Shear resistance in stainless steel plate girders with transverse and longitudinal stiffening

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Shear resistance in stainless steel plate girders with transverse andlongitudinal stiffeningImma Estrada ∗, Esther Real, Enrique MirambellDepartment of Construction Engineering, Universitat Politècnica de Catalunya, UPC, Jordi Girona 1-3, Campus Nord UPC, blg. C1, 207. 08034 Barcelona, Spain

a r t i c l e i n f o

Article history:Received 17 December 2007Accepted 2 July 2008

Keywords:Stainless steelMaterial non-linearityShear bucklingUltimate shear capacityExperimental testsNumerical analysis: FEM, Eurocode 3 Part1-4

Rotated Stress Field method

a b s t r a c t

This paper summarizes and presents main results of the investigation conducted in the Departmentof Construction Engineering of the UPC dealing with shear behaviour of stainless steel plate girders.Initial shear buckling stress together with ultimate shear capacity of these structural elements have beenevaluatedwith special attentionpaid to the effect of including stiffeners, both transverse and longitudinal.The studies conducted, both numerical and experimental tests, have permitted the development of newand simple design expressions to determine more accurately the initial shear buckling stress in stainlesssteel web panels and the ultimate capacity of plate girders considering the presence of a rigid or non-rigidend post.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Although the initial cost of stainless steel products is approxi-mately four times that of the equivalent carbon steel product, usingawhole-life costing approach tomaterial selection and consideringthe additional benefits in terms of durability, fire resistance [1] andrecyclability, it becomes a far more attractive option and often themost economic solution [2]. Thus, the increasing interest in stain-less steel as structural material shows the need to develop specificdesign rules considering the particular behaviour of this materialin order to achieve an optimal use of stainless steel in construction,since the current ones historically included into codes have turnedout to be clearly conservative. To this end, a research work beganin 1996 at the Department of Construction Engineering of the UPC,to deal with shear resistance of plated structures, understandingthat it was an important topic to work on.

The usual slender design of the web panels in plate structuresoften makes these structural elements susceptible to instabilityphenomena, i.e. web buckling. This situation makes it imperativeto evaluate accurately the shear buckling strength in orderto optimize their design. Historically, elastic shear buckling insteel plates has been determined assuming that web panelsare simply supported at the juncture between flanges and web.This assumption has turned out to be conservative since thegeometrical properties of the plate girder modify the boundary

∗ Corresponding author. Tel.:+34 93 405 41 56; fax: +34 93 405 41 35.E-mail address: [email protected] (I. Estrada).

conditions and influence the web behaviour in shear. Moreover,the analysis of web buckling on steel plates implies also thestudy of the material non-linearity effects, which are clearly moreevident in non-linear materials like stainless steel.

The instability phenomena occurring in theweb of plate girdersdoes not mean the ultimate collapse of the structure. In fact, onceit buckles, the web loses the capacity to carry any additionalcompressive loading, but experience and extensive investigationhave demonstrated that a new resistant mechanism is developedgiving to the structural element an additional capacity commonlyknown as postcritical resistance. This reserve of strength mustbe considered when designing steel structures to make the finaldesign optimum. The end of this postcritical resistant mechanismis given by the collapse of the plate girder when its ultimatecapacity is reached.

During the postcritical range, since compressive direct stressescan not increase anymore, the additional loading is resisted by anincrease of the tensile stresses with the formation of a diagonaltension field that can be schematically assumed as a Pratt truss,where the chords and the posts are provided by the flanges andthe transverse stiffeners. Significant postbuckling strength maybe mobilized because of this diagonal tension action. In the caseof stainless steel, special attention must be paid into the effectthat the material non-linearity has in all this resistant mechanism.Furthermore, the effect of the rigid and non rigid condition ofthe end post has been in-depth evaluated since it is one of theparameters clearly getting into play in the design of plate girders.

Specifically observed in the shear resistance topic duringthis research [3,4], the conservative character of the current

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Please cite this article in press as: Estrada I, et al. Shear resistance in stainless steel plate girders with transverse and longitudinal stiffening. Journal of ConstructionalSteel Research (2008), doi:10.1016/j.jcsr.2008.07.013

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formulation included in the design specifications related tostainless steel elements does not allow taking advantage ofstainless steel as structural material to achieve its full potential inconstruction. Thus, further experimental and numerical analyseshave been conducted during last decades in order to predict moreaccurately the actual ultimate capacity of stainless steel structures.In UPC studies carried out by Real [5,6] weremainly focused on theformulation of the ultimate shear capacity of stainless steel plategirders based on the Tension Field method. Likewise investigationconducted by Olsson [7] at the University of Lulea concluded witha proposal of a new design method for stainless steel plate girdersbased on the Rotated Stress Field model. This research intendsto be a new step forward and its final objective has been theproposal of simple analytical expressions to determine the initialshear buckling stress value in stainless steel plate girders, speciallyinteresting when evaluating the corresponding serviceability limitstate, and its ultimate shear capacity taking into account thegeometry of the end post.

2. Shear buckling and ultimate shear capacity in a nonlinearmaterial like stainless steel

2.1. Current expressions for shear design

In stainless steel structures shear buckling rarely occurs whollyin the elastic range so its behaviour becomes clearly influencedby the material non-linearity resulting in a lower capacity thanpredicted by a bilinear model, which most of times is theconstitutive equation considered. The first experimental workknown to address the shear resistance of stainless steel memberswas conducted by Carvalho, Van den Berg and Van der Merwe atRand Afrikaans University [8].

The analytical expression to obtain the elastic critical shearbuckling stress of a rectangular plate was first given by Timo-shenko [9]:

τcri = kπ2E

12(1 − υ2)

(twd

)2

(1)

an expression widely known, where E is the modulus of elasticity,ν is the Poisson’s ratio and tw and d are, respectively, the thicknessand the depth of the web panel. The coefficient k, known as shearbuckling coefficient, depends upon the boundary conditions andthe aspect ratio of the web panel, a/d, where a is the distancebetween two adjacent transverse stiffeners. Although the notionof real boundary condition at the juncture between web andflanges to be somewhere between simply supported and fixedhas been recognised from early days, the boundary condition hasbeen arbitrarily and conservatively assumed as simply supported,mainly due to the lack of means to evaluate it in a rational manner.

When determining the initial shear buckling stress of a stainlesssteel plate, which is one of the main objects of the work presentedin this paper, there are two important factors to consider thatare not included in the formulation given by the classic theoryformulation (1). First of all, the effect of the material non-linearitythat, for stainless steel, comes into play even for low stress levels.And secondly, the effect of the real boundary conditions of theweb panel on the whole shear response mechanism, recentlyanalysed by Lee and Yoo [10] and by Mirambell and Zárate [11]for rectangular and tapered carbon steel plates respectively.

In most current standards the effect of the material non-linearity is introduced in the formulation of the initial shearbuckling stress by including a plasticity reduction factor ηnl tothe elastic formulation given by Timoshenko. Former ENV-1993-1-4 [12] included an expression for determining the initial shearbuckling stress τbb, graphically presented in following Fig. 1.

