HYBRID STEEL PLATE GIRDERS SUBJECTED TO PATCH LOADING

Transcription

1 SDSS Rio 21 STABILITY AND DUCTILITY OF STEEL STRUCTURES E. Batista, P. Vellasco, L. de Lima (Eds.) Rio de Janeiro, Brazil, September 8-1, 21 HYBRID STEEL PLATE GIRDERS SUBJECTED TO PATCH LOADING Rolando Chacón*, Enrique Mirambell* and Esther Real* *Construction Engineering Department. Universitat Politècnica de Catalunya, Barcelona, Spain s: Keyords: Hybrid girders, patch loading, EN Abstract. The structural case o hybrid girders subjected to patch loading is treated identically than the one or homogenous specimens in EN The EN ormulation is based upon a theoretical model hich consists o a plastic resistance F y partly reduced by a resistance unction χ F, this latter accounting or instability. The EN ormulation predicts that the ultimate load capacity o steel plate girders subjected to patch loading increases, among others, ith the hybrid grade y/ y. In this ork, an appraisal o the theoretical and numerical responses o hybrid and homogenous specimens subjected to patch loading is presented. Some peculiarities concerning the treatment o hybrid girders are pinpointed though. It is numerically demonstrated that the moment capacity o the langes (and thus, y/ y) does not play any role in the resistance o girders predicted in EN Accordingly, a design proposal hich enhances the current ormulation is provided. 1 INTRODUCTION A girder is deemed as being hybrid hen it is abricated ith dierent steel strengths or the lange and eb panels. In hybrid design, the nominal yield strength o one or both langes is larger than the nominal yield strength o the eb. This type o girder is popular as the girder yields a greater lexural capacity at loer cost and eight compared to a homogeneous girder [1]. On the other hand, patch loading phenomena has been idely analyzed since the early sixties. Experimental and theoretical analyses have pinpointed the typical ailure mechanisms o girders subjected to patch loading and consequently ultimate load predictions are noadays available [2]-[4]. Broadly speaking, to magnitudes have been given to describe the resistance o members subjected to this sort o loads. The ormer deines a plastic resistance F y o the member hereas the latter, an elastic critical load F cr. The ormer has been generally obtained by limit analysis hereas the latter, by theoretical ormulae properly calibrated ith numerical simulations. The actual situations to hich these members are subjected lie inside a blurred transition beteen yielding and instability. It is ell knon that the root square o the ratio beteen the plastic resistance F y and the elastic critical load F cr is commonly reerred to as the slenderness parameter. Admittedly, there exists a direct relation beteen this slenderness and the actual ailure mode. This relation has been labeled in the European guidelines or the design o plated structural elements EN [5] as the resistance unction. The patch loading phenomena has been harmonized to this procedure. Despite the vast amount o research devoted separately to both hybrid girders and the patch loading ield, the research ork that matches both subjects is rather scant [6]-[8]. In this paper, the resistance o hybrid steel plate girders subjected to patch loading is dealt ith simultaneously. Theoretically, it is demonstrated that the EN ormulation predicts the resistance o plate girders subjected to patch loading as a monotonic increasing unction ith, among other parameters, the lange yield strength y (and consequently, ith the hybrid grade y/ y). Numerically, a vast study aimed at comparing numerical and theoretical results is presented. In this study, it is ound that the results obtained ith the 739

