PARTIAL INTERACTION ANALYSIS WITH SHEAR-LAG EFFECTS OF COMPOSITE BRIDGES: A FINITE ELEMENT IMPLEMENTATION FOR DESIGN APPLICATIONS

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1 Advnced Steel Construction Vol. 7, No. 1, pp (11) 1 PARTIAL INTERACTION ANALYSIS WITH SHEAR-LAG EFFECTS OF COMPOSITE BRIDGES: A FINITE ELEMENT IMPLEMENTATION FOR DESIGN APPLICATIONS Frizio Gr 1, Ginluc Rnzi nd Grzino Leoni 3,* 1 Deprtment of Architecture Construction nd Structures, Universit Politecnic delle Mrche, Ancon, Itly School of Civil Engineering, The University of Sydney, Sydney, NSW6, Austrli 3 School of Architecture nd Design, University of Cmerino, Ascoli Piceno, Itly *(Corresponding uthor: E-mil: grzino.leoni@unicm.it) ABSTRACT: This pper presents numericl model for the nlysis of composite steel-concrete ems with prtil interction to ccount for the deformility of the sher connection. The proposed pproch is cple of cpturing the structurl response produced y sher-lg effects nd y the time-dependent ehviour of the concrete. The verstility of the FE formultion is demonstrted for wide rnge of relistic ridge rrngements, e.g. from twin-deck girders to cle-styed ridges. The ccurcy of the pproch is vlidted ginst the results otined from more refined models generted with shell elements using commercil finite element softwre. For ech ridge typology considered, oth deformtions nd stresses re clculted to provide greter insight into the structurl performnce. Prticulr ttention is plced on the determintion of the effective width to e used for design purposes nd on the stress distriution induced in the concrete component, together with their vrition with time due to creep nd shrinkge. Keywords: Cle styed ridges, Composite ridge decks, Creep, Effective width, Sher-lg, Shrinkge, Steel-concrete memers 1. INTRODUCTION Steel-concrete composite decks re used in mny ridge typologies. In ddition to multi-spn continuous em rrngements, widely used in viducts nd flyovers, composite decks re lso comined with other structurl elements to chieve more rticulted systems, such s rch ridges nd cle styed ridges, when longer spns re required. In the modelling of these structures it is common to use simple em theory, ccording to which plne cross-sections remin plne fter loding, even if such pproch is not relistic s non-uniform stress distriutions usully rise in the sl (sher-lg effect) reducing its effective width. These cn lso rise due to the ppliction of concentrted lods, e.g. nchorge of prestressing cles or stys. When deling with sher-lg effects interntionl design guidelines recommend the use of the effective width method. While this pproch is very efficient nd useful for simple em lyouts, its fundmentl ssumptions re violted when pplied to more rticulted structurl rrngements. In design sitution such structures my e nlysed using refined models generted with shell or solid finite elements, in which cse oth model development nd post-processing re very time-consuming. As required in interntionl guidelines (e.g. Eurocode 4 [1]), designer needs to evlute the effective cross-section of ridge girder to determine its resistnce. Unfortuntely, simple rules for the definition of the sl effective widths re ville only for simple sttic cses. Another importnt spect reltes to the fct tht the effective width is time-dependent when longitudinl concentrted forces re pplied to the steel em or to the concrete sl [].