Table 1Initial shear buckling stress (τbb) proposed by Real [5,6]

λw slenderness τbb

λw ≤ 0.4(fy/

√3)

0.4 < λw ≤ 0.9 [1 − 0.7 (λw − 0.4)](fy/

√3)

0.9 < λw ≤ 2.2[

3.9−λw

2.1+2.8λw

] (fy/

√3)

λw > 2.2(1/λ2

w

) (fy/

√3)

Fig. 1. Initial shear buckling stress for stainless steel plates.

Values obtained by the application of this design expressionwere demonstrated to be clearly conservative. This conservativecharacter ismainly due to the lack of experimental results availablein stainless steel plateswhen stated Part 1–4 of the Eurocode 3wasfirst published in its ENV version.

Following the philosophy of the plasticity reduction factorsbased on material’s tangent modulus, further analysis wasconducted by Real [5] in order to obtain a closer approximationto the actual value of the initial shear buckling stress in simplysupported stainless steel plates taking special attention to thematerial non-linearity effects. Thus, working with a plasticityreduction ηnl =

√Gt/G0 which was demonstrated to give a

more accurate approximation to the actual behaviour of platesloaded in shear, the curve presented analytically in Table 1, wherethe slenderness is calculated as λw =

√τplτcri

=d/tw

37,4ε√k, was

proposed [5,6] to determine the initial shear buckling stress insimply supported stainless steel plates.

Results given by this new expression reproduce accuratelythe effect of the material non-linearity inherent to stainless steelbehaviour as it can be observed in Fig. 1, where it is shown it givesa better approximation to the actual behaviour of simply supportedstainless steel plates than the expression included in ENV-1993-1-4 [12] did.

As for the ultimate shear capacity, different theories have beendeveloped to analyse it in carbon steel plate girders, Höglund’sRotated Stress Field method [13] is the one that is provided inthe current parts of Eurocode 3 (Fig. 2). This method distinguishesbetween the behaviour of beams with rigid and non-rigid endpost to evaluate the contribution from the web to shear bucklingresistance. Those girderswith rigid endposts are supposed to reachhigher ultimate loads.

Current Eurocode 3 Part 1.4 [14], which deals specificallywith stainless steel structures, also provides an adaptation of theRotated Stress Field method to determine shear buckling strengthof plate girders mainly based in Olsson’s adaptation of the RotatedStress FieldMethod [7]. This newmethod, which has been recentlyincluded in the standard, gives much more accurate predictionsof the ultimate shear capacity of stainless steel plate girders than


ARTICLE IN PRESSI. Estrada et al. / Journal of Constructional Steel Research ( ) – 3

Fig. 2. State of stress in the web of a beam with transverse stiffeners at the ends only Rotated Stress Field Method.

Table 2Contribution from the web χw to the shear buckling resistance (EN 1993-1-4 [14])

λw χw

λw ≤ 0.5 1.2λw > 0.5 0.11 +

0.64λw

−0.05λ2w

the Simple Postcritical method ‘‘historically’’ considered in theEuropean code in its ENV version did. Analytical expressions forboth web and flange contributions formulated in this proposal arepresented in Table 2 and Eqs. (2) and (3) respectively:

The contribution of flanges is calculated fromEq. (2), expressiongiven by the application of the principle of virtual work to thecollapse mechanism:

χf =bf t2f fyf

√3

c · twd · fyw

[1 −

[Ms

MfR

]2](2)

being

c =

(0.17 +

1.6 · bf t2f fyftwd2fyw

)· a. (3)

At this point it is important to note that the analyticalexpression proposed by Olsson nowadays included in the EN -1993-1-4 [14] to obtain the distance c , which gives the positionof the plastic hinge, differs from the one proposed by Höglund forcarbon steel in only one factor: 0.17 for stainless steel as opposedto 0.25 for carbon steel. It is important to outline this is a numericalvalue obtained experimentally in both cases.

Although this proposal has meant a notable step forwardinto the design of stainless steel plate girders, it dismisses thedistinction between panelswith rigid and non-rigid end post. Thus,further analysis can be conducted in this direction.

2.2. Stiffening and influence of the real boundary conditions

The efficient design of a plate girder web normally requiresthe use of transverse and longitudinal web stiffeners. Economycommands the designer to balance the cost of fittings and weldingof stiffeners on a thin plate against the price of a thicker unstiffenedplate.

The predominant stiffening of plate girders subjected to shearis by transverse stiffeners. But the slenderness limitations forlong span structures subjected to high loads require, sometimes,longitudinal stiffening of the web plates; without this, anuneconomical web thickness would be necessary. The main objectof these longitudinal stiffeners is to increase the initial shearbuckling stress of the plate girder, reducing the geometrical non-linear effects in theweb but, as a consequence, its ultimate capacityalso gets higher.

The role of any intermediate transverse stiffener is to keepflanges from being pulled in by the tension band forces as well asproviding the adequate boundary conditions for adjacent panelsunder the critical shear buckling load and up to failure. In the endpanels the dominating force is shearwhich after buckling is carriedby the tension band anchored into the end post.

Apart from transverse stiffeners, when longitudinal stiffenersare included in the designs, the behaviour of the reinforced plateelement becomesmore complex. One of themain questions is howthe longitudinal stiffeners behave in the postbuckling range.

For plate web subjected mainly to shear the optimumposition of a longitudinal stiffener is at mid-depth, so the tworesulting subpanels buckle simultaneously if bending effects arenot considered. Anyway, there’s always a weakest panel wherebuckling first occurs. As the aspect ratio of the panel changes withthe presence of the longitudinal stiffener, the initial shear bucklingstrength of the element gets higher. As mentioned, increases inultimate capacity are also observed.

Although the design expressions to determine the initial shearbuckling stress included in current codes have been developedassuming simply supported boundary conditions on the edgesof the web panel, it is quite clear to demonstrate that the realboundary conditions of the web plate are not the ones of a simplysupported panel. In fact, assuming this unreal boundary condition,the web design tends to be too conservative in many cases.

In order to illustrate this situation, the table included inFig. 3 shows the theoretical classical shear buckling coefficientsfor rectangular plates assumed to be simply supported in itsall four edges (kss), compared with the buckling coefficientsconsidering two opposite edges delimited by transverse stiffenersbeing simply supported and the other two, those in the juncturewith flanges, assumed as fixed (ksf ). Formulation used to determinethe mentioned coefficients can be found in Timoshenko [9] andBulson [15] for kss and ksf respectively. As can be observed, with anincrease of the aspect ratio, differences between theoretical valueskss and ksf get more considerable and the assumption of simplysupported edges gets highly conservative.

However, as demonstrated by the experience, in a web panelincluded in a plate girder, the real boundary condition at thejuncture between web and flanges is somewhere between simplysupported and fixed. The graph included in Fig. 3 is presentedbelow to demonstrate this scenario. When considering realjunction between web and flanges, the presence of thicker flangesincreases the initial shear buckling stress by stiffening the edgesof the web panels so that their effective boundary conditionsare closer to clamped boundaries instead of simple boundariesordinarily assumed in classical stability analysis. Then, it is clearthat, in order to accurately calculate the initial shear bucklingstress, the boundary conditions of the web panel need to beproperly determined. Thus, further analysis in this direction hasbeen conducted.



Fig. 3. Shear buckling coefficients. The effect of the boundary conditions.

(a) 1st experimental campaign. R&NR end post. (b) 2nd experimental campaign. Longitudinal stiffening.

Fig. 4. Test configuration and specimens.

