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Design Rules Plate Eurocode Maquoi | Buckling | Strength Of Materials
Design Rules Plate Eurocode MaquoiUploaded by Tran Khanh LeRelated InterestsBucklingStrength Of MaterialsStandard DeviationYield (Engineering)Stress (Mechanics)Rating and Stats0.0 (0)Document ActionsDownloadShare or Embed DocumentEmbedView MoreCopyright: Attribution Non-Commercial (BY-NC)List price: $0.00Download as PDF, TXT or read online from ScribdFlag for inappropriate contentJournal of Constructional Steel Research 57 (2001) 279–311 www.elsevier.com/locate/jcsr
´ Bernt Johansson a, Rene Maquoi b, Gerhard Sedlacek
˚ ˚ Division of Steel Structures, Lulea University of Technology, SE-971 87 Lulea, Sweden 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. 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 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
* Corresponding author. Tel.: +49-241-80-5177; fax: +49-241-888-8140. E-mail address: stb@stb.rwth-aachen.de (G. Sedlacek).
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
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 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 test results. Several models for each failure mode have been checked with calibrations 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
Table of contents of Eurocode 3 Part 1.5.
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 effective area of unstiffened plates. The single formula from EC3-1-1 has been retained although it has not been harmonised with the slenderness limit between 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. Design of stiffened plates for direct stress 2.1. General 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 transverse stiffeners are usually parts of transverse bracings of the crosssection 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 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.
For longitudinally stiffened panels two extreme cases concerning the stiffening are identiﬁed in EC3-1-5. Those edges are so-called loaded edges. the behaviour of the subpanels has to be checked independently. the scope of which is plates subject to uniaxial direct stresses only. For design purposes.
. In addition. 2. / Journal of Constructional Steel Research 57 (2001) 279–311
In both cases the length a of the panel is equal to the distance between the transverse stiffeners deﬁning this panel. The case of a few unequally spaced longitudinal stiffeners in the compression zone Smearing of the stiffness would be too rough an approach in this case and is therefore not recommended. The case of plates subject to general biaxial loading is not included at the present time. The longitudinal edges of a panel are denoted unloaded edges. including possible interaction with bending. with
Fig. 2. A subpanel is an unstiffened plate having the length a of the panel to which it belongs and a width b. However. some account should be taken of the discrete location of the longitudinal stiffener(s).1. there are rules for patch loading. The wording loaded/unloaded refers to loading by direct stresses. the multiple stiffener approach is generally accepted when the number of stiffeners is at least three. Johansson et al. 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. or better.1. 2. 2. see Fig. The case of equally spaced multiple stiffeners in the compression zone 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.1. Instead. That is in relation to a third principle.
Components of a stiffened plate.282
B. It will be used especially when designing so-called web plate elements.2. That is a second principle of EC3-1-5. 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.
/ Journal of Constructional Steel Research 57 (2001) 279–311
the scope of the design rules of EC3-1-5. Johansson et al. is computed using the wellknown Winter formula used in EC3-1-1: r 1/lp 0.B. Unstiffened panels or subpanels This section refers to a subpanel according to Fig. The effectiveness r depends on a single parameter. 3. Unstiffened subpanel/panel.
. the relative plate slenderness lp. It is equally applicable to an unstiffened panel for which b should be replaced by B. 3. and for detrimental effects of unavoidable structural and geometrical imperfections.2. which is deﬁned as:
Fig.22/l2 1 p (3)
This formula accounts for favourable effects resulting from post-buckling plate behaviour. 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. on the other. The latter are devoted to normal structures. 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: Nu (bt)efffy (1) where (bt)eff is the effective cross-sectional area of the unstiffened (sub)panel and fy is the material yield stress. the characteristic resistance is described and no partial safety factors appear. termed the effectiveness of the cross-section. on the one hand. The possible effect of plate buckling is clearly introduced as a penalty on the gross cross-sectional area bt 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. For simplicity. 2.
in Eq. a. In practice. should the plate possibly be stiffened. which amounts to 4 for a simply supported long plate subject to uniform compression but depends on the aspect ratio a=a/b of the subpanel or a/B of the (unstiffened) panel. the conservative assumption of simply supported edges is usually made. this limit does not coincide with the limit between section classes 3 and 4. which indicates that the relative plate slenderness of a given plate increases with the material yield stress (fy.284
B. Johansson et al. This inconsistency is expected to be remedied in the ﬁnal version of Eurocode 3. y. / Journal of Constructional Steel Research 57 (2001) 279–311
(fy/scr)
Any elastic critical stress scr is commonly written as: scr kssE (5)
where ks is the buckling coefﬁcient and sE the so-called reference Eulerian stress: sE p2Et2/12(1 v2)b2 189800(t/b)2 (in N/mm2) More explicitly. the buckling coefﬁcient ks. (7). any non-linear distribution of longitudinal direct stresses across the plate can be characterised by the stress ratio y. the yield stress factor e=√(235/fy). stiffening properties) (8) (7) (6)
. the designer is free to take the edge restraint into account if the value is justiﬁed.4e ks) and it involves: the width-to-thickness ratio (b/t). that is to plate buckling what column slenderness is to column buckling. the relative plate slenderness lp writes: lp (b/t)/(28.673. That is due to the difﬁculty in assessing the magnitude of edge restraints. Finally. the buckling coefﬁcient for the most general case is a function: ks ks (boundary conditions. ratio of the extreme edge direct stresses. Unfortunately. the governing parameter. The validity of this formula has been extended to any type of boundary and longitudinal loading conditions by introducing the relevant buckling coefﬁcient ks. In addition. However. According to the Winter formula. As a result. then the properties of the stiffening would also affect the plate critical buckling stress. i. to be expressed in N/mm2). for a plate of constant thickness. the penalty r applicable to the gross crosssection is seen to start (r 1) when the relative plate slenderness lp exceeds 0.e. which creates a discontinuity in the design rules. except when a plate edge is free.
