Journal of Constructional Steel Research

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1 Journl of Constructionl Steel Reserch 67 (2011) Contents lists ville t ScienceDirect Journl of Constructionl Steel Reserch journl homepge: Review Review on the modelling of joint ehviour in steel frmes Concepción Díz, Pscul rtí, rino Victori, Osvldo. Querin, Deprtment of Structures nd Construction, Technicl University of Crtgen, Cmpus urll del r, Crtgen (urci), Spin School of echnicl Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom r t i c l e i n f o s t r c t Article history: Received 12 August 2010 Accepted 24 Decemer 2010 Keywords: Steel joint nlysis Semi-rigid joints Joint ehviour representtion oment rottion curve Bem-to-column joints Steel portl frmes were trditionlly designed ssuming tht em-to-column joints re idelly pinned or fully rigid. This simplifies the nlysis nd structurl design processes, ut t the expense of not otining detiled understnding of the ehviour of the joints, which in relity, hve finite stiffness nd re therefore semi-rigid. The lst century sw the evolution of nlysis methods of semi-rigid joints, from the slope-deflection eqution nd moment distriution methods, to mtrix stiffness methods nd, t present, to itertive methods coupling the glol nd joint structurl nlyses. Studies gree tht in frme nlysis, joint rottionl ehviour should e considered. This is usully done y using the moment rottion curve. odels such s nlyticl, empiricl, experimentl, informtionl, mechnicl nd numericl cn e used to determine joint mechnicl ehviour. The most populr is the mechnicl model, with severl vrinces (e.g. Component ethod). A summry is given of the dvntges nd disdvntges nd principl chrcteristics of ech model. Joint ehviour must e modelled when nlysing semi-rigid frmes, which is ssocited with mthemticl model of the moment rottion curve. Depending on the type of structurl nlysis required, ny moment rottion curve representtion cn e used; these include liner, iliner, multiliner nd nonliner representtions. The most ccurte representtion uses continuous nonliner functions, lthough the multiliner representtion is commonly used for mechnicl models. This rticle reviews three res of steel joint reserch: (1) nlysis methods of semi-rigid joints; (2) prediction methods for the mechnicl ehviour of joints; (3) mthemticl representtions of the moment rottion curve Elsevier Ltd. All rights reserved. Contents 1. Introduction Anlysis methods of semi-rigid joints ethods for modelling the rottionl ehviour of joints Experimentl testing Empiricl models Frye nd orris model Krishnmurthy model Kukreti model Attioge nd orris model Fell, Piluso nd Rizzno model Anlyticl models Chen et l. model Yee nd elchers model echnicl models Numericl models Informtionl models themticl representtion of moment rottion curve Corresponding uthor. Tel.: , (moile); fx: E-mil ddresses: conchi.diz@upct.es (C. Díz), pscul.mrti@upct.es (P. rtí), mrino.victori@upct.es (. Victori), O..Querin@Leeds.c.uk (O.. Querin) X/$ see front mtter 2011 Elsevier Ltd. All rights reserved. doi: /j.jcsr

2 742 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) Stiffness, resistnce nd shpe fctor-sed formultions Liner model Biliner model ultiliner model Nonliner model Formultion sed on curve fitting y regression nlysis Conclusions Acknowledgements References Nomenclture ep fc j t wt c f c wc c ws c wt d 1 d 3 d d g e p f y f y g g 1 g 3 g h I t I w k t l m n n p f t t ep t f t f t i t p t s end-plte width em flnge nd we in compression lest-squres coefficients olts in tension em we in tension column flnge in ending column we in compression column we in sher column we in tension distnce etween the middle lines of the legs djcent to the em flnges distnce etween the centre of the we ngles nd the middle line of the set ngle leg djcent to the em flnge olt dimeter verticl distnce etween the olt centrelines end-plte in ending yield stress of the se mteril olt yield stress guge of column flnge olts distnce etween the nut edge nd the middle line of the top ngle leg djcent to the em distnce etween the nut edge nd the middle line of the we ngle leg djcent to the em guge distnce etween the two olts in row em height inerti moment of the leg djcent to the column fce of the top ngle inerti moment of the leg djcent to the column fce of the we ngle distnce etween the heel of the top ngle nd the toe of the fillet ngle length numer of curve-fitting constnts (Eq. (28)); numer of points etween two elementry prts of the φ curve (Eq. (27)) shpe fctor which chrcterizes the knee of the moment rottion; shpe prmeter determined using the method of lest squres for differences etween the predicted moments nd the experimentl test dt (Eq. (30)) numer of olts per ngle leg on column flnge pitch of the olt (distnce from top of the flnge to the centreline of the olt) ngle thickness end-plte thickness column flnge thickness em flnge thickness ngle thickness end-plte thickness set ngle thickness t t t w t w Cpitl letters A C C 1,2,3 C j D k E H [φ] K K φ K f K φ,p K φ,y L s L t L w 0 i j j,p j,rd j,y P i W Greek letters α i α φ i φ k φ φ 0 φ Cd top ngle thickness we ngle thickness em we thickness gross cross-sectionl re of the olt regression prmeter curve-fitting constnts curve-fitting prmeter otined from liner regression (Eq. (28)); modelling prmeters otined y liner regression nlysis (Eq. (29)) curve-fitting prmeter otined from liner regression modulus of elsticity Heviside s step function prmeter depending on the geometricl nd mechnicl properties of the structurl detil (Eq. (1)); rottionl stiffness joint rottionl stiffness joint rottionl stiffness joint plstic rottionl stiffness; strin-hrdening connection stiffness (Eq. (28)) post-yielding rottionl stiffness length of the set ngle length of the top ngle length of the we ngle joint moment reference ending moment initil moment joint moment; upper ound moment of the jth prt of th curve (Eq. (27)) joint plstic moment joint moment resistnce yielding moment joint geometric prmeter em section modulus coefficients otined in such wy s to give good fit to the curve (Eq. (3)) regression prmeter (Eq. (25)); scling fctor for numericl stility (Eq. (28)); shpe prmeter determined using the method of lest squres for differences etween the predicted moments nd the experimentl test dt (Eq. (30)) initil rottion strting rottion of the kth liner component of the φ curve joint rottion joint permnent rottion joint rottionl cpcity