2.3. Proposal to develop new design expressions

Taking advantage of the numerical results, one of the goals ofthis research has been to propose a simple analytical expressionto determine the initial shear buckling stress value in stainlesssteel plate girders. To tackle this problem analytically the followingpresented methodology has been carried out.

As starting point, the analytical expression presented by Real[5,6] is taken to determine the shear buckling stress of a simplysupported stainless steel plate. The expression of the proposedcurve can be generically written as:

τbb = f (λw) · τpl (4)

where f (λw) is a function of the slenderness of the web paneldefined in four intervals and τpl = fy/

√3, being fy is the yield

stress of thematerial. Multiplying and dividing by τcri, elastic shearbuckling strength in Eqs. (1) and (4) can be rewritten as:

τbb = f (λw) · τplτcri

τcri, and consequently, τbb = f (λw) ·

τpl

τcriτcri (5)

and knowing that, by definition λ =

√τplτcri

, the above formula can

be expressed as τbb = f (λw) · λ2w · τcri.

Taking η = f (λw) · λ2w and writing the classic expression of

τcri previously presented in Eq. (1), the value of the critical shearbuckling in stainless steel plates, from now on τbss, can be writtenas:

τbss = η · krss ·π2E

12(1 − υ2)

(twd

)2

(6)

Table 3Coefficient η to evaluate the initial shear buckling stress in stainless steel plates

λw , slenderness η

λw ≤ 0.4 λ2w

0.4 < λw ≤ 0.9 [1 − 0.7 (λw − 0.4)] λ2w

0.9 < λw ≤ 2.2[

3.9−λw

2.1+2.8λw

]λ2

w

λw > 2.2 1.0

where η factor considers intrinsically the main effect of thestainless steel material non-linearity as it is defined fromthe expression proposed by Real for simply supported panels.Analytical expressions of the η parameter are presented in Table 3.

However, krss is the buckling coefficient factor, where maineffects of the geometry and boundary conditions of the web platewill be basically included. Thus, as the material non-linearityeffects are mostly comprised in the factor η, the main objective ofthe numerical analysis conducted has been to evaluate the effectof the geometry conditions in the shear buckling coefficient.

Closer predictions will be achieved with the new proposal. It isimportant to underline that the proposed methodology does notpretend to state that a complete dissociation of the effects of thematerial non-linearity and the effects of the geometry (boundaryconditions) is possible to be conducted. In fact, the mentionedproposal intends to be developed in order to achieve an easilyapplicable design expression to accurately predict the initial shearbuckling stress in stainless steel plate girders in order to predictthe critical shear load.

To determine the ultimate shear capacity, a redefinition of theRotated Stress Field method formulation adapted to stainless steel



Table 41st experimental programme

Girder L (mm) a (mm) d (mm) tw (mm) tf (mm) ts1 (mm) ts2 (mm) λw

nr700ad15 2360 1050 700 4 20 20 – 2.085r700ad15 2360 1050 700 4 20 20 20 2.085nr600ad2 2660 1200 600 4 20 20 – 1.894r600ad2 2660 1200 600 4 20 20 20 1.894nr500ad25 2760 1250 500 4 20 20 – 1.625r500ad25 2760 1250 500 4 20 20 20 1.625nr400ad325 2860 1300 400 4 20 20 – 1.329r400ad325 2860 1300 400 4 20 20 20 1.329

Geometrical details of the girder specimens.

Fig. 5. First experimental programme: Rigid and Non-rigid end post. Geometry of the tested girders. Rigid and non-rigid end post.

Fig. 6. Second experimental programme. Longitudinal stiffening Geometry of thetested beams.

taking into account the rigid or non-rigid condition of the end postis conducted.

3. Methodology: Experimental and numerical studies

3.1. Brief description of the experimental programmes conducted

To better understand the response of stainless steel plategirders loaded in shear, two experimental campaigns were carriedout at the Structural Technology Laboratory of the Departmentof Construction Engineering, UPC. After the experimental andnumerical research conducted in Real [5] and as result of theadvance of knowledge in the behaviour of stainless steel plategirders failing in shear, the following step to get into in this issuewas the study of the influence of the rigidity of the transverse andlongitudinal stiffeners on the behaviour of the plate elements. Theprimary design variables were, in all cases, the slenderness and theaspect ratio of the web panel (a/d) as general key parameters thatcharacterize shear behaviour. Moreover, as the first experimentalcampaign was focussed on the analysis of the influence of therigidity conditions of the end posts, the condition of rigid or non-rigid end post became a primary variable too. On the other hand,during the second experimental campaign, the main goal wasto study the effect of introducing longitudinal stiffening, so therigidity of the longitudinal reinforcement was the most importantfeature to consider. A total of 15 stainless steel plate girders weretested during both programmes (see Fig. 4).

This first test programme consisted of eight two-panel beamsto focus the analysis on the rigid and non-rigid geometry of theend post in an extreme panel. The girders chosen were designed inpairs. Both beams in a pair had the same geometrical and materialcharacteristics but one girder was designed with rigid end posts(named from now as r-beams) and the other one exhibited nonrigid end posts (nr-beams). This means that the effects of rigidend posts can be isolated in the tests and directly compared. Fig. 5shows the arrangement for the two types of girder. The conditionof rigid end post was taken from the formulation of the RotatedStress Field Method included in Eurocode 3, Part 1–5 [16] for platecarbon steel structures.

The geometry,main characteristics, and the general loading andsupport arrangement of the tested beams are presented in Fig. 5and Table 4. The girders were chosen with depths ranging from700 mm down to 400 mm and aspect ratios of the web panelranging from 1.5 to 3.25.

The second experimental programme sought to study theresponse of stainless steel plated girders under service loadsand up to failure, with particular emphasis on the influence ofthe longitudinal stiffening in the development of the resistantmechanism. With the introduction of longitudinal stiffeners, allplate girders design cases are going to be covered in this studyabout the analysis of the behaviour of stainless steel plate girderssubjected to shear. Five plate girders, each with one longitudinalstiffener, were tested.

At this stage of the study, other than web slenderness and theaspect ratio of the panel, the primary variable was the rigidity ofthe longitudinal stiffener. Therefore, all the beams were designedidentical except for the dimensions of the longitudinal stiffener(bls, tls), that were deliberately varied giving a wide range ofrigidities of this element. Consequently, different behaviours ofthe element would be observed depending on the longitudinalreinforcement rigidity. The longitudinal stiffener was attached atmid-depth. The web was thus divided into two sub-panels ofequal depth, this being the optimum stiffener positioning for apredominant shear loading. The geometry and general schemeof all the designed beam specimens are presented in Fig. 6 thatfollows:

The nominal dimensions of the five tested beams are given inTable 5. It can be observed that the overall dimensions of the beams



Table 52nd experimental programme

Girder tsl (mm) bsl (mm) γ ∗ γ γ /γ ∗ Ratio

lw-25.8 8 25 51.64 9.57 0.19 Weakli-40.8 8 40 51.64 31.75 0.61 Intermediatels1-50.8 8 50 51.64 56.46 1.09 Strongls2-100.8 8 100 51.64 341.51 6.61 Strongls3-50.20 20 50 51.64 102.65 1.99 Strong

Geometry and ratio of longitudinal stiffening of girders. L: 2360 mm, a: 1050 mm, d: 700 mm, tw: 4 mm, tf : 20 mm, tw 4 mm, tst : 20 mm.