c) 1 0 x 1 (10)
It is understandable that scr.c. owing to approximations and simpliﬁcations included in the procedures described in Sections 2. However. it can be suggested. to simplify the process without a signiﬁcant penalty on the results by adopting simply: rc cc but not smaller than rp. is the value of rp. a plate where the buckling coefﬁcient no longer depends on the plate aspect ratio.p. the effectiveness rc must increase from cc and approach rp when x increases. This interpolation is empirically based but its suitability was supported by calibration: rc (rp cc)x(2 x) cc (9)
The interpolation is governed by a factor x that measures the vicinity of the elastic critical plate buckling stress scr.
.3. only the behaviour of the stiffened panel as a whole is of concern.c according to: x (scr.3. the application of which is mainly governed by the number of longitudinal stiffeners in the compression zone of the stiffened panel. rp and cc is discussed in Sections 2.3.c. 0 must be adopted as a lower bound.p should not be smaller than scr.3. An individual (unstiffened) subpanel is treated in Section 2.1.2. Therefore. Effectiveness of the stiffened panel The 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 plate type element in the range of small slenderness. two distinct design approaches may be contemplated.p/scr. i. and the value cc for the column-like behaviour. The behaviour of any stiffened panel lies somewhere between these two limits. the effectiveness of the stiffened panel is obtained by a simple interpolation between the value of the effectiveness rp for the plate-like behaviour. 2.3. scr. On the other hand. which may largely exceed the elastic critical plate-buckling load. in a revised version of EC3-1-5. In order to prevent the parameter x from being negative. Longitudinally stiffened panels In this section. / Journal of Constructional Steel Research 57 (2001) 279–311
2. on the one hand. computed for a very long stiffened plate. For design purposes. Because expression (10) does not reﬂect a monotonic decrease when x increases.1. As mentioned in Section 2.3.B. where rp. x has 1 as an upper bound. this requirement is not necessarily fulﬁlled. While plate-like behaviour exhibits a signiﬁcant post-buckling resistance.2 and 2. Johansson et al.3. How to assess the values of scr.2 and 2. on the other.p to the elastic critical column buckling stress scr. the elastic critical column buckling load is an upper bound of the resistance.e.3.
multiple stiffeners are equally spaced or not far from being such.3.2.286
B. ratio of the cross-sectional area Asl of the longitudinal stiffeners without any contribution of the plate to the cross-sectional area Ap (=Bt) of the plate aspect ratio edge stress ratio. i. of the actual stiffened panel to the second moment of area Ip(=Bt3/12(1 v2)) of the plate for longitudinal bending relative cross-sectional area. 4 (compression is taken as positive).p.p 2((1+a2)2+g) if a (1 g)0. each is especially applicable to speciﬁc types of stiffened panels.25 a2(y+1)(1+d) 4(1+ 1+g) ks.p where: g=Ix/Ip d=Asl/Ap relative ﬂexural stiffness. 4. 2.
.e. appropriate charts [4. Multiple longitudinal stiffeners—Concept of equivalent orthotropic plate The basic idea is to smear the ﬂexural stiffness of the longitudinal stiffeners across the plate width.5] or simply by the following approximate expressions: ks.3. may be obtained by any means: computer analysis. Usually.2. The buckling coefﬁcient for the stiffened panel.25 (11a)
a=a/B y=s2/s1
Fig.1.e. Then the properties of the equivalent orthotropic plate may be assumed uniformly distributed across the width. ratio of the second moment of area Ix. Plate-like behaviour Two methods are speciﬁed. Johansson et al. s1 and s2 being respectively the larger and the smaller edge stresses. (y+1)(1+d) if a (1 g)0. see Fig. / Journal of Constructional Steel Research 57 (2001) 279–311
2. referred to as the equivalent orthotropic plate in the following. that would lead to the substitution of the actual discretely stiffened plate by an orthotropic plate. designated as ks. Conceptually.
