PLASTIC DESIGN AND SEISMIC RESPONSE OF KNEE BRACED FRAMES

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1 Advanced Steel Constructon Vol. 5, No. 3, pp (2009) 343 PLASIC DESIGN AND SEISMIC RESPONSE OF KNEE BRACED FRAMES Mara Antonetta Cont 1, Lug Mastrandrea 2 and Vncenzo Pluso 3,* 1 Department of Cvl Engneerng, Unversty of Salerno, va Ponte Don Melllo, Fscano (SA), Italy 2 Department of Cvl Engneerng, Unversty of Salerno, va Ponte Don Melllo, Fscano (SA), Italy 3 Full Professor, Department of Cvl Engneerng, Unversty of Salerno, va Ponte Don Melllo, Fscano (SA), Italy *(Correspondng author: E-mal: v.pluso@unsa.t) Receved: 15 July 2008; Accepted: 20 July 2008 ABSRAC: In ths paper a desgn methodology amng at the development of a collapse mechansm of global type for sesmc resstant knee braced frames s presented. he proposed methodology s based on the assumpton that the beam, brace and knee sectons are known, whle the column sectons consttute the unknowns of the desgn problem. he desgn requrements are derved by means of the knematc theorem of plastc collapse. In partcular, column sectons are obtaned by mposng that the mechansm equlbrum curve correspondng to the global mechansm has to le below those correspondng to all the undesred mechansms wthn a dsplacement range compatble wth the local ductlty supply of knee elements. he proposed desgn procedure has been mplemented n a computer program and appled to desgn some knee braced frames. Successvely, statc and dynamc non-lnear analyses have been carred out amng at the evaluaton of the performance of the desgned braced frames n terms of collapse mechansm developed under sesmc forces, energy dsspaton capacty and global and local ductlty demands. Keywords: Knee braced frames; collapse mechansm; capacty desgn; lmt desgn; non-lnear analyses; equvalent moment; global ductlty; local ductlty 1. INRODUCION Knee braced frames (KBFs) consttute an nnovatve bracng soluton. hey are obtaned by stffenng moment resstng frames by means of dagonal braces whch are not connected to beam-to-column jonts, but are restraned by means of short elements, namely knee, whch span, at each storey, between the column and the beam (Fgure 1). he brace allows to lmt nterstorey drfts, whle the knee element dsspates the earthquake nput energy by means of transverse cyclc deformatons n shear and/or bendng. From ths pont of vew, KBFs combne a lateral stffness smlar to that of concentrcally braced frames (CBFs) wth a ductle behavour smlar to that of eccentrcally braced frames (EBFs). he man advantage wth respect to EBFs s that damage s concentrated n a secondary member whch can be easly replaced after destructve earthquakes. he purpose of ths study s to llustrate a desgn methodology able to guarantee a collapse mechansm of global type for KBFs. As unversally recognsed, the global mechansm s the man goal of the desgn process of a sesmc resstant structure. In fact, among all the possble mechansms, t allows the maxmum global ductlty and energy dsspaton capacty. Even though modern codes provde the desgner wth the awareness of the mportance of the collapse mechansm typology, the suggested desgn rules are only able to avod soft storey mechansms, wthout developng the global one. hs s the case of member herarchy crtera, proposed for dfferent structure typologes, whch are not able to grasp the dstrbuton of nternal actons to be developed to assure a collapse mechansm of global type. he Hong Kong Insttute of Steel Constructon

2 344 Plastc Desgn and Sesmc Response of Knee Braced Frames In addton, all code member dmensonng rules are focused on the capacty desgn prncples, whle local ductlty supples are assured by means of technologcal suggestons, such as the use of web stffeners n the case of members subjected predomnantly to shear acton (lke lnks n EBFs). Conversely, there are no explct rules to lmt local ductlty demands,.e. to avod the premature fracture of the dsspatve zones. he proposed desgn methodology s amed at satsfyng both these desgn requrements. he targets are represented on one hand by the development of a global mechansm for KBFs, leadng to a satsfactory global ductlty, and on the other hand by the local ductlty control. hs s obtaned by means of the lmt analyss, whch allows to defne approprate desgn condtons and contemporary provdes useful tools to account for local ductlty demands. hs methodology has already been developed n the case of moment resstng frames (Mazzolan and Pluso [1]) and eccentrcally braced (Mastrandrea et al. [2]), provdng satsfactory results, so that t s proposed as a general desgn approach for sesmc resstant structures. Wth reference to KBFs, n Fgure 1 the possble collapse mechansms for KBFs are depcted; among these the global one s to be acheved because t maxmzes the energy dsspaton capacty and the global ductlty supply. hese mechansms, nvolvng the knee elements as dsspatve zones, are named type a mechansms. Wth reference to the portal KB-frame, other mechansm typologes, namely type b and type c, are depcted n Fgure 2 (Cont [3]). her pattern of yeldng s charactersed by the nvolvement of undesred non-dsspatve zones, such as columns and beams, so that they are responsble of reduced energy dsspaton capacty when compared to type a mechansms. he am of the proposed procedure s to provde a global mechansm of a type for mult-storey KB-frames. F ns F ns F n s F ns F k F k F k F k m m m h ns F 2 F2 F 2 F2 h m F 1 F 1 F 1 F 1 h 2 h 1 GLOBAL MECHANISM YPE 1 MECHANISM YPE 2 MECHANISM YPE 3 MECHANISM Fgure 1. Collapse Mechansms for a KB-Frame ype a Mechansm