Fig. 7. Boundary conditions assumed in the structural analysis of shear buckling resistance.

are the same as the beam r700ad15 tested in the first experimentalcampaign. The supporting capability of the longitudinal stiffenersis measured by their relative flexural rigidity γ . However, thereexists a theoretical minimum value for the relative flexural rigidityγ which ensures that the stiffener under considerationwill behaveas a rigid one. The corresponding value of γ is called the optimumrigidity of the stiffener and known as γ ∗. Thus, the parameter γ ∗

is the value of the minimum rigidity required to ensure that thestiffener remains straight and limits the buckling to the adjacentsub-panels of the web calculated from the linear buckling theory.This optimumrigidity γ ∗ is defined as a function of the loading caseand the position of the longitudinal stiffener [17].

The ratio of γ /γ ∗ is then the key parameter in this experimentalcampaign. Table 5 summarises the relative rigidity together withthe ratio γ /γ ∗ of each of the tested girders. As the nomenclatureindicates in thementioned Table, these beams have been designedcovering a wide range of relative rigidity of the longitudinalstiffener in relation to the optimum rigidity γ ∗ resulting from thelinear theory of web buckling. Thus, beamswith weak longitudinalreinforcement (lw), intermediate reinforcement (li) and strongreinforcement (ls) were tested.

Different behaviours, depending on these stiffness ratios, wereexpected to be observed during the tests.

Tomonitor the behaviour of the beams during the developmentof the laboratory test, the applied loads, strains at key points, anddisplacements were measured using different sets of instrumen-tation, uniaxial and triaxial strain gauges and linear displacementtransducers. All the variables were monitored continuously by thedata acquisition system. A complete description of the tests can befound in [3].

3.2. Numerical analysis

Different numerical analyses were conducted during theinvestigation and the code used to carry out this structuralanalysis is Abaqus [18]. Since the knowledge of real behaviourof the structures analysed was the object of the research, theentire numerical tests conducted included were geometrically andmaterially non-linear as it is described in detail in [3].

A deep analysis of the behaviour observed during the experi-mental tests was conducted together with a numerical simulation

of the tests. The results of this part of the analysis can be foundin [4].

Later on, a numerical study was carried out to evaluate theinfluence that the different geometric design parameters of a plategirder have on the initial shear buckling stress of the web panelto finally achieve general conclusions that have permitted theproposal of simple new analytical expressions to estimate theinitial shear buckling stress in stainless steel plates containedin plate girders. The complete and detailed description of thenumerical analysis conducted together with all the results canbe found in [3,4]. Plate girder segments between two adjacenttransverse stiffeners were modelled as shown in Fig. 7, includingthe boundary conditions defined in the same figure to matchclosely the behaviour of the entire girder under loading. Thisgeometry was elected in order to isolate as much as possible thebehaviour of a web panel, specifically the response up to the shearbuckling strength, butwithout neglecting the effect induced by thepresence of the flanges.

The main design variables whose influence has been analysedare: the web slenderness and its aspect ratio (a/d), the ratiobetween the flange width and the web depth, the flangeslenderness, and the ratio between flange and web thickness.

After the evaluation of the initial shear buckling stress, a newnumerical analysis dealing with the ultimate capacity of stainlesssteel plate girders was carried out as a new step forward into theanalysis of the shear behaviour of these structural elements. Theelected geometry to conduct the analysis regarding the ultimateshear capacity is presented in Fig. 8, which corresponds to a plategirder with two adjacent web panels defined by the presence oftransverse stiffeners. As can be observed in the mentioned Figure,and in order to isolate the effect of the presence of a rigid endpost, two general groups of geometries were numerically testedduring the whole development of this study: rigid and non-rigidend post geometries. As indicated by the given designation, non-rigid girders were designed to not fulfil the condition of rigid endpost given by the formulation derived from the Rotated StressField model, whereas the rigid ones effectively were designed toaccomplish it. Thus, the numerical investigation was conducted onplate girders, tested in identical pairs with the only difference onthe stiffness of the end post.



a. Rigid end post geometry. Double stiffener. b. Non-rigid end post geometry. Single stiffener.

Fig. 8. Numerical models for the analysis of ultimate capacity.

Fig. 9. Shear buckling coefficients for different aspect ratios. Effects of the tt/tw ratio.

Apart from the nature of the end post of the tested girders (rigidandnon-rigid), the influence of the aspect ratio and the slendernessof the web panel was also evaluated, both widely known to be themost important factors determining the shear response of plategirders. Therefore, a series of plate girders including web panelswith aspect ratios ranging from 0.5 up to 3.0 were tested. At thesame time, for each series of girders defined by the value of theiraspect ratio, the variation of slenderness has been also obtained byintroducingdifferentweb thickness ranging from3up to 12mm.Atthis point it is important to emphasise that the maximum value ofthe aspect ratio analysed has been 3.0 since formuch higher valuesof this parameter the mode of failure observed in the web panelswas not the one characteristic of the shear mode of failure.

All the structural elements analysed were numerically testedreproducing the loading and boundary conditions of the plategirders tested during the experimental campaigns conducted inthis research [3,4]. Therefore, all the elements were modelled assimply supported beams with a point load applied at mid-span toconsequently obtain a constant shear law in each of thewebpanels.

In all cases, the material stress–strain curve was approachedin the numerical simulation by the composed Ramberg–Osgoodcurve first proposed by Mirambell and Real [19].

4. Relevant results

4.1. About shear buckling

After the first stage of the numerical study, it was concludedthat the most important parameter defining the real boundary

conditions of a web panel in a plate girder was the ratiobetween flange and web thickness: tf /tw . Then, and from theprior knowledge that the aspect ratio and the web slenderness aretwo key parameters defining the response of web panels loadedin shear, a number of geometries where analysed by combiningdifferent aspect ratios andweb depths to cover awide range ofwebslenderness with a deliberate variation of the ratio tf /tw takingvalues from 1 up to 10. Fig. 9 shows the additional shear bucklingcapacity (in terms shear buckling coefficient, directly related withthe value of the initial shear buckling stress) introduced as theratio between flange and web thickness is increased. Each graphpresents the values of the shear buckling coefficients of thenumerical specimenswith different aspects ratios and slenderness.In the presented curves it can be observed how the value of theshear buckling coefficients in panels with low ratios betweenflange and web thickness (tf /tw = 1) are close to the valuesobtained for a simply supported panel (kss). Incremental increasesin the ratio tf /tw are translated into increases on the shear bucklingcoefficients values. For the highest tested levels of the ratio flangeto web thickness (tf /tw = 10), the shear buckling coefficientsare close to the results classically obtained for a panel simplysupported on two edges defined by transverse stiffeners and fixedin the edges defined by the junctionwith flanges (ksf ). In particular,it has been found that when tf /tw is much less than 2.0, theeffective support condition is close to a simple support, while forcases with tf /tw larger than 2, the effective support condition getscloser to a clamped support. The assumption that the flange-web



(a) a/d = 0.5. (b) a/d = 1.0.

(c) a/d = 2.0. (d) a/d = 3.5.

Fig. 10. Shear buckling coefficient vs tt/tw for different aspect ratios.

juncture is simply supported is, therefore, overly conservative,particularly in plate girders with high aspect ratios.