Deﬁnition of the stress ratio y. i.
the stiffeners are said to be fully effective. a fourth principle of EC3-1-5.o (Aefffy/Ascr.p) (bAfy/scr.22/l2 1 p. One or two stiffeners in the compression zone—concept of equivalent column on an elastic foundation The following procedure is especially dedicated to situations where both the number and the location of the longitudinal stiffeners result from a notably non-uniform distribution of direct stresses. In the case of such closed class 4 stiffeners the local buckling should be considered in the same way as for subpanels of the plate.B. Johansson et al. However. the relative plate slenderness lp. on the one hand. closed stiffeners.o 0. Possibly a plate can be ﬁtted with notably unequally spaced multiple stiffeners. e. the wall elements of the stiffeners do not exhibit local buckling and that.p is no longer based on the concept of an equivalent orthotropic plate but on one of an equivalent column
. When deﬁning the relative plate slenderness lp.g.p ks. Hence.p is: scr. as in a web element. A special procedure is suggested which accounts for the discrete nature of the stiffening in a simple way. (6) with B instead of b. In that case. / Journal of Constructional Steel Research 57 (2001) 279–311
The above expressions imply that.3. may very well have class 4 sections because the local buckling of the walls of the stiffener will not trigger a collapse. In EC3-1-5. The full effectiveness of any stiffener can be achieved by complying with deemed-to-satisfy requirements. becomes: lp. Then use shall be made of computer simulations or charts [5]. that is.o (14)
The symbols lp. The equivalent orthotropic plate is characterised by an effectiveness ro for the plate-like behaviour: ro 1/lp. The elastic critical plate-like buckling stress scr. 2. trapezoidal boxes.psE (12)
where sE is given by Eq. 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. no instability of any stiffener in its whole (stiffener tripping) occurs before the stiffened panel reaches its ultimate strength.o and ro are used instead of the EC3-1-5 symbols lp and r for the sake of clarity.2. which leads to a smaller value than if the whole plate had been considered in case the stresses change sign. The elastic critical plate buckling stress scr. on the other hand.2. bA is calculated only for the compressed part of the plate. which should be prohibited.p) (13)
where bA=Aeff/A. Plate buckling initiated by tripping of open stiffeners is likely to result in a sudden and so-called catastrophic type of collapse.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.
and the elastic foundation is represented by the plate. For sake of simplicity. including its elastic foundation. can be satisfactorily accounted for. must be determined so that both the number and location of the stiffeners.
. / Journal of Constructional Steel Research 57 (2001) 279–311
Fig. see Fig. possible stiffeners in the tension zone are fully disregarded. 6. on the one hand. Accordingly. Case of one stiffener When there is only one longitudinal stiffener in the compression zone.3. 2. on the other.
Fig. 6.2.3. The properties of the equivalent column. see Fig. respectively. Johansson et al. 5.
supported by an elastic foundation.p.288
Physical model of a compression strut on an elastic foundation. and the behaviour of the plate sheet in the direction transverse to the stiffeners. The elastic critical column buckling stress of this equivalent column is used as an approximation of scr. 5. the location of the equivalent column is that of the stiffener. the single stiffener divides the width B of the panel into two subpanels of width b1 and b2.
Notional cross-sectional area of equivalent column.
To account for possible simul2 1 2 taneous column buckling of both stiffeners.3. Johansson et al. Case of two stiffeners When there are two stiffeners in the compression zone. This may be the case if a closed stiffener is used. This consists of the effective parts of the plate adjacent to the stiffener and if the stiffener is partially effective only. It is composed of the gross cross-sectional area Asl of the stiffener and a notional crosssectional area of the plate sheet. the buckling length of the equivalent column would be equal to the distance a between the transverse stiffeners.4. see Fig. that is determined as follows from both subpanels adjacent to the stiffener. the buckling length ac of the equivalent column will be smaller than the distance a. on the other hand. In accordance with the physical model.2. each of these stiffeners is considered assuming that the other one acts as a rigid support. 7.p is given by Eq. second moment of area Isl about an axis through the centroid and parallel to the plate sheet). the procedure described above is applied three times. In addition. it is found to be: ac 4. due account shall be taken of an effective cross-sectional area of the stiffener. on the one hand. (16) with b1=b∗ and 1 b2=b∗ and B=B∗ is taken as the sum of b∗ and b∗. In a ﬁrst step.p scr. The value of scr. of the compressive force in the stiffener is disregarded in the following. / Journal of Constructional Steel Research 57 (2001) 279–311
The gross cross-section of the elastically founded equivalent column is used for the determination of the section properties (cross-sectional area A. 6: half the width of the subpanel when fully in compression.p 1.33 Islb2b2 1 2 t3B
0.B. The effective cross-sectional area of the equivalent column is used for the computation of bA. In the absence of any elastic foundation. a second step is required. Owing to the plate effect.05E Islt3B Ab1b2 if a ac (16a) (16b)
p2EIsl Et3Ba2 if a ac 2 2 Aa 4p (1−v2)Ab2b2 1 2
2. see Fig. one third of the width of the sole compressed part of the subpanel when stresses change from compression to tension. and that simple supports have been conservatively assumed for the stiffener. the variation.p is given as scr. That is done
The elastic critical column buckling stress that is taken as an appraisal of scr. It is noted that the latter are designed so as to be rigid. over the length a.
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.3. The whole procedure provides the designer with three values of scr.c p2EIx. This buckling stress appears as a characteristic of the compression part of the stiffened panel. 7. Johansson et al. 7: the cross-sectional area A is the sum of those computed earlier for the individual stiffeners. the second moment of area Isl is the sum of those computed earlier for the individual stiffeners. / Journal of Constructional Steel Research 57 (2001) 279–311
Fig. When there is a signiﬁcant gradient of the direct stresses along the length of the
.c/Aca2 (17)
where Ixc is the second moment of area for longitudinal bending and Ac the gross area of the equivalent column. In that step. of which the lowest one should be selected.