2 Prtil Interction Anlysis with Sher-Lg Effects of Composite Bridges : A Finite Element Implementtion for Design Applictions In this context, the finite element formultion dopted in this pper predicts the non-uniform distriution of stresses within the sl due to sher-lg effects, even in complex ridge rrngements, while mintining the ese of use of stndrd line elements commonly used for frme nlyses. The proposed technique is powerful tool for the designer tht cn (i) nlyse the structure without worrying out the effective width definition nd (ii) perform the cross-sectionl design clcultions using the effective sl width determined from the pproprite sttic scheme nd lod lyout. The fundmentl model ws lredy proposed y the uthors in [3], while in this pper the formultion is used to del with wide rnge of ridge rrngements nd to ccount for time effects. Its use is proposed in comintion with common truss nd frme elements to enle the modelling of complex three-dimensionl ridge rrngements. The pplictions to relistic cses re discussed with prticulr emphsis to creep nd shrinkge effects tht hve gret influence on the stress sttes nd on the definition of the sl effective width.. OVERVIEW OF THE ANALYTICAL AND NUMERICAL MODELS OF THE DECK A symmetric composite steel-concrete twin-girder ridge deck is considered (Figure 1). A locl reference system {O; x 1, x, x 3 } is introduced y plcing the plne x 1 -x on the symmetry plne of the cross-section in such wy tht x 1 nd x re longitudinl nd verticl xes of the deck, respectively. The origin O is locted t n ritrry loction even if it my e conveniently plced t the centroid of the steel ems. The cross-section is considered to e rigid in its own plne (x -x 3 ) nd the interfce sher connection is ssumed to prevent seprtion nd lterl slips etween the steel em nd the sl while permitting reltive longitudinl displcements etween them. The kinemtic model for deck suected to flexurl nd xil ctions in the plne x 1 -x is riefly outlined in the following to provide etter insight into the results presented in the pplictions. B x x 3 B 1 O Figure 1. Steel-concrete Twin Girder Cross-section Bsed on these considertions, nd neglecting the sher deformility of the steel ems, the displcement of generic point of the cross-section plced t x 1 is defined y the three components x x u x u1 1 1 x,x3 Ac d1 x1,x,x3 (1) u 1 x1 x1 xux1 x3 x1 x,x3 A

3 Frizio Gr, Ginluc Rnzi nd Grzino Leoni 3 x 1,x,x 3 u x 1 d (1) x,x,x d (1c) where the prime denotes the derivtive with respect to x 1, A c nd A re the cross-sections of the concrete sl nd the steel em, respectively, u 1 nd u re the longitudinl nd verticl displcements of the steel em mesured t the level of the reference xis x 1, is the longitudinl slip etween the steel ems nd the reinforced concrete sl nd is the function descriing the wrping in the sl tht is modulted long the deck xis y the intensity sher-lg function [4, 5]. Given the prticulrity of the ssumed deck cross-section, it is pproprite to dopt function formed y three second order prolic rnches [6] 3 B x 3 B1 x B x B B B B B x x B1 x3 B1 x B B B x B x () () x B B B1 x3 B (c) B B B 3 tht descrie rigorously the wrping in the cse of ems suected to pure sher given the position of the min longitudinl ems. Under the ssumption of liner-elstic ehviour for the mterils forming the cross section nd for the sher connection, displcement-sed finite element hs een developed [7] nd implemented for the modelling of composite ems with complex sttic schemes [3]. A locl reference system is introduced for ech element with the origin locted t node i (Figure ). If x 1 is the longitudinl xis of the element, oriented from oint i to oint, xes x nd x 3 complete n ortho-norml reference system. The nodl displcements of the finite element re defined ccording to the positive directions of the locl reference xes. For ech node, three displcement components nd three rottion components re considered. Two dditionl degrees of freedom re introduced to define the em-sl interfce slip nd the sher-lg function previously defined (Figure ). These lst two generlized displcements re sclr quntities nd thus do not depend on the dopted reference system. Generlised stress resultnts re defined corresponding to ech of the displcements considered, i.e. three force components nd three moments, the longitudinl sher force t em sl interfce nd the sl i-moment (Figure ). As for the displcements, the lst two generlized ctions re sclr quntities.

4 4 Prtil Interction Anlysis with Sher-Lg Effects of Composite Bridges : A Finite Element Implementtion for Design Applictions The generlized nodl displcements nd forces of element e re grouped in the vectors T i u1 i ui u3i 1 i i 3i i i u1 u u3 1 3 s (3) T i f1 i f i f3i m1 i mi m3i qi i f1 f f3 m1 m m3 f q (3) Bsed on the dopted mteril properties, the stiffness reltionship of the line element cn e expressed s f e K e se, where K e depicts its stiffness mtrix. When modelling three-dimensionl structurl systems, this line element cn e oriented freely. Adopting n ortho-norml glol reference system nd introducing nd ( = 1,,3) s unit vectors of the glol nd locl reference systems respectively, the rottion opertor R e cn e expressed using the following mtrices i x m i x m u 1i u i 1i u 3i i u 3i * * i i 3 u 3 u 1 f 1i f i m 1i f * * 1 x 3 x 1 m 3i f 3i i q i * * i m 3 f 3 f 1 * q * m 1 x 3 x 1 () () Figure. Composite Line Element: () Nodl Displcements; () Nodl Forces Cle element Rigid link Deck Composite element line element Figure 3. Comintion of Composite Line Element with Rigid Links nd Truss/Frme Elements R R I R e R ; R I R 1 3 ; I (4,,c) Bsed on these, the nodl displcement nd force vectors re trnsformed from the locl to the glol reference system sed on se Rese nd fe Re fe. In similr mnner, lso the stiffness reltionship cn e expressed in glol coordintes using e T e K R K R. e e

