Master Thesis – Plate buckling in design codes

Plate buckling in design codes The difference between NEN 6771 and NEN-EN 1993-1-5

Master Thesis

M.J.M. van der Burg

Master Thesis – Plate buckling in design codes M.J.M. van der Burg

2

Plate buckling in design codes The difference between NEN 6771 and NEN-EN 1993-1-5

Delft University of Technology

The Netherlands

Faculty of Civil Engineering and Geosciences

Department of Design and Construction

Section of Buildings and Civil Engineering Structures

Movares Netherlands B.V.

Division of Infrastructure

Section of Constructions

M.J.M. van der Burg

Student number: 1220934


3

BOARD OF EXAMINATION:

Prof. ir. F.S.K. Bijlaard Delft University of Technology




Ir. R. Abspoel Delft University of Technology




Dr. ir. M.A.N. Hendriks Delft University of Technology



Section of Structural Mechanics

Ing. B.H. Hesselink Movares Netherlands B.V.

Division of Infrastructure

Section of Constructions

Ir. L.J.M. Houben Delft University of Technology


Secretary


4

Preface

This thesis is made as a final project of my Civil Engineering master education at Delft University of

Technology, where I attended the Structural Engineering master. Most of the work for this thesis has

been done at Movares Netherlands B.V. between August 2010 and February 2011. I am very grateful

for the opportunity to do an internship at large company, which was a very educational experience

and a good addition to the knowledge gained at the university itself.

The subject of this thesis is plate buckling in steel structures and how various design codes treat this

subject. It is interesting to see how such a complicated phenomenon can be approached with

relatively simple methods, which still can give accurate design capacities. However, the different

methods also give different capacities. In this thesis it is clarified where these differences originate

from and which methods should be chosen in varying situations.

I would like to thank prof. ir. Bijlaard, ing. Hesselink, ir. Abspoel and dr. ir. Hendriks. Not only for their

tips and guidance during the research for this thesis, but also for proof-reading the final result.

May 2011,

Mark van der Burg


5

Abstract

Slender steel plates loaded in compression will buckle out of plane before their cross-sectional

capacity is reached. Precisely determining the failure load in an analytical way is an impossible

exercise. Therefore many researchers have put effort into finding a simple design method, which

would still predict the plate buckling load with the required accuracy. Many different methods have

been developed, some of which have been adopted in the currently valid design code in the

Netherlands. Some preliminary calculations showed that the Eurocode (NEN-EN 1993-1-5) gives

much more plate buckling capacity then the Dutch code (NEN 6771). This observation was the start

of this thesis, of which the main goal is to find and explain the differences between the two design

codes. A secondary goal is to formulate an advice as to what method in the Eurocode is the best to

use.

Most of the simple methods for plate buckling can be classified as either an effective cross-section

method or a reduced stress method. There are fundamental differences between these two

methods, which are analyzed and clarified in this thesis. It is found that in general the effective cross-

section method delivers a higher buckling capacity, but this method is also more labour-intensive. An

important element in every method is the reduction factor to the stress or steel area that is used.

Again different researchers came up with different reduction factors, so the influence of those

reduction factors is also analyzed.

With the use of some example cross-sections the difference in plate buckling capacity between NEN

6771 and NEN-EN 1993-1-5 in practical design situations is explored. Indeed the Eurocode always

gives more capacity, which was to be expected when analyzing the methods in detail. Later in the

thesis the plate buckling capacities have been verified using a finite element calculation in Ansys. In

all studied cases the design capacity according the Eurocode matched with the capacity according the

calculations in Ansys.

The fundamental principle of the effective cross-section method is the deformation capacity of

slender steel plates. This deformation capacity is needed in built-up members, to let slender plates

remain at capacity while more stocky plates deform further to their own buckling load. Ansys

calculations have been used in this thesis to confirm this fundamental phenomenon.

In analyzing the several plate buckling verification methods in the design codes it came to attention

that there are no requirements given for plate edges. This is in sharp contrast to stiffeners, which are

analyzed rather thoroughly. With the help of more finite element calculations to clarify this subject, it

was found that the global buckling verification ensures stability of the edges of plates in built-up

members.

As a final subject all understanding gained while working on this thesis is combined in giving a

guideline as to what method in NEN-EN 1993-1-5 is best used in what situation. Which method is

best depends on the goal of the plate buckling analysis.


6

Table of contents:

Chapter 1: Introduction ...............................................................................................................9

1.1 Design codes .......................................................................................................................9

1.2 Plate structures ................................................................................................................. 10

1.3 Buckling ............................................................................................................................. 11

1.4 Verification methods ......................................................................................................... 12

1.5 Formulation of problem..................................................................................................... 13

Chapter 2: Verification methods ................................................................................................ 14

2.1 Effective cross-section vs. reduced stress method ............................................................. 14

2.1.1 Assumptions .............................................................................................................. 14

2.1.2 Procedure .................................................................................................................. 15

2.1.3 Difference in normal force capacity ............................................................................ 15

2.1.4 Difference in bending moment capacity ..................................................................... 15

2.1.5 Conclusion ................................................................................................................. 16

2.2 Reduction factors .............................................................................................................. 17

2.2.1 Euler .......................................................................................................................... 17

2.2.2 Von Kármán ............................................................................................................... 21

2.2.3 Winter ....................................................................................................................... 23

2.2.4 NEN 6771 ................................................................................................................... 23

2.2.5 NEN-EN 1993-1-5 ....................................................................................................... 24

2.2.6 Overview ................................................................................................................... 25

2.3 Verification methods in design codes................................................................................. 26

2.3.1 Inventory of methods ................................................................................................ 26

2.3.2 NEN 6771 ................................................................................................................... 26

2.3.3 NEN-EN 1993-1-5 ....................................................................................................... 27

2.3.4 Step by step guide...................................................................................................... 29

2.3.5 Bending moment capacity differences ....................................................................... 33

2.3.6 Normal force capacity differences .............................................................................. 35

2.3.7 Conclusion ................................................................................................................. 37


7

Chapter 3: Design codes applied to example cross-sections ...................................................... 38

3.1 Capacities according design codes ..................................................................................... 38

3.1.1 I-shaped girder........................................................................................................... 39

3.1.2 Box girder .................................................................................................................. 40

3.1.3 Capacity differences ................................................................................................... 40

3.2 Influence of web slenderness ............................................................................................ 42

3.2.1 I-shaped girder........................................................................................................... 42

3.2.2 Box girder .................................................................................................................. 44

3.2.3 Conclusion ................................................................................................................. 44

3.3 Stiffeners according design codes ...................................................................................... 45

3.3.1 Capacity at given dimensions of plate and stiffener .................................................... 45

3.3.2 Minimum stiffener dimensions for full Winter capacity .............................................. 46

3.3.3 Plate and stiffener dimensions needed for given capacity .......................................... 47

3.3.4 Conclusion ................................................................................................................. 49

Chapter 4: Finite element calculations....................................................................................... 50

4.1 Objective ........................................................................................................................... 51

4.2 Procedure .......................................................................................................................... 52

4.3 Verification of design code capacities ................................................................................ 55

4.3.1 Single plate in compression ........................................................................................ 55

4.3.2 Very slender plate in compression ............................................................................. 61

4.3.3 Single plate in bending ............................................................................................... 62

4.3.4 Conclusion ................................................................................................................. 64

4.4 Capacity of built up sections .............................................................................................. 65

4.5 Deformation capacity ........................................................................................................ 66

Chapter 5: Requirements for plate edges .................................................................................. 68

5.1 Hypothesis......................................................................................................................... 69

5.2 Outline .............................................................................................................................. 69

5.3 Sensitivity analysis ............................................................................................................. 70

5.3.1 Element and load step size ......................................................................................... 70

5.3.2 Shape of imperfection ................................................................................................ 71

5.4 Elastically supported plate ................................................................................................. 73

5.5 I-shaped member .............................................................................................................. 76

5.6 Conclusion ......................................................................................................................... 81


8

Chapter 6: Conclusion and recommendation ............................................................................. 82

6.1 Conclusion ......................................................................................................................... 82

6.2 Recommendation .............................................................................................................. 82

6.3 Future research ................................................................................................................. 83

Chapter 7: References ................................................................................................................ 84

Annex A.1 Plate buckling capacity of a plate in bending ............................................................. 87

Annex A.2 Plate buckling capacity of a compressed plate ........................................................... 88

Annex B.1 Cross-sectional capacities calculated with codes ....................................................... 89

Annex B.2 Longitudinal stiffeners according codes ..................................................................... 92

Annex C.1 Log-file Ansys of single plate ..................................................................................... 94

Annex C.2 Log-file Ansys of elastically supported plate .............................................................. 97

Annex C.3 Log-file Ansys of I shaped member ............................................................................ 98

Annex D.1 Mathcad sheet for NEN-EN 1993-1-5......................................................................... 99


9

Chapter 1: Introduction

Within the field of civil engineering a structural engineer deals with the design and analysis of load

bearing structures. These structures can be anything from a simple beam or column to a complete

bridge or building. Structural engineers should ensure that their designs meet every given design

criteria. Most of these criteria are focused on safety and serviceability. The safety aspect means that

a structure should not collapse under the expected loads. When these loads are exceeded a structure

should warn that it is about to collapse. Serviceability means that under the design loads with safety

factor 1.0, displacements and vibrations should be kept under certain limits, to ensure safe and

comfortable use of the structure.

Figure 1.1: Some general examples of structural engineering.

From top left to bottom right: Burj Khalifa, in Dubai. Golden Gate Bridge, in California.

Wembley Stadium, in London. Millau Viaduct, in France. Falkirk Wheel, in Scotland.

1.1 Design codes

The loads acting on a structure can include wind, snow, people, traffic, earthquake loads, self weight

and so on. Not only in the loads but also in the materials used in construction there is a lot of

variation. Even though most structures are built from either steel, concrete or a combination of the

two, all kinds of different varieties of steel and concrete exist. Design codes have been composed to

provide a guideline, not only as a tool to design a structure, but also for the authorities to check

whether a design is safe enough before giving a building permit. In these design codes magnitudes of

relevant loads on a structure are stated, as well as properties of materials and methods to analyze a

structure.


10

Design codes are usually made by a committee of experts. Such a committee translates complicated

(statistical) data regarding load levels, research in material engineering and different calculation

methods into less complicated design graphs and formulas. An important part of this work is to

derive safety factors to ensure a design, made with a simplified analysis, still has the desired level of

safety against collapse. Once it is proven that the methods in the design standards provide a reliable

design, they can be used in practice by engineers. Now these engineers don’t have to prove why a

certain method, load or strength is applicable, since it is already done for them. In some situations a

more refined analysis might give a less conservative, cheaper design. If however something that is

not in the building code is used, the engineer would have to give proof that the method is correct

and applicable.

Most countries have their own institute to manage the development of new building standards. In

the Netherlands this is the ‘Nederlands Normalisatie-instituut’, which is also known as NEN (short for

‘NEderlandse Norm’). NEN is a neutral party which keeps track of the standards that are needed,

initializes development of standards and publishes them. These standard have a widely various field

of application, not only building and structural engineering, but anything from consumer goods to

paper sizes or sound levels of headphones.

All countries close to the Netherlands have similar agencies: the United Kingdom has the BSI Group

(British Standards Institution), Germany has the DIN (Deutsches Institut für Normung), Belgium the

NBN (Bureau voor Normalisatie or Bureau de Normalisation) and France the AFNOR (Association

Française de Normalisation).

Since the formation of the European Union there has been a desire for more cooperation between

European industries. The difference in national codes was a barrier in this process. As a solution the

CEN (Comité Européen de Normalisation or European Committee for Standardization) was formed to

create European Standards (ENs) which would eventually replace national standards. For example: in

the Netherlands plate buckling in steel structures was calculated according NEN 6771, but since 31

March 2010 it has been withdrawn and EN 1993-1-5 should be used.

1.2 Plate structures

Many small or medium sized steel structures use hot rolled profiles with a cross-section in the shape

of an I, H of U. These shapes are effective both in bending and in shear. Fabricating the profiles is

done by guiding steel which is still hot after casting through a series of rollers who successively press

it in the desired shape. Well known hot rolled profiles are IPE, HE and UNP sections.

Figure 1.2: Commonly used hot-rolled sections

IPE HE UNP


11

A big advantage of these hot-rolled profiles is their low price compared to built-up members.

Disadvantages are the fact there is a maximum size in which they can be fabricated and the low

resistance against torsion. Both problems can be solved by creating a cross-section built up from flat

steel plates. These plates are welded together to form a single member. It is now possible to create

larger cross-sections, or make box sections for torsion members. In principle these sections are more

expensive due the amount of welding needed.

Plate members are widely used in for example bridges or large span roof structures. A first reason is

that the required cross-sectional dimensions might be larger than the largest available hot rolled

section. Secondly the cross-section properties can be adjusted to the forces in the member. Where

large forces are present, for instance the bending moment at the middle of a span, thicker flanges

can be used. This creates a more efficient member, compared to a rolled member with uniform

dimensions throughout its length. Another option is to keep the flange thickness constant, while

varying the web height to follow the bending moment distribution. This can lead to aesthetically

pleasing solutions.

1.3 Buckling

Buckling is an effect that occurs in structures which have a high stiffness in one direction and a low

stiffness in another. When a compressive load in the stiff direction is gradually increased, the

structure suddenly collapses without warning. Structures which are susceptible to buckling are for

instance slender columns in buildings or structures built up from thin plates.

A very simple example of buckling is a plastic coffee cup, which hardly deforms when gently pressing

down on it. However, if the pressure is further increased the cup suddenly buckles and shows

excessive deformation. Failure is finally caused by crippling of the cup. Even though a coffee cup is a

curved structure and therefore actually shows shell buckling, it is a clear analogue for plate buckling.

The same experiment can be done for column buckling with for instance a drinking straw.

Slowly applying load Still undeformed Buckling Crippling

Figure 1.3: Simple example of shell buckling

In a load-displacement diagram buckling is clearly shown. At first a structure behaves linear

elastically along the primary load path. When the applied load reaches the buckling load there is a

bifurcation point, where the structure starts to follow the secondary load path. This secondary path

has a lower stiffness compared to the primary path, which explains the sudden increase in

deformation. With columns this secondary load path is almost horizontal, while for most plate


12

buckling case it still moves up. This means that a buckled plate can still have some capacity left after

initial buckling, also called the post buckling capacity. See also figure 1.4 for a schematic example of a

load-displacement diagram when buckling occurs.

Figure 1.4: Load-displacement diagram for elastic buckling (left: plate, right: column)

There are many ways to determine the theoretical elastic buckling load. For instance: the direct

equilibrium method, the virtual work method or via potential energy. These methods assume a

perfectly straight structure, without any initial deformation or residual stresses. In practice however

no structure is perfect. After fabrication, either by hot rolling or welding individual plates, every

beam or column will have an ever so slight bow or twist. As a result of uneven cooling of the steel

there will also be some residual stresses in the material. Both imperfections cause buckling to occur

at a lower stress than the theoretical elastic buckling stress, so design formulas have been developed

to quantify the influence of the imperfections. These formulas are mostly based on tests with

imperfect members.

1.4 Verification methods

When analyzing a steel structure preferably all stresses should be known exactly as they appear in

the real structure. This way a structural engineer would be able to check for every location if the

structure can carry the load. The problem with this method is that it is impossible to exactly calculate

the stress distribution, due to all kinds of local effects around welds or corners which introduce a

local stress peak. Since steel is a very ductile material these local stress peaks will not cause collapse.

When locally the yield stress is reached, redistribution occurs until a state of equilibrium is reached.

Even though a member under tension may already have local yielding at a small load, still the final

capacity will be the cross-sectional area times the yield stress.

In practice this ensures that beams and columns can be modeled by line elements. When the

structural forces (bending moment, normal and shear force) are calculated the most severely loaded

cross-sections are known. These cross-sectional forces can now be verified to the cross-sectional

capacities, which can be determined either elastic or plastic.

In case of a cross-section built up from slender plates, the cross-sectional capacity is lower than the

expected elastic or plastic capacity, due to local plate buckling. An extensive (finite element)

calculation can be carried out to find this local plate buckling capacity, as shown later in this thesis,

Bifurcation point

Load

Displacement

Secondary load path

Elastic

buckling load

Primary load path

Bifurcation point

Load

Displacement

Secondary load path

Elastic

buckling load

Primary load path


13

but for daily practice this takes too much time. Also in the time when finite element calculations had

not been developed yet, plate buckling capacities where needed. Therefore methods have been

developed to reduce the cross-sectional capacity to account for local plate buckling. These generally

work by determining a reduction factor, which can then be applied over either the steel area or the

yield stress. In chapter three these methods are analyzed.

1.5 Formulation of problem

The problem researched in this thesis is the difference in the methods for determining plate buckling

capacities between NEN 6771 and NEN-EN 1993-1-5. A preliminary calculation showed that the

Eurocode can give a much higher capacity, in some cases up to several dozens of percent. The goal of

this research is to find out why this difference exists and if it is correct. In chapter 2 an inventory is

made of the different methods available in both design codes and where they originate from. In

chapter 3 it is investigated what the results are for some example cross-sections, to get a view of the

difference in capacity in commonly used cross-sections. The fourth chapter deals with verifying the

calculations according the codes with a finite element calculation made in Ansys.

Within the Eurocode, which is required in the Netherlands nowadays, a user can choose different

methods to calculate the plate buckling capacity of a cross-section. These give different values for

the capacity and have a different workload for the structural engineer. This thesis will give a

recommendation what method to choose in what situation, regarding both capacity and workload.

The methods compared are chapter 4 and 10 of NEN-EN 1993-1-5 and chapter 13 of NEN 6771.

The last main subject is the requirements in the design codes for plate edges. In both NEN and

Eurocode many requirements are given regarding stiffeners. A small sized stiffener will either not be

allowed or give very large reductions to the adjacent plate parts. A plate edge on the other hand is

considered as a perfect hinged support to a plate, even if this edge is formed by a very small flange.

This problem is addressed in chapter 5 using finite element calculations in Ansys.


14

Chapter 2: Verification methods

Many different methods to determine the plate buckling capacity of steel plate cross-sections exist.

All these methods can be divided in two main groups: effective cross-section or effective width

methods (Dutch: doorsnede-reductiemethodes) and reduced stress methods (Dutch: spannings-

reductiemethodes). Within a method there still is a choice in the reduction factor to be used. This

chapter starts with an inventory of the different methods, before investigating how these differences

influence the calculated capacity of a steel cross-section.

2.1 Effective cross-section vs. reduced stress method

2.1.1 Assumptions

The effective cross-section method and the reduced stress method have a fundamental difference in

the approach for determining an approximation for the real stress distribution. The reduced stress

method checks at which stress level a plate part buckles, if a cross-section is built up from multiple

plate parts the lowest stress is governing for the entire cross-section. When the stress does not

exceed the critical stress, class 3 section properties can be assumed. The usual cross-section

verifications can be used, only with the yield stress multiplied by a reduction factor.

The effective cross-section method reduces the area of the cross-section in the parts affected by

plate buckling. This method was first introduced by von Kármán (1932), see also chapter 2.2.2. The

biggest difference with the reduced stress method is that the method assumes a buckled plate part

remains at capacity, so other plate-parts can also reach the buckled state. Due to this load shedding

there is in general a higher plate buckling capacity in a cross-section than when using the reduced

stress method. This effect is best visible in cross-sections with large differences in slenderness

between the individual plate parts.

Figure 2.1 shows the assumed stress-distribution for an I-shaped cross-section loaded by a bending

moment for both discussed methods.