3 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso 345 kb2 kt2 kt1 ' kt2 b2 kb1 kt1 b1 ' br type b type c Fgure 2. Collapse Mechansms for a KB-Frame ype b and ype c Mechansms In the proposed desgn methodology, t s assumed that the beam, brace and knee sectons are known, whle the column sectons consttute the unknowns of the desgn problem. In partcular accordng to capacty desgn, knee elements, whch represent dsspatve members, are szed consderng the nternal actons due to the code specfed sesmc horzontal forces, whle the brace sectons are desgned to prevent bucklng under the maxmum nternal actons whch the fully yelded and stran-hardened knee elements are able to transmt. In addton, the beam sectons are desgned to resst vertcal loads. herefore, the unknowns of the desgn problem are consttuted by the column sectons whch have to be defned so that the mechansm equlbrum curve correspondng to the global mechansm has to le below those correspondng to all the undesred mechansms wthn a dsplacement range compatble wth the local ductlty supply. It means that, accordng to the upper bound theorem, the global mechansm s the only mechansm whch can be developed wthn a dsplacement range compatble wth the local ductlty supples (Mazzolan and Pluso [4]). In ths paper, the procedure has been developed wth reference to one bay KB-frames where the brace connectons can be ether contnuous or pnned. For dfferent KBF layouts the same procedure can be appled by properly accountng for the actual geometrcal confguraton. 2. BEHAVIOUR OF KNEE ELEMENS he study of the plastc behavour of knee elements shows several analoges wth lnks n EBFs. In fact, both knee elements and lnks are subjected to transverse cyclc deformatons, so that they are charactersed by relevant shear actons (short elements). As a consequence, smlarly to the case of lnks, the ultmate behavour of knee elements s nfluenced by ther length. In fact, the nteracton between shear force and bendng moment becomes more and more relevant as the member length ncreases, up to a value for whch shear acton becomes almost neglgble (long elements). In partcular, snce the dagonal subdvdes the knee element nto two dfferent parts, the whole dsspatve member can be compared wth two lnks, each one of length equal to e/2, beng the brace-to-knee jont generally located at mdspan. Startng from ths consderaton, the well known lnk classfcaton (Kasa and Popov [5]) can be adopted also for knee elements, so that t s possble to recognse three knee typologes: short knees, ntermedate knees and long knees (Cont et al. [6]). In partcular, the plastc nteracton doman provded by Neal [7] s assumed: M M f M p M f 2 V V p 2 1 for M f M M p (1)

4 346 Plastc Desgn and Sesmc Response of Knee Braced Frames V V p for M M f (2) beng M p and V p the plastc moment resstance and the plastc shear resstance of the cross secton, and M f the contrbuton of the flanges to the plastc moment of the secton. he nterpretaton of testng results s usually based on the evaluaton of the plastc shear resstance as: V p = 0.6 f y (d 2 t f ) t w (3) where f y s the yeld stress, d the knee secton depth, t f and t w are the thcknesses of the flange and of the web, respectvely, accordng to the Amercan practce. In addton, stran-hardenng effects are accounted for on the bass of the expermental tests provded by Clement [8], whch reveal an ncrease of the plastc resstance equal to 70% n average. hs result s qute smlar to the case of lnks n EBFs, for whch, on the bass of several expermental tests performed worldwde (Popov and Engelhardt [9]; Ozak et al. [10]), a 50% ncrease of the resstance s recognsed, so that an overstrength factor of 1.5 s usually adopted n desgn practce (CEN [11]). herefore, wthn capacty desgn methodologes, ths overstrength has to be properly accounted for n predctng the nternal actons transferred by the lnk to the non-dsspatve zones of the structure, amng to ther dmensonng so that yeldng cannot occur before lnks develop ther ultmate resstance. Followng these ssues, on the bass of the smlarty between knee elements and lnks, an overstrength factor has to be adopted also n the case of KBFs. In fact, the dmensonng of non-dsspatve zones cannot neglect ths contrbuton, due to the consequent amplfcaton of the nternal actons transferred by knee elements. Wth ths am, an amplfcaton factor of 1.7 can be adopted for KBFs, as suggested by Clement [8]. As a consequence, n ths paper the ultmate nteracton doman s obtaned by means of an homothetc expanson of the plastc one usng the amplfcaton factor 1.7 (V u = 1.7V p, M u = 1.7M p, M fu = 1.7M f ). In order to justfy ths knd of approach, t s useful to remark that, due to the dscussed smlarty, the well known ultmate condtons for lnks (Engelhardt and Popov [12]) can be extended also to knee elements, leadng to the followng relatonshps: Vu ψv p ψ M u e 2V 2 p for M p e (4) V p M V u u ψ M M u 2 e 2 p for e M V p p (5) whle a lnear nterpolaton, dependng on the e/2 value, s necessary n the case of ntermedate knee elements. he coeffcent s the overstrength factor related to the stran-hardenng, whch assumes the value 1.5 n the case of lnks of EBFs and the value 1.7 n the case of knee elements as prevously dscussed. In Fgure 3 a comparson wth the proposed homothetc formulaton s depcted, showng a good agreement. hs knd of approach has been already used and valdated n the case of lnks of EBFs (Mastrandrea et al. [13]).

5 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso 347 In partcular, the KB-frames analysed n ths paper are charactersed by short knees,.e. knees for whch e/2 1.6M p /V p (Kasa and Popov [5]). he chosen scheme s charactersed by the followng propertes: the dagonal axs ntersects the beam-column jont; dagonal s connected to the knee mdspan secton, so that the brace and the knee have the same nclnaton and, generally, they are not perpendcular. Wth reference to the collapse mechansm of a portal frame (Fgure 4), the followng relatonshps between the plastc rotaton of each member and the plastc rotaton of the column can be obtaned: H e sen br (6) 2H esen kb 1 H e sen kb 2 2H H e sen e sen 2H e sen (7) kt 1 H e sen kt 2 H e sen 2H e sen 2 2 (8) b L L d (9) Fgure 3. Plastc V-M Interacton Dagram and Comparson between Ultmate Interacton Domans. Regardng the poston d of the plastc hnge developed wthn the beam span (Fgure 4), due to the superposton of vertcal loads and horzontal forces, the maxmum bendng moment s attaned at the abscssa (Mazzolan and Pluso [1]): d ecos for q q lm 4M b (10) L e cos 2 M b d L 2 for q qlm (11) q beng M b the plastc moment of the beam.