3 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) Introduction Steel portl frmes were trditionlly designed ssuming tht em-to-column joints re idelly pinned or fully rigid. The use of the idelly pinned condition implies tht no moment cn e trnsmitted etween the em nd the column; this mens tht the connections hve no rottionl stiffness nd cnnot trnsmit moments lthough they do trnsmit xil nd sher forces to the ttched memers (Fig. 1()). On the other hnd, fully rigid joints hve rottionl comptiility nd therefore trnsmit ll form of lods etween em nd column (Fig. 1()). An importnt spect of the nlysis of these joints is tht their ehviour is decoupled from the nlysis of the structure. Although this simplifies the nlysis nd structurl design processes; it comes t the expense of not eing le to otin detiled understnding of the ehviour of the joint. In relity, joints hve finite stiffness nd re therefore semi-rigid (Fig. 1(c)). In the lst century, nlysis methods of semi-rigid joints evolved considerly to otin the true structurl response. Strting in the 1930 s with the slope-deflection eqution nd moment distriution methods, the 1960 s with the mtrix stiffness methods, nd nowdys, with complex itertive nlysis methods which couple the structurl nlysis with tht of the joint. The true ehviour of joint cn e incorported within the glol nlysis of the structure y using the moment rottion curve ( j φ), (Fig. 2). This is chieved y determining the mechnicl properties of the joint in terms of its rottionl stiffness (K j ), moment resistnce ( j,rd ), nd rottionl cpcity (φ Cd ), strting from their geometricl nd mechnicl properties. There re severl models which cn e used to determine the mechnicl ehviour of joints, these re: nlyticl, empiricl, experimentl, informtionl, mechnicl nd numericl. The most populr of these is the mechnicl model, which hs severl vrinces, the most populr eing the Component ethod, Eurocode 3 [1]. This method considers joint s set of individul sic components, which llows the determintion of the moment resistnce nd stiffness chrcteristics of ll the different components of the joint. These joint ehviour models need to e incorported into structurl nlysis pckges in order to then e le to nlyse nd design the joint. To chieve this, mthemticl expressions re required which llow for the rottionl deformtion, rottionl stiffness nd design moment resistnce to e esily incorported into the glol structurl nlysis. This rticle provides stte-of-the-rt review of three res of steel joint reserch: (1) nlysis method of semi-rigid joints; (2) prediction methods for the mechnicl ehviour of joints; nd (3) mthemticl representtions of the moment rottion curve. 2. Anlysis methods of semi-rigid joints The first studies on semi-rigid joints were crried out in 1917, when Wilson nd oore [2] investigted the stiffness of riveted joints in steel structures. But it ws not until the 1930 s tht studies egn into the reltionship etween the moment nd rottion of semi-rigid joints nd their overll effect on steel structures. These cn e seen in the reports of The Steel Structures Reserch Committee [3 5] (UK), Young nd Jckson [6] (Cnd) nd Rthun [7] (USA). Since then, there hve een numerous experimentl nd theoreticl studies into the ehviour of semirigid steel joints (riveted, olted nd welded) nd their effect on the overll structure. Btho nd Rown [4] proposed grphicl method, clled emline, which ws used to determine the end restrint provided y ech joint. To pply this method, requires the use of the experimentlly clculted moment rottion curve. Bker [4] nd Rthun [7], were the first to pply the slope-deflection [8] nd the moment distriution [9] methods to the nlysis of semi-rigid joints. Between 1936 nd 1950, most of the reserch ws focused on the ppliction of these methods to the nlysis of structures with semi-rigid joints. The most notle pulictions re those of Bker nd Willims [5], Johnston nd ount [10], Stewrt [11] nd Sourochnikoff [12]. By the 1960s, the mtrix stiffness method of structurl nlysis utilising computers hd een estlished. onforton nd Wu [13] were the first to incorporte the effects of semi-rigid connections into the mtrix stiffness method in This ws chieved y modifying the em stiffness mtrices to tke the semi-rigid connection effects into ccount in the frme nlysis. Similr procedures were lso proposed y Livesley [14], nd Gere nd Wever [15], t out the sme time. In these nlysis methods, liner j φ reltionship ws ssumed nd the liner semi-rigid connection fctor Z = φ/ ws used to modify the em stiffness mtrices [16]. The dynmic ehviour of semi-rigid frmes ws investigted y Lionerger nd Wever [17] in 1969 nd y Suko nd Adms [18] in In these nlyses the connection elsto-plstic ehviour ws modelled y equivlent springs. In 1978, the Europen Convention for Constructionl Steelwork (ECCS) pulished Report 23 on the Europen recommendtions for steel construction [19]. This report formed the sis of the current Eurocode 3. These recommendtions replced the method of llowle stresses y the limit stte method, which is sed on proilistic concepts of sfety nd the use of enhncement lod fctor for the nlysis of structurl resistnce nd stility insted of the trditionl reference to llowle stresses. In 1981, oncrz nd Gerstle [20] proposed new pproximtion to the nlysis of semi-rigid frmes sed on modifiction of the sic mtrix stiffness technique. Bsed on the studies of the ECCS, in 1984 the Commission of the Europen Community pulished the first version of the Eurocode 3 [21]. In this document, the joints re clssified s rigid nd semirigid for elstic liner nlysis nd with full- or prtil-strength for elstic plstic nlysis. However, they neither consider their use, nor how to model them. The code ws pulished on tril sis (Europen Pre-Stndrd, ENV) inviting comments from its users s well s professionl, scientific, stndrds nd technicl orgnistions. Their comments nd suggestions were used to develop the finl code (Europen Stndrd, EN). In 1989 this work ws trnsferred to the Europen Committee for Stndrdiztion (CEN). In 1983, Jones et l. [22] presented revised review of the nlysis of frmes with semi-rigid joints. This work ws extended y Nethercot in [23,24], where he proposed different pproches nd improvements for the nlysis of semi-rigid frme y dopting the sic mtrix stiffness technique. In 1987, Lui nd Chen [25], nd Goto nd Chen [26], proposed methods for the nlysis of semi-rigid frme sed on mtrix stiffness nlysis, using smll computers. On the sme yer, the ECCS [27] creted the Working Group TWG 8.2, to study the influence of semi-rigid connections on the overll frme ehviour. The results of this study helped to estlish the Technicl Committee for Structurl Connections (TC10) of the ECCS to look t the ehviour of connections. The Eurocode hs evolved [28], nd finlly in y 2005, the Eurocode 3 [1] ws pulished. It ws exclusively dedicted to ll types of joints, including semi-rigid ones, where the response of joint is dependent on the geometric nd mechnicl properties of its components, using the component method. This code of prctice is collection of decdes of reserch in steel structures. Other interntionl codes of prctice which lso consider joint ehviour re those of the USA in AISC-ASD [29], LRFD [30], AISC- ASD/LRFD [31] nd Chin in GB [32].