Figure 2.1: Assumed stress distributions

d = fy;d

d < fy;d

fy;d

Effective cross-section method Reduced stress method

eN

d = buck;d


15

2.1.2 Procedure

In general the procedures for the effective cross-section and the reduced stress method have

similarities, but also some key differences. The short lists of fundamental steps below clarify these

differences.

Effective cross-section method: Reduced stress method: - Divide cross-section in plate parts - Calculate reduction factor for each part - Multiply area of each plate part with its

own reduction factor - Use effective part of cross-section in

verifications

- Divide cross-section in plate parts - Calculate reduction factor for each part - Lowest factor is governing - Design stress equals governing factor

times yield stress - Use design stress in cross-section

verifications

It is now clear that the fundamental difference between the two methods is the assumption whether

the cross-sectional capacity is reached when the first plate part buckles or not. In the effective cross-

section method load shedding is allowed, until all plate parts reach the buckled state. An advantage

of using this load shedding effect in the cross-section is that in general a higher capacity is calculated.

This capacity will be a more realistic value, since load shedding does occur in cross-sections and

buckled plate parts maintain their carrying capacity when buckled. Further attention to this

phenomenon is paid in chapter 5.4. A disadvantage is the fact that when using an effective cross-

section method one has to recalculate the cross-sectional properties several times. When using a

frame program different models have to be made for different checks. In both situations this brings

extra workload to the engineer.

In daily practice a good procedure might be that a design is made using the reduced stress method,

taking advantage of the quicker calculation. If at some locations the design has an insufficient local

buckling capacity, instead of adding material a more refined calculation could be made using the

effective cross-section method.

2.1.3 Difference in normal force capacity

For a single unstiffened plate the effective cross-section and the reduced stress method should in

principle give the same capacity, as long as the reduction factor used is equal. For a load case of pure

compression this is easily calculated. The reduced stress method will multiply the reduction factor

with the yield stress, while the effective width method multiplies it with the width of the plate. These

procedures lead to the same plate buckling load:

Fbuckling = ( · fy) · b · t = fy · ( · b) · t or: Fbuckling = d · b · t = fy · Aeff

2.1.4 Difference in bending moment capacity

In a pure bending load case the effective width method will give a slightly higher capacity. The

assumed stress distribution (see figure 2.1) is closer to the real stress distribution, which gives a

higher capacity. Also the shift of the neutral axis is taken into account. Since the plate now has a gap


16

in it, no simple formulas can be used as was the case with pure compression. First the location of the

neutral axis has to be determined using the first moment of area of the two parts. To calculate the

second moment of area with regard to the neutral axis, the rule of Steiner has to be used. These

more complicated formulas have been put in a spreadsheet, to plot the difference in capacity. Figure

2.2 shows that when a reduction factor of around 0.6 is applicable, the effective width method gives

about 10% more capacity than the reduced stress method using the same reduction factor.

Figure 2.2: Extra capacity when using effective cross-section method

Keep in mind that this difference in capacity for a plate in bending is only due to the assumed stress

distribution as shown in figure 2.1, and completely independent of the reduction factor . The

reduction factor itself has a much larger influence, which is shown in chapter 2.2.

2.1.5 Conclusion

In chapter 2.1 a comparison has been made between the effective cross-section and the reduced

stress method. The reduction factor has a large influence on the ultimate capacity, but in this

comparison an equal reduction factor for both methods is used, to clarify the difference caused by

the assumed stress distribution in the buckled state.

For a single plate loaded in compression the buckling load will be identical. When bending is

regarded the effective cross-section gives a slightly higher capacity. In a practical situation a plate

loaded in bending will be a web in a girder. Since the web only has a small contribution in the

bending moment of an I or box section the total bending moment capacity of the cross-section will

hardly be influenced by this effect.

An effect which does influence the capacity of a cross-section quite a lot is that when using the

effective cross-section method a buckled plate part remains at capacity when further loaded. The

cross-sectional capacity is no longer governed by the weakest link. This plays a larger role as the

difference in plate slenderness increases, but nevertheless the reduced stress method will always

give a lower or equal buckling load.

0

5

10

15

0,5 0,6 0,7 0,8 0,9 1,0

Per

cen

tage

[-]


17

2.2 Reduction factors

Both the effective cross-section and the reduced stress method cannot be used without a way to

determine the reduction factor. In history many different forms of reduction factors have been

developed. In this chapter these are presented and their origin explained. NEN 6771 uses a formula

based on the critical Euler buckling stress, while NEN-EN 1993-1-5 uses a formula first introduced by

Winter, which is based on the effective width theory by von Kármán. These formulas all calculate the

reduction factor () using the relative slenderness (rel) of the plate. The relative slenderness is

defined using only geometrical and material properties:

plate slenderness

specific slenderness

relative slenderness

2.2.1 Euler

The critical Euler plate buckling stress ( ) is the starting point for every method, since it is used to

determine the relative plate slenderness. It is the stress level at which buckling occurs in a perfect

plate. No imperfections are taken into account, and linear elastic material is assumed. The critical

stress can be derived in many ways, for instance via direct equilibrium equations or a virtual work

method. In this thesis a potential energy method is used. The definition of the plate, its properties,

relevant axis and displacements can be found in the following figure.

Figure 2.3: Definition of a single plate

u

nxx

x

y

z

w

x u

v

y

a

b

z w

E, , t


18

The most general form of the potential energy equation is:

In other words, the elastic potential energy is the half of strain times stress, summed up over every

direction and integrated over the entire structure. This equals to the area under a linear stress-strain

curve. The potential energy lost by the load is equal to the work done by the load, which is force

times displacement.

For a minimum of potential energy for every δu it should hold that:

In this case the elastic potential energy is due to bending of the plate. For a very small plate part with

dimensions dx and dy loaded by distributed bending moments along its edges the following holds for

bending in x, bending in y and twisting:

For relating moment to curvature these expressions are available from general plate theory:

Adding the three components, integrating over the area of the plate and substituting the moments

we get the total elastic potential energy of the plate:

x

y

z

dx

dy mxx


19

The potential energy lost by the load is given by:

The next step is to assume a displaced shape due to the buckling of the plate. One half sine wave in y

direction and m half waves in x direction are used. This equates to:

Using this displaced shape implies that a must be larger than b, i.e. a large aspect ratio. Because the

buckles are perfectly sinusoidal, only thin plates are covered.

All necessary information for calculating the buckling load is present. The next step is to obtain the

partial differentials and substitute them in the potential energy equations.

Substituting and simplifying gives elastic potential energy:


20

Substituting and simplifying gives potential energy in the load:

This makes the total potential energy in the structure:

There is a minimum of potential energy when the first order variation is set to zero:

Since the second order variation will lead to the same equation as above, the equilibrium is classified

as indifferent. Further elaboration gives the critical plate buckling stress:

This expression for the critical elastic buckling stress can be applied for all types of plates. The factor

k is called buckling factor, and is used to take into account the influence of the load case, support

conditions and aspect ratio. The here derived buckling factor is valid for a simply supported plate,

loaded by pure compression. The influence of the aspect ratio can be found in figure 2.4.


21

Figure 2.4: Influence of aspect ratio on buckling factor

In many design codes the aspect ratio is not taken into account in the plate buckling calculations. For

a simply supported plate in compression it is taken as 4.0, since the error made is relatively small and

on the safe side. In the table below the standard buckling factors for more situations are given.

Internal element Outstand element

Compression 4.0 0.43

Bending 23.9 23.8 (compression at support)

0.85 (tension at support)

Table 2.1: Buckling factors

The critical buckling stress can directly be used in a reduced stress method. In some codes however it

is preferred to give a formula for a reduction factor expressed in the relative plate slenderness. In the

case of the Euler critical stress this leads to:

This reduction factor 1/2 is widely known as the Euler hyperbola.

2.2.2 Von Kármán

Many researchers showed that the actual ultimate carrying capacity was much larger than the critical

load. This was most obvious with the more slender plates. Stockier plates have an ultimate load

based on yielding, which can be smaller than the critical plate buckling load. Slender plates on the

other hand showed an extra post buckling capacity after passing the initial buckling load.

0

1

2

3

4

5

6

7

8

0,0 1,0 2,0 3,0 4,0 5,0

k [-

]

a[-]

m=1 m=3

m=4

m=5

m=2


22

The elastic critical buckling stress in 2.2.1 is derived using the assumption that the stress distribution

stays linear. What actually happens in a buckled plate is stress redistribution away from the buckled

part. The real stress distribution is a complex combined action of membrane stress induced by the

load, bending stress due to the buckled shape and shear stress because of the rotation at the corners

of the plate. There have been developed methods for analyzing a plate with this approach, for simple

design purposes however these are too advanced to use.

In 1932 Theodore von Kármán introduced the concept of effective width. He stated that (at a given

thickness) a fictitious plate with the width of beff would have the critical stress equal to the yield

stress. If the actual plate has larger width, the capacity would be the same as that of the fictitious

plate. In a plate the real stress distribution is approximated, or replaced, with two strips which

describe the load carrying effective width of the plate.

Figure 2.5: Stress distribution after buckling,

the von Kármán assumption for effective width.

With the above assumed stress distribution the reduction factor can be derived by replacing the

critical stress with the yield stress and the actual width with the effective width.

Although von Kármáns method gained reputation as a reliable way to determine the ultimate load of

a plate, the method was still based on plates without initial imperfections. When compared to test

results it was only valid for large b/t ratios, where initial imperfections have a smaller influence.

However, von Kármán still is the first researcher to propose a reduction factor function and has had a

large influence on all simplified design methods for plate buckling.

b

cr

x

a

fy

beff/2

beff/2


23

2.2.3 Winter

Since no manufactured plate is without initial deformation and residual stresses many researchers

worked at adapting von Kármáns formula. The function introduced by George Winter in 1947 is one

of the more widely spreads in design codes. Winter conducted many experiments on cold formed

steel sections and came up with:

The first suggestion was made with a coefficient of 0.25, but was later changed to the 0.22 used now.

It is interesting to see how small the difference is between Winters experimentally based function

and the original theoretical function by von Kármán. Even though much research has been conducted

since 1947, and the fact that this formula is based only on cold formed sections, it is still the most

widely used nowadays.

2.2.4 NEN 6771

In the Dutch NEN code the main method uses formulas based on the Euler critical buckling stress. To

take into account the influence of imperfections on the buckling capacity there is a reduction with

respect to the Euler hyperbola in the relative slenderness area 0.7 < rel < 1.291. Since the ultimate

load bearing capacity is higher than the critical load, post critical strength is allowed where 1.291 <

rel < 2.5. At a larger relative slenderness the Euler hyperbola is used again. The reduction factor is:

= 1.0 if: 0 < rel ≤ 0.7

= 1.474 – 0.677 rel if: 0.7 < rel < 1.291

=

if: 1.291 ≤ rel ≤ 2.5

=

if: 2.5 < rel

Taking in account the post critical strength is not allowed if one of the longitudinal edges is

supported by a stiffener. The rules for stiffeners do not take this enlarged load into account. Even if

they would the philosophy is that what will be gained in the plate field by allowing post critical

strength is lost in dimensioning the stiffeners to this larger load. So for plates supported by

longitudinal stiffeners the reduction factors are:

= 1.0 if: 0 < rel ≤ 0.7

= 1.474 – 0.677 rel if: 0.7 < rel ≤ 1.291

=

if: 1.291 < rel

NEN 6771 chapter 13 was developed in the years 1978 until 1985, as a successor of the VOSB

(Voorschrift Ontwerp Stalen Bruggen, or Regulation Design Steel Bridges). At the time of the

development the formula by Winter, with the higher ultimate load, was already known. Nevertheless


24

for the main method there has been chosen to use formulas based on critical load, because the NEN

6771 was mainly intended for steel bridges. These designs would be governed by fatigue, so a refined

way of calculating plate buckling would not give more economical designs, since it is not the

governing criterion. To prevent being too conservative the small amount of post critical strength was

added. For structures without fatigue problems the method by Winter wás added to the code,

however more modestly (art. 10.2.4.2.3) between other sub articles and not as the main method in a

separate chapter (art. 13).

2.2.5 NEN-EN 1993-1-5

In the Eurocode the formulas by Winter have been used. A small adjustment to the original formula

has been made, to make it dependent of , the stress distribution in the plate. This factor is

determined by dividing the smallest stress in the plate by the largest, while taking compressive

stresses as positive. For a plate loaded in compression equals unity and the formula is identical to

the original by Winter. A plate loaded by bending, 1, has a slightly higher capacity.

For plate parts with a free longitudinal edge (outstand element) the reduction factor is:

Using the ultimate load according Winter in bridge structures could give problems regarding fatigue.

In the normal fatigue verifications the peak stresses are higher because the steel plates will be

thinner. In proper fatigue verifications it should become clear whether this is a problem or not. These

calculations however do not take into account the deformations out-of-plane of a steel plate. At for

instance a web-flange connection in a variably loaded I beam there is high fatigue sensitivity in the

welds if the web needs the buckled mode to take up the maximum loads. This phenomenon is called

web breathing and NEN-EN 1993-2 takes it into account using:

If only normal stresses are present it is clear that they are allowed to be ten percent higher than the

critical stress. Or expressed in a reduction factor:

x,Ed,ser ≤ 1.1 · cr = · fy


25

2.2.6 Overview

Multiple reductions factors have been discussed, but the best way to compare them is graphically. In

figure 2.6 the black lines represent the base formulas, which disregard any imperfection. The blue

lines represent the different reduction factors in NEN-EN 1993-1-5 and the red lines are for NEN 6771

both with and without post critical strength. In NEN 6771 there is a correction factor 1/C, which can

slightly adjust the reduction factor (higher or lower) in some cases, but his is not shown in the graph.

Figure 2.6: The various discussed reduction factors

In figure 2.7 it can be found that the largest difference in reduction factor between NEN-EN 1993-1-5

and NEN 6771 occurs around rel = 1.9. For a plate in pure compression Eurocode will give about 30

percent extra capacity, for a plate in bending this is around 38 percent.

Figure 2.7: Extra capacity in Eurocode compared to NEN

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

1,2

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

[-

]

[-]

Von Karman (1/λ)

Euler (1/λ²)

EN 1995-1-3; Ψ = 1

EN 1995-1-3; Ψ = -1

Web Breathing

NEN 6771; incl.

NEN 6771; excl.

0,95

1,00

1,05

1,10

1,15

1,20

1,25

1,30

1,35

1,40

0,5 1,0 1,5 2,0 2,5

E

uro

cod

e/ N

EN [

-]

rel [-]

Ψ = 1

Ψ = 0

Ψ = -1


26

2.3 Verification methods in design codes

2.3.1 Inventory of methods

To calculate the ultimate bearing capacity of a plate field one needs to choose whether to use the

effective cross-section or the reduced stress method. Besides that, a choice must be made between

all the different reduction factors. Not all design codes have chosen to use the same method as their

default method.

As shown in chapter 2.2.4 in NEN 6771 the default method is the reduced stress method, with the

reduction factors based on the critical stress according Euler. Even though the effective cross-section

method and reduction factors according Winter are allowed, they are not prominently presented.

In NEN-EN 1993-1-5 again both reduced stress and effective cross-section method are allowed. Since

the effective cross-section method is presented in chapter 4, called “plate buckling”, it is clear this is

the preferred method. The reduced stress method is presented in chapter 10, which consists of only

two pages. Both these methods however work with the reduction factors according Winter, as

discussed in chapter 2.2.5. As shown in figure 2.7 this can lead to much extra capacity compared to

the stress reduction method in NEN 6771. Since deformation out-of-plane can become a problem

using these high capacities the web breathing article is introduced for variably loaded structures.

At first hand the stress reduction methods in both design codes look completely different. In NEN

6771 a buckling stress is determined, which is then used instead of the yield stress in the usual cross-

section verifications. In NEN-EN 1993-1-5 the complete stress field of the entire structure is verified

in a single formula. Two values have to be determined (ault and acrit), which is usually done by a

computer calculation. The big advantage of this second method is the speed of the calculation,

provided a computer program is available to calculate the a values. If however the equivalent stress

or von Mises criterion is used, the simple cross-section verifications can still be deducted.

2.3.2 NEN 6771

In article 13.8.1 of NEN 6771 the most important verification for a plate loaded in compression is

given. The formulas (note that ‘plooi’ is Dutch for plate buckling) in this article combined state:

main formula

Where is equal to the reduction factor as presented in chapter 3.2.4 of this thesis and the

correction factor C is determined according articles 13.8.1.1 through 13.8.1.3 of NEN 6771.


27

2.3.3 NEN-EN 1993-1-5

Article 4 of NEN-EN 1993-1-5 contains the effective cross-section method. In article 4.4 the general

formula for effective area is given:

Where is the reduction factor for plate buckling, as presented in chapter 2.2.5 of this thesis and Ac

is the steel area in compression. However there is slightly different notation in the Eurocode

formulas. The relative plate slenderness, rel, is referred to as , not to be confused with the plate

slenderness p. This is due to the fact that in NEN-EN 1993-1-1 the relative slenderness is defined as

, which could be for any kind of element. To emphasize NEN-EN 1993-1-5 works with plates the

index ‘p’ is added. Compare also to the definition of the specific, plate and relative slenderness at the

start of chapter 2.2. The relative plate slenderness used in the Eurocode is simplified to:

When the reduction factor is determined it depends on the stress distribution in the plate how it

must be applied. Table 2.2 and 2.3 show how Eurocode defines the effective width in different

situations.

Table 2.2: Internal compression elements


28

Table 2.3: Outstand compression elements

These layouts of effective areas applied to an I-shaped beam will give the effective cross-sections as

in figure 2.8. The parts marked in black are ineffective. The white parts can be used for cross-

sectional verifications or other structural calculations.

Figure 2.8: Effective area of an I-shaped cross-section

d = fy;d

d < fy;d

Loaded in bending Loaded in compression

d = fy;d


29

In article 10 of NEN-EN 1993-1-5 the reduced stress method is described. The main formula which

governs the plate buckling effect is given as:

In which is the minimum load amplifier for the design loads to reach the characteristic value of

resistance of the most critical point of the plate. Or in other words, it is the yield stress divided by the

largest stress present, since resistance is governed by yielding and the structure is supposed to

behave linear elastic without plate buckling. The reduction factor is determined with the same set

of formulas as in the effective cross-section method, only now the relative slenderness is defined as:

Where is the minimum load amplifier for the design load to reach the critical load of the plate

under the complete stress field. Calculating for simple plate structures can be done by hand, but

for large structure a finite element buckling (eigen value) analysis is ideal. Since plate buckling in real

structures is not equal to the critical load, the extra step through the equations by Winter is made, to

take into account imperfections and post buckling strength.

In a previous chapter it was already stated that this method with all stresses combined is analogue to

the method where separate cross-section verifications are used. This can be shown for any situation

using the von Mises criterion, but for simplicity we can also assume stress is present in only one

direction. This state can be used for pure bending or compression situations, where only normal

stresses are present. The verification formula now becomes:

For the reduction factor the following derivation can be made:

The relative slenderness is determined in the same manner as in the other methods, and the

reduction factor is multiplied with the yield stress to give a maximum allowable stress. Except for the

formulas relating and (Euler vs. Winter) there is no difference in the reduced stress methods

given in NEN 6771 and NEN-EN 1993-1-5.

2.3.4 Step by step guide

In table 2.4 a step by step guide is presented, to give a guideline and overview of the methods

described in the last two subsections. Each step identifies where in the relevant code it can be found.


30

NEN 6771 Reduced stress method

NEN-EN 1993-1-5 Reduced stress method

NEN-EN 1993-1-5 Effective cross-section method

Remarks

1 Divide the cross-section in separate plate parts.

Divide the cross-section in separate plate parts.