6 348 Plastc Desgn and Sesmc Response of Knee Braced Frames q F kt2 d x b kb2 kt1 b kb1 e br H L Fgure 4. Collapse Mechansms for a KB-Frame ype a Mechansm Furthermore, wth reference to the collapse mechansm of a mult-storey KB-frame (Fgure 5), t s useful to pont out that the plastc rotaton of braces provded by Eq. 6 s vald only for the frst storey ( br.1 ). In fact, t s easy to recognze that the plastc rotatons of dagonals of second and upper storeys are affected by the rgd rotaton of the column. he followng relatonshps are obtaned: br h. 2h e sen = 2, 3,, n s (12) beng h the nterstorey heght, e the length of the knee and the nclnaton of the brace of the th storey, respectvely. br.4 br.3 br.2 br.1 Fgure 5. Global Collapse Mechansms for a KB-Frame Eqs. 7-8 show that plastc rotatons of the two knees (bottom and top) are dfferent. Furthermore, the rotaton values are also dfferent at the two ends wthn a sngle knee. In the case of domnant shear behavour (short knee,.e. e/2 1.6 M p /V p smlarly to Kasa and Popov [5]) the prevously dscussed rotatons represent angular dstortons under the plastc shear acton V p. Nevertheless, a member subjected to unform shear acton s theoretcally charactersed by a unform angular dstorton along ts length. herefore, the crcumstance for whch dfferent values of the rotaton arse at the ends of the member leads to the concluson that the deformaton s partally due to the

7 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso 349 shear acton and partally due to the bendng moment. In other words, even a short knee s subjected to moment-shear nteracton, whch s yet neglected n Eq. 2. For ths reason, n the case of short knees an average value of the plastc deformaton for each knee porton can be approxmately adopted: kb1 kb2 H 4H 3 e sen kb 2 e sen 4H 2 e sen (13) kt kt1 2 kt2 4 H H e sen e sen 4H 2 e sen (14) and t can be assumed that the ultmate behavour of the member s governed only by the shear force. It s useful to note that, startng from Eqs , the followng value s obtaned for the mean value of the plastc deformaton of the entre knee element: kb kt H k (15) 2 e sen Eq. 15 s partcularly useful, because t provdes, at least wthn the rgd-plastc analyss, a measure of the plastc engagement of the knee element as a functon of the lateral dsplacement,.e. the column rotaton, of the entre structure. Conversely, n the case of an ntermedate knee, moment-shear nteracton becomes sgnfcant so that plastc deformatons provded by Eqs. 7-8 actually represent a combnaton of plastc shear deformaton and plastc moment rotaton. An analytcal procedure to compute the ultmate resstances and the plastc deformatons of ntermedate knee elements, accountng for the nteracton between bendng moment and shear force, has been developed by the Authors wthn the framework of plastc desgn (Cont et al. [6]). However, n ths paper reference s made to the case of short knees. As a concluson, wth reference to the proposed KBF layout, Eqs. 6, 9, 12, 13 and 14 consttute useful tools for lmt desgn applcatons, and they are appled wthn the desgn procedure descrbed n the followng Sectons. 3. HE PROPOSED DESIGN MEHODOLOGY he am of the proposed desgn methodology s to allow the dmensonng of KB-frames developng at collapse a mechansm of the global type. herefore, the desgn requrements are derved to avod the partal mechansms depcted n Fgure 1 (type 1, type 2 and type 3 mechansms) and the less dsspatve mechansm typologes depcted n Fgure 2. o ths scope, the knee elements are recognsed as dsspatve zones, so that the desgn procedure s amed to assure the engagement of all of them n the collapse mechansm. On the contrary, all other members have to reman n elastc range, except for the frst order columns and dagonals (f not pnned at the ground) whch partcpate to the knematc mechansm. he procedure s based on the assumpton that knee, beam and brace cross sectons are known, snce they are desgned on the bass of the code specfed sesmc actons as n the followng clarfed, whle sectons of the columns consttute the unknowns of the desgn problem. he desgn condtons are defned by mposng that the mechansm equlbrum curve correspondng to the

8 350 Plastc Desgn and Sesmc Response of Knee Braced Frames global mechansm has to lay below those correspondng to all the undesred mechansms wthn a dsplacement range compatble wth the local ductlty supply. Accordng to the upper bound theorem, these condtons guarantee that the true collapse mechansm of the structure s represented by the global falure mode. 3.1 Desgn of Dsspatve Zones, Beams and Braces Accordng to capacty desgn, knee sectons are desgned on the bass of nternal actons arsng from the sesmc forces prescrbed by the codes. In partcular, knee elements are dmensoned wth reference to a smplfed scheme assumng that the axal nternal forces of braces of KBF are approxmately equal to those of braces of a concentrcally braced frame wth the same geometry. herefore, the axal force of the brace of the generc storey s evaluated as: N d. Q cos (16) where Q s the sesmc shear force of the th storey and s the brace nclnaton. he desgn value of shear force of the knee porton s obtaned by means of the approxmate translatonal equlbrum equaton of the knee-to-brace jont n the drecton orthogonal to the knee: V N d. knee. Sd. sen2 (17) 2 hs value underestmates the actual shear forces of the knee elements. herefore, t s suggested the applcaton to the value provded by Eq. 17 of an amplfcaton factor equal to Smlarly, the desgn axal force of the brace s obtaned by maxmsng Eq. 17. In fact, beng non-dsspatve zones, braces have to be dmensoned on the bass of the maxmum nternal actons that dsspatve zones are able to transmt, whch means that the desgn axal force of each brace s obtaned by assumng that the correspondng knee s yelded and stran-hardened up to ts ultmate resstance, so that t s equal to: 2 Vknee. u. N d. Sd. (18) sen2 where V knee.u. = 1.7 V knee.p. s the ultmate shear force of the knee porton accountng for stran-hardenng (Clement [8]). he brace sectons are desgned to prevent bucklng under the axal force gven by Eq. 18. Fnally, the beam sectons are desgned n order to resst to vertcal loads. 3.2 Equlbrum Curves of Analysed Mechansms Where not specfed, symbols adopted n the followng formulas are referred to the notaton reported n Appendx. Wth reference to Fgure 1, t s possble to recognse that, for a vrtual rotaton value d of columns and dagonals nvolved by the mechansm, the total nternal vrtual work s provded by the followng relatonshp:

9 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso 351 W tr( C R C) Br Rd 2B Rb K knr d kn (19) where the symbol tr denotes the trace of the matrx. he external work, due to vertcal and horzontal loads, s expressed by the followng equaton: W e F s q D v d where s the knematcally admssble multpler. By means of the vrtual work prncple, the knematcally admssble multpler of horzontal forces, for the mechansms depcted n Fgure 1, s gven by: (20) tr( C R C ) B r R d 2B F s R b K kn R kn q D v (21) he second-order work due to vertcal loads s provded by the followng relatonshp (Mazzolan & Pluso, 1997): W v V s H 0 d (22) therefore, the slope of mechansm equlbrum curve s gven by: 1 H 0 V F s s (23) By means of Eqs. 21 and 23, the mechansm equlbrum curve s expressed by: (24) c = he followng notaton wll be used to denote the parameters of the equlbrum curve of the consdered mechansms: (g) and (g) are, respectvely, the knematcally admssble multpler of horzontal forces and the slope of the softenng branch of the - curve, correspondng to the global type mechansm; m (t) and m (t) have the same meanng of the prevous symbols, but they are referred to the m th mechansm of tth type. In the case of global type mechansm (Fgure 1), as all the storeys partcpate to the collapse mechansm, the shape vector of the horzontal dsplacements s gven by s (g) = h. In addton, all knee elements are nvolved so that the knematcally admssble multpler s gven by: (g) M c1 I B r R (g) d 2B (g) R b (g) F s K n R (g) kn q D (g) v (25) where I denotes the unty vector of order n c.

10 352 Plastc Desgn and Sesmc Response of Knee Braced Frames Furthermore, because all the storeys partcpate to the global mechansm, H 0 s equal to h ns, and the slope (g) s obtaned from Eq. 23 for s = s (g) and H 0 = h ns. Wth reference to the m th mechansm of type 1, the shape vector of the horzontal dsplacements can be wrtten as: (1) s m h 1, 2 3 h, h,..., h m, h m, h m where the frst element equal to h m corresponds to the m th component. he knematcally admssble multpler correspondng to the m th mechansm of type 1 s gven by: (26) c1 r (1) d m cm 1I 2 (1) sm (1) b m n M I B R M B R K R q D (1) (27) m F In addton, only the frst m storeys partcpate to the collapse mechansm, so that H 0 = h m. As a consequence, the slope of the mechansm equlbrum curve s stll computed through Eq. 23, but assumng s = s m (1) and H 0 = h m. he expressons of the knematcally admssble multplers m (2) and m (3), and of the slopes m (2) and m (3), of the mechansm equlbrum curves correspondng to type 2 and type 3 mechansms, respectvely, can be obtaned by means of the same procedure. 3.3 Desgn Condtons for Falure Mode Control Accordng to the upper bound theorem of lmt desgn, the cross-sectons of columns have to be defned so that the knematcally admssble horzontal force multpler correspondng to the global type mechansm s the mnmum among all the knematcally admssble multplers. hs condton s suffcent to guarantee the desred mechansm only n the case of rgd-plastc behavour of the structural materal, so that no dsplacements are developed untl the collapse mechansm s reached. Conversely, the actual behavour s elastoplastc, and t s charactersed by sgnfcant dsplacements before the collapse mechansm s completely actvated. hese dsplacements are responsble of second order effects whch cannot be neglected n the desgn process. herefore, to account for second order effects, the desgn condtons have to be defned by mposng that the mechansm equlbrum curve correspondng to the global type mechansm has to le below those correspondng to all the other mechansms wthn a dsplacements range compatble wth the plastc deformaton capacty of members (Fgure 6). hs represents the extenson of the knematc theorem of plastc collapse to the mechansm equlbrum curve concept. Consequently, the desgn condtons are provded by the followng equatons: (1) kn m (1) v m ( g) ( g) u ( t) m ( t) m u (28) for m = 1, 2,, n s and t = 1, 2, 3.

11 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso 353 Fgure 6. Mechansm Equlbrum Curves It s useful to note that the dsplacement u has to be compatble wth the local ductlty supply of dsspatve zones and t assumes the meanng of desgn dsplacement, so that the proposed desgn methodology leads both to falure mode and local ductlty control. Substtutng the values of (g), (g), m (1) and m (1) n Eq. 28, the condtons to be satsfed to avod type 1 mechansms are obtaned. In partcular, the followng desgn condtons are obtaned ( m = 1, 2,, n s ): c1 r (g) d M I B R 2B (g) Rb (g) F s K n R (g) kn q D (g) v V s F s (g) (g) h u n s (1) Mc1I Br Rd Mc 1I 2B m m F s (1) m R (1) b m K n R (1) kn m q D (1) v m V F (1) sm (1) sm h u m (29) In the same way, the condtons to be satsfed n order to avod type 2 and type 3 mechansms can also be obtaned. he plastc secton modul of columns, whch consttute the unknowns of the desgn problem, can be determned satsfyng smultaneously these 3n s desgn condtons. Moreover, t s necessary to guarantee the development of type a mechansm, rather than type b or type c. hs s made by means of addtonal desgn rules, whch are stll based on the applcaton of the knematc theorem of plastc collapse. In partcular, by comparng the collapse confguratons depcted n Fgure 2 and Fgure 4, t s easy to recognse that the deformed shapes are smlar (.e. the same top-sway dsplacement s obtaned for each structure f the same column rotaton s consdered), so that the external vrtual work due to an horzontal force F s gven by: W e. F F F H (30) ndependently of the mechansm typology. herefore, by applyng the vrtual works prncple, the followng relatonshp s obtaned for the horzontal multpler (t = a, b, c): t t t t t W We. q We. F We. q F H (31) (32)