4 744 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) c Fig. 1. Joint types ccording to their ehviour, where φ is the ngulr rottion etween the em nd the column: () pinned; () rigid; nd (c) semi-rigid. There is currently gret rnge of studies of steel frmes with semi-rigid connections: Jsprt [33], Jsprt nd quoi [34], Weynnd et l. [35], Chen [36], Brhm nd Jsprt [37], Ashrf et l. [38], Crero nd Byo [39], Byo et l. [40], d S. Vellsco et l. [41], Ashrf et l. [42], Yng nd Lee [43], Fell et l. [44], d Silv et l. [45], Dniũns nd Urons [46], Sekulovic nd Nefovsk-Dnilovic [47], Bel Hdj Ali et l. [48], Ihddoudène et l. [49], ehrin et l. [50], Drío [51], etc. These studies were concerned with two principl themes [52]: (1) the evlution of the mechnicl properties of the joints in terms of rottionl stiffness, moment resistnce nd rottion cpcity, nd (2) the nlysis nd design procedures for frmes including rottionl joint ehviour. All studies gree tht when crrying out structurl nlysis of ny frme, the rottionl ehviour of the joints must e considered. It is evident tht the prediction of the joint ehviour y mens of one of the ove methods hs to e generlly ccompnied y mthemticl representtion of the moment rottion curve, which is necessry to e used s input dt in computer progrms for the structurl nlysis of semirigid frmes. In the next section, ll methods for the prediction of the joint rottionl ehviour s well s their mthemticl representtion will e explined. 3. ethods for modelling the rottionl ehviour of joints To properly model the em-to-column joint ehviour, the moment rottion curve for the joints is required. Fortuntely there re mny models which cn e used to predict it. The most commonly used models re included here, grouped into: nlyticl, empiricl, experimentl, mechnicl, numericl nd informtionl models. The lst of which is the most recent. Other clssifictions cn e found in the work of Nethercot nd Zndonini [53], Fell et l. [54] nd Jsprt [52] Experimentl testing The most ccurte knowledge of the joint ehviour is otined through experimentl tests, ut this technique is too expensive for everydy design prctice nd is usully reserved for reserch purposes only [54]. In 1917, Wilson nd oore [2] performed the first experiment to ssess the rigidity of steel frme connections. Since then, experimentl testing hs een continued. Prior to 1950, most connection tests were focused on riveted joints: Btho [3]; Btho nd Rown [4]; Btho nd Lsh [5]; Young nd Jckson [6]; Rthun [7]. After 1950, high strength olts were used extensively in steel construction. A lrge numer of tests were mde nd reported, llowing for the genertion of severl dt nks. The informtion required from ech test usully includes: the geometric nd mechnicl properties of ech component which mkes up the joint, the moment rottion curve, the rottionl stiffness (K j ) nd moment resistnce ( j,rd ) s well s the nme of the reserchers. The four most importnt dt nks re: 1. Goverdhn dt nk. The first one to e developed, in 1984 [55], hs the results of 230 tests from the USA crried out etween 1950 nd It includes tests on the following connection typologies: doule we ngle connections, single we ngle/plte connections, heder-plte connections, endplte connections nd top nd set ngle connections with or without we ngles. 2. Nethercot dt nk. The first Europen dt nk on steel connections ws developed in Nethercot [56,57] exmined more thn 70 experimentl studies collecting more thn 700 individul tests y other reserchers [58]. The connection typologies include those exmined y Goverdhn s well s T-stu connections with nd without we ngles. 3. Steel connection dt nk. In the USA, the work of Goverdhn [55] ws followed y tht of Kishi nd Chen [59,60] who prepred dt nk collecting experimentl tests from ll over the world crried out from 1936 to They compiled results from over 303 tests. In ddition, they developed the Steel Connection Dt Bnk (SCDB) progrm for the recovery of ll the experimentl dt nd the formultion of mthemticl reltionships for the curve fitting of experimentl moment rottion ehviour [61,62]. In 1995, Adll nd Chen [63] dded the results of 46 dditionl experimentl tests of steel em-to-column joints. The tests collected in the progrm SCDB re contined, ccording to the following connection typologies: single ngle we 1 clet/plte connections, doule ngle we clet connections, top nd set ngle clets connections with or without we ngles, extended nd flush end-plte connections nd heder-plte connections. 4. SERICON dt nk. Developed y Ared Recherches [64] nd Achen University [65], includes only Europen test results [66]. It lso contins tests from single joint components nd tests on composite connections. This dt nk ws extended into the SERICON II dtse y Cruz et l. [67]. The use dt nk is minly devoted to the vlidtion of models, imed t the prediction of the joint ehviour from its geometricl nd mechnicl properties, rther thn to dily design prctice. In fct, the designer hs only low proility of finding in the dt nk the specific structurl detil of the joint studied, due to the gret vriety of connection typologies, geometricl properties nd stiffening detils of pnel zone [54]. 1 Also referred to s we ngle.

