Journal of Constructional Steel Research 57 (2001) 279–311 www.elsevier.com/locate/jcsr
New design rules for plated structures in Eurocode 3 Bernt Johansson a, Rene´ Maquoi b, Gerhard Sedlacek c,* a
˚ University of Technology, SE-971 87 Lulea˚ , Sweden Division of Steel Structures, Lulea˚ University b 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. Supplementary rules for planar plated structures without transverse loading have been developed together with the Eurocode 3-2 Steel bridges. It covers stiffened and unstiffened plates in common steel bridges and similar structures. This paper presents the background and justification of some of the design rules with focus on the ultimate limit states. The design rules for buckling of stiffened plates loaded by direct stress are presented and explained. For shear resistance and patch loading the new rules are briefly derived and compared with the rules in Eurocode 3-1-1. Finally, the statistical calibration of the rules to tests is described. © 2001 Elsevier Science Ltd. All rights reserved. Keywords: Steel structures; Plated structures; Design; Plate buckling; Stiffened plates; Shear buckling;
Patch loading
1. Introd Introduction uction
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 Part 1.5 (EC3-1-5) [1]. It has been drafted in close co-operation with the project team PT2 preparing the steel bridge code and it contains rules for stiffened or unstiffened plated structures. These rules are not specific for bridges, which is the reason for
* E-mail Corresponding Correspond ing author. Tel.: 49-241-80-5177; fax: +49-241-888-8140. + -aachen.de
[email protected] stb@stb .rwth-aac hen.de (G. Sedlacek). Sedlacek ). address: 0143-974X/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 9 7 4 X ( 0 0 ) 0 0 0 2 0 - 1
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making them a part of EC3-1, which contains general rules. The table of contents shown in Fig. 1 gives an overview of EC3-1-5. In addition there is an informative annex containing formulae for elastic buckling coefficients, which has been included for the convenience of the designer. Such coefficients may alternatively be found in handbooks or by computer calculations. All verifications are presented in Section 2 of EC3-1-5. For the ultimate limit states (ULS) the requirements are the same as in EC3-1-1. For the serviceability limit states (SLS) no requirements are given, only methods for finding stresses, etc. The requirements depend on the particular application; for instance, requirements for bridges are found in EC3-2 [2]. The focus of this paper is Section 4 of EC3-1-5, which contains methods for finding the resistance to plate buckling in ULS. The objective of the paper is to present the scientific background to the rules. First the mechanical models behind the rules are explained and references to source documents are given. All such models include simplifications, which had to be justified by calibration of the rules against testt res tes result ults. s. Sev Severa erall mod models els for eac each h fai failur luree mod modee hav havee bee been n che checke cked d wit with h cal caliibrations according to Annex Z of EC3-1-1 [3] and the ones chosen to be included in 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. Fig
Table Tab le of conte contents nts of Euroc Eurocode ode 3 Part Part 1.5.
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in so-called hybrid girders. No detail rules are given for the design of such girders but in all design rules, subscripts (f for flange and w for web) indicate the relevant yield strength. Although the rules may look unfamiliar to many engineers they are in fact only a new combination of rules from different European countries. For the time being they represent a set of useful rules for common plated structures. However, the rules are not complete in the sense that any type of plated structure is covered. There are also details that may be improved with existing knowledge but the time and funds available for the work have not allowed this. One such item is the formula for the effect eff ective ive are areaa of uns unstif tiffen fened ed pla plates tes.. The sin single gle form formula ula fro from m EC3 EC3-1-1 -1-1 has bee been n retain ret ained ed alt althou hough gh it has not bee been n har harmon monise ised d wit with h the sle slende nderne rness ss lim limit it bet betwee ween n cross section classes 3 and 4. A set of formulae for different boundary conditions and states of residual stresses should be developed but this has to wait until the ENversion is prepared.
2. Desi Design gn of sti stiffen ffened ed plates plates for direct stress stress
2.1.. Gene 2.1 General ral
Plates resisting predominantly direct stresses are used as flanges and webs of plateand box-girders. The distinction between flange and web is sometimes questionable. The definition used here is that a flange is subject to a distribution of direct stresses that is not very far from being uniform (no account being taken in this respect of shear-lag effects). A web is subject to a distribution of direct stresses with a significant gradient and most often a change from tension to compression. For very wide plates used as webs or flanges, it is sometimes more economical to stiffen a relatively thin plate than to increase the plate thickness in order to avoid any stiffening. A plate is normally first stiffened transversally, i.e. by stiffeners transverse to the direction of longitudinal stresses, and, when necessary, by additional longitudinal stiffeners. When the distribution of compressive stresses is quasi uniform, the longitudinal stiffeners are equally spaced. If not, the stiffeners are located in an optimum manner in order to combine efficiency and economy. The tra transv nsvers ersee sti stiffen ffeners ers are usu usuall ally y par parts ts of tra transv nsvers ersee bra bracin cings gs of the cro crosssssection of the structure and for this purpose they are normally stiff in bending. There is some advantage for them being designed to fulfil this requirement. Such transverse stiffeners are denoted rigid when when they constitute nodal lines for plate buckling under the action of compressive stresses. That is a first principle of the design rules of EC3-1-5. Accordingly, the amount of efforts devoted to check plate buckling is substantially reduced and facilitated. Possible instability is restricted to: buckling of the whole panel for unstiffened panels; buckling of unstiffened subpanels or buckling of the whole panel for longitudinally stiffened panels.