5 Frizio Gr, Ginluc Rnzi nd Grzino Leoni 5 It is worth noting tht in the cses of composite line element suected only to flexurl ctions in its verticl plne, eing the composite element chrcterised y only 1 dof, the reltive 1x1 stiffness mtrix needs to e expnded to dimension 16x16 y dding rows nd columns of zeros to comply with the three-dimensionl prolem. Oviously, this results in n ill-conditioned prolem solved y imposing specil constrints to the element to limit rigid motions in the horizontl plne nd for torsionl rottions. The freedoms of the proposed finite element re presented in Figure. This element cn e connected to conventionl frme nd truss elements using stndrd ssemling procedures. Although different choices of nodl displcements cn e used to descrie the displcement field of the proposed finite element, for the selection dopted in this pper (Figure ), the element end freedoms ville for ssemling to other conventionl line elements (i.e. frme nd truss elements) consist of the verticl deflection, rottion nd xil displcement of the steel oist. The em-sl interfce slip nd the sher lg function cnnot e constrined to freedoms of conventionl line element nd, s consequence, no interction cn occur directly with the concrete sl. Such restriction cn e esily overcome y dopting set of independent end displcements in the formultion of the element which includes the xil displcement in the concrete component. The connectivity of the proposed element reflects common prctice in which the connections mong steel elements re usully preferred. For clrity, the possile connectivity used in the modelling of cle styed ridge is depicted in Figure 3, where the stys re connected lterlly to the line element y mens of rigid links. In ddition to the usul externl restrints which prevent the six displcement components, other two kinds of restrints, le to prevent the interfce slip nd the sl wrping, my e specified for the composite line element. The restrints on the slip cn e used to model deck with rigid sher connections wheres the restrint on the sl wrping cn simulte the presence of rigid structurl elements (e.g. end trnsverse ems). The cse of flexile trnsverse ems limiting the sl wrping cn e depicted y defining specil generlized springs. 3. VALIDATION OF THE FINITE ELEMENT FORMULATION The ccurcy of the composite line element depicted in Figure hs een vlidted in the cse of relistic ridge. In the proposed ppliction, the ridge on the Nive River uilt in Byonne hs een used s cse study [8]. This ridge consists of four-spn continuous configurtion nd ll relevnt geometric properties re specified in Figures 4 nd 5. Mteril properties for the steel, concrete nd sher connection re: steel elstic modulus E s = 1 GP, concrete elstic modulus E c = 349 MP, concrete Poisson's rtio ν =.15, nd sher connection stiffness ρ = 1 kn/mm. The sl is supported y cntilevered cross ems plced t spcing of 3.78 m. The results clculted modelling the ridge with the composite line elements re compred with those otined with shell model (Figure 6) implemented with SAP [9]. In prticulr, 13 line elements re used to descrie the ridge deck with totl of 173 degrees of freedoms, significntly smller thn the 5375 degrees of freedom required y the shell element model.

6 6 Prtil Interction Anlysis with Sher-Lg Effects of Composite Bridges : A Finite Element Implementtion for Design Applictions E s = 1 MP = 1 kn/mm E c = 349 MP =.15 Figure 4. Bridge on Nive River: Cross-section Top flnge 11 x We Bottom flnge 13 x Figure 5. Bridge on Nive River: Thickness of Steel Pltes within the Bems 3.1 Comprisons Figure 6. Bridge on Nive River: Shell Finite Element Model of the Deck A verticl uniformly distriuted lod is pplied to the top side of the sl. In the dopted numericl clcultions, line lod is specified when using the line element while surfce distriuted lod is dopted in the shell element model. The selected cse study is useful to provide insight into the influence of the cross ems, lso connected to the concrete component, on the overll sher-lg ehviour of the sl.