2 EN provisions do not reproduce satisactorily the trends obtained numerically. Alternatively, a design proposal in accordance ith the procedure implemented in EN and aimed at correcting the aorementioned anomaly is proposed. 2 EN PLATED STRUCTURAL ELEMENTS 2.1 Hybrid girders The European design rules EN overtly consider the hybrid girder usage. These rules recommend a maximum value o Φ h= y/ y<2,. The treatment o hybrid girders is identical to one o homogeneous prototypes except or the olloing remarks. First, as the resistance to direct stresses o plate girders is calculated by using the eective area o the cross-section, in the particular case o hybrid design, y must be used in determining the eective area o the eb or plate buckling purposes. Second, or the particular case o hybrid plate girders it is indicated that the potential yielding o the eb must be taken into account in direct stresses veriications. 2.2 Patch loading Regarding the veriication to patch loading, the general approach currently included in EN is based upon a plastic resistance F y hich is partially reduced by means o the resistance unction χ F. The plastic resistance is based upon the mechanical model suggested in [2]. In eq. (1), l y is the yieldprone eectively loaded length and is calculated rom geometrical and mechanical properties o the girders by using equations (2), being l y not greater than the distance beteen adjacent transverse stieners a. χ F takes instability into account by means o equations (3) and (4) and can be obtained ith eq. (5). The buckling coeicient k F or longitudinally unstiened panels is given in (6). F Rd χf Fy χf y l y t χf y a t = = (1) γ γ γ M1 M1 M1 l y = s + 2 t s b h ( 1 m m ) s 2 t 2 y + + = ,2 a 1 2 s y t t (2) y l y t λ F = (3) F cr 3 t F cr =,9 k F E (4) h k,5 χ F = 1, (5) λ F 2 h F = (6) a In eq. (2), it is observed that l y is an increasing unction ith the term m 1 (hich includes the hybrid ratio Φ h= y/ y). As a result, F Rd is an increasing unction ith Φ h accordingly. 74

3 3 NUMERICAL MODEL 3.1 Description A numerical model implemented in Abaqus [9] as used systematically as a simulation tool. In this ork, ully nonlinear analyses ere perormed on ideally modeled steel plate girders using quadrilateral shell elements or the eb, langes and stieners. These models ere based upon the olloing numerical eatures. Geometrically, S4 shell elements ound in Abaqus libraries ere employed or the simulations. Materially, an elastic-perectly plastic constitutive equation as adopted or all cases. The used nonlinear solution strategy as the arc-length method (based on the modiied Riks algorithm). The used iterative procedure as the Neton-Raphson method. Eigenvalue extractions ere also perormed or the sake o obtaining elastic critical loads as ell as the elastic critical shapes typically associated to ailure modes. All numerical simulations ere perormed by olloing the current European guidelines EN Annex C. In a previous ork perormed by the authors, the validity o these guidelines hen developing numerical models on plate girders subjected to patch loading as pointed out [1]. The main conclusions pinpointed in such ork are given herein. Firstly, the initial shape o the geometry o the girders can be based upon the 1 st critical eigenmode. Secondly, the largest amplitude o this critical shape can be scaled to a value equaling 8% o the abrication tolerances. Eventually, the structural imperections must be included in the orm o residual stress patterns. Further inormation concerning those topics can be ound [6]-[8] and [1]. 3.2 Numerical database Numerical simulations ere perormed on a single panel centrically loaded ith a patch. The panel as locally loaded up to ailure. The load as introduced as a pressure on the top lange ithin the load length s s. The panels ere modeled as simply supported (a beam ith support reactions) ith additional restrains in all lange corners. Transverse stieners ere provided in the bearing sections. The numerical database as constructed by varying geometrical and material parameters. Table 1 summarizes the set o variations, hich resulted in an amount o 192 specimens. Four dierent groups orm the rameork o the sample. Each group consists o a eb panel presenting a given value o h. Within each group, three dierent distances beteen transverse stieners a ere studied. Likeise, to dierent values o t studied or each case leading to varying the eb slenderness ithin each group. The hybrid parameter Φ h= y/ y as varied systematically. The eb yield strength as held constant hereas the lange yield strength as accordingly increased rom Φ h= y/ y =235/235=1, (homogenous girder) to Φ h= y/ y =46/235=1, Numerical results Numerical results o 192 plate girders subjected to concentrated loads are available or all the depicted specimens [6]-[7]. Ultimate load capacities F u,num as ell as elastic critical buckling loads F cr,num ere obtained in all prototypes. In this paper, or the sake o conciseness, these results are presented only graphically in igure 1. This plot represents the resistance unction curve deined in EN (Eq. 6) together ith the results obtained numerically. These numerical results are plotted in the orm o F u,num/f y (being F u,num the ultimate load obtained ith the numerical model and F y the theoretically predicted plastic resistance according to Eq. (1). At irst glance, all dots corresponding to numerical results are located above the theoretical EN resistance unction. This inding is structurally sound and on the saety side. By contrast, in previous orks perormed by the authors [6]-[8], it as ound that the structural response o girders subjected to patch loading dier considerably hether the calculated eectively loaded length l y is less than or greater than the distance beteen transverse stieners a. The ormer case is typically associated ith eb olding as the ailure mode hereas the latter, to a ailure mode involving both eb olding and lange yielding. In the developed numerical database, 148 girders belong to the ormer category (largely spaced transverse stieners, l y<a). This category represents the vast majority o 741