12 354 Plastc Desgn and Sesmc Response of Knee Braced Frames t t W W t e. q F H Accordng to the knematc theorem of plastc collapse, the requrements to be fulflled to avod undesred collapse mechansms are obtaned by mposng that the knematcally admssble horzontal force multpler correspondng to type a collapse mechansm s less than those correspondng to type b and type c collapse mechansms. herefore, the followng equatons have to be satsfed: a b and a c (33) leadng to the requred desgn crtera, respectvely: M c H e sen sen M H e b L d L d M d e sen 2H e sen (34) e sen H e sen Vu H H H e sen L d ql L d (35) M c M d M b H e sen 2H e sen 2sen e sen H e sen L d 2 whch are obtaned by means of the plastc rotatons depcted n Fgure 2 (Cont et al. [6]). Wth reference to a mult-storey KBF, n order to avod mechansms charactersed by the alternaton of type a, type b and type c confguratons at dfferent levels, t s suffcent to check that Eqs are satsfed for each storey of the structural system. herefore, takng nto account all the prevously dscussed condtons and requrements, handlng the prevous formulatons and pontng out attenton on technologcal condtons, smlarly to the case of moment-resstng frames (Mazzolan and Pluso [1]), t s possble to defne a desgn algorthm whch allows, by means of an teratve process, to obtan the desgn soluton n terms of column plastc moments (Cont et al. [14]). Due to the presence of the axal forces, these moments represent a reduced value f compared to the resstance provded n smple bendng. hen, the desgn column sectons are obtaned by couplng, for each column, the determned plastc moment wth the axal force occurrng n the collapse state. hs axal force can be estmated by means of smplfed equlbrum equatons. In partcular, the column axal force s provded by the vertcal translatonal equlbrum equaton of column-to-knee jont and s equal to: n N s c Vknee. u. j j. cos (36) j Amng at the evaluaton of ts accuracy, the proposed desgn procedure has been mplemented nto a computer program and appled to dmenson an adequate number of KBFs whose sesmc response has been nvestgated by means of non-lnear statc and dynamc analyses. In partcular, reference s made to the case of short knees, whch s the most frequent, due to the length of such members. In next Sectons the results of these analyses are presented and dscussed.

13 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso NON-LINEAR KNEE MODEL: HE EQUIVALEN MOMEN CONCEP In order to carry out non-lnear analyses for KBFs, the post-elastc behavour of each element has to be properly modelled. o ths scope, due to ther shear plastc nvolvement, the geometrcal nclnaton of the knees ntroduces some dffcultes n ther non-lnear modellng. In fact, plastc shear deformatons for each knee porton are the result of both horzontal and vertcal dsplacements occurrng at the member ends. Several computer programs, as for example DRAIN-2DX (whch has been used to carry out dynamc non-lnear analyses), do not provde the user wth an adequate predefned element n ther lbrary for modellng a non-lnear behavour n shear, partcularly n the case of nclned members. herefore, to easly smulate the actual non-lnear behavour of a knee element t s useful to defne an equvalent element wth plastc hnges n smple bendng at ts ends. hs s obtaned by mposng the equvalence between the nternal work developed by each actual knee porton and the nternal work correspondng to the smplfed theoretcal model. he nternal vrtual work developed, for example, by the bottom knee porton s equal to: L. kb e 2 0 V kb dz V p kb e 2 (37) where kb s provded by Eq. 13. he nternal vrtual work developed by the equvalent model n smple bendng s gven by: L M eq. p. kb 2 kb (38) beng M eq.p the so defned equvalent moment n plastc condtons. he equvalence between Eq. 37 and Eq. 38 provdes the followng expresson: M eq. p V p e 4 (39) whch s easly obtaned also n the case of the top knee porton. he same procedure can be extended to the ultmate condtons, when the stran-hardenng effect nfluences the behavour of the member. o ths scope, Eqs are appled consderng the shear resstance V u = 1.7V p, so that the ultmate equvalent moment s obtaned: e 4 1. M eq. u Vu 7M eq. p (40) he Eqs allow to deal wth a model n whch the shear behavour of knee elements s taken nto account by means of an equvalent element exhbtng flexural behavour, easer to be handled. However, a valdaton of the approach has been performed by means of a complete fnte element model. In partcular, a knee element has been studed by means of SRAUS7 computer program. he cross secton s represented by an HEA 200 standard profle. It has been subdvded n several areas, both for the flanges and for the web, whle the fllet of the web-to-flange connecton has been neglected. An elastc-perfectly plastc law has been assgned to each of them, by assumng a yeld stress equal to 275 MPa. he length of the member s equal to 100 cm, and the edge restrants have been modelled as rgd. In order to smulate the nteracton wth the brace element, a rgd body has been assgned to the mddle of the knee element wth a length equal to 20 cm representng the

