Beams With Corrugated Webs

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LITERATURE STUDY

Master thesis: Girders with corrugated webs

H.G. de Hoop

Supervising Committee

Prof. Ir. F.S.K. Bijlaard

Dr. Ing. A. RomeijnIng. S. Wierda (Iv-Groep)

Ir. P.H.G. Feijen (GLP)

Papendrecht, The Netherlands

November 2003


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Literature study 2

TABLE OF CONTENTS

1 INTRODUCTION ....................................................... ........................................................ ............ 3

2 INTRODUCTION TO GIRDERS WITH CORRUGATED WEBS ........................................... 4

2.1 PROFILED STEEL SHEETING.................................................... ....................................................... 42.2 GIRDERS WITH CORRUGATED WEBS AND THEIR APPLICATION ................................................. ..... 42.3 WEB CONFIGURATIONS ................................................ ....................................................... ......... 62.4 MANUFACTURING AND MATERIALS................................................. ............................................. 72.5 WEB-FLANGE CONNECTION................................................... ....................................................... 82.6 ADVANTAGES AND DISADVANTAGES OF THE CORRUGATE WEBS.................................................. 92.7 CURRENT RESEARCH WORK ................................................... ..................................................... 10

3 MECHANICAL BEHAVIOUR..................................... ........................................................ ....... 11

3.1 BASIS FOR CALCULATION ....................................................... ..................................................... 113.2 BENDING ....................................................... ........................................................... ................... 113.3 NORMAL FORCES................................................. ....................................................... ................. 183.4 TRANSVERSAL FORCES.................................................. ....................................................... ....... 19

3.5 TORSION........................................................ ........................................................... ................... 19

4 FAILURE MECHANISMS AND CHECK-CRITERIA............................................................ 21

4.1 FLANGE BUCKLING....................................................... ........................................................ ....... 214.1.1 Global flange buckling......... ........................................................ ........................................ 214.1.2 Local flange buckling..... ............................................................ .......................................... 24

4.2 YIELD OF THE TENSION FLANGE ....................................................... ........................................... 264.3 WEB PLATE BUCKLING .................................................. ....................................................... ....... 27

4.3.1 Local buckling............................ ............................................................ .............................. 284.3.2 Global buckling.................... ............................................................ .................................... 304.3.3 Interaction between local and global buckling? ............................................ ...................... 314.3.4 Local transverse loads ................................................ ...................................................... ... 32

4.4 YIELD OF THE WEB PLATE ...................................................... ..................................................... 33

4.5 LATERAL TORSIONAL BUCKLING OF THE GIRDER ................................................. ....................... 334.6 BUCKLING OF THE GIRDER ..................................................... ..................................................... 364.7 TORSIONAL BUCKLING .................................................. ....................................................... ....... 374.8 DEFORMATION OF THE GIRDER ................................................ .................................................... 37

5 DIFFERENCES FLAT-CORRUGATED WEB GIRDER ........................................................ 38

5.1 I NTRODUCTION.................................................... ....................................................... ................. 385.2 GENERAL....................................................... ........................................................... ................... 385.3 MECHANICAL BEHAVIOUR ..................................................... ..................................................... 39

5.3.1 Basis of calculation..................................................... ...................................................... ... 395.3.2 Transversal moments ........................................................... ................................................ 39

5.3 ARITHMETIC METHODS.................................................. ....................................................... ....... 395.3.1 Moment capacity .................................................... ........................................................... ... 395.3.2 Transverse moments.............. ............................................................ ................................... 40

5.3.3 Flange buckling........................................................... ...................................................... ... 405.3.4 Local web buckling due to transverse forces ....................................................................... 415.3.5 Global web buckling due to transverse forces ..................................................................... 435.3.6 Buckling of the girder................................................. ....................................................... ... 435.3.7 Lateral torsional buckling of the girder ................................................ ............................... 43

5.4 APPLICATION FIELD...................................................... ........................................................ ....... 44

6 STANDARDS AND GUIDELINES ..................................................................... ........................ 45

6.1 I NTRODUCTION.................................................... ....................................................... ................. 456.2 DUTCH STANDARD – TGB 1990 ................................................ ................................................. 456.3 DAST-R ICHTLINIE 015........................................................ ....................................................... . 46

6.3.1 Global content of the standard.................... .................................................... ..................... 466.3.2 Section 4: Girders with trapezoidal corrugated webs.......................................................... 46

6.4 EUROPEAN STANDARD .................................................. ....................................................... ....... 476.4.1 Eurocode 3: Design of steelstructures ..................................................... ........................... 476.4.2 Annex D – Members with corrugated webs........................................................... ............... 47


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1 INTRODUCTION

This literature study is the first part of the master thesis: ‘Girders with corrugated webs’ and

is carried out for better understanding of the behaviour and use of these girders. With help of

this literature study an explorative analysis will be carried out during the next phase of thethesis and finally a software-tool can be developed.

Section 2 of this literature study will give a general introduction to girders with corrugated

webs. The girders are still used for more than thirty years in several application fields, which

will be described in this section. Also topics like the manufacturing, the web-flange

connection and the web configuration will come up.

The mechanical behaviour of girders with corrugated webs will be described in section 3.

After an explanation of the basis for calculation, the effects of the following load types will be

discussed: bending, normal forces, transversal forces and torsion.

Section 4 will deal with all relevant failure mechanisms of girders with corrugated webs. Here

not only the failure mechanisms will be described, but also the available check-criteria with

regard to these mechanisms.

For a clear understanding of girders with corrugated webs, the differences between plate

girders with flat plates and corrugated plates will be described in section 5.

In the last section an overview will be given of the most relevant standards and guidelines

relevant to the production and design of girders with corrugated web plates. With help of

these standards and guidelines the several case studies of the explorative analysis will be

carried out.


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2 INTRODUCTION TO GIRDERS WITH CORRUGATED WEBS

2.1 Profiled steel sheeting

The lightness of cold-formed thin-walled structures was formerly their most importantfeature and therefore they were used mostly in products where the weight saving was of

great importance. This kind of products was naturally needed in

especially transportation industries e.g. aircrafts and motor

industry.

The profiled sheeting types have been developed significantly

since the first profiled steel sheets. The first plates were very

simple and the stiffness of these was not very high. The

manufacturing process and the materials limited the shape of

the profiles to simply folded or corrugated shapes. From the

early 1970’s, the development of the shapes of sheeting profiles

and also better materials and manufacturing technologies leadto possibilities to provide more complex profiles. Also stiffeners

were added to flanges of the profile. This improved substantially

the load-bearing capacities of the developed new profiled steel

sheets. A huge range of profile types are available nowadays

used for structural purposes (floors, walls,

roofs, pipes, etc.) and for functional purposes.

2.2 Girders with corrugated webs and their application

To save weight, a Swedish company got the idea to fabricate plate-girders with corrugated

webs (see figure 2.2). These girders also could be used in structures where the weight

saving was of great importance. Girders with corrugated webs are marketed as a productfrom specialised fabricators or as one-off structures. One example of the former is the Dutch

company GLP Corrugated Plate Industry, manufacturer of trapezoidal corrugated webs with

unlimited lengths and girders as well. Other examples are Ranabalken, which has been on

the Swedish market for about forty years and the company Zeman & Co form Austria, which

is producing similar beams, but with sinusoidal corrugated webs.

