PLATE BUCKLING ACCORDING TO EUROCODE 3. COMPARISON OF THE EFFECTIVE WIDTH METHOD AND THE REDUCED STRESS METHOD

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1 SDSS Rio 00 STABILITY AND DUCTILITY OF STEEL STRUCTURES E. Batista P. Vellasco L. de Lima (s.) Rio de Janeiro Brail September PLATE BUCKLING ACCORDING TO EUROCODE 3. COMPARISON OF THE EFFECTIVE WIDTH METHOD AND THE REDUCED STRESS METHOD Pro. Jose M. Simon-Talero* Mrs. Ana Caballero ** * Civil Engineer ETS Ingenieros de Caminos (UPM-España) Torroja Ingenieria S.L. ** Civil Engineer Imperial College (London) s: jsimontalero@torroja.org ; ana.caballero05@imperial.ac.uk Kewords: plate buckling reduced stress method eective width method steel structure stabilit Abstract. The Eurocode 3 section -5 (EC3-5) on plated structures presents two approaches to calculate the ultimate resistance o compressed plated elements aected b plate buckling (class ): the eective width method and the reduced stressed method. Given its compleit the latter o these methods has not been the subject o much research until now. Thereore the scope o its applicabilit as well as its relative advantages and disadvantages with respect to the other methods remain to a large etent uncertain. The aim o the present paper is to contribute to a better understanding o the background and practical use o these methods. To that end a brie introduction to the background o these methods as well as their application to non-stiened and stiened plated structures is irst included. Then the article reproduces a comparative evaluation o these methods based on the parametric stud carried on the paper Estudio comparativo de los métodos de cálculo propuestos en los artículos 0 del EN []. Finall the main conclusions o this stud are summarised at the end o the article. INTRODUCTION Eurocodes are produced in the EU to harmonise dierent tpes o structures and building materials. Steel structures are included in Eurocode 3 (EN 993). Section -5 o Eurocode 3 [] on plated structures presents two approaches to calculate the ultimate resistance o compressed plated elements aected b plate buckling (class ): the eective width method and the reduced stressed method. Given its compleit the latter o these methods has not been the subject o much research until now [3] []. Thereore the scope o its applicabilit as well as its relative advantages and disadvantages with respect the other methods remain to a large etent uncertain. The paper Estudio comparativo de los métodos de cálculo propuestos en los artículos 0 del EN [] which can be downloaded rom [A] contributes to a better understanding o the implications o using these alternative calculation methods and covers a comparative evaluation o the methods. This article summaries the main points o the paper in order to give a concise but clear introduction to the subjcet. To that end the background o each method is irst presented in particular the undamental concepts o the reduced stress method. This document covers the case o non-stiened as well as stiened plates. Then the results o the application o each method to a particular case are reproduced. Finall the comparative evaluation o the eective width and reduced stress methods is achieved b a parametric stud. The parameters and conclusions o this stud are summaried at the end o the article. The parametric stud was perormed using an EXCEL spreadsheet which can be downloaded rom [A]. 05

2 Pro. José M. Simón-Talero Mrs Ana Caballero BACKGROUND THEORY OF THE EC3-5 APPROACHES. The eective width method a) Non-stiened plates This approach was irst developed b Von Karman (93) and subsequentl modiied b Winter (97). It is included in section o Eurocode 3 part -5. The eective width approach is based on the act that a plate subjected to in-aial compressive stresses will go through a post-buckling stage where higher stresses will be transerred to the stier areas o the plate due to the membrane eect (Figure ). Thereore the strength o a plate aected b buckling can be assumed to be that o an imaginar plate o reduced breadth b e not subjected to buckling. According to this the plate will ail when compressive stresses reach the maimum strength (ield strength ) o the reduced plate. Figure : Stress transer The behavior o a plate under compression stresses depends to a large etent on the ratio o its principal dimensions. Considering a thin plate o length a width b and thickness t mm (ab<<t) i a is o the same magnitude as b (square shaped plate) its response will be similar to that o a compressed plate and will be thereore governed b the plate-buckling theor. However i a is signiicantl smaller than b its response will be better described b the column buckling theor [5] [6]. These responses are known as the plate and the columns mechanisms respectivel. The theor o this mechanisms is ull covered on section.. a and.. b o EN (EC3-5) []. In act the actual response o a compressed plate alls between these two mechanisms and the greater inluence o one above the other depends purel on the plate s dimensions. The EC3-5 takes this act into consideration and suggests an interpolation ormula between the two mechanisms in order to obtain a more realistic value o the plate s strength: c ( c ) ( ) c () where and c are the reduction actors or the plate and column mechanisms respectivel and is a measure between the ical strength o the plate mechanism and that o the column mechanism given b cr p () cr c where crp and crc are the ical stresses according to the plate mechanism and the column mechanism respectivel. Figure represents the interpolation between the two mechanisms according to 05

