is a curved box girder bridge which is stayed with cables
connected to two steel pylons. Some simplifications will be made in the design and
structural behaviour to be able to preform a parametric study within the time frame.

5.1 Method

The box girder bridge is studied with the the EWM analytically and compared with
the RSM in combination with FEM. A stress- and buckling analysis with the FE-
program SOFiSTiK is preformed just as with the I-beam in chapter 4. The analyses
generates dimensionless eigenvalues and Von Mises stresses which are used as input
data in the RSM hand calculations. The uniformly distributed load, Qd, is then
changed until the utilization is just below one. The bridge is then investigated with
varying measurements of the sub parts of the bridge in order to achieve a deeper
understanding of how the two methods are related to each other with different
geometries. This parametric study is preformed with a parametric design tool in
Grasshopper.

5.1.1 Description

The box girder bridge planned at Oceanpiren in Helsingborg is a walking and cycling
bridge. A cable-stayed bridge with a s-shaped bridge deck as in figure 5.1

29



5. Box section

Figure 5.1: Oceanpiren

A cross section of the box girder bridge is presented in figure 5.2 below. There are
vertical stiffeners both in the top plate and the inclined web plates to stiffen for both
positive and negative bending moments. The top plate and the webs are extended
past the side plates and connects with the end inclined end plates. Transverse
stiffeners connects the top plate, side plates, webs and transverse stiffeners along the
span. The holes in the transverse stiffener at the side plate corners and longitudinal
stiffener ends are there to prevent stress concentrations and therefore increase the
fatigue life. The top plate is inclined by 1.5% from the middle to the side ends for
drainage purposes.

1.5%

Figure 5.2: Cross section of box girder

The actual cross section dimensions varies along the span but a typical section is
used as starting point for the calculations. The layout of the cross section is as previ-
ously mentioned simplified, the extensions of the top plate and webs are drawn back
to the side plates. The holes in the transverse stiffener and the inclination of the
top plate is ignored. These simplifications will not affect the structural behaviour
with regard to short time buckling strength which is covered in this thesis.

The bridge span is reduced to a 18 meter, straight, simply supported box girder
with dimension as presented in figure 5.3 and table 5.1 below. The inclination of

30



5. Box section

the webs are 21.12◦ from the bottom plate as the actual design. The transverse
stiffener distance sts is constant at 2 meters through out the span except for by the
supports where an extra stiffener is placed at a distance, sts0 0.5 meter from the
support. The material properties which varies with the thickness are presented in
Table 5.2.

bd

sso

hts

tso

td

ts

tw

tsutb

bb0.5
ssu

ssu0

hso

hsu

hwxbb

hwy

hw

hs

Figure 5.3: Simplification of cross section

Measurements of the beam
Length [mm] Length [mm]

bd 3500 ssu0 20
sso 350 ssu 440
td 10 tsu 12
tso 16 hsu 120
hso 140 bb 1000
hs 389.2 tb 16
ts 12 tts1 18
tw 12 tts2 12
hw 1340 tts3 8
hwy 482.8 sts 2000
hwx 1250 sts0 500

Table 5.1: Measurements of the beam

Plate with thickness: fy [MPa] E [GPa]
t<40mm 390 210
t>40mm 335 210

Table 5.2: Material properties of the beam

31



5. Box section

5.1.2 FE-Model
The geometry and boundary conditions are generated with Rhino with its algo-
rithmic modeling plugin Grasshopper. The line along the supports are fixed in all
directions on both sides except for one side which is free to move along the bridge
in x-direction. The geometry is then interpreted in SOFiSTiK as a 3D system with
a combination of quadrilateral and triangular shell elements for the analyses.

The structural model as well as boundary conditions are presented in 5.4 below.
The supports are created with support lines placed at the very end of the sort edges
of the bottom plate. No other boundary conditions are needed on this structure
in order to obtain sufficient stability in the structure and prevent from instability
problems, in contrary to the I-beam in Chapter 4. The layout of the transverse
stiffeners along the span can be seen trough the transparent plate element in Figure
5.4a below.

(a) FE model of the box girder

(b) Left BC (c) Right BC

Figure 5.4: FE-Model of the bridge girder

The stress analysis is a linear elastic analysis where the maximum Von Mises stresses
are calculated for the most critical parts of the structure. An advantage of using a
linear analysis is that a unit load can be applied to the structure and the final load
capacity can be interpolated without rerunning the program. The buckling analy-
sis is a material-linear (but geometrical-non-linear) eigenvalue calculation which is
solved with a simultaneous vector iteration. The linear nature of this procedure

32



5. Box section

makes it possible, just as the stress analysis, to use a unit load which is then inter-
polated up to the ultimate capacity.