In 1966 in Sweden, the first variant of the thin walled welded plate girder with a very great

slenderness has been fabricated. This girder had a vertical corrugated web, also for low

weight reasons. Especially for bridge building they thought that the application of these

girders would be very useful. But it was only in 1986 that the first composite bridge with

corrugated webs was constructed. The idea of the steel-concrete bridge was the

construction firm Campenon Bernard. This first bridge, the ‘Cognac bridge’, was constructed

with tubular members with in situ casted pre-tensioned concrete flanges. The advantage of

Figure 2.2: Girder with corrugated web

Figuur 2.1: Facade with profiled steel

Fig 2.3: Composite bridge with corrugated webs.


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using corrugated webs in structures with pre-stressed concrete is a lower loss of pre-

stresses, because of the low longitudinal stiffness of the corrugated web. The Cognac bridge

is builded within the framework of an innovation project, where economical considerations

were of overriding importance. The next few years several similar bridges are built, like the

‘Maupre bridge’, the ‘Asterix bridge’ and the ‘Dole bridge’ in Europe, but also several inJapan. The Maupre bridge has a steel tubular member filled with concrete as lower flange.

Figure 2.4 shows a similar bridge, recently built in South Korea.

Later on, these webs were not only used for bridges constructed with tubular members, but

also for other types of bridges, like the plate girder bridge showed in figure 2.5.

In the United States, girders with corrugated webs are more and more widely used for bridge

building. Many manufacturers are producing such kind of girders. Young j. Paik patented the

girder and marketed it to builders in 1970’s, and with Sumitomo, a Japanese company that

manufactures girders with corrugated webs, the girders were manufactured with PACO

Engineering Corporation as the exclusive U.S. distributor. Pennsylvania Department of

Transportation (PennDOT) is sponsoring research, adopting corrugated webs, to realise

additional benefits from High Performance Steel. It intends to construct a demonstrationbridge with girders with corrugated webs.

Bridges with corrugated webs have been used at least once in Sweden as well, although it

was a temporary bridge.

The other important application field became the use as roof girder in the industry and high-

rise building. Especially in Germany, but also in Sweden, several halls are built with portals

of girders with corrugated webs (see figure 2.6).

Because of the high strength-to-weight ratio, the span lengths could be wider and less

columns are necessary. Nowadays this is the main application field of these girders.

Other application fields of (girders with) corrugated webs are: cranebrackets, craneways,

feed silos/ tanks and transportation structures (side panels of trains, containers and trucks).

Figure 2.4: Bridge in South Korea Figure 2.5: Trapezoidal plate girder

Figure 2.6: Halls built with girders with corrugated webs (left: US Army headquarters in Germany


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2.3 Web configurations

The webs of the girders can be corrugated to different types of profiles. Figure 2.7 shows not

only the trapezoidal webs, but also a sinusoidal web and a swallow’s tail-web. Also other

configurations are possible, like the cell-formed webs and webs with rolled stiffeners. Theserolled stiffeners cause a better buckling behaviour.

The aesthetics of the different configurations is totally different, but also the fabrication, the

mechanical behaviour and the costs of the different types have to be taken in account when

making a choice for one of them.

One of the possible configurations is the sinusoidal profiled web (see figure 2.7). In addition

to benefits in production technology, the sinusoidal corrugation has the advantage over

trapezoidal profiling of eliminating local buckling of the flat plate strips.

To increase the shear strength, the plates of ‘le viaduc de franchissement de la vallee de la

Marne’ in France, are made of circular hollow sections welded together with flat panels.

Figure 2.7: Web configurations

Figure 2.8: Web shape of ‘le viaduc de franchissement de la vallee de la Marne (France)


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For most of the configurations still a great lack of knowledge exists. The available

information mostly deals with trapezoidal webs and also this master thesis will only deal with

trapezoidal profiled girders. Figure 2.9 shows a cross section of a trapezoidal corrugated

girder with the variable quantities. An increase of the angle α between the diagonal and the

parallel strip of the web, causes an increase of the bending stiffness of the girder, but it alsoleads to a higher use of steel. The angle is also an important factor in the local and global

buckling behaviour of the web girder. An other important quantity of the cross section is the

width of the web bt, because it has an influence on magnitude of the transversal moments on

the flanges. The webs are available with a thickness tw in the range of 2…15 mm, which is

governed by the fabrication process. The maximum depth of the web is 3 m, a limitation

given by the welding machine. These figures would make it possible to fabricate girders with

bt/tw = 1500, though that’s beyond the maximum ratio given by the German standard: DASt-

Richtlinie 015.

2.4 Manufacturing and materials

Cold-formed steel members can be manufactured e.g. by folding, press-braking or cold-

rolling. The trapezoidal webs are manufactured using cold-forming. Also the cylindrical

members are manufactured by cold rolling from flat steel webs. Cold-rolling technique gives

good opportunities to vary the shape of the profile (see figure 2.10).

During the cold-forming process varying stretching forces can induce residual stresses,

which can significantly change the load-bearing resistance of a section. Favourable effects

can be observed if residual stresses are induced in parts of the section which act in

compression and, at the same time, are susceptible to local buckling.

The most common steelmaterial that is used in

profiled steel webs is cold-

formed structural steel. For

corrosion protection, the

girders with corrugated webs

can be hot chip zinc coated,

but the girders can also be

hot-galvanised without

difficulty. Normally is made

use of S235 or S355 steel.

The use of higher strength

material for the flanges is

possible, but in terms of Fig. 2.11: stress-strain curves cold-formed structural web

Figure 2.10: Corrugating of the webFigure 2.9: Cross section of a trapezoidal web


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statics, this is only meaningful in exceptional cases.

The mechanical properties are dependent on the rolling direction so that yield strength is

higher transversally to the rolling direction. In figure 2.11, typical stress-strain curves of cold-

formed structural web steel with nominal yield strength of 350 N/mm2 at room temperature

both longitudinally and transversally to rolling direction are shown. For this thesis will bemade use of the yield strength longitudinally to the rolling direction.

2.5 Web-flange connection

Smith (1992) performed four tests on two girders with corrugated webs, which were welded

to the flange using intermittent welding. He found that the connection between the flange and

the web is critical for the shear strength as the weld used in the test was subjected to high

strength and web was easily ruptured at this point before it reached its buckling strength.

Smith suggested that intermittent welding of the corrugated webs to the flange is not

advisable.

Normally the corrugated web is connected to the flanges with a single continuous fillet weld,which is important for the competitiveness. According to the standards, the connection

between the flanges and a flat web should be carried out with two fillet welds on both sides

of the web. Figure 2.12 shows how the transversal moments in the web can be transferred to

the flanges. According to the German recommendations (DAST – Richtlinie 015), it is also

allowed to use a single continuous weld. Probably, the reason for this is the profiled shape of

the web. Figure 2.13 shows how the transversal moments in the web could be transferred to

the flange. According to fabricators of the girders, there is some talk of burning in of the

weld, so it’s not a perfect fillet weld, but a (partial) penetrated weld. However, this argument

can only be put forward when standardised.

At the institute for steelbuilding, TU Braunschweig, research has been done to connections

between flanges and trapezoidal profiled webs [11]. One of the connections is the bolted

connection showed in figure 2.14. This connection could be economic favourable, because

of the lower labour costs. However, this depends on the circumstances of the fabricator.

Web

Flange

Flange

Figure 2.12: Double welded web Figure 2.13: Transversal momentsfor a sin le welded corru ated web

Figure 2.14: Bolted connection web-flanges


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2.6 Advantages and disadvantages of the corrugate webs

The advantages of the corrugated webs are:

- thinner webs are possible;- lower dead weight of the structure;

- a better buckling behaviour;

- aesthetics of the structure?