Procedure for two stiffeners in the compression zone. see Fig.p.3. is subject to uniform compression and has a buckling length equal to the length of the stiffened panel. the lumped stiffener is located at the position of the resultant of the forces in the individual stiffeners. assuming that this part is released from any support along its longitudinal edges. both stiffeners are lumped into a single one having the following properties.290
by means of an intuitive conservative trick. Column-like behaviour The elastic critical column-like buckling stress scr. That clearly appears as a set of conservative assumptions. 2. scr.
c/Ac) (20)
and e is the largest of the distances from the neutral axis of the stiffened panel to the centre of the plate or the centroid of the one-sided longitudinal stiffeners (alternatively of either set of stiffeners when both-sided stiffeners). distinction is made between types of stiffener sections according to: Curve b (a0=0.
.34) for hollow section stiffeners
Fig. 8[7]. and the nature of built-up section (the stiffeners being welded onto the plate).09/(i/e)] where: i (Ix.c 0. 8. The initial out-ofstraightness accounted for is 1/500 of the buckling length.B. which is done by increasing the imperfection coefﬁcient ae to [6]: ae a0 [0.c (19)
with f=0. see Fig. on the one hand.c then writes: lp. due allowance is made for a geometric imperfection larger than 1/1000 of the buckling length (the latter is the one covered implicitly by the regular European column buckling curves). Johansson et al. on the other hand. / Journal of Constructional Steel Research 57 (2001) 279–311
compressed part of the stiffened panel.
Distances e1 and e2.5[1+ae(lp.2)+l2 ]. which then becomes smaller than the length a of the stiffened panel. Because of the better stability of closed section stiffeners. undue conservatism can be avoided by reducing appropriately the buckling length.c (bAfy/scr.c) (18)
The effectiveness cc for the column-like behaviour is given by the reduction factor for column buckling given as: cc 1 f+ (f2−l2 ) p. The relative column slenderness lp. p.c Because of the non-symmetry about the buckling axis due to one-sided longitudinal stiffeners.
of the part of the equivalent orthotropic plate located in the compression zone.o (Asl)t ( bt)t (21)
where (Asl)t is the gross cross-sectional area of all the stiffeners located in the tension zone. Johansson et al.o of the part of the equivalent orthotropic plate located in the tension zone: At. / Journal of Constructional Steel Research 57 (2001) 279–311
Curve c (a0=0. a more conservative strength.49) for open section stiffeners.c is the effective cross-sectional area of all the stiffeners located in the compression zone. In EC3-1-5 the maximum strength in shear is put to 0. The effective cross-sectional area Aeff. 0. where Ac.7fy for steels of grade S355 and lower.c
where (Asl)eff. In EC3-1-1 there are special rules for rolled beams for which a shear area
. Hence. It is a tension ﬁeld theory that is capable of predicting ¨ the resistance of short as well as long panels and it replaces the two methods in EC3-1-1. Effective cross-sectional area of the stiffened panel The effective cross-sectional area of the stiffened panel is composed of: The gross cross-sectional area At.
2. Strain hardening and the contribution from the ﬂanges makes it possible to utilise higher resistance without excessive deformations.6fy. For higher grades the strain hardening is less pronounced and there are no test results available. is proposed.4.o accounts for possible plate buckling of the subpanels: Ac.1.
3. At a certain slenderness the plate reaches its yield resistance but this does not necessarily mean the maximum resistance.c. and (Sbt)t is the gross cross-sectional area of all the subpanels that are fully in tension. General The resistance of slender plates to shear is based on the rotated stress ﬁeld theory as proposed by Hoglund [8]. For very wide ﬂanges there is a further reduction of the effective area with respect to shear lag according to EC3-1-5. and (Sbt)eff.o rcAc. Design of plates for shear 3.c ( bt)eff.c is the effective cross-sectional area of all the subpanels that are fully or partially in compression.o (Asl)eff.3.292
This is simply that the plateau is shifted in relation to the ratio between the partial safety factors.
. Rotated stress ﬁeld theory for 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]. First. for which the conditions of equilibrium are s1 t/tan(j) (23)
Fig. That is another way of taking the increased resistance into account and it can not be combined with the above mentioned increased strength which refers to the geometrical web area. The reason for this is a study of the statistical distribution of yield strength and geometrical properties of beams. respectively. This would be the rational solution but in the meantime a temporary solution has been introduced for the shear resistance. The state of stress in the web caused by a shear force must be such that no vertical stresses appear at the edges. State of stress in a slender girder web after buckling. It was later widened to include such stiffeners [9]. This result is quite new and there has been no time for re-calibrating the resistance functions for various instability modes in order to use the same partial safety factor.B. The state of pure shear that may exist for low loads rotates as shown in Fig. The partial safety factors gM0 and gM1 have been suggested to have different values in EC3-2: 1. 9. consider a girder with a slender web and widely spaced transverse stiffeners. Vertical stiffeners are supposed to be widely spaced and no vertical stresses act on the ﬂanges. / Journal of Constructional Steel Research 57 (2001) 279–311
larger than web area is deﬁned.2. 3. 9.0.1.0 and 1. which justiﬁes gM0=1. Johansson et al.
