Computer calculations of a complex steel bridge verified by model investigations

r I

,

, RIJKSWATERSTAAT

COMMUNICATIONS

No. 23

Computer Calculations

of a Complex Steel Bridge

verified by Model Investigations

(1

-

by IR. TH.H. KAYSER aod IR.J. BINKHORST

1975

.

.,.

RIJKSWATERSTAAT COMM UNICA TIONS

COMPUTER CALCULATIONS

OF A COMPLEX STEEL BRIDGE

VERIFIED BY MODEL INVESTIGATIONS

(APPLIED AT THE STEEL BRIDGE

SPANNING THE RIVER WAAL NEAR EWIJK)

by

IR. TH. H. KA YSER Data Processing Division of Rijkswaterstaat, Rijswijk IR. J. BINKHORST Department of Bridges of Rijkswaterstaat, Voorburg (upto 1-4-74)

Aacorresponaeflce s!zouta (je aaaresseafO

RIJKSWATERSTAAT

DIRECTIE WATERHUISHOUDING EN WATERBEWEGlNG

THE HAGUE - THE NETHERLANDS

The views in this article are the authors' own.

Contents

Page

5 Summary

6 Introduction

7 2 Description of the superstructure

7 2.1 General

7 2.2 Construction of the box girder

7 2.3 The main diaphragms

8 2.4 Erection

11 3 Overall view of the study

13 4 Methods for calculating box girders

14 5 Calculationmodel ofthe KOKER programme

14 5.1 Strain and stress representation

15 5.2 Interpretation of the stress results

16 5.3 Outline of the structure

16 5.4 Transverse diaphragms

17 6 Model investigation

17 6.1 Description of the model

18 6.2 Loading

19 7 _{Stress resuIts of the model investigation and the computer calculations}

19 7.1 Outline

19 7.2 (Jx-stresses 20 7.3 (Jy-stresses 20 7.4 1"Xy-stresses

21 7.5 1"xy in the web of the main diaphragm

22 8 Explanation of the stress distribution observed

22 8.1 Causes

22 8.2 Deformable transverse diaphragm

23 8.3 Transverse contraction

page 25 25 25 26 27 29 29 29 30 30 30 30 31 31 31 32 33 34 35 37 9 9.1 9.2 9.3 9.4 10 10.1 10.2 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 12 13

Tbe displacements according to tbe model and tbe computation Comparison of the displacements

Reduced flexural stiffness Shear deformation

Internal-external work approach Constructive modifications of tbe model Constructive improvements

New simplified model Results of tbe improved model

Model for the extra longitudinal diaphragm crx-stresses

cr y-stresses

"Cxy-stresses

Effect of the longitudinal diaphragm

Model with a longitudinal diaphragm and greater plate thickness Computer programme accepted

Concluding remarks Acknow1edgement Literature

Summary

The Department of Bridges of the Netherlands Ministry of Transport, Water Control and Public Works (Rijkswaterstaat) has recently designed a large cable-stayed box girder steel bridge which is now being built over the river Waal near Ewijk.

After the design phase several critical building phases were investigated using a finite element programme designed for box girder-type problems. Though the overall pattern of the stress distribution as given by the computer programme was expected, thc order and magnitude ofthe peak stresses gave rise to the need for an extra verification of these results. It was decided therefore to make a model investigation, the results of which confirmed the outcome of the computer programme.

Not only is a comparison of both results given but, in addition, the behaviour of the complex steel bridge is discussed from an analytic and engineering point of view in the article.

1

Introduction

National Highway 75 is to be built across the river Waal, one of the most important rivers of the Netherlands, in the vicinity of Ewijk (approx. 10 km west of Nijmegen) (see Figure 1).

A cable-stayed steel bridge has been designed by the Department of Bridges of Rijks-waterstaat, to enable National Highway 75 to span the Waal (see Figure 2), and the bridge is to have a length of 1055 mand a main span of 270 m.

For a number of reasons, a very wide box girder was chosen with a bracket on both sides. The box girder is supported at each pier by two bearings placed at a distance of 6.20 and 9.60 m. The distances were made as short as possible in order that optimal use may thus be made of the good soil data and the light weight of the steel bridge when designing the concrete substructure.

Itis of the utmost importance when designing such an enormously wide box girder that the exact stress distribution in the box girder be known, primarily in those com-ponents under compression; the safety against coUapse of the broad plate panels under compression is highly dependent on the stress distribution.

In order to gain a deeper insight into this phenomenon use was made of a finite element programme developed by and under the supervision of the Data Processing Division (Dienst Informatieverwerking, DIV) of the Rijkswaterstaat, to perform a ca1culation of the box girder near the supports. This part of the box girder was chosen because the greatest deviation from the linear theory is expected at this point and, in addition, the lower flange is under pressure at this point in two perpendicular directions. A judgment as to the stability required is only possible here with a good insight into the actual stress distribution.

The results of the computer ca1culations did not agree with what was expected on the basis of preliminary design ca1culations. The differences between the two methods of ca1culations were rather great and there was no obvious explanation for this phenomenon.

This is whyitwas decided to verify the computer ca1culation.

Partly because of the absence of other finite element programmes with the required properties a decision was taken to carry out a model investigation. The Dutch research institute TNO (The Central Organisation for Applied Scientific Research in the Netherlands) - I.B.B.C. (Instituut voor Bouwmaterialen en Bouwconstructies or lnstitute for Building Materials and Building Structures), was appointed to carry out a study.

2

Description of tbe superstructure

2.1 General

The overall length of the bridge is 1055 m, which is divided over span lengths of 75-90-90-90-105-270-105-85-85-60 m (see Figure 3). The main girder is formed bya steel box approx. 25 m wide with a maximum height of 3.50 m. The upper side ofthe box, together with the brackets extending on both sides, forms the roadway which has a total breadth between the parapets of 36.4 m. The roadway goes from the heart of the box under a cross-slope of I : 50.

The main girder is stayed in the main span by high-strength steel cables located in the heart of the main girder. The stays are supported above the river piers by steel towers which have a tapered box-shaped cross-section.

2.2 Construction of the box girder

The main principle to be kept in mind where the design of the box girder is concerned is that it should be almost entirely welded. The panels forming the box are chosen in such a way that a considerable element of repetition is present. During fabrication in the workshop and during erection at the site optimal use can therefore be made of the possibilities of series production.

The basic element chosen is a panel 2.40 m wide and 15.0 m long. Each panel is composed of a steel plate with a thickness which varies between 10 and 25 mm and which is stiffened longitudinally by welding to it 4, 3 or 2 troughs, respectively. The roadway panels have 4 stiffeners, the bottom flange panels 3 stiffeners and the web panels 2 stiffeners. The thickness of the steel plate is dependent on the location of the panel in the box. This classification is given in Figure 4 for a frequently occurring section. The box is provided with diaphragms composed of a steel plate 8 mm thick and stiffeners welded on one side, every 5 m in transverse direction. At the site of the bearings and the introduction ofthe cable forces main diaphragms will be found which are much more solidly constructed than the normal diaphragms. The plate thicknesses of the box are also much larger than normal at these points.