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In both cases the length a of the panel is equal to the distance between the transverse stiffeners defining this panel, see Fig. 2. Those edges are so-called loaded edges. The width B of the panel is the distance between its boundaries to adjacent panels or possibly between one such boundary and a free edge. The longitudinal edges of a panel are denoted unloaded edges. A subpanel is an unstiffened plate having the length a of the panel to which it belongs and a width b . The wording loaded/unloaded refers to loading by direct stresses. That is a second principle of EC3-1-5, the scope of which is plates subject to uniaxial direct stresses only. The case of plates subject to general biaxial loading is not included at the present time. However, there are rules for patch loading, including possible interaction with bending. For longitudinally stiffened panels two extreme cases concerning the stiffening are identified in EC3-1-5. 2.1.1. The case of equa 2.1.1. equally lly spaced multiple multiple sti stiffe ffener nerss in the com compres pressio sion n zon zonee When the behaviour of the longitudinally stiffened panel as a whole is considered the number of longitudinal stiffeners located in the compression zone is sufficiently large to justify the smearing of their flexural stiffness across the panel width. In addition, the behaviour of the subpanels has to be checked independently. For
design purposes, the multiple stiffener approach is generally accepted when the number of stiffeners is at least three. 2.1.2. The case of a few unequall 2.1.2. unequallyy spa spaced ced longitud longitudina inall sti stiffe ffeners ners in the compression zone Smearing of the stiffness would be too rough an approach in this case and is therefore not recommended. Instead, some account should be taken of the discrete location of the longitudinal stiffener(s). This method enables the designer to analyse special situations where the widths of the subpanels are very different because of a steep stress gradient across the panel width. It will be used especially when designing so-called web plate elements. That is in relation to a third principle, or better, with
Fig.. 2. Fig
Compon Com ponent entss of a stiffene stiffened d plate. plate.
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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 web would need a large number of unequally spaced longitudinal stiffeners in the compression zone. 2.2. Unstif Unstiffened fened panels or subpanels subpanels
This section refers to a subpanel according to Fig. 3. It is equally applicable to an unsti unstiffened ffened panel for which b should be replaced by B . For simpl simplicity icity,, the characteristic resistance is described and no partial safety factors appear. The resistance of an unstiffened subpanel of width b and thickness t is conventionally given by the squash load of the effective cross-sectional area ( bt )eff : N u(bt )eff f y
(1)
where (bt )eff is the effective cross-sectional area of the unstiffened (sub)panel and f y is the mate material rial yield stress. The possi possible ble effect of plate buckling buckling is clearly introduced introduced as a penalty on the gross cross-sectional area bt rather rather than on the magnitude of the stress at the ultimate limit state. The effective cross-sectional area (bt )eff of a subpanel is a part r(1) of the gross cross-sectional area (bt ): ): (bt )eff r(bt )
(2)
where r, termed the effectiveness of the cross-section, is computed using the wellknown Winter formula used in EC3-1-1:
r1/ l p0.22/ l p21
(3)
This form This formula ula acc accoun ounts ts for fav favoura ourable ble eff effect ectss res result ulting ing from pos post-b t-buck ucklin ling g pla plate te behaviour, on the one hand, and for detrimental effects of unavoidable structural and geometrical imperfections, on the other. The effectiveness r depends on a single parameter, the relative plate slenderness l p, which is defined as:
Fig. 3.
Unstiffene Unstif fened d subpan subpanel/pan el/panel. el.
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l p ( f y / s s cr cr )
(4)
Any elastic critical stress s cr cr is commonly written as:
s cr ss E cr k s
(5)
where k s s is the buckling coefficient and s E the so-called reference Eulerian stress: E
s E p 2 Et 2 /12(1v2)b2189800(t / b)2 (in N/mm2)
(6)
More explicitly, the relative plate slenderness l p writes:
l p(b / t t)/(28,4 ) /(28,4e k k s s)
(7)
and it involves:
the width-to-thickness ratio (b / t t ), ), that is to plate buckling what column slenderness is to column buckling, i.e. the governing parameter;
y the ), which indicatesyield that stress the relative slender e=√(235/ f with nessyield of a stress given factor plate increases the material ( f y, toplate be expressed in N/mm2); the buckling coefficient k s s, which amounts to 4 for a simply supported long plate subjec sub jectt to uni unifor form m com compre pressi ssion on but dep depend endss on the asp aspect ect ratio a =a / b of the subpanel or a / B of the (unstiffened) panel.
According Accord ing to the Win Winter ter for formul mula, a, the pen penalt alty y r applic applicabl ablee to the gros grosss cro crosssssection is seen to start ( r1) when the relative plate slenderness l p exceeds 0.673. Unfortunately, this limit does not coincide with the limit between section classes 3 and an d 4, wh whic ich h cr crea eate tess a di disc scon onti tinu nuit ity y in th thee de desi sign gn ru rule les. s. Th This is in inco cons nsis iste tenc ncy y is expected to be remedied in the final version of Eurocode 3. Thel validity thistions formula has been extended toant any typeing of coeffi boundary longitudina tudinal loading loadi ng of conditions condi by introd introducing ucing the relev relevant buckling buckl coefficien cienttand k s s, in Eq. (7). In practice, except when a plate edge is free, the conservative assumption of simply supported edges is usually made. That is due to the difficulty in assessing the mag magnit nitude ude of edg edgee res restra traint ints. s. Howe However ver,, the designer designer is fre freee to tak takee the edg edgee restraint into account if the value is justified. In addition, for a plate of constant thickness, any non-linear distribution of longitudinal direct stresses across the plate can be characterised by the stress ratio y , ratio of the extreme edge direct stresses. Finally, should the plate possibly be stiffened, then the properties of the stiffening would also affect the plate critical buckling stress. As a result, the buckling coefficient for the most general case is a function: k s s k s s (boundary conditions, a , y , stiffening properties)
(8)
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2.3. Longi Longitudina tudinally lly 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 Section 2.1, two distinct design approaches may be contemplated, the application of which is mainly governed by the number of longitudinal stiffeners in the compression zone of the stiffened panel. 2.3.1. Effect Effectiveness iveness of the stiffened stiffened panel Thee fa Th fact ct th that at a pl plat atee pa pane nell is st stif iffe fene ned d in th thee lo long ngit itud udin inal al di dire rect ctio ion n ma make kess it itss behaviour more like that of a column type structural element than that of a plate type element in the range of small slenderness. While plate-like behaviour exhibits a significant post-buckling resistance, which may largely exceed the elastic critical plate-buckling load, the elastic critical column buckling load is an upper bound of the 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 simple
interpolation between the value of the effectiveness r p for the plate-like behaviour, on the one hand, and the value c c for the column-like behaviour, on the other. This interpolation is empirically based but its suitability was supported by calibration:
rc( r p c c)x(2x) c c
(9)
The interpolation is governed by a factor x that measures the vicinity of the elastic critical plate buckling stress s cr,p cr,p to the elastic critical column buckling stress s cr,c cr,c according to:
x(s cr scr r ,c)1 0 x1 cr , p / s c
(10)
It is understandable that s cr,p should not beinsmaller than s cr,cdescribed . However, to approximations and simplifications included the procedures in owing Sections 2.3.2 and 2.3.3, this requirement is not necessarily fulfilled. In order to prevent the parameter x from being negative, 0 must be adopted as a lower bound. On the other hand, the effectiveness rc must increase from c c and approach r p when x increases. Therefore, x has 1 as an upper bound. How to assess the values of s cr,p cr,p, s cr,c cr,c, r p and c c is discussed in Sections 2.3.2 and 2.3.3. Because expression (10) does not reflect a monotonic decrease when x increases, it can be suggested, in a revised version of EC3-1-5, to simplify the process without a significant penalty on the results by adopting simply:
rc c c but not smaller than r p,
where r , is the value of r , computed for a very long stiffened plate, i.e. a plate p p where the buckling coefficient no longer depends on the plate aspect ratio.
