design rules plate eurocode maquoi

Journal of Constructional Steel Research 57 (2001) 279–311www.elsevier.com/locate/jcsr

New design rules for plated structures inEurocode 3

Bernt Johanssona, ReneMaquoi b, Gerhard Sedlacekc,*

a Division of Steel Structures, Lulea˚ University of Technology, SE-971 87 Lulea˚, Swedenb MSM, Department of Civil Engineering, University of Liege, B-4000 Liege, Belgium

c Institute of Steel Construction, RWTH Aachen, D-52074 Aachen, Germany

Received 18 June 1999; received in revised form 6 July 2000; accepted 31 August 2000

Abstract

This paper gives an overview of Eurocode 3 Part 1.5 Design of Steel Structures. Supplemen-tary rules for planar plated structures without transverse loading have been developed togetherwith the Eurocode 3-2 Steel bridges. It covers stiffened and unstiffened plates in commonsteel bridges and similar structures. This paper presents the background and justification ofsome of the design rules with focus on the ultimate limit states. The design rules for bucklingof stiffened plates loaded by direct stress are presented and explained. For shear resistanceand patch loading the new rules are briefly derived and compared with the rules in Eurocode3-1-1. Finally, the statistical calibration of the rules to tests is described. 2001 ElsevierScience Ltd. All rights reserved.

Keywords:Steel structures; Plated structures; Design; Plate buckling; Stiffened plates; Shear buckling;Patch loading

1. Introduction

New design rules for plated structures have been developed by CEN/TC250/SC3(project team PT11). The result of the work is the ENV-version of Eurocode 3 Part1.5 (EC3-1-5) [1]. It has been drafted in close co-operation with the project teamPT2 preparing the steel bridge code and it contains rules for stiffened or unstiffenedplated structures. These rules are not specific for bridges, which is the reason for

* Corresponding author. Tel.:+49-241-80-5177; fax:+49-241-888-8140.E-mail address:[email protected] (G. Sedlacek).

0143-974X/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0143 -974X(00)00020-1

280 B. Johansson et al. / Journal of Constructional Steel Research 57 (2001) 279–311

making them a part of EC3-1, which contains general rules. The table of contentsshown in Fig. 1 gives an overview of EC3-1-5. In addition there is an informativeannex containing formulae for elastic buckling coefficients, which has been includedfor the convenience of the designer. Such coefficients may alternatively be found inhandbooks or by computer calculations.

All verifications are presented in Section 2 of EC3-1-5. For the ultimate limitstates (ULS) the requirements are the same as in EC3-1-1. For the serviceabilitylimit states (SLS) no requirements are given, only methods for finding stresses, etc.The requirements depend on the particular application; for instance, requirementsfor bridges are found in EC3-2 [2].

The focus of this paper is Section 4 of EC3-1-5, which contains methods forfinding the resistance to plate buckling in ULS. The objective of the paper is topresent the scientific background to the rules. First the mechanical models behindthe rules are explained and references to source documents are given. All such mod-els include simplifications, which had to be justified by calibration of the rules againsttest results. Several models for each failure mode have been checked with cali-brations according to Annex Z of EC3-1-1 [3] and the ones chosen to be includedin EC3-1-5 are those giving the lowest scatter and the most uniform safety.

EC3-1-5 explicitly permits the use of different steel grades in flanges and webs

Fig. 1. Table of contents of Eurocode 3 Part 1.5.

281B. Johansson et al. / Journal of Constructional Steel Research 57 (2001) 279–311

in so-called hybrid girders. No detail rules are given for the design of such girdersbut in all design rules, subscripts (f for flange and w for web) indicate the relevantyield strength.

Although the rules may look unfamiliar to many engineers they are in fact onlya new combination of rules from different European countries. For the time beingthey represent a set of useful rules for common plated structures. However, the rulesare not complete in the sense that any type of plated structure is covered. There arealso details that may be improved with existing knowledge but the time and fundsavailable for the work have not allowed this. One such item is the formula for theeffective area of unstiffened plates. The single formula from EC3-1-1 has beenretained although it has not been harmonised with the slenderness limit betweencross section classes 3 and 4. A set of formulae for different boundary conditionsand states of residual stresses should be developed but this has to wait until the EN-version is prepared.

2. Design of stiffened plates for direct stress

2.1. General

Plates resisting predominantly direct stresses are used as flanges and webs of plate-and box-girders. The distinction between flange and web is sometimes questionable.The definition used here is that aflangeis subject to a distribution of direct stressesthat is not very far from being uniform (no account being taken in this respect ofshear-lag effects). Aweb is subject to a distribution of direct stresses with a signifi-cant gradient and most often a change from tension to compression.

For very wide plates used as webs or flanges, it is sometimes more economicalto stiffen a relatively thin plate than to increase the plate thickness in order to avoidany stiffening. A plate is normally first stiffened transversally, i.e. by stiffeners trans-verse to the direction of longitudinal stresses, and, when necessary, by additionallongitudinal stiffeners. When the distribution of compressive stresses is quasi uni-form, the longitudinal stiffeners are equally spaced. If not, the stiffeners are locatedin an optimum manner in order to combine efficiency and economy.

The transverse stiffeners are usually parts of transverse bracings of the cross-section of the structure and for this purpose they are normally stiff in bending. Thereis some advantage for them being designed to fulfil this requirement. Such transversestiffeners are denotedrigid when they constitute nodal lines for plate buckling underthe action of compressive stresses. That is afirst principle of the design rules ofEC3-1-5. Accordingly, the amount of efforts devoted to check plate buckling is sub-stantially reduced and facilitated. Possible instability is restricted to:

O buckling of the whole panel forunstiffened panels;O buckling of unstiffened subpanels or buckling of the whole panel forlongitudi-

nally stiffened panels.


