ELASTIC CRITICAL LOAD OF PLATES AND PLATE GIRDERS SUBJECTED TO PATCH LOAD

Poslednjih decenija problemi stabilnosti i postkritičnog ponašanja limenih nosača značajno privlače pažnju. Intenzivno je istraživano ponašanje limenih nosača (zavarenih I poprečnih preseka) pri dejstvu lokalizovanog opterećenja, odnosno pod uticajem delimično raspodeljenog opterećenja po nožici, i to na mestima gde neposredno ispod zadatog opterećenja ne postoje vertikalna (poprečna) ukrućenja. Važnost ovog problema porasla je sa opštim trendom izbegavanja vertikalnih ukrućenja izuzimajući preseke iznad oslonaca, i u slučaju postojanja pokretnog opterećenja. Sem kranskih nosača opterećenih točkovima krana, važan primer ovakvog opterećenja odnosi se na prevlačenje konstrukcije preko privremenih ili stalnih oslonaca, tokom montaže kontinualnih čeličnih mostova. Analiza ponašanja nosača pri dejstvu lokalizovanog opterećenja obuhvata određivanje raspodele lokalnih normalnih napona u rebru nosača, određivanje elastične kritične sile izbočavanja i određivanje granične nosivosti nosača. Granična nosivost igra bitnu ulogu pri proračunu. Naučnici iz mnogih zemalja bavili su se teorijskim i eksperimentalnim istraživanjima ovog problema tokom


INTRODUCTION
The stability problems and ultimate load behaviour of steel plate girders have attracted a lot of attention during the last decades.The behaviour of the plate girder (welded I-girder) subjected to patch load or partially distributed load on the flange in the plane of a web, without a vertical (transverse) stiffener bellow the load was also intensively investigated.This problem has got the importance with a general trend to avoid vertical stiffeners except at supports and also in the case of moving loads.Except for crane girders loaded by crane wheels, a remarkable realistic load case in which this situation arises is the launching phase of multi-span steel plate girder bridges during construction over temporary or permanent supports.
In the analysis of the behaviour of the patch loaded girder the attention is directed towards the distribution of the local direct stresses under the load in the web, the elastic critical load of the web panel and the ultimate load of the girder.For the development of design procedures, the ultimate load is of a great importance.The research workers in many countries have investigated theoretically and experimentally this problem over Isidora Jakovljevic, MSc, University of Belgrade, Faculty of Civil Engineering, Bulevar kralja Aleksandra 73, 11000 Belgrade, Republic of Serbia, isidora@imk.grf.bg.ac.rsSasa Kovacevic, MSc, School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, USA, sasa.kovacevic@wsu.eduAssis.Prof. Nenad Markovic, PhD, University of Belgrade, Faculty of Civil Engineering, Bulevar kralja Aleksandra 73, 11000 Belgrade, Republic of Serbia, nenad@grf.bg.ac.rs poslednjih pedeset godina.Međutim, i značajan broj eksperimenata i složenog teoretskog rada nije pružio potpun uvid u rešenje problema, tako da su i dalje širom sveta u toku značajna istraživanja različitih segmenata ove oblasti.
In the initial phases of investigations, attempts were made to relate the ultimate load to the elastic critical (buckling) load [8], [5], [15].Very soon it was found that generally, it was not possible.Also, analytical procedures were not successfully applied, as the problem is very complex, depending on many interrelated parameters.As a corollary, a majority of the suggested solutions were of empirical nature, based on experimental researches.The development of computer programs based on the finite element (FE) method and increased capabilities of computers enabled the development of "numerical experiments" that are nowadays widely applied.
With the development of Eurocodes for the design of civil engineering structures, a new approach was applied to harmonize solutions of the patch load problem with other stability problems.The main elements in the determination of the ultimate patch load are elastic critical load, yield load, and reduction factor.The determination of the elastic critical (buckling) load got again the importance, as it is needed for the determination of the nondimensional slenderness F λ according to the following equation [3]: gde su: ly -efektivna opterećena dužina, tw -debljina rebra, fyw -granica razvlačenja rebra, Fcr -kritična sila izbočavanja, definisana kao: where are: ly -the effective loaded length, tw -the web thickness, fyw -the yield strength of the web, Fcr -the critical buckling force, defined as: gde su: kF -koeficijent izbočavanja, E -modul elastičnosti, hw -visina rebra.
The solution applied in Eurocode 3 for design of steel structures [3] uses simplified expressions for the buckling coefficients.For example, for a plate girder under patch load applied on the top flange, between two vertical stiffeners on a distance a, the buckling coefficient should be obtained using Eq.3: Kao što se može primetiti, izraz iz jednačine 3 ne uzima u obzir dužinu na kojoj je aplicirano poprečno opterećenje.Naveden izraz za određivanje koeficijenta izbočavanja usvojen je uprošćenjem izraza koji je predložio Lagerkvist [8]: As it could be observed, the given expression in Eq. 3 does not account the distribution length of an applied patch load.This expression for determination of buckling coefficient is adopted by simplifying the expression proposed by Lagerqvist [8]:  [9] upoređeni s prediktivnim vrednostima koje daje Evrokod 3 pokazali su [10] da standard predviđa izrazito konzervativne rezultate.Ovaj rad je pokušaj da se daju izvesna poboljšanja navedenih procedura.
Unlike Eq. 3, expression 4 includes a ratio between flange and web stiffness and a ratio between load length and web depth.
The investigation of patch load has started at the Faculty of Civil Engineering University of Belgrade initially in the eighties within an international cooperation with the University of Cardiff (with T. M. Roberts) and with the Czech Academy of Sciences in Prague (with M. Skaloud).In the last period, the collaboration was continued with the Faculty of Civil Engineering University of Montenegro in Podgorica (with D. Lučić and his collaborators), where intensive researches were also carried out [2].The results obtained in the experimental research [9] compared with the procedures applied in Eurocode 3 had shown [10] significantly conservative results given by Eurocode 3.An attempt is made in this paper to give some improvements to that procedure.
Elastic critical load of an isolated web plate for different boundary conditions is compared with an elastic critical load of an I-girder.Moreover, the experimentally determined ultimate load is compared with the procedure given in Eurocode 3 , as well as with the modified Eurocode procedure including the suggested correction for determination of the elastic critical load accounting the distribution length of an applied patch load.
Nowadays, numerical analysis techniques are widely used in research involving structural steel and in analyses and designs of steel structures and elements.The FE method based numerical analysis is the most popular computational tool in this field and it has been successfully applied in many papers regarding the critical load of plate girders under different loading conditions.The commercial multi-purpose FE analysis software Abaqus was used for the numerical analysis [1].In order to get a better insight into the problem, geometry and loading of the girders were considered according to those applied in the accompanying experimental research [9].