- erection cost is reduced, since the corrugation in the web provides a higher

resistance against bending about the weak axis, none of the auxiliary lifting

equipment normally needed is required;

- when used in combination with prestressed concrete (composite bridges): a

lower loss of pre-stresses, because of the low stiffness of the web in

longitudinal direction (see figure 2.15);

- decrease of necessary transverse and longitudinal stiffeners, because of the

dfgg high out-of-plane stiffness of the web (see figure 2.16).

Possible disadvantages of building with corrugated webs are:

- a lack of knowledge and standards;

- a lack of software packages for the design;

- longer delivery times and appropriate minimum order conditions apply;

- aesthetics of the structure?

Figure 2.15: stiffness in longitudinal direction

Figure 2.16: Out-of-plane stiffness web


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2.7 Current research work

A wide range of different kind of research activities concerning corrugated steel webs is

going on in several countries. Most of the studies are based on both experimental test

results and usually also modelling results produced with some finite element modellingprograms. Usually the aim is to analyse the mechanical behaviour of the girders, for the

development of design rules. So at TU Berlin, Aschinger obtained his doctorate with a thesis

on the bearing capacity of girders with trapezoidal corrugated webs. Later on an

investigation is carried out into the interaction between flexural buckling and plate buckling

on the web crippling capacity of trapezoidal sheeting and into the different buckling

coefficients. Also in Sweden a lot of investigation has been done into the shear resistance,

flange buckling, etc. The work in Sweden has mainly been done at Chalmers university and

at the Royal institute of technology in Stockholm.

The scope of some investigations is to increase the load-bearing capacity of corrugated

webs, with different web configurations. There are also some on-going research projects(especially in Japan) concerning the application of corrugated webs in composite bridges.

In Australia, e.g. in Queensland University of Technology, and also in several other

universities, there are on-going investigations into the behaviour of girders with corrugated

webs under variable loading. Preliminary results from limited tests indicate that the

resistance to fatigue can be higher for girders with corrugated webs than for conventionally

stiffened girders. One of these projects will determine the behaviour of bridge girders with

corrugated webs under variable loading and establish the fatigue strength for such girders.

At this moment, the fatigue durability of the Shinkansen Bridge with corrugated steel web

(Japan) is investigated.

Also the materials, coatings and the manufacturing technology are developed constantly.


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3 MECHANICAL BEHAVIOUR

3.1 Basis for calculation

As a result of its trapezoidal profiling, the

web has a very low stiffness in longitudinal

direction (see figure 3.1). So the web does

not participate in the transfer of longitudinal

normal stresses from bending. This means that:

In static terms, the girder with corrugated web corresponds to a lattice girder

in which the bending moments and the normal forces are transferred only via the flanges,

while the transverse forces are only transferred through the diagonals and verticals of the

lattice girder (in this case the corrugated web). Among others, A. Bergfelt [4] has confirmedthis assumption with a laboratory research and a finite element analysis. Figure 3.2 shows

the normal stresses in the flanges and the transversal stresses in the web as a result of a

laboratory research with help of strainmeters.

3.2 Bending

Figure 3.3 shows the mechanics scheme of a single span girder with trapezoidal corrugated

web, loaded with a local force. This local force causes shear stresses in the web and normal

stresses in the flanges. In figure 3.4 an infinitesimal part of the web in interaction with theflanges is regarded. The shear stresses in the web cause extra shear stresses in the

flanges. Figure 3.5 shows these stresses and the resulted shear forces T1 (x) and T2 (x).

Figure 3.1: Longitudinal stiffness of the web

Fig 3.4: Shear stresses in web/ flanges Fig 3.5: Shear stresses/ forces upper flange

Fig 3.2: Normal/shear stresses in the web Figure 3.3: Mechanics scheme


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When the flange is schematised as a girder, the following loads on the flange exist (see

figure 3.6):

- A horizontal component (in x-direction) from the shear force in the diagonal parts of

the web F x (x):

2

12 )()(

a

aw xT x F x

(w = length of corrugation)

- A horizontal component (in y-direction) from the shear force in the diagonal parts of

the web F y (x):

1)()( aw

b x F x F t x y

- Transversal moments around the z-axe M z,1 (x) as a result of the eccentrical shear

forces T 1 (x):

2)()( 11,

t z

b xT x M

Figure 3.6a: Loads on the upper flange (in XY-plane)


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Because of the thickness of the flanges, the shear forces also cause bending moments

around the y-axe and torsion moments around the x-axe. Figure 3.6b shows a scheme of the

same upper flange, but now for the cross section in XZ-plane.

The following bending moments and torsion moments exist:

- A bending moment M y,1 (x) around the y-axe as a result of the shear force T 1 (x):

2)()( 111,

t xT x M y

(t 1 = thickness of the flange)

- A bending moment M y,2 (x) around the y-axe as a result of the x-component of the

shear force F x (x) in the diagonal parts of the web:

2)()( 12,

t x F x M x y

- A torsion moment M x,2 (x) around the x-axe as a result of the y-component of the

shear force F y (x) in the diagonal parts of the web:

2)()( 112,

t xT x M x

Figure 3.6b: Loads on the upper flange (in XZ-plane)


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To find mathematical expressions for the loads on the flanges, an equation for the shear

force Tx is necessary:

dxh

xV xT

dxh

xV x

dxt x xT

w

z

w

z

w

)(

)()()(

)()(

The transverse force could have a parabolic function, so a lot of calculations are necessary.

This can be prevented with help of the following equation:

dx xV x M

z y

)()(

Figure 3.7 shows the V- and M-line of a

part of a corrugated web. So the

following equation for T(x) can be

obtained:

w

y

w

y y

h

x M

h

x M x x M xT

)(

)()()(

Now equations can be found for all

loads on the flanges. Because of the

analogy, only the force Fy (x) and

bending moment Mz,1 (x) will be worked

out:

1

,,)(

awb

h M M x F t

w

P yQ y

y

and:

2)(

,,

1,t

w

Q y R y

z

b

h

M M x M

Figure 3.7: V- and M-line of one corrugation length


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Now it is possible to formulate the expressions for points at any place. For the points A,B,C

and D (see figure 3.8 on the next page) the expressions are given.

1

1

,

)()(

aw

b

h

a M awa M F t

w

end yend y

B y

1

11

,

)())(2(

aw

b

h

wa M aawa M F t

w

end yend y

D y

2

)0()(,1,

t

w

yend y

A z

b

h

M a M M

2

)()( 1,1,

t

w

end yend y

C z

b

h

awa M wa M M

etc.

Because of the regularity of the distances, general formula can be given for the loads on the

flanges. Again the formula for Fy (x) and Mz,1 (x) are given:

)(

))(())(()(1

1awh

bwia M wiawa M x F

w

t end yend y y

For I is 0…n-1 (n = number of diagonal parts of the web)

w

t end yend y z

h

bawia M wia M x M

2))(())(()( 11,

For I = 1…n-1 (n = number of diagonal parts of the web)

Figure 3.8: The points A,B,C and D on the flange


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Furthermore, attention have to be paid to the end field of the girder (a end). Figure 3.9 shows

the possible end fields and in figure 3.10 the shear force T2 (x) and its components Fx,E (x)

and Fy,E (x) are given.