In both cases the lengtha of the panel is equal to the distance between the transversestiffeners defining this panel, see Fig. 2. Those edges are so-calledloaded edges.The width B of the panel is the distance between its boundaries to adjacent panelsor possibly between one such boundary and a free edge. The longitudinal edges ofa panel are denotedunloaded edges. A subpanel is an unstiffened plate having thelengtha of the panel to which it belongs and a widthb. The wording loaded/unloadedrefers to loading by direct stresses. That is asecond principleof EC3-1-5, the scopeof which is plates subject to uniaxial direct stresses only. The case of plates subjectto general biaxial loading is not included at the present time. However, there arerules for patch loading, including possible interaction with bending.

For longitudinally stiffened panels two extreme cases concerning the stiffeningare identified in EC3-1-5.

2.1.1. The case of equally spaced multiple stiffeners in the compression zoneWhen the behaviour of the longitudinally stiffened panel as a whole is considered

the number of longitudinal stiffeners located in the compression zone is sufficientlylarge to justify the¿smearingÀ of their flexural stiffness across the panel width.In addition, the behaviour of the subpanels has to be checked independently. Fordesign purposes, the multiple stiffener approach is generally accepted when the num-ber of stiffeners is at least three.

2.1.2. The case of a few unequally spaced longitudinal stiffeners in thecompression zone

Smearing of the stiffness would be too rough an approach in this case and istherefore not recommended. Instead, some account should be taken of the discretelocation of the longitudinal stiffener(s). This method enables the designer to analysespecial situations where the widths of the subpanels are very different because of asteep stress gradient across the panel width. It will be used especially when designingso-called web plate elements. That is in relation to athird principle, or better, with

Fig. 2. Components of a stiffened plate.


the scope of the design rules of EC3-1-5. The latter are devoted to normal structures,thus excluding implicitly girders of very high depth where the stiffening of the webwould need a large number of unequally spaced longitudinal stiffeners in the com-pression zone.

2.2. Unstiffened panels or subpanels

This section refers to a subpanel according to Fig. 3. It is equally applicable toan unstiffened panel for whichb should be replaced byB. For simplicity, the charac-teristic resistance is described and no partial safety factors appear.

The resistance of an unstiffened subpanel of widthb and thicknesst is conven-tionally given by the squash load of the effective cross-sectional area (bt)eff:

Nu5(bt)efffy (1)

where (bt)eff is the effective cross-sectional area of the unstiffened (sub)panel andfyis the material yield stress. The possible effect of plate buckling is clearly introducedas a penalty on the gross cross-sectional areabt rather than on the magnitude of thestress at the ultimate limit state.

The effective cross-sectional area (bt)eff of a subpanel is a partr(#1) of the grosscross-sectional area (bt):

(bt)eff5r(bt) (2)

wherer, termed the effectiveness of the cross-section, is computed using the well-known Winter formula used in EC3-1-1:

r51/lp20.22/l2p#1 (3)

This formula accounts for favourable effects resulting from post-buckling platebehaviour, on the one hand, and for detrimental effects of unavoidable structural andgeometrical imperfections, on the other.

The effectivenessr depends on a single parameter, the relative plate slendernesslp, which is defined as:

Fig. 3. Unstiffened subpanel/panel.


lp5Î(fy/scr) (4)

Any elastic critical stressscr is commonly written as:

scr5kssE (5)

whereks is the buckling coefficient andsE the so-called reference Eulerian stress:

sE5p2Et2/12(12v2)b25189800(t/b)2 (in N/mm2) (6)

More explicitly, the relative plate slendernesslp writes:

lp5(b/t)/(28,4eÎks) (7)

and it involves:

O the width-to-thickness ratio (b/t), that is to plate buckling what column slendernessis to column buckling, i.e. the governing parameter;

O the yield stress factore=√(235/fy), which indicates that the relative plate slender-ness of a given plate increases with the material yield stress (fy, to be expressedin N/mm2);

O the buckling coefficientks, which amounts to 4 for a simply supported long platesubject to uniform compression but depends on the aspect ratioa=a/b of thesubpanel ora/B of the (unstiffened) panel.

According to the Winter formula, the penaltyr applicable to the gross cross-section is seen to start (r,1) when the relative plate slendernesslp exceeds 0.673.Unfortunately, this limit does not coincide with the limit between section classes 3and 4, which creates a discontinuity in the design rules. This inconsistency isexpected to be remedied in the final version of Eurocode 3.

The validity of this formula has been extended to any type of boundary and longi-tudinal loading conditions by introducing the relevant buckling coefficientks, in Eq.(7). In practice, except when a plate edge is free, the conservative assumption ofsimply supported edges is usually made. That is due to the difficulty in assessingthe magnitude of edge restraints. However, the designer is free to take the edgerestraint into account if the value is justified. In addition, for a plate of constantthickness, any non-linear distribution of longitudinal direct stresses across the platecan be characterised by the stress ratioy, ratio of the extreme edge direct stresses.Finally, should the plate possibly be stiffened, then the properties of the stiffeningwould also affect the plate critical buckling stress. As a result, the buckling coef-ficient for the most general case is a function:

ks5ks (boundary conditions,a, y, stiffening properties) (8)


2.3. Longitudinally stiffened panels

In this section, only the behaviour of the stiffened panel as a whole is of concern.An individual (unstiffened) subpanel is treated in Section 2.2. As mentioned in Sec-tion 2.1, two distinct design approaches may be contemplated, the application ofwhich is mainly governed by the number of longitudinal stiffeners in the compressionzone of the stiffened panel.

2.3.1. Effectiveness of the stiffened panelThe fact that a plate panel is stiffened in the longitudinal direction makes its

behaviour more like that of a column type structural element than that of a platetype element in the range of small slenderness. While plate-like behaviour exhibitsa significant post-buckling resistance, which may largely exceed the elastic criticalplate-buckling load, the elastic critical column buckling load is an upper bound ofthe resistance.