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2.3.2. PlatePlate-like like behaviour behaviour Two methods are specified; each is especially applicable to specific types of stiffened panels. 2.3.2. 2.3 .2.1. 1. Mul Multip tiple le lon longit gitudi udinal nal sti stiffe ffener ners—C s—Conce oncept pt of equ equival ivalent ent ort orthot hotrop ropic ic pla plate te
basic ideaConceptually, is to smear thethat flexural stiffness thesubstitution longitudinalofstiffeners across theThe plate width. would lead toofthe the actual discretely stiffened plate by an orthotropic plate, referred to as the equivalent orthotropic plate in the following. Usually, multiple stiffeners are equally spaced or not far from being such. Then the properties of the equivalent orthotropic plate may be assume ass umed d uni uniform formly ly dis distri tribut buted ed acr across oss the wid width. th. The buc buckli kling ng coe coeffic fficien ientt for the stiffened panel, designated as k s s ,p , may be obtained by any means: computer analysis, appropriate charts [4,5] or simply by the following approximate expressions: 2((1+a 2)2+g ) k s if a a (1g )0.25 s , p 2 a (y +1)(1+ +1)(1+d ) k s s , p
where:
g = I x / I p
4(1+ 1+ 1+g )
if a a (1g )0.25
(11a)
(11b)
(y +1)(1+ +1)(1+d )
d = Asl / A p
a =a / B y =s 2 / s s1
relative flexura relative flexurall stiff stiffness ness, i.e. ratio of the second moment of area I x,oftheactualstiffenedpaneltothesecondmomentofarea I p(= Bt 3 /12(1v2)) of the plate for longitudinal bending relative relati ve crosscross-section sectional al area, i.e. ratio of the cross-sectional area Asl of the longitudinal stiffeners without any contribution of the plate to the cross-sectional area A p (= Bt ) of the plate aspect ratio edge stres stresss ratio, s 1 and s 2 being respectively the larger and the smaller edge stresses, see Fig. 4 (compression is
taken as positive).
Fig.. 4. Fig
Definiti Defi nition on of the the stres stresss ratio ratio y .
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The above expressions imply that, on the one hand, the wall elements of the stiffeners do not exhibit local buckling and that, on the other hand, no instability of any stiffener in its whole (stiffener tripping) occurs before the stiffened panel reaches its ultimate strength. In that case, the stiffeners are said to be fully effective; that is, a fourth principle of EC3-1-5. The full effectiveness of any stiffener can be achieved by complying with deemed-to-satisfy requirements. bucklingcatastrophic initiated by type tripping of open stiffeners is likely to result in a sudden Plate and so-called of collapse, which should be prohibited. However, closed stiffeners, e.g. trapezoidal boxes, may very well have class 4 sections because the local buckling of the walls of the stiffener will not trigger a collapse. In the case of such closed class 4 stiffeners the local buckling should should be consid considered ered in the same way as for subpanels subpanels of the plate 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 equivalent orthotropic plate is varying linearly across the panel width may look more appropriate than a uniform distribution. Then use shall be made of computer simulations or charts [5]. The elastic critical plate-like buckling stress s cr,p cr,p is:
s cr s, ps cr , pk s E E
(12)
where s E is given by Eq. (6) with B instead of b. When defining the relative plate slenderness l p,o of the equivalent orthotropic plate it should be taken into account that the critical stress is referred to the gross crosssectional area A and yield load to the effective cross-section Aeff . Hence, the relative plate slenderness l p, becomes:
l p,o ( Aeff f y / As cr scr r , p) ( b A f y / s cr , p) c
(13)
where b A= Aeff / A. In EC3-1-5, b A 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 stresses change sign. The equivalent orthotropic plate is characterised by an effectiveness ro for the plate-like behaviour:
ro1/ l p,o0.22/ l p2,o1
(14)
The symbols l p,o and ro are used instead of the EC3-1-5 symbols l p and r for the sake of clarity. 2.3.2.2. 2.3.2. 2. One or two stiffener stiffenerss in the comp compres ressio sion n zon zone—co e—concep nceptt of equ equiva ivalen lentt column on an elastic foundation The following following procedu procedure re is especial especially ly dedicated dedicated to situations where both the number and the location of the longitudinal stiffeners result from a notably non-uniform distribution of direct stresses, as in a web element. A special procedure is suggested which accounts for the discrete nature of the stiffening in a simple way. The elastic critical plate buckling stress s is no longer based on cr,p the concept of an equivalent orthotropic plate but on one of an equivalent column
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Fig.. 5. Fig
Physic Phy sical al model of a com compre pressio ssion n strut on an elastic elastic foundati foundation. on.