Divide the cross-section in separate plate parts.

The next steps will have to be repeated for each individual plate part.

2 Determine Determine Determinefor each compressed flange

= 2 / 1

3 Determine k;x

[fig. 43 + 44] Determine k[table 4.1 + 4.2]

Determine k[table 4.1 + 4.2]

Eurocode table is a little more straightforward.

4a DetermineE [eq. 13.6-4]

4b Determinei;k;x [eq. 13.6-1]

4c Determineplaat;rel [eq. 13.7-1]

Determinep [art. 4.4.2]

Determinep [art. 4.4.2]

Eurocode takes a more direct approach.

5 Determineplooi;rel [art. 13.7.2]

Determine [art. 4.4.2]

Determine [art. 4.4.2]

EC gives choice from 2 formulas, NEN in total 5.

6a Determine whether interaction with column buckling is present [art. 4.5.3 + 4.5.4]

Determine whether interaction with column buckling is present [art. 4.5.3 + 4.5.4]

Only for small a

6b Determine C;x

[art. 13.8.1]

Choose from 4 situations.

7 Determineplooi;d with

the smallest plooi;rel [eq. 13.8-3a]

Determinemax with the

smallest [art. 10.5]

Determine the effective area of the flanges [table 4.1 + 4.2]

8 Determine elastic cross-sectional properties (Atot and Wel)

Determine elastic cross-sectional properties (Atot

and Wel)

Determine elastic cross-sectional properties with effective flanges

9 Determine capacity

(using Atot, Wel and plooi;d)

Determine capacity

(using Atot, Wel and max) Determinein web

10 Determine k,pand (web only) [tabel 4.1 + 4.2] and [art. 4.4.2]

11 Determine whether interaction with column buckling is present [art. 4.5.3 + 4.5.4]

12 Determine the effective area of the web [table 4.1 + 4.2]

13 Determine effective cross-sectional properties (Aeff and Wel;eff)

14 Determine capacitity (using Aeff, Wel;eff and fy;d)

Table 2.4: Step by step guide for codified plate buckling design

Using the effective cross-section method in NEN-EN 1993-1-5 there are another two options. Either a

single effective cross-section is determined for the actual value of , or both for bending and normal

force a separate effective cross-section is calculated. See also the flow chart on the next page.


31

Table 2.5: Flow chart for plate buckling in NEN-EN 1993-1-5

Determine stresses in

structure

(1st order, linear elastic)

Stress reduction method

Ch. 10

Effective cross-section method

Ch. 4 up to 7

Aeff for normal force

verification (=1)

Weff for bending

moment verification

(=-1)

Single effective cross-

section for bending

moment and normal force

no

yes

no yes

Verification of cross-

sections:

NEN-EN 1993-1-5

Art. 4.6 en 7

Start

Determine with actual

acting forces

Use effective cross-

sections in frame

program

Use gross cross-

sections in frame

program

Verification of cross-

sections and (lateral

torsional) buckling:

NEN-EN 1993-1-1

using Aeff, Ieff, Weff

Determine forces.

1st order

(geometrical linear)

Determine forces.

2nd order

(GNL incl. imperfections)

Has eN

changed?

Aeff=lim·Agross

islim < 0.5?

Choose

verification method

Formula

10.1 satisfied

Increase

cross-sections

no

Verification according

NEN-EN 1993-1-1 using

class 3 properties

yes


32

As a part of this thesis the effective cross-section method in NEN-EN 1993-1-5 has been implemented

in a Mathcad sheet. As long as the geometry of a cross-section is as assumed in the sheet, the plate

buckling capacity can be calculated. In annex D.1 one of the sheets is given as an example.

After a detailed analysis of the different methods studied in this chapter some remarks can be made:

- The reduced stress method in NEN 6771 and NEN-EN 1995-1-5 are very similar. Eurocode has

a more compact way of formulating the equations, but the only major difference is in the use

of the reduction factor. This can be the only explanation for any deviation between plate

buckling capacities.

- For more elaborate structures the method in NEN-EN 1993-1-5 will give an advantage in

efficiency, provided that software is used which calculates and .

- The effective cross-section method will generally give a higher workload for the engineer, but

also more cross-sectional capacity. The workload is caused by the fact that first the effective

flanges need to be determined, before finding the stress distribution in the web. This makes

that the cross-sectional properties have to be determined several times. The higher capacity

is mainly due the fact that each plate part is multiplied with its own reduction factor, instead

of choosing the lowest as governing for the entire cross-section.

- The forces in a statically undetermined structure are depending on the stiffness of the

members. If these members are built up from plates, plate buckling may reduce the stiffness.

When calculating with the effective cross-section method this effect is taken into account.

Displacements will generally be larger and better predict the behavior of the real structure. A

disadvantage is that the iterative procedure which must be made can be quite time

consuming. In NEN-EN 1993-1-5 it is however stated this is only relevant when < lim = 0.5,

where lim can also be given in the National Annex.


33

2.3.5 Bending moment capacity differences

Different calculation methods will give different plate buckling capacities. Figure 2.9 gives the

bending moment capacity of a plate with an increasing width, b. The thickness is kept constant and

the length of the field, a, is supposed to be long enough not to have an influence in the buckling

capacity.

These properties give a quadratic increasing elastic capacity, since Mel,d = ⅙ fy,d t b2. Since Winters

formula is inversely proportional to the relative slenderness, the capacity calculated by the Eurocode

increases linear after initial buckling. The critical stress declines quadratic with the relative

slenderness, combined with a quadratic increase of moment of area gives a constant plate buckling

bending moment.

In this example the thickness is chosen to be 10 mm, the yield stress is 235 N/mm2. The width of the

plate field starts at 0 mm or rel = 0.0, while it stops at 3600 mm or rel = 2.6. As a check if the graph is

correct a plate with relative slenderness rel = 2.0 is calculated manually. In annex A.1 this calculation

can be found.

Figure 2.9: Bending moment capacity for several methods

It is clear that especially for more slender plates the difference in capacity can be very large. The

difference between the two dotted lines is exactly 10%, which can be related to the factor 1,1 found

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6

Mb

uck

ling

[kN

m]

rel [-]

Effective cross-section methods

EC: Reduced stress method

EC: Web breathing

NEN: Red. stress method, with post critical strenght

NEN: Red. stress method, no post critical strengh

Elastic capacity

b

x a

t, fy

1666 kN

1429 kN

1039 kN


34

in the web breathing article. The penalty for having a variable loaded structure is quite steep, the

capacity falls back from the green line to the dotted red line.

Part of this thesis is about understanding why NEN-EN 1993-1-5 gives more plate buckling capacity

then NEN 6771. Therefore it is interesting to know how big the difference is in the first place. In

figure 2.10 the capacities of a plate in bending calculated by Eurocode are divided by the capacity

according NEN 6771. This relates to figure 2.9 as the difference between the blue line on the one

hand and the red and green on the other.

Figure 2.10: Capacity of Eurocode relative to NEN 6771 (plate in bending)

The only difference between both reduced stress methods is due to the choice in reduction factor,

therefore the dotted line in figure 2.10 is identical to the graph for = -1 in figure 2.7, where only

the reduction factors were compared.

The solid line in figure 2.10 shows that an NEN-EN 1993-1-5 calculation can give up to 60% extra

capacity compared to a NEN 6771 calculation. The difference is caused by using the effective cross-

section method instead of the reduced stress method and again the different reduction factor.

Multiplying the graphs in figure 2.2 and 2.7 therefore gives the same result as in figure 2.10, which

was made by dividing the graphs shown in figure 2.9.

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,8

0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6

Md

,EC

/ M

d,N

EN [-

]

rel [-]

EC: Effective cross-section method



35

2.3.6 Normal force capacity differences

While in the last subsection the bending moment capacities where compared, in this section the

same scheme is used to compare the normal force capacity. Again the thickness is kept constant,

while plate width ‘b’ is increased and length ‘a’ is long enough not to have influence in the plate

buckling capacity.

Since the normal force is calculated by Nel = fy t b, the elastic capacity shows a linear increase. The

effective cross-section methods approach a constant value, which is limited by the effective width

the chosen thickness can support, as shown first by von Kármán. The larger the slenderness, the less

influence initial imperfections have and the closer the capacity approaches the theoretical capacity

by von Kármán. The critical buckling load used in the other methods declines as the slenderness

becomes larger. The critical stress decreases quadratic, while the area increases only linear. Even

though more steel area is present, the buckling load becomes smaller.

In this example the thickness is chosen to be 24 mm, the yield stress is 235 N/mm2. The width of the

plate field increases from zero to 4800 mm or rel = 3,6. As a check if the graph is correct a plate with

relative slenderness rel = 2.0 is calculated manually. In annex A.2 this calculation can be found.

Figure 2.11: Bending moment capacity for several methods

Since both reduced stress method and effective cross-section method will give the same result for a

compressed plate, the red line in figure 2.11 represents a total of three of the discussed methods.

These are both methods in NEN-EN 1993-1-5 and the effective cross-section method in NEN 6771.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 3,2 3,4 3,6

F bu

cklin

g[k

N]

rel [-]

Effective cross-section methods and EC red.str.meth.

EC: Web breathing

NEN: Red. stress method, with post critical strenght

NEN: Red. stress method, no post critical strenght

Elastic capacity

b

x a

t, fy

5289 kN

6842 kN


36

Figure 2.12: Capacity of Eurocode relative to NEN 6771 (plate in compression)

The graph in figure 2.12 is obtain by dividing the red line from figure 2.11 by the blue line, and

represents the extra capacity calculated by NEN-EN 1993-1-5 compared to NEN 6771. Since the only

difference is in the choice of reduction factor, the graph is again identical to the one in figure 2.7.

In the area around ≈ 3.0 the buckling load according Eurocode is almost 2,5 times higher than what

NEN predicts. This deviation might seem extremely high but can be explained very well using the

contents of this chapter.

0,95

1,00

1,05

1,10

1,15

1,20

1,25

1,30

1,35

0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6

F d,E

C3

/ F d

,NEN

[-]

rel [-]


37

2.3.7 Conclusion

The differences between NEN 6771 chapter 13 and NEN-EN 1993-1-5 chapter 4 as found in this

chapter can be summed up to:

- Eurocode is based on the ultimate load according Winter, NEN uses formulas based on the

critical load.

- This choice can give up to dozens of percent extra capacity when using the Eurocode.

- When drafting NEN 6771 the choice of using initial buckling theory was deliberately made,

since the code was meant for bridge design. These are loaded variably and therefore

sensitive for fatigue, which eliminates the need for a refined plate buckling analysis.

- When designing bridges according to NEN-EN 1993-1-5 the buckling stress drops very fast

after around rel ≈ 1.3, so the most economic structures will have members with plates below

this slenderness.

Fundamental differences between the reduced stress and the effective cross-section method:

- Using the reduced stress method the weakest link in the cross-section is governing for the

maximum allowable stress.

- Using the effective cross-section method a buckled plate part will remain at capacity when

deformed, so other parts can also be loaded up to buckling. This distributing of the load to

the stronger parts is called load shedding.

- Load shedding (and with it the difference between reduced stress and effective cross-section

method) has more influence on the total capacity in cross-section with large differences in

slenderness between individual plate parts.


38

Chapter 3: Design codes applied to example cross-sections

This chapter examines what the impact of the different methods found in the previous chapter on

the plate buckling capacity of some example cross-sections is. These examples are chosen such that

they represent the order of magnitude used in large bridges in the Netherlands. Not just the capacity

is investigated, but also the workload using a certain method and the influence of web-flange

slenderness ratios. The last part is about the application of stiffeners.

3.1 Capacities according design codes

In this subsection two example cross-sections are analyzed using both NEN 6771 and NEN-EN 1993-1-

5. These sections are an I-shaped girder and box girder, the dimensions are found in figure 3.1. The

length of the girders is assumed to be long enough not to be of influence to the plate buckling

capacity. Global (or column) buckling is not taken into account.

Figure 3.1: Dimensions of example cross-sections

The plate buckling capacities to be compared are for loading with a normal force, bending moment

around the weak axis and around the strong axis. Furthermore the difference in ULS (ultimate limit

state) and SLS (serviceability limit state) is distinguished. This implies that a total of 36 calculations is

made, of which as example three can be found in the annex B.1.

A capacity for the ULS is determined using the normal plate buckling calculations, for NEN 6771 this

means the reduced stress method including post buckling strength. The calculation for the SLS

according the Eurocode is made with the web breathing article. Since the capacity is severely

reduced using this article, it might be the governing situation in a bridge structure, even though no

load factors and only the frequent loads are used. To make a fair comparison with NEN 6771 the post

buckling capacity has been disregarded. Using only the critical stress comes down to not allowing the

web to buckle, and thus preventing breathing fatigue in the web-flange connection.

hw = 2000

tw = 15

tf = 24

bf = 800

hw = 1000

tw = 15

tf = 20

bf = 1000

All dimensions in: [mm]

ht = 2048

ht = 1015

bt = 1020


39

3.1.1 I-shaped girder

In table 3.1 the results of the capacity of the I-shaped girder can be found. The workload is the total

for all six calculations with the same design code.




SLS ULS SLS ULS SLS ULS

Normal force [kN]

2579 2579 3215 7795 3215 13908

Bending strong axis [kNm]

11838 11838 13861 13082 13861 13254

Bending weak axis [kNm]

1488 1488 2280 1550 2280 1524

Workload

4 pages 2.0 hours

4 pages 1.9 hours

5 pages 2.75 hours

Table 3.1: Plate buckling capacities of I-shaped girder

In the calculations according NEN 6771 no plate part was in the proper range to adopt post buckling

strength. Therefore the capacities found for ULS are equal to those for the SLS. The SLS capacities for

both Eurocode calculations are made with the web breathing article, and therefore the same. They

have been added in both columns to make a fair comparison between the two methods.

Remarkable is that some of the SLS capacities are higher than the ULS capacities. This is a result of

the way the web breathing article is formulated. In this case it means that the chosen cross-sectional

properties are such that web breathing is not governing for that load case.

Even though it has been said several times that the effective cross-section method will give more

capacity that the reduced stress method, when bending around the weak axis is regarded it is the

other way around. Due to the presence of the web, the neutral line will hardly shift when a part of

the flange becomes ineffective. Consequently the distance from neutral line to extreme fiber is still

large, and yielding reached sooner. Figure 3.2 gives a graphical representation of this effect.

Figure 3.2: Bending around the weak axis

Elastic capacity Reduced stress method Effective cross-section method


40

3.1.2 Box girder

Also for the box girder the hand calculations were carried out. The results can be found on the next

page in table 3.2. For some example calculations see annex B.1. In contrast to 3.1.1 here there are

some small differences in the column for NEN 6771, which means there was a plate part which

utilized post buckling capacity.




SLS ULS SLS ULS SLS ULS

Normal force [kN]

10563 11004 13167 14612 13167 16728

Bending strong axis [kNm]

6101 6101 8360 6541 8360 7268

Bending weak axis [kNm]

3712 3869 4082 4530 4082 5916

Workload

4 pages 1.75 hours

4 pages 1.6 hours

5 pages 2.25 hours

Table 3.2: Plate buckling capacities of box girder

The workloads have been determined as if every calculation would have been the first one. Many

times a relative slenderness or other parameter from a previous calculation could have been used,

but to keep the comparison unbiased this is not exploited. Again some of the web breathing

capacities are much higher than the normal plate buckling capacities.

In bending the webs of the box girder are not sensitive to plate buckling. This made the final step in

the Eurocode calculations much faster. Otherwise the second moment of area of a cross-section with

a gap in the web had to be calculated, which can be time consuming. Even with this advantage the

Eurocode calculations took more time to complete.

3.1.3 Capacity differences

The absolute values of the calculated capacities differ pretty much, but are therefore hard to

compare accurate. In table 3.3 four comparisons have been made, similar to situations an engineer

might come across in practice. In each column can be found what happens to a capacity when the

calculation is switched from an ‘old’ method to a ‘new’ one. These switches are:

1: Using article 10 of NEN-EN 1993-1-5 (reduced stress method) instead of NEN 6771.

2: Using article 4 of NEN-EN 1993-1-5 (effective cross-section method) instead of NEN 6771.

3: Using article 10 of NEN-EN 1993-1-5 instead of article 4.

4: Using the web breathing article of NEN-EN 1993-2 instead of NEN 6771 without post critical

strength.


41

1: 2: 3: 4:

Method 1: NEN 6771 NEN 6771 EC: reduced stress m.

NEN (SLS)

Method 2: EC: reduced stress m.

EC: eff. cross-section m.

EC: eff. cross-section m.

EC (SLS)

I-shaped girder:

Normal force + 202.2 % + 439.3 % + 78.4 % + 24.7 %

Bending strong axis + 10.5 % +12.0 % + 1.3 % + 17.1 %

Bending weak axis + 4.2 % + 2.4 % - 1.7 % + 53.2 %

Box girder:

Normal force + 32.8 % + 52.0 % + 14.5 % + 24.7 %

Bending strong axis + 7.2 % + 19.1 % + 11.1 % + 37.0 % Bending weak axis + 17.1 % + 52.9 % + 30.6 % + 10.0 %

Table 3.3: Difference in capacity

When regarding the type of load the I-shaped beam is intended for (bending around the strong axis),

the Eurocode gives around ten percent extra capacity. When using the Eurocode and choosing

between the reduced stress and effective cross-section the difference in capacity is only 1.3 percent,

which might not be worth the extra workload.

The box girder loaded in compression has up to fifty percent extra capacity. Also choosing the

effective cross-section method delivers almost fifteen percent extra capacity over the reduced stress

method.

Even though these comparisons confirm the large differences between NEN 6771 and NEN-EN 1993-

1-5, they are only valid for the example cross-sections found in figure 3.1 and not for all girders. In

the next chapter the influence of the web slenderness is taken into account to get a more general

image of the difference between the design codes.


42

3.2 Influence of web slenderness

In chapter 3.1.3 about the differences in calculated capacities of the two example profiles big

discrepancies showed up. For a more general understanding the slenderness of thickness of the web

is varied. Looking at the basic assumptions for the calculation methods this parameter should have a

large influence.

3.2.1 I-shaped girder

In this subsection the same example profile as shown in figure 3.1 is taken, only the thickness of the

web plate is varied between 0 and 40 millimeter. For the I-shaped girder only the bending around the

strong axis is taken, since this is what the cross-section is designed for. The results can be found in

figure 3.3, where also the capacities from table 3.1 are found by checking tw = 15.

Figure 3.3: Bending moment capacity

The most fundamental difference between the reduced stress method and the effective cross-

section method is whether the weakest link is found to be governing, or that a buckled plate remains

at capacity to give other plate parts the opportunity to also reach the buckling load. In the graph this

is illustrated clearly, both stress reduction methods have a capacity reduced to zero as the web

becomes thinner. The effective cross-section will always have the bending moment capacity of the

effective flanges, even if there is no web. Of course this is not a realistic design situation, but is

clearly illustrates the effect of the assumption.

Where the reduction of the web is equal to that of the flange, the capacities of both Eurocode

methods are almost equal, which is to be expected. The small difference is in the bending moment

capacity of the web, but this is not much compared to the entire cross-section.

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

22000

24000

0 5 10 15 20 25 30 35 40

Mb

uck

llin

g[kN

m]

tw [mm]

EC: Effective cross-section methodEC: Reduced stress methodEC: Web-BreathingNEN 6771Elastic capacity

13861 kNm

13254 kNm

13082 kNm

11838 kNm


43

Another clear conclusion from figure 3.3 is that web breathing only is of influence in slender webs.

This effect is reinforced by the fact that only the frequent load level has to be checked against the

web breathing capacity.