5.1.3 Convergence study

The convergence study on the box section is preformed by studying both the Von
mises stress in a single point in the middle of the girder as well as the buckling
eigenvalue as the mesh size decreases. The same load is applied for every mesh
refinement and the outcome is plotted and presented in Figure 5.5 and 5.6 below.

337

338

339

340

341

342

0.020.030.040.050.060.070.080.090.1

Vo
n 

m
is

es
 st

re
ss

 in
 th

e 
m

id
dl

e 
 [M

Pa
]

Mesh size [mm]

Figure 5.5: Von mises convergency of the boxsection

An average element size of 60 mm seems to be sufficient for the Von mises stresses by
just analysing the slope of the curve since no significant change in the result occurs
below that. The percentage difference is although below 1 % already between 90
and 80 mm which indicates that the stress analysis is not that sensitive to the mesh
size. It is indeed important in this model to not refine the mesh more than necessary
because of the already vast computational time so a mesh size of 90 mm is sufficient
for the stress analysis. The analyses are computed on the same model so the other
convergence study could be governing.

33



5. Box section

2.2

2.3

2.4

2.5

2.6

0.030.040.050.060.070.080.090.1

Bu
ck

lin
g 

ei
ge

nv
al

ue
 [-

]

Mesh size [mm]

Figure 5.6: Egenvalue convergency of the boxsection

The buckling eigenvalue does not show much of a convergence pattern before the
50 mm element size. The calculation time reached about 40 minuets for the 40 mm
mesh size which should be regarded as a maximum limit since the analysis should
run about 50 times so the refinement stopped there. The slope does decrease a bit
between the element sizes 50 and 40 mm but the percentage change is 1.3%. The
1% convergence limit is not reached but the computational time limit is, hence a
mesh size of 40 mm has been chosen for the model.

34



5. Box section

5.1.4 Parametric study
Rhino and the plug-in tool grasshopper is used to effectively generate a geometry
where parameters easily can be changed. The geometry is then analyzed with a
stress analysis and a buckling eigenvalue analysis in the software Sofistik. Both an-
alyzes are linear so that a unit less load can be applied. Since the geometry most
often changes during projects there is a need of a general code with the possibility to
make easy changes regarding for example its measurements and amount of stiffeners.
A parametric tool which can be used for different box girder bridges can streamline
a designers workflow significantly and more room for optimization is possible.

The design tool made in grasshopper is not fully generalized but merely quite
adapted to the parametric study preformed in this project. Other design changes
are possible but the following parameters will be investigated in this thesis:

Figure 5.7: Illustration of the studied parameters

Figure 5.7 illustrates which parameters that will be investigated and in what order.
Since the structure is simply supported and subjected to a uniformly distributed
load only the plates above the center of gravity will be in compression. The study
is concentrated on those plates since they will affect the capacity directly which is
not the case for example the bottom plate.

The measurements that should be investigated for every parameter are presented in
table 5.3 below. The parameters are investigated one by one while the others are set
to the original measurements. The original measurements are presented in bold in
the table. The top plate is in cross section class 4 in the original design but as the
thickness increases the class changes to 3 for the whole section. That means that

35



5. Box section

the calculation methods are unnecessary since no reduction will be needed. That
investigation is only meant to verify that the two methods approaches each other
in that section. The top plate is in cross section class 4 for all other studies so the
whole cross section is to, regardless of the other measurements.

1. Thickness of the top plate [mm]
4 6 8 10 12 14 16 18

2. Thickness of the top plate stiffeners [mm]
2 4 6 8 10 12 14 16 18

3. Thickness of the side plates [mm]
1 3 4 6 8 10 12 14 16

4. Height of side plates [mm]
389.2 470 550 630 710 790 870 950 1030

5. Amount of vertical stiffeners
5 6 7 8 9 10 11 12

Table 5.3: Investigated parameters and thicknesses

5.2 Result & Discussion
The already mentioned parameters with different measurements in table 5.3 are in-
vestigated with the EWM and the RSM. Generally the methods coincides quite well,
but for some thicknesses there are a significant difference in capacity and a changed
behaviour in the buckling analysis. The results from the different parameters are
presented one by one in section 5.2.1 to 5.2.5 below. Both the capacity and the
buckling analyses are discussed thoroughly for every parameter.