which has to be anchored at the girder ends. If the end-post consists of a single plate the resistance to shear will be less than predicted by Eq. (27). (27) is shown in Fig.294
B. The resistance actually used in the design according to EC3-1-5 is reduced slightly compared with Eq. / Journal of Constructional Steel Research 57 (2001) 279–311
t tan(j)
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 s2 tcr p2E t2 w kt 12(1−v2)hw2 (25)
The ultimate strength of a web is assumed to be reached when it yields according to the von Mises yield criterion s2 s2 s1s2 fyw 1 1 From Eqs. The force has to be resisted by the end-post if the full strength should be developed. (23)–(26) the shear resistance can now be solved to tu fv in which fv fy/ 3 lw fv tcr
1 1 − 4l4 2 3l2 w w
Eq. The reason for this is the resulting tension in the web. 10 together with some test results for a girder with widely spaced vertical stiffeners. (27) in order to allow for scatter in the test results and also for
. 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. (23) and (24) is uniform over the depth leads to the following expression for the tensile force Nt hwtwfv 1 l2 w lw tu fv
This force is larger than the actual force because the state of stress close to the ﬂanges will be more like pure shear. Assuming that the state of stress as given by Eqs. Johansson et al.
S275 and h=1. / Journal of Constructional Steel Research 57 (2001) 279–311
Fig.83/lw 1. Shear resistance according to rotated stress ﬁeld theory together with test results for girders with widely spaced vertical stiffeners. This has been done by curve ﬁtting using simpler expressions than in Eq. Longitudinal stiffeners may be ﬂexible. 10.20gM1/gM0 for S235.3. that is.B. The design resistance is given by VRcd cvfywdhwtw/ 3 where cv=cw+cf.83/lw
. Contribution of stiffeners The inﬂuence of stiffeners is accounted for by their increase of the critical stress.08 lw h=1. that is. they form nodal lines in the buckling pattern.37/(0.83h lw 1. 3. (27). and requirements for stiffness and strength are given in EC3-1-5. Transverse stiffeners are assumed to be rigid.7+lw) S355 Non rigid end-post h 0.83/lw 0. cw is found in Table 1 and cf will be discussed later.08 1. they deform under buckling.83h 0.
the systematic deviation for girders with non-rigid end-posts.05gM1/gM0 for S420 and S460 Rigid end-post h 0. It is clear from test results that the effect of longitudinal stiffeners will be overestimated
Table 1 Contribution from the web to shear resistance cw according to EC3-1-5 lw lw 0. Johansson et al.
which is quite close to the resistance for a non-rigid end-post according to EC3-1-5. Comparison between resistance to shear without contribution from ﬂanges according to EC31-1 and EC3-1-5 assuming gM1=gM0. There is less post-critical strength in a web with ﬂexible stiffeners than in a plain web. The resistance according to EC3-1-5.
. 11. is the same in both diagrams because the inﬂuence of the panel length is reﬂected only by its inﬂuence on the slenderness parameter lw. / Journal of Constructional Steel Research 57 (2001) 279–311
if the theoretical critical stress is used for calculating the slenderness parameter lw.34(a/hw)2 ktsl ktsl 9 hw a
2 3/4 1/3
(30) (31) 2.34 4(hw/a)2 ktsl kt 4 5. The draw-
Fig. This is also true for the simple post critical resistance according to EC3-1-1. Johansson et al. assuming that the stiffeners are rigid. This comparison assumes that the ﬂanges do not contribute. 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. shown by solid curves. In addition to the check for buckling of the whole stiffened panel there is a check for buckling of the largest subpanel. A comparison between the resistance to shear according to EC3-1-1 and EC3-15 is shown in Fig. 11.1 Isl tw hw (32)
Isl 2 twhw
In (32) Isl denotes the sum of the second moments of area of all longitudinal stiffeners. 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. This reduction has been considered in the following approximate formulae for the buckling coefﬁcient from Annex A3 of EC3-1-5 kt 5.296
Johansson et al.B. After some simpliﬁcations the contribution from the ﬂanges can be expressed as cf bft2fyf 3 f ctwhwfyw 0. 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. 13. including girders with no intermediate transverse stiffeners. which gives fair estimates of the resistance for any length of the panel. Contribution from ﬂanges The intermediate vertical stiffeners prevent the ﬂanges from moving towards each other. 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. The resistance according to EC3-1-5 is compared with the one according to EC3-1-1. 3.
. This does not reﬂect the real behaviour of a girder. 12. Another advantage of the new rules is that they are simpler to use. / Journal of Constructional Steel Research 57 (2001) 279–311
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. The simple post critical method does not take the contribution of the ﬂanges into account.6bft2fyf f a th2 fyw w
An example of the resistance to shear including the effect of the ﬂanges is shown in Fig.
Fig.Rd
1.8. This effect is taken into account by adding a tension ﬁeld that can be supported by the ﬂanges acting as beams supported by the stiffeners according to Fig. 12.4.25 1 MSd Mf.
Tension ﬁeld supported by the ﬂanges.