NUMERICAL SIMULATION
Introductory it was stated that the purpose of this paper is the comparison of the elastic critical loads for an isolated web plate, varying boundary conditions, with critical loads of an I-girder.The two plates schematically presented in Fig. 1, labelled as Model 1 with an aspect ratio a/hw = 1 and Model 2 with an aspect ratio a/hw = 2 (a = 1000 mm), were investigated.The web plate thickness is tw = 4 mm.The length of an applied uniform load ss was varied.The degree of freedom 2 is only constrained in the vertical edges while the degree of freedom 3 is constrained in all edges.Three different cases were analysed: (a) simply supported plate with no additional constraints, uklještena duž horizontalnih ivica (sprečen je stepen slobode 4), (c) ploča uklještena duž svih ivica (duž vertikalnih ivica sprečen je stepen slobode 5, a duž horizontalnih, stepen slobode 4).
(b) plate simply supported on the vertical and clamped on the horizontal edges (degree of freedom 4 is constrained), (c) plate clamped along all edges (degree of freedom 5 is constrained in the vertical edges and the degree of freedom 4 in the horizontal edges).
For the sake of brevity, through the rest of the paper, these cases will be marked as SS, CS, and CC, respectively.
In the same vein, the two models of an I-girder according to the previously described Model 1 and Model 2 are displayed in Fig. 2. The flange width and thickness were set to 120 mm and 8 mm, respectively.The boundary conditions are according to the experimental procedure described in [9].The considered material for all cases is homogenous with an elastic modulus of E = 210 GPa and Poisson's ratio of ν = 0.3.For the FE analysis, a general-purpose four-node quadrilateral shell element with reduced integration and six degrees of freedom per node S4R from the Abaqus element library was used.According to the mesh convergence study performed on the isolated web plate of Model l, the adopted finite element size is 5 mm for all numerical runs.The variation of the buckling coefficient due to change of the finite element size in the case of Model 1 for a simply supported plate is graphically presented in Fig. 3(a).
The buckling coefficient depends on boundary conditions, loading type and an aspect ratio a/hw.On the other hand, it is not influenced by plate thickness or material properties.However, for a different plate thickness applied on a simply supported plate of Model 1, a slight variation in the results could be noticed, as charted in Fig. 3(b).
this change of boundary conditions does not affect the value of kF.On the contrary, in the case of Model 1 and a longer uniform load, the difference in the results comes up to 25 %.Expectedly, those boundary conditions do not play a decisive role for higher aspect ratios since the applied load is localized and for an aspect ratio a/hw ≥ 2 their influence is negligible.