2.3 The main diaphragms

to one anotner. These me u e a type of main diapnragm W IC IS oeateO at ffie lTooa plains piers. This place in the box girder rorms the subject or the study on wilieh

lh.is article is based.

ln Figure 5 the conslruction or the main diaphragm is given together with a ponion

or the box girder. Tile main diapilragm consists or a plate with welded horizontal and

vertical stiffeners (a number of longitudinal and transverse splices were inserted for transport and erection purposes).

Tilis girder, which has the rorm or the inside or the box girder, is placed transversely

againsl the inside of the box. This means that the trough sliffeners are intcrrupted at the site of the main diaphragm.

Arter the main diaphragm ilas been wclded the trough stiJfeners are again made continuous by welded pieces oftrough whieh arc 300 mm long. The verticallongitudi-nal diaphragms are also welded at Ihe intersection or the web and the botlom flange'. Bearings are placed under the heavy stiJfeners in the figure to the lert or the wclded splices in the web.

2.4 Ereerion

All parts or the bridge, i.e. panels, diaphragms (such as are indicated in Figure 4),

etc. are carried over the road. The box: girder is divided up ioto 71 sections, cach with 1The function of the longitudinal diaphragms will hedisCllsscd later.

Photo 2. Segment en route 10 constructÎoll sÎle.

a Icngth of 15 m. The sections arc put together in a tcmporary workshop set up on the

slopc to the bridge which has been made suitable as an assembly yard. The temporary workshop has a lcngth of 120 mand a width of approx. 22 m (see Photo I). Whcn a

section is ready il is carried on bogies from the temporary workshop and pUI on a heavy-wcighl carrier. which moves the section to the construction sitc (sec Photo 2). Onec it has reached lhis location the seclion is hoislCd up onto two launching bogies and driven on launching girders which (see Photo J) arc lowered by rods while the

section is kept in a horizontal position on the launching bogies (sec Pholo 4). Then

thc section is adjusted to the previous scction and welded to it.

The maximum length ofthe free cantilever of the box is approx. 56 m which makes it ncccssary tO lIse temporary supports. Tlle erection mClhod used is shown in Figure

6. Work is being begun on the north si de and will heextended up 10 the middle of

lhe main span.

Thereaftcr, lhe temporary workshop and equipment are removed to the South Slde. afterwhich crection takes place therc, in the same way as on the ofth Side, until the

bridge can be closed.

3

Overall view of the study

After the preliminary design stage a linear-elastic calculation was made of a part of the box with the finite element programme KOKER which was developed for the final design by the Data Processing Department of Rijkswaterstaat. The initial calcula-tions were made for an erection stage which proved representative for the dimension-ing of the boxes near the supports, i.e. the free cantileverdimension-ing of the box from a pier or temporary support. A diagram of the superstructure with the dimensions and loads is given in Figure 7. The cross-section above the support piers is considered in the calculation as a symmetrical cross-section. In the preliminary design stage the box is calculated in accordance with the beam theory. The box is then considered longi-tudinally as a girder clamped in above the bearings with free cantilevering. The stresses which have thus been calculated 10ngitudinaIIy are drawn in Figure8. Based on ideal-ised conditions for the cIamping, according to the beam theory we do not see any stresses in the transverse direction. In reality the reactions are practiced in a concen-trated manner on two points under the main diaphragm, whereby stresses occur in transverse direction at the support. In such cases, at the main diaphragm, the box is also considered to be a girder according to the beam theory, which conveys the reactions from the box in transverse direction to the two bearings (see Figure 9).

The stresses thereby created are dependent on the effective width of fiange consider-ed (b).

In addition to this approach which considers the box longitudinally and in transverse direction as a beam, we also have to take into account the so-caIIed "shear lag effect" [14] and the effect of the transverse contraction.

These effects are included in the calculation using the KOKER programme. The results from using the KOKER programme in this situation showed considerable deviations near the pier or were completely contradictory, such as was the case with the shear distribution. As mentioned in the introduction this was a justification for verifying the computer results by an independent investigation.

Other computer programmes developed by third parties produced very disappointing results so that the performance of an experimental model investigation proved to be the correct procedure.

The model investigation of a, though somewhat simplified, box was accompanied by a computer calculation with the KOKER programme of an identical simplified box. The results obtained from both methods are thereby directly comparable. In the foIIowing discussion a comparative study of the two methods wiU be extensively treated anditwiII be seen that both produced identical results. At the same time an

explanation will be given, wherever possible, of the difference between the "beam theory approach" and the computer calculation. lthas appeared from this study that a calculation using the KOKER computer programme produces reliable results.

4

Methods for calculating box girders

The calculation of box girder bridges is often made with the help of the conventional beam theory as outlined in Figure 8 when drawing up a preliminary design. In point of fact these assumptions do not taUy and deviant stress distributions and side-effects will have to be borne in mind. This phenomenon has been recognised for some time now and has constituted an area to which extensive study has been devoted. Vlasov's work [I] has enjoyed wide renown. Blaauwendraad shows in [2] how, in certain cases, we may analytically resolve the behaviour of single·ceU or two eeU box girders. A semi-analytie approach has also been developed by Y. K. Cheung et al. [3,4,5 and6]. In aecordance with the fini te strip method the box is divided longitudinally into narrow strips (see Figure 10). An analytie approach was taken to its longitudinal behaviour and this was combined with a discrete method in transverse direetion. Using the latter a c1ear picture may be obtained of the behaviour of the box with deformable cross-sections. Computer programmes based on this method have been developed at Berkeley and other places and are being used.

Nonetheless, when using such a programme, we are soon eonfronted with such pro-blems as the ehangeable longitudinal stiffness and the description of complicated load patterns. They should be used more as laboratory instruments.

Ithas been clear for some time that the Data Processing Department should have ealculating equipment available which would enable the engineers at Rijkswaterstaat to verify the designs of box girder bridges. Calculation models based on the finite element method are particularly attractive for computer programmes in view of the flexibihty of sueh programmes. We shall not go into the background of the method and refer those readers interested to the literature [8, 9 and 10].

5

Calculation

model of

the KOKER programme

The Data Processing Division began the development in 1970 of the KOKER Pro-gramme on which a report has already appeared [11]. We shall briefiy explain the basic assumptions of the programme.

If we consider a cross-section of a box before and after a load has been installed we see that with the 3 degrees of freedom, v, w, and rp, the complete deformation of the cross-section can be described (Figure 11). Longitudinally, the degrees of freedom u and ware adequate, thereby enabling an approach to the deflection line through a polygon of straight line segments. The use of these degrees of freedom, u, v, wand rp implies that we are neglecting the f1ange's own f1exural stiffness longitudinally. Where steel bridges sueh as the one at Ewijk are concerned this is of no importance whatso-ever, but also in the case of concrete box girder bridges the flexural stiffnesses of the f1ange itself cancel that of the total structure.

5.1 Strain andstress representation

The rectanguJar plane (stress) element is used with eight degrees of freedom for the plane stress part ofthe implemented plate element. To adapt this element as a web of a box it has been modified to what is known in finite element circles as the "constant shear element".

Ithas been shown in Figure 12a what kind of strain representation may be expeeted in this element. The strainExis constant in the x direction and may proceed in a linear

fashionin the y-direction.Eyis constant in the y-direction and linear in the x-direction.