5.2.1 Top plate thickness
The thickness of the top plate has quite a large impact on the capacity of the
bridge and the buckling behaviour. The capacities of the box girder are compared
with varying top plate thickness calculated with the EWM versus the RSM with
eigenvalues and von mises stresses from Sofistik.

5.2.1.1 Capacity comparison

The original box section consist of a 10 mm thick top plate which resulted in a
lower capacity with the RSM compared to the EWM according to figure 5.8. Both
moment- and shear capacity are plotted for EWM as with the I-beam in chapter 4
but the interaction is not relevant for in this structure so that is left out. The plot
includes vertical lines which marks the boundaries of the different buckling modes,

36



5. Box section

a, b and c. The buckling modes and their influence will be presented in the next
section.

Figure 5.8: Load capacity of the box girder bridge with the EWM and the RSM
with varying top plate thickness.

The yellow curve which represents the capacity with the RSM is, as mentioned be-
fore, lower bound for all thicknesses of the top plate. The curve does follow the blue
moment capacity curve from the EWM for thinner top plates and the green shear
capacity curve for thicker top plates. This seems reasonable since the two EWM
curves cross each other in the middle. But the yellow RSM curve is furthest away
from the two EWM curves close to that intersection as if the method is less accu-
rate when the governing force changes. This less accurate part also happens to be
within the limits of buckling mode b so the answer might be in the behaviour of the
structure. The I-beam in chapter 4 had a close relation between its buckling mode
and governing slenderness curve. The slenderness curves have been compared for
this structure as well and the result is presented in figure 5.9 below. Just as with the
capacity plot, two vertical lines are added to show where the buckling mode changes.

37



5. Box section

Two slenderness curves are very close. One of them is corresponding to plate like
buckling of an equivalent plate, which basically means a fictional plate which is
equvalent to the combination of the top plate and all longitudinal stiffeners conected
to it. The other slenderness curve is corresponding to column buckling of the lon-
gitudinal stiffeners. It does not matter mathematically which one is used in the
calculations since they are almost the same but the buckling analysis will reveal
which one that corresponds to the actual behaviour.

Figure 5.9: Slenderness with varying top plate thickness. Buckling mode a, b and
c.

Besides the two very close slenderness curves in the EWM there are a third which
is governing for the thinner top plates. This is a slenderness that corresponds to
a situation where a part of the top plate buckles between two stiffeners. This is a
quite special situation which occurs when the top plate is so thin compared to the
stiffeners that they wont help stiffen it up. The stiffeners acts in stead as supports
where the top plate is free to move in between.

The non-governing parts of the EWM slenderness curves has been removed in figure
5.10 below to make it easier to compare them to the RSM slenderness curve. The
grey curve that represents column buckling of the stiffeners is entirely removed due
to its similarity to the slenderness curve of the equivalent plate.

38



5. Box section

Figure 5.10: Slenderness with varying top plate thickness. Buckling mode a, b, c

The combined EWM curve and the RSM curve are now looking very similar to each
other. The RSM curve is, just as the capacity, below EWM for every thickness but
a little bit closer for thicker top plates. The increased difference in capacity that
occurred in the middle of the span earlier observed in figure 5.8 does not seem to
have much to do with the slenderness. No such patterns occurs in the slenderness
comparison and the slenderness curves does not change between buckling mode b
and c. It might therefor be that the slenderness and the buckling modes are’nt
that close related for this structure and that the actual buckling mode can tell more
about why there is a gap in the middle of the capacity plot.

5.2.1.2 Buckling analysis

The structure buckles in three different ways through out the parametric study of
the top plate thickness. The buckling modes are presented in figure 5.11 below. For
the thinnest top plates (4-10 mm) the top plate buckles locally between stiffeners
by plate like buckling (figure 5.11a). The second buckling mode (figure 5.11b) at
the thicknesses 12 and 14 mm is global plate like buckling of the top plate and the
longitudinal stiffeners together. The third and final buckling mode (figure 5.11c)
occurs at top plate thicknesses 16 mm and over. It is shear buckling of the web
between the two end stiffeners by the support.

39



5. Box section

(a) Buckling mode a

(b) Buckling mode b

(c) Buckling mode c

Figure 5.11: Three different buckling modes with varying top plate thickness.

To get a clear view of the results, the buckling behaviour for each top plate thickness
with its respective buckling mode are presented in table 5.4 below.