Comparing both graphs mentioned in Fig. 9, it can be observedthat shear buckling coefficients obtained for web panels withthe same aspect ratios and flange to web thickness ratios, butdifferent slenderness, turn to be quite similar. Slight differences areobserved since, asmentioned before, the effect of thematerial non-linearity and boundary conditions interact in the whole resistantmechanism development and it is not possible to completelyseparate both effects.

It is also interesting to observe that the evolution of the shearbuckling coefficients from values close to the classical simplysupported coefficient (kss) to the simply supported-fixed one (ksf )is analogous for all the tested web panels with aspect ratios equalor higher than 1.0. Different behaviour has been observed in webpanels with aspect ratios lower than 1.0. Thus, separate analysis ofthe two geometries have been conducted.

Fig. 10 illustrates the obtained values of the shear bucklingcoefficients for the different ratios between flange and webthickness tested. On the one hand, in the case of the aspect ratiolower than 1.0 (see Fig. 10a) it is observed that the value ofthe shear buckling coefficient of a simply supported plate is notreached until a value of the ratio tf /tw around 3.0 is achieved. Fromthat point, an increase of the shear buckling coefficient is observed

since higher values of tf /tw mean boundary conditions closer tothe simply supported-fixed conditions perfectly considered forthe determination of the coefficient ksf . For highest levels ofthe ratio between flange and web slenderness the values of theshear buckling coefficients are demonstrated to be similar to thecoefficient ksf . At this point it is important to outline that in thecase of ratios a/d lower than 1.0 the difference between the valuesof kss and ksf is about 5%, which means that the assumption ofconsidering a simply supported panel, without taking into accountthe real boundary conditions of the web panel given by thepresence of flanges, do not introduce any significant error whendetermining the shear buckling coefficient. In fact, the higher theaspect ratio is, the larger the difference between both coefficientskss and ksf becomes, and consequently, the higher is the errormadewhen assuming a simply supported panel when analysing theinitial shear buckling stress of awebplate included in a plate girder.Consequently, special interest has been taken in the evaluation ofthe results for aspect ratios a/d higher or equal than 1.0.

However, as mentioned before, results obtained for web panelswith aspect ratios equal and higher than 1.0 all show homotheticbehaviours (see Fig. 10b, c and d). In this case, the relationshipbetween shear buckling coefficients and the ratio tf /tw is clearlynon-linearwith a pronounced change of tendency for a value of theratio between thickness around 2.0. For this value of tf /tw , which



Table 6Evaluation of the effect of longitudinal stiffeners in shear buckling strength

Girder bls (mm) tls (mm) γ /γ ∗ λpanel λsubpanel Buckling start τcrit,Abaqus (MPa) kAbaqus

ls-0.0 0 0 0.00 2.17 1.32 Panel 33.33 11.57ls-20.8 20 8 0.29 1.91 1.32 Panel 48.80 17.30ls-30.8 30 8 0.82 1.85 1.32 Panel 73.97 26.51ls-35.8 35 8 1.22 1.83 1.32 Panel 79.31 28.60ls-40.8 40 8 1.73 1.80 1.32 Panel 80.86 29.36ls-50.8 50 8 3.08 1.74 1.32 Subpanel 82.27 9.18ls-60.8 60 8 4.95 1.68 1.32 Subpanel 83.11 9.28ls-70.8 70 8 7.39 1.63 1.32 Subpanel 83.67 9.34ls-100.8 100 8 18.74 1.45 1.32 Subpanel 84.24 9.40ls-60.14 60 14 7.17 1.63 1.32 Subpanel 83.95 9.37ls-120.14 120 14 43.75 1.24 1.32 Subpanel 89.16 9.95

Geometry tested Web panel 1000 × 1000 mm, 4 mm thickness γ ∗= 13.50.

(a) Buckling in the overall depth panel, ls-30.8. (b) Buckling starting in the subpanel, ls-100.8.

Fig. 11. Shear buckling start. Observed modes of response.

can be taken as a minimum value to consider when designingrational plate girders, the value of the shear buckling coefficient isclearly above the corresponding one assuming simply supportededges (kss). In fact, in all the cases analysed, the correspondingshear buckling coefficient for a ratio tf /tw equal to 2.0 is around25%–30% higher than kss. Also observed in all the numericallytested aspect ratios, for a ratio between flange and web thicknessequal to 10.0 (higher values would be rarely taken in engineeringplate girders design), the value obtained for the shear bucklingcoefficient is around 95%–100% of the value of ksf .

Examination of over 70 hypothetical plate girders models hasrevealed that the effective support condition at the flange andweb panel juncture depends primarily upon the ratio of the flangethickness to web thickness, tf /tw.

Tomaximise the strength/weight ratio of a plate girder in orderto obtain further economy and efficiency in design, slender websare often reinforced by longitudinal, in addition to transverse,stiffeners. The main function of such longitudinal stiffeners hasbeen demonstrated to be the increase of the initial shear bucklingstress of the web panel. In fact, an effective stiffener willremain straight and consequently subdividing the web panel andlimiting the buckling to the smaller sub-panels. The option to uselongitudinal stiffeners is specially appealing when talking aboutstainless steel, since the saving in material can be significant, andcan result a cost-effective solution.

An evaluation of the influence of the presence of the longitudi-nal stiffener as a function of its relative bending stiffness (γ /γ ∗) inthe shear buckling strength has been conducted. Thus, the relativebending stiffness was taken as a key parameter and the geometry

of all the web panels analysed was the same only varying delib-erately the longitudinal stiffener size. As shown in Table 6, wherea summary of the geometry and results obtained is presented, awide range of γ /γ ∗ was analysed by playing with the width andthickness of the longitudinal stiffener (bls, tls).

Fig. 11 shows the load-displacement curves for two of theelements tested where the two possible situations are illustrated.In Fig. 11a, which plots the results of a web panel with a ratio γ /γ ∗

below 1.0, it can be observed that the out-of-plane displacementsinitiate in the centre of the web panel even though the presence ofthe longitudinal stiffener is crossed by the buckling wave. Thus, inthis case it can be affirmed that the shear buckling phenomenon isdeveloped through the overall web panel. On the contrary, in theresults plotted in Fig. 11b (γ /γ ∗ much higher than 1.0), it can beseen that important out-of-plane displacements are developed inthe centre of the upper subpanel while the longitudinal stiffenerremains practically undisplaced. It can be then stated that the shearbuckling phenomenon is limited by the presence of longitudinalstiffener and it develops in the subpanel.

Table 6 summarises (in its column named ‘‘buckling start’’)which of the above exposed situations has been detected in eachof the tested web panels. From the results it is easily observedthat the change of response occurs for a ratio of γ /γ ∗ close to3.0. For web panels longitudinally reinforced but with a relativestiffness lower than 3.0, the longitudinal stiffener is not able toavoid the buckling wave developing in all the web depth and shearbuckling progresses through thewholeweb panel. On the contrary,for values of the relative longitudinal stiffening γ /γ ∗ higher than3.0, the presence of the longitudinal stiffener creates a nodal line



(a) Complete web panel, a/d = 1.0. γ /γ ∗ < 3.0. (b) Subpanel of the web, a/d = 2.0.γ /γ ∗≥ 3.0.

Fig. 12. Shear buckling coefficients in function of γ /γ ∗ .

which makes the shear buckling phenomenon develop first in oneof the sub-panels.