14 356 Plastc Desgn and Sesmc Response of Knee Braced Frames nteracton zone due to the brace. Along ths part, a monotoncally ncreasng transversal load has been appled, so that the non-lnear force-dsplacement curve has been carred out. In Fgure 7 and Fgure 8 the FEM model and the dstrbuton of the Von Mses deal stresses at frst yeldng are depcted. It s useful to note that at frst yeldng an deal stress equal to f y s almost reached on the whole web panel (Fgure 8). Fgure 7. SRAUS7 FEM Model he descrbed complete FEM model has been compared wth the equvalent model n smple bendng prevously ntroduced. o ths scope, a scheme geometrcally equal to the prevous one has been analysed by means of SAP2000 computer program. In ths case, the knee element has been modelled by means of three beam-column elements, two representng the knee portons and one representng the rgd body n the mddle. he post-elastc behavour has been taken nto account by means of plastc hnges n smple bendng at the ends. he plastc hnges have been calbrated wth an elastc perfectly-plastc moment-rotaton curve, by means of the Eq. 39, n whch the plastc shear resstance calculated by means of the Eq. 3 s equal to kn. In ths way, the comparson between the two models s performed under the same consttutve law. In Fgure 9 a comparson between the force-dsplacement curves of the two models s depcted. It shows a satsfactory agreement, provdng the valdaton of the equvalent model n smple bendng. In fact, the scatters between the two curves are very small, both n elastc and plastc ranges. It s useful to note that the dfference n the transton zone between elastc and plastc behavour s due to the fact that the non-lnear analyss carred out by means of SAP2000 s based on a lumped plastcty approach, whle n the case of SRAUS7 the spreadng of plastcty s step-by-step developed as far as the dsplacement ncreases. Fgure 8. SRAUS7stress Map at the Frst Yeldng

15 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso 357 SRAUS7 model SAP2000 model F [kn] [mm] Fgure 9. Force-Dsplacement Curves of the Knee Models herefore, by means of Eqs , the post-elastc behavour of a knee element can be modelled by ntroducng two beam-column elements wth plastc hnges at ther ends. In partcular, an elastc-plastc wth stran-hardenng b-lnear moment-rotaton relatonshp can be assgned to each plastc hnge. he elastc range develops untl the condton ( y, M eq.p ) s reached, beng y the frst yeldng rotaton, whch s calculated as the rotaton correspondng to the attanment of the moment M eq.p at both ends of the member smultaneously. he second branch, accountng for stran-hardenng, termnates at the pont ( u, M eq.u ), where u s the ultmate monotonc rotaton, snce the monotonc behavour represents the envelope of the hysteretc cycles. Accordng to the smlarty wth lnks of EBFs, the value of 0.2 radans (Engelhardt and Popov [12]) s chosen for the ultmate monotonc deformaton of each knee porton. 5. SAIC AND DYNAMIC NON-LINEAR ANALYSES Non-lnear analyses have been performed n order to evaluate the sesmc response of the desgned structures, so that the accuracy of the proposed procedure can be tested. Statc and dynamc non-lnear analyses have been carred out wth dfferent purposes. Statc non lnear analyses provde useful nformatons concernng the energy dsspaton capacty of each structure and ts global ductlty. In addton, the pattern of yeldng obtaned from these analyses can be drectly compared wth the global falure mode representng the man goal of the proposed desgn methodology whch, even though based on rgd-plastc analyss, operates wthn the framework of statc approaches. As t s well recognsed that statc approaches do not account for hgher mode effects, dynamc non-lnear analyses are preferred to predct the actual pattern of yeldng for the structure,.e. the actual collapse mechansm, and the local ductlty demands. Desgned KBFs are charactersed by the same structural scheme (3 bays, wth the KBF n the mddle one; bay span L = 6 m; knee length e = 1 m; nterstorey heght h = 3 m, Fgure 10) and dfferent number of storeys (from 4 to 10). In partcular, wth reference to the consdered buldng layout, t s assumed that only KBFs resst to horzontal forces, whle all other beams and columns are only able to carry the vertcal loads, snce all the beam-to-column connectons and the column-bases outsde of the KB-frame are pnned.

16 358 Plastc Desgn and Sesmc Response of Knee Braced Frames Nevertheless, the proposed procedure could be extended also to the case of dual systems, n whch moment resstng frames are combned wth knee braced frames. o ths scope, t s suffcent to combne the desgn procedures developed for each structural typology (Mazzolan and Pluso [1]); ths means that the desgn equatons descrbed n the prevous Sectons have to be updated n order to account for the nternal works due to all the beams and the columns, external to the KBF, nvolved n each mechansm typology. In addton, the axal forces n the columns have to take nto account for the contrbuton of the framed bays, whch s sgnfcant due to the shear forces transmtted by the beams when plastc hnges are developed at ther ends. However, snce both moment resstng frames and knee braced frames are nvolved n sesmc forces absorpton, partcular care has to be taken n the determnaton of the respectve rates. In fact, as far as the rate of sesmc shear actng on the KB-frame ncreases the dmensonng of knee elements becomes more and more severe, leadng to a stronger braced structure. Consequently, the desgn of a dual system should be obtaned by means of a calbraton of the sesmc shear dstrbuton, n order to obtan an effectve sesmc response wth the mnmum structural weght. Researches n ths feld are currently n progress by the Authors wth reference to dual systems consttuted by the combnaton of moment resstng frames and dfferent brace typologes. Wth reference to Fgure10, the characterstc values of the vertcal loads actng on the floors of the analysed scheme are equal to 3kN/m 2 and 2kN/m 2 for dead and lve loads, respectvely. he structural materal adopted for all the members s S 275 steel grade (f y = 275MPa). he desgn horzontal forces have been determned accordng to Eurocode 8, assumng a peak ground acceleraton equal to 0.35g, a sesmc response factor equal to 2.5, a behavour factor q equal to 6, and an mportance factor I equal to 1.0 (CEN [11]). herefore, an horzontal force dstrbuton accordng to the frst vbraton mode s assumed for the desgn process. Nevertheless, ths assumpton does not consttute a restrcton of the method. In fact, the desgn approach could be carred out by means of an teratve process, n whch the trangular dstrbuton of horzontal forces represents only the frst tentatve dstrbuton for dmensonng the structure. Successvely, by means of a modal analyss, the vector F can be updated by means of a combnaton of the most relevant vbraton modes, and the desgn process can be appled agan. However, snce dynamc non-lnear analyses drectly account for the nfluence of all the vbraton modes, the correspondng results provde the actual response of the structure, and they consttute a measure of the accuracy of the proposed procedure even when t s led accordng to the frst egenmode lke n ths work. Fgure 10. Buldng Layout