Buckling mode Buckling behaviour Top plate thickness
(a) Local normal stress buckling 4, 6, 8, 10 [mm]

of the top plate.

(b) Global normal stress buckling 12, 14 [mm]
of the top plate and stiffeners.

(c) Shear buckling of the web. 16, 18 [mm]

Table 5.4: Buckling modes with varying top plate thickness.

40



5. Box section

Buckling mode a and b corresponds well to the governing slenderness curves obtained
by the EWM but mode c does not. No slenderness that corresponds to shear buckling
is governing in the EWM calculations. It is though within buckling mode c that the
shear capacity is starting to be governing so it would be logical for the shear buckling
slenderness to be governing in that area. But shear capacity, shear buckling and
shear buckling slenderness are three different things that not always relates entirely.
The buckling analysis seems, in this case, to have anticipated a buckling mode where
the hand calculated did not. The calculations are based on stiff end supports which
the design assures according to the codes but the buckling mode should maybe not
occur between the end stiffeners in that case. The two methods agrees quite well
within this buckling mode after all so even though this is odd, the capacity miss
match can not depend on this. A guess could instead be that the methods differ the
most when the behaviour of the structure changes vastly. In this case, from buckling
in the top plate in the middle of the span to shear buckling in the web in the end of
the span. The moment- and shear capacity are calculated separate in the EWM but
in the RSM a combined capacity, based on the whole structure, is calculated. This
could be an affecting factor to why the method is more conservative in the section
where the change of behaviour occurs.

5.2.2 Top plate stiffener thickness
The original thickness of top plate stiffeners is 16 millimetres. In this section thick-
nesses between 5-18 millimetres are studied. The top plate stiffener thickness has
a large impact on the behaviour of the box girder bridge because of it’s stiffness
contribution to the top plate which is the weakest part in the original design.

5.2.2.1 Capacity comparison

The stiffness contribution from the stiffeners to the already weak top plate decreases
as they decrease in size which is presented in figure 5.12 below. The shear capacity
is never governing in this study because its not effected by the decrease of stiffener
thickness. As for the previous investigation the RSM is lower bond for every cal-
culation in the study. It occurs three different buckling modes which are further
discussed in the next section but the boundaries are presented in figure 5.12 and
labeled a, b and c.

The capacity difference between the methods differs from 24 to 14% from a 6 to
a 18 mm stiffener thickness which indicates that the methods gets closer when
the stiffeners increase in thickness. There are two reasons to this, firstly the cross
section gets closer to a class 3 section so that the reduction with regard to buckling
approaches zero in both methods. This was the same case in the previous section
where the section actually reached class 3 for the thickest top plate measurements.
The methods did in fact not receive the exact same capacity within the class 3 cross
section. The other reason is that the top plate, with its connecting stiffeners, is
the weakest part of the cross section in the original design. So an increase in both
top plate thickness and stiffener thickness should result in a equalization in capacity

41



5. Box section

between the sub plates within the section. As mentioned in section 2.2.1 the EWM
lack the ability to account for load shedding between sub plates within the section
so a more even capacity distribution would decrease the impact of this inability.

Figure 5.12: Load capacity with EWM and RSM with eigenvalues from Teddy for
different top plate stiffener thickness.

The capacity drops significantly in the RSM calculation compared to the EWM
when the top plate stiffener is set to a 4 mm thickness. This might be related to
the buckling mode since this thickness is the only one within buckling mode a. The
slenderness have been plotted in figure 5.13 for both methods to compare and see
if this might explain this capacity drop. The EWM has two governing slenderness
curves for this investigation. The blue curve which is governing from 4-12 mm
stiffener thickness is related to local buckling of one stiffener. The orange curve
which is governing for 14 mm stiffeners and thicker is related to buckling of the
equivalent plate. The later of the two correlates well with the RSM slenderness
curve except for a flatter inclination between 4 and 6 mm stiffeners. The blue curve
which is a lot steeper is governing through all three buckling mode changes so the
capacity drop at 4 mm stiffener thickness does not seem to have much to do with the
slenderness. The difference in the slenderness between the two methods does in fact

42



5. Box section

not seem to correlate with the over all difference in capacity. A high slenderness
should result in a weak section with high reduction due to the increased risk of
buckling. So the very high slenderness in the EWM for lower thicknesses does not
affect its capacity very much since the capacity in the EWM is quite steady through
out the whole study. Since the slenderness doesn’t correlate very well to the buckling
modes the answer could lay within the buckling analysis in the next section.