The behaviour of any stiffened panel lies somewhere between these two limits.For design purposes, the effectiveness of the stiffened panel is obtained by a simpleinterpolation between the value of the effectivenessrp for the plate-like behaviour,on the one hand, and the valuecc for the column-like behaviour, on the other. Thisinterpolation is empirically based but its suitability was supported by calibration:

rc5(rp2cc)x(22x)1cc (9)

The interpolation is governed by a factorx that measures the vicinity of the elasticcritical plate buckling stressscr,p to the elastic critical column buckling stressscr,c

according to:

x5(scr,p/scr,c)21 0#x#1 (10)

It is understandable thatscr,p should not be smaller thanscr,c. However, owing toapproximations and simplifications included in the procedures described in Sections2.3.2 and 2.3.3, this requirement is not necessarily fulfilled. In order to prevent theparameterx from being negative, 0 must be adopted as a lower bound. On the otherhand, the effectivenessrc must increase fromcc and approachrp whenx increases.Therefore,x has 1 as an upper bound. How to assess the values ofscr,p, scr,c, rp

andcc is discussed in Sections 2.3.2 and 2.3.3.Because expression (10) does not reflect a monotonic decrease whenx increases,

it can be suggested, in a revised version of EC3-1-5, to simplify the process withouta significant penalty on the results by adopting simply:

rc5cc but not smaller thanrp,`

whererp,` is the value ofrp, computed for a very long stiffened plate, i.e. a platewhere the buckling coefficient no longer depends on the plate aspect ratio.


2.3.2. Plate-like behaviourTwo methods are specified; each is especially applicable to specific types of stiff-

ened panels.

2.3.2.1. Multiple longitudinal stiffeners—Concept of equivalent orthotropic plateThe basic idea is to smear the flexural stiffness of the longitudinal stiffeners across

the plate width. Conceptually, that would lead to the substitution of the actual dis-cretely stiffened plate by an orthotropic plate, referred to as theequivalent ortho-tropic plate in the following. Usually, multiple stiffeners are equally spaced or notfar from being such. Then the properties of the equivalent orthotropic plate may beassumed uniformly distributed across the width. The buckling coefficient for thestiffened panel, designated asks,p, may be obtained by any means: computer analysis,appropriate charts [4,5] or simply by the following approximate expressions:

ks,p52((1+a2)2+g)a2(y+1)(1+d)

if a#(11g)0.25 (11a)

ks,p54(1+Î1+g)

(y+1)(1+d)if a.(11g)0.25 (11b)

where:

g=Ix/Ip relative flexural stiffness, i.e. ratio of the second moment ofareaIx, of the actualstiffenedpanel to the secondmomentof areaIp(=Bt3/12(12v2)) of the plate for longitudinal bending

d=Asl/Ap relative cross-sectional area, i.e. ratio of the cross-sectionalareaAsl of the longitudinal stiffeners without anycontribution of the plate to the cross-sectional areaAp (=Bt)of the plate

a=a/B aspect ratioy=s2/s1 edge stress ratio, s1 ands2 being respectively the larger

and the smaller edge stresses, see Fig. 4 (compression istaken as positive).

Fig. 4. Definition of the stress ratioy.


The above expressions imply that, on the one hand, the wall elements of the stiffenersdo not exhibit local buckling and that, on the other hand, no instability of any stiff-ener in its whole (stiffener tripping) occurs before the stiffened panel reaches itsultimate strength. In that case, the stiffeners are said to be fully effective; that is, afourth principleof EC3-1-5. The full effectiveness of any stiffener can be achievedby complying with deemed-to-satisfy requirements. Plate buckling initiated by trip-ping of open stiffeners is likely to result in a sudden and so-called catastrophic typeof collapse, which should be prohibited. However, closed stiffeners, e.g. trapezoidalboxes, may very well have class 4 sections because the local buckling of the wallsof the stiffener will not trigger a collapse. In the case of such closed class 4 stiffenersthe local buckling should be considered in the same way as for subpanels of the plate.

Possibly a plate can be fitted with notably unequally spaced multiple stiffeners.Then the assumption that the distribution of the flexural properties of the equivalentorthotropic plate is varying linearly across the panel width may look more appropri-ate than a uniform distribution. Then use shall be made of computer simulations orcharts [5].

The elastic critical plate-like buckling stressscr,p is:

scr,p5ks,psE (12)

wheresE is given by Eq. (6) withB instead ofb.When defining the relative plate slendernesslp,o of the equivalent orthotropic plate

it should be taken into account that the critical stress is referred to the gross cross-sectional areaA and yield load to the effective cross-sectionAeff. Hence, the relativeplate slendernesslp, becomes:

lp,o5Î(Aefffy/Ascr,p)5Î(bAfy/scr,p) (13)

wherebA=Aeff/A.In EC3-1-5,bA is calculated only for the compressed part of the plate, which leads

to a smaller value than if the whole plate had been considered in case the stresseschange sign.

The equivalent orthotropic plate is characterised by an effectivenessro for theplate-like behaviour:

ro51/lp,o20.22/l2p,o#1 (14)

The symbolslp,o andro are used instead of the EC3-1-5 symbolslp andr for thesake of clarity.

2.3.2.2. One or two stiffeners in the compression zone—concept of equivalent col-umn on an elastic foundation The following procedure is especially dedicated tosituations where both the number and the location of the longitudinal stiffeners resultfrom a notably non-uniform distribution of direct stresses, as in a web element. Aspecial procedure is suggested which accounts for the discrete nature of the stiffeningin a simple way. The elastic critical plate buckling stressscr,p is no longer based onthe concept of an equivalent orthotropic plate but on one of an equivalent column


Fig. 5. Physical model of a compression strut on an elastic foundation.

supported by an elastic foundation, see Fig. 5. The elastic critical column bucklingstress of this equivalent column is used as an approximation ofscr,p.

The properties of the equivalent column, including its elastic foundation, must bedetermined so that both the number and location of the stiffeners, on the one hand,and the behaviour of the plate sheet in the direction transverse to the stiffeners, onthe other, can be satisfactorily accounted for.