The angle of shearing yxy is assumed to be constant over the entire element in the

"constant shear element", and to be equal to that in the centre of the element. The strains are determined by differentiating the chosen displacement functions. From this, stresses are determined according to the relationship

1

(J'j

11

Y

(J E Y 1

T:

y

=

I - y

2

0 0

Ify

=

0 the same representation is true for the stresses as for the strains. This is no

longer the case wheny #- 0, especially where considerable stresses exist in two

The plate element in Figure 12b only allows for bending in thc y-direction and this behaviour may be described with beams in the y-direction and infinite stiff girders in the x-direction. The torsional reaction is described with a constant torsion panel.

5.2 Interpretation of tbe stress results

The results of the stresses are given for every element by the program me in the four angular points. In particular, the reflection of the stresses in plane shows considerable discontinuities at the element boundaries, which is not surprising in view of the con-stant description of(Jx in the x-direction anda yin the y-direction. We know that the

values are reliable in the middle of these sides.Itis usual thus to ave rage the stresses in the four elements around a joint. This is only correct if we are dealing in two directions with a regular interval between elements such as will be shown. In Figure 13 elements 1-111 are shown around a joint. Here we see that strains Ex and E_{y can be}

determined in joint 5 as

E/ = (E/5

+

E/8) * .5

E/ = (E/5

+

E/6)

* .

5

With these strains, it can be seen with regard to stress (Jx that:

If we determine the stress as an ave rage of theaxstress such as is indicated in angular

point 5 by the various elements we then see that:

I ax 11 ax E (J III = (E 58

+

VE 45) - - - : 0 x x y 1 _ y2 a IV = (E 58

+

VE 06)

~_

x x y 1 _ y2

(J I

+

(J 11

+

(J 111

+

(JIV

a 5= x x .~x~~_

x 4

IS

The average of the four element stresses produces the same result as the direct determination of the stress from the actual strain in joint 5. The same explanation is valid for stresses (Jy and rxY. This kind of verification of the results is only good if

the measurements of the elements are constant in x and y directions. As long as the measurements are approximately the same in a particular direction we arc not, how-ever, making any great errors.

5.3 Outline of the structure

When the Bridge Department requested that a calculation be made for the box girder bridge at Ewijk the programme had only just been completed. This request was one of the reasans why the programme was extended with new features and was more adapted to the requirements and wishes of the user. Special mention may be made of the calculation of orthotropic elements and transverse diaphragms.

Figure 14 gives a diagram of the cross-section of the box. The flanges and webs are longitudinally provided with stiffeners. Figure 15 shows how they are translated into orthotropic elements.

5.4 Transverse diaphragms

lnclusion of diaphragms in the computations creates no problems in the programme if the transverse cross-section is rectangular. But a trapezium is nceded for the trans-verse cross-section ofthe Ewijk bridge (see Figure 16). Such an element was not avail-able at the moment when the calculation had to be made. For this reason we trans-formed a rectangular constant shear element to the shape of the trapezium trans-formed by the four nodes (see Figure 17). The average deformation is thus weil described and the trapezium element is also in balance.

6

Model investigation

6.1 Description of the model

The model investigation was performed by the TNO Institute for Building Materials and Building Structures. The data of the model and the measurement results given here have been taken from [12] and [13]. To enable a rapid construction of the model and also to cut down expenses the model was simplified with respect to its real-life counter-part. The upper flange of the real box proceeds from mid-way under a cross-slope of 1:50 towards the outside (Figure 14). The incline has been omitted from the model, and the orthotropy of the flanges and webs, produced by box-shaped longitudinal stiffeners, has not been included. This dedsion was taken after a computer calculation for an isotropic model showed that the surprising stress distribution continues to occur even then.

The model was made of Perspex and represents the erection stage during which the bridge was been extended for 30 m beyond a pier. In Figure 18 a sketch will be found of this erection phase at the moment when the fol1owing section which is 15 m long is being hinged. The stress in the box itself and in the transverse diaphragm at the site of the pier are the reasons why this investigation was carried out. The model has been built on a scale of 1 : 50. The transverse cross-section of the model is seen in Figure 19. The thickness of the wall of the box has been made uniform so that in reality the plate thickness amounts to .016 m. The diaphragm at the site of the bearing has a comparable plate thickness of .024 m. The remaining transverse diaphragms are located at intervals of 5 m from one another and have a thickness of .008 m. At the pier the bridge is supported by two supports at a distance of 4.80 from the mid-point. For the purposes of this investigation the cross-section above the support pier was used as a symmetrical cross-section. Use is made hereofin the model when the loading is introduced. Strain measurements were made by"means of strain gauges in trans-versal cross-sections 1, 2 and 3 of Figure 20. Cross-section 1is located at the symmetric-al diaphragm. Cross-section 2 is hsymmetric-alfway between this main diaphragm and the first diaphragm of .008 m. Cross-section 3 is located 5 m fr om the bearing, near the first transverse diaphragm. The measurement sites are indicated in Figure 20. Strain gauges have also been placed in the longitudinal direction ofthe bridge in a few cross-sections. The strains have been determined at every point in three directions in the transverse cross-sections both above and below the plate sections.

6.2 Lomting

The model was supported where the real loading was applied, and the loading was introduced at the site ofthe supports. The loading cases have been given in Figure 2l. The fiTSt loading case is a pure bending moment of 150,000 KNm.

A bending moment of 150,000 KNm and a shear force of 5,000 KNm in the sym-metrical cross-section are obtained with two point loads in the second loading case. Finally, the behaviour was investigated under a torsional moment of 120,000 KNm. The model has also been calculated with the KOKER programme for the above mentioned loading cases. These computer calculations and the measurements will now be discussed and compared. Whenever possible we shall attempt to explain the stress distribution observed.

7

Stress results of the model investigation and the computer

calcula-tions

7.1 Outline

The measurement and calculation yield nearly identical results for the loading cases with a pure bending and torsional moment. For both loading cases the stresses did not deviate from what can be expected from the elementary theory. For our purposes, the results which are interesting are seen in loading case two, where two point loads, which cause a bending moment plus a shear force at the pier, are applied at the ex-tremities. The resuJts are given as follows:

(Jx : stress in the longitudinal direction of the box (Jy : stress in the transverse direction of the box

, xy: shear stresses in the box.

The distribution of these kinds of stresses in accordance with the beam theory has al ready been given for this loading case in Figure 8.

The longitudinal stresses (Jx were calculated by dividing the bending moment by the

moment of resistance of the entire cross-section of the bridge. The shear stresses, xy are the direct result of this distribution of (Jx stresses because of considerations of

balance.

In similar manner the(Jy stresses also result once again in the circumferential direction

from the distribution of, xy' If, xy is longitudinally constant, as is so in the loading case seen, (Jy is zero over the entire cross-section. The circumference of the walls of

the box girder do not therefore change longitudinally.

In Figure 9 the shear stress distribution is considered, in line with the beam theory, as loading on the transverse diaphragm above the bearings .The moments which thus result in the diaphragm, the shear forces and the normal forces have been indicat-ed in the figure in question.