17 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso 359 he desgn dsplacement d ( u = d ) has been determned as the value correspondng to the maxmum allowable engagement of the more nvolved knee element,.e. ts ultmate rotaton u. Wth reference to Eq. 15, ths leads to the followng relatonshp: ek sen k d hn s u hn h s (41) k hk 1 mn where u s assumed equal to 0.08 rad for short knee n cyclc loadng lke the case of short lnks of eccentrcally braced frames (Engelhardt and Popov [12]). For each examned KBF both the scheme wth pnned column-to-brace jonts and the scheme wth contnuous column-to-brace jonts have been consdered (Cont [3]). In ths paper, for sake of shortness, only the results concernng 8-storey KBF wth pnned brace-to-column jont are presented. In partcular, the applcaton of the desgn procedure descrbed n the prevous Sectons (wth u = 32 cm) leads to the standard HEB profles reported n able 1 for each member. Statc non-lnear analyses have been performed by means of SAP2000. he structure has been modelled by means of non-lnear elements wth possble plastc hnges located at the ends of each member. In partcular columns, dagonals and beams have been modelled usng hnges accountng for axal force-bendng moment nteracton whle the shear deformatons of each knee porton are accounted for by means of shear hnges. In fact, SAP2000 computer program s suppled wth a plastc hnge n pure shear, referred to the local axes of the member, so that the nclnaton of the knee element wth respect to the global reference system s not relevant, and the use of the prevously dscussed non-lnear model n equvalent bendng s not necessary. he shear versus plastc dsplacement dagram has been modelled through a rgd-hardenng behavour. he hardenng branch s developed between the values V p and V u = 1.7V p, whose correspondng ultmate dsplacement s gven by u e/4, as obtaned by assumng that each plastc hnge s related to an half knee element porton (2 plastc hnges both for bottom and top knee portons, 4 plastc hnges for all the member), so that t s related to the half part of the nternal work provded by Eq. 37. able 1. 8-Storey KBF Desgned Sectons storey Knee Brace Beam Column 1 HEB 300 HEB 300 HEB 160 HEB HEB 280 HEB 300 HEB 160 HEB HEB 280 HEB 300 HEB 160 HEB HEB 260 HEB 280 HEB 160 HEB HEB 240 HEB 280 HEB 160 HEB HEB 220 HEB 260 HEB 140 HEB HEB 180 HEB 240 HEB 140 HEB HEB 120 HEB 180 HEB 120 HEB 140 he push-over analyses have been led under dsplacement control accountng for both mechancal and geometrcal non-lneartes. Moreover, for each step of the analyss, out-of-plane stablty checks for compressed elements have been executed. In Fgure 11 the push-over curve and the dstrbutons of plastc hnges for a top-sway lateral dsplacement both equal and exceedng the desgn value are reported. hese fgures underlne that accordng to the goal of the proposed desgn procedure, yeldng occurs for all the knee elements at the desgn dsplacement, so that satsfactory energy dsspaton capacty and adequate nelastc performances are acheved. Furthermore, the dstrbuton of plastc hnges for dsplacement values exceedng the desgn one shows the development of a collapse mechansm of global type. Smlar results have been obtaned for all the examned KBFs.

18 360 Plastc Desgn and Sesmc Response of Knee Braced Frames In addton, ncremental dynamc non-lnear analyses (IDA) have been carred out for the desgned KBFs. Dynamc non-lnear analyses allow to predct the actual sesmc response of the examned structures n terms of actually developed collapse mechansm, energy dsspaton capacty and local ductlty demands. hese analyses have been carred out by means of DRAIN-2DX computer program consderng three dfferent sesmc motons: two smulated records matchng Eurocode 8 nelastc desgn response spectrum, wth A and C subsol class respectvely (CEN [11]), and an hstorcal one (olmezzo 1976, PGA = 0.313g). he results comng from these analyses have been post-processed to perform out of plane stablty check of the members. he structure has been modelled as an assembly of non-lnear elements charactersed through specfc hysteretc consttutve laws. In partcular columns, braces and beams have been modelled usng beam-column elements. For these members the bendng moment versus plastc rotaton dagram has been modelled through an elastc-perfectly plastc behavour. he axal force-bendng moment nteracton doman has been defned accordng to the plastc doman provded by Eurocode 3 (CEN, 2005). he modellng of knee elements has been carred out, as prevously dscussed, by means of equvalent plastc hnges n smple bendng. Amng at the evaluaton of the sesmc response of the desgned KBFs, for each ground moton, the plastc hnge dstrbuton, the maxmum absolute and relatve dsplacements and the maxmum plastc deformaton demands for each structural element have been evaluated. In partcular, n order to valdate the results provded by the statc non-lnear analyss, the results concernng 8-storeys KBF wth pnned column-to-brace jont are heren presented. In partcular, reference to the Eurocode 8 smulated ground moton for subsol class A (EC8-A) s made (cm) push-over curve = 28.8cm = 69.0cm Fgure Storeys KBF Push-Over Results