4 realistic structural cases o steel plate girders subjected to concentrated loading in, or instance, bridge applications. The conclusions presented in this paper are related to this category only. Table 1. Numerical database Group Numerical database variations I II III Web yield strength y (N/mm 2 ) Flange yield strength y (N/mm 2 ) h (mm) a (mm) t (mm) s s (mm) Flange dimensions (mm 2 ) 8x6 9 x 8 1 x 8 12 x 1 Stiener thickness (mm) Girders per group Total number o numerical simulations 192 1,6 Resistance curve. EN ormulation 1,4 χ F 1,2 1,8,6 EN Group Group I Group II Group III,4, Slenderness λ Figure 1. Resistance unction and numerical results. 4 ANALYSIS OF THE RESULTS The structural response o hybrid steel plate girders subjected to patch loading is analyzed on a sample o typical load-displacement plots extracted or 16 girders o the hole numerical database (namely, only girders ith geometrical proportions such as l y<a). All plots are dran rom values o the actual applied load and, in addition, the vertical displacement o a node located at the mid-span crosssection o the top lange. Figure 2 displays our dierent plots. In each one, our curves corresponding to girders ith identical geometry are displayed. Fundamentally, the only varying parameter ithin each plot is the nominal yield strength o the lange y (and consequently, the Φ h= y/ y ratio). Each plot corresponds to a sample o girders excerpted rom each group ( to III). Three additional remarks 742

5 eature the selected sample. Firstly, all girders present an aspect ratio a/h =3,. Secondly, the eb slenderness is dierent rom one group another. Girders belonging to the group are airly stocky (h /t =83,33) hereas girders belonging to group III are airly slender (h /t =266,66). Thirdly, iit is orth mentioning that the geometric proportions o the girders (eb, lange and stieners proportions) are dierent rom one group another. Numerically obtained ultimate loads F u are indicated or all cases. 3 Group. a/h =3,; h /t =83,33 t =12mm, h =1mm 3 Group I. a/h =3,, h /t =166,66 t =12mm, h =2mm Load (kn) /235. Fu=1679 kn 275/235. Fu=1679 kn 355/235. Fu=1679 kn 46/235. Fu=1679 kn Load (kn) /235. Fu=1947kN 275/235. Fu=1947kN 355/235. Fu=1947kN 46/235. Fu=1947kN Displacement (mm) Group II. a/h =3,; h /t =2 t =15mm, h =3mm Displacement (mm) Group III. a/h =3,; h /t =266,66 t =15mm, h =4mm Load (kn) /235. Fu=2711 kn 275/235. Fu=2712 kn 355/235. Fu=2712 kn 46/235. Fu=2712 kn Displacement (mm) Load (kn) /235. Fu=282 kn 275/235. Fu=282 kn 355/235. Fu=282 kn 46/235. Fu=282 kn Displacement (mm) Figure 2. Structural response o girders subjected to concentrated loads. All girders present a similar load-delection plot ith a monotonically increasing branch as ell as a peak load at hich the load-bearing capacity is exhausted (F u). Subsequently, a sotening branch is observed in all cases. Surprisingly, the inluence o the lange yield strength (and thus, the hybrid ratio Φ h= y/ y) is negligible. For each series o girders in hich Φ h is varied ranging rom Φ h= to Φ h=1,96, the shape o the curves as ell as the magnitudes o the ultimate load coincide. This act is observable or all groups (insistently, solely in girders ith l y<a). Moreover, an appraisal o EN ormulation hen obtaining ultimate loads o hybrid specimens subjected to patch loading is presented. For the sake o illustration, F Rd (theoretical) and F u (numerical) values are standardized to the ultimate load obtained ith a geometrically equivalent 743