With help of the found equations:

1

,,)(

aw

b

h

M M x F t

w

P yQ y

y

and:

2)(

,,

1,t

w

Q y R y

z

b

h

M M x M

the equations for the components of the shear force in the end

field (Fx,E (x); Fy,E (x) and the following moment Mz,2 (x)) can be

determined. These equations are also valid for the place where

the transverse forces change sign (an example of this is given in

figure 3.11).

i

i

w

y

i

i E x E y

a

b

h

x M

a

b x F x F

)()()( ,,

i y

w

y

i y E x z eh

x M e x F x M ,,,2,

)()()(

Figure 3.9: Possible endfields

Figure 3.10: Shear force in endfield Fi ure 3.11: Shear forces at V z


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Figure 3.6b shows the bendingmoments My,1 (x) and My,2 (x) on theupper flange,

)(5,0)( 1,, 1 x M x M yT y

)(5,0)( 2,, x M x M y F y x

and the torsion moments around the x-

axe Mx,2 (x).

)(5,0)( 2,, x M x M x y F x

These bending moments also cause

normal stresses in the upper flange,

which are made visible in figure 3.13.

Now a formula for the total normal

stresses can be obtained:

1,

1,

1,

1,

0,

)()()(

y

y

i

z

z

i

y

y

xW

x M

y I

x M

z I

x M

x

Figure 3.13: Normal stresses in the upper flange

Figure 3.12: M-lines of the upper flange


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3.3 Normal forces

Girders with corrugated webs are also used in structures, where it has to transfer normal

stresses (portals, pillars, etc.) Because of the low longitudinal stiffness of the corrugated

web, the normal forces are transferred via the flanges.

21 A A

N x

in which:

222

111

t b A

t b A

Figure 3.14 shows the mechanics scheme of a girder loaded with a normal force and of the

upper flange loaded with half of the normal force. The normal forces cause strains in the

flanges, which are hindered by the corrugated web. (See constraint forces F1, F2 en T1.)

The points of application of the constraint forces are on the insides of the flanges, so these

forces cause moments in the flanges. The derivation of the expression for the normal

stresses in the upper flange is analogous to the derivation of the normal stresses in

paragraph 3.2:

1,

1,

1,

1,

21 y

y

i

z

z

xW

M y

I

M

A A

N

in which:

25,0 1

11,

t

T M y respectively: 25,0 1

1,

t

F M x y

Figure 3.14 Girder and its upper flange loaded with normal force


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3.4 Transversal forces

The transverse forces are only transferred

through the diagonals and verticals of the

corrugated web.

w

z z

A

xV x

)()(

with: www t h A

3.5 Torsion

When a girder with a trapezoidal web is loaded by a torsionmoment MT (see figure 3.16),

shear forces arise in the sections between the web and the flanges. Figure 3.17 shows the

shear forces and their directions for the upper- and lower flange. The flanges are varying

under compression and under tension.

With help of figure 3.18, in which the transferring shear forces over the web are made visible,

a deformation figure of the total girder can be made (figure 3.19).

Figure 3.15: Shear stresses in web as result of transverse force

Fig 3.16: Girder with torsionmoment Fig 3.17: Shear forces in upperflange (a), lower flange(b)

Fig 3.18: Shear forces T x and T z in the web Fig 3.19: Deformation of the girder


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The total deformation of the girder subjected to torsion can be subdivided into four basic

deformations:

1. Shear deformation of the parallel parts of the web by Tx.

2. Flange bending around the y-axe by Tz.

In which: z z T

T T C

22

3. Bending of the total girder around the y-axe, as a result of Tx.

4. Bending of the flange around the z-axe, as a result of Tx.

Figure 3.20: Shear deformationof a arallel art

Figure 3.21: Vertical deformation of the upper flange

Figure 3.22: Bending of the total crosssection around the y-axe


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Literature study 21

4 FAILURE MECHANISMS AND CHECK-CRITERIA

4.1 Flange buckling

In determining the normal bearing force ofthe flanges, a distinction must be made

between tensile and compressive stresses.

In the context of compressive stresses, the

stability of the flange must be taken into

account. A distinction must be made here

between the global and local stability of the

flange. Paragraph 4.1.1 deals with global

flange buckling and 4.1.2 with local flange

buckling.

4.1.1 Global flange buckling

According to Annex D.2.1 of prEN 1993-1-5, the design buckling resistance of the

compression flange is derived as follows:

0

,11

,,

M

r y

d Rb

f t b N

where: r y f , includes the reduction due to transverse moments in the flanges.

0

,

)(4,01

M

y

z xT

T yr y

f

M f

f f f

is the reduction factor for lateral buckling according to 6.3 of EN 1993-1-1

z M is the transversal moment in the flange. (see next pages)

2

11

6

bt

M z X

is the maximum longitudinal stress as a result of . z M

0,10 M (see NAD-NVN-ENV 1993-1-1:1995)

Finally, the compression flange should be verified against buckling as follows:

0,1,,

,

d Rb

d E

N

N

Figure 4.1: Local flange buckling


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Literature study 22

Where the design value of the compression force is:

21

,,

,

21

11,,

5,05,0 t t h

M N

A A

A N

w

d E y

d E d E

Determination of the reduction factor for lateral buckling

The reduction factor for lateral buckling can be determined in accordance with 6.3 of EN

1993-1-1: ‘Buckling resistance of members’. The value of the reduction factor for the

appropriate non-dimensional slenderness should be obtained from:

)(

)(49,0

9,93

1

)2,0(15,0:

)0,1(1

,11

,

1

1

,

2

22

forcecritical elasticthe f t b N

factor onimperfactithe

f

E

gyrationof radiustheisi

lengthbuckling theis L

i

L

N

f A

Where

r ycr

r y

cr

cr

cr

r y

(For slenderness 2,0 or for crit d E N N 04,0, buckling effects may be ignored and

only cross sectional checks apply.)

Determination of z M With help of figure 4.2, the expression for z M can be found:

Figure 4.2: Loads on the flange

bt


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Literature study 23

)(5,0)(1, x F w x M M y z z

In paragraph 3.2 the formula for )(1, x M z and )( x F y are obtained:

)(

))(())(()(1

31

awh

awia M wiawa M x F

w

end yend y y

For I is 0…n-1 (n = number of diagonal parts of the web)

w

end yend y z h

aawia M wia M x M

2))(())(()( 311,

For I = 1…n-1 (n = number of diagonal parts of the web)

However, to determine the exact value of Mz, the influence of the end fields of the girders

and of the local loads have to be taken in account. For these reasons, too many calculations

are necessary to determine this exact value. Aschinger [2] has done some FEM-analyses to

define some practical formula for Mz, which come close to the real values:

)max()max( 1, z y z M m F f M

where:

)max()max(

)max()max(

)max(22

)max(

2)max()max(

,2

131

311,

z

w

t wt

ww

z y y

z

w

t w

ww

z z

V h

bt b

t h

V T F

V h

baat a

t h

V aT M

and where the factors f and m can be obtained from table 4.1.

Load figuration Factor f Factor m

0,13 1,50

0,065 0,60

0,065 0,50

Table 4.1: The factors f and m


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Literature study 24

The Dutch standard

According to the European standard, the compressed flange should be verified against

buckling around the z-axe. Analogous to this arithmetic method, the check can be carried out

with help of 12.1 of NEN 6771 as follows:

0,1,,,1,

,,,1

d ucbuc z

d sc

N

N

Where: d sc N ,,,1 is the design value of the compression force.

r yd uc f t b N ,11,,,1 )( is the elastic critical force of the compression flange.

r y f , can be calculated analogous to the European Standard!

buc z , is the buckling factor for buckling around the weak z-axe.

For the determination of the buckling factor is referred to 12.1.1 of NEN 6770. The buckling

factor depends on the relative slenderness, which can be calculated with:

u

z

E z

d uc

rel z F

N

,

,,

,

where:

r y

d u

f

E

,

z

buc z

z i

l , is the slenderness with regard to buckling around the z-axe.

in which: buc z l , is the buckling length with regard to the z-axe.