2.3.2.3. Case of one stiffenerWhen there is only one longitudinal stiffener in thecompression zone, the location of the equivalent column is that of the stiffener.For sake of simplicity, possible stiffeners in the tension zone are fully disregarded.Accordingly, the single stiffener divides the widthB of the panel into two subpanelsof width b1 and b2, respectively, and the elastic foundation is represented by theplate, see Fig. 6.

Fig. 6. Notional cross-sectional area of equivalent column.


The gross cross-section of the elastically founded equivalent column is used forthe determination of the section properties (cross-sectional areaA, second momentof areaIsl about an axis through the centroid and parallel to the plate sheet). It iscomposed of the gross cross-sectional areaAsl of the stiffener and a notional cross-sectional area of the plate sheet, that is determined as follows from both subpanelsadjacent to the stiffener, see Fig. 6:

O half the width of the subpanel when fully in compression;O one third of the width of the sole compressed part of the subpanel when stresses

change from compression to tension.

The effective cross-sectional area of the equivalent column is used for the compu-tation ofbA. This consists of the effective parts of the plate adjacent to the stiffenerand if the stiffener is partially effective only, due account shall be taken of an effec-tive cross-sectional area of the stiffener. This may be the case if a closed stiffeneris used.

In the absence of any elastic foundation, the buckling length of the equivalentcolumn would be equal to the distancea between the transverse stiffeners. It is notedthat the latter are designed so as to be rigid, on the one hand, and that simple supportshave been conservatively assumed for the stiffener, on the other hand. In addition, thevariation, over the lengtha, of the compressive force in the stiffener is disregarded inthe following. Owing to the plate effect, the buckling lengthac of the equivalentcolumn will be smaller than the distancea. In accordance with the physical model,it is found to be:

ac54,33FIslb21b2

2

t3B G0.25

(15)

The elastic critical column buckling stress that is taken as an appraisal ofscr,p isgiven as

scr,p51.05EÎIslt3B

Ab1b2if a$ac (16a)

scr,p5p2EIsl

Aa2 1Et3Ba2

4p2(1−v2)Ab21b2

2if a#ac (16b)

2.3.2.4. Case of two stiffenersWhen there are two stiffeners in the compressionzone, the procedure described above is applied three times, see Fig. 7.

In a first step, each of these stiffeners is considered assuming that the other oneacts as a rigid support. The value ofscr,p is given by Eq. (16) withb1=b∗

1 andb2=b∗

2 and B=B∗ is taken as the sum ofb∗1 and b∗

2. To account for possible simul-taneous column buckling of both stiffeners, a second step is required. That is done


Fig. 7. Procedure for two stiffeners in the compression zone.

by means of an intuitive conservative trick. In that step, both stiffeners are lumpedinto a single one having the following properties, see Fig. 7:

O the cross-sectional areaA is the sum of those computed earlier for the individ-ual stiffeners;

O the second moment of areaIsl is the sum of those computed earlier for the individ-ual stiffeners;

O the lumped stiffener is located at the position of the resultant of the forces in theindividual stiffeners.

The whole procedure provides the designer with three values ofscr,p, of which thelowest one should be selected.

2.3.3. Column-like behaviourThe elastic critical column-like buckling stressscr,c is defined as the Euler stress

for out-of-plane buckling of an equivalent column represented by the part of thestiffened plate that is in compression.

scr,c5p2EIx,c/Aca2 (17)

where Ixc is the second moment of area for longitudinal bending andAc the grossarea of the equivalent column. This buckling stress appears as a characteristic of thecompression part of the stiffened panel, assuming that this part is released from anysupport along its longitudinal edges, is subject to uniform compression and has abuckling length equal to the length of the stiffened panel. That clearly appears as aset of conservative assumptions.

When there is a significant gradient of the direct stresses along the length of the


compressed part of the stiffened panel, undue conservatism can be avoided by reduc-ing appropriately the buckling length, which then becomes smaller than the lengtha of the stiffened panel.

The relative column slendernesslp,c then writes:

lp,c5Î(bAfy/scr,c) (18)

The effectivenesscc for the column-like behaviour is given by the reduction factorfor column buckling given as:

cc51

f+Î(f2−l2p,c)

(19)

with f=0.5[1+ae(lp,c20.2)+l2p,c].

Because of the non-symmetry about the buckling axis due to one-sided longitudi-nal stiffeners, on the one hand, and the nature of built-up section (the stiffeners beingwelded onto the plate), on the other hand, due allowance is made for a geometricimperfection larger than 1/1000 of the buckling length (the latter is the one coveredimplicitly by the regular European column buckling curves). The initial out-of-straightness accounted for is 1/500 of the buckling length, which is done by increas-ing the imperfection coefficientae to [6]:

ae5a01[0.09/(i/e)] (20)

where:

i5Î(Ix,c/Ac)

and e is the largest of the distances from the neutral axis of the stiffened panel tothe centre of the plate or the centroid of the one-sided longitudinal stiffeners(alternatively of either set of stiffeners when both-sided stiffeners), see Fig. 8[7].

Because of the better stability of closed section stiffeners, distinction is madebetween types of stiffener sections according to:

Curveb (a0=0.34) for hollow section stiffeners

Fig. 8. Distancese1 and e2.


Curvec (a0=0.49) for open section stiffeners.

2.3.4. Effective cross-sectional area of the stiffened panelThe effective cross-sectional area of the stiffened panel is composed of:

O The gross cross-sectional areaAt,o of the part of the equivalent orthotropic platelocated in the tension zone:

At,o5(Asl)t1(Obt)t (21)

where (Asl)t is the gross cross-sectional area of all the stiffeners located in thetension zone, and (Sbt)t is the gross cross-sectional area of all the subpanels thatare fully in tension.

O The effective cross-sectional area

Aeff,c,o5rcAc,o (22)

of the part of the equivalent orthotropic plate located in the compression zone,whereAc,o accounts for possible plate buckling of the subpanels:

Ac,o5(Asl)eff,c1(Obt)eff,c

where (Asl)eff,c is the effective cross-sectional area of all the stiffeners located inthe compression zone, and (Sbt)eff,c is the effective cross-sectional area of all thesubpanels that are fully or partially in compression.