4. Because of space limitation only the most common case is dealt with here. Design for patch loading 4. 14. 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. Model for patch loading resistance The design rules in EC3-1-5 cover three different cases of patch loading. 13. The design procedure includes the following parameters: a yield resistance Fy a slenderness parameter l=√Fy/Fcr where Fcr is the elastic buckling force
. 4. / Journal of Constructional Steel Research 57 (2001) 279–311
Fig. see Fig. The new rules also cover a wider range of load applications and steel grades. Resistance to shear for a girder with bf=25tf. tf=3tw and fyw=fyf=355 MPa. crippling and buckling have been merged into one veriﬁcation. The new rules use the same format as other buckling rules.298
B. Johansson et al. MSd=0 and assuming gM1=gM0. General 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].1.2. The three veriﬁcations in EC3-1-1 for crushing.
15.B.5 1 l (39)
The mechanical model according to Fig. The
Fig. 15 is used for the yield resistance. (38) was originally proposed [10] but during the drafting of EC3-1-5 it was simpliﬁed to c(l) 0. The characteristic resistance is written as FR Fyc(l) and the parameters are written as Fy fywtwly Fcr kF pE t 12(1−v2)hw 0.
a resistance function c=c(l) which reduces the yield resistance for l larger than a certain limiting value.
The expression in Eq. Deﬁnition of parameters. 14. Patch loading. / Journal of Constructional Steel Research 57 (2001) 279–311
Mechanical model for the yield resistance for patch loading. Johansson et al.47 1 l
c(l) 0.
Mi. a more realistic case.5 the bending moment has no inﬂuence on the patch load resistance. (37) was determined on the basis of the results from an FE analysis. which shows the quotient between the two resistances as a function of ﬂange thickness over web thickness. 16. is calculated under the assumption that the ﬂange alone contributes to the resistance. the effective loaded length ly. 10.8 1. 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 increased. where the inﬂuence of these parameters are included can be found in Ref. The right diagram for a loaded length of 0.2 times the web depth. (45). Johansson et al. The FE analysis included the inﬂuence from the stiffness of the ﬂanges as well as the length of the applied load and expressions for kF. The left diagram for zero loaded length shows a fairly large difference between the two design methods. / Journal of Constructional Steel Research 57 (2001) 279–311
mechanical model has four plastic hinges in the ﬂange and the plastic moment resistance for the inner plastic hinges. it is assumed that a part of the web contributes to the resistance. from which it follows that when the ratio Ms /MR 0. For the outer plastic hinges. 15 is given by ly ss 2tf(1 where m1 fyfbf fywtw hw tf
m1+m2)
m2 0. shows a large difference only for a stocky web. This inﬂuence is accounted for by Eq. Fs Ms 0. With a simpliﬁed expression for Mo.4 FR MR (45)
The resistance to patch loading according to EC3-1-5 is compared with that of EC31-1 in Fig.02
The buckling coefﬁcient kF in Eq. for the model in Fig. Mo.
. These expressions were simpliﬁed in EC3-1-5 to kF 6 2 hw a
ss kF 2 6 6 hw
It is also necessary to consider the interaction with bending moment.300
1. General 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. mR is the mean value of the resistance. 17.
5. To guarantee that the distribution of the action effects S and the resistance R have a sufﬁcient safety distance a safety index b is deﬁned in EC1-1 as follows: b mR−mS s2 +s2 R S 3. The safety requirement for a structure is deﬁned by the criterion [Rd] [Sd] 0 where [Rd] and [Sd] are design values. Johansson et al. To deﬁne the design values in Eq. sS is the standard deviation of the action effect and sR is the standard deviation of the resistance.8 (46)
where mS is the mean value of the action effect. Calibration of design rules versus test results 5. / Journal of Constructional Steel Research 57 (2001) 279–311
Fig. Eq. which uses the following deﬁnitions and assumptions. 16. It is assumed that both the action effects S and the resistance R of a structure are subject to statistical normal distributions. (47). see Fig. Comparison between patch loading resistance according to EC3-1-5 and EC3-1-1 for a girder with bf/tf=25 and a large distance between vertical stiffeners. (46) may be expressed by (47)
.B. which are characterised by mean values “m” and standard deviations “s”.
7 the design values of the action effects and of the resistances can be described independently from each other and a more
. 17. Johansson et al. / Journal of Constructional Steel Research 57 (2001) 279–311
Fig.8 and aS=0.302
sR s +s
−sS s2 +s2 R S
With the notations aR sR s2 +s2 R S sS s2 +s2 R S
it is possible to express the design values as Rd mR aRbsR Sd mS aSbsS (49) (50)
With the approximations aR=0. Statistical distribution of the action effects S and the resistances R.
Therefore. see Fig. the actual distribution corresponds to a unimodal ¯ normal function as assumed and the statistical data (b and Sd) are determined with the standard formulae provided in Annex Z of EC3-1-1. the mean value correction factor ¯ b for the resistance function rt and the standard deviation Sd for the deviation term d can be determined. 17. has to be established. For the case that the plot shows a curved line the relevant normal distribution at
Fig. This is an arithmetic description of the inﬂuence of all relevant parameters x on the resistance r which is investigated ¯ by tests.