12

12 1

11

3

11 b

t b

bt

A

I i z z

is the radius of gyration.

4.1.2 Local flange buckling

Local flange buckling will be influenced by the geometry of the web in a favourable waycompared to the flat web. So normally, this failure mechanism will not be normative.

According to annex D.2.1 of prEN 1993-1-5, the rules for plate elements without longitudinal

stiffeners in 4.4 (1) and (2) of prEN 1993-1-5 can be used with appropriate buckling

coefficients. The effective area of the compression flange with the gross area A c should be

obtained from:

11,1 t b A A ceff

where is the reduction factor for buckling. This factor may be taken as follows:

0,1188,02

p

p


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Literature study 25

with:

k

t b f

cr

y p

4,28

where:

r y f ,

235 (f y,r in N/mm

2)

t is the thickness of the compression flange

k is the buckling factor:

Determination of the flange slenderness p

To calculate the flange slenderness, the buckling factor should be determined. The bucklingfactor depends on the appropriate width b. There are several proposals of which b has to be

taken into account. Johnson and Cafolla [8] suggested that the average outstand of the

flange could be used for the appropriate width b if:

14,0

2 11

3

baw

aw

It is not stated what to do if this criterion is not fulfilled, but presumably the idea is to use the

largest outstand.

The design rules for Ranabalken (see paragraph 2.2) states that the outstand should betaken as:

mmb

b 302

1

In order to cover a wide range of different patterns of corrugations, two checks are needed

according to annex D.2.1 (2) of prEN 1993-1-5. The highest flange slenderness from these

checks should be taken into account.

Check (a)

For a very long corrugation in combination with a narrow flange, there is a possibility that the

largest outstand will govern the buckling. However, the flange will be supported by the

inclined parts of the web and a safe approximation of the relevant length should be a = a1 +

2a3. The buckling coefficients of such a plate, assuming conservatively a hinged support

along three edges of the web, is:

2

43,0

a

bk

with: b is the largest outstand from weld to free edge

31

2 aaa


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Literature study 26

0

,22

,,2

M

r y

d R

f t b N

Check (b)

For a geometry with a corrugation width of the same order as the flange width, the flange will

buckle in a mode of rotation around the centreline of the web, but with a stronger restraint

than for a girder with a flat web. This restraint will depend on the stiffness of the flange andthe flexibility of the web. The buckling coefficient will vary in the range 0,43…1,25 and for

simplicity a fixed value of 0,55 is accepted. So for this check, the slenderness parameter will

be defined with the following input:

255,0 1

bband k

Finally, the compression flange shall be verified against buckling in the same way as it has to

be done for global flange buckling:

0,1,,

,,1

d Rb

d E

N

N

Where:

0

,,1

,,

M

r yeff

d Rb

f A N

21

,,

,

2,1

,1

,,15,05,0 t t h

M N

A A

A N

w

d E y

d E

eff

eff

d E

With help of the Dutch standard

The check can also be carried out with help of 12.1.1 of NEN 6771, explained in paragraph

4.1.1, with the difference that an effective width of the compressive flange should be taken

into account. For calculation of the effective width can be made use of the European

standard, but also of the German standard: DASt-Richlinie 015, paragraph 4.2.3.

1

,1,

1

2407,30 b

f t b

d y

eff

This equation has been derived with help of buckling coefficient k = 0,6.This value may be used (instead of k

= 0,43 assuming conservatively a hinged support

along three edges and one free edge), because of the trapezoidal shape of the web plate.

4.2 Yield of the tension flange

In case of normal stresses in the flange, the load carrying capacity of the flanges is derived

as follows:


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Literature study 27

Where: r y f , includes the reduction due to transverse moments in the flanges.

0

,

)(4,01

M

y

z xT

T yr y

f

M f

f f f

z M is the transverse moment in the flange (see paragraph 4.1).

Finally, the tension flange shall be verified against yielding as follows:

0,1,,2

,,2

d R

d E

N N

Where the design value of the normal force is:

21

,,

,

21,

2,,2

5,05,0 t t h

M N

A A

A N

w

d E y

d E

eff

d E

4.3 Web plate buckling

The use of corrugated webs makes it possible to use slender web plates. For high shear

stresses, two main buckling modes exist; one local governed by the largest flat panel and

one global involving more than one corrugation. Figure 4.4 shows these buckling modes and

an intermediate form: zonal buckling.

Figure 4.4: buckling modes (from left to right): local, zonal and global


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Literature study 28

According to annex D.2.2 of prEN 1993-1-5, the shear resistance due to buckling may be

taken as follows:

ww

M

w y

cd R t h

f

V

1

,

,3

Where: 0,11 M (see NAD-NVN-ENV 1993-1-1:1995)

buckling global for factor reduction

buckling local for factor reduction

g c

l c

c

,

,

4.3.1 Local buckling

Local buckling is governed by the largest flat panel with width amax (the maximum of a1 and

a2). For this buckling mode the folding lines keep straight.

The reduction factor l c, for local buckling may be calculated from:

0,19,0

15,1

,,

l cl c

Where the slenderness l c, may be taken as:

3,

,,

l cr

w yl c

f

The critical stress for local buckling cr,l can be derived with help of the mechanics scheme

of the largest flat panel of the corrugated web, shown in figure 4.5.

2

max

, 83,4

a

t E wl cr

with: 21max ,max aaa

With help of the German and Dutch standards

According to equation (410) of DASt-Richtlinie 015, the shear resistance due to local

buckling may be taken as follows:

wwd bww M

k y

l d R t ht h

f

V

,

,

,, 60,035,0

Figure 4.5: mechanics scheme largest flat panel


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Literature study 29

Peil [11] has done a thorough investigation into reduction factors with regard to the local

shear strength of girders with corrugated webs. It’s concluded that the reduction factor can

be increased safely up to 0,85. So, only with regard to local buckling the shear resistance

can be calculated with the following equation:

wwd bd R t hV ,, 85,0

The design buckling shear stress should be determined with the following slenderness (412):

l cr

d yrel plaat l c

f

,

,,,

3

The rest of the calculation can be carried out in accordance with 13.7 of NEN 6771, where:

E l k il cr k ,,,

in which: 35,5

(see paragraph 5.3.4)

2

max2

2

112

w

d

d E

t

a

E

with: 21max ,max aaa

The design buckling shear stress can be calculated as follows:

C

plooi

d plooi ,

where:

291,11210,10937,0

291,101

,,

,

rel plaat rel plaat

rel plaat

if

if C

3

,,

,

d w y

rel plooi plooi

f

in which:

291,11

291,17,0677,0474,1

7,000,1

,

2

,

,,

,

,

rel plaat rel plaat

rel plaat rel plaat

rel plaat

rel plooi

if

if

if


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Literature study 30

4.3.2 Global buckling

The reduction factor g c, for global buckling may be calculated from:

0,15,0

5,12,

,

g c

g c

Where the slenderness g c, may be taken as:

3,

,,

g cr

w y g c

f

The critical stress for global buckling cr,g can be derived from the orthotropic plate theory

[13] and is given by:

4 3

2,

4,32 y x

ww

g cr D Dht

where:

)2(312

)(

112

21

2

21

2

3

ncorrugatiooneof areaof moment nd aabt

I

lengthunfolded aa s

ncorrugatiooneof lengthw

w I E D

s

wt E D

t w y

y

y

w x

Although it is more correct to include the factor 21 in the denominator of the expressionof D x , according to prEN 1993-1-5 the factor can be omitted from the formula.