Having classified the type of shear buckling response observed(panel or subpanel), the corresponding value of the shear bucklingcoefficient was calculated for each element from the value of thecritical shear buckling stress obtained with Abaqus (τcrit,Abaqus).Results of the shear buckling coefficients obtained are presentedin Table 6 and graphically shown in Fig. 12, where both cases ofresponse are differentiated, in relationship with the ratio γ /γ ∗.

Results in Fig. 12a show clearly how an increase in thelongitudinal stiffening, measured with the parameter γ /γ ∗, istranslated into an important increase in the shear bucklingcoefficient and consequently, into an important improvement ofthe shear buckling strength of the web panel. Numerical results(presented as diamonds) are compared with the shear bucklingcoefficient of the web panel obtained by applying the proposedanalytical expressions (Eqs. (7) and (8)) taking into account theactual boundary conditions and the material non-linearity effectsin the stainless steel web panel but neglecting the presence of thelongitudinal stiffener. An improvement of the results is necessaryto be conducted in order to add the effect of the longitudinalstiffening in the formulation. Fig. 12b illustrates the values ofthe shear buckling coefficients in those cases with ratios γ /γ ∗

higher than 3.0 where the buckling phenomenon initiated in oneof the sub-panels, which present an aspect ratio equal to 2.0.The numerical results (diamonds again) are compared with theshear buckling coefficient assuming simply supported boundaryconditions in four edges of the web panel (kss) that, as widelydiscussed in previous sections, turns out to be highly conservative.Moreover, the black squares show the obtained values with theformulation presented as a proposal in previous section wherethe actual features of the web sub-panel (geometry and materialproperties) are taken into consideration (Eqs. (7) and (8)). Resultsshow how this new proposal is adequate in this situation sinceit approximates accurately the actual results and it means animprovement to the Classical Theory to determine the initial shearbuckling stress in a web panel of a stainless steel plate girder.

4.2. Related to ultimate shear capacity

This section includes the ultimate shear capacity resultsobtained during the first step of the numerical analysis where allthe elements tested were composed of the same stainless steel

Fig. 13. Evaluation of the analytical formula to predict the position of the plastichinges, c.

grade. Further analysis on the effect of the material grade wasconducted and it is presented in [4].

The very first step in the analysis conductedwas the verificationof the analytical expression to obtain the distance c given byOlssonwhich is included in current Eurocode 3, Part 1–4 [14]. Fig. 13,where analytical values and Abaqus results of this parameter arecompared, makes evident the accuracy of the analytical expressionproposed by Olsson to predict the position of the plastic hingesduring the postcritical range. Thus, the formulation resulting fromthe application of the principle of virtual work during the collapsemechanism development to obtain the flange contribution in theshear resistant mechanism in Eq. (2) has been applied by using theanalytical expression of the distance c as starting point to evaluatethe web contribution in the whole shear resistance.

Having validated and assumed as accurate the analyticalformula to predict the flange contribution, the resistance given bytheweb is calculated for each tested panel by subtracting the flangecontribution applying previous Eq. (2) as schematically indicatedin Fig. 14.

Fig. 15 shows the results obtained for the plate girders withaspect ratio equal to 1.0 together with the curve proposed byOlsson to predict the web contribution in stainless steel plategirders (analytical expressions in previous Table 2). Note that the



Fig. 14. Schematic of the methodology of analysis adopted during the study of the ultimate capacity.

Fig. 15. Web contribution (χw) in plate girders with aspect ratio a/d = 1.0.

curve corresponding to the Olsson’s proposal is unique since nodistinction is made between geometries with rigid and non rigidend post.

In this figure it is noticed that for a given geometry (definedby the aspect ratio and the web slenderness) the condition ofrigid end post is clearly translated into an increase of the webstrength contribution.Moreover, although being an important stepforward into the design of stainless steel plates loaded in shear, thedesign method proposed by Olsson is still markedly conservative.Analogous results for the series of aspect ratios 0.5, 1.5, 2.0 and3.0 [20] as it can be observed in Fig. 16. Global analyses of theobtained results allow the achievement of interesting conclusions.

In themajor part of the analysed elements it is observed that theincrease of strength given by the rigid end post is more evident thehigher theweb slenderness is. This circumstance turns to be logicalsince high values of slenderness indicate higher potential postcrit-ical reserve to develop by the structural element, and it is knownthat it is during the postcritical range when the bending stiffnessof the end post is required to adequately resist the bending intro-duced by the tension field stresses developed in the web panel.

On the other hand, further and detailed examination of theevolution of the results obtained while increasing the value of theaspect ratio allows concluding that the condition of rigid end postis more effective the lower the aspect ratio is. At this point ofthe analysis it is important to outline that, this situation is due tothe fact the higher the aspect ratio is, the condition given to theend post gets less influential in the whole behaviour of the webpanel since the percentage of the perimeter constrained gets lower.Anyway, it would be interesting further analysis in this directionsince it seems clear that the rigid or non-rigid condition should bedefined taking into account the aspect ratio of the web panel.

Moreover it is also interesting to observe that panels with thesame values of web slenderness but different aspect ratios exhibitdifferent strength given by the web contribution. In fact, highervalues of the aspect ratio give less contribution of the web panel(χw) to the whole resistant mechanism. As a consequence, andbeing evident by the different degree of influence of the conditionof rigid or non rigid end post for the different aspect ratios tested,it seems logical to define the design expressions to predict the webcontribution in the shear strength of stainless steel plate girders asa function of the aspect ratio of the analysed web panel.

5. Proposal of new design expressions

5.1. Initial shear buckling stress and ultimate shear capacity

Based on the numerical investigation conducted briefly pre-sented in Section 4, it is proposed to determine the initial shearbuckling stress in a stainless steel web panel of a plate girder τbssas:

τbss = η · krss ·π2E

12(1 − υ2)

(twd

)2

where η is a coefficient which mainly takes into account thematerial non-linearity of stainless steel. It can be obtained fromTable 3.



Fig. 16. Web contribution (χw) in plate girders with different aspect ratios. Austenitic grade.

krss is the shear buckling coefficient for stainless steel girders de-fined next which takes into account the real boundary conditionsof the web panel.

tw , d are the thickness and the depth of the web panel.E, ν are respectively the Young’s modulus and the Poisson’s

coefficient of the material.λw, which is the web slenderness as indicated in Section 2.1.The simple equations given below are proposed to be used for

the determination of the shear buckling coefficients in stainlesssteel web panels included in girders taking into account the realboundary conditions of the web panel. Different formulations areproposed for aspect ratios lower and higher than 1.0.

For web panels a/d ≥ 1.0:

krss = 0.5kss + (0.05kss + 0.2ksf ) ·tftw

tftw

< 2.0 (7a)

krss = (0.6kss + 0.4ksf ) +0.55ksf − 0.6kss

8·

(tftw

− 2)

,

krss ≤ ksf ;tftw

≥ 2.0. (7b)

For web panels a/d < 1.0:

krss = 0.8kss +0.2kss

3·tftw

tftw

< 3.0 (8a)

krss = kss +ksf − kss

7·

(tftw

− 3)

, krss ≤ ksftftw

≥ 3.0 (8b)

where the values of the coefficients kss and ksf are obtained fromthe classical formulation for plates loaded in shear developed byTimoshenko and Bulson respectively for plates simply supportedon four edges and for plates clamped on two opposite edges andsimply supported on the other two.