For very wide flanges there is a further reduction of the effective area with respectto shear lag according to EC3-1-5.

3. Design of plates for shear

3.1. General

The resistance of slender plates to shear is based on the rotated stress field theoryas proposed by Ho¨glund [8]. It is a tension field theory that is capable of predictingthe resistance of short as well as long panels and it replaces the two methods inEC3-1-1.

At a certain slenderness the plate reaches its yield resistance but this does notnecessarily mean the maximum resistance. Strain hardening and the contributionfrom the flanges makes it possible to utilise higher resistance without excessivedeformations. In EC3-1-5 the maximum strength in shear is put to 0.7fy for steelsof grade S355 and lower. For higher grades the strain hardening is less pronouncedand there are no test results available. Hence, a more conservative strength, 0.6fy, isproposed. In EC3-1-1 there are special rules for rolled beams for which a shear area


larger than web area is defined. That is another way of taking the increased resistanceinto account and it can not be combined with the above mentioned increased strengthwhich refers to the geometrical web area.

The partial safety factorsgM0 andgM1 have been suggested to have different valuesin EC3-2: 1.0 and 1.1, respectively. The reason for this is a study of the statisticaldistribution of yield strength and geometrical properties of beams, which justifiesgM0=1.0. This result is quite new and there has been no time for re-calibrating theresistance functions for various instability modes in order to use the same partialsafety factor. This would be the rational solution but in the meantime a temporarysolution has been introduced for the shear resistance. This is simply that the plateauis shifted in relation to the ratio between the partial safety factors.

3.2. Rotated stress field theory for plain web

The rotated stress field theory was first developed for girders with slender webswith stiffeners at the supports only and for girders with transverse stiffeners butwithout horizontal stiffeners [8]. It was later widened to include such stiffeners [9].

First, consider a girder with a slender web and widely spaced transverse stiffeners.The state of stress in the web caused by a shear force must be such that no verticalstresses appear at the edges. The state of pure shear that may exist for low loadsrotates as shown in Fig. 9, for which the conditions of equilibrium are

s15t/tan(j) (23)

Fig. 9. State of stress in a slender girder web after buckling. Vertical stiffeners are supposed to bewidely spaced and no vertical stresses act on the flanges.


s252t tan(j) (24)

An observation from tests is that the compressive stress remains close to the criticalshear stress and this is used as an assumption in the theory

s252tcr52ktp2E

12(1−v2)t2whw

2 (25)

The ultimate strength of a web is assumed to be reached when it yields accordingto the von Mises yield criterion

s211s2

12s1s25fyw (26)

From Eqs. (23)–(26) the shear resistance can now be solved to

tufv

5

3Î4

lw!!1 −1

4l4w

−1

2Î3l2w

(27)

in which

fv5fy/Î3

lw5! fvtcr

Eq. (27) is shown in Fig. 10 together with some test results for a girder with widelyspaced vertical stiffeners. It is clear that the solid dots representing tests of girderswith rigid end-posts fit very well with the prediction, while the tests with non-rigidend-posts do not. The reason for this is the resulting tension in the web, which hasto be anchored at the girder ends. Assuming that the state of stress as given by Eqs.(23) and (24) is uniform over the depth leads to the following expression for thetensile force

Nt5hwtwfvF 1l2

w

2Slw

tufvD2G (28)

This force is larger than the actual force because the state of stress close to theflanges will be more like pure shear. The force has to be resisted by the end-post ifthe full strength should be developed. If the end-post consists of a single plate theresistance to shear will be less than predicted by Eq. (27).

The resistance actually used in the design according to EC3-1-5 is reduced slightlycompared with Eq. (27) in order to allow for scatter in the test results and also for


Fig. 10. Shear resistance according to rotated stress field theory together with test results for girderswith widely spaced vertical stiffeners.

the systematic deviation for girders with non-rigid end-posts. This has been done bycurve fitting using simpler expressions than in Eq. (27). The design resistance isgiven by

VRcd5cvfywdhwtw/Î3 (29)

wherecv=cw+cf. cw is found in Table 1 andcf will be discussed later.

3.3. Contribution of stiffeners

The influence of stiffeners is accounted for by their increase of the critical stress.Transverse stiffeners are assumed to be rigid, that is, they form nodal lines in thebuckling pattern, and requirements for stiffness and strength are given in EC3-1-5.Longitudinal stiffeners may be flexible, that is, they deform under buckling. It isclear from test results that the effect of longitudinal stiffeners will be overestimated

Table 1Contribution from the web to shear resistancecw according to EC3-1-5

lw Rigid end-post Non rigid end-post

lw,0.83h h h0.83h,lw,1.08 0.83/lw 0.83/lw

1.08,lw 1.37/(0.7+lw) 0.83/lw

h=1.20gM1/gM0 for S235, S275 and S355h=1.05gM1/gM0 for S420 and S460


if the theoretical critical stress is used for calculating the slenderness parameterlw.There is less post-critical strength in a web with flexible stiffeners than in a plainweb. This is dealt with by reducing the second moment of area of the longitudinalstiffeners to one third of the actual value when calculating the critical stress. Thisreduction has been considered in the following approximate formulae for the buck-ling coefficient from Annex A3 of EC3-1-5

kt55.3414(hw/a)21ktsl (30)

kt5415.34(a/hw)21ktsl (31)

ktsl59Shw

a D2S Isl

t2whwD3/4

but not less than2.1twSIsl

hwD1/3

(32)

In (32) Isl denotes the sum of the second moments of area of all longitudinal stiff-eners. In addition to the check for buckling of the whole stiffened panel there is acheck for buckling of the largest subpanel, assuming that the stiffeners are rigid.