5 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) Fig. 2. oment rottion ( j φ) curve. Other experiments include the work of Popov nd Tkhirov [68], who crried out two tests on olted lrge seismic steel em-to-column joints. Girão et l. [69] evluted 8 tests to ssess the ductility of extended end-plte connections. Girão et l. [70] crried out 32 tests on olted T-stu connections mde up of welded pltes. Girão nd Bijlrd [71] crried out experiments to study the ehviour of high strength steel end-plte connections nd in [72] the experimentl ehviour of high strength steel we sher pnels. Crero nd Byo [73], nlysed the semi-rigid ehviour of three-dimensionl steel em-to-column joints sujected to proportionl loding. Shi et l. [74] crried out 5 experiments of em-to-column olted extended end-plte joints to develop n nlyticl model to otin the rottionl stiffness nd the moment rottion curve of joint. Piluso nd Rizzno [75] did n experimentl nlysis nd modelling of olted T-stus under cyclic lods. Fig. 3. Geometricl prmeters for the Frye orris polynomil representtion of end-plte connections without column stiffeners Empiricl models Empiricl models re sed on empiricl formultions which relte the prmeters of the mthemticl representtion of the moment rottion curve to the geometricl nd mechnicl properties of em-to-column joints. These formultions cn e otined using regression nlyses of dt which cn e derived in different wys such s: experimentl testing, prmetric nlyses developed y mens of Finite Element (FE) models, nlyticl models or mechnicl models. The min disdvntge of this type of model is tht it is only pplicle to joints whose chrcteristics mtch those used to generte the model. It is lso not possile to determine how ech prmeter of the joint ffects its overll performnce. Five common models re descried next Frye nd orris model The Frye nd orris model [76] is sed on n odd-power polynomil representtion of the moment rottion curve, Eq. (1). φ = C 1 (K) + C 2 (K) 3 + C 3 (K) 5 (1) where K is prmeter depending on the geometricl nd mechnicl properties of the structurl detil, nd C 1, C 2 nd C 3 re curve-fitting constnts. For exmple, for the end-plte connections without column stiffeners of Fig. 3, the curve-fitting constnts re given y Eq. (2). C 1 = ; C 2 = ; C 3 = ; K = d 2.4 g t 0.4 p t 1.5 f. The min drwck of this formultion is tht, in some cses, the slope of the moment rottion curve cn ecome negtive for some vlues of [77]. This is physiclly unrelistic nd cn cuse (2) Fig. 4. Extended end-plte connections with four olts in the tension zone for the Krishnmurthy model [88]. numericl difficulties in the nlysis of semi-rigid frmes using the tngent stiffness formultion. To solve this prolem, Azizinmini et l. [78] proposed different formultion of the prmeter K, Eq. (3). K = P α 1 1 Pα 2 2 Pα n n (3) where P i re geometric prmeters of the joint nd the α i re the coefficients otined to give good fit to the curve. This model ws used in severl studies to investigte the effect of semi-rigid joints on steel frme structures: Picrd et l. [79]; Altmn et l. [80]; Goverdhn [55]; Kmeshki nd Sk [81]; Hdinfrd nd Rzni, [82]; Hylioglu nd Degertekin [83]; Prh et l. [84] Krishnmurthy model Krishnmurthy [85,86] crried out wide prmetric study y mens of the FE ethod (FE) to study the rottionl ehviour of end-plte connections. Experimentl tests, limited to 5 prototypes, were used to djust some of the prmeters of the model nd confirm the numericl results. The two-dimensionl (2D) plne stress numericl model ws for plne prllel to the em we. Five experiments were used to correlte this model [87]. This method ws further developed

6 746 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) prmeters (ϕ 0, 0, n, K ϕ,p ) which re relted to the geometricl properties of connection, Eq. (6). ϕ 0 = t g l h n = t g l h n n = t g l h n K ϕ,p = t g l h n where t is the ngle thickness (mm), g the guge of column flnge olts (mm), l is the ngle length (mm), h is the em depth nd n is the numer of olts per ngle leg on column flnge. The units ϕ 0, 0 nd K ϕ,p re rdins, kn, nd kn m/rd, respectively. (6) Fig. 5. Structurl detil of flush end-plte connection nlysed [89]. to the cse of extended end-plte connections with four olts in the tension zone (Fig. 4), leding to the development of n empiricl model sed on the simple power representtion of the moment rottion curve [88], Eq. (4). φ = C α (4) α = 1.58; C = 1.4βµp2.03 f ep ; β = t1.03 f ; A 0.36 tep 1.38 h 1.30 t 0.26 w W 1.58 µ = 1.0 fy 0.38 f 1.20 y where W is the em section modulus, f y is the yield stress of the se mteril, f y is the olt yield stress nd A is the gross crosssectionl re of the olt. These prmeters re independent of the geometry of the column s this ws considered in the FE model. For this reson, the moment rottion curve is for the connection nd not the joint Kukreti model Kukreti extended the method of Krishnmurthy y crrying out new prmetric study of flush end-plte connections without column stiffeners (Fig. 5). Kukreti et l. [89] lso used the FE to otin the power model of Eq. (5). φ = C α (5) α = 1.58; C = p f h t w t f d g ep tep where the lengths re in inches nd the moments in kip-ft. This method ws lter pplied to study of the extended endplte connection where eight olts re locted in the tensile zone nd the end-plte is stiffened y mens of reinforcing ri [90]. Empiricl models, sed on the power of the moment rottion curve, re le to ccurtely predict the initil rottionl ehviour of the connection, rther thn the whole moment rottion curve. There is significnt sctter etween the predicted nd experimentl moment rottion curves for high vlues of plstic deformtions [91] Attioge nd orris model Attioge nd orris [92] proposed new model sed on lortory experimentl results nd the mthemticl representtion of Golderg nd Richrd [93], to predict the moment rottion curve for doule we ngle connections. This model requires four Fell, Piluso nd Rizzno model The empiricl model of Fell et l. [94] for the prediction of the flexurl resistnce nd rottionl stiffness of extended end-plte em-to-column joints ws developed y mens of mechnicl model [95,96] sed on the component method from the Eurocode 3 [28] Anlyticl models Anlyticl models use the sic concepts of structurl nlysis: equilirium, comptiility nd mteril constitutive reltions, to otin the rottionl stiffness (K j ) nd moment resistnce ( j,rd ) of joint due to its geometric nd mechnicl properties Chen et l. model Chen nd his collegues worked extensively to predicting the response of joint from its geometricl nd mechnicl properties. The work on joints with the semi-rigid connections with ngles is presented in [97 99]. For top nd set ngles with doule we ngles connections (Fig. 6) the initil stiffness is given y Eq. (7). 