To conclude this section, Fig. 17 is presented in order toevaluate the improvement that of the presented design expressionover current methods to obtain the value of the shear bucklingcoefficients in stainless steel plates included in girders taking intoaccount the actual features of the analysed panel.

In this Figure, it can be first observed that the proposedequations, although still slightly conservative, yield more accuratepredictions of the shear buckling coefficients than the assumptionof a simply supported web panel included in most design codes.This historically presumed hypothesis of simply supported edgeshas been demonstrated to be clearly more conservative in webpanels with aspect ratio higher than 1.0. In fact, for aspect ratiosmuch lower than 1.0, the assumption of four simply supportededges does not lead to over conservative results and it is theauthors opinion that it could be maintained in design expressions.

It is also important to notice that a minimum ratio betweenflange and web thickness exists in order to assume at least thesimply supported condition. As results indicate, this minimum



Fig. 17. New proposal to determine shear buckling coefficients krss . Analytical vs. numerical results. Goodness of the proposal.

ratio is tf /tw equal to 3 for aspect ratios a/d < 1 and tf /tw equal to1.0 for aspect ratios a/d > 1. Special care has to be taken in case ofaspect ratios lower than 1.

When web panels are longitudinally stiffened, two differentbehaviours can be expected:

(a) The shear buckling develops through the whole web panel.Then, the shear buckling coefficient can be obtained from

klrss = krss + 9(da

)24

√(Islt3wd

)3

but not less than2.1tw

3

√Isld

(9)

where klrss is the shear buckling coefficient of the longitudinallystiffened web plate.

krss is the shear buckling coefficient of the web panel defined inEqs. (7) and (8).

Isl is the actual second moment of area of the longitudinalstiffener.

Then, the critical shear buckling stress (τbss) can be obtainedfrom Eq. (6) taking klrss= krss. The geometrical parameters toconsider to obtain the shear buckling stress must be the ones ofthe whole web panel.

This behaviour has been observed to be developed in webpanels of a/d = 1 and with only one longitudinal stiffener locatedat mid-depth for a value its the relative bending of the stiffness ofthe web panel γ /γ ∗ lower than 3.0.

(b) The shear buckling develops in the weakest subpaneldefined by the longitudinal stiffener. In this case, the shearbuckling coefficient can be obtained from previous Eqs. (7) and (8)introducing the dimensions of the corresponding sub-panel andtaking tf = (tf +tls)/2, being tf and tls are respectively the thicknessof the flange and the longitudinal stiffener defining the sub-panelobject of the analysis. This behaviour has been observed to bedeveloped inweb panels of a/d = 1 andwith only one longitudinalstiffener located at mid-depth for a value its the relative bendingof the stiffness of the web panel γ /γ ∗ higher than 3.0.

Dealing with ultimate capacity and based on the numericalinvestigation presented and discussed in previous sections, newdesign expressions are proposed below to predict the ultimateshear capacity of stainless steel plate girders. The ultimate shearcapacity (Vu) of a stainless steel plate girder is given by thecontribution of flanges (χf ) and the contribution of the web panel(χw). Thus, harmonized with Eurocode 3, Part 1–5 [15], it can be



Table 7Proposal of design expressions to evaluate the contribution of the web (χw)

Aspect ratio λw Rigid end post χw Non-rigid end post χw

a/d ≤ 1.0 λw ≤ 0.38 1.2 1.20.38 > λw ≤ 0.78 0.48 +

0.35λw

−0.03λ2w

0.48 +0.35λw

−0.03λ2w

λw > 0.78 0.48 +0.35λw

−0.03λ2w

0.33 +0.55λw

−0.095λ2w

a/d > 1.0 λw ≤ 0.47 1.2 1.2λw > 0.47 0.2 +

0.6λw

−0.06λ2w

0.2 +0.6λw

−0.06λ2w

written as:

Vu = Vw + Vf =(χw + χf

)· d · tw

fy√3. (10)

The contribution of theweb (χw) is givenby thedesign formulaepresented in Table 7 that follows, where different expressionsare given for aspect ratios lower (or equal) and higher than 1.0.Although themaximum theoretical value of the function governingthe contribution of thewebχw is 1.0, in the case of stainless steel itis justified the consideration of the strain hardening phenomenon,and that is the reasonwhy χw maximum value has been defined tobe 1.2.

At this point it is important to outline that the increase instrength given by the rigid condition of the end post is only takeninto account when designing plate girders with web panels withaspect ratios lower than or equal to 1.0.

The contribution of flanges (χf ), if the flanges are not fullyutilised to withstand bending, i.e. Ms < MfR, may accurately bepredicted according to following equation:

χf =bf t2f fyf

√3

c · twd · fyw

[1 −

[Ms

MfR

]2](11)

and

c =

(0.17 +

1.6 · bf t2f fyftwd2fyw

)· a. (12)

To finish this section, Figs. 18 and 19 illustrate, respectively fora/d ≤ 1.0 and a/d > 1, the predicted contribution of the webaccording with formulation presented in Table 7, together with allthe numerical tests results obtained. The accuracy of the proposeddesign expressions to determine the contribution of the web in theultimate shear strength in stainless steel plate girders presented isshown in these Figures. As can be observed, the design expressionshave been formulated as a lower bound of the numerical resultsobtained during the development of the numerical analysis.

The proposal of these new design expressions, although stillconservative for medium values of the aspect ratio (a/d = 1.5or 2, 0), means another step forward into the achievement of anoptimum design of the stainless steel plate elements since theypredict the actual capacity of these elements more accurately thanthe design expressions included in current codes and design guidesdealing with stainless steel structures.

5.2. Evaluation: Comparison with experimental results

Table 8 summarises the predictions of the initial shear bucklingload made by the classic theory, the ENV 1993-1-4 [12] and theone given by the new proposal of shear design. In the case of theultimate shear capacity, Table 9 presents the results obtained bythe application of the Simple Postcritical Method included in ENV1993-1-4 [12] to predict the ultimate shear resistance, togetherwith the results obtained from the design procedure included inEN 1993-1-4 [14] which is the one developed by Olsson [7], andthe ones given by the new shear proposal.

Fig. 18. Design curves proposed to obtain the web contribution in stainless steelplate girders. a/d ≤ 1.0.

Fig. 19. Design curves proposed to obtain the web contribution in stainless steelplate girders. a/d > 1.0.

From the results presented it can be observed that with theproposed design expressions, a better correlation with test resultsof the girders assayed at the Structural Technology Laboratoryof the UPC is achieved. A comparative analysis considering allthe predictions evaluated allows stating that the new sheardesign method means an important step forward into the efficientdesign of stainless steel structures, since a better accuracy in thepredictions is clearly observed.

6. Summary and conclusions

The lack of structural specifications and design guides ac-curately predicting the actual behaviour of stainless steel asstructural material has led to a limited use of this new material inbuilding and civil engineering despite exhibiting excellent aesthet-ics andmechanical properties. This situation has promoted, all overthe world, investigation and research programmes dealing withthe structural performance of stainless steel in order to later on de-velop newdesign expressions to achieve amore efficient use of thismaterial in structural applications in construction. The final pur-pose of the research reported herein is to make a new contributionin this direction. Specifically, to improve the fundamental under-standing of the behaviour of stainless steel plate girders loaded inshear considering both transversely and longitudinally stiffening.