7 Frizio Gr, Ginluc Rnzi nd Grzino Leoni 7 For clrity, the results otined using the composite line elements re reported with continuous lines while those otined with the shell element model re illustrted with mrkers (circles nd tringles). Figure 7 depicts the deflection nd the longitudinl stresses in the ottom flnge of the steel girder nd Figure 8 shows the longitudinl norml stress in the middle plne of the sl for selected set of cross-sections plced long the ridge. Bsed on these comprisons it is shown tht the use of the composite line elements provides very good representtion of the structurl response clculted with the more refined shell element model. Smll discrepncies etween these results occur only for the concrete sl stresses t the cross-sections 1 nd in Figure 8 locted over nd close to the end support, respectively. This is ustified y the fct tht, t these sections, the locl trnsverse ehviour ffects the longitudinl ehviour of the girder nd the dopted model cptures this loclised stress diffusion only in simplified wy. Despite this, these discrepncies re not significnt since t these loctions the stresses re smll nd do not govern the design. When considering the evlution of the deflections, errors higher thn 1% re otined y compring the vlues clculted with the proposed numericl model with those of the shell element model plotted with dotted line with tringles (Shell ). This discrepncy ws lredy oserved in [7] for simpler ridge rrngements nd ws ustified y the sher deformility of the steel components tht is neglected in the proposed numericl formultion. The sme finite element model is re-utilised for this nlysis while incresing the sher modulus of the mterils dopted for the shell elements. The results of this ltter model hve een referred to in the following s 'Shell ' to distinguish them from those clculted using the initil FE model denoted s 'Shell '. The discrepncies oserved for the deflection using model 'Shell ' ecome negligile when using model 'Shell ' s shown with the dotted line nd circulr mrks in Figure 7, in which cse the ltter line ecomes coincident with the continuous curve representing the proposed numericl model. It is worth noting tht this chnge in the sher modulus lso ffects the discrepncy of the concrete stress sttes t cross-section 4 (Figure 8), ecoming less remrkle s the sher deformility of the em ecomes negligile. 5 kn/m u [mm] - -6 () Proposed method Shell Shell [Nmm - ] 5 () Figure 7. Bridge on Nive River: () Deflection of the Deck () Longitudinl Stresses in the Bottom Flnge of the Steel Girders

8 8 Prtil Interction Anlysis with Sher-Lg Effects of Composite Bridges : A Finite Element Implementtion for Design Applictions In regrds to the occurrence of sher-lg effects, the presence of cross ems does not ffect the overll response. This is prtilly due to the fct tht cross ems do not constitute restrint ginst the sl wrping ecuse of their low lterl nd torsionl stiffness. As the concrete stresses re evluted t the mid-height of the sl, these re only ffected y the memrne ction of the sl nd not y its ending one tht is more influenced y the presence of the cross-ems. Of course this is limit of the proposed model tht nevertheless mintins its vlidity in descriing the ehviour of the ridge nd confirms to e good tool for the verifiction of the composite cross-sections. It is worth noting tht, if necessry, ending effects of the sl my e esily evluted with prtil models (or lso with ville chrts) nd superimposed to the memrne effects to evlute the mximum stresses t the top or ottom plne of the sl c [Nmm - ] Proposed method Shell Shell Figure 8. Bridge on Nive River: Longitudinl Stresses in the Middle Plne of the Sl 4. APPLICATION TO A CABLE STAYED BRIDGE A cle-styed ridge is considered to outline the cpilities of the proposed composite finite element, when comined with other truss nd frme elements, to model complex ridge systems. The cse study considered is shown in Figure 9. This consists of steel-concrete composite deck supported y two reinforced concrete pylons, higher thn 7 m, nd y 8 couples of stys pretensioned to control the deck deflection nd the horizontl displcements of the pylons. Two nchor piers re plced t the ridge ends. The reinforced concrete sl of the deck is.3 m thick nd is sustined y cross ems supported y the two longitudinl min girders. The spcing etween