3EI t d 2 1 K ϕ = g 1 g t2 t I i = L it 3 i 12 3EI w d 2 3 g 3 g t2 w (7) where I t nd I w re the inerti moments, Eq. (8), of the leg djcent to the column fce of the top ngle nd of the we ngle, respectively; t i is the thickness of the ngles; g 1 nd g 3 re the distnces etween the nut edge nd the middle line of the ngle leg djcent to the em, g 1 is referred to the top ngle nd g 3 to the we ngles; d 1 is the distnce etween the middle lines of the legs djcent to the em flnges; d 3 is the distnce etween the centre of the we ngles nd the middle line of the set ngle leg djcent to the em flnge. The ultimte ending moment is given y Eq. (9). j,u = f y L s t 2 s 4 + V pt (g 1 k t ) 2 (8) + V pt d 2 + 2V p d 4 (9) where L s nd t s re the length nd thickness of the set ngle, k t is the distnce etween the heel of the top ngle nd the toe of the fillet, nd d 2 nd d 4 re given y Eqs. (10) nd (11). d 2 = d + t s 2 + k t (10) d 4 = 2V pu + f ytw 2 L w + t s 3 V pu + f ytw 2 + L I. (11) 2

7 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) Fig. 6. Top nd set ngle connection with doule we ngles nd the geometricl prmeters of ngles. The prmeters V pu, V pt nd V p re otined using Eqs. (12) (14). 4 2Vpu + g c k 2Vpu = 1 (12) f y t w t w f y t w 4 2Vpt + g 1 k t 2Vpt = 1 (13) f y L t t t t t f y L t t t V p = V pu + f ytw 2 L w (14) 2 where L t nd L w re the lengths of the top ngle nd of the we ngles, respectively. These reltionships were comined non-dimensionlly [100,101], to provide the influence of the min geometricl prmeters on the rottionl ehviour of connections with ngles. Their use within design procedure sed on dvnced nlysis methods hs een shown [102]. The min prolem with the Chen nd Krishnmurthy models is tht they do not consider the deformtion of the column. The ssumption eing tht the support to connection is rigid Yee nd elchers model In 1986, Yee nd elchers [103] proposed mthemticl model tht could predict the moment rottion reltionships of olted extended end-plte eve connections, using the connection dimensions. The model represents physiclly sed pproch to the prediction of moment rottion curves, tking into ccount the possile filure modes nd the deformtion chrcteristics of the connection elements. The model included five deformtion nd six modes of filure. The deformtions re: (1) end-plte flexure; (2) column flnge flexure; (3) olt extension; (4) column we pnel sher deformtion; nd (5) column we compression. And the filures re: (1) olt filure (tension); (2) formtion of end-plte plstic mechnism; (3) formtion of column flnge plstic mechnism; (4) sher yielding of the column we; (5) uckling of the column we; nd (6) we crippling. The rottionl stiffness of joint is otined y comining the elstic displcements of the different components of the joint. The limiting moment cpcity depends on the strength of the weker djoining section. Johnson nd Lw [104] developed with similr pproch method for predicting the initil stiffness nd plstic moment cpcity of flush end-plte connections. Pirmoz et l. [105] proposed semi-nlyticl model of otining the moment rottion ehviour of olted top set ngle connections under comined xil tension nd moment loding sed on the dt nk, creted using FE simultion echnicl models echnicl or spring models [54,95,96,103,106] represent the joint y using comintion of rigid nd flexile components, which re modelled y mens of stiffness nd resistnce vlues otined from empiricl reltionships. The nonlinerity of the response is otined y mens of inelstic constitutive lws used for the spring elements. Fig. 7 shows the mechnicl model used y Fell [54] for the extended end-plte em-to-column joint. To develop mechnicl model three steps re required: (1) identify the components of the joint tht will provide significnt deformtion nd filure of the joint; (2) determine the constitutive lws for ech component of the joint using nlyticl, experimentl or numericl mens, nd (3) ssemle ll of the components together to produce the moment rottion curve for complete joint. This procedure is very flexile s it cn e pplied to joints of ny type: olted or welded, nd where specific effects cn e introduced, such s: olt pretensioning or plstic hrdening, etc. This is ecuse ll tht is required re the constitutive ehviour of the components which mke up the joint. The firsts to introduce this type of model were Wles nd Rossow [107] in 1983 to simulte the ehviour of doule we ngle connection with n pplied ending moment nd xil lod (Fig. 8). The joint ws modelled using two rigid rs connected y homogeneous continuum of independent nonliner springs. An importnt chrcteristic of this model ws tht it included n xil lod. Kennedy nd Hfez [108] used this model to represent heder-plte connections. Chmielowiec nd Richrd [109] extended this model to predict the ehviour of ll types of cleted connections suject to ending nd sher. Since then, significnt reserch hs een crried out using mechnicl models to study the ehviour of joints nd to introduce their effect in the nlysis of structure. Fell et l. [54] developed the progrm JRC to evlute the moment rottion curve for welded connections, olted end-plte connections nd olted connections with ngles. Pucinotti [110] proposed model for top nd set nd we ngle connection sed on simplifiction of the model in prt J of the Eurocode 3 [28]. A model for joints under ending nd xil lods ws proposed y Simões d Silv nd Girão [111], Simões d Silv et l. [112], s well s y Urons nd Dniũns [113], Sokol et l. [114] nd del Svio et l. [115]. Byo et l. [40] proposed n improvement to the Eurocode 3 model y introducing component-sed finite dimensioned elstic plstic 4-node joint element which tkes into ccount the ctul size of the joint, its deformtion chrcteristics, including those of the pnel zone, locl phenomen nd ll the internl forces tht concur t the joint. Crero nd Byo [116] proposed model to