Figure 5.13: Slenderness with different top plate stiffener thicknesses.

5.2.2.2 Buckling analysis

The buckling analysis resulted in three buckling modes as mentioned before, these
are presented in figure 5.14 and table 5.5 below. All three modes occur in the center
of the span. The first buckling mode, mode a, is local buckling of one longitudinal
top plate stiffener. The second mode, buckling mode b, is local buckling of several
top plate stiffeners and the third mode, mode c, is normal stress buckling of the
top plate. This correlates to the two slenderness curves obtained in the previous
section but the slenderness related to buckling of one or several stiffeners is governing
through out all three of the buckling mode so the overlaps does not match each other
very well. The mesh is kept in figure 5.14a and 5.14b to be able to see the contours
of the top plate stiffeners which otherwise blends in with the surrounding plates.

43



5. Box section

(a) Buckling mode a

(b) Buckling mode b

(c) Buckling mode c

Figure 5.14: Three different buckling modes of the bridge with varying top plate
stiffener thickness.

Buckling mode Buckling behaviour Thickness [mm]
(a) Normal stress buckling 4

of one longitudinal stiffener.

(b) Normal stress buckling 6
of all longitudinal stiffeners.

(c) Normal stress buckling 8, 12, 14, 16
of the top plate.

Table 5.5: Buckling modes with varying top plate stiffener thickness.

44



5. Box section

One could argue that buckling mode a and b are the same buckling mode but
since something changes in the capacity between these two modes it may be wise
to separate them. Both modes only consist of one thickness each and the rest of
the thicknesses belongs to mode c so this only occurs when the stiffeners are very
thin in comparison to the rest of the cross section. A possible reason for the big
capacity difference when the stiffeners are as thin as possible actually has to do
with how it buckles. In the EWM a check for the stability of the stiffener it self
needs to be fulfilled. The stiffener needs to be prone to torsional buckling which is
not the case for the 4 mm stiffener since the check is not passed and it’s clear in
the buckling analysis that the stiffener experiences torsional buckling. The check
is actually not passed for the 6 mm stiffener either but since the buckling mode is
spread out through all stiffeners, it seems that the instability problem does not really
come in to play so to speak. The EWM does however not account for this instability
of the stiffener so the capacity should maybe be decreased but this is not suggested
in [SIS, 2011b] since a stiffener that does not pass the check is quite useless and
should be avoided. In the RSM however there is no such suggested design check to
be made so the weak stiffener is accounted for which probably is the reason for the
earlier mentioned difference change between the two methods capacities.

5.2.3 Side plate thickness
The original side plate thickness is 12 millimetres. The capacity is compared for
thicknesses between 1-16 millimetres. The side plates has great impact of the shear
capacity so this investigation will probably result in a change of governing capacity
in the EWM calculation.

5.2.3.1 Capacity comparison

The results from the side plate thickness study is different compared to the previ-
ous results since the RSM is higher bond for some thicknesses which has not been
the case before. The result is presented in figure 5.15 below. Vertical lines have
been included to show the buckling mode changes throug out the study, just as
for the previous results. Once again three different buckling modes has been iden-
tified, thieve been labeled a, b and c and will be further discussed in the next section.

The RSM gives higher capacity for almost all thicknesses where the shear capacity
is governing in the EWM, which is at side plate thicknesses from 1 to 8 mm. The
moment capacity is governing for side plates thicker than that and the EWM gives
higher capacity in that section. The shape of the RSM curve follows both EWM
curves quite well which has been the case for the previous investigations as well. In
section 5.2.1, where the top plate thickness where investigated, a change in governing
capacity where also the case. But an inconsistency between the two method where
identified in the section where the overlap between shear- and moment capacity
occurred. This inconsistency does not occur in this investigation. The slenderness

45



5. Box section

curves or buckling analysis in this investigation may explain why that is not the
case.

Figure 5.15: Load capacity of the box girder bridge with the EWM and the RSM
with varying side plate thickness.

The slenderness are plotted for both methods and presented in figure 5.16 below.
The EWM has two governing slenderness curves, slenderness for plate like buckling
of the side plate for thinner side plates and slenderness for plate like buckling of
the equivalent plate (top plate and longitudinal stiffeners) for thicker side plates.
The side plate slenderness is almost governing for all thicknesses but slenderness of
the equivalent plate is just a little bit higher from 10 mm side plates and higher.
The yellow RSM slenderness curve is following the shape of the two EWM curves as
it should but it’s usually lower than EWM which is not the case in this investigation.