RESULTS AND DISCUSSION
The aim of this chapter is to highlight the obtained numerical results of the buckling coefficient considering an isolated web plate and I-girder.The results for all described numerical models are listed and discussed thoroughly.Furthermore, new expressions for the determination of the buckling coefficients, as a function of patch load length, are presented.The purpose of the proposed expressions is to improve the ultimate load calculated using the procedure given in Eurocode 3.
Table 1 and Table 2 show a comparison summary of the numerical results and the buckling coefficients obtained by Eurocode 3 (Eq.3) and according to Lagerqvist's expression (Eq.4), while a graphical representation is shown in Fig. 5.One can instantaneously see that the SS plate for both models and for all lengths of an applied patch load gives extremely low values of the critical loads.Introducing the clamped constraint on the horizontal edges (CS plate) the kF values are improved but still below the kF of the Igirder, especially for Model 1. Boundary conditions that lead to the results closest to the I-girder behaviour are clamped edges, both vertical and horizontal (CC plate).Later on in this chapter, relations between kF values for the CC plate and for the I-girder are discussed.
It is noteworthy to observe another interesting point regarding the clamped boundary conditions for large values of ss.It is clear from Fig. 5 that the rigid transversal stiffeners (see Fig. 2) influence the buckling coefficient more than clamped constraint whereas for small values of ss their influence is not present since the load is localized.Additionally, the clamped constraint on the vertical edges (the difference between the CC and CS plate) reveals the fact that constrained rotations along these edges do not have an influence on the  Sa slike 5, može se još uočiti da izraz koji daje Evrokod 3 vodi do konstantnog koeficijenta izbočavanja pri promeni dužine raspodeljenog opterećenja.Za veće dužine ss, razlike u rezultatima prema Evrokodu 3 i onim dobijenim za I-nosač jesu značajne.S druge strane, Lagerkvistov izraz ima linearan trend rasta.Međutim, predložena jednačina ne daje dobro podudaranje s numerički dobijenim rezultatima za I-nosač, posebno u slučaju Modela 2, gde koeficijenti izbočavanja prema Lagerkvistu čak nisu na strani sigurnosti.
S obzirom na opisane nedostatke jednačina 3 i 4 pri buckling coefficient for an aspect ratio a/hw ≥ 2, as stated before.Conversely, they influence the buckling coefficients with a uniform factor for an aspect ratio a/hw = 1 (Model 1) for all values of ss.A further parametric study should be made in order to make an airtight conclusion about this boundary condition and its influence on the buckling coefficient.It could be also observed in Fig. 5 that the Eurocode expression gives constant values of buckling coefficient although the load length is varied.For large values of ss, differences in Eurocode 3 and I-girder results are significant.On the other side, the Lagerqvist's expression has a linear trend line.However, the proposed equation d oes not give a good fit to numerically obtained results for I-girder, especially in the case of Model 2, when the buckling coefficients according to Lagerqvist are not even on the safe side.određivanju koeficijenta izbočavanja za analizirani Inosač, predlaže se korišćenje modifikovanih izraza za određivanje kF.Definisane su polinomske funkcije koje dobro aproksimiraju dobijene diskretne vrednosti koeficijenta kF za I-nosač u funkciji dužine raspodeljenog poprečnog opterećenja.Uzimajući u obzir jednostavnost proračunskih procedura u Evrokodu 3, poželjno je koristiti polinome prvog ili drugog reda.Na primer, sledeće jednačine daju dobru aproksimaciju koeficijenta kF za I-nosač: za Model 1: Due to the described drawbacks of Eq. 3 and Eq. 4 for prediction of the buckling coefficient for the analysed I-girder models, we propose obtaining kF through modified expressions.Polynomial functions of load length are defined to fit the obtained discrete values of kF for the I-girder.Keeping in mind the simplicity of the design procedure in Eurocode 3, a first or second order polynomial function is desirable.The following forms give a good approximation for the kF values of the Igirder: for Model 1: Na slici 6 grafički je predstavljeno poređenje koeficijenata izbočavanja za I-nosač, dobijenih numeričkom analizom i aproksimiranih koeficijenata prema predloženim izrazima.Treba imati na umu da su ove vrednosti koeficijenata određene samo za slučajeve analizirane geometrije.Potrebno je sprovesti detaljnu parametarsku analizu kako bi se odredili jedinstveni opšti izrazi u funkciji geometrijskih parametara, tj.odnosa širine i visine rebra, debljina rebra i nožice, širine nožice, itd.
A graphical comparison of the numerically obtained buckling coefficient of the I-girder and coefficients approximated by the proposed expressions is shown in Fig. 6.One should bear in mind that these values for the constants are determined only for the analysed geometries.A detailed parametric study is necessary in order to obtain a unique general expression as a function of geometric parameters, i.e. web aspect ratio, thickness of the web and flange, flange width, etc.
It is interesting to speculate on whether the assumed stress distributions for the analysed cases of Model 1 and Model 2, are in correlation with the aspect ratios a/hw.In other words, the hypothesis is that stress distribution is not only influenced by flange thickness or stiffness, but also by the length of the zone between vertical stiffeners.For making final conclusions in this direction, more numerical simulations of different girder geometries should be obtained.
Values of the experimentally obtained ultimate loads and corresponding ultimate loads according to the procedure applied in Eurocode 3 are pictorially compared in Fig. 8 and numerically in Table 5.The ultimate loads according to Eurocode 3 are calculated following two approaches.Firstly, the values are obtained according to the current procedure for determination of the buckling coefficient presented in Eq. 3 (with kF not dependent on the loading length ss).Secondly, the Eurocode defined procedure is modified by using Eq. 5 and Eq. 6 for kF determination proposed in this paper.
Noticeably, the ultimate loads calculated by the current Eurocode procedures are considerably lower than the experimentally obtained ultimate loads and the difference is more pronounced for higher patch load lengths.In other words, the Eurocode design prediction gives rather conservative values for the ultimate load.However, the proposed modification of the Eurocode procedure with a simple correction regarding only the buckling coefficient improves predictive values, especially for longer loading lengths.