A comparison between the resistance to shear according to EC3-1-1 and EC3-1-5 is shown in Fig. 11. This comparison assumes that the flanges do not contribute.The resistance according to EC3-1-5, shown by solid curves, is the same in bothdiagrams because the influence of the panel length is reflected only by its influenceon the slenderness parameterlw. This is also true for the simple post critical resist-ance according to EC3-1-1, which is quite close to the resistance for a non-rigidend-post according to EC3-1-5. This is because EC3-1-1 does not have any require-ments other than that there should be a stiffener at the end of the girder. The draw-

Fig. 11. Comparison between resistance to shear without contribution from flanges according to EC3-1-1 and EC3-1-5 assuminggM1=gM0.


back of the tension field method (it overestimates the resistance for short panels andunderestimates it for long panels) has been eliminated by the method for a rigid end-post in EC3-1-5, which gives fair estimates of the resistance for any length of thepanel, including girders with no intermediate transverse stiffeners. Another advantageof the new rules is that they are simpler to use.

3.4. Contribution from flanges

The intermediate vertical stiffeners prevent the flanges from moving towards eachother. This effect is taken into account by adding a tension field that can be supportedby the flanges acting as beams supported by the stiffeners according to Fig. 12. Thisis a much smaller tension field than that of EC3-1-1 because the rotated stress fieldalready catches the post buckling resistance of the web alone.

After some simplifications the contribution from the flanges can be expressed as

cf5bft2f fyfÎ3

ctwhwfywF12S MSd

Mf.RdD2G (33)

c5S0.2511.6bft2f fyf

th2wfyw

Da (34)

An example of the resistance to shear including the effect of the flanges is shownin Fig. 13. The resistance according to EC3-1-5 is compared with the one accordingto EC3-1-1. The simple post critical method does not take the contribution of theflanges into account. The tension field method does take the effect into account butin such a way that the effect disappears when the slenderness is lower than 0.8. Thisdoes not reflect the real behaviour of a girder.

Fig. 12. Tension field supported by the flanges.


Fig. 13. Resistance to shear for a girder withbf=25tf, tf=3tw and fyw=fyf=355 MPa,MSd=0 and assuminggM1=gM0.

4. Design for patch loading

4.1. General

The rules for the resistance of a web to patch loading are new in the Eurocodecontext and have been developed by Lagerqvist [10] and Lagerqvist and Johansson[11]. The new rules use the same format as other buckling rules.

The three verifications in EC3-1-1 for crushing, crippling and buckling have beenmerged into one verification. The new rules also cover a wider range of load appli-cations and steel grades. The rules have been checked for steel grades up to S690and there is no longer any need for the special formula for S460 in Annex D ofEC3-1-1.

4.2. Model for patch loading resistance

The design rules in EC3-1-5 cover three different cases of patch loading. Becauseof space limitation only the most common case is dealt with here, see Fig. 14. Thedesign procedure includes the following parameters:

O a yield resistanceFy

O a slenderness parameterl=√Fy/Fcr whereFcr is the elastic buckling force


Fig. 14. Patch loading. Definition of parameters.

O a resistance functionc=c(l) which reduces the yield resistance forl larger thana certain limiting value.

The characteristic resistance is written as

FR5Fyc(l) (35)

and the parameters are written as

Fy5fywtwly (36)

Fcr5kF

p2E12(1−v2)

t3whw

(37)

c(l)50.0610.47l

#1 (38)

The expression in Eq. (38) was originally proposed [10] but during the drafting ofEC3-1-5 it was simplified to

c(l)50.5l

#1 (39)

The mechanical model according to Fig. 15 is used for the yield resistance. The

Fig. 15. Mechanical model for the yield resistance for patch loading.


mechanical model has four plastic hinges in the flange and the plastic moment resist-ance for the inner plastic hinges,Mi, is calculated under the assumption that theflange alone contributes to the resistance. For the outer plastic hinges,Mo, it isassumed that a part of the web contributes to the resistance. This assumption is basedon the observations from the tests that the length of the deformed part of the webincreased when the web slenderness increased. With a simplified expression forMo,the effective loaded lengthly, for the model in Fig. 15 is given by

ly5ss12tf(11Îm1+m2) (40)

where

m15fyfbf

fywtw(41)

m250.02Shw

tfD2

(42)

The buckling coefficientkF in Eq. (37) was determined on the basis of the resultsfrom an FE analysis. The FE analysis included the influence from the stiffness ofthe flanges as well as the length of the applied load and expressions forkF, wherethe influence of these parameters are included can be found in Ref. 10. Theseexpressions were simplified in EC3-1-5 to

kF5612Shw

a D2

(43)

kF5216ss

hw

#6 (44)

It is also necessary to consider the interaction with bending moment. This influenceis accounted for by Eq. (45), from which it follows that when the ratioMs /MR#0.5 the bending moment has no influence on the patch load resistance.

Fs

FR

10.8Ms

MR

#1.4 (45)

The resistance to patch loading according to EC3-1-5 is compared with that of EC3-1-1 in Fig. 16, which shows the quotient between the two resistances as a functionof flange thickness over web thickness. The left diagram for zero loaded length showsa fairly large difference between the two design methods. The right diagram for aloaded length of 0.2 times the web depth, a more realistic case, shows a large differ-ence only for a stocky web.


Fig. 16. Comparison between patch loading resistance according to EC3-1-5 and EC3-1-1 for a girderwith bf/tf=25 and a large distance between vertical stiffeners.

5. Calibration of design rules versus test results

5.1. General

The new design rules provided in EC3-1-5 were calibrated versus test results bya statistical evaluation according to Annex Z of EC3-1-1, which uses the followingdefinitions and assumptions.

It is assumed that both the action effectsS and the resistanceR of a structure aresubject to statistical normal distributions, which are characterised by mean values“m” and standard deviations “s”, see Fig. 17.

To guarantee that the distribution of the action effectsSand the resistanceR havea sufficient safety distance a safety indexb is defined in EC1-1 as follows:

b5mR−mS

Îs2R+s2

S

$3.8 (46)

wheremS is the mean value of the action effect,mR is the mean value of the resist-ance,sS is the standard deviation of the action effect andsR is the standard deviationof the resistance.