The slope change of the graph for both reduced stress methods around tw = 15 is caused by the fact

that above this web thickness the flange is governing. The maximum allowable stress is not increased

after this point, so the only increase in bending moment is possible because the area of steel

increases. Since the contribution of the web to the bending moment capacity is rather small the

slope is not very steep.

In the effective cross-section method it is never either the web or the flange which is governing for

the capacity. The slope change around tw = 22 is because the reduction factor for the web has

turned to unity. After this point again the increase in bending moment capacity is only due to

increase in steel area.

The difference between the methods can be found in figure 3.4, were two situations have been

calculated. The first is using the reduced stress method according NEN-EN 1993-1-5 instead of NEN

6771. This extra capacity is always present since nowadays the Eurocode has replaced the NEN. The

second line is for the choice of method within the Eurocode. This is a choice engineers must make

when calculating the capacity of a structure.

Figure 3.4: Difference in bending moment capacity

In the spreadsheet file with which the graphs were produced other dimension of for instance the

flanges can be substituted. The location of slope changes, slopes, capacities and differences all

change, but there always is a point where difference between the Eurocode methods is almost zero.

This happens when the slendernesses are such that web ≈ flange. If a simple reduced stress method

calculation is preferred, this could be a good principle to still have a high capacity in the cross-

section.

1,0

1,1

1,2

1,3

1,4

1,5

1,6

5 10 15 20 25 30 35

Met

ho

d 2

/ M

eth

od

1

tw [mm]

1: NEN reduced stress method 2: EC reduced stress method

1: EC reduced stress method 2: EC effective cross-section method


44

3.2.2 Box girder

The same comparison as in 3.2.1 has also been made for the normal force capacity of the box girder.

The result is very similar and therefore only the two figures are shown, the same remarks as before

still apply.

Figure 3.5: Normal force capacity of box girder

Figure 3.6: Difference in normal force capacity

3.2.3 Conclusion

The most significant conclusion made in this chapter is that by wisely choosing your web-flange

slenderness ratio a reduced stress method calculation can give a capacity similar to a more

complicated effective cross-section method.

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

0 5 10 15 20 25 30 35 40

F bu

clkl

ing

[kN

]

tw [mm]



EC: Web breathing

NEN 6771

Elastic capacity

1,0

1,1

1,2

1,3

1,4

1,5

1,6

5 10 15 20 25 30 35

Met

ho

d 2

/ M

eth

od

1

tw [mm]

1: NEN reduced stress method 2: EC reduced stress method

1: EC reduced stress method 2: EC effective cross-section method

13167 kN

11004 kN

16728 kN

14612 kN


45

3.3 Stiffeners according design codes

If a steel plate does not have enough capacity to carry the load, it is not unavoidable to increase the

thickness of the plate. Adding one or more stiffeners will also increase the capacity of the plate. In

many structures adding stiffeners is in the end the cheapest way to built. Even though stiffeners can

cost quite a lot to fabricate, the structure itself will be lighter. Which option gives the most

economical design cannot be known for sure in advance.

In NEN 6771 longitudinal stiffeners are regarded as columns loaded in bending and compression.

They are loaded with the plate buckling stress of the adjacent plates, and can either meet the

requirements or not. If a stiffener does not meet the requirements the size must be increased, or the

length decreased by adding extra transverse stiffeners. In NEN-EN 1993-1-5 another approach is

taken. Depending on the dimensions of the stiffener an extra reduction factor c is determined and

multiplied with the area of the neighboring plate parts. A stiffener and plate combination which is

not allowed by NEN 6771 can still be analyzed with NEN-EN 1993-1-5.

Since in NEN 6771 it is stated that loading until the full post critical strength of a plate gives such a

load on the stiffener, the advantage won in the plate is immediately lost in the stiffener. Therefore

we would expect that according Eurocode the stiffeners should be larger, to carry the full load. In

three different situations this is investigated. First a given set of dimensions is used to determine the

buckling capacity. In the second part it is investigated if a stiffener can be designed to fully make use

of the plate buckling capacity by Winter. In the third part a certain design load is assumed and plates

and stiffeners are found to match the load.

3.3.1 Capacity at given dimensions of plate and stiffener

Longitudinal stiffeners are regarded as columns loaded in compression and bending in NEN 6771. The

length of the plate now has a large influence on the final capacity of the plate and stiffener. The first

situation investigated is the calculated capacity at some given dimensions. These dimensions can be

found in figure 3.7.

Figure 3.7: Dimensions of plate and strip stiffener

The plate and stiffener have been analyzed according both NEN 6771 and NEN-EN 1993-1-5, of which

an example can be found in annex B.2. The results are found in table 3.4 on the next page.

4000 mm

1000 mm

1000 mm Plate: t = 12 mm

Stiffener: t = 12 mm

h = 180 mm

Steel grade: S355


46

180·12 Buckling normal force [Nc;u;d;total];[kN]

Area steel plate [As;plate];[mm2]

Area stiffener [As;lv];[mm2]

Total area [As;total];[mm2]

Average stress

[cr];[N/mm2]

NEN 2526 24000 2160 26160 96.56

EC3 3464 24000 2160 26160 132.4

Table 3.4: Normal force capacities of plate and stiffener

In the calculation according NEN 6771 the unity check of the longitudinal stiffener was 0.74, which

means that the stiffener is larger strictly necessary. In table 3.5 the results are found for a stiffener of

170·12 mm. This gives a unity check of 1.002. Since the plate is the same, the total capacity drops

with N = A · cr = 12 · 10 · 96.56 · 10-3 = 11.6 kN. The capacity according the Eurocode has a larger

decline, since not only the steel area decreases, but also the average stress decreases because the

plate is less supported than before.





Average stress

[cr];[N/mm2]

NEN 2514 24000 2040 26040 96.56

EC3 3315 24000 2040 26040 127.3

Table 3.5: Normal force capacities of plate and stiffener

From the last two tables it can be seen that the Eurocode calculates a higher capacity when a

longitudinal stiffener is applied. However this is mainly due to the fact that plates itself already have

a higher capacity when using Winters method.

3.3.2 Minimum stiffener dimensions for full Winter capacity

The buckling capacity of the plate itself, calculated in 3.3.1 in accordance with NEN-EN 1993-1-5, is

reduced because of the presence of the stiffener. The buckling load of the plate could be higher if the

longitudinal stiffener would provide better support. If the stiffener would provide full support, the

buckling load by Winter could be reached in the plate. This would be as much as 0.485 · 2000 · 12 ·

355 · 10-3 = 4132 kN, even without the extra contribution of the area of the stiffener.

The dimensions needed for the longitudinal stiffener can be deducted from article 6.3.1.2 of NEN-EN

1993-1-1. To get perfect support, we set the reduction factor cto 1.0 and find rel;lv = 0.2. But to get

the relative column slenderness this low we would need a stiffener measuring 800 by 60 millimeters.

This would make the total steel area of the stiffener twice as large as that of the plate it is supposed

to support. The design philosophy of a stiffener (adding a relatively small steel area to get a relatively

large increase in buckling capacity) is completely gone. So when using rationally sized stiffeners, the

maximum plate buckling load by Winter will never be reached.

The conclusion to be drawn is that there is some truth in the philosophy (no post critical strength

when using stiffeners) in NEN 6771. Indeed it is not possible to design a stiffener which enables a

plate to reach the buckling capacity by Winter. In the stiffeners it is more than lost what is gained in

the plate itself. Nevertheless Eurocode does give a higher capacity for a given set of dimensions.


47

When a longitudinal stiffener of 400 · 42 mm is chosen the reduction factor is equal to = 0.970. Still

not an economical design, but it does reflect what is needed to guarantee near perfect support to

the plate.





Average stress

[cr];[N/mm2] EC3 9855 24000 16800 40800 242

Table 3.6: Buckling capacity with very large stiffener

The total buckling load is much larger than the previously calculated 4132 kN, which was for only the

plate and perfect support. The area of the stiffener is loaded with the yield stress, because its

dimensions make it not sensitive to buckling. Since the stiffener is still 40% of the total area this more

than doubles the ultimate load. Compared to the stiffener of 170 by 12 millimeters the structure is

now 1.6 times as heavy, but nearly 3 times as much load can be carried. This is reflected in the

average stress which becomes 3 / 1.6 = 1.9 times as large. Even though this is quite an academic

situation, it does reflect the possibilities by the Eurocode method.

An interesting question is whether this 400 by 42 mm longitudinal stiffener really adds capacity to

the structure. Therefore it is compared to a single unstiffened plate, with the same total area. The

stiffener is as it were smeared out over the entire plate width. The thickness of this plate is 40800 /

2000 = 20.4 mm, and the width is 2000 mm. In table 3.7 the capacity can be found according both

NEN and the Eurocode.

t = 20.4 mm b = 2000 mm

Buckling normal force [Nc;u;d;total];[kN]

Average stress

[cr];[N/mm2] NEN 6771 4248 104

EC 1993-1-5 6126 150

Table 3.7: Buckling capacity of equivalent plate

With the same amount of steel still 60% of the load can be carried. Taking into account the

fabrication costs of a stiffener the second option will be more economic. Again the Eurocode will

generate more capacity than NEN when using small stiffeners, but increasing the size will indeed cost

more than is gained.

3.3.3 Plate and stiffener dimensions needed for given capacity

So far a given set of dimensions was analyzed to compare the capacities in the design codes. In this

subsection a total load of 5000 kN is assumed and dimensions to carry the load are sought. This order

is more similar to what a structural engineer would come across, so it is a good way of comparing the

design codes. The length and total width of the plate field are equal to before, but the plate

thickness, number of stiffeners and their dimension are not set. To make an accurate comparison no

standard sizes for plate thickness of stiffener are used. Figure 3.8 shows the dimensions of the plate.


48

Figure 3.8: Dimensions of plate and stiffener(s)

Designing this structure using NEN 6771 the procedure is quite straight forward. First a plate

thickness and number of longitudinal stiffeners has to be chosen, with this the critical stress can be

calculated. Using the critical stress the dimensions of the stiffener(s) can be determined. The total

capacity is the total area times the critical stress. If the capacity is not what was needed, the plate

thickness can be adjusted and the calculation remade.

Using NEN-EN 1993-1-5 first a plate thickness is chosen. By adjusting the size of the stiffener any

capacity can be created. So if the goal is a fixed total capacity, like in this example, an infinite number

of structures can be designed. This is in contrast to NEN 6771, where only a single structure is found.

Due to the amount of formulas needed in the Eurocode calculation it is hard to figure out what the

most economical design is.

Design code

Number of stiffeners

Plate thickness [t];[mm]

Dimensions of stiffener(s) [hst·tst];[mm·mm]

Steel area [As];[mm2]

Relative steel area

Ultimate load [Nc;u;d];[kN]

Average stress

[b];[N/mm2] NEN 0 22.5 0·0 {= 0} 45000 1.630 4997 111.0

NEN 1 15.0 190·20 {= 3800} 33800 1.225 5099 150.9

NEN 2 11.0 190·19 {= 7220} 29220 1.059 5333 182.5

EC3 0 18.3 0·0 {= 0} 36600 1.326 4988 136.3

EC3 1 16.0 13.0

145·16 {= 2320} 210·22 {= 4620}

34320 30620

1.243 1.109

4910 5090

143.1 166.2

EC3 2 13.0 10.0

160·16 {= 5120} 190·20 {= 7600}

31120 27600

1.128 1.000

5056 4941

162.5 179.0

Table 3.8: Dimensions of plate stiffener needed for Nu = 5000 kN

In table 3.8 the results are shown for the plate with stiffeners and a length of 4000 mm. Since the

length has a large influence in the buckling load of the stiffeners, the calculations are repeated with a

length a = 2000 mm. The dimensions of the unstiffened plates do not changes, since the aspect ratio

a has gone from two to one, which is of no influence on the buckling load. In the stiffened plates

with an equal thickness having half the length also means having about half the stiffener area. In

table 3.9 the results of this second calculation can be found.

4000 mm

1000 mm

1000 mm Plate: t = … mm

Stiffener: t = … mm

h = … mm

Steel grade: S355


49

Design code

Number of stiffeners

Plate thickness [t];[mm]

Dimensions of stiffener(s) [hst·tst];[mm·mm]

Steel area [As];[mm2]

Relative steel area

Ultimate load [Nc;u;d];[kN]

Average stress

[b];[N/mm2] NEN 0 22.5 0·0 {= 0 } 45000 1.891 4997 111.0

NEN 1 15.5 141·10 {= 1410} 32410 1.362 5221 161.1

NEN 2 11.5 135·10 {= 2700} 25700 1.080 5127 199.5

EC3 0 18.3 0·0 {= 0} 36600 1.538 4988 136.3

EC3 1 16.0 13.0 10.0 0.0

105·11 {= 1155} 160·16 {= 2560} 250·25 {= 6250} 380·38 {= 14440}

33155 28560 26250 14440

1.393 1.200 1.103 0.607

5012 4962 5016 5126

151.2 173.7 191.1 355.0

EC3 2 13.0 10.0 8.0

115·12 {= 2760} 160·16 {= 5120} 195·20 {= 7800}

28760 25120 23800

1.208 1.055 1.000

5159 5198 5100

179.4 206.9 214.3

Table 3.9: Dimensions when a = 2000 mm

By dividing the length of the plate field in half less steel is needed to carry the load of 5000 kN. This is

also reflected in the average stress which is higher. If this will also be the most economical design is

not certain, since now a transverse stiffener is needed halfway the original plate.

In both situations discussed in this section NEN-EN 1993-1-5 gives a lighter design with higher

average stresses. If however the structure has a variable load and web breathing should be taken

into account, capacities similar to that of NEN 6771 are found.

Where NEN 6771 automatically allows only reasonable structures, following the EC3 formulas exactly

a plate thickness of zero millimeters can be chosen. In this case the slender plate is replaced by a

stocky column which carries the entire load. However in reality the longitudinal stiffener needs

lateral support by the plate, to support the weak direction. Since this is not checked in the Eurocode

some common sense is needed while designing stiffened plate structures.

3.3.4 Conclusion

Between NEN 6771 and NEN-EN 1993-1-5 there is a fundamental difference in the approach when

calculating the capacity of stiffened plates. Where in NEN 6771 a stiffener might be too small to give

support to the plate, EC3 just calculates the extra reduction and gives a capacity. This is because NEN

6771 uses a method with rigid stiffeners, where perfect support is needed, while NEN-EN 1993-1-5

uses a flexible stiffener method, which can always be used.

The assumption was that a plate calculated according Eurocode needs a larger stiffener, because

post critical strength is taken into account. When starting with an equal plate thickness this is true,

but the larger stiffener and higher stress also give a higher design load. When designing a structure

for a certain fixed load, the post critical stress used in EC3 ensures a lighter structure can be built. So

even while the hypothesis is in principle true, in practical situations NEN is more conservative.


50

Chapter 4: Finite element calculations

In the plate buckling methods discussed so far, only the ultimate capacity is calculated. Visualized in a

load-displacement diagram this will give a horizontal line, since deformations are not known. Using a

finite element calculation the entire behavior of the plate under a certain type of loading can be

analyzed.

In figure 4.1 a schematic view of the expected load-displacement diagrams is shown. Whether effects

like plasticity, initial imperfections and second order effects are taken into account decides which

graph is followed. The real capacity will be found if all the effects are put into the calculation.

Figure 4.1: Load-displacement diagram

In figure 4.2 the displacement out-of-plane is given. This illustrates the difference between columns

and plate very well. When the critical load is reached and the displacement out-of-plane starts to set

in, a plate still can increase the load and therefore it is stable in its buckled shape. A column on the

other hand has almost no increase in capacity and the equilibrium will quickly become instable. The

plate has a secondary method to stabilize itself, namely membrane stresses. A column can only

generate bending stresses when it is deformed out-of-plane, which are not as affective in handling

the second order effects.

F x,d

[kN

]

ux [mm]

Ultimate load (Winter)

Critical load (Euler)

Linear elastic

ux

x

y

Second order plastic

Second order elastic,

with initial imperfections

Second order elastic,

no imperfections


51

Figure 4.2: Out-of-plane load-displacement diagram

If a plate without initial imperfections is regarded, the critical load is clearly distinguished in the load-

displacement diagram. When the critical load is reached the plate suddenly buckles which is seen as

a sharp slope change of the graph. There is a clear distinction between the primary load path and the

secondary, this point is called the bifurcation point. When however a plate with a certain amount of

initial imperfection is analyzed this clear bifurcation point is not visible. The slope gradually changes

and approaches the secondary behavior. If only an analysis of an imperfect plate is available it is

quite hard to determine the critical load accurately. This is probably the background of the factor 1,1

found in the web breathing article. Since the critical load is not clearly found in an imperfect plate the

committee drafting the Eurocode found the displacements at 10% above the real critical load was the

start of web breathing problems.

Comparing the ultimate plate buckling load by Winter with a finite element calculation is more

straightforward. The ultimate load is the very top of the load-displacement diagram, while its

location may be depending on the initial imperfections, it will at least always be clearly visible.

4.1 Objective

This thesis so far was mostly about inventorying the plate buckling methods in the design codes

which apply to the Netherlands. The differences have been calculated and where possible clarified by

researching the background of the methods. However some questions have formed that could not be

answered immediately. Three of these will be answered using finite element calculations using

Ansys. The questions are:

For various cross-section types the plate buckling capacity has been calculated according

NEN 6771 as well as NEN 1993-1-5. How well do these capacities match with a more

advanced finite element calculation?

In the effective cross-section method it is assumed that a very slender plate will stay at its

buckling capacity when further deformed, so other plate parts in the cross-section may also

reach their buckling load. How correct is this assumption?

The edge of a plate is always modeled as a hinged support, which stays perfectly straight

during loading so membrane stresses can develop. A longitudinal stiffener which is designed

F x,d

[kN

]

No imperfections

With imperfections

uy [mm] uy [mm]

No imperfections

With imperfections

F x,d

[kN

]

Critical load


52

to give the same perfect support needs to be extremely large, chapter 4.3.2 demonstrated

this. Normally the edges of a plate are formed by other plates, for example the edges of a

web plate in an I-shaped beam are formed by the flanges. This leads to the question whether

a flange can actually give this assumed perfect support to the web. No requirements at all

are given in the Eurocode and NEN for the edges of a plate. A flange with tiny dimensions

would still be a perfect edge for the web plate even though it obviously cannot provide the

out-of-plane support the web needs. The question to be answered by the finite elements

calculations is: why are there no requirements given for plate edges or might they be implied

in another article?

4.2 Procedure

The answers to the above questions will be sought using the finite element software Ansys. This is a

general purpose finite element package, with not only structural capabilities but also for instance

magnetism, electricity, heat or flow problems can be addressed. For this thesis only the structural

part is used. A big advantage of Ansys is the scripted method of giving input. For instance geometry

can be put in parametric, so that when another thickness is analyzed this parameter is changed and

the script run again. There is no need to manually redefine the elements, re-mesh the structure and

reapply the boundaries and loads, since the script does this for you in a matter of seconds. A

disadvantage is that it takes some experience to get the script right, but this does not weigh up to the

advantage gained in having a correct script to investigate the influence of parameters. Defining and

meshing a rectangular plate in Ansys can for instance be done like this:

a=4000 (define parameters a and b, length and width of plate) b=1000

na=80 (define parameters na and nb, the number of elements in each direction)

nb=20

k,1,0,0,0 (define four keypoints, ‘k’ is the command for keypoint, followed by the

k,2,b,0,0 keypoint number and the coordinates in x, y and z) k,3,b,a,0

k,4,0,a,0

a,1,2,3,4 (‘a’ generate an area between the given keypoints)

lsel,s,tan1,x,1 (‘lsel’ selects lines, in this case those in x direction) lsel,a,tan1,x,-1

lesize,all,,,nb (divides all selected lines and associated area in the given amount)

lsel,s,tan1,y,1 (the same for the y direction) lsel,a,tan1,y,-1

lesize,all,,,na

amesh,all (automatically meshed all areas, using the latest defined element)

Some complete Ansys scripts, stored as a *.log file, can be found in annex C.