supported by an elastic foundation, see Fig. 5. The elastic critical column buckling stress of this equivalent column is used as an approximation of s cr,p cr,p. The properties of the equivalent column, including its elastic foundation, must be determined 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, on the other, can be satisfactorily accounted for. 2.3.2.3. Case of one stiffener 2.3.2.3. stiffener When there there is only only one longitud longitudinal inal stiffen stiffener er in the compre com pressi ssion on zon zone, e, the loc locati ation on of the equ equiva ivalen lentt col column umn is tha thatt of the stiffener stiffener.. For sake of simplicity, possible stiffeners in the tension zone are fully disregarded. Accordingly, the single stiffener divides the width B of the panel into two subpanels of width b1 and b2, respectively, and the elastic foundation is represented by the
plate, see Fig. 6.
Fig. 6. Notion Notional al cross-sec cross-sectional tional area of of equivalent equivalent column. column.
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The gross cross-section of the elastically founded equivalent column is used for the determination of the section properties (cross-sectional area A, second moment of area I slsl about an axis through the centroid and parallel to the plate sheet). It is composed of the gross cross-sectional area Asl of the stiffener and a notional crosssectional area of the plate sheet, that is determined as follows from both subpanels adjacent to the stiffener, see Fig. 6: half the width of the subpanel when fully in compression; 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 computation of b A. This consists of the effective parts of the plate adjacent to the stiffener and if the stiffener is partially effective only, due account shall be taken of an effective cross-sectional area of the stiffener. This may be the case if a closed stiffener is used. In the absence of any elastic foundation, the buckling length of the equivalent column would be equal to the distance a between the transverse stiffeners. It is noted that the latter are designed so as to be rigid, on the one hand, and that simple supports have been conservatively assumed for the stiffener, on the other hand. In addition, the variation, over the length a , of the compressive force in the stiffener is disregarded in the following. Owing to the plate effect, the buckling length ac of the equivalent column will be smaller than the distance a. In accordance with the physical model, it is found to be:
2 2 I sl slb1b2 ac4,33 t 3 B
0.25
(15)
The elastic critical column buckling stress that is taken as an appraisal of s cr,p cr,p is given as 1.05 E I slslt 3 B
s cr cr , p
Ab1b2
if a a ac
p 2 EI slsl
Et 3 Ba2 s cr if a a ac cr , p Aa2 4p 2(1−v2) Ab21b22
(16a)
(16b)
2.3.2.4. 2.3.2. 4. Cas Casee of two stiffene stiffeners rs When there there are two two stiffeners stiffeners in the the compression compression zone, 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 one acts as a rigid support. The value of s cr,p cr,p is given by Eq. (16) with b1=b1 and b b and B B is taken as the sum of b and b . To account for possible simul2 1 2 = 2 column = buckling of both stiffeners, taneous a second step is required. That is done ∗
∗
∗
∗
∗
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Fig.. 7. Fig
Proced Pro cedure ure for two stiffen stiffeners ers in the compress compression ion zone. zone.
by means of an intuitive conservative trick. In that step, both stiffeners are lumped into a single one having the following properties, see Fig. 7: the cross-sectional area A is the sum of those computed earlier for the individual stiffeners; the second moment of area I sl sl is the sum of those computed earlier for the individual stiffeners; the lumped stiffener is located at the position of the resultant of the forces in the individual stiffeners.
The whole procedure provides the designer with three values of s cr,p cr,p, of which the lowest one should be selected. 2.3.3. Column Column-like -like behaviour behaviour The elastic critical column-like buckling stress s cr,c cr,c is defined as the Euler stress for out-of-plane buckling of an equivalent column represented by the part of the stiffened plate that is in compression. 2 s cr Aca2 cr ,cp EI x,c /
(17)
where I xc is the second moment of area for longitudinal bending and Ac the gross area of the equivalent column. This buckling stress appears as a characteristic of the compression part of the stiffened panel, assuming that this part is released from any support along its longitudinal edges, is subject to uniform compression and has a buckling length equal to the length of the stiffened panel. That clearly appears as a setWhen of conservative assumptions. there is a significant gradient of the direct stresses along the length of the
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compressed part of the stiffened compressed stiffened panel panel,, undue conse conservati rvatism sm can be avoided by reducing appropriately the buckling length, which then becomes smaller than the length a of the stiffened panel. The relative column slenderness l p,c then writes:
l p,c ( b A f y / s scr r ,c) c
(18)
The effectiveness c c for the column-like column-like behaviour behaviour is given by the reduction factor factor for column buckling given as:
1
c c f+ (f2− l p2,c)
(19)
with f=0.5[1+a e( l p,c0.2)+ l p2,c]. Because of the non-symmetry about the buckling axis due to one-sided longitudinal stiffeners, on the one hand, and the nature of built-up section (the stiffeners being welded onto the plate), on the other hand, due allowance is made for a geometric imperfection larger than 1/1000 of the buckling length (the latter is the one covered implicitl implic itly y by the reg regula ularr Eur Europe opean an col column umn buc buckli kling ng cur curves ves). ). The ini initia tiall out out-of-ofstraightness accounted for is 1/500 of the buckling length, which is done by increasing the imperfection coefficient a e to [6]:
a ea 0[0.09/(i / / e)]
(20)
where: i ( I x,c / Ac)
and e is the largest of the distances from the neutral axis of the stiffened panel to thee ce th cent ntre re of th thee pl plat atee or th thee ce cent ntro roid id of th thee on onee-si side ded d lo long ngit itud udin inal al st stif iffe fene ners rs (alternatively of either set of stiffeners when both-sided stiffeners), see Fig. 8[7]. Becaus Bec ausee of the bet better ter sta stabil bility ity of clo closed sed sec sectio tion n sti stiffe ffener ners, s, dis distin tincti ction on is mad madee between types of stiffener sections according to: Curve b (a 0=0.34) for hollow section stiffeners
Fig. Fi g. 8.