8 748 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) Fig. 7. echnicl model for the extended end-plte em-to-column joint [54]. Fig. 8. echnicl model for we ngle connections [107]. clculte the stiffness in three-dimensionl steel em-to-column joints for oth mjor nd minor xes. Simões d Silv [117] proposed generic model for steel joints under generlized loding. Lemonis nd Gntes [118], proposed model sed on the component method for olted connections with end-pltes nd with ngles. Simões d Silv et l. [119] proposed mechnicl model to evlute the ehviour of cruciform flush end-plte em-to-column steel joints t elevted tempertures. The component method [1] is hyrid nlyticl mechnicl method (Fig. 7). It consists of modelling joint s n ssemly of extensionl springs (components) nd rigid links, where ech spring represents specific prt of joint with its own strength nd rigidity, dependent on the type of loding. The ehviour of the joint is otined y knowing the mechnicl nd geometricl properties of ech component of the joint. It produces good results when the joint is cting primrily in ending with miniml xil loding Numericl models Numericl simultion strted to e used for severl resons: (1) s mens of overcoming the lck of experimentl results; (2) to understnd importnt locl effects which re difficult to mesure with sufficient ccurcy, e.g. prying nd contct forces etween the olt nd the connection components; nd (3) to generte extensive prmetric studies. FE Anlysis (FEA) is idelly suited to determine the rottion of joint; however such nlysis is still computtionlly expensive. The moment rottion curve is the result of the complex interction etween the different elements of joint. The nlysis of steel joints requires the introduction of geometricl nd mteril nonlinerities of the elementry prts of the connection; olt prelod nd its response under generl stress distriution; interction etween olts nd plte components: i.e., shnk nd hole, hed or nut contct; compressive interfce stresses nd friction resistnce; slip due to olt-to-hole clernce; vriility of contct zones; welds; imperfections. Currently the FE llows for the introduction into the model of: lrge deformtions, plsticity, strin-hrdening, instility effects, the representtion of lrge strin nd/or displcements, contcts etween pltes nd pre-stressing of olts [120,121]. In 1972 Bose et l. [122] crried out the first FE study of welded em-to-column joints, which included: plsticity, strin hrdening nd uckling. The results otined compred fvourly with ville experimentl results. Since then, severl reserchers hve used the FE to investigte joint ehviour. In 1976, Krishnmurthy nd Grddy [87] were the first to model three-dimensionl (3D) joints. They used n eight-node rick element to model the end-plte connection. The nlysis included contct etween the different joint elements nd preloded olts. However due to the limited computtionl power t the time, the 3D model ws only used to develop correltion fctor etween the two-dimensionl (2D) nd 3D results to enle the prediction of the more ccurte 3D vlues from the less expensive 2D results (Fig. 9). A similr process ws proposed y Kukreti et l. [89], to generte the moment rottion curve for olted endplte connection, otining very good results. Kukreti et l. [91] developed hyrid 2D 3D FE model for teehnger connections, using 3D FE for the tee-flnge, olt heds nd olt shnks, nd 2D FE elsewhere else.

9 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) Fig. 9. 2D mesh of the end-plte connection with 581 nodes nd 508 elements [87]. Chsten et l. [123] studied lrge extended unstiffened endplte connections with eight olts t the tension flnge (fourolts wide). FE ws used, using shell elements for the end-plte nd em flnges nd plne stress elements for the em we. Contct etween the end-plte nd column flnge ws modelled to determine the prying force. Geeken et l. [124] studied extended end-plte connections using shell elements. The chrcteristics of their model re: olts with simplified geometry; plne stress nlysis; nonliner strin displcement reltionship. For the cse of friction etween end-plte nd screw hed, only the limit cses when completely stick nd frictionless slip were considered; nd the friction etween the flnge nd end-plte ws neglected. Sherourne nd Bhri [125,126] developed FE to investigte the ehviour of steel olted end-plte connections. Where the end-plte, em nd column flnges, wes, nd column stiffeners were represented s plte elements with ech olt shnk modelled using six spr elements. Three-dimensionl interfce elements were used to model the oundry etween the column flnge nd the ck of the end-plte tht my mke or rek contct. Bursi nd Jsprt modelled T-stu connections [127] nd isolted extended end-plte connections [128,129] (Fig. 10). They crried out severl models using 3D rick elements nd contct elements. They considered the effect of: element type, preloding, different constitutive reltionships, nd friction coefficient. Their results compred fvourly to test results. Troup et l. [130] used FEA to crete numericl model of T-stu nd n extended end-plte connection. Simplified iliner stress strin curves for the steel sections nd olt shnk were dopted. teril nonlinerity ws considered for steel memers nd connecting components, together with geometric nonlinerity due to the chnging re of contct etween the fces of the endplte or T-stus. An encourging correltion etween the model nd experimentl tests ws oserved showing good comprison of the stiffness in oth thick nd thin plte conditions. Bhri nd Sherourne [131] developed detiled 3D FE to study 8-olt unstiffened extended end-plte connections using primrily shell elements (Fig. 11). Neither the olt hed or nut were included in the model, insted the end-plte nd column flnge thicknesses were incresed round the olt hole. The olt shnk ws represented using truss elements connecting corresponding nodes etween the end-plte nd column flnge. The contct etween the column flnge nd ck of the end-plte ws modelled using 3D interfce elements. Sumner et l. [132] lso used FE to develop 4- nd 8-olt extended unstiffened moment end-plte connections, otining Fig D model of the extended end-plte connection [129]. Fig D model using shell elements of n extended end-plte connection [131]. very good correltion etween theirs nd test results. Their model included: solid eight-node rick elements for the em section nd column flnge, which included plsticity effects; solid twentynode elements for the olts nd end-plte; nd contct elements etween the end-plte nd the rigid column flnge. Swnson et l. [133] presented the results of FE investigtion of the ehviour of T-stu flnges (Fig. 12). Two types of models were used; 3D T-stu model consisting of rick nd wedge elements nd severl 2D T-stu flnge models consisting of rectngulr nd tringulr elements. All models incorported