7.2 (Jx-stresses

(Jx-stresses have been plotted graphically in Figure 22 as they appear fr om the measure

ments and calculations in cross-sections 1, 2 and 3. Itcan be seen in cross-section 1 that strong peak stresses develop in the upper flange at the web and at the point where the bottom flange meets the web. Both the computation and the measurement model remain in the linear-elastic area, so that stresses greater than the true yield stresses

fiom steel may occur. Above the pier support the middIe parts ofthe upper and Jower flange become partly detached from the stress transfer in accordance with what is known about shear lag in the literature. At a distance of 2.5 m from the pier supports these stress concentrations are less and at a distance of 5 m there is a stress pattern which corresponds with what we would expect on the basis of the beam theory. The stress concentration at the webs, in the vicinity of the pier supports, was known, but the extent of these stresses was nonetheless a surprise.

In Figure 23 we find the distribution of(Jx longitudinally. The stresses are given in the

upper flange where the vertical web is joined and in the lower flange at the intersection with the web. For the purposes of comparison the line for (Jx is also indicated just as

we calculated it following the beam theory. We see that a peak stress occurs with respect to this value, which may become a factor 2-2,5 times greater.

The resuJts for loading case 2 for stress(Jy in transverse direction are plotted in Figure

24. We again find in cross-section 1 in the transition from the web to the lower fiange a high stress concentration with stresses of the same size as the(Jx-stresses. This means

that considerable compression stresses occur in the lower flange and the web in two directions, which is quite unfavourable as far as the stability is concerned. Here, too, we again see a smooth stress distribution for (Jy after 2.5 m.

7.4 Txy-stresses

Both (Jx and (Jy stresses correspond to what was expected although the peak stresses

were quite strong. The stress distribution of TXY' on the other hand, varies greatly

from what is predicted by the beam theory (Figure 25). The shear stress in the upper fiange in cross-section 1 reacts in just the opposite way from what is known from the elementary beam theory (Compare with Figure 8). This is even more the case in the 10wer flange, where the shear stress increases sharply from the mid-point on up to where the web begins. At this point the shear stress sign suddenly changes.

The total vertica1 component of the compound shear stresses in a transverse cross-section is expected to yie1d continuously the shear force to be transmitted. This equili-brium control tallies with the computer calculation in every cross-section. In the model measurements the vertica1 balance could not always be found again. There are a number ofreasons for this which are mentioned in [12]. The most important was that an equivalent stress representation was found in the model which was similar to that of the calculation. In particular, the change in the Txy sign where the lower fiange meets the web also showed up again in the model.

Inthe shear stress representation we also see that at some distance from the main diaphragm the stress distribution follows the beam theory. At 5 m from the bearing this is almost the case.

7.5 't"xyin the web of the main diaphragm

The shear stress and the resulting computations for the shear force in the main dia-phragm have been plotted in Figure 26 in accordance with the computer calculations. A part of the 2500 kN great shear force is transferred in the tapering part. The rest is transmitted by way of compression stresses (jyin the web and near the bend given

to the web of the main diaphragm. Here, too, the computer calculation indicates complete balance.

8

EX]ltanatioD

of me stress distnoufion observed

8.1 Causes

The stress distribution in the box girder near the bearing is particularly surprising with respect to the shear stresses. Inthe upper and lower flanges the shear stress has a sign which is the opposite of the result of the beam theory and on the transition from the lower flange to the slanting web there is a large increase in the shear stress, of course. We were intrigued about this behaviour of the box girder and were eager to learn the cause.

There are three causative factors in this interplay of forces. In any case we have the well-known phenomenon ofwhat is called shear lag in the Iiterature.Inaddition, there is an effect of the transverse contraction and, thirdly, we suspected that the behaviour was determined to a considerable degree by the fact that the transverse diaphragm at the bearing is deformable. We shall now successively discuss the three cases that one of the causes is present while the other two are not. Then it is clear what happens then when all three effects simultaneously occur.

8.2 Deformahle transverse diaphragm

The effect ot the rigidity of the transverse diaphragm is illustrated in Figure 27, where the box girder was ca1culated four times, with the main diaphragm being 0.008, 0.08 and 0.80 m thick, respectively, and the fourth time infinite stiffness was assumed. The shear stress distribution is plotted for these four box girders for cross-section I.

H may be clearly seen in this figure how the stiffness of the main diaphragm affects the shear stresses in the box girder. As the diaphragm becomes stiffer the distribution shifts increasingly in the direction of the solution offered by the beam theory. There , remains, however, a difference and there still exists a lag in the shear stress

distribu-tion, even though it has been reduced to a quarter of the original value.

For an explanation of the effect of the stiffness of the transverse diaphragm we take the stress distribution of the beam theory as our point of departure and claim the absence of transverse contraction. The transverse diaphragm must carry off a force of 2500 KN from the slanting and vertical walls to the bearings and will thereby be-come deformed as is sketched in Figure 28a.

The fibres have a tendency to become longer in the upper part of the diaphragm and those in the lower part of the diaphragm shorter. The diaphragm is, however, an

entity with the box flanges which resists these deformations of the transverse diaphragm Both components will therefore exert forces on one another in the form of shear stresses. We have seen that for the upper and lower flange the direction of activity of the shear stress is the opposite of what is found according to the beam theory (Figure 8). As the transverse diaphragm b('comes weaker the shear stresses will become greater, a situation which is illustrated c1early in Figure 28b. The diaphragm reacts as a girder with a given effective width of the flange B from the upper and lower flange for trans-rnitting the shear force from the webs to the bearing. A thick transverse diaphragm takes up much of the total moment itself, so that the stresses(Jy remain slight in the flange. With a thin diaphragm the flanges must make a large contribution to the total moment and (Jy is much greater in the flanges. The shaded areas in Figure 28 are a measure for the shear stresses between the flange and the web, from which it may be immediately seen that with a stiff diaphragm weaker shear stresses occur and the solution is more compatible with the beam theory.

A sketch ofthe above-mentioned flange width B is given in Figure 29, enabling another phenomenon to become visible. The effective width of the flange B of the transverse diaphragm must be near zero in the transition from the lower flange to the slanting web and from the slanting web to the vertical web. At the site ofthe transitions(Jythere must always be zero. A sketch appears in Figure 29 of the shear stresses which then occur at the link-up of the lower flange and the slanting web at the transverse diaphragm. Once again we see in this figure that the shear stresses in the lower flan ge are the opposice of what the beam theory tells us and also that a great decrease in rxy takes place near the intersection of the lower flange and the slanting web. This also occurs as we have seen in the model investigations. The entire phenomenon is accom-panied by considerable stresses (Jy in the diaphragm at the site of the intersection from the lower flange to the slanting web, for the bending moment in the transverse diaphragm must be carried there in its entirety by the diaphragm alone.

8.3 Transverse contraction

Of the remaining difference between the computer results and the beam theory the greatest part is caused by transverse contraction. We start with the beam theory solution with the assumption of a undeformable diaphragm. Compression stresses(Jx

predominate longitudinally in the lower flange and in the web. Deformations are prevented in a circumferential direction by a stiff diaphragm. Compression stresses(Jy thus appear as a result of the transverse contraction. Stresses (Jy also have this symbol because of the susceptibility to deformation of the transverse diaphragm, so that this effect works in the same direction because of transverse contraction. And(Jy must be zero for this effect as well at the intersection of the lower flange and the web, so that a rxy distribution is caused again as can be seen in the sketch in Figure 29.