6 homogeneous prototype (in all cases, the material yield strength in these homogenous prototypes is y= y=235 N/mm 2 ). Namely, the plotted ratio is equal to F hyb / F hom. This ratio is plotted against the variation o the lange steel grade Φ h= y/ y. Figure 3 shos the evolution o this ratio as long as Φ h= y/ y is increased or both numerical and EN results. 32 specimens are included in such appraisal. It is observed that the EN ormulation predicts a considerable increment o ultimate load capacity o the girders as the yield strength o the lange is increased. Dierent maximum increments are observed or each case (Group 15%; Group III 1%). The numerical model, hoever, neither predicts the same results nor the same trend. The ultimate load capacity is maintained constant as the Φ h= y/ y ratio is increased. The present EN ormulation leads to a structural anomaly and must be evaluated. F hyb / F Hom 1,15 1,1 1,5 Group Num. h/t=12 EN h/t=12 Num. h/t=83,33 EN h/t=83,33 F hyb / F Hom 1,15 1,1 1,5 Group I Num. h/t=166,66 EN h/t=166,66 Num. h/t=1 EN h/t=1,95,95 F hyb / F Hom,9,8 1,3 1,8 2,3 y / y Group II Num. h/t=2 1,15 EN h/t=2 Num-h/t=12 EN h/t=12 1,1 1,5 F hyb / F Hom,9,8 1,3 1,8 2,3 y / y 1,15 1,1 1,5 Group III Num. h/t=266,66 EN h/t=266,66 Num-h/t=133,33 EN h/t=133,33,95,95,9,8 1,3 1,8 2,3 y / y,9,8 1,3 1,8 2,3 y / y 5 DESIGN PROPOSAL Figure 3. Numerical results vs. EN predictions. The numerical studies on hybrid steel plate girders subjected to patch loading hitherto presented have shon that or girders ith largely spaced transversal stieners (l y<a), the contribution o the lange yield strength y upon the patch loading resistance seems to be negligible. The current ormulation o EN takes this ratio into account in such a ay that, the greater the ratio Φ h = y/ y, the higher the ultimate load capacity o the girders. Presently, the term m 1 included in Eq. (2) is a linear unction o Φ h= y/ y. One irst attempt or correcting this anomaly is to take the y/ y ratio equal to 1, in the current expression or F Rd, according to EN This attempt leads to reshaping the eectively loaded length l y into eq. (8), in hich m 1 is shited by m 1*. This modiication o the ormulation has been tested both structurally and statistically in [6]-[8]. The ormer veriication has permitted to dra limits to the ormulation (depicted in eq (8)) and the 744

7 latter, to evaluate the corresponding enhancement this ne ormula gives to the saety margin o the ormulation. * y l = ss + 2 t 1+ m * 1 + m 2 ss 2 t = + 1+ b t h +,2 t 2 h i t b 12,5 t (8) In this paper, the ratio F u,num / F Rd is used or evaluation o the ormulation (in this assessment, the partial actor is treated as γ M1=). F u,num / F Rd represents the saety margin o the ormulation. Figure 4 (a) shos the variable F u,num / F Rd as a unction o Φ h = y/ y. Each value o y/ y includes a number o numerical results (n=37). All populations it to a normal distribution. It is noticeable that as long as y/ y is increased, the scatter gradually moves don (the arro in the plot shos such trend). Remarkable statistical inormation extracted rom each sample is presented in Table 2. This inormation gives hints about the trends observed ithin the plots. From Table 2, one can point out signiicant results. First, or each studied value o y/ y, the mean value or n=37 decreases ith y/ y. Second, the sample standard deviation S x and the sample variation V x remain nearly constant in all cases and third, maxima and minima values o the sample decrease gradually ith y/ y. According to the results obtained, one can observe a certain dependency o the statistics upon the y/ y ratio (undamentally, the mean and the extremes). These statistics, hich are essentially estimators o the saety margin, decrease monotonically. This act may be structurally detrimental and undesirable. F u,num / F u,en , 1,8 1,6 1,4,8,6 b y m1 =,4 t y,2,,8 1,4 1,6 1,8 2, F u,num / F u,m1* 2, 1,8 1,6 1,4,8,6,4,2, m * 1 = b t,8 1,4 1,6 1,8 2, y / y y / y Figure 4. F u,num/f Rd vs y/ y. (a) based upon m 1 (b) based upon m 1*. Table 2. Statistical values or the saety margin variable F u,num/f Rd. Values based upon m 1. n y / y X Sx Vx Maximum Minimum 37 1,42,157,111 1,73 1, ,17 1,39,155,111 1,7 1, ,51 1,36,156,115 1,65 1,7 37 1,96 1,32,149,113 1,59 1,3 Table 3. Statistical values or the saety margin variable F u,num/f Rd. Values based upon m 1*. n y / y X Sx Vx Maximum Minimum 37 1,42,159,112 1,73 1, ,17 1,42,158,111 1,74 1, ,51 1,43,162,113 1,74 1, ,96 1,43,157,11 1,74 1,14 745