The critical stress for global buckling is valid for simply supported long plates. In [13],

Höglund gives a solution for restrained rotation along the edge. For fully clamped edges the

coefficient 32,4 increases to 60,4. But this assumption is hard to believe, because it is not

very likely that the flanges are rigid enough to provide a complete rotational restraint for such

a stiff plate as a corrugated web.

With help of the German and Dutch standards

In accordance with equation (410) of DASt-Richtlinie 015, the shear resistance due to global

buckling may be taken as follows:


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Literature study 31

wwd bww

M

d y

g d R t ht h f

V ,,

,, 60,035,0

The design buckling shear stress should be determined with the following slenderness (414):

otherwise f

for f

g cr

d y

l cr

g cr

g cr

d y

g cr

.

,

,

,

,

,

,

3

0,25,03

2

The design buckling shear stress can be calculated in accordance with 13.7 of NEN 6771, as

presented for local buckling in paragraph 4.3.1.

4.3.3 Interaction between local and global buckling?

Model Leiva

The main concern of Leiva in [9] is the interaction between local and global buckling, which

is based on observation tests. His idea is to consider this interaction by defining a combined

critical stress:

nn g cr nl cr cr 1

,,3,

Leiva discussed only n =1, but the equation has been written more general for later use.

This model of Leiva forms the basis of the ‘Combined model’, in which the combined critical

stress is used in combination with the reduction factor:

0,19,0

2,1

3,3,

cc

Model Höglund

The model of Höglund [13] has two separate checks, one for local and one for global

buckling. The reduction factors for local and global buckling, respectively, are given by:

0,166,1

64,1

,,

l cl c

0,15,0

5,12

,

,

g c

g c

The reasoning behind the two checks is that for local buckling a post-critical strength

(redistribution of stresses when the critical buckling stress is reached) is expected, which

should not be present in case of global buckling. This is reflected by c,l appearing linear and

c,g appearing squared in the reduction factors.


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Literature study 32

Further, Höglund has discussed the restraint from flanges to global buckling and he suggests

an increase of the buckling coefficient to 40 instead of 32,4 if a certain stiffness criterion is

met.

For giving rules to EN 1993-1-5, Johansson [7] has evaluated the above-mentioned modelsamong others. This evaluation shows that the theory of Höglund gives the best results,

especially for global buckling. For this reason, in the European standard is also mentioned

that two separate checks are sufficient and the reduction factor for global buckling of

Höglund can be used.

However, the reduction factor for local buckling starts the reduction at zero slenderness,

which is considered quite unlikely. For some low value of the slenderness, the shear yield

resistance of the web should be reached. For this reason, in prEN 1993-1-5 the reduction

factor for local buckling has been replaced with help of the model of Leiva:

0,1

9,0

15,1

,

,

l c

l c

4.3.4 Local transverse loads

In annex D of prEN 1993-1-5 is noted that for transverse loads the rules in section 6:

’Resistance to transverse loads’ can be used as a conservative estimate.

For unstiffened and stiffened webs the design resistance to local buckling under local

transverse loads should be taken as:

1

,

, M

weff w y

d R

t L f

F

where: eff L is the effective length for resistance to transverse forces:

y F eff l L

where: yl is the effective loaded length, see 6.5 of prEN 1993-1-5.

F is the reduction factor due to local buckling, see 6.4.

For the determination of the reduction factor F should be made use of the following

buckling coefficient:

2

max

26

a

hk w F with: 21max ,max aaa

The verification should be performed as follows:

0,1,

,

2

d R

d E

F

F (Compressive stresses are taken as positive.)


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Literature study 33

With help of the Dutch standard

The verification mentioned above could also be performed in accordance with the Dutch

standard, paragraph 14.2.2 of NEN 6770, in which the design resistance to local buckling

under transverse loads should be determined as follows:

f f

w

w

f

d ywd ud Rt h

c

t

t

t

t f E t F F

23125,0 ,

2

,2,, (at the supports)

f f

w

w

f

d ywd ud Rt h

c

t

t

t

t f E t F F

235,0 ,

2

,2,, (in other cases)

where: c is the effective length for transfer of the force.

and: 2,02

f t hc

4.4 Yield of the web plate

The shear stresses are only transferred through the diagonals and verticals of the corrugated

web. So, for corrugated webs the design resistance for shear should be taken as:

3

,

,,

d yw

d u z

f AV

with: www t h A

The corrugated web will be verified against yielding as follows:

0,1,,

,,

d u z

d s z

V

V where: d s z V ,, is the design shear force.

4.5 Lateral torsional buckling of the girder

It is not easy to determine the lateral torsional buckling resistance of a girder with corrugated

webs. For that reason only a few investigators have paid some attention to this failuremechanism and still no satisfactory solution has been found. So the standards doesn’t deal

with this topic and most of the constructors only verify the compression flange against lateral

buckling around the strong axe. According to NEN 6771 this procedure should be followed

when the following condition is fulfilled:

500010

23

12

g w

f

l bt

t h

However, in case of girders with corrugated webs, the thickness of the web should be

replaced by an alternative higher thickness tw

*. Further investigation into appropriate

restrictions is necessary.


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Literature study 34

According to 12.2.2 of NEN 6771 a laterally unrestrained beam subjected to bending around

the y-axe shall be verified against lateral-torsional buckling as follows:

0,1,,

,max,,

d u ykip

d s y

M

M

where: d s y M ,max,, is the design value of the moment around the y-axe.

22,min 21,,22,,11,, t t h f t b f t b M w f r y f r yd u y

kip is the lateral torsional buckling factor

According to Lindner (1990), a reduction factor for lateral torsional buckling k should be

applied for girders with corrugated webs. This reduction factor k can be used analogous to

the lateral torsional buckling factor:

4

1

1

M

kip k

Where M is the relative slenderness ratio for bending, which should be determined in the

same way as it should be done in accordance with 12.2-4 of NEN 6771:

ke

d u y

rel M

M

M ,, (If 4,0rel , then this check is not necessary.)

Determination ke M

The differential equation for description of lateral torsional buckling of an I-girder, loaded with

a constant bending moment, is:

0,,0

2

2

4

4

z d

d s

t d wad I E

M

dx

d I G

dx

d I E

This equation results into the following lowest value of the moment where lateral torsional

buckling will take place:

wad z d

g

t d z d

g

d s I E I E l

I G I E l

M 2

2

,,0

The first part of the equation refers to the torsional rigidity and the second part to the warping

rigidity of the girder. In NEN 6771 the second part is neglected, however for girders with

corrugated webs with usual dimensions this second part also has to be taken into account.

For that reason the theoretic elastic lateral torsional buckling moment can be defined as

follows:


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Literature study 35

wad z d

g

t d z d

g

red ke I E I E l

I G I E l

C k M

2

2

In which: red k is a reduction factor to be determined with 12.2.5.2 of NEN 6771

C is a coefficient to be determined with 12.2.5.3, mindful of the recentdfgdmodifications (NEN 6771:2000/A1:2001)

g l is the length of the girder between the gaffes.

z I is the second moment of inertia with regard to the z-axe

t I is the torsion moment of inertia

23213112

1t bt b I z

3

22

3

11

33

3

1

3

1

3

1

3

1t bt bt ht b I

wwt

Single span plate girder

For a single span plate girder, loaded with a constant bending moment, the following formula

can be derived:

t d g

wad t d z d

g

ki

I Gl

I E I G I E

l

M 2

2

1

Lindner and Aschinger [2] studied lateral torsional behaviour of girders with corrugated webs

and found that the torsional section constant J for a girder with a corrugated web doesn’t

differ from that of a beam with a flat web, but that the warping section constant is different.