9 Frizio Gr, Ginluc Rnzi nd Grzino Leoni 9 stys nchorges is 18 m. The cylinder strength of the concrete used for the sl nd the pylons is f ck = 45 N/mm ; the elstic constnts E s = 165 N/mm nd E = 1 N/mm re considered for the stys nd the structurl steel, respectively S1 S14 S15 S8 P1 P Stys 1, 14-15, 8-3, 11-13, 16-18, , 19-5 N of strnds Seven-wires strnds - nominl dimeter of 15.7 mm Figure 9. Generl Lyout nd Dimensions of the Cle-styed Bridge Considered The proposed finite element is used for the composite deck wheres common truss nd em elements re used for the stys nd the pylons, respectively; in order to ensure the correct positioning of the stys t the sides of the ridge, rigid links hve lso een used in the model. Short- nd long-term nlyses re crried out considering the following loding conditions: (i) uniformly distriuted lod (UDL) of 15 kn/m, representing the self-weight nd other permnent lods; (ii) UDL nd shrinkge; nd (iii) trffic lods (TL) pplied t the centrl spn. The long-term nlyses re crried out y mens of the Effective Modulus Method s recommended in [1]. With this pproch n effective elstic modulus E eff (t,t ) is dopted for the concrete when considering the time-dependent ehviour which is clculted s: E eff t, t t t, Ec 1 t (5) L in which E c (t ) is the concrete elstic modulus clculted t the time of first loding t, (t,t ) depicts the creep coefficient defining the creep occurred t time t for stress first pplied t time t, nd ψ L represents the creep multiplier which depends on the type of loding nd its vlues re sed on Europen guidelines [1]. In prticulr, the vlue for ψ L used for sustined lods nd prestressing of stys is 1.1. The effects of concrete shrinkge re evluted using ψ L vlue of.55 nd creep coefficient (t,t s ) defined with respect to time t s, which is the instnt t which shrinkge is supposed to strt (end curing). The creep coefficients nd the shrinkge ction re defined ccording to Eurocode [1]. Results re presented in terms of deflections, stress stte in the steel ems nd concrete sl nd effective width of the sl. Figure 1 shows the results otined considering oth short- nd long-term ctions clculted t 8 nd dys, respectively. In prticulr, the results of the short-term nlysis crried out re reported long the whole ridge wheres long-term results re reported with nd without shrinkge ction in the left nd right sides of the digrms, respectively. The deck deflection is depicted in Figure 1. The short-term results re chrcterised y null displcements t the stys nchorge s consequence of the prestressing. Long-term deflections

10 1 Prtil Interction Anlysis with Sher-Lg Effects of Composite Bridges : A Finite Element Implementtion for Design Applictions slightly increse s creep nd shrinkge develop. It is worth noting tht the configurtion of zero-deflection, chieved in the instntneous nlysis lncing the self-weight of the deck y mens of the stys prestressing, is lost in the long-term even if the modifictions re not importnt since they re within 8 mm. UDL u [mm] () [Nmm - ] 8 dys Without shrinkge dys With shrinkge () c [Nmm - ] 1.6 (c) B eff /B.8 (d).6.4. Figure 1. Permnent Actions: () Deck Deflection; () Longitudinl Norml Stresses in the Top nd Bottom Flnges of the Steel Bem; (c) Longitudinl Norml Stresses t Mid-height of the Sl; (d) Effective Width over the Actul Deck Width

11 Frizio Gr, Ginluc Rnzi nd Grzino Leoni 11 UDL N s [kn] -5 () M s [knm] 3 15 () Without shrinkge 8 dys dys With shrinkge Figure 11. Permnent Actions: () Axil Force nd () Bending Moment of the Steel Bem The stresses otined for the top nd ottom steel flnges re illustrted in Figure 1 while Figure 1c plots the stresses clculted t mid-height of the sl long the ridge length t the loction of the steel memer xis nd long the centre-line of the deck. The sl effective width is determined sed on the clculted stress distriutions nd is illustrted in Figure 1d. The stress distriutions in the steel girders (Figure 1) re determined sed on the xil force tht, s expected, is mximum ner the pylons nd minimum t the deck ends nd t mid-spn of the centrl spn (Figure 11), nd on the ending moment tht vries etween ech couple of stys ccording to prolic curves (Figure 11). Due to the long-term ehviour of the concrete, the compressive xil forces grow in time. In prticulr, the effect of shrinkge is quite remrkle s shown in the right side digrm of Figures 1 nd 11. Ner the longitudinl steel memers the concrete stress distriution hs distinguished vrying pttern which tends to smoothen when moving towrds the centre-line of the cross-section (Figure 1c). This demonstrtes the effects produced y the concentrted forces exerted y the stys nchorges. The diffusion mechnism cptured y the model is such tht for cross-sections close to the sty nchorge, the higher stress vlue re otined long the longitudinl ems while for cross-sections plced etween two consecutive sty nchorges, the stress mximum vlue is otined t the centre-line of the sl. When considering long-term effects, the concrete compressive stresses tend to reduce with time; in prticulr, the effects of shrinkge produces tensile stresses for lrge deck sections where prestressing due to stys ecomes less significnt. This is prticulrly evident from the contour digrms of the sl tht highlight sections chrcterised y peks of tensile stresses for the sl sections in the proximity of stys nchorges (Figure 1).