Dist Di stan ance cess e1 and e2.
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Curve c (a 0=0.49) for open section stiffeners.
2.3.4. Effect Effective ive cross-sectional cross-sectional area of the stiff stiffened ened panel The effective cross-sectional area of the stiffened panel is composed of:
The gross cross-sectional area At,o of the part of the equivalent orthotropic plate located in the tension zone:
At ,o( Asl)t (
bt )t
(21)
where ( Asl)t is the gross cross-sectional area of all the stiffeners located in the tension zone, and ( S bt bt )t is the gross cross-sectional area of all the subpanels that are fully in tension. The effective cross-sectional area Aeff ,c,o rc Ac,o
(22)
of the part of the equivalent orthotropic plate located in the compression zone, where Ac,o accounts for possible plate buckling of the subpanels: Ac,o( Asl)eff ,c(
bt )eff ,c
where ( Asl)eff,c is the effective cross-sectional area of all the stiffeners located in the compression zone, and ( S bt bt )eff,c is the effective cross-sectional area of all the subpanels that are fully or partially in compression. For very wide flanges there is a further reduction of the effective area with respect to shear lag according to EC3-1-5.
3. Desi Design gn of plates plates for shear
3.1.. Gene 3.1 General ral
The resistance of slender plates to shear is based on the rotated stress field theory as propos proposed ed by Ho¨ glund [8]. It is a tensi tension on field theor theory y that is capab capable le of predic predicting ting the resistance of short as well as long panels and it replaces the two methods in EC3-1-1. At a certain slenderness the plate reaches its yield resistance but this does not necess nec essari arily ly mea mean n the max maximu imum m res resist istanc ance. e. Str Strain ain har harden dening ing and the con contri tribut bution ion from the flan flanges ges mak makes es it pos possib sible le to uti utilis lisee hig higher her res resist istanc ancee wit withou houtt exc excess essive ive deformations. In EC3-1-5 the maximum strength in shear is put to 0.7 f y for steels of grade S355 and lower. For higher grades the strain hardening is less pronounced y and there are no test results available. a more strength, 0.6 f area , is proposed. In EC3-1-1 there are specialHence, rules for rolledconservative beams for which a shear
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larger than web area is defined. That is another way of taking the increased resistance into account and it can not be combined with the above mentioned mentioned increased increased strength which refers to the geometrical web area. The partial safety factors g M M 0 and g M 1 have been suggested to have different values M in EC3-2: 1.0 and 1.1, respectively. The reason for this is a study of the statistical distribution yield is strength and and geometrical of beams, which justifies g M 0=1.0. Thisofresult quite new there hasproperties been no time for re-calibrating the resistance functions for various instability modes in order to use the same partial safety factor. This would be the rational solution but in the meantime a temporary solution has been introduced for the shear resistance. This is simply that the plateau is shifted in relation to the ratio between the partial safety factors. 3.2.. Rot 3.2 Rotate ated d str stress ess field theory theory for plain plain web
The rotated stress field theory was first developed for girders with slender webs with stiffeners at the supports only and for girders with transverse stiffeners but without horizontal stiffeners [8]. It was later widened to include such stiffeners [9]. First,, consi First consider der a girder with a slend slender er web and widely spaced spaced trans transverse verse stiffeners. stiffeners. The state of stress in the web caused by a shear force must be such that no vertical stresses appear at the edges. The state of pure shear that may exist for low loads rotates as shown in Fig. 9, for which the conditions of equilibrium are
s 1t /tan(j )
(23)
Fig. 9. Sta Fig. State te of stress stress in a slender slender gir girder der web after bucklin buckling. g. Vertical Vertical stiffene stiffeners rs are sup suppos posed ed to be widely spaced and no vertical stresses act on the flanges.
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s 2t tan( tan(j )
(24)
An observation from tests is that the compressive stress remains close to the critical shear stress and this is used as an assumption in the theory 2
2
s 2t cr p E 2 t w2 cr k t t
12(1−v )hw
(25)
The ultimate strength of a web is assumed to be reached when it yields according to the von Mises yield criterion
s 21s 21s 1s 2 f yw
(26)
From Eqs. (23)–(26) the shear resistance can now be solved to
t u
3
f v
4
lw
in which
1 −
1 4
1
−
4 lw
(27)
2 3 l2w
f v f y / 3
t
lw
f v cr cr
Eq. (27) is shown in Fig. 10 together with some test results for a girder with widely spaced vertical stiffeners. It is clear that the solid dots representing tests of girders with rigid end-posts fit very well with the prediction, while the tests with non-rigid end-posts do not. The reason for this is the resulting tension in the web, which has to 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 the tensile force N t t hwt w f v
l l 1
2
w
t u
w
f v
2
(28)
This force is larger than the actual force because the state of stress close to the flanges will be more like pure shear. The force has to be resisted by the end-post if the full strength should be developed. If the end-post consists of a single plate the resistance to shear will be less than predicted by Eq. (27). The resistance actually used in the designfor according to the EC3-1-5 is reduced slightly compared with Eq. (27) in order to allow scatter in test results and also for
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Fig. 10. She Fig. Shear ar resistanc resistancee acc accord ording ing to rot rotate ated d str stress ess field theory theory toge togethe therr wit with h tes testt res results ults for gir girder derss with widely spaced vertical stiffeners.