An accurate calculation of plate buckling loads needs two types of nonlinearities in the analysis. The

two types are geometrical nonlinear (second order effects or equilibrium in displaced state) and

physical nonlinear (plasticity). The plate itself needs to be modeled accurately so it can have bending

stresses trough thickness, membrane stresses in the buckles and shear stress where it is twisted.


53

Since the finite element models can be quite complex, it is hard to find the errors in modeling when

an unexpected result is obtained. Therefore the model is built up in steps, while checking if the

results of each step are correct. This has the extra advantage that it is clear that for each

characteristic of the plate what the effect on the load-displacement diagram is. In table 4.1 the

different steps taken to build the final model can be found. In 4.3.1 the results of each step are

discussed briefly.

Geometric

behavior

Physical

behavior

Imperfections Geometry and objective

1 linear linear no Single unstiffened plate [elastic behavior]

2 linear non-linear no Single unstiffened plate [plastic behavior]

3 non-linear linear no Single unstiffened plate [critical buckling load]

4 non-linear linear yes Single unstiffened plate [elastic buckling behavior]

5 non-linear non-linear yes Single unstiffened plate [elastic-plastic ultimate load and

deformation capacity of very slender plates]

6 non-linear non-linear yes I-shaped and box girders [check of design code capacities]

7 non-linear non-linear yes I-shaped and box girders [check of plate edge requirements]

Table 4.1: Step by step building of finite element model

The elements used in the Ansys models are of the type SHELL181. This element type is the most

suitable for the analysis of plate buckling, since it is the most general. The element has four nodes,

each nodes is capable of all six displacements. These are the translations in the x, y and z direction

and rotations around the x, y and z axis. Furthermore the equations governing the element are

suitable for large displacements and strains as well as having geometrical and physical nonlinear

capabilities. Having such a complete element increases the time needed for an analysis, so for many

simple situations reduced elements have been made. In plate buckling however all these aspects play

a role, so the most elaborate element is the only choice.

Figure 4.3: Left: SHELL181 geometry

Right: defining an element

The nonlinear material behavior can be modeled in several ways. These include linear or nonlinear

hardening of softening during plasticity, or a fully nonlinear representation of the real stress strain

diagram. In the calculation for this thesis plasticity is modeled with a constant yielding plateau, with

Defining an element in a log file:

t=10

et,1,SHELL181

R,1,t

The first command line stores

the thickness as a variable, the

second line defines element

type 1 as SHELL181 and the third

line couple the thickness to

element type 1.


54

fy = 355 N/mm2. A perfectly horizontal plateau can give problems with the convergence of the

analysis, therefore after yielding the Young’s modulus is Ey = E/10000 = 210000/10000 = 21 N/mm2.

Figure 4.4 shows the stress-strain diagram as given by Ansys, and the code to create it.

Figure 4.4: Left: stress-strain diagram in Ansys

Right: defining and plotting a material

Finite element calculations can generally be made in two ways, load controlled or displacement

controlled. With the plate buckling problem we expect the load in the plate to drop after the

ultimate load is reached. Using a fixed amount of displacement which is slowly added while

monitoring the support reaction is the easiest way of getting a good record of the post critical

behavior. The biggest advantage of a displacement controlled calculation is that the load does not

have to decline after the ultimate capacity is reached, which ensures easier modeling and better

convergence.

The imperfections can be divided in two general parts, the initial deformation of a plate and its

residual stresses, both resulting from the production process. In annex C to NEN-EN 1993-1-5 there is

a guideline to add all imperfections as a single equivalent initial deformation. This is useful since

deformations are easier to add then residual stresses. The imperfections should be identical to the

plate buckling mode, which is a sinusoidal shape. According to the Eurocode the equivalent initial

imperfection should be the smallest plate dimension (length or width) divided by 200. In Ansys an

eigenvalue analysis is made to determine the shape of the imperfection, the original geometry can

be updated using this shape to add an imperfection.

Since the value of the imperfection used in annex C to NEN-EN 1993-1-5 is tuned to give a design

load, the capacities calculated by Ansys can be compared to the capacity according Winter, because

this method also determines a design load.

The calculation has multiple non-linearities in it, therefore the convergence criteria of importance.

Both an energy and a force based convergence criteria has been used, to ensure close approximation

of the non-linear effects.

Defining and plotting a

material:

mp,ex,1,2.1e5

mp,prxy,1,0.3

tb,biso,1

tbdata,1,fy,21

tbpl,biso,1


55

4.3 Verification of design code capacities

4.3.1 Single plate in compression

The most basic case is the single unstiffened plate field. In this first section a relatively stocky plate is

analyzed. The dimensions of the plate are a*b*t = 2000*1000*20, which makes the plate sensitive to

buckling, but with the critical load higher than the ultimate load (NWinter < NEuler). All edges are simply

supported, so the displacement of the edges in z-direction is set to zero. The nodes at y = 0 are

supported in y-direction, the sum of support reactions of these nodes are taken as the normal force

in the plate. The nodes at y = a are given a fixed displacement which is added in steps. In figure 4.5 all

characteristics can also be found. The complete log-file of the Ansys calculation is found in annex C.1.

Figure 4.5: Characteristics of unstiffened plate field

Since this is the first finite element calculation the steps given in table 4.1 are briefly discussed. Later

in this thesis only the final results are shown. In figure 4.10 the results of the five plate models are

given to clarify the influence of the added characteristics.

FE model 1

This is the linear elastic model, so the calculations in Ansys are easily verified. Even though

calculating the elastic behavior is not something a finite element method is needed for, it is a good

way of checking whether boundary conditions, loads, geometry and other modeling features are

done correctly.

Ansys: y = 0.105e10 N/m2

Analytical: y = E · = E · (l / l ) = 210000 · (10 / 2000) = 1050 N/mm2

Ansys: Ny;total = 0.210e08 N

Analytical: Ny;total =y · As = 1050 · 20 · 1000 = 21.0 · 106 N

Since no yielding is taken into account in the analysis the calculated stress can be higher than the

yield stress of 355 N/mm2.

b

a

y

x z

uy

Dimensions:

a = 2000 mm

b = 1000 mm

t = 20 mm

Boundary conditions:

all edges: dz = 0

y = 0: dy = 0

y = a: dy = 10

x = y = 0: dx = 0


56

FE model 2

This model does include plasticity, but without any imperfections or geometrical nonlinearities.

Ansys: Ny;total = 0.71019e7 N

Analytical: Ny;total = fy · As = 355 · 20 · 1000 = 7.100 · 106 N

There is a very small difference in normal force, which is due to the Young’s modulus not being zero

when yielding and some numerical errors.

FE model 3

Using an eigenvalue analysis the exact critical buckling load is calculated. A disadvantage is that using

this type of analysis no load-displacement diagram is found. The displacement controlled calculation,

geometrical nonlinear but without imperfections, does not converge due to the lack of initial

imperfection. Using an imperfection approaching zero (and an increasingly small step size to ensure

stability) the critical buckling load is still found.

In figure 4.6 the difference is shown between a calculation without imperfections and one with a

very small one. The chosen imperfection was e0 = 0.15 mm, which naturally is very small for a plate of

1000x2000x20 mm. The little imperfection is needed to get the second order effects started, its size

is mainly determined by element size and load step size. The consequence of the imperfection is that

no exact bifurcation point is found, but using an extrapolation of the post critical area the critical

buckling load can still be approximated.

Figure 4.6: (a) no imperfections, clear bifurcation point

(b) very small imperfection, extrapolation needed

Ansys (eigenvalue analysis): Ncr = 6074 kN

Ansys (displ. contr. analysis): Ncr = 5902 kN

Analytical: Ncr =

= 303.7 · 1000 · 20 · 10-3 = 6074 kN

There is a three percent difference in the displacement controlled calculation and the analytical one.

This is caused by the influence of the initial imperfection. Using e0 = 0.08 mm and decreasing the load

step size even further the critical load becomes 5970 kN. It is reasonable to assume the critical load

approaches 6074 kN as the initial imperfection approaches 0 mm.

F y [

kN]

uz [mm] (a) (b)


57

FE model 4

This model is the same as the previous one, only the initial imperfection has a larger value. According

to annex C to NEN-EN 1993-1-5 the initial imperfection should be e0 = b/200 = 5 mm, which would

cover both fabrication tolerances and residual stresses. The results of the calculation do not show a

clear critical load, but they gradually transit to the secondary load path. As a comparison not only the

prescribed 5 mm of imperfection is shown in figure 4.7, but also some other values. The

displacement used in the graph is the maximum out-of-plane displacement of the plate.

Figure 4.7: Elastic plate buckling behaviour with different initial imperfections

FE model 5

This model has all non-linearity’s implemented and can be used to determine the maximum plate

buckling capacity of a single unstiffened plate. The initial imperfection is taken as the prescribed 5

mm by the Eurocode. The complete calculation written as Ansys input can be found in annex C.1. The

next two figures show the sinusoidal mode shape and the stress distribution in a plate with aspect

ratio a = 2. The represented stress in figure 4.9 is the von Mises stress in the middle plane of the

plate, which clearly shows how the force flows around the buckled part.

Figure 4.8: First mode shape with two sinusoidal buckles

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 5 10 15 20 25 30 35 40

F y,t

ota

l[k

N]

uz [mm]

1: 0.15 mm

2: 2.0 mm

3: 5.0 mm

4: 10.0 mm

Euler


58

Figure 4.9: Stress distribution, respectively at 0.3∙Fmax , 0.6∙Fmax and 1.0∙Fmax

The capacities are:

Ansys: Fmax = 5297 kN

Eurocode: FWinter = 5232 kN

At first hand the results are very close together, with the Ansys result slightly higher. However the

Ansys result is very sensitive to the initial imperfection. The fact that the two calculated capacities

above match that good is because the cold formed sections used by Winter to derive the formula had

an equivalent imperfection around the b/200 value used in the Eurocode.

Figure 4.10 shows the results of the calculation, again with the out-of-plane displacement and

several values of initial imperfection. Figure 4.11 shows the load-displacement diagram with the

displacement taken as the shortening of the plate, which is the displacement in y direction of the

nodes at y = a. In both graphs the elastic behavior has been added in gray to show the influence of

plasticity.


59

Figure 4.10: Elastic-plastic buckling behavior (in gray: elastic)

Remarkable is the fact that right after the maximum is reached the load falls back rather quickly.

Whether in the case of a built up cross-section the capacities of slender and stockier plates can just

be added certainly needs to be investigated.

Figure 4.11: Elastic-plastic load-displacement diagram

0

1000

2000

3000

4000

5000

6000

7000

8000

0 5 10 15 20 25 30

F y,t

ota

l[k

N]

uz [mm]

1: 0.15 mm

2: 2.0 mm

3: 5.0 mm

4: 10.0 mm

Euler

Winter

0

1000

2000

3000

4000

5000

6000

7000

8000

0 1 2 3 4 5 6 7 8

F y,t

ota

l[k

N]

uy [mm]

1: 0.15 mm

2: 2.0 mm

3: 5.0 mm

4: 10.0 mm

Winter


60

Figure 4.12 shows the load-displacement graphs for the five discussed methods. The assumptions

made in each method are clearly reflected in the graph.

Figure 4.12: Load-displacement diagrams of the five models

Concluding it can be said that the ultimate load is estimated very well by the method of Winter, but

there is a large influence of the initial imperfection. Using the imperfection given in NEN-EN 1993-1-5

the finite element calculation in Ansys provided 1.2 % more capacity than the method by Winter.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 1 2 3 4 5 6 7 8 9 10

F y,t

ota

l[k

N]

uy [mm]

Model 1

Model 2

Model 3

Model 4

Model 5

Euler

Winter


61

4.3.2 Very slender plate in compression

The plate analyzed had a critical load higher than the ultimate load. The exact same Ansys

calculations as before have been made again, only this time with more slender plate. Only the model

including geometrical and physical imperfections has been used, since these calculate the ultimate

capacity. The dimensions and predicted capacities now are:

Dimensions: a*b*t = 2000*1000*10

Capacities: FEuler = 759 kN

FWinter = 1474 kN

The Ansys analysis gave an ultimate capacity of 1559 kN (with e0 = 5 mm), which is 5.8 % more than

according Winter. Figures 4.13 and 4.14 reflect the buckling behavior of this slender plate. The post-

critical strength is visible since after the critical value the load still increases, even if it is with less

stiffness. Striking is that in the very slender region the ultimate load is no longer depending on the

initial imperfection, since the load is carried mostly by membrane stresses.

Figure 4.13: Load-displacement diagram of slender plate

Figure 4.14: Out-of-plane behavior of slender plate

0

200

400

600

800

1000

1200

1400

1600

1800

0 1 2 3 4 5 6

F y,t

ota

l[k

N]

uy [mm]

1: 0.05 mm

2: 2.0 mm

3: 5.0 mm

4: 10.0 mm

Winter

Euler

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25 30 35 40

F y,t

ota

l[k

N]

uz [mm]

1: 0.05 mm

2: 2.0 mm

3: 5.0 mm

4: 10.0 mm

Winter

Euler


62

The influence of the slenderness on the ultimate capacity can be found in figure 4.15, which shows

the relative buckling load as a function of the slenderness. Here it can be seen that the method by

Winter has the best estimation of the buckling load of a imperfect plate. Since a finite element

calculation needs dimensions in order to calculate the buckling load, the vertical axis is made

dimensionless by dividing by the buckling load according von Kármán. This is the theoretical ultimate

load of a perfect (no imperfections) plate, given by: Fbuckling = (1/rel) As fy.

Figure 4.15: Buckling load of compressed plate

From figure 4.15 it can be concluded that for every plate slenderness the finite element calculation

gives more capacity than NEN-EN 1993-1-5. The largest difference is found in the more slender

plates. In the range of the slenderness mostly applied in practice, rel <2.0, the difference is smaller.

4.3.3 Single plate in bending

Unstiffened plates in bending have been analyzed the same way as the compressed plates in section

4.3.1 and 4.3.2. The results support the same general concept, in a relative stocky plate the ultimate

capacity is depending on the initial imperfections, where in more slender plates it does not. Also here

the imperfection suggested by NEN-EN 1993-1-5 annex C gives results close to the capacity

calculated by the method of Winter.

The analysis was harder to complete, since a lot of deformation was focused in a single point. This

was because very soon after buckling, crippling started to occur and that caused large strains and

displacements. Crippling was found in the upper left area of the plate, or in figure 4.16 around x = 0

and y = b. This crippling effect is also the cause why the load drops very fast after reaching the

ultimate load. Even though a displacement controlled analysis was made, due to the crippling

convergence was much harder to achieve that in the previous sections. This resulted in finer meshes,

smaller step sizes and larger computation times than the previous analyses. Despite that, the analysis

using e0 = 0.5 mm never converged after reaching the ultimate load.

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

F bu

cklin

g/

F vo

n K

arm

an[-

]

rel [-]

Critical load (Euler)Ultimate load (Winter)NEN 6771AnsysElastic load


63

Figure 4.16: Unstiffened plate in bending

Figure 4.17: Load-displacement diagram

Figure 4.18: Out-of-plane behavior in bending

Using the initial imperfection given in NEN-EN 1993-1-5 the Ansys calculation gave an ultimate

capacity of 3005 kNm, while the method by Winter predicts a capacity of 2866 kNm. The finite

element calculation gives 4.8 % extra capacity compared to the method by Winter.

0

500

1000

1500

2000

2500

3000

3500

0 1/1000 2/1000 3/1000

Mto

tal [

kNm

]

[mm-1]

1: 0.5 mm2: 3.0 mm3: 5.0 mm4: 10.0 mmWinterEuler

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40

Mto

tal[k

Nm

]

uz [mm]

1: 0.5 mm2: 3.0 mm3: 5.0 mm4: 10.0 mmWinterEuler

b

a

y

x z

z Dimensions:

a = 4000 mm

b = 2000 mm

t = 15 mm


all edges: dz = 0

x = 0: dx = 0

x = a: rotz = 6/1000

x = y = 0: dy = 0


64

The analysis of the plate in bending is also performed with a more slender plate (t = 10 mm). The

other dimensions are kept the same as in figure 4.16. As with the plates in pure compression the

dependence on the initial deformation of the ultimate load is reduced severely. The capacity

according Winter is 1427 kNm, with Ansys calculating 5.0 % more, or 1498 kNm.

4.3.4 Conclusion

Four different single unstiffened plates have been analyzed using Ansys, two different slendernesses

both in compression and bending. In each situation the capacity calculated by Ansys was higher than

the capacity by NEN-EN 1993-1-5, ranging from 1.2 % until almost 6 %. Also figure 4.15 shows how

the Eurocode slightly underestimates the capacity for any given slenderness. Assuming the finite

element calculation is more advanced, the simple methods in the Eurocode are on the safe side. On

the other hand there is a dependence of the ultimate capacity on the initial imperfection, which can

make a difference of a couple percent of the total load.


65

4.4 Capacity of built up sections

In this subsection the capacity of the box girder as calculated in chapter 3.1 is verified. The

dimensions of the cross-section and previously calculated capacities are given again in figure 4.19.

Two questions have to be answered here, whether the total capacity of the example cross-section

was calculated correctly in the design codes, and if a very slender plate can remain at capacity until a

more stocky plate reached buckling.

Figure 4.19: Dimensions and capacities of box girder

In figure 4.20 the load-displacement diagram as calculated by Ansys is shown, including the elastic

path to show the critical load and the second order path to show the influence of plasticity. The

ultimate capacity is 13435 kN, which is 0,1 % more than the Eurocode capacity of 13416.

Figure 4.20: Dimensions and capacities of box girder

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0 1 2 3 4 5

F y,t

ota

l[k

N]

uy [mm]

Ansys

Critical load (Euler)

NEN: reduced stress method

EC3: reduced stress method

EC3: effective cross-section method

hw = 1000

tw = 10

tf = 20

bf = 1000

All dimensions in: [mm]

ht = 1015

bt = 1020

Critical (Euler):

NEN 6771:

reduced stress method:

NEN-EN 1993-1-5:

effective cross-section method:

reduced stress method:

4560 kN

6229 kN

13416 kN

9046 kN


66

The influence of the web slenderness on the capacities calculated with the different methods was

also determined. This is a very clear way of comparing the methods and since Ansys works with input

given in a script it is quite easy to recreate the graph. In figure 4.21 the Ansys calculations are given,

which confirms that capacity of built up members is calculated most accurately by the Eurocode

effective cross-section method.

Figure 4.21: Ansys verification in influence web slenderness

4.5 Deformation capacity

In chapter 4.3 two plates have been analyzed using Ansys, both were 1000 mm wide, but the

thickness was either 10 or 20 mm. Their respective ultimate capacities were 1559 and 5297 kN.

These plates have been combined into a box girder in chapter 4.4, were Ansys calculated an ultimate

capacity of 13435 kN. Since the expected value was 2∙(1559+5297)=13712, the deformation capacity

of the slender plate seemed to be sufficient.

To investigate whether for each slenderness ratio this would be true, a large number of plate

dimensions have been put in the Ansys log file. In figure 4.22 the results are compared by plotting

the normalized load (Fy,total/Fmax,elastic) against the principle strain (l/l). Here the fundamental

principle behind any effective cross-section method becomes visible. Due to the influence of the

initial imperfections, the stiffness of a plate decreases, ensuring every plate will reach the ultimate

load around the same axial strain. In a built-up cross-section all plates undergo the same

deformation, so each single plate can reach its plate buckling capacity.