12 1 Prtil Interction Anlysis with Sher-Lg Effects of Composite Bridges : A Finite Element Implementtion for Design Applictions S1 S14 UDL S15 S8 P1 P S1 S S3 S4 S5 S6 S7 P1 dys 8 dys dys 1 P1 S8 S9 S1 S11 S1 S13 S14 sym 8 dys Figure 1. Permnent Actions: Contour Plots of Stresses in the Reinforced Concrete Sl due to Permnent Actions Prticulr ttention is required when nlysing the results in Figure 1d where the sl effective width is plotted. In the short term, it my e oserved tht the fluctution reflects the vrition of the compressive stresses in the sl nd tht, excepting for sl sections t the middle of the centrl spn nd t the ridge ends, the effective width remins within.9 nd 1. times the rel sl width; the very low vlues otined for the other sections previously mentioned (less thn.4) re due to the fluctution of stress compred with the low overll compression of the sl. It is worth highlighting tht the loclised minimum vlues oserved re not relevnt from design viewpoint s relted to ridge segments long which the stress levels re reltively low. The long-term effects on the definition of the sl effective width re strongly chrcterised y the concrete shrinkge; this produces significnt reduction of the effective width (down to.7) lso for sections chrcterised in the short-term nlyses y moderte effects of stress concentrtions. This is due to shrinkge which is responsile for inducing tensile stresses, lmost uniformly distriuted within the whole sl except for the end sections, tht superimpose onto stresses due to permnent lods nd stys pretensioning. In this process shrinkge does not modify the fluctution t the lterl em loctions ut minly reduces the overll xil force in the sl y mgnifying the stress concentrtion. Tle 1 shows the effects of the different lod cses considered on the stys pretensioning forces. It is evident tht the long-term effects re such tht only smll vritions of the internl forces re induced in the stys.

13 Frizio Gr, Ginluc Rnzi nd Grzino Leoni 13 Tle 1. Axil Forces in the Stys due to Prestressing nd Permnent Actions 8 dys dys dys Stys Without With shrinkge shrinkge [kn] [kn] (%) [kn] (%) 1, (-1.9) 3559 (-6.3), (-.7) 3314 (-.6) 3, (+.1) 943 (+.6) 4, (+.5) 658 (+.7) 5, (+.7) 81 (+.5) 6, (+.4) 189 (+.) 7, (-1.9) 147 (-9.1) 8, (-1.4) 1456 (-7.3) 9, (+.4) 184 (.) 1, (+.5) 54 (+1.1) 11, (+.4) 6 (+1.) 1, (-.) 958 (+.4) 13, (-.8) 357 (-.9) 14, (-1.5) 3864 (-.8) Figure 13 shows the results otined y considering trffic lods t the centrl spn. According to the Lod Model 1 of Eurocode 1- [11], four notionl lnes re considered within the deck width; this results in uniformly distriuted lod of 5 kn/m nd in couple of concentrted lods of 7 kn. On the left side of the digrms, the results re superimposed on the short-term stresses due to permnent lods wheres on the right side the sme re superimposed on long-term stresses inclusive of shrinkge. Also in this cse it is interesting to highlight the differences etween the short- nd long-term ehviours, which re strongly chrcterised y shrinkge, nd how the effective width is sensitive to the loding conditions. Figure 14 shows contour plot of stresses t mid-height of the concrete sl. 5. CONCLUSIONS This pper presented n implementtion for design pplictions of composite line finite element sed on prtil interction theory. The element is suited for cpturing the sher-lg effects nd its use is suitle for composite steel-concrete decks for which the ssumption of preservtion of the plne cross-section is not vlid due to the lrge widths of the sls when compred to their thickness. Models of complex ridges my e esily developed without considertions of the definition of the effective cross-section when comining the use of this element with other line elements (trusses nd frme ems). The proposed formultion is le to ccount for sher-lg effects in complex ridges where simplified methods re not pplicle. Bsed on this formultion, the effective widths to e used in design re clculted consistently for ech lod comintion nd the produced results cn e used y designer following stndrd procedures.