the systematic deviation for girders with non-rigid end-posts. This has been done by curve fitting using simpler expressions than in Eq. (27). The design resistance is given by V Rcd c v f ywd hwt w / 3
(29)
where c v= c w+ c f . c w is found in Table 1 and c f will be discussed later. 3.3. Contr Contributio ibution n of stiffe stiffeners ners
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 the buckling 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 is clear from test results that the effect of longitudinal stiffeners will be overestimated Table 1 Contribution from the web to shear resistance c w according to EC3-1-5
lw
Rigid end-post
lw0.83h h 0.83h lw1.08 0.83/ l w 1.08 lw 1.37/(0.7+ lw) h=1.20g M g M M 1 / g M 0 for S235, S275 and S355 h=1.05g M 1 / g g M 0 for S420 and S460
Non rigid end-post
h
0.83/ lw 0.83/ lw
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if the theoretical critical stress is used for calculating the slenderness parameter lw. There is less post-critical strength in a web with flexible stiffeners than in a plain web. This is dealt with by reducing the second moment of area of the longitudinal stiffeners to one third of the actual value when calculating the critical stress. This reduction has been considered in the following approximate formulae for the buckling coefficient from Annex A3 of EC3-1-5 k t t 5.344(hw / a)2k t t slsl k t t 45.34(a / hw)2k t t slsl
hw k t t sl 9 sl a
2
I sl sl t 2whw
3/4
2.1 I slsl but not less than t w hw
(30)
(31)
1/3
(32)
In (32) I slsl denotes the sum of the second moments of area of all longitudinal stiffeners. In addition to the check for buckling of the whole stiffened panel there is a check 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 EC3-15 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 both diagrams because the influence of the panel length is reflected only by its influence on the slenderness parameter lw. This is also true for the simple post critical resistance according to EC3-1-1, which is quite close to the resistance for a non-rigid end-post according to EC3-1-5. This is because EC3-1-1 does not have any requirements other than that there should be a stiffener at the end of the girder. The draw-
Fig. 11. Compa Comparison rison between between resistanc resistancee to shear without contributi contribution on from flanges flanges according according to EC31-1 and EC3-1-5 assuming g M M 1=g M 0. M
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back of the tension field method (it overestimates the resistance for short panels and underestimates it for long panels) has been eliminated by the method for a rigid endpost in EC3-1-5, which gives fair estimates of the resistance for any length of the panel, including girders with no intermediate transverse stiffeners. Another advantage of the new rules is that they are simpler to use. 3.4. Contr Contributio ibution n from flanges
The intermediate vertical stiffeners prevent the flanges from moving towards each other.. This effect is taken into account other account by adding a tensi tension on field that can be support supported ed by the flanges acting as beams supported by the stiffeners according to Fig. 12. This is a much smaller tension field than that of EC3-1-1 because the rotated stress field already catches the post buckling resistance of the web alone. After some simplifications the contribution from the flanges can be expressed as
c f
b f t f 2 f yf 3 ct h f
1
2
w w yw
c
M Sd Sd
(33)
M
f . Rd
1.6b f t f 2 f yf 0.25 2 a thw f yw
(34)
An example of the resistance to shear including the effect of the flanges is shown in Fig. 13. The resistance according to EC3-1-5 is compared with the one according to EC3-1-1. The simple post critical method does not take the contribution of the flanges into account. The tension field method does take the effect into account but in such a way that the effect disappears when the slenderness is lower than 0.8. This does not reflect the real behaviour of a girder.
Fig.. 12. Fig
Tension Ten sion field field support supported ed by the flange flanges. s.
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Fig.. 13. Res Fig Resista istance nce to to shear shear for a girde girderr with b f =25t f f , t f f =3t w and f yw yw= f yf =355 MPa, M Sd Sd =0 and assuming g M M 1=g M 0. M
4. Desi Design gn for pat patch ch loading loading
4.1.. Gene 4.1 General ral
The rules for the resistance of a web to patch loading are new in the Eurocode context 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 been merged into one verification. The new rules also cover a wider range of load applications and steel grades. The rules have been checked for steel grades up to S690 and there is no longer any need for the special formula for S460 in Annex D of EC3-1-1. 4.2.. Mod 4.2 Model el for patch loading loading resistance resistance
The design rules in EC3-1-5 cover three different cases of patch loading. Because of space limitation only the most common case is dealt with here, see Fig. 14. The design procedure includes the following parameters:
F y aa slenderness yield resistance parameter l=√F y / F F cr cr where F cr cr is the elastic buckling force
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Fig.. 14. Fig
299
Patch Pat ch loading. loading. Defini Definition tion of paramet parameters ers..
a resistance function c = c ( l) which reduces the yield resistance for l larger than a certain limiting value.
The characteristic resistance is written as F RF y c ( l)
(35)
and the parameters are written as
F y f ywt wl y
p 2 E t 3w
F cr cr k F F 12(1−v2)hw
0.47
c ( l)0.06
l
1
(36) (37) (38)
The expression in Eq. (38) was originally proposed [10] but during the drafting of EC3-1-5 it was simplified to 0.5
c ( l)
1
(39)
l The mechanical model according to Fig. 15 is used for the yield resistance. The
Fig.. 15. Fig
Mechan Mec hanica icall model for the yield resistan resistance ce for patch loading loading..
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mechanical model has four plastic hinges in the flange and the plastic moment resistance for the inner plastic hinges, M i, is calculated under the assumption that the flange flan ge alo alone ne con contri tribut butes es to the res resist istanc ance. e. For the out outer er pla plasti sticc hin hinges ges,, M o, it is assumed that a part of the web contributes to the resistance. This assumption is based on the observations from the tests that the length of the deformed part of the web increased when the web slenderness a simplified expression for M o, the effective loaded length l y, for theincreased. model in With Fig. 15 is given by l yss2t f (1 m1+m2)
(40)
where f yf b f m1 f ywt w
(41)
2
hw m20.02 t f
(42)
The buckling coefficient k F F in Eq. (37) was determined on the basis of the results from an FE analysis. The FE analysis included the influence from the stiffness of the flanges as well as the length of the applied load and expressions for k F F , where thee in th influ fluen ence ce of th thes esee pa para rame mete ters rs ar aree in incl clud uded ed ca can n be fo foun und d in Re Ref. f. 10 10.. Th Thes esee expressions were simplified in EC3-1-5 to
hw k F F 62 a
2
ss k F 6 F 26 hw
(43)
(44)
It is also necessary to consider the interaction with bending moment. This influence is accounted for by Eq. (45), from which it follows that when the ratio M s / M R 0.5 the bending moment has no influence on the patch load resistance. F s M s 0.8 1.4 F R M R
(45)
The resistance to patch loading according to EC3-1-5 is compared with that of EC31-1 in Fig. 16, which shows the quotient between the two resistances as a function of flange thickness over web thickness. The left diagram for zero loaded length shows a fairly large difference between the two design methods. The right diagram for a loaded length 0.2 times the web depth, a more realistic case, shows a large difference only for of a stocky web.