3 Pro. José M. Simón-Talero Mrs Ana Caballero Figure : Interpolation between the column behaviour and the plate behaviour b) Stiened plates Plate girder The irst step in the calculation o the ical strength o a compressed plate girder b appling the eective width approach is to stud the buckling eects in ever individual plate that conorms the plate girder and its stieners. This is known as local plate-buckling stud. Subsequentl the stud o the platebuckling phenomenon o the whole plate takes place. To this end the eective width method reers to the interpolation between the column mechanism and the plate mechanism [7] [8]. The column mechanism establishes imaginar cuts in the plate between two stieners that eliminates its parts aected b buckling. Thus the remaining parts o the plate along with each stiener can be treated as a column under an aial compressive orce. This mechanism assumes that columns do not receive an support rom the adjacent one but are able to buckle in a direction normal to their plane (Figure 3a). The plate mechanism is based on the act that these cuts do not eist and thereore the column will be supported b the adjacent columns (Figure 3b). Figure 3a: Column mechanism Figure 3b: Plate mechanism The reduction actor or the plate mechanism is determined rom: 0055(3 ) where is the slenderness o the plate epressed as: A c () cr p 053

4 Pro. José M. Simón-Talero Mrs Ana Caballero Ac being the ratio between the eective area and the gross area o the stiened plate in order to take into account the eects o plate buckling in the parts between the stieners. The term crp represents the ical stress according to the plate mechanism which in Anne A o EN is given b: cr p k p E where ( ) k cp when ( )( ) a) k cp when (6b) b The column mechanism will appl the European curves or buckling in which the coeicient given 009 i / e takes the values =09 or =03 depending on whether the stieners are open or closed. In the column mechanisms calculations each longitudinal stiener will be treated as a separate column since the are simpl supported between two transversal stieners. To conclude the eective width approach is based on an interpolation between the two mechanisms using equation (8) ielding the inal reduction actor c. This actor is applied to the eective area o the plate girder (i.e. the resultant area o the plate girder ater the local plate-buckling stud) to obtain the total eective area: Webs A e bc e T p c Ac e loc t (8) The application o the eective width method to stiened webs remains analogous to that o the plate girder ecept or a number o adjustments arising rom its linearl varing stress distribution: in the local plate-buckling analsis the reduction actor applied to each subplate will depend on its particular stress distribution. Thereore the dimensions o the subplates parts adjacent to each stiener will var along the web. The EN [] presents the eective width o the subplates as a unction o the stress distribution in Table Reduced stress method a) Non-stiened plates This method is briel covered in section 0 o EC3-5 as an alternative to the eective width method. The reduced stress method states that the strength o a plate aected b buckling can be obtained b appling a reduction actor to the entire section with no reduction due to plate buckling. In this wa the plate could be treated as a class 3 element without carring an reduction in its dimensions. The reduction actor is a unction o the coeicient ultk relating the applied stress to the ield stress o the 05

5 055 Pro. José M. Simón-Talero Mrs Ana Caballero plate. The latter ma be obtained b one o the established stresses eria like the Von Mises erion which is the erion adopted in the EC3-5. In contrast with the eective width method the reduced stress method depends on both the normal and the shear stress. This increases the compleit o its application to compressed plates speciall in the case o webs since the concentrate the majorit o the shear stresses. This eplains its limited o use in common engineering practice. The coeicient ultk which when applied to the design stress leads to ielding o the most compressed point o the plate can be obtained rom: 3 k ult (9) In an analogous manner the coeicient given as the scaling actor that when applied to the design stress ields the ical stress is determined rom: / (0) The slenderness o the plate will be given b the ratio o these coeicients: k ult ma () Subsequentl the reduced stress method accounts or the eects o plate buckling b introducing a reduction actor which can be either the minimum o the reduction actor obtained b the interpolation ormula c and the reduction actor due to shear stresses v or the value obtained b interpolating between these actors. The inal veriication o this approach is epressed as ollows: M k ult Rd () b) Stiened plates Plate girder The application o the reduced stress method to a stiened plate girder proves to be relativel simple due to the act that it is onl subjected to normal compressive stresses. Thereore the coeicient ultk will onl be a unction o the applied normal stress k ult (3) And the coeicient is given b / ()