An extremely high slenderness factor of about 15 is measured at the thinnest side
plate (1mm). The slenderness factor usually lays below 2 but both methods receives
higher values than that for thicknesses below 3 mm. That is perhaps not very
surprising since 3mm is very thin for a plate that is 2 m long. The capacity for
those measurements are quite low as well so there is no doubt that such thin side
plates are quite ineffective. There is not much of a change in the slenderness when
the side plates are set to thicknesses at 8 mm and higher. This is also visualized in
the capacity plot where no significant change to the capacity occurs when the side
plates are thicker than 8 mm in both methods. This is interesting since they are 12
mm thick in the original design. It is although important to ad that this thesis only

46



5. Box section

focuses on short time buckling capacity and no other lode case than a uniformly
distributed load is calculated so there is probably other affects to the side plates
that governs the design.

Figure 5.16: Slenderness with varying side plate thickness.

There is no inconsistency to the slenderness curves except for the very high slender-
ness in RSM for very thin side plates. But there seems to be nothing that explains
the difference from this investigation to the top plate investigation where an in-
crease in the capacity difference between the methods occurred in the overlap from
moment- to shear capacity. perhaps the buckling analysis can reveal why this case
is different to the top plate study.

5.2.3.2 Buckling analysis

An interesting buckling behaviour is achieved with an varying side plate thickness
as presented in figure 5.17 and table 5.6. Shear buckling near the support occurs
with side plates thicknesses below 4 mm (mode a). Normal stress buckling of the
side plate in the middle of the span occurs at thicknesses within 6-8 mm (mode b).
The third and final buckling mode (c) occurs at thicknesses higher than 8 mm and
is local buckling of the top plate between stiffeners in the middle of the span but
towards one of the side plates. The third buckling mode moves towards the middle
of the section as the side plate gets thicker but it is still normal buckling of the top
plate.

47



5. Box section

(a) Buckling mode a

(b) Buckling mode b

(c) Buckling mode c

Figure 5.17: Three different buckling modes of the bridge with varying side plate
thickness.

Buckling mode Buckling behaviour Thickness [mm]
(a) Shear buckling of the side plate. 1, 2, 3, 4

of the side plate

(b) Local normal stress buckling 6, 8
of the side plate.

(c) Local normal stress buckling 10, 12, 16
of the top plate.

Table 5.6: Buckling modes with varying side plate thickness.

There are some differences to the top plate study to be observed. Firstly the shear
buckling mode occurs in the side plate in stead of the web. The side plate is totally

48



5. Box section

in compression and the web is mostly in the tension zone so the transition from shear
buckling to normal stress buckling is quite abrupt. The web is also further away
from the top plate where the next buckling mode occurs than the side plate. The
buckling mode occurs after the second vertical stiffener from the support in stead
of in between the two end stiffeners. This out rules the possible scenario where the
assumption of a rigid end support in the EWM does not account for that buckling
mode. Another difference is that the transition between buckling modes seems a
little bit more smooth. In this case the change from shear buckling near the sup-
port to normal force buckling in the middle happens in the same sub plate. In the
top plate study the change of buckling mode type was combined with a change of
buckled sub plate. All buckling modes are local in this study as well which indicates
some type of consistency.

All of the observed differences above can have more or less impact to why the side
plate study gets a more consistent RSM capacity curve than the top plate study.
These indications is of course hard to observe without a parametric study and a
typical designer might not change the design to this extent. A designer is probably
not able to see these transitions between buckling modes and weather they are
smooth or not. It is therefore hard to conclude any guidance for where to use any
of the methods from this particular study. But the methods are quite close in both
the top plate study and the side plate study when shear force is governing in the
EWM which could be a more helpful finding for this purpose. Perhaps the next two
studies can be helpful to further investigate this.

5.2.4 Side plate height
The side plate height study is unique in the context since the change of side plate
height results in a change of the total height of the cross section. The height ratio
between the web and the side plate changes and the neutral axis moves from the
web to the side plate. So the side plate which until now has been entirely in the
compression zone will experience some tension as well. The original side plate height
is 389.2 millimetres and this study covers heights up to 1030 millimetres which is
almost three times as high.