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Fig. 16.
301
Comparison Compar ison between between patch loading loading resistance resistance according according to EC3-1-5 EC3-1-5 and EC3-1-1 EC3-1-1 for a girder
with b f / t t f =25 and a large distance between vertical stiffeners.
5. Cal Calibr ibrati ation on of des design ign rules versus versus tes testt resu results lts
5.1.. Gene 5.1 General ral
The new design rules provided in EC3-1-5 were calibrated versus test results by a statistical evaluation according to Annex Z of EC3-1-1, which uses the following definitions and assumptions. It is assumed that both the action effects S and and the resistance R of a structure are subject to statistical normal distributions, which are characterised by mean values “mTo ” and standard deviations “s ”, ”, see 17. effects S and guarantee that the distribution ofFig. the action and the resistance R have a sufficient safety distance a safety index b is defined in EC1-1 as follows:
b
m R−mS
s s +s
3.8
2
2
R
S
(46)
where mS is the mean value of the action effect, m R is the mean value of the resistance, s S S is the standard deviation of the action effect and s R R is the standard deviation of the resistance. The safety requirement for a structure is defined by the criterion [ Rd ][S d d] 0 S d ] are values where [ Rd ] and design in values. To define the [design Eq. (47), Eq. (46) may be expressed by
(47)
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Fig.. 17. Fig
Statis Sta tistic tical al distribut distribution ion of the action action effects effects S and and the resistances R.
s R
bs R R s s R 2 +s 2S
m R
mS
−s S S
bs S S
s s +s 2
2
R
S
0
(48)
With the notations
a R
s R R
s s +s
a S S
2
2
R
S
s S S
s +s s 2
2
R
S
it is possible to express the design values as Rd m Ra R R bs R
(49)
S d d mS a S S bs S S
(50)
R With and a S = 0.7 the designfrom values of the action a be =0.8 and ofthe theapproximations resistances can described independently each other and aeffects more
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detailed investigation of the design value of the resistance can be carried out using the statistical procedure given in Annex Z of EC3-1-1. In a first step of this procedure a resistance function r t t =g R( x¯ ), the so called design ¯ ), model for the resistance, has to be established. This is an arithmetic description of the influence of all relevant parameters x¯ ¯ on the resistance r which is investigated by tests. tes ts. By compar com paring ing the streng ength th values ues from the mean resistance resistanc e fun functi ction on r t factor using measured input data with test str results r eval , see Fig. 18, the value correction b¯ for the resistance function r t t and the standard deviation S d d for the deviation term d can be determined. This gives the following formula describing the field Rb¯ r t t d
(51)
In most cases the probabilistic density distribution of the deviation term d cannot be described by a single normal distribution as is assumed in Fig. 17. It may be represented by a non-normal distribution, which may be interpreted as a composition of two or more norma normall distri distributio butions. ns. Therefore, Therefore, the density distribution distribution for the resistance is checked by plotting the measured probability distribution on Gaussian paper. If the plot shows a straight line, the actual distribution corresponds to a unimodal ¯ and S d ) are determined with normal function as assumed and the statistical data (b¯ the 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. Fi g. 18 18..
Plot Pl ot of r r er t t values, mean value correction b¯ and standard deviation S d d of the deviation term d .
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the design point is determined by a tangent to the lower tail of the measured distribution, see Fig. 19. The statistical data b¯ and S d d of the relevant normal distribution are then determined from the tangent approach to the actual distribution. In general, the test population is not representative of the total population of struc¯ b and not tures therefore is only used to determine the mean deviation the scatterand value S d d of the design model. To consider scattervalue effects of parameters sufficiently represented by the test population the standard deviation of the resistance has to be increased. To this end, in addition to the standard deviation S d d , the following variation coefficients are taken into account for the yield strength and geometrical values:
n fy = 0. 0.07 07 for st stre reng ngth th f y nt = 0. 0.05 05
forr th fo thic ickn knes esss t
nb = 0.00 0.005 5 for width width b nh = 0.00 0.005 5 for dept depth hh These variation coefficients are combined with the standard deviation S d d according to Eq. (52)
r ei / r rti -values on Gaussian paper and definition of the relevant normal distribution at the Fig. 19 Fig. 19.. Pl Plot ot of r design point.
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s R (ni)2+S 2d
(52)
Using a log normal distribution for R the characteristic value Rk of the resistance function may be represented by the 5% fractile value and can be obtained from Eq. (53) Rk b¯ m R exp(1.64s R0.5s R2 )
(53)
Also, the design value Rd of the resistance function may be defined by 2 Rd b¯ m R exp(a R R bs R0.5s R R)
(54)
where a R b = 0.8·3.8 = 3.04. The g M M -value of the resistance function is obtained from the ratio of the characteristic value to the design value Rk Rd
g M
(55)
In most cases instead of a 5% fractile value Rk a value Rnom with nominal values for the input parameters is used as characteristic value. To consider Rnom instead of Rk a modified partial safety factor g M * is used from:
g M k g g M
∗
(56)
where k = Rnom / Rk . For the resistance functions for plate buckling k may be expressed by: 2 exp(−2.0s fy−0.5s fy ) 0.867 k b¯ exp(−1.64s R−0.5s R2 ) b¯ exp(−1.64s R−0.5s R2 )
(57)
The procedure explained above is used in the following to determine the g M values for the resistance functions for plate buckling due to compression stresses, shear buckling and buckling due to patch loading. Where g M is not in compliance with the standard value g M =1.10 used for stability checks, the function Rnom is subsequently modified to reach the standard value g M . ∗
∗
∗
∗
5.2.. Cal 5.2 Calibr ibrati ation on of the design rules for shea shearr buc buckli kling ng
The design rules for shear buckling were checked versus test results according to the procedure given in Section 5.1. For the statistical evaluation the test results were obtained obtai ned from a data bank given by Ho¨ glund [9] which contains contains 166 test result resultss for the following types of steel plate girders:
girders girders with with transverse stiffeners at support only; intermediate stiffeners;
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girders with longitudinal and transverse stiffeners.