They carried out tests at the Technical University of Berlin to determine the warping section

constant wa I of girders with trapezoidal corrugated webs:

2

2

min

E

l c I I

g

wwawa

where:2

2,1,

2,1,min m

z z

z z

wa h I I

I I I

22: 21

t t hhwhichin wm

wu

hbc mt w

8

22

2,1,

2,1,

2

1

32

1 6002:

y y

y ym

w

m

I I

I I

E a

wh

t aG

huwhichin


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Literature study 36

4.6 Buckling of the girder

According to 1.3 of DASt-Richtlinie 015, at most one of the flanges is assumed to be under

compression. It’s not clear that this assumption is also valid for girders with corrugated webs.

Even Eurocode 3 doesn’t give any restrictions to the normal forces in girders with corrugatedwebs. For that reason this assumption is not adopted in this analysis.

Analogous to 12.1 NEN 6771, a girder with corrugated web under compression (columns of

a portal, etc.) shall be verified against buckling around the weak axe as follows:

0,1,,,

,,

d ucbuc z

d sc

N

N

where: d sc N ,, is the design value of the compression force.

d w yd uc f t bt b N ,,2211,, )( is the design buckling resistance of the girder.

buc z , is the buckling factor for buckling around the weak z-axe.

For determination of the buckling factor is referred to 12.1.1.4 of NEN 6770. The buckling

factor depends on the relative slenderness, which can be calculated with:

u

z

E z

d uc

rel z F

N

,

,,

,

where:

d y

d u

f

E

,

z

buc z

z i

l , is the slenderness with regard to buckling around the z-axe.

in which: buc z l , is the buckling length with regard to the z-axe.

A

I i z z is the radius of gyration with regard to the z-axe.

Because of the small thickness, the contribution of the web to the second

moment of inertia around the weak z-axe will be overlooked. When only theflanges are taken in account, the following formula can be derived for the

second moment of inertia:

23213112

1t bt b I z

However, for A the total area of the girder should be taken into account:

ww t hw

st bt b A 2211

(s is the unfolded length and w is the length of one corrugation.)


37/48


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Literature study 38

5 DIFFERENCES FLAT-CORRUGATED WEB GIRDER

5.1 Introduction

In this chapter, an explanation on differences between girders with corrugated webs andgirders with standard plates will be presented. After a short description of several general

differences, a more extensive study will be made of the mechanical differences (5.3) and the

differences between the arithmetic methods and standards to use (5.4). Finally, it will be tried

to gain an insight into the economic favourable application field of girders with corrugated

webs.

5.2 General

Since the 1970’s, girders with corrugated webs are fabricated and at this moment building

with these girders is still in the making. In the first instance, the corrugated webs were

fabricated because of their good stability behaviour. Especially high corrugated websshowed a much better buckling behaviour and so webs with a very small thickness could be

made possible. For portals of halls, bridges and high-rise building, the low dead weight can

be a great advantage over the standard plate girders.

Because of the profile of the web, the fabrication costs for material (per kilogram) and

welding (per meter) are higher than for a flat plate girder. On the other hand, no transverse

and longitudinal stiffeners are necessary and the web is connected to the flanges with only a

single continuous fillet weld. An important difference with the (double welded) flat plate girder

is that it’s not permitted to use these girders under variable loading.

An other important difference forms the much lower stiffness of the corrugated web in

longitudinal direction. This low stiffness can be very useful in composite bridge building. Forbridges constructed with tubular members with in situ casted pre-tensioned concrete flanges,

the low longitudinal stiffness of the web causes a lower loss of pre-stresses.

For a long time the standard plate girders have been

used and a lot of experience has been obtained. So it’s

very easy for the structural engineer to make use of the

flat plate girder.The use of girders with corrugated webs

is still in the initial period and there is still a lack of

necessary information and experience. Even the

necessary standards and adequate software packages

are not available. An other disadvantage over the use of standard plate

girders can be the longer delivery times and minimum

orders conditions of girders with corrugated webs.

Finally, the aesthetics of the structure plays a part in the

possible use of girders with corrugated webs. Fig. 5.1: Aesthetic appearance


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Literature study 39

5.3 Mechanical behaviour

5.3.1 Basis of calculation

In paragraph 3.1 is concluded that, in static terms, the girder with corrugated webcorresponds to a lattice girder, in which the bending moments and the normal forces are

transferred only via the flanges, while the transverse forces are only transferred through the

diagonals and verticals of the lattice girder (in this case the corrugated web).

Because of the participation of the flat plate in the transfer of longitudinal normal stresses

from bending, the flat plate girder does not correspond to a lattice girder. The transverse

forces are only transferred through the plate, but the bending moments and the normal

forces are also partially taken up by the plate. So buckling of the web plate, due to these

normal forces and bending moments, shall be checked, while for corrugated webs only

buckling due to transverse forces shall be checked.

5.3.2 Transversal moments

For a girder with a corrugated web, the shear stresses in the web cause extra shear forces

(Fx, Fy), bending moments (My,1, My,2, Mz,1) and a torsion moment (Mx,2) in the flanges (par.

3.2). Because of the transversal moments in the flange, the yield stress will be reached

earlier in the flanges. This can be taken into account with help of a reduced yield stress.

Normal forces in the girder also cause strains in the flanges, which are hindered by the

corrugated web. So, also due to normal forces, transversal moments exist in the flanges.

For the flat plate girder, these transversal moments does not exist and it’s not necessary to

calculate with a reduced yield stress.

5.3 Arithmetic methods

5.3.1 Moment capacity

According to annex D.2.1 of prEN 1993-1-5, the bending moment resistance may be derived

from:

1

,11

0

,11

0

,22

, ;;min M

wr y

M

wr y

M

wr y

d R

h f t bh f t bh f t b M

So, the bending moment resistance of girders with corrugated webs, is determined by the

normal force resistance of the flanges.For flat plate girders also participation of the web plate has to be taken in account. Due to

the normal forces and bending moments buckling of the plate could happen. The calculation

of the moment capacity of a welded flat plate girder can be carried out with the cross-

section-reduction method in accordance with 10.2.4.2.3 of NEN 6771;

the effective width of the web can be calculated as follows:

11

bbeff

where:d

d

,1

,2

1

is the ratio between the lowest and the highest stress in the web.


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Literature study 40

whb

2

,

, 22,0

rel

rel

is the reduction factor.

in which:

xk i

d y

rel

f

,,

,

,

where: E x xk i k ,,,

The buckling coefficient can be determined with table 2.

For 01 :

ee

ee

bb

bb

6,0

4,0

2,

1,

With help of the effective cross-section, the bending moment resistance of the girder can be

determined:

e y

e yd s x

d s z

I M

,

,,,

,

in which: e y I , is the moment of inertia with regard to the y-axe.

e y

z ,

is the distance between the neutral line and the top of the girder.

5.3.2 Transversal moments

According to the German and European standards the effect of transversal moments can be

taken into account with help of a reduced yield stress:

0

,

)(4,01

M

y

z xT

T yr y

f

M f

f f f

Where: z M is the transverse moment in the flange. (see next pages)

0,10 M (see NAD-NVN-ENV 1993-1-1:1995)

5.3.3 Flange buckling

To prevent the compression flange of a girder with a corrugated web from local and global

buckling, it should be verified against buckling around the z-axe. For a standard flat plate

girder, the effective width of the compression flange should be determined with bucklingcoefficient k

= 0,43.