The safety requirement for a structure is defined by the criterion

[Rd]2[Sd].0 (47)

where [Rd] and [Sd] are design values.To define the design values in Eq. (47), Eq. (46) may be expressed by


Fig. 17. Statistical distribution of the action effectsS and the resistancesR.

3mR2sR

Îs2R+s2

S

bsR423mS2−sS

Îs2R+s2

S

bsS4$0 (48)

With the notations

aR5sR

Îs2R+s2

S

aS5sS

Îs2R+s2

S

it is possible to express the design values as

Rd5mR2aRbsR (49)

Sd5mS1aSbsS (50)

With the approximationsaR=0.8 andaS=0.7 the design values of the action effectsand of the resistances can be described independently from each other and a more


detailed investigation of the design value of the resistance can be carried out usingthe statistical procedure given in Annex Z of EC3-1-1.

In a first step of this procedure a resistance functionrt=gR(x), the so called designmodel for the resistance, has to be established. This is an arithmetic description ofthe influence of all relevant parametersx on the resistancer which is investigatedby tests. By comparing the strength values from the resistance functionrt usingmeasured input data with test resultsre, see Fig. 18, the mean value correction factorb for the resistance functionrt and the standard deviationSd for the deviation termd can be determined. This gives the following formula describing the field

R5brtd (51)

In most cases the probabilistic density distribution of the deviation termd cannotbe described by a single normal distribution as is assumed in Fig. 17. It may berepresented by a non-normal distribution, which may be interpreted as a compositionof two or more normal distributions. Therefore, the density distribution for the resist-ance is checked by plotting the measured probability distribution on Gaussian paper.If the plot shows a straight line, the actual distribution corresponds to a unimodalnormal function as assumed and the statistical data (b and Sd) are determined withthe standard formulae provided in Annex Z of EC3-1-1.

For the case that the plot shows a curved line the relevant normal distribution at

Fig. 18. Plot ofre2rt values, mean value correctionb and standard deviationSd of the deviation termd.


the design point is determined by a tangent to the lower tail of the measured distri-bution, see Fig. 19.

The statistical datab andSd of the relevant normal distribution are then determinedfrom the tangent approach to the actual distribution.

In general, the test population is not representative of the total population of struc-tures and therefore is only used to determine the mean value deviationb and thescatter valueSd of the design model. To consider scatter effects of parameters notsufficiently represented by the test population the standard deviation of the resistancehas to be increased. To this end, in addition to the standard deviationSd, the followingvariation coefficients are taken into account for the yield strength and geometricalvalues:

nfy = 0.07 for strengthfy

nt = 0.05 for thicknesst

nb = 0.005 for widthb

nh = 0.005 for depth h

These variation coefficients are combined with the standard deviationSd accordingto Eq. (52)

Fig. 19. Plot ofrei/rti-values on Gaussian paper and definition of the relevant normal distribution at thedesign point.


sR5ÎO(ni)2+S2d (52)

Using a log normal distribution forR the characteristic valueRk of the resistancefunction may be represented by the 5% fractile value and can be obtained fromEq. (53)

Rk5bmR exp(21.64sR20.5s2R) (53)

Also, the design valueRd of the resistance function may be defined by

Rd5bmR exp(aRbsR20.5s2R) (54)

whereaRb = 0.8·3.8= 3.04.The gM-value of the resistance function is obtained from the ratio of the character-

istic value to the design value

gM5Rk

Rd

(55)

In most cases instead of a 5% fractile valueRk a valueRnom with nominal valuesfor the input parameters is used as characteristic value. To considerRnom instead ofRk a modified partial safety factorgM* is used from:

g∗M5DkgM (56)

whereDk=Rnom/Rk.For the resistance functions for plate bucklingDk may be expressed by:

Dk5exp(−2.0sfy−0.5s2

fy)b exp(−1.64sR−0.5s2

R)5

0.867b exp(−1.64sR−0.5s2

R)(57)

The procedure explained above is used in the following to determine theg∗M valuesfor the resistance functions for plate buckling due to compression stresses, shearbuckling and buckling due to patch loading. Whereg∗M is not in compliance with thestandard valueg∗M=1.10 used for stability checks, the functionRnom is subsequentlymodified to reach the standard valueg∗M.

5.2. Calibration of the design rules for shear buckling

The design rules for shear buckling were checked versus test results according tothe procedure given in Section 5.1. For the statistical evaluation the test results wereobtained from a data bank given by Ho¨glund [9] which contains 166 test results forthe following types of steel plate girders:

O girders with stiffeners at support only;O girders with transverse intermediate stiffeners;


O girders with longitudinal and transverse stiffeners.

Only 150 of 166 tests could be used for the statistical evaluation because some ofthese specimens did not fail by shear buckling.

Since the procedure of the design model for shear buckling depends on thearrangement of stiffeners, the available test results were subdivided into subsets, seeFig. 20. For these individual subsets the statistical evaluations were carried out andthe statistical results are presented in Fig. 21. The figure shows that for all subsetsg∗M-values lower than 1.10 were determined so that ag∗M-value of 1.10 can be appliedin the design modelg∗M.

The sensitivity of the design model to the variation of the yield strengthfyw waschecked by plotting the ratioVei/Vti versus the yield strength of the webfyw (Fig.22). Owing to the small variation of the mean values ofVei/Vti the conclusion canbe drawn that the influence offyw is adequately considered in the design model.

5.3. Calibration of the design rules for patch loading

The statistical evaluation for the design model of patch loading was carried outwith test results, which were obtained from a data bank given by Lagerquist [10].The data bank contains test results for welded and rolled I-girders, which were loadedby patch loading, end patch loading or opposite patch loading, see Fig. 23.

Fig. 20. Subsets of test results.


Fig. 21. g∗M-values for the design model of shear buckling.

Fig. 22. Sensitivity plot forfyw.


Fig. 23. Different cases of patch loading.