8.4 SItear lag

The deformation of the transverse diaphragm and the transverse contraction explain most of the great discontinuity in the shear stress at the site of the intersection of the lower flange and the web. Nevertheless, even a smaIl discontinuity still occurs near an infinitely stiff diaphragm and the absence of transverse contraction. Shear lag is the only possible explanation for this. Because of the stiff diaphragm the circum-ferential stress (Jy is of course zero at the site of the diaphragm, but compression stresses(Jy may weIl occur at some distance from the diaphragm. They occur because of theTxytrend in the longitudinal direction of the bridge. A sketch is given in Figure 30 of the course of the compression-stressed(Jy in the lower flange. For the sake of balance in the (Jy direction a distribution of rxy pressures is thus needed in the section along the transverse diaphragm. These shear stresses which are, moreover, slight, appear to show a high incidence at the intersection of the lower flange and the web.

9

The displacements according to the model and the computation

9.1 Comparison of the displacements

The vertical displacements from the girder have been measured in the model for the loading cases of the constant moment and the point loads at the extremity. The dis-placements have been plotted longitudinally in both cases in Figure 31. The displace-ments at the end of the girder correspond closely for the measurement and the compu-tation.In addition, the displacement was indicated for both cases as computed

accord-ing to the well-known beam theory. For the constant moment we see a small difference only at the extrernity between the beam theory and reality, which is caused by the fact that in the calculation and the model the moment is introduced as a torque which acts on the web as two opposite forces. Only at a given distance from the final cross-section, at the point where the load is applied, wil! the stress distribution be complete1y initiated for the constant bending moment.

When loading with a point load at the end, a displacement is seen which in reality is a factor 1.7 times greater than could be expected on the basis of the beam theory. This magnification of the displacement by a factor of 1.7 may be explained, on the one hand, by a smaller flexural stiffness of the box at the bearing and, on the other hand, by the shear deformation in the flanges and web. These effects will now be successively explained.

9.2 Reduced f1exural stiffness

When considering the(Jxwe saw that in the vicinity ofthe supports the middle section of the flanges withdraws from the (Jx transfer. This phenomenon, which is known as shear lag, causes the box to have less flexural stiffness in that area. This leads to a situation in which, near the bearing, a greater curvature develops which produces extra vertical displacements in the rest of the box. The stress representation for (Jx

longitudinally has again been given in Figure 32. The shear lag develops in the last 5 m. Proceeding from several assumptions we calculate the deflection at the extremity. We assume the normal flexural stiffness of the box to be EI.

In the cross-section at the bearing(Jxis approximately 2 times larger than would be expected on the basis of the beam theory , so that the flexural stiffness is apparently half as great at this point. We assume that it has been reduced to 0.5 EI.

with a 1engthI,which is 10aded with a point load P. The deflection at the extremity is equal to the static moment of the reduced moment area with respect to the free outer end of the beam.

For a constant EI, this becomes:

PI P/3

°

=.1..

-·1·1-.1=

.333-B 2 EI 3 EI

Because of the reduced EI at the bearing there is also:

PI PI3

°

= .1. . -

.

.l.I,uI= .

079-2 EI 6 18 EI

For the purpose of our box we must calcu1ate the deflection as:

PI3 PI3 •4125000303

OB

=

(.333

+

.079)-

=

.412 -

=

=

.12 m

EI EI 2. 110 2. 16 This value has been given in Figure 31b.

9.3 Shear deformation

The vertical displacement at the outer end is also influenced by shear deformation. The shear stress distribution according to the beam theory at a shear force of 5000 kN is shown again in Figure 33. How the shearing angle y has been distributed over the section is shown in Figure 34. Because of the shear force deformation the cross-section of the outer end tends to become deformed. We first act as though this does not present any problems and allow the deformations to develop while we keep the box in a horizontal position (imaginary experiment). In Figure 35 we have folded out the various parts of the box. With respect to point A the displacements of the remaining points have been plotted, such as they can be calculated from the shearing angles from Figure 34.

Itis known thatwarping occurs in every cross-section as a result ot the shear deformation. From Figure 35 we can derive that the upper flange moves forward and the lower flange moves backward.

In the symmetrical cross-section above the bearing warping is completely prevented. This means first of all that the cross-section must be built up here from straight line segments. On the upper flange, webs and 10wer flanges an equilibrium system of stresses will therefore have to develop to straighten out the deformations once again. The dotted line in Figure 35 indicates this. The sum of the shaded areas should be

about zero. This equilibrium system of stresses is causing the peak stresses in cross-section 1. So it is the prevented warping of the cross-section that is responsible for the resulting(Jx-distribution and not what is generally known as shear lag.

Not only should the points ofthis cross-section!ie on a single surface but this surface must, moreover, remain vertical because of the necessity of maintaining symmetry. If we fold back the components of the box from Figure 35 then the surface of the cross-section appears to farm an angle with the vertical direction which is equivalent to 0, the difference in horizontal displacement between the upper and lower fianges, divided by the heighth of the box. To get a vertical cross-section here the box has to rotate on an angle rp which is equal to:

6.810-3

rp= 3.25

This produces an extra displacement at the terminal cross-section as a result of the shear deformation :

6.810-3

3.25 x 30 =.063 m.

The total displacement is now, as a result of the flexion and shear deformation: 0= OB

+

OD= .12

+

.063 m= . 183 m (see Figure 31b).

Itwas calculated that a displacement of. 167 had occurred. The displacement calcu-lated here is therefore too great. That is correct of course. The shear stress distribution assumed does not exist over the entire length.

After the cross-section at the extremity a transitional area exists in which the shear stress distribution caused by the loading has to be built up. This is also the case at the clamped end, where, for other reasans, a completely different shear stress distribu-tion is seen.

9.4 Internal-external work approach

The displacement by shear deformation can also be computed in another way than was done in Figure 35. We are referring to the use of an internal-external work approach which is finer to use in aetual praetice. Hereby we ean derive the stiffness against shear force deformation. Use is made of the equation:

IJ

.2

D

2 d A =

-G GA_D

G AD

A

-shear stress sliding modulus effective surface

surface of the transverse cross-section

The integral which represents the accumulated internal work may be simply calculated from the shear stress distribution of Figure 33. The only unknown then isAD'which is the effective cross-section, which in this case is equal to:

(5000)2 2

-7.-S-*-1O-;;8 = .032 m

The shear angle Y can then be calculated.

o

5000 3

Y = - - = 8 = 2. OS

10-GAD •75 10 .032

As vertical displacement resulting from the shear deformation we find then at the extremity:

OD=Y 1=30*2.OS.1O-3 _=.062m

10

Constructive modifications of the model

10.1 Construetive improvements

The results of the model measurement corresponded very dosely with the results of the computer calculation, such as may be ascertained from the previously given stress distribution. The box responded therefore in the same way as has already been indica-ted by the finite element model. The stress distribution observed with the high peak stresses at the site of the transition from the lower flange into the web caused the bridge designers to take constructive measures in order to lower the peak stresses. To this end, extra longitudinal diaphragms were added at the lower part of the box over a relatively short length as indicated in Figure 36. In the model discussed earlier the box was made using isotropic plates with a thickness of 16 mm over the entire length of the model. In reality the box is not composed at all points of the same plate thickness. In particular, a greater thickness is used at the support for the lower flange and the web. The result has been investigated in the model by using, near the support, a thick-ness of 32 mm over a length of 2.25 m on both sides. This has also been indicated in Figure 36.