19 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso 361 In Fgure 12 the maxmum requred plastc deformatons are reported for ncreasng values of the peak ground acceleraton (PGA). In addton, Fgure 13 provdes the values of the plastc deformatons for all the members (n the fgure plastc rotatons have to be scaled for the factor 10-2 ). In Fgure 14 the maxmum absolute dsplacements for dfferent values of PGA are depcted n non-dmensonal form by means of the roof dsplacement angle (n symbols RDA = /H). In addton, n the same Fgure 14 the maxmum nter-storey drft ratos are shown (n symbols MIDR = max /h). Smlar results have been obtaned for all the analysed structures and for all the examned ground motons (showng that the most evdent dfferences n terms of sesmc response occur for EC8-C ground moton) and lead to the followng observatons (Cont [3]). Frst of all, the desgned structures are charactersed by the fulflment of the desgn goal,.e. the development of a collapse mechansm of global type. In fact, the dstrbuton of plastc hnges shows that all the dsspatve zones,.e. all the knee elements, are nvolved n the collapse mechansm (Fgure 13), so that large dsspaton capacty and global ductlty are acheved. Furthermore, for ncreasng values of PGA, yeldng occurs n the other members (beams and base columns) accordng to the predcton of the desgn methodology, and a global collapse mechansm develops. In addton, regardng local ductlty demands, the maxmum values of plastc rotatons have to be compared wth the cyclc local ductlty supply provded by short knees (.e. 0.08rad). Wth reference to Fgure 12, knee elements reach the lmt value of 0.08 rad for a PGA value equal to about 1.0g. Smlar results have been obtaned for the other analysed structures, where the ultmate knee rotaton s reached for hgh values of PGA (generally between 0.7g and 1.0g). hs result ponts out that the KBFs desgned accordng to the proposed desgn procedure are able to wthstand severe earthquake explotng ther large global ductlty wthout the fracture of the dsspatve zones, unless very hgh values of PGA are reached. hs s a relevant result, snce local ductlty demands for short members, such as knee elements or short lnks n EBFs, are adversely affected by the nfluence of the geometrcal confguraton knee A/g Fgure Storeys KBF IDA Maxmum Plastc Knee Deformatons (EC8-A) Fnally, t s useful to pont out some consderatons about the rregular behavour of the curves depcted both n Fgure 12 and n Fgure 14. In fact, t s possble to recognze a sgnfcant varaton of the curve slopes wthn the PGA range [0.35g, 0.55g]. Frst of all t s necessary to underlne that not all the analysed structures show the same knd of response, even f several of them are charactersed by non-monotonc behavours.

20 362 Plastc Desgn and Sesmc Response of Knee Braced Frames It s well known that IDA curves often provde counter-ntutve results, as thoroughly dscussed by Vamvatskos and Cornell [16]. In fact, segments of softenng or hardenng can be provded after the ntal stffness of the structure,.e. regons n whch the slope of the curve decreases wth hgher ntensty measures or other regons where such slope ncreases. hs means that the structure can experence an acceleraton responsble of damage accumulaton, but at the followng tme t s subjected to a deceleraton that can be powerful enough to momentarly stop the damage accumulaton or even reverse t. So the IDA curve s charactersed by lower damage measures and t becomes a non-monotonc functon of the PGA. In partcular, the examned case s analogous to the case of severe hardenng (Vamvatskos and Cornell [16]). In fact, the system shows an hgh response for a gven ntensty level, but exhbts the same or lower response for hgher sesmc ntenstes due to excessve hardenng. Wthn ths ssue, t s the pattern and tmng of the accelerogram rather than just the ntensty that make the dfference. As the earthquake s scaled up, weak response cycles n the early part of the response tme-hstory become amplfed enough to generate yeldng, so that the structure propertes result modfed when the stronger cycles are reached. he extreme case of hardenng leads to the case n whch a structure reaches the global collapse for an ntensty measure but not for an hgher one, reappearng wth hgh response but stll n an equlbrum state. hs ssue s known as structural resurrecton. 6. CONCLUSIONS hs paper has been focused on several ssues concernng the desgn of knee braced frames amng at the attanment of satsfactory sesmc performances wth the prmary goal of developng a collapse mechansm of the global type. Frst, the behavour of knee dsspatve zones located n knee elements has been brefly dscussed, pontng out the relevant aspects concernng ther plastc nvolvement. hese results are partcularly useful n the development of a desgn procedure whch s the natural extenson to KB-structures of a desgn methodology already suggested for sesmc-resstant frames (Mazzolan and Pluso [1]) and for eccentrcally braced frames (Mastrandrea et al. [2]). herefore, ths methodology consttutes a general desgn approach whch allows the control of the falure mode for any type of sesmc-resstant structure. In partcular, the procedure starts from the knowledge of the dsspatve zones (.e. knee elements), whch are dmensoned n order to resst the nternal actons due to the desgn horzontal loads. hen, accordng to capacty desgn, by means of smplfed equlbrum equatons, the brace cross sectons are dmensoned on the bass of the maxmum nternal actons that the knee elements, at the ultmate lmt state, are able to transmt. In addton, the beam cross sectons are dmensoned on the bass of the vertcal loads actng on the floor. Fnally, the column cross sectons are obtaned by applyng the knematc theorem of plastc collapse (takng nto account second order effects due to vertcal loads), whch allows to derve the desgn condtons to be satsfed to guarantee a global collapse mechansm for the structure. In addton, n ths procedure, a target desgn dsplacement of the structure s selected accordng to the maxmum allowable plastc rotaton for the dsspatve zones, so that also the local ductlty demands are under control. Push-over analyses carred out for an adequate number of desgned KBFs have ponted out the goodness of the proposed desgn procedure. In fact, for all the desgned KBFs all the dsspatve zones are nvolved n the collapse mechansm, leadng to large global ductlty.

21 Mara Antonetta Cont, Lug Mastrandrea and Vncenzo Pluso 363 Furthermore, the actual sesmc response of the desgned KBFs has been studed by means of IDA. he results stll show the accuracy of the desgn procedure, n fact all the desgned structures actually exhbt a global mechansm and are charactersed by hgh global ductlty and energy dsspaton capacty. In addton, the local ductlty demands are compatble wth the correspondng supples, even for hgh PGA values rangng between 0.7g and 1.0g. hs means that the proposed procedure s able to control both the collapse mechansm typology and the local ductlty demands E E PGA = 0.25g PGA = 0.35g PGA = 0.55g plastc rotatons have to be scaled for the factor PGA = 0.70g 6.4E-3 PGA = 0.85g 0.12 PGA = 1.00g 0.06 Fgure Storeys KBF IDA Plastc Hnges Envelope (EC8-A)