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

0 5 10 15 20 25 30 35 40

F bu

clkl

ing

[kN

]

tw [mm]



EC: Web breathing

NEN 6771

Elastic capacity

Ansys calculation


67

t [mm]

a [mm]

b [mm]

1 40 2000 1000

2 40 5000 2000

3 40 4000 2000

4 20 2000 1000

5 30 3500 2000

6 15 2000 1000 7 30 4000 2000

8 10 2000 1000

Figure 4.22: Loss of stiffness in slender plates

Concluding it can be said that the deformation capacity in a slender plate is not the classical plastic

deformation, which is normally regarded in a steel structure. Plates in buckling lose their stiffness as

soon as buckling sets in, so the ultimate load is always reached around the same amount of

deformation.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1F y

,to

tal /

(A

s*f y

)

l/l

12345678


68

Chapter 5: Requirements for plate edges

A steel member can be made up from several plate elements, which together form the cross-section.

In the calculation of the plate buckling capacity the various subpanels can be regarded separately.

The line along which the subpanel was originally connected to another plate part is a ‘plate edge’.

Figure 5.1: Stiffeners and edges in a cross-section

When calculating the plate buckling capacity of a stiffened cross-section, there are many

requirements given in various available design codes concerning longitudinal and transverse

stiffeners. Since a buckled plate develops membrane stresses when it is loaded in-plane, a load out-

of-plane exists on the stiffeners. These stiffeners must give adequate support to the plate in the out-

of-plane direction while also having to withstand the in-plane compressive loading itself. All plate

edges have to remain straight for the membrane stresses to fully develop.

In Eurocode 3 this problem is solved by reducing the area of a plate part which is supported by a

longitudinal stiffener. The amount of reduction depends on the geometry of the stiffener. Due to this

reduction the design load of the structure decreases. This decreased design load ensures the stresses

in the actual structure do not exceed the value at which the stiffener loses its ability to support the

plate. In NEN 6771 another approach is chosen. A stiffener is loaded by the plate buckling stress of

the adjacent subpanel, assuming simply supported conditions when determining the plate buckling

stress. The stiffener is then checked regarding it as a column loaded in bending and compression. If it

does not satisfy the requirements the size of the stiffeners or the plate thickness itself needs to be

increased. On top of these two methods for checking the out-of-plane properties of a stiffener, there

are more requirements given. These regard for instance the torsional buckling capacity and

maximum permitted initial deformation.

For the edges which are not formed by stiffeners on the other hand, there has not been given any

requirement. A plate edge is just a simply supported line which stays perfectly straight during

loading. When considering a box girder loaded in compression this naturally feels to be a good

assumption. The webs and flanges are perpendicular to each other and can therefore give good

support in de out-of-plane direction of the other plate element.

Plate edges

Web: three subpanels

Longitudinal stiffeners

Steel cross-section


69

When considering an I-shaped member this assumption might seem less logical. Now the flanges

have to support the web out of its plane when it buckles. If these flanges would be of small

dimensions it is easy to see they cannot give enough support to the web, even though according the

plate buckling verification methods it is still okay. If the web would continue past the flanges, they

suddenly become longitudinal stiffeners and would not meet the requirements, or lead to a very high

reduction of the area of the web plate. This leads to the question why flanges in compression don’t

have to comply with the requirements for stiffeners, and if this is correct or not.

5.1 Hypothesis

To give the study some guidance an assumption for the outcome is needed:

When calculating the plate buckling capacity of a cross-section built up from steel plates by regarding

the different subpanels separately, the plate edges (the location where the other subpanels are

connected) can indeed be assumed as simply supported. This assumption can be made because the

out-of-plane support capacity of the edges is determined by the global column buckling of the

complete member.

In practice this would mean that a local plate buckling capacity of the cross-section is determined

assuming 100% effective edges. In a situation where these edges can’t be regarded fully effective,

since they are below the critical stiffness, an extra reduction in capacity is given by the global

buckling of the member.

In the rest of the chapter this hypothesis is tested using finite element modeling in Ansys.

5.2 Outline

In this part of the chapter an outline is given for the study of the edges. The study is divided in three

different parts which are described here.

In the first part some sensitivity analyses are made. Apart from element and step size the precise

shape of the imperfection is also studied. This is an important parameter in the calculations and

therefore it is good to know how sensitive the calculations are to this shape. The main reason for this

is that when modeling in Ansys it is not possible to add a certain imperfection in a single step. The

buckling mode shape of the web is not perfectly sinusoidal when flanges are attached. Therefore we

need to know how accurate this imperfection must be modeled to get satisfying results, but in

general more accuracy means more time to build the model. This is because the individual parts have

to be uncoupled while determining the imperfect shape, and then be coupled again for the analysis

of the plate buckling capacity.

In the second part a single unstiffened plate is modeled. The simple support which normally

constrains the long edge is now replaced by a distributed spring. By varying the spring stiffness and

monitoring the change in collapse load, the influence of the supporting edge is studied. See also

figure 5.2 on the next page.


70

Figure 5.2: Boundary conditions and loads

In the third part an I-shaped cross-section is studied. This is mostly the same model as in figure 5.2,

only the out-of-plane support of the plate is now given by flanges instead of the distributed elastic

foundation. The flanges are chosen in class 3 or lower, mainly to keep modeling of the structure

relative simple by excluding an extra imperfection.

5.3 Sensitivity analysis

5.3.1 Element and load step size

The element size and load step size play an important role in the finite element calculation. When

using a displacement controlled, geometrical and physical nonlinear calculation the collapse load will

increase while decreasing the load step size. As a result of the smaller steps the top of the load-

displacement curve is not ‘stepped over’. With a larger step size the values itself may be as accurate

as with a small step size, because of extra iterations in each load step, but the iterations do not show

in the results table in Ansys so the collapse load may be missed.

Increasing the number of elements will decrease the collapse load, which makes for a more realistic

solution. Due to the extra number of elements the computation time on the other hand increases

dramatically. In my example they went from a couple of minutes to well over an hour. This

completely limits the ability to quickly study different situations, while the decrease in collapse load

was only a couple percent. The goal here is to find an element size which gives reasonably accurate

results, combined with manageable computation times. There is no golden rule to reach this and it

depends mainly on experience and some trial and error.

An example is given in figure 5.3, where the same plate structure is calculated three times. The first

calculation is just to have a load-displacement graph to verify the effect of the changes. The second

calculation adds a five times smaller load step size to the first, which makes the load-displacement

curve much smoother, since intermediate iterations are more regular and stored in the final results.

The third analysis is made by using half the element size of the second calculation.


71

Figure 5.3: Influence of element and load step size

In most calculations the load step size needs to be quite small to ensure convergence. To see

whether it is small enough for an accurate result of the collapse load, you to reference it to the

previous and the next value in the load-displacement curve. The difference between these two is a

measure for the accuracy of the collapse load itself.

The element size is chosen such that computation time is limited to around five minutes. In all

calculations a constant element size is maintained, so any inaccuracy will constantly be the same and

not influence the results.

5.3.2 Shape of imperfection

The main goal in this subsection is to determine how complicated the model has to be to get

accurate results. In Ansys adding imperfection cannot be done with the click of a button, but requires

programming with more than a single step.

In the regular plate model the buckling mode shapes can be calculated using an eigenvalue analysis.

The lowest mode shape can then be scaled to match the required amount of imperfection and the

original geometry updated according to this buckled shape. Using the lowest mode shape makes sure

the lowest collapse load is found.

In a cross-section built up from more than one single steel plate this method is not applicable. Only

the most slender plate will deform in the eigenvalue analysis and the rest of the plates will not get an

initial imperfection. This problem can be get round by fixing all but one plate in the out-of-plane

direction, then calculation its imperfect shape and deleting the extra constraints. This procedure

needs to be repeated for each plate part.

The problem with this method is that the ‘free’ plate, from which the buckled shape is calculated, has

fixed edges. This leads to an imperfect shape which may not give the lowest collapse load. Another

problem is the number of sinus waves in the imperfection. When regarding a plate with an aspect

ratio of two (plate length is two times width), the first mode shape should have two sinus shaped

imperfections. When the edges have been fixed the first mode shape has three sinus waves.

The influence of this shape is studied by using the regular plate model, with the rotation around the y

axis as optional constraint. Six calculations are made to compare the different situations.

Load

Displacement

Top

of

load

cu

rve

Displacement

1: Starting point

2: 1/5 load step size

3: 1/2 element size

+0.2%

-1.6%


72

Figure 5.4: Constraints in plate model

Four different shapes of the imperfection are tested, with and without rotational constraint and for

both situations the shape with two and three sinusoidal buckles.

Figure 5.5: Mode shape 1 and 2 (roty = free)

Figure 5.6: Mode shape 1 and 3 (roty = 0)

The results can be found in table 5.1, for each situation is given how the roty constraint of the long

edge is varied. The corresponding shapes of the initial imperfection can be found in figures 5.5 and

5.6. The dimensions of the plate are a·b·t = 2000·1000·5 mm. Which makes the plate buckling

capacity according Winter is 389.7 kN.


73

‘roty’ in making mode shape

Mode shape no. (no. of waves)

‘roty’ in nonlinear analysis

Collapse load [kN]

Relative collapse load [-]

1 free 1 (2) free 445.3 1.000

2 free 2 (3) free 438.8 0.985

3 fixed 3 (2) free 449.0 1.008

4 fixed 1 (3) free 447.4 1.005

5 fixed 3 (2) fixed 599.3 1.346

6 fixed 1 (3) fixed 601.8 1.351

Table 5.1: Collapse loads for different shapes of imperfection

Since the plate is very slender it is expected that the collapse loads according Ansys are higher than

the plate buckling capacity according Winter (see also chapter 4.3.1). The interesting conclusion from

this study is that the actual shape of the imperfection does not have a very large influence on the

collapse load. Another remarkable result is that the lowest collapse load is not reached with the

lowest elastic buckling mode, although again the difference is very small.

When the rotational degree of freedom of the long edge is still fixed during the non linear analysis, a

higher collapse load is reached. This is expected because in transverse direction there the plate is

clamped, instead of simply supported. In practice this capacity can’t be used, because the cross-

section will alternate the direction of the buckles so no fixed edges can be found.

Figure 5.7: Plate buckled cross-sections

5.4 Elastically supported plate

To get a clear picture what the influence of the support condition of the long edge is, a distributed

spring is added to the plate model. The boundary conditions can be found in figure 5.8. The

dimensions of the plate are a · b · t = 4000 · 1000 · 10. The spring stiffness (kz) is variable, since this is

the parameter we are interested in. The initial imperfections are chosen according Eurocode 1993-1-

5 Annex C, which states that the local imperfection must be b/200 = 5 mm and the global bow

imperfection of the edges must be a/200 = 20 mm. The a value is chosen to be four, so no

unexpected stiffness introduced at the support of the short edges of the plate has an influence in the

plate buckling capacity.


74

Figure 5.8: Boundary conditions

For the two extreme values for kz (kz = 0 and kz = ∞) the load bearing capacity can be calculated using

the methods in NEN-EN 1993-1-5. The finite element calculations can later be confirmed using these

values.

When kz = 0 the plate can be regarded as a column which can be checked according NEN-EN 1991-1-1

article 5.3.1:

With k = ∞ the plate buckling capacity can be calculated according NEN-EN 1993-1-5 article 4.4:

lk = 4000 mm

As = 1000 · 10 mm2

I = 12-1

· 1000 · 103 mm4


75

The log-file of the calculation in Ansys can be found in annex C.2. Results are listed in the following

table and figure.

k [N/mm2] 0 0.002 0.02 0.2 0.6 2 6 20 60 200 2000 20000

Fmax [kN] 11.6 17.4 63.9 180 322 548 900 1305 1450 1506 1520 1522

Table 5.2: Plate capacity with various kz

Figure 5.9: Plate capacity versus spring stiffness diagram

From the figure it can clearly be seen that in the beginning an increase in spring stiffness leads to an

increase in collapse load. This is to be expected since the plate element is better supported and more

membrane stresses can develop. Failure is governed by a global column buckling mode. After the

critical value of the spring stiffness is exceeded, no more increase in collapse load occurs. Here the

failure mode is plate buckling, which is independent of the spring stiffness. The capacity for k=0

according to the Eurocode is confirmed by the Ansys calculation, although it is too small to be

properly plotted in the graph.

It is clearly noticeable that a small area of interaction between the two failure modes occurs. If there

would not have been any interaction the transition between k=10 and k=100 would have had a sharp

angle.

Figure 6.10: Region with interaction

0

200

400

600

800

1000

1200

1400

1600

0,0001 0,01 1 100 10000

F d,b

uck

ling

[kN

]

kz [N/mm2]

Nmax Ansys

Nmax Eurocode

0

200

400

600

800

1000

1200

1400

1600

0,1 1 10 100 1000

F d,b

uck

ling

[kN

]

k [N/mm2]


76

5.5 I-shaped member

In this part of the chapter the elastic support is replaced by steel flanges. This creates an I-shaped

member, from which the length will be varied. When the length is small the plate buckling failure

mode will be governing, while at a bigger member length the global buckling capacity will be smaller.

In principle the same procedure as with the spring supported plate can be followed, and similar

results can be expected. The geometry and boundary conditions of the member can be found in

figure 5.11.

Figure 5.11: Boundary conditions and geometry

Two important choices have been made to keep the modeling in Ansys relatively simple. Since the

model is a 3D plate model, it is easier to give the ends of the members a fixed support. The nodes at

y=0 are set to uy = ux = uz = 0 while the nodes at y = l are set to ux = uz = 0 and uy as the prescribed

displacement. The buckling length of the member is now of course half of the system length. The

other choice is to use flanges of class three or lower. This ensures only two instead of three

imperfections must be added. The two remaining imperfections are a global imperfection in the

shape of the buckling mode of the member, as seen in figure 5.11, and a local plate buckling

imperfection of the web as seen in figure 5.5. Since the member is not supported anywhere except at

the ends, global buckling will take place in the direction of the weak axis, which gives displacements

in the z-direction. If a global imperfection is given in the positive z-direction, the local imperfection of

the web at y = ½ l must be in the same positive z-direction. This gives the least favorable situation for

the member. If opposite directions are used collapse loads up to 20% higher can be found, but of

course these are not realistic for the structure. Imperfections are added according to Eurocode 1993-

1-5 Annex C, which states that global bow imperfection must be l / 200 and local plate imperfection

hw / 200.

If the hypothesis is correct a member with a small length will have stable plate edges and therefore

reach the plate buckling capacity according Winter. A member with a bigger length will have unstable

edges and the collapse load will now be governed by global buckling. These two failure modes can be

determined using NEN-EN 1993-1-1 and 1993-1-5.

z

y

x

z

web: hw·tw = 2000·15 [mm2]

l

flanges: bf·tf = 550·24 [mm2]


At y=0: ux = 0

uy = 0

uz = 0

At y=l: ux = 0

uz = 0

uy


77

Local buckling capacity according NEN-EN 1993-1-5:

Global buckling capacity according NEN-EN 1993-1-1:


78

The hypothesis would suggest that when the relative column slenderness ( ) is smaller than 0.2 the

local plate buckling mode will be governing. If it is larger the plate edges become unstable and global

buckling is the leading failure mode. With the chosen cross-section this transition point will occur at a

member length of around 4000 mm, or a buckling length of 2000 mm.

The Ansys calculation log file can be found in Annex C.3. The resulting collapse loads for different

lengths are summarized in table 5.3. Figure 5.12 and 5.13 shows how the shape of the buckles is

affected by the global displacement. Given here is the displacement of the web in the out-of-plane

direction, at the moment the ultimate load is reached. The buckles are no longer sinusoidal and

equal in size.

Figure 5.12: Out-of-plane displacement of web for l = 8000 mm

Figure 5.13: Out-of-plane displacement of web for l = 6000 mm

l

[mm]

Fy;max [kN]

l

[mm]

Fy;max [kN]

l

[mm]

Fy;max [kN]

l

[mm]

Fy;max [kN]

1500 12714 4800 12147 6900 11754 12000 10212

2000 12626 4900 12115 7000 12012 15000 8819

2500 12542 4900 12313 7500 11865 20000 6748

2800 12471 5000 12289 8000 11711 25000 5050

3000 12683 5500 12163 8900 11400 30000 3898

3500 12550 6000 12028 9000 11641 40000 2530

4000 12400 6500 11879 10000 11279 50000 1639

Table 5.3: Collapse loads according Ansys calculations

These results are plotted together with the solution according to the Eurocode in figure 5.14. At first

glance the two lines seem to match, which confirms the hypothesis. At the location of where global

buckling starts to play role however (rel = 0.2), some interaction seems to take place in the results

according the Ansys calculation.


79

Figure 5.14: Collapse load while increasing l

Figure 5.15: Detail of region with interaction

The jumps in the Ansys results can be explained by the number of buckles present in the

imperfection of the web plate. The buckling factor is always taken as k= 4.0 in the Eurocode, while

in reality it varies with the aspect ratio a and the number of buckles. Even then the collapse load

should not have jumps but only changes in direction. The jumps are created because the aspect ratio

where a mode shape (shape of imperfection) succeeds another one is not necessarily the aspect ratio

where the two succeeding mode shapes have the same collapse load. This effect is also seen in table

5.1. In figure 5.16 the influence of the number of buckles is shown in even more detail.

0 10000 20000 30000 40000 50000

0

2000

4000

6000

8000

10000

12000

14000

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4

l [mm]C

olla

pse

load

[kN

]

rel [-]

Eurocode

Ansys

0 2000 4000 6000 8000 10000

11000

11250

11500

11750

12000

12250

12500

12750

13000

0,0 0,1 0,2 0,3 0,4 0,5

l [mm]

Co

llap

se lo

ad [k

N]

rel [-]

Eurocode

Ansys


80

Figure 5.16: Influence of number of buckles (m) on k and Ansys results

Note that in this study no member lengths smaller than 1500 mm have been studied. In the region of

very small member lengths the collapse load would have become higher than the collapse load

according Winter. This can be seen in the left graph of figure 5.16, where for a < 1 the value of k

increases. In Eurocode 1993-1-5 this effect is accounted for by article 4.5.4, about the column type

buckling which can occur in plates with small a’s.

Except for a small region around rel = 0.4 the collapse loads calculated in Ansys are consistently

smaller than what the Eurocode predicts. The difference expressed in per cent can be found in figure

5.17.

-4

-2

0

2

4

6

8

10

De

viat

ion

[%

]

l [mm]

Figure 5.17: Deviation Eurocode in relation to Ansys

m=1

m=2

m=3

m=4

m=5


81

5.6 Conclusion

Even though some deviation exists, the general idea in the hypothesis is correct. Several reasons can

be found why the Ansys results are consistently below the Eurocode values:

- Inaccurate a- and 0-values.

Here ais not the aspect ratio, but the parameter in the buckling curve in the Eurocode. Only a couple

of values can be chosen, according the buckling curves a0, a, b, c, and d. Intermediate values would

give a better fitting. The following values are permitted:

a0 a b c d

a 0.13 0.21 0.34 0.49 0.76

0 0.2 0.2 0.2 0.2 0.2

Table 5.4: Buckling parameters

If a = 0.69 and 0 = 0.29 would have been chosen, the difference for larger length members would be

significantly smaller. A big disadvantage is that now an extra curve would be needed to account for

the interaction, making the verification code more complex. Most likely a choice for a relatively

simple formula has been made in the Eurocode, while making sure the deviation stays within certain

reliability limits.