Important improvements have been achieved during thedevelopment of the analysis presented in this technical paper



Table 8Initial shear buckling load

Girder λw Vcri,test (kN) Vcri,predicted (kN) Vcri,test/Vcri,predicted

Classic ENV 1993-1-4 Proposal τbss Classic ENV 1993-1-4 Proposal τbss

R&NR

nr700ad15 2.085 150.00 115.89 116.41 149.96 1.29 1.29 1.00r700ad15 2.085 150.00 115.89 116.41 149.96 1.29 1.29 1.00nr600ad2 1.894 180.00 120.43 117.37 157.30 1.49 1.53 1.14r600ad2 1.894 180.00 120.43 117.37 157.30 1.49 1.53 1.14nr500ad25 1.625 175.00 136.32 120.99 167.87 1.28 1.45 1.04r500ad25 1.625 180.00 136.32 120.99 167.87 1.32 1.49 1.07nr400ad325 1.329 175.00 162.95 120.60 175.02 1.07 1.45 1.00r400ad325 1.329 175.00 162.95 120.60 175.02 1.07 1.45 1.00i500ad1 1.3 280.00 212.91 153.97 194.37 1.32 1.82 1.44i500ad15 1.48 237.50 162.25 134.01 179.30 1.46 1.77 1.32

LS

lw-25.8 1.85 350.36 142.3 138.30 202.03 2.46 2.53 1.73li-40.8 1.74 382.36 161.1 151.5 276.95 2.37 2.52 1.38ls1-50.8 1.67 414.52 174.3 159.9 341.83 2.38 2.59 1.21ls2-100.8 1.38 421.83 255.7 200.0 341.83 1.65 2.11 1.23ls3-50.20 1.59 421.53 192.6 170.5 356.73 2.19 2.47 1.18

Comparison of the proposed shear formulation and the procedures included in ENV 1993-1-4 [12] with test results of the experimental campaign.λw is the slenderness of the web panel.Vcri,test is the experimental critical shear load.Vcri,pred Classic is the initial shear buckling load obtained by applying the Classical Theory.Vcri,pred ENV 1993-1-4 is the initial shear buckling load obtained by applying formulation included in ENV 1993-1-4 [12].Vcri,pred Proposal is the initial shear buckling load obtained by the proposal presented in this research.R&NR are the girders specimens tested during the First experimental campaign.LS are the girders specimens tested during the Second experimental campaign.

Table 9Ultimate shear load

Girder λw Vplast (kN) Vu,test (kN) Vu,predicted (kN) Vu,test/Vu,predicted

SPM ENV1993-1-4 RSFM EN1993-1-4 Proposal SPM RSFM Proposal

R&NR

nr700ad15 2.085 487.30 309.21 193.05 228.94 268.24 1.60 1.35 1.15r700ad15 2.085 487.30 327.17 193.05 228.94 268.24 1.69 1.43 1.22nr600ad2 1.894 417.69 260.65 176.76 204.56 241.92 1.47 1.27 1.08r600ad2 1.894 417.69 262.92 176.76 204.56 241.92 1.49 1.29 1.09nr500ad25 1.625 348.07 228.05 162.61 187.29 217.76 1.40 1.22 1.05r500ad25 1.625 348.07 236.54 162.61 187.29 217.76 1.45 1.26 1.09nr400ad325 1.329 278.46 217.90 146.46 170.19 192.87 1.49 1.28 1.13r400ad325 1.329 278.46 215.33 146.46 170.19 192.87 1.47 1.27 1.12i500ad1 1.3 348.07 357.33 185.34 262.18 275.84 1.93 1.36 1.30i500ad15 1.48 348.07 279.47 171.45 218.02 227.12 1.63 1.28 1.23

LS

lw-25.8 1.85 488.2 350.36 207.3 244.3 283.75 1.69 1.43 1.23li-40.8 1.74 488.2 382.36 216.0 253.5 292.10 1.77 1.51 1.31ls1-50.8 1.67 488.2 414.52 221.5 259.5 297.61 1.87 1.60 1.39ls2-100.8 1.38 488.2 421.83 248.8 292.4 327.41 1.70 1.44 1.29ls3-50.20 1.59 488.2 421.53 228.6 267.5 304.92 1.84 1.58 1.38

Comparison of the proposed shear formulation and the procedures included in ENV 1993-1-4 [12] and EN 1993-1-4 [14] with test results of the experimental campaign.Vplast =

fy√3dtw .

Vu,test is the experimental ultimate shear load.Vu,pred SPM is the ultimate shear load obtained by applying the Simple Postcritical Method included in ENV 1993-1-4 [12].Vu,pred RSFM is the ultimate shear load obtained by applying formulation included in EN1993-1-4 [14].Vu,pred Proposal is the ultimate shear load obtained by the proposal presented in this research.

dealing with the shear buckling strength in stainless steel webpanels of plate girders.

A simple design expression is proposed to determine the shearbuckling strength in stainless steel web panels. This expressionconsiders the effects of the material non-linearity together withthe actual boundary conditions of the web panel given by twodefined parameters: η and krss, the shear buckling coefficient.Expressions given to obtain η will quantify mainly the effect of thematerial non-linearity whereas the proposed analytical expressionof krss will reproduce main effect of the boundary conditions.

After confirming the assumption that simply supported webpanels is clearly conservative, numerical models have

demonstrated that the actual support condition at the juncture be-tween web and flanges is closer to a clamped support. Design ex-pressions to obtain the shear buckling coefficients considering thereal effect of the boundary conditions have been proposed.

Last but not least, it is the authors’ opinion that the proposedmethodology and expressions to determine more realistically theshear buckling strength in stainless steel plate girders can be withno difficulty extended to carbon steel and other non-linear metalmaterials employed in construction.

On the other hand, another important step forward into theanalysis of the shear behaviour of stainless steel plate girders,this paper has dealt with the ultimate shear capacity of these



structural elements. Although highly improved in its last version,the conservative character of the current design rules includedin Eurocode 3, Part 1–4 [12] to predict the ultimate strengthof stainless steel plate girders since the condition of rigid andnon-rigid end post is not considered. A complete numerical andexperimental study has been conducted.

Relevant conclusions achieved during the numerical analysisdevelopment have led to the proposal of new design expressionsbased on the Rotated Stress Field model developed by Höglund topredict the ultimate shear strength in stainless steel plate girdersby considering the effect of the condition of rigid and non-rigidend post in the geometry of the plate girder. Different analyticalexpressions have been proposed for aspect ratios equal to or lowerthan 1.0, where the rigid condition has been demonstrated to beefficient, and for aspect ratios higher than 1.0, where no significantdifferences between rigid and non-rigid geometries have beenobserved.

Acknowledgements

This research work was carried out under the financial supportprovided by the Spanish Ministry of Science and Technologyand the Spanish Ministry of Education and Science as partof the Research Projects MAT 2000-1000 and BIA2004-04673respectively. The first author received support from a grantawarded by the Universitat Politècnica de Catalunya sinceJanuary 2001 to December 2004. The authors also appreciate thecollaboration of Inoxcenter and URSSA providing the materialrequired to do the experimental tests.

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Documents

Shear resistance in stainless steel plate girders with transverse and longitudinal stiffening