10.2 Newsimplified model

In order to make a study of the various above-mentioned aspects, a new and simplified model was made. This was possible (measurements were made at fewer places and the profile of the box has become shorter) because only a detail of the box (the peak stresses in the lower plate) was important. The cross-section of the model remained identical to that of the original model. The length was reduced by one-half. As was the case in the first model, the measurements were made using strain gauges. The study was restricted to a determination of the stresses occurring at loading case 2 (moment

+

transverse force at the bearing). The number of points where measurements were made was also reduced. Calculations have also been made for this model with the KOKER programme. Use of another plate thickness over the length of 2.25 m did not present any problem for the programme. It was not possible to input the extra longitudinal diaphragm in the same way as was done in the model.ltwould have cost too much time to change the programme. Therefore, the longitudinal diaphragm was represented in the finite element model as shown in Figure 37. The longitudinal diaphragm is used over the entire height, in contrast to what was the case in the model. Looking at the results which will now be discussed, we shall see that extension of the extra longitudinal diaphragm to the upper flange does not result in any essential deviation with respect to the model results.

11

ResuIts of the improved model

11.1 Model for the extra longitudinal diaphragm

Although the results ofthe model measurements are only available in the lower flange and in the web for cross-sections 1 and 2, we are nonetheless including, for the discussion on the effect of the improvements made, the complete stress distribution over cross-sections 1, 2 and 3, such as they were calculated with the programme. This wil1 be done in order to give a picture of the changes in the stress distribution in these parts of the box. First of all, we shall turn our attention to the stress distribu-tion in the box with the extra longitudinal diaphragm.

11.2 (Tx-stresses

In Figure 38 the distribution of stress(Txfor cross-sections 1, 2 and 3 is plotted graphic-ally as has been calculated by the computer. Where the results of the measurements are available they have also been included. In the very first place, we should like to point out that model and computation results once again show a wide measure of concordance. In additionitappears that the longitudinal stresses(Txare nearly equal in cross-sections 1 and 2 in the bottom of the box. By addition of a longitudinal dia-phragm we displace as it were the clamped point. We see in the upper flange in cross-section 1 a low stress concentration at the longitudinal diaphragm, which, in reality, will not occur in view of the fact that the longitudinal diaphragm does not connect with the upper flange. The effect ofthe stress distribution for(Txcan only be thoroughly appreciated by a comparison of the results with and without the longitudinal dia-phragm for cross-sections 1 and 2 in Figure 39.

The extreme peak stress seen in the original model disappears in cross-section l,and the(Txstresses seen in cross-section 2 as a result of the extra longitudinal diaphragm become higher than was originally the case. The box responds in fact to the relatively rigid diaphragm as a sort of clamping.

11.3 (Ty-stresses

The distribution of the stress in transverse direction(Jy has been graphically plotted

for the various cross-sections in Figure 40. Here too we see that the stresses are equally strong in cross-sections 1 and 2 on the bottom surface of the box. The situation is even such that in cross-section 2 stress (Jy is greater in the bend between the lower flange and web than in cross-section 1. We shall return to this point during the

dis-cussion of shear stresses. By way of comparison we again show in Figure41 the stress distribution for (Jyin the models with and without longitudinal diaphragm. Here too

the stress peaks in cross-section I have been considerably reduced by the addition of the longitudinal diaphragm, whereas in cross-section 2 the stresses have as a conse-quence increased. The evolution of(Jy in the webs and the lower flanges is closely

connected with that of the shear stresses L xy'This wiU be discussed later on.

11.4 Lxy-stresses

An outline of the evolution forL xyin the model with the extra longitudinal diaphragm

has been given in Figure 42. Results of such measurements are not available. We now see that in cross-section I, the shear stress sign is the opposite of what we would have expected on the basis of the beam theory, not only in the upper and lower flanges but also in the web. The change in the shear stresses from positive to negative is now seen again in cross-section 2. In order to show the significant difference between the evolution of the shear scress in the model with and without longitudinal diaphragm, these distributions have been indicated in Figure 43 for cross-sections land 2.

11.5 Effect of the longitudinal diaphragm

The alteration in the evolution of(JyandL xyin the first and second cross-sections is

the result of the fact that the total shear force present in the bended web of the box as a result of the extra longitudinal diaphragm can be transmitted in an entirely different way to the bearing. Figure 44indicates the way in which the shear force is transmitted to the bearing. The extra longitudinal diaphragm alone accounts for 1780 kN of the total shear force of 2500 kN which is to be transmitted. The remaining 720 kN is transmitted by way of the bended web to the web of the main diaphragm above the bearing. By adding this extra longitudinal diaphragm the box is in a position to transmit a considerabie part ofthe shear force to the longitudinal diaphragm before the last cross-section.

Before the last cross-section, the web can transmit vertical forces to the longitudinal diaphragm by way of(Jystresses which absorbs them as shear stresses. Relatively high

stresses accompany them in cross-section 2 in transverse direction, as \Ve have pre-viously seen. Moreover, the (Jy stresses in cross-section I are considerably reduced

because of the fact that a much smaller moment now has to be transrnitted at this point.

11.6 Model with a longitudinal diaphragm and greater plate thickness

With the results such as we have seen for the model with the longitudinal diaphragm and the explanations for the modifications in the stress distribution which then occurs for(Jx'(JyandL xy'the results of the introduction of a thicker plate in the lower flange

and the web combined with {he IongitudinaI diaphragm wiJl hardly produce any surprises. For the sake of completeness we shaJl give these results in Figures 45,46 and 47.

As far as the stress representation for (Jxis coneerned we see that in eross-seetion 1

the stress is redueed by the extra thiekness of the plate. The same thing ean be seen for (Jx and Txy in eross-section 1. In this connection we should like to add that the

stresses in cross-seetion 2 are valid for the area in whieh the thiekness of the plate in the lower f1ange and the web is again equal to .016.

11.7 Computer programme accepted

The goal of the model investigacion was to verify the results obtained from the KOKER computer programme. As can be seen from all previous comparisons of stress distribution, model measurements and computer calculations show a wide measure of agreement. This has led to a situation in which the finite element model in the programme has now been completely accepted by the designers as part of their equipment for analysing and detailing box structures. In the foJlowing chapter we shall discuss the way in which the results of the programme are being developed and put to use.

12

Concluding remarks

After the previously described study, the KOKER element programme has been used on many occasions for the calculation of superstructures of the Waal Bridge at Ewijk. By using this programme the stresses computed are compared with admissible stresses and are used for the verification oftht" stability ofthe elements under compression [14]. Determination of the definitive profiling of a particular part of the box is, however, a repetitive process, in which the calculation must be repeated just as long as is neces-sary until the stresses calculated fulfil all the criteria.