According to the data base and the various design models the test results weresubdivided into the following subsets:

Data set 1: Patch loadingData set 2: End patch loadingData set 3: Opposite patch loading

For these subsets the statistical evaluations were carried out, and a summary of thestatistical results is presented in Fig. 24. The figure shows that for all three subsetsa g∗M-value of 1.10 is justified.

In Fig. 25 a sensitivity plot is given for the slenderness parameterl which showsthat the scatter of the ratioFei/Fti is only slightly influenced by the slenderness para-meterl.

Fig. 24. g∗M-values for the design model of patch loading.


Fig. 25. Sensitivity plot for the slendernessl.

5.4. Calibration of the design rules for buckling of stiffened plates

The calibration of the design model for plate buckling was carried out with testresults for multiple longitudinally stiffened steel plate girders in compression whichwere obtained from a literature study. Unfortunately, not all of the collected testscould be used to check the design model, because either some relevant data werenot given in the test reports or the tests were carried out with additional initial imper-fections which are not considered in the design model.

Finally 25 tests were applicable to calibrate the design model. In these tests thelongitudinal stiffeners were designed as bulb flats, flats or angles, see Table 2. Someof these stiffeners do not fulfil the design recommendations given in EC 3 Part 1.5,because they are not fully effective (class 4 section).

In this case the design resistance of a longitudinal stiffened steel plate was determ-ined by taking into account the local buckling of both subpanels and stiffeners.

In addition, the shifting of the neutral axis of the stiffened steel plate due tolocal buckling was considered using the interaction formula for bending and axialcompression which is provided in EC3-1-1, see Eq. (58).

Ns

rcAcfy1

kyNSeN

Wefffy#1.0 (58)

where:

ky512myNS

rcAcfy#1.5


Table 2Test results for longitudinal stiffened steel plates in compression

Authors Test No of tests Stiffener types

Dorman and Dwight 3 (TPA3) 4 All tests with bulb flats[12]

4 (TPA4)7 (TPB3)8 (TPB8)

Scheer and Vayas [13] 1 (III A 50-70) 3 All tests with bulb flats2 (III A 75-70)3 (III A 75-100)

Fukumoto [14] B-1-1 6 All tests with flatsB-1-1rB-2-1B-3-1C-1-4C-2-1

Lutteroth [15] All tests 12 9 tests with flat plates3 tests with flat plates and angles

my5lc(2bM,y24),0.9

bM,y51.1

The statistical evaluation was carried out for the different groups of tests given inTable 2. The results of the statistical evaluations are presented in Fig. 26 and showthat ag∗M-value of 1.10 can be applied in the design model.

Fig. 26. Statistical results for the design model of plate buckling.


6. Conclusions

The first common European pre-standard considering plate buckling has been pub-lished for trial application. In addition to widening the scope to stiffened plates italso includes some improvements of the present rules in EC3-1-1. The design ruleshave been verified by calibrations to tests, which include also steel grades abovethe present limit of S460. This will facilitate a future widening of EC3 to highersteel grades.

The test application shows that this first version of EC3-1-5 can be improved inseveral respects. The first author has already received several questions and remarksindicating the need for clarification and corrections. Further comments and remarksare welcomed and they may be sent to the first author for consideration in the EN-version of EC3-1-5.

References

[1] Eurocode 3 Design of steel structures. Part 1.5 General rules. Supplementary rules for planar platedstructures without transverse loading. ENV 1993-1-5:1997.

[2] Eurocode 3 Design of steel structures. Part 2 Steel Bridges. ENV 1993-2:1997.[3] Eurocode 3 Design of steel structures. Part 1.1 General rules and rules for buildings. ENV 1993-

1-1:1992.[4] Kloppel/Scheer Beulwerte ausgesteifter Rechteckplatten, W. Ernst & Sohn (Tables V/1.1 to V/5.2

pp. 96–105).[5] Kloppel/Moller Beulwerte ausgesteifter Rechteckplatten, Band II, W. Ernst & Sohn (Tables Q001

to Q005, pp. 130–139).[6] Rondal J, Maquoi R. Formulations d’Ayrton-Perry pour le flambement des barres me´talliques. Constr

Met 1979;4:41–53.[7] Jetteur PH, Maquoi R, Massonnet CH, Skaloud M. Calcul des aˆmes et semelles raidies des ponts

en acier. Constr Met 1983;4:15–28.[8] Hoglund T. Design of thin plate I-girders in shear and bending with special reference to web buck-

ling. Bull. 94, Division of Building Statics and Structural Engineering, Royal Institute of Technology,Stockholm, 1981.

[9] Hoglund T. Strength of steel and aluminium plate girders—shear buckling and overall web bucklingof plane and trapezoidal webs. Comparison with tests. Tech. Report, Dept. Structural Engineering1995:4, Steel Structures. Royal Institute of Technology, Stockholm.

[10] Lagerqvist O. Patch loading, resistance of steel girders subjected to concentrated forces. Doctoralthesis 1994:159 D, Department of Civil and Mining Engineering, Division of Steel Structures, Lulea˚University of Technology.

[11] Lagerqvist O, Johansson B. Resistance of I-girders to concentrated loads. J Constr Steel Res1996;39(2):87–119.

[12] Dorman AP, Dwight JB. Tests on stiffened compression panels and plate panels. In: Conference onsteel box girder bridges. London: Institute of Civil Engineers. p. 63-75.

[13] Scheer J, Vayas I. Traglastversuche mit ausgesteiften Blechfeldern unter allseitiger Navierscher Lag-erung und konstanter Stauchung der Endquerschnitte. Schluβbericht Nr. 6036-1, Institut fu¨r Stahlbau,TU Braunschweig, 1982.

[14] Fukumoto Y. Ultimate compressive strength of stiffened plates. In: Specialty conference on metalbridges. New York: ASCE, 1974.

[15] Balaz I. Ausgesteifte Druckgurte von Kastentra¨gerbrucken. Stahlbau 1987;5:145–54.

design rules plate eurocode maquoi

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