14 14 Prtil Interction Anlysis with Sher-Lg Effects of Composite Bridges : A Finite Element Implementtion for Design Applictions TL UDL u [mm] () dys dys [Nmm - ] 1 5 () c [Nmm - ] (c) B eff /B.8 (d).6.4. Figure 13. Trffic Lods: () Deck Deflection; () Longitudinl Norml Stresses in the Top nd Bottom Flnges of the Steel Bem; (c) Longitudinl Norml Stresses t Mid-height of the Sl; (d) Effective Width over the Actul Deck Width

15 Frizio Gr, Ginluc Rnzi nd Grzino Leoni 15 S1 S14 TL S15 S8 UDL P1 P m m 1 S1 S S3 S4 S5 S6 S7 P1 m dys 8 dys m 1 P1 S8 S9 S1 S11 S1 S13 S14 sym dys 8 dys Figure 14. Trffic Lods: Contour Plots of Stresses in the Reinforced Concrete Sl due to Permnent Actions With respect to refined finite element models (e.g. using shell elements) the method hs the dvntge of reducing the mount of degrees of freedom without loss of precision in evluting stress sttes. The precision of the method ws demonstrted y pplying the use of the deck finite element to rel ridge nd compring the results with those given y refined shell finite elements. The proposed method produces very good results for wht concern stresses in the steel ems nd in the concrete sl while lower level of precision is otined in the evlution of deflections due to the sher strins of the steel ems tht re neglected in the model. The presence of cross ems, even if connected to the reinforced concrete sl, does not ffect the results. A cle styed ridge suected to different ctions (permnent lods, shrinkge nd trffic lods) hs then een considered nd nlysed. Both short- nd long-term nlyses were crried out to demonstrte the importnce of long-term effects nd the difficulties tht designer could hve in choosing dequte sl effective widths s these depend sensily on the loding lyouts nd re time dependent. The limittion of the proposed formultion relies on its inility to ccount for torsion nd trnsverse ending ut the uthors do not feel tht this compromises the usefulness of the proposed element which intends to complement the nlysis tools ville to ridge designers. ACKNOWLEDGEMENT The work crried out in this rticle y the second uthor ws supported y Discovery Proect wrded under the Austrlin Reserch Council's funding scheme nd through reserch grnt wrded y the Commonwelth through the Austrli-Kore Foundtion of the Deprtment of Foreign Affirs nd Trde.

16 16 Prtil Interction Anlysis with Sher-Lg Effects of Composite Bridges : A Finite Element Implementtion for Design Applictions REFERENCES [1] EN 1994-, EUROCODE 4: Design of Composite Steel nd Concrete Structures - Prt : Generl Rules nd Rules for Bridges, Europen Committee for Stndrdiztion, 5. [] Dezi, L., Gr, F., nd Leoni, G., Effective Sl Width in Prestressed Twin-girder Composite Decks, Journl of Structurl Engineering, ASCE, 6, Vol. 13, No. 9, pp [3] Gr, F., Rnzi, G. nd Leoni, G., Anlysis of the Sher Lg Effect in Composite Bridges with Complex Sttic Scheme y Mens of Deck Finite Element, Interntionl Journl of Steel Structures, 8, Vol. 8, pp [4] Reissner, E., Anlysis of Sher Lg in Box Bems y the Principle of the Minimum Potentil Energy, Qurterly of Applied Mthemtichs, 1946, Vol. 4, No. 3, pp [5] Dezi, L., Gr, F., Leoni, G. nd Trntino, A.M., Time Dependent Anlysis of Sher-lg Effect in Composite Bems, Journl of Engineering Mechnics, 1, Vol. 17, No. 1, pp [6] Dezi, L., Gr, F. nd Leoni, G., Sher-lg Effect in Twin-girder Composite Decks, Steel nd Composite Structures, 3, Vol. 3, No., pp [7] Gr, F., Leoni, G. nd Dezi, L., A Bem Finite Element Including Sher Lg Effect for the Time-dependent Anlysis of Steel-concrete Composite Decks, Engineering Structures, 9, Vol. 31, pp [8] Kretz, T., Michotey, J. nd Svetchine, M., 5ème pont sur l Nive, Bulletin Ponts Metlliques, SETRA, 199, Vol. 15, pp (in French) [9] SAP, Computer nd Structures, Inc. CSI Anlysis Reference Mnul, Berkeley, Cliforni, 4. [1] EN , EUROCODE : Design of Concrete Structures - Prt 1-1: Generl Rules nd Rules for Buildings, Europen Committee for Stndrdiztion, 4. [11] EN 1991-, EUROCODE 1: Action on Structures - Prt : Trffic Lods on Bridges, Europen Committee for Stndrdiztion, 3.