Figure 5.16: Results with a = 0.69 and 0 = 0.29

- Too much imperfection according Annex C.

The magnitude of the imperfection which has to be added according Annex C of Eurocode 1993-1-5 is

hw / 200. This imperfection includes geometrical imperfection as well as residual stress. It appears

this value is rounded off to a safe margin. Maybe a detailed study of the real imperfections can lead

to a more favorable value and to higher collapse loads.

- Rotation of flanges.

The buckling of the web plate causes the flanges to rotate. Since the ends of the flanges are fixed, a

twisting deformation occurs and strains are introduced in the flanges. Due to these strains collapse is

reached sooner than in perfectly flat flanges.

l [mm]

Co

llap

se lo

ad [k

N]

rel [-]

Eurocode

Ansys

-3-2-101234567

0 10000 20000 30000 40000 50000

Dev

iati

on

[%]

l [mm]


82

Chapter 6: Conclusion and recommendation

6.1 Conclusion

This thesis started with the question why NEN-EN 1993-1-5 can give much higher plate buckling

capacities then NEN 6771. By analyzing the methods used in both codes, and the principles behind

these methods the differences in capacity can be explained clearly. The NEN 6771 uses the reduced

stress method, with a reduction factor based upon the critical Euler plate buckling stress. For slender

plates this critical stress is always lower than the stress at the ultimate failure load. The NEN-EN

1993-1-5 on the other hand uses the effective cross-section method, first introduced by von Kármán,

and the reduction factor according Winter. Since this reduction factor is calibrated using many tests

on steel specimens, it predicts the ultimate load very well. When calculating with the reduced stress

method in NEN-EN 1993-1-5 the same reduction formula is used. Due to this the calculated capacity

is higher than in a NEN 6771 reduced stress method calculation. Nevertheless, NEN 6771 also

contained the effective cross-section method with Winters reduction formula, so strictly speaking

there is no extra capacity gained in using NEN-EN 1993-1-5. However in a variably loaded structure

web-breathing might be governing over the effective cross-section method and a stress based upon

the critical stress should be used.

The ultimate capacities calculated by the effective cross-section method were verified using finite

element calculations in Ansys. The finite element models made in Ansys had both geometrical and

physical nonlinearities, in order to find the ultimate load of thin steel plates. As well for single plates

as for built up sections the capacities of finite element and design code calculations match very

precise. The deformation capacity needed in built-up sections was found to be a loss of stiffness due

to second order effects. This ensures that when steel plates of different slenderness are combined in

a single section, each plate will reach its own buckling capacity at around the same axial strain. This is

the fundamental principle behind the effective cross-section method.

Plate edges do not have any requirements given for either in NEN 6771 or in NEN-EN 1993-1-5. Using

finite element calculations the hypothesis that the stability of the edges is implied in the global

stability of the member was found to be correct. If global stability is guaranteed, the edges of a plate

will also remain stable. However in between the stable edges the plate can still experience local plate

buckling. If the edges cannot support the adjacent plates, the local plate buckling capacity will not be

reached and global buckling is governing for the capacity of the member.

6.2 Recommendation

Finally a recommendation is given as to what method to choose when calculating a plate buckling

capacity using NEN-EN 1993-1-5. The reduced stress method given in chapter 10 of NEN-EN 1993-1-5

is a quick procedure to determine if a structure is below its plate buckling capacity, especially when

appropriate software can be used. Also for a quick assessment of the plate buckling capacity of a

given cross-section, this is very good method. The capacity will however always be lower than the


83

capacity calculated with the effective cross-section method of chapter 4 of NEN-EN 1993-1-5, but if

the plate slendernesses are chosen such that the reduction factors are of similar value, the capacities

will also be close together. If on the other hand such an option for the slenderness is not available,

the effective cross-section method will give a higher capacity. How much percent of extra capacity

depends on the cross-sectional properties, but several dozens is quite possible. Plate buckling in a

member will be accompanied with a loss of stiffness. If this loss of stiffness is expected to have an

influence in the force distribution in the structure, the effective cross-section method must be used

to take it into account.

6.3 Future research

This thesis dealt only with the failure load of slender steel plate cross-sections. At some points the

fatigue sensitivity of variably loaded structures was mentioned. Keep in mind that web-breathing is

caused by an out-of-plane deformation, while fatigue verifications check in-plane stress cycles. A

possibility for future research is how the fatigue verifications are dealt with in the design codes when

a structure sensitive for plate buckling is regarded. The effective area is in this case smaller than the

gross area, which will increase local stress peaks and therefore decrease fatigue life. Since many

effects play a role at the same time it would be interesting to have a better understanding of the

fatigue life of plate buckling sensitive structures. In the Netherlands several bridges have suffered

from fatigue damage, so it is also a very current subject.

The imperfection of the steel plates has a large influence on the final carrying capacity. In all methods

treated in this thesis the imperfection is implemented in some way. NEN 6771 has a linear cut-off to

the critical Euler stress, the reduction factor in Winters method is calibrated to a large number of

tests on cold-formed sections and in a finite element calculation the imperfection is added directly in

the geometry. A first thing which would be interesting to analyze is how the calibration by Winter to

von Kármáns formula relates to the b/200 value used in NEN-EN 1993-1-5 and if this can be an

explanation for any differences in capacity that is found. A second, more applied research direction

can be to the plates fabricated in the steel shops. For instance whether the imperfections of the

actual produced plates match with the fabrication tolerances given in the codes, or if they can be

much higher or lower.


84

Chapter 7: References

[1] Abspoel, R., Bijlaard, F.S.K., Stability of Steel Plate Structures, Delft University of Technology,

Lecture notes CT4121, 2005.

[2] Vrouwenvelder, A.C.W.M., Structural stability, Delft University of Technology, Lecture notes

CT5144, 2003.

[3] Gardner, L., Nethercot, D.A., Designers’Guide to EN 1993-1-1 – Eurocode 3: Design of steel

structures, general rules and rulesfor buildings, London, 2005.

[4] Hendy, C.R., Murphy, C.J., Designers’Guide to EN1993-2 – Eurocode 3: Design of steel

structures, Part 2: Steel Bridges, London, 2007.

[5] Beg, D., et al, ECCS Eurocode Design Manual: Design of Plated Structures. Eurocode 3: Design

of Steel Structures. Part-1-5: Design of Plated Structures, Ljubljana, 2010.

[6] Johansson, B., et al, Commentary and Worked Examples to EN 1993-1-5 “Plated Structural

Elements”, Luxembourg, 2007.

[7] Timoshenko, S.P., Gere, J.M., Theory of Elastic Stability, Stanford, 1963.

[8] Clarin, M., Plate Buckling Resistance – Patch Loading of Longitudinally Stiffened Webs and

Local Buckling, Luleå University of Technology, Doctoral Thesis 2007:31, 2007.

[9] Bloom, F., Douglas, C., Handbook of Thin Plate Buckling and Postbuckling, New York, 2001.

[10] Falzon, B.G., Aliabadi, M.H., Buckling and Postbuckling Structures, London, 2008.

[11] Yu, W.W., Cold-Formed Steel Design – Third Edition, New York, 2000.

[12] Ghersie, A., et al., Metallic Cold-Formed Thin-Walled Members, London, 2002.

[13] Paik, J.K., Thayamballi, A.K., Ultimate Limit State Design of Steel-Plated Structures, London,

2002.

[14] Subramanian, N., Design of Steel Structures, New Delhi, 2008.

[15] Daley, C., Hermanski, G., Ship Frame Research Program – Investigation of finite element

analysis boundary conditions, Ocean Engineering Research Center, Report TR-2005-05, 2005.

[16] Sedlacek, G., et al., Leitfaden zum DIN Fachbericht 103 – Stahlbrücken, Aachen, 2003.


85

[n1] NEN 6700; Technische Grondslagen voor Bouwconstructies – TGB 1990 Algemene basiseisen,

Delft, 1991.

[n2] NEN 6770; Staalconstructies – Basiseisen en basisrekenregels voor overwegend statisch

belaste constructies – TGB 1990, Delft, 1997.

[n3] NEN 6771; Technische Grondslagen voor Bouwconstructies – TGB 1990 – Staalconstructies –

Stabiliteit, Delft, 2000.

[n4] NEN 6788; Het Ontwerpen van Stalen Bruggen – Basiseisen en Eenvoudige Rekenregels (VOSB

1995), Delft, 1995.

[n5] NEN-EN 1990 (nl); Eurocode – Grondslagen van het Constructief Ontwerp, Delft, 2002.

[n6] NEN-EN 1993-1-1 (nl); Eurocode 3: Ontwerp en Berekening van Staalconstructies – Deel 1-1:

Algemene regels en regels voor gebouwen, Delft, 2006.

[n7] NEN-EN 1993-1-5 (en); Eurocode 3: Ontwerp en Berekening van Staalconstructies – Deel 1-5:

Constructieve Plaatvelden, Delft, 2006.

[n8] NEN-EN 1993-2 (en); Eurocode 3: Ontwerp en Berekening van Staalconstructies – Deel 2:

Stalen Bruggen, Delft, 2007.

[n9] DIN-Fachbericht 103; Stahlbrücken; Ausgabe März 2003.

[n10] NEN-EN 1090-1 (en); Constructieve Delen van Staal en Aluminium – Deel 1: Eisen voor

Conformiteitsbeoordeling van dragen delen, Delft, 2009.

[n11] NEN-EN 1090-2 (en); Het vervaardigen van Staal- en Aluminiumconstructies – Deel 2:

Technische Eisen voor Staalconstructie (corrected), Delft, 2008.

[w1] Eurocodes in Nederland, by NEN.

www.eurocodes.nl (sept. 2010)

[w2] LUSAS engineering analysis software company, case studies.

http://lusas.co.uk/case/bridge/critical_buckling_analysis.html (sept. 2010)

[w3] Eurocodes Expert, by the Institition of Structural Engineers.

www.eurocodes.co.uk (sept. 2010)

[w4] ANSYS – Simulation Driven Product Development

www.ansys.com (okt. 2010)

http://www.eurocodes.nl/

http://lusas.co.uk/case/bridge/critical_buckling_analysis.html

http://www.eurocodes.co.uk/

http://www.ansys.com/


86

List of annexes A.1 Plate buckling capacity of a plate in bending A.2 Plate buckling capacity of a compressed plate B.1 Cross-sectional capacities calculated with codes B.2 Longitudinal stiffeners according codes C.1 Log-file Ansys of single plate C.2 Log-file Ansys of elastically supported plate C.3 Log-file Ansys of I shaped member D.1 Mathcad sheet for NEN-EN 1993-1-5


87

Annex A.1 Plate buckling capacity of a plate in bending

In this annex the plate buckling capacity of a single plate in bending is determined, as support of

chapter 3.3.5. The same plate is analyzed using NEN 6771, the reduced stress method in NEN-EN

1993-1-5 and the effective cross-section method in NEN-EN 1993-1-5.


88

Annex A.2 Plate buckling capacity of a compressed plate

In this annex the plate buckling capacity of a single plate in compression is determined, as support of

chapter 3.3.6. The same plate is analyzed using NEN 6771, the reduced stress method in NEN-EN

1993-1-5 and the effective cross-section method in NEN-EN 1993-1-5.


89

Annex B.1 Cross-sectional capacities calculated with codes

This annex shows how a manual calculation using NEN-EN 1993-1-5 and NEN 6771 would look, for

several different cross-sectional shapes. The given calculations on the following pages are for:

- I –shaped member, normal force capacity, according NEN 6771.

- I –shaped member, bending around strong axis, according NEN 6771.

- I –shaped member, normal force capacity, according NEN-EN 1993-1-5, effective cross-

section method.

- I –shaped member, bending around strong axis, according NEN-EN 1993-1-5, effective cross-

section method.

- I –shaped member, bending around weak axis, according NEN-EN 1993-1-5, effective cross-

section method.


92

Annex B.2 Longitudinal stiffeners according codes

This annex shows how a manual calculation using NEN-EN 1993-1-5 and NEN 6771 would look, for a

single plate with longitudinal stiffener. The given calculations are for:

- A given plate and longitudinal stiffener, according NEN 6771.

- A given plate and longitudinal stiffener, according NEN-EN 1993-1-5.

- The dimensions of the stiffener needed to get the full plate buckling capacity according

Winter.


94

Annex C.1 Log-file Ansys of single plate

This annex gives the used log file for a single plate loaded in compression. The comments added to

clarify the commands are in Dutch.

/BATCH,LIST

/filnam,Plaatmodel

/title,Plastisch GNL Plaatmodel

/units,mpa

/nerr,1,99999999

! Dit rekenblad is geldig voor de volgende constructie:

!

! ____ - rechthoekige stalen plaat

! | | - alle randen scharnierend opgelegd

! | | - zuiver druk op zijde 'b'

! | |

! | | a

! | |

! | |

! |____|

! b

t=10 ! plaatdikte (mm)

a=2000 ! lengte van de plaat (mm)

b=1000 ! breedte van de plaat (mm)

na=40 ! aantal elementen in a richting

nb=20 ! aantal elementen in b richting

e=5 ! initiële vervorming (mm)

umax=10 ! maximale verplaatsing (mm)

nmax=40 ! aantal load substeps

fy=355 ! vloeispanning (N/mm2)

/prep7 ! Preprocessor mode

et,1,SHELL181 ! SHELL181 element op id1

R,1,t ! Definieer dikte

mp,ex,1,2.1e5 ! E-modulus voor elastisch materiaal 1

mp,prxy,1,0.3 ! Poisson ratio voor mat. 1

tb,biso,1 ! Bilineair isotropisch hardening

tbdata,1,fy,21 ! met na vloeien E/10000

tbpl,biso,1 ! plot ter controle

/wait,1

k,,0,0,0 ! Keypoints definiëren

k,,b,0,0

k,,b,a,0

k,,0,a,0

kplot

a,1,2,3,4 ! Definieer oppervlakken

lsel,s,tan1,x,1 ! Deelt lijnen op tot elementgrootte

lsel,a,tan1,x,-1

lesize,all,,,nb

lsel,s,tan1,y,1

lsel,a,tan1,y,-1


95

lesize,all,,,na

amesh,all ! Auto mesh

nmiddenboven=node(0.5*b,a,0) ! node nummer midden boven, voor plotten van

! verplaatsing

save

finish

/solve ! In oplossingsmodus

antype,static ! Static analysis

pstress,on ! Prestress effects

nsel,s,loc,x,0 ! langsranden opgelegd als scharnier.

d,all,uz,0

nsel,r,loc,y,0

d,all,ux,0

nsel,s,loc,x,0

nsel,r,loc,y,a

d,all,ux,0

nsel,s,loc,x,b

d,all,uz,0

nsel,s,loc,y,0 ! Onderrand scharnier

d,all,uy,0

d,all,uz,0

nsel,s,loc,y,a ! Bovenrand scharnier

d,all,uz,0

nsel,s,loc,y,a ! Kleine verplaatsing bovenrand

d,all,uy,-0.1

solve

finish

/solve

antype,buckle ! Buckling analysis

bucopt,lanb,1 ! Eerste plooivorm

mxpand,1

solve

finish

/prep7

/inquire,myjobname,jobname

allsel

upgeom,e,1,1,'%myjobname(1)%',rst ! Update geometrie met eerste mode-shape

nsel,s,loc,z,e

*get,nmidmid,node,,num,min

save

finish

/solve


96

antype,static

nlgeom,on ! GNL rekenen staat aan

outres,all,all ! alle resultaten opslaan

nsel,s,loc,y,a ! verplaatsing bovenrand

d,all,uy,-umax

nsubst,nmax,nmax,nmax ! verdeel in nmax substeps

kbc,0 ! linear toenemende belasting

solve

save

finish

/post1 ! Post processing modus

allsel ! selecteer alle nodes

set,last ! Haal resultaten laatste stap op

/dscale,all,10 ! Schaal verplaatsingen = 10 op 1

shell,mid ! Selecteer middenvlak

plnsol,s,y,2 ! Plot sigma yy

finish

/wait,6

/post26 ! modus met ook tijd

numvar,200 ! 200 variabelen beschikbaar

allsel

nsol,200,nmiddenboven,u,y,Uyneg ! variabele 2 is de verplaatsing u

filldata,199,,,,-1,0 ! slaat -1 op als variabele

prod,2,200,199,,Uy ! product geeft mooie grafiek

nsel,s,loc,y,0 ! Sommatie voor f_tot als var.3

*get,nmin,node,0,nxth

*get,ntot,node,,count

rforce,3,nmin,f,y

*do,ndo,2,ntot

*get,nmin,node,nmin,nxth

rforce,196,nmin,f,y

add,3,3,196,,Fy_tot

*enddo

nsol,198,nmidmid,u,z,Uz_echt ! verplaatsing z in het midden

filldata,197,,,,e,0

add,4,198,197,,Uz

/axlab,x,Verplaatsing Uz [mm]

/axlab,y,Totale Belasting Ftot [N]

xvar,4

plvar,3

/wait,4

/axlab,x,Verplaatsing Uy [mm]

xvar,2

plvar,3

finish


97

Annex C.2 Log-file Ansys of elastically supported plate /BATCH,LIST

/filnam,Plaatmodel_f

iles

/title,Plastisch GNL

Plaatmodel

/units,mpa

/nerr,1,99999999

t=10

a=4000

b=1000

na=80

nb=20

e=5

e2=20

k=100

umax=12

nmax=60

fy=355

/prep7

et,1,SHELL181

R,1,t

et,2,COMBIN14

R,2,k

mp,ex,1,2.1e5

mp,prxy,1,0.3

tb,biso,1

tbdata,1,fy,21

tbpl,biso,1

/wait,1

k,,0,0,0

k,,b,0,0

k,,b,a,0

k,,0,a,0

kplot

a,1,2,3,4

lsel,s,tan1,x,1

lsel,a,tan1,x,-1

lesize,all,,,nb

lsel,s,tan1,y,1

lsel,a,tan1,y,-1

lesize,all,,,na

amesh,all

nsel,s,loc,x,0

nsel,a,loc,x,b

nsel,u,loc,y,0

nsel,u,loc,y,a

cm,lr,node

nmiddenboven=node(0.