5.2.4.1 Capacity comparison

The capacity curves are plotted and presented in figure 5.18 and as for most of
the previous results, the RSM is lower bond for all calculated measurements. the
boundaries of the buckling modes are once again added to the plot but this time
there are four different buckling modes, a, b, c and d along the studied designs. The
moment capacity in the EWM is governing for lower side plate heights but increases
in a linear manner as the side plate increases because of the increase of moment of
inertia within the cross section. The shear capacity in the EWM however behaves
quite irregular where an increase in the capacity occurs for the lower side plates.
The capacity starts to decreases as buckling mode b emerges and then flattens out
along mode c and then increases a little bit again by mode d. So it seems that the

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shear capacity in the EWM is quite close related to the governing buckling mode in
this study, this will be further analyzed in the next section. The RSM curve, on the
other hand, do not seem to be as influenced by the buckling mode. The inclination
of the curve does changes by buckling mode b where the capacity flattens out a
bit so there is some relation to the buckling mode there as well. The methods are
closest when the side plate is as short as possible and furthest away when the EWM
changes from moment- to shear capacity. Maybe the methods approaches each other
again for side plates higher than 1030 mm but the capacity increase wearers of in
both methods so the design gets less material effective for higher side plates.

Figure 5.18: Load capacity of the box girder bridge calculated with the EWM and
the RSM for different height of the side plates.

A comparison between the slenderness has been made for this study as well to be able
to identify any difference in the methods there. The result is plotted and presented
in figure 5.19. The EWM received two different slenderness curves. Slenderness
related to plate like buckling of the equivalent plate (top plate and stiffeners) is
governing at the original side plate height (398.2 mm) and at 470 mm. All side
plates higher than that has a governing slenderness related to plate like buckling of
the side plate.

The RSM curve follows a bi linear relation that is very close to the inclination of
the two EWM curves. The inclination change occurs at about one side plate height
later than the intersection of the two EWM curves. This inclination change happens
to be where the buckling mode changes from a to b which was the case with the

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capacity curve as well. This indicates that the change from buckling mode a to b is
quite critical for the RSM calculations. The EWM does not have the same relation
between capacity and slenderness but the buckling mode seems to affect more in
this case. Note that the result from the buckling analysis is only used in the RSM
calculations but the behaviour should apply for the EWM as well and the relation
between them gets even more interesting because of this fact.

Figure 5.19: Slenderness with different side plate heights.

5.2.4.2 Buckling analysis

The buckling analysis resulted, as earlier mentioned, in four different buckling modes
which are presented in figure 5.20 and table 5.7 below. The first buckling mode,
mode a, is local normal stress buckling of the top plate between stiffeners in the
middle of the span. This buckling mode is known by now since it occurs in the
original design which is a part of all parametric studies. It’s governing at the side
plate height 389.2, 470 and 550 mm. The second mode, mode b, the same as mod a
but in combination with local normal stress buckling of the side plate in the middle
of the span. This mode occurs at a side plate height of 630 mm solely. Buckling
mode c (at 710 and 790 mm) and d (above 790 mm) is one and two shear buckling
modes respectively. Even though the two last modes are different modes there are
not much of a difference between them. They occur at the same location as well
as element and both are in fact shear buckling but since the EWM capacity curve
seemed to change direction in this area they are kept separated.

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5. Box section

(a) Buckling mode a

(b) Buckling mode b

(c) Buckling mode c

(d) Buckling mode d

Figure 5.20: Three different buckling modes of the bridge with varying side plate
height.

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5. Box section

Buckling mode Buckling behaviour Thickness [mm]
(a) Local normal buckling 389.2, 470, 550

of the top plate.

(b) Local normal stress buckling 630
of the top- and side plate.

(c) Shear buckling 710, 790
of the side plate.

(d) Shear buckling of the side plate 870, 950, 1030
in two end sections.

Table 5.7: Buckling modes with varying side plate thickness.

The capacity in the RSM seems to be affected as soon as the side plate is involved
in the buckling analysis since that’s the major difference between buckling mode a
and b. The change from normal stress buckling to shear buckling does not seem
to change the resulting RSM capacity significantly. The same goes for the moment
capacity in the EWM which does not seem to be affected by any buckling mode.
Another situation is identified in the shear capacity with the EWM. The capacity
increase is affected by the entrance of side plate buckling but the capacity decrease
occurs when shear buckling is governing the structure. But the decrease wears off,
and even changes to an increase as one extra mode is added. It’s quite reasonable
that the capacity can decrease when the structure changes element, location and
type of the governing buckling mode. But that the capacity increase again when the
same buckling mode spreads out more is perhaps not an intuitive guess. Maybe the
post buckling reserve increases as the possible tension bands increases in numbers.
In that case, the capacity won’t increase further if not a third buckling mode will
appear and increase the post buckling capacity even more. The increase wont be
very high and the design not particularly material effective though. This is not very
important to the comparison since both methods probably will behave the same with
those high side plats. The most interesting part of this investigation is how the two
methods cope with different buckling modes and overlaps from moment capacity as
discussed earlier.

5.2.5 Distance between vertical stiffeners
Last but not least the capacity with a bridge with changed vertical stiffener distance
is studied. The original distance is two metres which results in 8 equally spaced
stiffeners along the span. The study consists of distances from 1.38 to 3.6 meters
which means 12 to 5 equally spaced stiffeners respectively. One increase in space
between the stiffeners means one less vertical stiffener along the span. This will
result in a wider and wider span for the top plate and it’s connecting longitudinal
stiffeners.

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5. Box section

5.2.5.1 Capacity comparison

The comparison between the two methods in the vertical stiffener study is presented
in fig 5.21 below. The boundaries for the buckling modes are added as for the other
studies. Three buckling modes occurred for this study where the first two, a and
b, alternated for the shorter distances and a third, mode c, where governing for the
wider spans. The buckling modes will be further discussed in the next section. The
EWM calculation resulted in a moment capacity curve as the shear capacity where
never governing. The EWM result were extended below 1.35 m and above 3.6 m
as the calculations where not as tedious as the FE-computational time in the RSM
calculations. The RSM result however is lower bond as for every other comparison
where moment capacity is governing in the EWM. The RSM curve seems to be
more constant through the shorter stiffener spans than the EWM curve which is
decreasing quite linearly through out all calculated distances. A sudden increase in
the capacity occurs in the RSM at a stiffener distance of 2.25 m which is just before
the change to buckling mode c. Through out the wider stiffener distances, within
buckling mode c, the RSM capacity starts to decrease in about the same manor as
the EWM curve.

Figure 5.21: Load capacity with EWM and RSM with eigenvalues and stresses
taken from Teddy for different vertical stiffener distance.

There are two odd things about the RSM capacity, the first constant part of the
curve and the sudden increase by the 2.25 m distance calculation. A decreasing

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slope like the EWM capacity seems more reasonable as an increasing span should
lower the capacity. Hopefully an answer to this can be found in the slenderness
curves or the buckling analysis.

The result from the slenderness calculations are presented in figure 5.22 below. The
EWM has three governing slenderness curves, two of them which are almost the
same and a third which is governing for the lower distances. The two very similar
curves are slenderness for both plate like- and column like buckling of the equivalent
plate, in orange and green respectively. The third slenderness is plotted in blue
and corresponds to plate like buckling of the stiffener solely. The EWM capacity
is rather unaffected by the fact that the slenderness changes from a constant to a
rapidly increasing curve but then again the slenderness does not always correlate
very well with the capacity. The RSM slenderness curve on the other hand has a
shape that is almost an exact upside down reflection of the capacity curve. The
slenderness is constant until 2.25 m stiffener distance where it drops down a bit
to then increase. The drop at 2.25 is not as big as the increase in the capacity
in the same point but it’s not a one to one relation so a small slenderness change
can result in a bigger capacity increase. But it’s still not very clear why there is a
slenderness drop at that distance nor why the slenderness is constant before that.
The question will be saved to the buckling analysis but it can be concluded that the
slenderness curves of EWM does not match the capacity as good as for the RSM.
The Slenderness curves match each other’s shape quite well however, besides where
the slope change occurs.

Figure 5.22: Slenderness with different vertical stiffener distance.

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5.2.5.2 Buckling analysis

As earlier mentioned, the buckling analysis resulted in three different buckling modes
which are presented in figure 5.23 and table 5.8 below. The two first ones are
both local normal stress buckling of the top plate but mode a occurs when there
is a transverse stiffener in the middle of the span so that it buckles on both sides.
Buckling mode b is of course when there are an even amount of stiffeners so that it
only buckles in one section of the beam. Mode c is a global normal stress buckling
of the top plate along with the longitudinal stiffeners. Mode a and b occurs up to a
2.25 m stiffener distance and mode c above that.

(a) Buckling mode a

(b) Buckling mode b

(c) Buckling mode c

Figure 5.23: Three different buckling modes of the bridge with varying distance
between transverse stiffeners.

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5. Box section

Buckling mode Buckling behaviour Distance[m]
(a) Local normal buckling of the top plate 1.5, 1.8, 2.25

in two mid sections