6 Pro. José M. Simón-Talero Mrs Ana Caballero Webs Webs concentrate the majorit o the shear stresses o a section. Thereore in addition to the compleit o the variant distribution o normal stress the incorporation o the shear stresses in calculations appears as a new diicult. The coeicient ultk will in this case be epressed as a unction o the design normal stress and the shear stress : and the coeicient remains ult k 3 (5) / (6) 3.- APPLICATION TO A PARTICULAR CASE (REFERENCE CASE) In order to illustrate all the steps involved in the eective width method and in the reduced stress method the have been applied to a particular case o plate girder and webs issued rom a common bridge cross-section. The dimensions o these elements are represented in igures and 5 respectivel. Figure : Plate girder dimensions (mm) Figure 5: Web dimensions (mm) 056

7 Pro. José M. Simón-Talero Mrs Ana Caballero The detailed calculations as well as the EXCEL spreadsheet used to perorm these calculations [] can be downloaded rom [A]. The results issued rom this particular stud are summaried in Table. Table : results or the application o the eective width method and the reduced stress method to a particular eample o plate girder and web (reerence case) Eective width method Reduced stress method Plate girder A A c e c e Rd A A b t bruta c etremos Web W W e bruto Rd Rd c PARAMETRIC STUDY In order to evaluate the accurac o both methods a parametric stud has been perormed. In this stud [] each method is applied to a number o dierent smmetries and dimensions o stiened plate girders and webs. As it has been said all these calculations have been perormed using an EXCEL spreadsheet which can be downloaded rom [A]...- Case studies The parametric stud includes cases studies based on eamples o plate girders and 8 eamples o webs derived b varing the thickness o the reerence case. Ever case stud is divided in groups o analsis: 3 or the plate girder and or the webs. Each group covers the application o each method when varing a single dimension parameter rom the reerence case: the width b length a and number o stieners N will be the varing parameters or the plate girder and the height h or the web s case. For each group the eective area A ce (or eective modulus W ce in the case o webs) and the reduced strength ultd will be obtained. Finall the comparison o the two methods will ollow rom A the analsis o a number o graphs showing the variation o the results ratio c e / Abruta varing each dimension. /( / ) when Plate girder The stud cases are based in a plate girder o thickness 6 0 and 5 mm respectivel. For all the stud cases the values or the varing parameters o each group are: - width b: rom b=000mm to b=5000 at 00mm intervals and rom b=5000mm to b=0000 at 500mm intervals - length a: rom a=000mm to a=0000 at 000mm intervals - Number o stieners N: N= Webs The parametric stud includes 8 case studies considering webs o thickness t w = and 30mm. Two tpes o langes were considered: langes o dierent dimensions (using the dimensions o the reerence case) and langes o equal dimensions. The stud will involve onl one group o analses with the height o the web h as the varing parameter. This parameter will take values rom h=500 to h=000mm at 50mm intervals. It is important to note that the stieners dimensions also var as the height o the web increases as can be seen in the results. ult d M 057