5*b,a,0)

save

finish

/solve

antype,static

pstress,on

nsel,s,loc,x,0

d,all,uz,0

nsel,s,loc,x,b

d,all,uz,0

nsel,s,loc,y,0

d,all,uy,0

d,all,uz,0

d,all,ux,0

nsel,s,loc,y,a

d,all,uz,0

d,all,ux,0

nsel,s,loc,y,a

d,all,uy,-0.1

solve

finish

/solve

antype,buckle

bucopt,lanb,1

mxpand,1

solve

finish

/prep7

/inquire,myjobname,j

obname

allsel

upgeom,e,1,1,'%myjob

name(1)%',rst

nsel,s,loc,z,e

*get,nmidmid,node,,n

um,min

save

finish

/solve

antype,static

pstress,on

cmsel,s,lr

ddele,all,uz

f,all,fz,1

nmidrand=node(0,0.5*

a,0)

nsel,s,loc,y,a

d,all,uy,0

solve

finish

/prep7


obname

allsel

*get,urand,node,nmid

rand,u,z

upgeom,e2/urand,1,1,

'%myjobname(1)%',rst

cmsel,s,lr

f,all,fz,0

*get,lraantal,node,,

count

*get,lrlaag,node,,nu

m,min

ngen,2,(nb+1)*(na+1)

,all,,,0,0,-5000,1

*do,teller,1,lraanta

l

type,2

real,2

e,lrlaag,lrlaag+(nb+

1)*(na+1)

*get,lrlaag,node,lrl

aag,nxth

*enddo

nsel,s,node,,(nb+1)*

(na+1)+1,(nb+1)*(na+

1)*2

d,all,uz,0

d,all,ux,0

d,all,uy,0

allsel

save

finish

/solve

antype,static

nlgeom,on

outres,all,all

nsel,s,loc,y,a

d,all,uy,-umax

nsubst,nmax,nmax,nma

x

kbc,0

solve

save

finish

/post1

allsel

set,last

/dscale,all,10

shell,mid

plnsol,s,y,2

finish

/wait,6

/post26

numvar,200

allsel

nsol,200,nmiddenbove

n,u,y,Uyneg

filldata,199,,,,-1,0

prod,2,200,199,,Uy

nsel,s,loc,y,0

*get,nmin,node,0,nxt

h

*get,ntot,node,,coun

t

rforce,3,nmin,f,y

*do,ndo,2,ntot

*get,nmin,node,nmin,

nxth

rforce,196,nmin,f,y

add,3,3,196,,Fy_tot

*enddo

nsol,195,nmidrand,u,

z,Uz_echt

filldata,194,,,,e2,0

add,5,195,194,,Uz_ra

nd

nsol,198,nmidmid,u,z

,Uz_echt

filldata,197,,,,e,0

add,4,198,197,,Uz_pl

aat

/axlab,x,Verplaatsin

g Uz [mm]

/axlab,y,Totale

Belasting Ftot [N]

xvar,4

plvar,3

/wait,4

xvar,5

plvar,3

/wait,4


g Uy [mm]

xvar,2

plvar,3

finish


98

Annex C.3 Log-file Ansys of I shaped member /BATCH,LIST

/filnam,Plaatmodel_f

iles

/title,Plastisch GNL

Plaatmodel

/units,mpa

/nerr,1,99999999

tw=15

hw=2000

tf=24

bf=550

l=8000

nw=20

nf=4

nl=l/100

ew=10

ef=l/200

umax=20

nmax=50s

fy=355

/prep7

et,1,SHELL181

R,1,tw

R,2,tf

mp,ex,1,2.1e5

mp,prxy,1,0.3

tb,biso,1

tbdata,1,fy,21

tbpl,biso,1

/wait,1

k,,0,0,0.5*bf

k,,hw,0,0.5*bf

k,,0,0,0

k,,hw,0,0

k,,0,0,-0.5*bf

k,,hw,0,-0.5*bf

k,,0,l,0.5*bf

k,,hw,l,0.5*bf

k,,0,l,0

k,,hw,l,0

k,,0,l,-0.5*bf

k,,hw,l,-0.5*bf

kplot

a,3,4,10,9

a,1,3,9,7

a,3,5,11,9

a,2,4,10,8

a,4,6,12,10

lsel,s,tan1,x,1

lsel,a,tan1,x,-1

lesize,all,,,nw

lsel,s,tan1,y,1

lsel,a,tan1,y,-1

lesize,all,,,nl

lsel,s,tan1,z,1

lsel,a,tan1,z,-1

lesize,all,,,nf

asel,s,area,,1

real,1

amesh,all

nsel,s,loc,x,0

nsel,a,loc,x,hw

cm,lr,node

nmiddenboven=node(0.

5*hw,l,0)

save

finish

/solve

antype,static

pstress,on

nsel,s,loc,y,0

d,all,uy,0

d,all,uz,0

d,all,ux,0

nsel,s,loc,y,l

d,all,uz,0

d,all,ux,0

cmsel,s,lr,node

d,all,uz,0

nsel,s,loc,y,l

d,all,uy,-0.1

solve

finish

/solve

antype,buckle

bucopt,lanb,5

mxpand,5

solve

finish

/post1

allsel

eplot

set,first

plnsol,u,z,0

/wait,5

finish

/prep7


obname

allsel

upgeom,ew,1,1,'%myjo

bname(1)%',rst

nsel,s,loc,z,ew

*get,nmidmid,node,,n

um,min

asel,s,area,,2,5

real,2

amesh,all

save

finish

/solve

antype,static

pstress,on

cmsel,s,lr,node

ddele,all,uz

f,all,fz,1

nmidrand=node(0,0.5*

l,0)

nsel,s,loc,x,0

nsel,a,loc,x,hw

d,all,ux,0

nsel,s,loc,y,l

d,all,uy,0

nsel,s,loc,y,0

d,all,uy,0

d,all,uz,0

d,all,ux,0

nsel,s,loc,y,l

d,all,uz,0

d,all,ux,0

solve

finish

/prep7


obname

allsel

*get,urand,node,nmid

rand,u,z

upgeom,ef/urand,1,1,

'%myjobname(1)%',rst

nsel,s,loc,x,0

nsel,a,loc,x,hw

nsel,u,loc,y,0

nsel,u,loc,y,l

ddele,all,ux

save

finish

/solve

antype,static

nlgeom,on

outres,all,all

nsel,s,loc,y,l

d,all,uy,-umax

nsubst,nmax,nmax,nma

x

kbc,0

solve

save

finish

/post1

allsel

set,last

/dscale,all,10

shell,mid

plnsol,s,y,2

finish

/wait,6

/post26

numvar,200

allsel

nsol,200,nmiddenbove

n,u,y,Uyneg

filldata,199,,,,-1,0

prod,2,200,199,,Uy

nsel,s,loc,y,0

*get,nmin,node,0,nxt

h

*get,ntot,node,,coun

t

rforce,3,nmin,f,y

*do,ndo,2,ntot

*get,nmin,node,nmin,

nxth

rforce,196,nmin,f,y

add,3,3,196,,Fy_tot

*enddo

nsol,195,nmidrand,u,

z,Uz_echt

filldata,194,,,,ef,0

add,5,195,194,,Uz_ra

nd

nsol,198,nmidmid,u,z

,Uz_echt

filldata,197,,,,ew,0

add,4,198,197,,Uz_pl

aat


g Uz [mm]

/axlab,y,Totale

Belasting Ftot [N]

xvar,4

plvar,3

/wait,4

xvar,5

plvar,3

/wait,4


g Uy [mm]

xvar,2

plvar,3

finish


99

Annex D.1 Mathcad sheet for NEN-EN 1993-1-5

As a part of this thesis the effective cross-section method has been implemented in a Mathcad sheet.

Here the sheet for an I-shaped cross-section is given, but also for a box girder and an unstiffened

plate a sheet has been made.

Plooitoetsing van een I-ligger, volgens Eurocode 3,H4 - Doorsnedereductiemethode

Informatie over dit rekenblad Dit rekenblad toetst een I-ligger zonder langs- of dwarsverstijvers volgens dedoorsnede-reductiemethode uit hoofdstuk 4 van NEN-EN 1993-1-5. De losse plaatvelden moeten dusvoldoen aan de eisen gesteld in art. 4.1.

De toetsing gaat uit van art. 4.3(3) en 4.4(4), een losse toetsing van normaalkracht en buigendmoment capaciteit. Een gecombineerde toetsing is ook toegestaan, alleen moet het effect van hetopschuiven van de neutrale lijn, eN, dan meegenomen worden door te itereren.

De hier bepaalde doorsnedengrootheden (Aeff, Ieff, Weff) mogen ook gebruikt worden voor kip enkniktoetsen volgens NEN-EN 1993-1-1.

Wanneer de constructie als geheel gevoelig is voor 2e orde effecten moeten deze meegenomen zijnin het berekenen van de belasting op het plaatveld.

Bij de invoer van de gegevens moeten alle geel gekleurde velden ingevuld worden. Deelasticiteitsmodulus en poissonfactor worden niet gevraagd, omdat deze in de Eurocode al oprespectievelijk 210.000 N/mm2 en 0.3 aangenomen zijn.Bij de belastingen word een drukkracht met positieve waarde ingevuld, de trekkracht negatief.Het buigend moment dient zo ingevuld te worden, dat de bovenflens gedrukt wordt. Als het buigendmoment negatief is, moet deze alsnog positief worden ingevuld, het profiel moet nu echter 'op zijnkop' ingevuld worden. De boven en onderflens wisselen dus van plek.

Aannamen over de doorsnede:- geen lokale belasting (patch loading).- geen "flange induced buckling"- geen belasting haaks op de plaatvelden.- buiging en dwarskracht alleen in de sterke richting.- geometrie is dus zoals hiernaast:

Figuur 1

Invoer gegevens

Geometrie van de doorsnede:Vul de parameters in zoalsaangegeven in figuur 1:

Materiaalgrootheden:Vloeispanning

Veiligheidsfactoren

Belastingen in de doorsnede:Buigend moment sterke as(ontwerpwaarde)

Normaalkracht(ontwerpwaarde)

Dwarskracht(ontwerpwaarde)

hw 2000 mm⋅:= bf1 800 mm⋅:= bf2 800 mm⋅:=

tw 15 mm⋅:= tf1 24 mm⋅:= tf2 24 mm⋅:=

htotaal hw tf1+ tf2+ 2.048 103× mm=:=

fy 355N

mm2⋅:= ε

235N

mm2⋅

fy

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠

0.814=:=

γM0 1.0:=

γM1 1.0:=

MEd 10500 kN⋅ m⋅:= (altijd positief teken, altijd druk in bovenflens!)(als het moment negatief is, profiel spiegelen)

NEd 2200 kN⋅:= (drukkracht positief)(trekkracht negatief)

VEd 200 kN⋅:=

Berekening bruto doorsnedegroothedenIn onderstaand deel worden de doorsnedengrootheden van de bruto doorsnede bepaald:

A hw tw⋅ bf1 tf1⋅+ bf2 tf2⋅+ 6.84 104× mm2=:=

S tf2 bf2⋅ 0.5⋅ tf2⋅ hw tw⋅ 0.5 hw⋅ tf2+( )⋅+ bf1 tf1⋅ htotaal 0.5 tf1⋅−( )⋅+ 7.004 107× mm3=:=

eSA

1.024 103× mm=:=

Ieigen112

bf2⋅ tf23⋅

112

tw⋅ hw3⋅+

112

bf1⋅ tf13⋅+:=

Isteiner tf2 bf2⋅ e 0.5 tf2⋅−( )2⋅ hw tw⋅ e 0.5 hw⋅− tf2−( )2⋅+ bf1 tf1⋅ htotaal e− 0.5 tf1⋅−( )2⋅+:=

I Ieigen Isteiner+ 4.933 1010× mm4=:=

WonderIe

4.817 107× mm3=:=

WbovenI

htotaal e−4.817 107× mm3=:=

Berekening normaalkracht capaciteit

Om deze capaciteit te bepalen word volgens art. 4.3(3) een spanningsverdeling in de doorsnededoor een zuivere normaalkracht aangenomen. Dit levert Ψ=1.

Plooifactoren volgens art. 4.4, tabel 4.1 en 4.2:

kσ.w 4.0:= kσ.f 0.43:=

Relatieve slankheden en reductiefactoren volgens art. 4.4(2):

λp.w

hw

tw

28.4 ε⋅ kσ.w⋅:= λp.f1

0.5bf1 0.5 tw⋅−

tf1

28.4 ε⋅ kσ.f⋅:= λp.f2

0.5bf2 0.5 tw⋅−

tf2

28.4 ε⋅ kσ.f⋅:=

λp.w 2.885= λp.f1 1.079= λp.f2 1.079=

ρw 1.0 λp.w 0.673≤if

λp.w 0.22−

λp.w2

λp.w 0.673>if

:=

ρw 0.32=

ρf1 1.0 λp.f1 0.748≤if

λp.f1 0.188−

λp.f12

λp.f1 0.748>if

:=

ρf1 0.765=

ρf2 1.0 λp.f2 0.748≤if

λp.f2 0.188−

λp.f22

λp.f2 0.748>if

:=

ρf2 0.765=

Het effectieve deel kan bepaald worden volgens tabel 4.1 en 4.2, hiermee is het effectieveoppervlak en de verschuiving van de neutrale lijn te bereken (zie ook figuur 2):

Figuur 2

Aeff ρw hw⋅ tw⋅ ρf1 bf1 tw−( )⋅ tf1⋅+ ρf2 bf2 tw−( )⋅ tf2⋅+ tf1 tf2+( ) tw⋅+:=

Aeff Aeff NEd 0≥if

A NEd 0<if

3.915 104× mm2=:= Aeff 3.915 104× mm2=

Seff.2 ρw hw⋅ tw⋅ 0.5 hw⋅ tf2+( )⋅ ρf1 bf1 tw−( )⋅ tw+⎡⎣ ⎤⎦ tf1⋅ htotaal 0.5 tf1⋅−( )⋅+ ρf2 bf2 tw−( )⋅ tw+⎡⎣ ⎤⎦ t⋅+:=

eeff.2Seff.2

Aeff:= eeff.2 1.024 103× mm=

eN e eeff.2−:= eN 0 mm=

Berekening buigend moment capaciteit

Om deze capaciteit te bepalen word volgens art. 4.3(3) een spanningsverdeling in de doorsnededoor een zuiver buigend moment aangenomen. Dit levert Ψ=-1 voor een symmetische doorsnede.Echter moet eerst de effectieve gedrukte flens bepaald worden, waarmee de Ψ voor het lijf bepaalddient te worden.

Figuur 3

Plooifactor volgens art. 4.4, tabel 4.2 (bij Ψ=1 voor bovenflens): kσ.f 0.43:=

Afmeting flens: c 0.5bf1 0.5 tw⋅−:= c 392.5 mm=

Relatieve slankheid en reductiefactor volgens art. 4.4(2):

λp.f1

c

tf1

28.4 ε⋅ kσ.f⋅:= λp.f1 1.079=

ρf1 1.0 λp.f1 0.748≤if

λp.f1 0.188−

λp.f12

λp.f1 0.748>if

:= ρf1 0.765=

Het effectieve deel van de flens kan bepaald worden volgens tabel 4.2:

ceff ρf1 c⋅:= ceff 300.309 mm=

beff.f1 2 ceff⋅ tw+:= beff.f1 615.618 mm=

Aeff.1 hw tw⋅ beff.f1 tf1⋅+ bf2 tf2⋅+ 6.397 104× mm2=:=

Met deze effectieve gedrukte flens kan Ψ voor het lijf bepaald worden:

Seff.1 tf2 bf2⋅ 0.5⋅ tf2⋅ hw tw⋅ 0.5 hw⋅ tf2+( )⋅+ beff.f1 tf1⋅ htotaal 0.5 tf1⋅−( )⋅+ 6.103 107× mm3=:=

eeff.1Seff.1

Aeff.1:= eeff.1 953.999 mm=

ht eeff.1 tf2−:= ht 929.999 mm=

hc hw ht−:= hc 1.07 103× mm=

Ψwht−

hc:=

Ψw 0.869−=

De plooifactor voor het lijf kan nu bepaald worden volgens art. 4.4, tabel 4.1:

kσ.w 7.81 6.29 Ψw⋅− 9.78Ψw2+ 0 Ψw> 1−≥if

5.98 1 Ψw−( )2⋅ 1− Ψw> 3−≥if

95.68 Ψw 3−<if

:= kσ.w 20.665=

Relatieve slankheid en reductiefactor volgens art. 4.4(2):

λp.w

hw

tw

28.4 ε⋅ kσ.w⋅:= λp.w 1.269=

ρw 1.0 λp.w 0.5 0.085 0.055 Ψw⋅−+≤if

λp.w 0.055 3 Ψw+( )⋅−

λp.w2

λp.w 0.5 0.085 0.055 Ψw⋅−+>if

:= ρw 0.715=

De effectieve doorsnede van de gehele I-ligger kan bepaald worden volgens tabel 4.1 en 4.2, ookkunnen de effectieve doorsnedegrootheden bepaald worden (zie ook figuur 3)

heff ρw hc⋅ 765.124 mm=:=

hw.1 0.4 heff⋅ 306.05 mm=:=

hw.2 0.6 heff⋅ ht+ 1.389 103× mm=:=

Aeff.2 hw.1 hw.2+( ) tw⋅ beff.f1 tf1⋅+ bf2 tf2⋅+ 5.94 104× mm2=:=

Seff tf2 bf2⋅ 0.5⋅ tf2⋅ hw.2 tw⋅ 0.5 hw.2⋅ tf2+( )⋅+ hw.1 tw⋅ htotaal tf1− 0.5 hw.1⋅−( )⋅+ beff.f1 tf1⋅ htotaal(⋅+:=

eeffSeff

Aeff.2906.921 mm=:=

Ieigen112

bf2⋅ tf23⋅

112

tw⋅ hw.23⋅+

112

tw⋅ hw.13⋅+

112

beff.f1⋅ tf13⋅+ 3.388 109× mm4=:=

Isteiner tf2 bf2⋅ eeff 0.5 tf2⋅−( )2⋅ hw.2 tw⋅ eeff 0.5 hw.2⋅− tf2−( )2⋅+ hw.1 tw⋅ htotaal eeff− 0.5 hw.1⋅− −(⋅+:=

Ieff Ieigen Isteiner+ 4.261 1010× mm4=:=

WeffIeff

htotaal eeff−3.734 107× mm3=:=

Berekening dwarskracht capaciteit

De dwarskrachtcapaciteit wordt bepaald aan de hand van de methode uit hoofdstuk 5. In artikel5.1(2) wordt een belangrijke parameter geintroduceerd:

Factor afhankelijk van staalsoort: η 1.20 fy 460MPa≤if

1.00 fy 460MPa>if

:= η 1.2=

Voor de rest van de berekening zijn een tweetal parameters uit Bijlage A (A.1.2) nodig.Hierbij is aangenomen dat er behalve bij de oplegging geen dwarsverstijvers aanwezigzijn, zodat de plooifactor meteen bekend is.

De eulerse kritieke plooispanning is σE 190000tw

hw

⎛⎜⎝

⎞⎟⎠

2

⋅N

mm21.069 107× Pa=:=

De plooifactor bedraagt kτ 5.34:=

De slankheid van het lijf volgt uit artikel 5.3(3):

λw 0.76fy

kτ σE⋅⋅:= λw 1.895=

In tabel 5.1 kan de reductiefactor χ opgezocht worden. Hierbij is aangenomen dat als er opdwarskracht wordt getoetst, deze zich bij de oplegging bevindt. Als het lijf gevoelig is voor plooi, moeter een dwarsverstijver bij de oplegging aanwezig zijn. Deze oplegging word nu aangenomen als een'non-rigid end post' (figuur 5.1(c)).

χ η λw0.83η

<if

0.83λw

0.83η

λw≤if

:= χ 0.438=

De dwarskrachtcapaciteit van het lijf is nu te bepalen met 5.2(1):

Vb.Rdχ fy⋅ hw⋅ tw⋅

3 γM12.692 106× N=:=

VerificatieMet 4.6(1) kan nu een verificatie uitgevoerd worden voor druk en buiging:

UC1: η1NEd

fy Aeff⋅

γM0

⎛⎜⎝

⎞⎟⎠

MEd NEd eN⋅+

fy Weff⋅

γM0

⎛⎜⎝

⎞⎟⎠

+:= η1 0.95=

Met 5.5(1) kan een verificatie voor dwarskwacht uitgevoerd worden:

UC2: η3VEd

Vb.Rd:= η3 0.074=

Interactie is van belang als meer dan de helft van de dwarskrachtcapaciteit gebruikt wordt, in dat gevalmoet artikel 7.1 toegepast worden.

UnityCheck "Voldoet" η1 1.0≤ η3 0.5≤∧if

"Voldoet niet" η1 1.0> η3 0.5≤∧if

"Zie art. 7.1" η3 0.5>if

:=UnityCheck "Voldoet"=

Master Thesis – Plate buckling in design codes

Documents

Transcript of Master Thesis – Plate buckling in design codes