10 750 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) Fig D solid T-stu model [133]. nonliner mteril chrcteristics, nonliner geometric ehviour, nd severl contct interctions. Citipitioglu et l. [134] presented different 3D models of olted connections with ngles, (Fig. 13), following the recommendtions of [129] on the FE selection. Contct etween ll prts ws modelled, including the effect of friction. Their results, similr to those of [129], confirmed tht the effect of friction on the initil stiffness of the joint ws negligile, lthough it ws slightly more on the plstic regions. The effect of olt pretension ws similr to tht of friction, lthough it could modify the ultimte moment of the joint y s much s 25%. Gntes nd Lemonis [135] developed n FE model for olted T-stu steel connections. teril nd geometric nonlinerities s well s contct nd friction were implemented, which ws vlidted y comprison with experimentl dt. The impct of olt length considered in the model ws investigted nd shown to e of primry importnce. Ju et l. [136] developed 3D elsto-plstic FE model to study the structurl ehviour of utt-type steel olted joints. The results showed tht the nominl cpcities of the olted connection clculted from the AISC specifiction nd using FE were similr. ggi et l. [137] crried out prmetric nlyses on the ehviour of olted extended end-plte connections using 3D FE models clirted to experimentl results. The models took into ccount: mteril nonlinerities, geometricl discontinuities, lrge displcements nd contct to ccount for geometric discontinuities. Comprisons etween numericl nd experimentl dt for the moment rottion curves, displcements of the end-plte, nd forces on olts showed stisfctory greement. Xio nd Pernetti [138] proposed severl models using shell FE sed on [130], where shell elements were shown to give equivlent results to solid 3D elements ut t frction of the solution time. Slip etween end-plte nd olt hed ws neglected. Contct elements were introduced etween the end-plte nd the column flnge to model the movement of end-plte wy from the column flnge. Tgw nd Gurel [139] used FE simultions to exmine the strength of steel em-to-column joints stiffened with olted chnnels. 3D eight-node structurl solid elements were used to model ll components of the joint, with pretensioned olts. Aolmli et l. [120] developed 3D FE model for flush endplte connections using 8-noded solid isoprmetric elements for the em, column, end-plte nd olts. Geometric nd mteril nonlinerity, contct nd pretension in the olts were considered. oreno [140] developed 3D FE model of flush nd extended end-plte olted connections. The model included the em, endplte, olts ends nd the column. Considering the interction etween: end-plte nd the column flnge; olts (hed nd nut) nd the column flnge; nd olts nd the end-plte. The olt shnks were modelled using truss elements. The nlysis incorported mteril nonlinerity for the pltes nd olts. The FE results were compred with numericlly predicted moment rottion curves, which corresponded to the experimentl tests crried out nd with the component method [1]. Crero [141] developed two extended end-plte connections models, following the FE recommendtions of Bursi nd Jspr [129]. One of the models used 8-node rick elements with incomptile modes, wheres the other model used truss elements for the olts nd shell elements for the end-plte, em nd column. Different strtegies were used for modelling contcts in the second model, such s gp elements. Both models produced good results, with slight underestimtion of the rottionl stiffness nd slight overestimtion moment resistnce when compred with experimentl results. Pirmoz et l. [142] studied the ehviour of olted top set ngle connections with we ngles sujected to comined sher force nd moment. Severl 3D prmetric FE models were used with geometric nd mechnicl properties used s prmeters. With ll of the connection components, such s em, column, ngles nd olts re modelled using solid elements. The contcts etween surfces were simulted y surfce-to-surfce contct elements. The results were compred with experimentl results with good greement. ohmdi-shooreh nd ofid [143] presented the results of severl prmetric nlyses on the initil rottionl stiffness of olted flush end-plte em splice connections using FE with 20-noded rick elements, mteril ehviour, geometricl discontinuities nd lrge displcements. The model ws verified for three cse studies from the literture with the predicted results compring well with reported dt. Lemonis nd Gntes [118] proposed methodology to estimte the moment rottion curve of structurl em-to-column joints sed on the component method. The cses exmined in this work included olted connections with end-pltes nd with ngles. The methodology ws found to e very stisfctory compred with experimentl tests nd dvnced FE models in terms of stiffness, strength nd rottionl cpcity. Di et l. [144] mde simultion study of 10 fire tests on restrined steel em column ssemlies using five different types of joints. Three-dimensionl solid elements were used in modelling the min structurl memers. The results demonstrted good greement etween numericl simultions nd experimentl oservtions. Díz [145] developed detiled 3D FE model to study the ehviour of em-to-column olted extended end-plte joints (Fig. 14). The em, column, extended end-plte, olts (hed, nut nd shnk) were ll modelled using 8-node rick elements with full integrtion nd incomptile modes. Contct elements were on ll contct surfces of the joint. The otined results were in good greement with the rel ehviour of joints, found in experimentl rests in the literture. The model ws used to develop metmodel for use into the design nd optimiztion of semi-rigid connections Informtionl models Informtionl models using Neurl Networks (NN), cn provide n lterntive to conventionl methods of determining the moment rottion curve y providing n inside reltionship in