8 Correspondingly, the ratio F u,num / F Rd,m1* is used or evaluation. Figure 4 (b) shos a plot in hich the populations (n=37 or each value o Φ h) are plotted against the y/ y ratio. A substantial change in the trend is observed regarding the inluence o y/ y. Table 3 shos similar statistics or the population in hich l y is based upon m 1*. In this case, all statistics are nearly constant or all groups. The mean and the extremes in all cases practically coincide. Accordingly, the aorementioned peculiarities vanish hen m 1 is shited by m 1*. According to the results obtained, the term m 1* corrects quite satisactorily the potential overestimation encountered ithin the numerical results. 6 CONCLUSIONS Numerical results presented in this paper have shon some aspects concerning the eect o the lange yield strength (and thus, the hybrid parameter Φ h= y/ y) on the resistance o steel plate girders subjected to patch loading. Numerically, it is predicted that or girders ith largely spaced transverse stieners (l y<a), there is null inluence o this ratio upon the ultimate load capacity o patch loaded girders. The current ormulation o EN takes this ratio into account in such a ay that, the greater the ratio y/ y is, the higher the ultimate load capacity o the girders. For the sake o correcting this anomaly, a modiication on the current EN ormulation that enhances the results quite satisactorily is proposed. Readably, it is suggested that m 1 be shited by m 1*. This ne ormula has been tested both structurally and statistically. The results lead to a satisactory improvement o the ormulation. The results obtained ith the upgraded coeicient m 1* are structurally sound and on the saety side. 7 ACKNOWLEDGMENTS The authors ish to grateully acknoledge the inancial support provided by Spanish Ministerio de Fomento, as a part o the Research Project 51/7 Integral bridges REFERENCES [1] Veljkovic, M., Jojansson B. Design o hybrid steel girders. Journal o Constructional Steel Research, 6(3-5), , 24. [2] Lagerqvist, O., Jojansson B. Resistance o I-girders to concentrated loads. Journal o Constructional Steel Research, 39(2), , [3] Roberts, R., Rockey, K. A mechanism solution or predicting the collapse loads o slender plate girders hen subjected to in-plane patch loading. Proc. Inst. o Civil Eng. 67(2), , [4] Elgaaly, M. Web design under compressive edge loads. Eng Journal 2(4) , 1983 [5] EN Design o Steel Structures. Part-1-5. Plated Structural Elements. CEN26. [6] Chacon, R., Resistance o hybrid steel plate girders to concentrated loads. Doctoral Thesis. Universitat Politècnica de Catalunya. 29 [7] Chacon, R., Mirambell, E., Real, E., Hybrid steel plate girders subjected to patch loading. Part-1 Numerical study. Journal o Constructional Steel Research. 66(5), [8] Chacon, R., Mirambell, E., Real, E., Hybrid steel plate girders subjected to patch loading. Part-2 Design proposal. Journal o Constructional Steel Research. 66(5), [9] Abaqus/Standard V.6.8. Simulia products. Dassault Systemes S.A. 21 [1] Chacon, R., Mirambell, E., Real, E., Inluence o designer assumed initial conditions o the numerical modeling o steel plate girders subjected to patch loading. Thin-Walled Structures. 47(4), ,