The entire process has been carried out for approx. 10 different parts of the box at the Ewijk bridge.

Many measurements have to be made therefore for which the designer must have the computation results at his disposal on short notice to enable a continuous design process to be made.

The use ofthe KOKER finite element programme has madeitpossible for the designer to build such a complicated, three-dimensional structure within areasonabie period of time.

The experience gained during the process of designmg the Ewijk bridge will be shortly incorporated in the programme in order to makeitas suitable as possible for use in the design phase of a bridge.

13

Aekno'Wledgemellt

The model measurements were performed at the TNO Institute for Building Ma-terials and Building Structures, under the direction of Mr. W. J. Beranek and Mr. C. Gouwens.

We should also like to expressOUf thanks to Dr.J. Blaauwendraad for his

Literature

[1] VLASOV, V. Z. - Thin walled elastic beams. Israel Program for Scientific Translations, Jerusalem,

1961.

[2] BLAAUWENDRAAD, J., KOETSVELD, M. J. VAN - Torsie en dwarsvervorming in rechthoekige koker zonder dwarsschot. DIV-nota, maart 1972.

[3] CHEUNG, Y. K. - The finite strip method in the analysis of elastic plates with two-opposite

simply supported ends. Proc. Inst. Civ. Eng. 40.1.7, 1968.

[4] CHEUNG, Y. K. - Finite strip method of analysis of elastic slabs. Proc. Am. Soc. Civ. Eng. 94,

EM 6,1968.

[5] KAYSER, TH. H. - Lastspreiding in enkele moderne brugtypen Il. TNO-IBBC rapport no. BI-70-20j07.1.311j07.1.312.

[6] KAYSER, TH. H. - (B)ruggespraak met de computerJ. Cement 1971, No. 6.

[7] MEYER,c.,SCORDELIS, A. C. - Analysis of curved folded plate structures. University of

Califor-nia, Berkeley Report No. UC SESH 70-8.

[8] BLAAUWENDRAAD, J., KOK, A. W. M. - Elementenmethode voor constructeur. Agon Elsevier, Amsterdam 1973.

[9] ZIENKIEWICZ, O. C. - The finite element method in Engineering Science, Mac Graw Hili, Londen, 1971.

[10] VELDPAUS, F. E. - Numeriek gereedschap ten behoeve van dunwandige balkkonstrukties. Diss. T.H. Eindhoven, 1973.

[11] BLAAUWENDRAAD, J., KAYSER, TH. H. - (B)ruggespraak met de computer Il. Cement 1971, No. 11.

[12] BERANEK, W. J., GOUWENS, C. - Onderzoek naar de spanningsverdeling in een stalen

koker-brug(I).TNO-IBBC rapport no. BI-71-360j06.2.147.

[13] GOUWENS,c.,BRASSINGA, H. E. - Onderzoek naar des~nningsverdelingin een stalen

koker-brug(Il).TNO-IBBC rapport no. B-72-96j06.2.147.

[14] MERRISON REPORT - Inquiry into the Basis of Design and Method of Erection of Steel Box Girder Bridges. Department of the Environment Scottish Development Department Welsh Office.

d .!S 0-G' ~ ....: Ij) .... ::l ~ ü:

N,l, . , -i-1600 ';'11000 ~i~ 1'lo.Q.üO

-:-~

WAAL NAP

Perspective sketch of the bridge. Figure 2. NAP

-3600~15100 5500>:20055 , i~

~-n7ÎT

~

..

r---

IIlI

L:z:=..