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Literature study 41

Local flange buckling will be influenced by the geometry of the web in a favourable way

compared to the flat web. The flange will be supported by the inclined parts of the web and

so, according to annex D.2.1 of prEN 1993-1-5, a safe approximation of the relevant length

should be a = a1 + 2a3. The buckling coefficients of such a plate, assuming conservatively a

hinged support along three edges of the web, is:

2

43,0

a

bk

with: b is the largest outstand from weld to free edge

31 2 aaa

For a geometry with a corrugation width of the same order as the flange width, the flange will

buckle in a mode of rotation around the centreline of the web, but with a stronger restraintthan for a girder with a flat web. This restraint will depend on the stiffness of the flange and

the flexibility of the web. The buckling coefficient will vary in the range 0,43…1,25 and for

simplicity a fixed value of 0,55 is accepted. So for this check, the slenderness parameter will

be defined with the following input:

255,0 1

bband k

According to DASt-Richtlinie 015, the effective width of the flange can be calculated with

help of buckling coefficient k = 0,6:

1

,1,

1

2407,30 b

f t b

d y

eff

5.3.4 Local web buckling due to transverse forces

Unlike the flat web plate, the corrugated web has two shear buckling modes; one local

governed by the largest flat panel and one global involving more than one corrugation.

Buckling factor

k According to 13.6 of NEN 6771, the critical stress for shear buckling in a flat plate can be

calculated as follows:

2

2

2

,

)1(12w

wd

d E E k i

t

b

E k

in which

k depends on the ratio between the length and the height of the web:


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Literature study 42

1435,5

135,5

4

2

2

for k

a for k

For local buckling of the corrugated web, the critical stress can be calculated with help of the

following equation (see paragraph 4.4.1):

2

max

, 83,4

a

t E wl cr 21max ,max aaawith

So, the used value of the buckling factor for local shear buckling of a corrugated web is:

35,5

)1(12

83,4

112

83,4

2

2

2

1

2

22

2

max,

d d

wd

w

E

l cr

a

t E

a

t E

k

For a long flat plate ( 1 ),

k also reaches the value 5,35. The difference between thebehaviour of the flat plate and the corrugated plate, must become clear in a different Euler

buckling stress.

Euler buckling stress E

The Euler buckling stress can be calculated with 13.6-4 of NEN 6771:

2

22

22

2

2

112

1

112

wd d

E

t E

b

t

b

E

For the flat plate: whb

For the corrugated web: maxab

The different width causes a great difference between the Euler buckling stress of the flat

and the corrugated web plate. The lower value of the width of the corrugated web causes a

much higher Euler buckling stress of the corrugated web. This forms the most important

difference between the flat plate girder and the girder with a corrugated web.


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Literature study 43

5.3.5 Global web buckling due to transverse forces

The critical stress for global buckling of a corrugated web is not calculated with help of a

buckling factor and the Euler buckling stress, but with the variable value:

4 3

2,

4,32 y x

ww

g cr D Dht

(see paragraph 4.4.2)

Because of the high orthotropic plate stiffness (Dx, Dy) of the corrugated web, the critical

stress for global buckling will be higher than for the flat plate girder.

5.3.6 Buckling of the girder

According to 12.1 of NEN 6771, a girder with a corrugated web and the flat plate girder

should be verified both to buckling around the weak z-axe as follows:

0,1,,,

,,

d ucbuc z

d sc

N

N

Because of the low longitudinal stiffness of the corrugated web, the normal stresses are only

transferred through the flanges. For this reason, the design buckling resistance of the girder

is:

d yd uc f t bt b N ,2211,, )(

While for the flat plate girder the whole cross-section can be used:

d ywwd uc f t ht bt b N ,2211,, )(

Due to these normal forces and possible bending moments, buckling failure of the plate

could happen. So only for the flat plate the buckling behaviour of the plate should be

checked in accordance with 10.2.4.2.3 of NEN 6771 (see also paragraph 5.3.1).

5.3.7 Lateral torsional buckling of the girder

Because of the small thickness of the corrugated web, the torsion stiffness has a low value.

However, the warping stiffness has a very high value (because of the higher warping sectionconstant and other common used dimensions), so the warping stiffness should also be taken

into account for calculation of the theoretic elastic lateral torsional buckling moment.

For girders with flat web plates, the lateral torsional buckling factor should be determined

analogous to the buckling factor, according to 12.1.1.4 of NEN 6770. For girders with

corrugated webs an alternative lateral torsional buckling factor can be used (par. 4.5):

4

1

1

M

kip


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Literature study 44

5.4 Application field

Usually, rolled I-sections are used in an economic favourable way for span lengths in the

range of 2…12 meter, with a ratio of h/l 1/30. For flat plate girders and comb girders, the

optimum span length is in the range of 8…20 meter, with a ratio of h/l 1/20 and for trussesthe optimum span length is in the range of 15…35 meter, with a ratio of h/l 1/10.

It is not easy to give a reliable estimation of the application range of girders with corrugated

webs. To know the economic application range, not only a girder with a corrugated web has

to be calculated, but for comparison also the competing girders, like the flat plate girder, a

truss and a comb girder. For all of these alternatives a lot of design choices have to be

made; A flat plate with a great thickness or application of transverse/longitudinal stiffeners?

The same design height or the optimum height? What kind of supports? Truss type? etc.

For a good estimation of the several girder types, it’s also necessary to do the calculation for

several span lengths.

For these reasons, the girder with a corrugated web will be compared with the competing

girders in a further stadium.

For a limited comparison between the flat plate girder and the girder with a corrugated web is

referred to [16], in which a calculation example is given for girders with a height of 2,0 meter

and a flange width of 350 mm. The girders in the example have a span length of 18,0 meter

and are loaded with two local forces of 500 kN. It is concluded that the transverse force

resistance of a girder with a corrugated web is ten times higher than the resistance of the flat

plate girder with the same web thickness. To reach the same strength, the web thickness of

the flat plate girder should be multiplied by 2,5.


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Literature study 45

6 STANDARDS AND GUIDELINES

6.1 Introduction

Based on analytical, numerical and experimental results obtained, several designrecommendations, guidelines, etc. have been published. However, the knowledge obtained

is limited to a few mainly European countries like Sweden and Germany. Table 6.1 shows an

overview of the standards and guidelines relevant to the production and design of girders

with corrugated webs.

Date Country Code number Title

1982 Sweden StBK N5 Light-Gauge Metal Structures

1991 Netherlands TGB 1990 Grondslagen bouwconstructies

- NEN 6702 Belastingen en vervormingen

- NEN 6770 Staalconstructies

- NEN 6771 Stabiliteit

- NEN 6772 Verbindingen

1990 Germany DASt-Richtlinie 015 Träger mit Schlanken Stegen

2002 Europe Eurocode 3 Design of steel structures

- Part 1.1 General rules/ rules for buildings

- Part 1.5 Plated structural elements

Only part 1.5 of Eurocode 3, the Swedish standard StBK N5 and the German DASt-Richtlinie

015 deal with girders with trapezoidal corrugated webs. For the general demands with regard

to the material properties, tolerances, etc, the European and the Dutch standard will be

taken into account. In the next paragraphs the contents of the above-mentioned standards

will be presented, except for the Swedish standard (unobtainable).

6.2 Dutch Standard – TGB 1990

In NEN 6700 ‘TGB 1990 – Algemene basiseisen’, all basic demands are given, which are

applicable for all kinds of structures irrespective of the sort of material. These fundamental

demands are specified in NEN 6702 ‘TGB 1990 – belastingen en vervormingen’ and in the

structure material bounded NEN 6770 ‘TGB 1990 – Staalconstructies’.

This standard with regard to st

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