11 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) Fig D FE models of olted connections with ngles [134]. Fig D FE of em-to-column extended end-plte joint [145]. the form of generliztions etween the prmeters involved. Therey otining more pproximte moment rottion curve y extrcting informtion directly from the experimentl results. Artificil NN (ANN) is n rtificil intelligence ppliction implemented y engineers to crry out design tsks. It hs een pplied to prolems of: predicting function pproximtion; clssifiction; filtering; structurl nlysis, design, dynmics nd control nd structurl dmge ssessment [146]. Informtionl NN formultions re eqution-free glol representtion. The purpose of curve fitting is to find the prmeters for mthemticl eqution, wheres NN modelling is to lern the ckground mechnics. Once this lerning is done, the neurl network cn e implemented into other structurl nlysis pltforms without further simplifiction nd clirtion chllenges [147]. Jdid nd Firirn [148] investigted the reltionship etween the ehviour of em column joints nd the geometricl shpe, mount nd size of steel reinforcement, fixed em nd column cross-sectionl dimensions nd concrete strength using ANN. Anderson et l. [149] used NN to predict the iliner pproximtion of the moment rottion curves of minor xis em-to-column flush end-plte joints; Stvroulkis et l. [150] to predict the glol moment rottion curve for single we ngle em-to-column joints. Dng nd Tn [151] proposed n inner product-sed hysteretic model for the ppliction to piezocermic ctutors; Yun et l. [152] s model for hysteretic ehviour of mterils; Yun et l. [153] s hysteretic mteril model to expedite lerning of the cyclic ehviour of connections. De Lim et l. [154] used NN to predict the flexurl resistnce nd initil stiffness of em-to-column steel joints, the results of which were consistent with experimentl nd design code reference vlues; Guzeley et l. [155] to estimte the rottion cpcity of wide flnge ems. The dtse used to trin the NN ws sed on 81 experimentl results from the literture. Pirmoz nd Golizdeh [156] nd Sljegheh et l. [157] used NN to estimte the ehviour of olted top set ngle connections with we ngles nd Kim et l. [147] to model the nonliner hysteretic cycle for olted em-to-column ngle joints in steel frmes. Another methodology to predict the moment rottion curve is Genetic Progrmming (GP). Cevik [158] ws the first to investigte the use of GP to determine the rottion cpcity of wide flnge ems. 4. themticl representtion of moment rottion curve In order to consider the ehviour of joint in the glol nlysis of structure, it is necessry to consider the mthemticl representtion of the moment rottion curve. This representtion cn e performed y mens of different reltionships nd levels of precision. Fig. 15 shows the different mthemticl representtions of the moment rottion curve: liner; () iliner; (c) multiliner (triliner); (d) nonliner.

12 752 C. Díz et l. / Journl of Constructionl Steel Reserch 67 (2011) c d Fig. 15. Different mthemticl representtions of the ( j φ) curve: () liner; () iliner; (c) multiliner (triliner); (d) nonliner. The moment rottion curve cn e represented mthemticlly in one of two wys [54]: (1) depending on prmeters with cler physicl mening (e.g. stiffness, resistnce) nd shpe fctor; nd (2) sed on no cler physicl mening s it is derived from regression nlysis, clled curve-fitting formultions. For full review of this topic, the reder is referred to Fell et l. [54] nd Eurocode 3 [1] Stiffness, resistnce nd shpe fctor-sed formultions The mthemticl representtion of the moment rottion curve depends on prmeters with physicl mening, such s the rottionl stiffness (K ), moment resistnce () nd shpe fctor n which chrcterizes the knee of the moment rottion [54] Liner model The liner model, Eq. (15), is the simplest to use ut it is the lest ccurte. It overestimtes the rigidity of the joint [159] nd is only dependent on the rottionl stiffness (K φ ) of the joint. Btho et l. [3,5], Rthun [7], onforton nd Wu [13], mongst others, used this model. j = K φ φ. (15) Biliner model This model depends on three prmeters, the: rottionl stiffness (K φ ); plstic moment ( j,p ); nd plstic rottionl stiffness (K φ,p ) of the joint, Eq. (16). Used y mny [17, ] nd implemented in FEA progrms it hs shrp chnge in rigidity nd the intersection of the two curves (Fig. 15()). Kφ φ for j = j j,p (16) K φ,p φ for j > j,p ultiliner model This model ws proposed to remedy the prolem of the iliner model. oncrz nd Gerstle [20] use triliner representtion with five prmeters, Eq. (17), the: rottionl stiffness (K φ ); first yielding moment ( j,y ); post-yielding rottionl stiffness (K φ,y ); plstic moment ( j,p ); nd plstic rottionl stiffness (K φ,p ) of the joint. Kφ φ for j j,y j = K φ,y φ for j,y < j < j,p (17) K φ,p φ for j,p j. The representtion proposed in Eurocode 3 [1] is divided into three segments (Fig. 16), lthough for elstic plstic nlysis, simplified iliner model is proposed. The first segment of the curve hs the liner ehviour of Eq. (15) up to the moment vlue of 2/3 j,rd, where j,rd is the design vlue of the joint plstic moment j,p. The second segment is nonliner ccording to Eq. (18) in the rnge of 2/3 j,rd < j < j,rd. j = K φ 1.5 j j,rd ξ φ (18) where ξ depends on the [1]: 2.7 welded, olted end-plte nd se-plte connections ξ = 3.1 olted ngle flnge clets. The lst segment is stright horizontl line representing plstic ehviour ( j = j,rd ). Other multiliner models cn e cn e found in the work of [105,115, ] Nonliner model This is the most ccurte model so fr. Proposed in 1943 y Rmerg nd Osgood [168], Eq. (19), depends on three prmeters: