DYNAMIC ANALYSIS OF SEMI-RIGID FRAMES

Transcription

1 Matematal and Computatonal pplatons, Vol., No., pp. -8, 5. ssoaton for Sentf esear DYNMIC NYSIS OF SMI-IGID FMS l Ugur Ozturk and Hkmet H. Catal Department of Cvl ngneerng, Dokuz ylul Unversty, 5, Izmr, Turkey strat-te dynam response of sem-rgd frames s studed y usng a omputer program. Te onneton flelty s modeled y lnear elast rotatonal sprngs. Havng te same geometry and ross-seton; sem-rgd frames, wt dfferent sprng oeffents, are eamned. Te redung oeffents and lateral rgdty values, representng te real eavor of frames, are determned for ea frame. To represent te real eavor, all deformatons of a frame are aounted for a dynam analyss. esponse aratersts of fve dfferent multstory frames are ompared wt referene to ter modal attrutes. Te study ndates tat onneton flelty tends to nrease vraton perods, espeally n lower modes, wle t auses vraton frequenes derease. Keywords- sem-rgd, redung oeffent, rotatonal sprng. INTODUCTION Defnng eavors of frames under dynam loads eatly takes a mportant plae n eartquake engneerng. In engneerng desgn, to know te real eavor of a struture s provded y determnng geometral, dampng, mass and onneton model well. In desgn purposes; strutures are desgned as avng rgd onnetons. However, te eavor of onnetons s not rgd. Strutures avng su flele onnetons n w onneton flelty eomes mportant are alled sem-rgd frames. Sem-rgd frames are frames for w te eam-to-olumn jonts are neter pnned nor rgd []. In realty all frames are sem-rgd, eause tere s not a frame w as truly pnned and perfetly rgd onnetons. For pratal desgn; two lassfaton systems were developed. Te lassfaton system y Bjorovde et al [] s ased on te response of a frame element, wle te oter lassfaton system y uroode [] s ased on te response of a frame. Tese lassfaton systems were developed y usng te results of many studes performed n last deades.. SMI-IGID FM MOD ND DUCING COFFICINT Te sem-rgd frame model used for te present study s sown n Fgure. Ts model nludes a eam wt moment of nerta I and lengt, and two olumns wt moment of nerta I, lengt and ross-seton. Te modulus of elastty s te same n all frame elements.

2 . U. Ozturk and H. H. Catal,I C θj C θk,,i,,i Fgure Sem-rgd frame model Te onnetons are modeled as rotatonal sprngs at eam-to-olumn jonts. ll deformatons are norporated n ts study. One an defne rgdty at te ends of frame element y te term of rgdty nde. For a onneton wt nge, rgdty nde s zero, and fleural moments do not our at te ends of a frame element. For a rgd onneton, ts value s nfnte, and fleural moments our at te ends of a frame element []. Fleural moments at te two ends for a frame element, wt sprng oeffents represented y C θj and C θk, an e gven y M jf = Cθ j Φ j ; M kf = Cθ k Φ k () were M jf and M kf are fleural moments, respetvely, at j and k ends of a frame element, Φ j and Φ k are rotatons, ourred y rotatonal sprngs. Te relatonsp etween sprng oeffents and rgdty nde an e wrtten y Cθ j Cθ k j = ; k = () were j and k are rgdty nde at two ends of a frame element, respetvely. otatons at two ends and aal dsplaement of a sem-rgd frame element gven y Fgure are element dsplaements. Fgure Dsplaements of a sem-rgd frame element

3 Dynam nalyss of Sem-gd Frames φ jyr and φ kyr are total rotatons at two ends of a sem-rgd element, φ jf ve φ kf are rotatons ourred wtout rotatonal sprngs at two ends of a sem-rgd element, respetvely. φ j and φ k an e wrtten y usng equaton () and equaton (). φ j = j jf M ; φ k = k kf M () Usng rotatonal sprngs, te stffness matr relatng rgdty nde at te ends s gven y equaton () []. [K yr t ] = () were ; ( ) = ; ( ) = ; ( ) = (5) + = j ; + = k () Te stffness matr of a sem-rgd olumn element n Fgure an e wrtten y [K f ]= (7) were ; + + = ; + = ; + = (8) Te stffness matr of a sem-rgd eam element n Fgure an e wrtten y

4 . U. Ozturk and H. H. Catal [K f ]= (9) Te struture stffness matr s otaned y assemlng te olumn and eam stffness matres desred aove aordng to onventonal stffness matr analyss proedure. One otans a stffness matr for te frame of Fgure. 5 θ θ Fgure Degrees-of-freedom By assumng tat and are equal, one an elmnate from te frame of Fgure.Te redued dsplaements are gven y Fgure. Te remanng stffness matr s a 55 matr.