~lif---i

__

J!l~n

;;c:=J

Z.l,

~

~~

/"/~/

~~~~~ ~ 75.000 _

r

90000

I

90000

_T

90000

I

105000

I

r

85000

r

60000 = 270.000 _-- _ • I. 105.000 85000 1.055.000 H.A.P~

~

'\~ooo

37110 15

oT

0, I/)

M1-==--=---=-S;:::~=.=J====r::===I===-:I1~--~rl:::==r:===:r:~~2"-

_

-Figure 5. Construction of the box girder at the site of amam. d·Jap ragm.h

.,..,.

~::::

t

I

~/~

__ - ---1

I / . r

-o

1

1

c

Figure 6. Overall view of erection work.

:4>

~,

SIMPLIFIED LOAD CASE ERECTION STAGE ~ 1.25

.1

14 3.25 ~ ... "'V

,

•

-~I' 0 0

0

IX) CD ~ <Ó tri <0

....

C\I (Y')

lr

.... "'V

.,

z

~

o

o

10 C\I

1======

30.00

~=====~ç::~~II~

~---.,;;:::::::-_---~

1---lr1.

5000 kN

z

~

o

o

10 C\I

<Tx

51.6

l_ _

~5000kN

CD

30 m

j

~

...

...

-

...

~

_...

..

...

-

...

_...

~

--3Q6

--48.6 N/mm 2 Nfmm 2

---

ioo----

h

36.0

p..---

L14B

t.--- 1----

4

4 - -

1--_-I

ft

150.8

I

I I

l'xy

51.6

I

_I

I

50 50 100 100 +

t

0 +

t

0 - - BEAM THEORY

/

/

I

~

,

/

/

,

T

/

- - -

- - - -- - - -

__ 1I

,

_{N/mm 2} J Wmm2 ~--

10----

~---~---~---1 0 - - -

-50 50 150 100 100 +

t

0

+

t

0

t

l.xy

1;:;:;:;:::::::::1

50.8 51.6 -17350 MOMENT 51.5 I---.,f---I--+-:-~J?o~ +1100 (tNm) SHEAR FORCE 2500 :::::::::::::::: ::::.:.:.:. :

.

(kN) 3600 NORMAL FORCE

...

,

.

1/2.(b) (kN)

w~

Figure 10. A box divided longitudinally in strips.

z

~u

w

y

[ +

-'Yxy

=

CONSTANT

Figure l2a. The strain-representation of the plate element.

mx=o

c

J

LJ

I

~xl

DD

ca

0

c=J

_m

xy

=

CONSTANT

7 4

.TIr

I

+

x

I

I

I

8 9

I

I

I

TI:

I

I

I

5~5----6--~Y

TI

1 2 3

---.~

-

--.~ ₂₅ 58

Ex

EX

1

1.

---~x 2 5 8

,"'

... ... 45 ...

Ey

...

:E;61

- - + y

4 5 6

I 8 40 4 50 5.30

I<f-~---'----~.~I---'---~...---~

Figure 14. The cross-section of the box girder.

b I _\

\\

"

Y d I • AREA OF STJFFENER

=

F F

+

b

*

d ro2 /ml Fx b Fy d ro2_/ml Fxy = de v) ro2/ml 2 Fv vd ro2_/ml Iy .-l·d3 _ro4_/ml 12 Ixy ---L.d3 _ro4 /ml 1 2 Nx

I

Fx Fv 0 0 0

l

ex

l

N y Fv Fy 0 0 0 ey

I~

1 E Nxy _{_ V2} 0 0 Fxy 0 0

'., J

L

my 0 0 0 Iy 0 xy mxy 0 0 0 0 I,y Xxy

Figure 16. The fust element-mesh of the main diaphragm. 10 _{r-_}

-1*

- -2tt

_02~a v2

r

y b x 3* _.4~

04~

v4

-30 1---'-v2-= v2+..9..(V2-V4) b

Figure 17. Transformation of the rectangular element to a trapezium

I·

30m

I~

12.90

+

5.30

-I

Î

t = 016 ₀₁₆ IC)_~

8

t = 016 C\l ~

I~

4.80

+

3.60

+

4.50

·1

Figure 19. The cross section of the model.

Figure 20. The Iocation of the strain gauges.

2 3

M(

)M

M

=

150000 kNm

CD

,ffT

_7frr

I.

60

~

-hff

~

2 P

7fn-

$

®

~

60

·1

~ ;5~

.1

HOOQ kN -p

1J

p

2::\

0

Jfti

_7frr

I.

60

·1

I.

;5':'

.1

P" 2500 kN

KOKER ----~ MODEL - o - - o - t : > . BEAM THEORIE N/mm 2 •

~

_~ I:;. x-0 0 - O -F-=: 2.5 5 7.5m

T:

l-:::-o- ___-"-'10 • 0 -0· 0 i f .

-!

/ /

//

<rx N/mm 2 200 100 200 100 300 500 400 +

!

0

I~

5000 kN

~I

30m

@:!

i

~®@

I.

/#

~I

/ ~/ l""'-"_=':_~ ,-

-J:/

'

---

----,

,-N/mm 2 N/mm 2 --"--

---

-

-~--==;;

-

':::

~

.-

.-- .--~ 50 50 150 150 100 100 + - - - - BEAM THEORY _ - - KOKER

>t---

MODEL

!

0

~~

" ' - - - - +

!

0 N/mm 2 ,...-;/~~

...

~-

-

-~--- -

,...--

t7

- - - I -If----..; - - - - l

~

----

- - I---I

.J. /

I~/

/!/~

!

--"- ... I "- _/

'

.... ~-- ~--~_ _ _ J

,

I

"

I I I 380 \ _I

,r--;,

_I "'lRn \ I

+

\ I N/mm 2

,

,

I \ 150 50 200 100 250 <ïx N/mm 2 <ïx 150 150 "-100 100

--50 50

t

0

t

0

v

LJ

50 25 ...~ 't

~

2500 kN kN 2500 D 250 kN

'50 +

1

0 50 Il N/mm2 _ ~ x. ~ I---~---)- x - "'x

~---'1t--

-~ -0- -

:a:.

-::-B=

.=

~

r-- -cr--~

~

IX-...0 - - 0 -

1--<>---1

x--x RIGID b---A 800 mm 150 100 50 +

1

0 50 100 N/mm 2 _.L::::...

/ )

..I

/

/

I

"cr

...

j "

j{"Á

~A1' ~ ,06"

I

~.J--> -0--':-eP' . _X--x _{_c} ~ -x--~ . /

P'"

/ - - - [ ~x x

....

lJ

Q-' .... /

~/

0 - - - 0 0----0 8 mm 80 mm

'txy 50 +

!

0 50 N/mm 2

---

---

ï

1 - - -

---

---._-

---~_....::::...

_-

_----

_{.... , . b~.,...--}

...--...

'txy 50 +

!

0 50 N/mm2 f - - - 1 - - -

rt

--

~----

---

.J{.-

--0-

o~~

~

~

50 +

t

0 50 NImrrf f - - -1--._--

~

---

f

----

~

----

-x'

---..,.

----x~~ --x--50 +

t

0

--

/ / / / 50 'txy N/mm2

~

i

!

I~

<D®~ 5000 klll

I.

30 m

.\

BEAM THEORY ~----* MODEL

----I

KOKER N/mm 2 50+---'---'----'---""-=-""""'-d 5 O , . . - - - - r - - - , - - - , - - - , - - - , 'txy •

t

0 ~=__+----+-~"*::t-_;__+_-____,.4 100 50 +

!

0 50 100 'txy N/mm 2

Nfmm 2 I II

---

....

-~_~:.:::...--

__

" I

-I I I I I I ---.J 50 50 100 +

!.

0 N/mm 2 -IE---_ ...

-... _-~

N

... ~""''''-~

-

-i I I Nfmm 2 50 50 100 +

!

0 50 f=~=+---+--~~>k ----+--~----+---+----1 50 - - - _ _~ _____'____ L ____'_____~c---~---' 100 ffy Nfmm~ +

!

0

+--_+-_+-_---+-_---+_:r::....l'":--+---_ _

____*'"~

[

:

:

@-

-:

7~'

50 T + I

!

0 50 ---CÎ_y N/mm 2

~i

:@@

!

In

5000 kN ~ 30m

_.1

KOKER ~---~ MODEL 50 -j---t---t---7'--:-J--+--150 CÎY

~

Nfmm 2 100+--~__+---+--_+_---+--~-_________1 100 - - ---- -~___+~---+--150 200

-0---.

2500 kN

Figure 28a. The forces acting upon the main diaphragm.

ay

Figure 28b. The shear-forces are depending of the thickness of the main diaphragm.

B

..._...

...

EFFICTIVE WIDTH OF BOnOM-FLANGE AND WEB

k+~

o

o

5 10 15 20 25 30

~

.05 .10

+

....

... I KOKER

*- -

- " , , * MODEL X - ' - ' - X BEAM THEORY .15 cl m .20 A M=15.0000

0::

ML2 2EI

~

~

150000 kNm 14 1=30m ..I FLEXURAL STIFFNESS=EI

~

30 25 20 15 10 ~m

o

5 .05

-r

'i-... ~I-... ... _I

_>

_BENDING ...

...

... .10

-I

I

~

....

""-,

~

I

Ó =.333 -PI3 Ó PI3 B=·412 -EI .15

r

-'ij

D

5HEAR FORCE DEFORMATION

.20

U

Tm B

-

-

KOKER ~--~ METING

~

.[k

5000 kN

- - -

BEAM THEORY _41=30~ FLEXURAL STIFFNESS=EI .0 0"-N

2000 3000 I

I

I

4000

~

4J,

5000 kN

~

1=30m

_~

5000 N/mm2

<T

_X M_/E1

...L

EI

E~

d l '••

M

•••••••

i l l · ·••• ••

·ill

•••••••

M!:!.![!ill[l[l:l~M[[[j[[[lli[[~[~[j[ill[~[~~[~B::[[[~[2[[[[[[[î[[[jjjjgff2j[{[llit\Z~~{2~~:}2~:~::E::::::::2::::::::z::-:.:,:::"".,::::",

.::::!=.

==----

n

~J---,,-i_l~_~~--=---=-=3

l

===~

Figure 32. The displacement ror loading case 2 calculated ror the box girder with reduced flexural

50.8

_ _I

51.6 51.6 39.6 48.6 2 36.0

:~~

~

l~

tsJ

39.6

Figure 33. The shear-stresses according to the beam theory.

4.8 10-4 6.8 10-4

I

'---6.9 10 -4 6.9 10-4 6.5 10- 4 5.3 10- 4

x x A

1""--+---'1""

D

---A

F

_---

_

-G /'" y ö=6.8m E 2.1 mm 4.0 mm 3.1 mm

3030 ~

..

~

~~~

2.50

~

o

C\l C\Ï

Figure 36. Constructive modifications of the model.