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. By comparing the strength values from the resistance function rt using measured input data with test results re.
¯ Plot of re rt values. Johansson et al. 18. which may be interpreted as a composition of two or more normal distributions. the density distribution for the resistance is checked by plotting the measured probability distribution on Gaussian paper. 18. If the plot shows a straight line. the so called design ¯ model for the resistance. This gives the following formula describing the ﬁeld ¯ R brtd (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. In a ﬁrst step of this procedure a resistance function rt=gR(x). mean value correction b and standard deviation Sd of the deviation term d. It may be represented by a non-normal distribution.B.
To consider scatter effects of parameters not sufﬁciently represented by the test population the standard deviation of the resistance has to be increased. ¯ The statistical data b and Sd of the relevant normal distribution are then determined from the tangent approach to the actual distribution. see Fig. in addition to the standard deviation Sd. the following variation coefﬁcients are taken into account for the yield strength and geometrical values: nfy = 0.07 for strength fy nt = 0. To this end. In general.304
B. / Journal of Constructional Steel Research 57 (2001) 279–311
the design point is determined by a tangent to the lower tail of the measured distribution. the test population is not representative of the total population of struc¯ tures and therefore is only used to determine the mean value deviation b and the scatter value Sd of the design model. 19. Johansson et al.
.05 for thickness t nb = 0.005 for width b nh = 0. 19. (52)
Fig.005 for depth h These variation coefﬁcients are combined with the standard deviation Sd according to Eq. Plot of rei/rti-values on Gaussian paper and deﬁnition of the relevant normal distribution at the design point.
5s2 ) b exp(−1.5s2 ) R (54) where aRb = 0.0sfy−0. Calibration of the design rules for shear buckling The design rules for shear buckling were checked versus test results according to the procedure given in Section 5. (53) ¯ Rk bmR exp( 1.64sR−0. the design value Rd of the resistance function may be deﬁned by ¯ Rd bmR exp(aRbsR 0. / Journal of Constructional Steel Research 57 (2001) 279–311
(ni)2+S 2 d
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. For the statistical evaluation the test results were obtained from a data bank given by Hoglund [9] which contains 166 test results for ¨ the following types of steel plate girders: girders with stiffeners at support only.64sR−0. 5.
. shear ∗ buckling and buckling due to patch loading.04. To consider Rnom instead of Rk a modiﬁed partial safety factor gM* is used from: g∗ M kgM k may be expressed by: (57) (56)
where k=Rnom/Rk.867 fy k ¯ ¯ b exp(−1. the function Rnom is subsequently ∗ modiﬁed to reach the standard value gM.8 = 3. Johansson et al.5s2 ) R Also.1.8·3. For the resistance functions for plate buckling
exp(−2.10 used for stability checks.5s2 ) R R
The procedure explained above is used in the following to determine the g∗ values M for the resistance functions for plate buckling due to compression stresses.64sR 0. girders with transverse intermediate stiffeners. The gM-value of the resistance function is obtained from the ratio of the characteristic value to the design value gM Rk Rd (55) (53)
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.B.2. Where gM is not in compliance with the ∗ standard value gM=1.5s2 ) 0.
Since the procedure of the design model for shear buckling depends on the arrangement of stiffeners. Owing to the small variation of the mean values of Vei/Vti the conclusion can be drawn that the inﬂuence of fyw is adequately considered in the design model. For these individual subsets the statistical evaluations were carried out and the statistical results are presented in Fig.306
B. 23. end patch loading or opposite patch loading. The ﬁgure shows that for all subsets ∗ g∗ -values lower than 1. 20. / Journal of Constructional Steel Research 57 (2001) 279–311
girders with longitudinal and transverse stiffeners.
Fig. see Fig.
. which were obtained from a data bank given by Lagerquist [10].
Subsets of test results. 22). Only 150 of 166 tests could be used for the statistical evaluation because some of these specimens did not fail by shear buckling.10 can be applied M in the design model g∗ . Calibration of the design rules for patch loading The statistical evaluation for the design model of patch loading was carried out with test results.10 were determined so that a gM-value of 1. M The sensitivity of the design model to the variation of the yield strength fyw was checked by plotting the ratio Vei/Vti versus the yield strength of the web fyw (Fig. 21. see Fig.3. 5. Johansson et al. The data bank contains test results for welded and rolled I-girders. which were loaded by patch loading. the available test results were subdivided into subsets. 20.
g∗ -values for the design model of shear buckling.
Sensitivity plot for fyw. / Journal of Constructional Steel Research 57 (2001) 279–311
Fig. 22. M
. 21. Johansson et al.
Fig. 23.308
B.10 is justiﬁed. 24.
g∗ -values for the design model of patch loading. Johansson et al.