5 Dynam nalyss of Sem-gd Frames 5 5 θ θ Fgure Te redued dsplaements {F} = [Ksf] { δ } () Te relatonsp etween deformatons and fores are gven y equaton.solvng te aove matr equaton for dsplaements eept and ak susttutng te result nto te frst row, te one-degree-of freedom system stffness relatonsp an e wrtten as F= α r () were s te lateral dsplaement,and F and α r are te lateral fore and redung oeffent respetvely.. DYNMIC NYSIS ND NUMIC STUDIS Te prmary ojetve of te present study s to nvestgate te dynam aratersts of sem-rgd frames and ow onneton flelty nfluenes tem. For a gven frame n Fgure, te equaton of moton for a sem-rgd frame n free vraton s gven y [ M ] { v.. }+[k]{ v }={} () were v.. and v are, respetvely, aeleraton and dsplaement of a struture. Te dynam aratersts of sem-rgd frame are determned y modal analyss. Te frequeny and perod of a vraton wll e nvestgated. Te nfluene of onneton flelty wll e studed. In te present study, -story sem-rgd frames avng four dfferent sprng oeffents and a rgd onneted frame were studed. Te sem-rgd model for te present analyss s gven n Fgure 5. ll frames ave te same geometry, ross-seton and materal property to ompare te nfluene of onneton flelty on dynam aratersts. Frst, te redung oeffents were determned y usng a omputer program. Ten, lateral rgdty values were alulated for ea frame. Te redung oeffents and perods are gven n Tale elow.

6 . U. Ozturk and H. H. Catal w = t C θ 5/5 m C θ w = t C θ 5/5 m C θ w = t C θ 5/5 m C θ.5 m.5 m.5 m 5 m Fgure 5 Sem-rgd model for te present analyss Tale edung oeffents Conneton model edung oeffent (α r ) ateral rgdty ( t / m ) Sem-rgd (C θ = tm/rd) Sem-rgd (C θ =5 tm/rd).9. Sem-rgd (C θ = tm/rd) Sem-rgd (C θ = tm/rd) gd Te results of te onduted analyss are gven for ea mod of vraton elow.

7 Dynam nalyss of Sem-gd Frames 7 Tale Dynam results of st mod Conneton model Frequeny ω ( rd / se ) Perod T ( se ) Sem-rgd ( tm/rd) Sem-rgd (5 tm/rd).8.9 Sem-rgd ( tm/rd).78. Sem-rgd ( tm/rd).88.5 gd.88.5 Tale Dynam results of nd mod Conneton model Frequeny ω ( rd / se ) Perod T ( se ) Sem-rgd ( tm/rd).97.8 Sem-rgd (5 tm/rd).57.5 Sem-rgd ( tm/rd) Sem-rgd ( tm/rd).87.9 gd.87.9 Tale Dynam results of rd mod Conneton model Frequeny ω ( rd / se ) Perod T ( se ) Sem-rgd ( tm/rd).97.8 Sem-rgd (5 tm/rd).577. Sem-rgd ( tm/rd) Sem-rgd ( tm/rd) gd CONCUSIONS sem-rgd frame was modeled y rotatonal sprngs. Te stffness matr was otaned y usng rgdty nde at te ends of a sem-rgd frame element. omputer program was wrtten to otan te redung oeffents from ts 55 stffness matr. Dynam analyss was performed for fve dfferent types of onneton. Te effets of onneton flelty were nvestgated. In a sem-rgd frame, an nrease n te rate etween lengt of ay and egt of story ( / ) auses redung oeffent and lateral rgdty derease, and n te same rate etween lengt of ay and egt of story ( / ), te redung oeffents for

8 8. U. Ozturk and H. H. Catal frames wt lower sprng oeffents are lower tan te redung oeffents for frames wt ger sprng oeffents. Te dynam eavor of a sem-rgd frame s dfferent from te dynam eavor of a rgd onneted frame. Sne te onneton flelty nfluenes te dynam aratersts of frames. Te study ndates tat onneton flelty tends to nrease perods, espeally n lower modes, wle t tends to derease te frequeny. FNCS. J.C. wkar and.m. u, Sesm analyss and response of multstory semrgd frames, Journal of ngneerng Strutures,, 5-, Bjorovde,.Colson and J. Brozett, lassfaton system for eam to olumn onnetons, SC Journal of Strutural ngneerng,,59-7,99. uropean Communty, Desgn of steel struture; part general rules and rules for uldngs, Brussels,Belgum,99.. H.H. Catal, Yapı ve deprem müendslğnde matrs yöntemler, İzmr,. 5. W.F.Cen, Jont flelty n steel frames, New York, lsever, W.F.Cen, Steel eam to olumn onnetons, New York, lsever, W.F.Cen, Y. Goto and. ew, Stalty desgn of sem rgd frames, New York, Wley, K. Copra, Dynams of strutures: Teory and applatons to eartquake engneerng, nglewood Clffs, NJ, Prente Hall, W. Cloug and J. Penzen, Dynams of strutures nd ed, New York, NY, MGraw-Hll,99...F. orenz, B. Kato and W.F. Cen, Counl on tall uldngs and uran atat, Sem rgd onnetons n steel frames, New York, 99...M. u,. opes, Dynam analyss and response of sem-rgd Frames. Journal of ngneerng Strutures,9(8),-5,997.