8 Pro. José M. Simón-Talero Mrs Ana Caballero..- Analsis o the stud s results In the parametric stud the loss o strength o a compressed element has been quantiied in the case o the eective width method and the reduced stress method respectivel b the ollowing ratios: A c e ult d Abruta / M (7a) (7b) Ac e / Abruta In order to compare the results o each method the ratio has been computed or each ult d /( / M) group o analsis and plotted in a graph against its corresponding varing parameter. Consequentl a ratio> implies that the eective width method ields higher estimations o post-buckling strength than the reduced stress method. Plate girder The ollowing graphs show the ratio o the reduced strengths obtained with the eective width and Ac e / Abruta the reduced stress methods when varing the width b and the length a o the plate. /( / ) ult d M (Ace/Abruta)/(ultd/(/M) t=mm 0. t=6mm t=0mm 0. t=5mm b (mm) (Ace/Abruta)/(ultd/(/M) t=mm t=6mm t=0mm t=5mm a (mm) Figure 6a: Plate girder. Results depending on the width (b) Figure 6b: Plate girder. Results depending on the length (a) Web The ollowing graphs show the ratio o the reduced strengths obtained with the eective width and Wc e / Wbruta the reduced stress methods when varing the height h o the webs. /( / ) ult d M (Wce/Wbruta)/(ultd/(/M)) tw = 6mm 3.0 tw = 8mm tw = 0mm.5 tw = mm.0 tw = 6mm tw = 0mm.5 tw = 5mm.0 tw = 30mm 0.5 h (mm) Figure 7a: Web with equal langes. Results depending on the height (h) (Wce/Wbruta)/(ultd/(/ tw = 6mm tw = 8mm tw = 0mm tw = mm tw = 6mm tw = 0mm tw = 5mm tw = 30mm h (mm) Figure 7b: Web with dierent langes. Results depending on the height (h) 058

9 Pro. José M. Simón-Talero Mrs Ana Caballero 5.- CONCLUSIONS 5..-Conclusions relatives to the plate girder case - As the width b o the plate girder increases (ig. 7a) the reduction actors obtained b each Ac e / Abruta method converges (i.e. 00 ) until b=6000mm. For higher values o b the ult d /( / M) slenderness o the plate becomes ecessive and the reduced stress method proves to be less conservative than the eective width method. This can be most clearl observed in the case o the slenderest plate girder o t =mm and b=0000mm or which the reduction actor ielded b the eective width method is 065 times that ielded b the reduced stress method. - When varing the length o the plate girder the opposite conclusion is derived: the ratio between the reduction actor remains constant up to a value o a=6000mm and then or higher values the eective width method gives less conservative values that the reduced stress method. In particular the reduction actor or the eective width method is 35 times that o the reduced stress method or the case o a plate girder o t =5mm and b=0000mm. - Varing the number o stieners implies varing the width o the subplates. In this case the outcomes o each method are ver similar since the slenderness o the subplates is never ecessive remaining within the range 05 b t 50.However or signiicantl small slenderness ( b t 0) the eective width method becomes too conservative ielding a reduction actor 059 times the reduction actor obtained b the reduced stress method 5..- Conclusions relatives to the web with dierent lange dimensions - The slenderest webs with thickness t w between 6 and 0mm are highl unstable against normal stresses and thereore their strength is signiicantl reduced when appling the reduced strength method. However the eective width method allows or the redistribution o the centre o gravit so that the strength is less reduced. Thereore this method leads to resistances up to 3 times that obtained with the reduced stress method or the case o h=000mm. - For thicker elements o thickness between t w = and 30mm the same conclusion is obtained. However the dierence between the two methods is less pronounced the reduction actor or the eective width case been onl 5 times that o the reduced strength case or h=000mm 6.- REFERENCES [A] [] Caballero A. and Simon-Talero J.M. Estudio comparativo de los métodos de cálculo propuestos en los artículos 0 del EN Final Year Report ETS Ingenieros de Caminos MADRID Imperial College LONDON 009 [] European Committee or standardiation EN Eurocode 3 Design o steel structures Part -5. Plated structural elements CEN Bruselas 006 [3] Asociación para la promoción técnica del acero APTA Manual de Proecto COMBRI. Puentes competitivos mitos de acero hormigón APTA España

10 Pro. José M. Simón-Talero Mrs Ana Caballero [] Johansson B. Maquoi R. Sedlacek G. Muller C. Beg D. Commentar and worked eamples to EN Plated Structural Elements JRC-ECCS Luembourg 007 [5] Millanes F. La leión en estructuras metálicas: Análisis de esueros control de secciones E.T.S. de Ingenieros de Caminos Canales Puertos Madrid 999 [6] Simón-Talero J.M. Introducción al cálculo de estructuras metálicas según el Eurocódigo 3 Madrid 000 [7] Quintero F. La piea aislada. Inestabilidad - Curso de Estructuras Metálicas II UNED Fundación Escuela de la iicación Madrid 00 [8] Simón-Talero J.M. Bases para el estudio de la inestabilidad de placas cargadas en su plano Jornadas COMBRI sobre puentes competitivos de acero mitos Burgos