Different cases of patch loading. / Journal of Constructional Steel Research 57 (2001) 279–311
Fig. M ¯ In Fig. The ﬁgure shows that for all three subsets a g∗ -value of 1. and a summary of the statistical results is presented in Fig. 25 a sensitivity plot is given for the slenderness parameter l which shows that the scatter of the ratio Fei/Fti is only slightly inﬂuenced by the slenderness parameter l. M
According to the data base and the various design models the test results were subdivided into the following subsets: Data set 1: Data set 2: Data set 3: Patch loading End patch loading Opposite patch loading
For these subsets the statistical evaluations were carried out.
see Eq.4. (58). Some of these stiffeners do not fulﬁl the design recommendations given in EC 3 Part 1. Sensitivity plot for the slenderness l.B. / Journal of Constructional Steel Research 57 (2001) 279–311
Fig. see Table 2.0 rcAcfy Wefffy where: ky 1 myNS 1.5 rcAcfy (58)
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. Unfortunately. Ns kyNSeN 1. 25. Calibration of the design rules for buckling of stiffened 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. In these tests the longitudinal stiffeners were designed as bulb ﬂats.5. ﬂats or angles. 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. In addition. Finally 25 tests were applicable to calibrate the design model. not all of the collected tests could be used to check the design model. the shifting of the neutral axis of the stiffened steel plate due to local buckling was considered using the interaction formula for bending and axial compression which is provided in EC3-1-1. Johansson et al.
B.y 4) 0.
Statistical results for the design model of plate buckling. The results of the statistical evaluations are presented in Fig.y 1. / Journal of Constructional Steel Research 57 (2001) 279–311
Table 2 Test results for longitudinal stiffened steel plates in compression Authors Dorman and Dwight [12] Test 3 (TPA3) No of tests 4 Stiffener types All tests with bulb ﬂats
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
All tests with bulb ﬂats
All tests with ﬂats
9 tests with ﬂat plates 3 tests with ﬂat plates and angles
¯ my lc(2bM.9 bM. 26. M
Fig. 26 and show that a g∗ -value of 1.10 can be applied in the design model. Johansson et al.
.1 The statistical evaluation was carried out for the different groups of tests given in Table 2.
Maquoi R. Ernst & Sohn (Tables V/1. [2] Eurocode 3 Design of steel structures.4:41–53. Doctoral ˚ thesis 1994:159 D.1 General rules and rules for buildings. [4] Kloppel/Scheer Beulwerte ausgesteifter Rechteckplatten. Ultimate compressive strength of stiffened plates. Steel Structures. ˆ [7] Jetteur PH. W. Dept. In: Conference on steel box girder bridges. Calcul des ames et semelles raidies des ponts en acier. 96–105). ENV 19931-1:1992. Traglastversuche mit ausgesteiften Blechfeldern unter allseitiger Navierscher Lagerung und konstanter Stauchung der Endquerschnitte. Royal Institute of Technology. [11] Lagerqvist O. Dwight JB.2 ¨ pp. ¨ TU Braunschweig. Constr Met 1979. Part 1. Ausgesteifte Druckgurte von Kastentragerbrucken. Department of Civil and Mining Engineering. Vayas I. 63-75. Maquoi R. ¨ [8] Hoglund T.B. Royal Institute of Technology. Strength of steel and aluminium plate girders—shear buckling and overall web buckling of plane and trapezoidal webs. / Journal of Constructional Steel Research 57 (2001) 279–311
6. resistance of steel girders subjected to concentrated forces. [12] Dorman AP. Tests on stiffened compression panels and plate panels. ENV 1993-2:1997.5 General rules. p. 94. New York: ASCE. 1981. Ernst & Sohn (Tables Q001 ¨ ¨ to Q005. 6036-1.4:15–28. Massonnet CH. Supplementary rules for planar plated structures without transverse loading. J Constr Steel Res 1996.5:145–54. Division of Steel Structures. ¨ [9] Hoglund T. [10] Lagerqvist O. 1974. [3] Eurocode 3 Design of steel structures. Report. Skaloud M. [15] Balaz I. Resistance of I-girders to concentrated loads.1 to V/5. Patch loading. pp. Stockholm. This will facilitate a future widening of EC3 to higher steel grades. Formulations d’Ayrton-Perry pour le ﬂambement des barres metalliques. Tech. [13] Scheer J. ENV 1993-1-5:1997. W. 1982. Structural Engineering 1995:4. Lulea University of Technology. Stockholm. Conclusions The ﬁrst common European pre-standard considering plate buckling has been published for trial application. Division of Building Statics and Structural Engineering. Stahlbau 1987.39(2):87–119. Further comments and remarks are welcomed and they may be sent to the ﬁrst author for consideration in the ENversion of EC3-1-5. Comparison with tests. In addition to widening the scope to stiffened plates it also includes some improvements of the present rules in EC3-1-1. Part 2 Steel Bridges. [14] Fukumoto Y.
[1] Eurocode 3 Design of steel structures. ´ [6] Rondal J. Institut fur Stahlbau. Schluβbericht Nr. [5] Kloppel/Moller Beulwerte ausgesteifter Rechteckplatten. The design rules have been veriﬁed by calibrations to tests. Constr Met 1983. The ﬁrst author has already received several questions and remarks indicating the need for clariﬁcation and corrections. Design of thin plate I-girders in shear and bending with special reference to web buckling. Bull. ¨ ¨
. which include also steel grades above the present limit of S460. 130–139). Johansson et al. In: Specialty conference on metal bridges. Johansson B. The test application shows that this ﬁrst version of EC3-1-5 can be improved in several respects. Band II. Part 1. London: Institute of Civil Engineers.
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