Only 150 of 166 tests could be used for the statistical evaluation because some of these specimens did not fail by shear buckling. Sinc Si ncee th thee pro proce cedu dure re of th thee de desi sign gn mo mode dell fo forr sh shea earr bu buck ckli ling ng de depe pend ndss on th thee arrangement of stiffeners, thesubsets available results were subdivided subsets, see Fig. 20. For these individual thetest statistical evaluations wereinto carried out and the statistical results are presented in Fig. 21. The figure shows that for all subsets g M -values lower than 1.10 were determined so that a g M -value of 1.10 can be applied in the design model g M . The sensitivity of the design model to the variation of the yield strength f yw was checked by plotting the ratio V ei Vtit i versus the yield strength of the web f yw (Fig. ei / V 22). Owing to the small variation of the mean values of V ei Vtit i the conclusion can e / i V be drawn that the influence of f yw is adequately considered in the design model. ∗
∗
∗
5.3.. Cal 5.3 Calibr ibrati ation on of the design rules for pat patch ch loa loadin ding g
The statistical evaluation for the design model of patch loading was carried out with 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 loaded by patch loading, end patch loading or opposite patch loading, see Fig. 23.
Fig.. 20. Sub Fig Subset setss of test test resu results lts..
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Fig. 21. g M -values for the design model of shear buckling. ∗
Fig.. 22. Fig 22.
Sensit Sen sitivit ivity y plot plot for for f yw.
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Fig.. 23. Fig
Differ Dif ferent ent cases cases of patch patch loadin loading. g.
According to the data base and the various design models the test results were subdivided into the following subsets: Data set Data set 1: 1: Data Da ta set set 2: Data Da ta set set 3:
Patch Patc h loa loadi ding ng End En d patc patch h load loadin ing g Oppo Op posi site te pa patc tch h load loadin ing g
For these subsets the statistical evaluations were carried out, and a summary of the statistical results is presented in Fig. 24. The figure shows that for all three subsets a g M -value of 1.10 is justified. In Fig. 25 a sensitivity plot is given for the slenderness parameter l¯ which shows that the scatter of the ratio F ei Ftit i is only slightly influenced by the slenderness parae / i F meter l. ∗
Fig. 24. g M -values for the design model of patch loading. ∗
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Fig.. 25. Fig
309
Sensiti Sen sitivit vity y plot for the the slender slendernes nesss l.
5.4.. Cal 5.4 Calibr ibrati ation on of the design rules for buckling buckling of sti stiffe ffened ned plates
The calibration of the design model for plate buckling was carried out with test results for multiple longitudinally stiffened steel plate girders in compression which were obtained from a literature study. Unfortunately, not all of the collected tests could be used to check the design model, because either some relevant data were not given in the test reports or the tests were carried out with additional initial imperfections which are not considered in the design model. Finally 25 tests were applicable to calibrate the design model. In these tests the longitudinal stiffeners were designed as bulb flats, flats or angles, 2. Some of these stiffeners do not fulfil the design recommendations givensee in Table 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 determined by taking into account the local buckling of both subpanels and stiffeners. In ad addi diti tion on,, th thee sh shif ifti ting ng of th thee ne neut utra rall ax axis is of th thee st stif iffe fene ned d st stee eell pl plat atee du duee to local buckling was considered using the interaction formula for bending and axial compression which is provided in EC3-1-1, see Eq. (58). N s k y N S Se N 1.0 rc Ac f y W eff f y eff
where: N k y1 r m A f S 1.5 c c y y
(58)
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Table 2 Test results for longitudinal stiffened steel plates in compression Authors
Test
Dorman and Dwight
3 (TPA3)
No of tests
Stiffener types
4
All tests with bulb flats
3
All tests with bulb flats
6
All tests with flats
[12]
4 (TPA4) 7 (TPB3) 8 (TPB8) Scheer and Vayas [13] 1 (III A 50-70) 2 (III A 75-70) 3 (III A 75-100) Fukumoto [14] B-1-1 B-1-1r B-2-1 B-3-1 C-1-4 C-2-1 Lutteroth [15] All tests
12
9 tests with flat plates 3 tests with flat plates and angles
m y l¯ c(2 b M , y4)0.9 b M , y1.1 The statistical evaluation was carried out for the different groups of tests given in Table 2. The results of the statistical evaluations are presented in Fig. 26 and show that a g M -value of 1.10 can be applied in the design model. ∗
Fig.. 26. Fig
Statis Sta tistic tical al results results for the design design model of plate plate buckling buckling..
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6. Conclu Conclusions sions
The first common European pre-standard considering plate buckling has been published for trial application. In addition to widening the scope to stiffened plates it also includes some improvements of the present rules in EC3-1-1. The design rules have been verified calibrations tests, which include also steel grades above the present limit ofby S460. This willtofacilitate a future widening of EC3 to higher steel grades. The test application shows that this first version of EC3-1-5 can be improved in several respects. The first author has already received several questions and remarks indicating the need for clarification and corrections. Further comments and remarks are welcomed and they may be sent to the first author for consideration in the ENversion of EC3-1-5.
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