CONCLUSIONS
Buckling of plates and I-girders under patch load is analysed in this paper.Elastic critical load of an isolated web plate for different boundary conditions is compared with an elastic critical load of an I-girder.
The connection between buckling of a clamped plate and an I-girder is observed.Accounting stress distribution through flange plate, it is possible to relate buckling of a clamped plate to an I-girder.Also, it is assumed that not only flange thickness or stiffness influences the stress distribution, but also aspect ratios a/hw and the length between vertical stiffeners.Conclusions in this direction may lead to the improvement of the definition of effective loaded length.
In order to improve the results from Eurocode for Kako bi se poboljšale vrednosti koje Evrokod daje za proračun čeličnih konstrukcija, predloženi su novi izrazi za određivanje koeficijenta izbočavanja.Predloženi izrazi obuhvataju određivanje koeficijenta izbočavanja i u funkciji dužine apliciranog poprečnog opterećenja, dok su dalje procedure za određivanje graničnog opterećenja nepromenjene u odnosu na Evrokod 3. Predloženi izrazi poboljšavaju prediktivne vrednosti granične nosivosti, posebno u slučajevima većih dužina jednakopodeljenog lokalizovanog opterećenja.Međutim, kako bi se objasnio uticaj dužine opterećenja na koeficijent izbočavanja i kako bi bila određena direktna veza između izbočavanja izolovanog rebra nosača i izbočavanja I-nosača, steel structures design, a new expressions for the determination of the buckling coefficient are proposed in this paper.The expressions include the calculation of the buckling coefficient also as a function of patch load length, while the procedure for obtaining the ultimate load is the same as in Eurocode 3. The p roposed expressions improve predictive ultimate load values, especially for longer patch load lengths.However, in order to elucidate the influence of the length of patch load on buckling coefficient and to make a straightforward connection with an isolated web plate considering different geometries of an I-girder, further analyses are required.uzimajući u obzir uticaj različitih geometrija nosača, neophodno je sprovesti dalje analize.
In a nutshell, the presented expressions enable a fruitful background for parametric analyses and production of a large number of numerical tests that could be used in order to better determine the ultimate loads of plate girders using the buckling coefficients.Furthermore, since the ultimate loads are still too far below the experimentally obtained ultimate loads, the present expressions can be also exploited in order to improve different elements currently present in the design procedure in Eurocode 3, i.e. the effective loaded length.

Isidora JAKOVLJEVIC Sasa KOVACEVIC Nenad MARKOVIC
The determination of the elastic critical load (buckling load) is an important element of the assessment of the ultimate load of plate girders according to Eurocode 3 for design of steel structures.An analysis of the critical load of a plate corresponding to the web of an I-girder for different boundary conditions and the critical load of the I-girder itself subjected to patch load is given in the paper.Conclusions regarding their correspondence are given.Also, experimentally determined ultimate loads of I-girders are compared with predictive values according to Eurocode 3 for the models used in this analysis.A modified procedure for the ultimate load determination is proposed by following the Eurocode 3 algorithm and changing only buckling coefficient.In order to improve the ultimate load prediction, it is suggested to calculate the buckling coefficient also as a function of patch load length.
Key words: buckling coefficient, elastic critical load, patch loading, plate buckling, plate girder buckling

Table 1 .
Buckling coefficients for different boundary conditions for Model 1

Table 4 .
Relation between buckling coefficients for CC plate and I-girder for Model 2

Table 5 .
Ultimate loads obtained experimentally, according to Eurocode 3 and modified Eurocode procedure