Research Note NONLINEAR ANALYSIS OF SEMI-RIGID FRAMES WITH RIGID END SECTIONS * S. SEREN AKAVCI **

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1 Iranan Jornal of Scence & Technology, Transacton, Engneerng, Vol., No., pp 7-7 rnted n The Islamc Repblc of Iran, 7 Shraz Unversty Research Note NONLINER NLYSIS OF SEMI-RIGID FRMES WIT RIGID END SECTIONS * S. SEREN KVCI ** Crova Unversty, Department of rchtectre, alcal, dana, Trey Emal: aavc@mal.c.ed.tr bstract Ths wor presents a compter-based analyss of sem-rgd steel frames. The geometrc nonlnearty of the strctre and the materal nonlnearty of the connectons are consdered n the analyss. The crtcal load has been searched as a stable load parameter for the loss of stablty of the system. Several examples are presented to demonstrate the valdty of the analyss procedre. Keywords Sem-rgd, nonlnear, plane frame, rgd-end secton, crtcal load. INTRODUCTION Connecton flexblty affects both the force dstrbton and the deformaton n the beams and colmns of a frame and mst be nclded n stctral analyss. The behavor of flexble connectons s sally descrbed by ther moment-rotaton crves, n whch the slope of the crve corresponds to the rotatonal rgdty of the connectons. Several analytcal and expermental stdes have been done on ths sbject n recent years [, 7]. In ths stdy, the nonlnear analyss of frames wth rgd end sectons and nonlnear sem-rgd connectons has been carred ot tang nto consderaton the effect of geometrc nonlnearty. The element matrces of members are formlated by sng the second order theory. The analyss s then carred ot by an ncremental procedre. In the analyss, the crtcal load of the system has been obtaned by sng a doble teraton process. The calclaton of the crtcal load s prely a search for the vale of a stable load parameter λ for the loss of stablty of the system.. ELEMENT STIFFNESS MTRIX ND STILITY NLYSIS In ths stdy, to descrbe the connecton behavor, dfferent mathematcal models have been proposed. The chosen connecton model to descrbe the nonlnear M-θ r crve of sem-rgd connectons s gven by Kng and Chen [] as follows: dm K t K dθr M M c () where K t s the tangent stffness of the connecton, K s the ntal connecton stffness, M s the ltmate bendng moment capacty of the connecton, M s connecton moment, and c s the shape factor. Receved by the edtors May, ; fnal revsed form Janary, 7. Correspondng athor

2 8 S. Seren avc The nonlnear stffness matrx of a homogenos, flexbly connected member havng rgd end sectons has been obtaned from the solton of the second order analyss, where axal force s taen nto accont as j where; j jj E L EI L EI L EI L EI L j E L EI L EI L EI L EI L jj E L EI L EI L () EI L EI L In expresson () ( + * N EI ) S + α EI ( + ) C α EI S * ( C) ( N L * ) S + *( αl) C α EI S * ( C) * ( αl) + * S ( N L * ) S + *( αl) C [ α EI( NL *( + )) + *( αl) ] S + *[ NL ( + ) * ] C + () where E s Yong s modls, s the cross-sectonal area of the member, I s the moment of nerta of the member, L s the length of the member, s the moment vale necessary for the sprng at the left spport to rotate one radan, s the moment vale necessary for the sprng at the rght spport to rotate one radan, N s the axal force n the member,c s [cos (αl)] for compressve and [cosh (αl)] for tensle axal force, S s [sn (αl)] for compressve and [snh (αl)] for tensle axal force, * s - for compressve and + for tensle axal force, and α L. The pont of frame nstablty occrs when the determnant of the overall stffnes matrx becomes eqal to zero. Det K( N ) (). INCREMENTL ITERTIVE SOLUTION METOD The comptatonal technqe, namely The Newton-Raphson method, sed to obtan the nonlnear response of a frame s shown graphcally n Fg.. The loadng s appled ncrementally to the frame and wthn each load step an teratve solton s performed ntl a reqred accracy s attaned. Fg.. The Newton-Raphson teraton procedre Iranan Jornal of Scence & Technology, Volme, Nmber October 7

3 Nonlnear analyss of sem-rgd frames wth 9 The Newton-Raphson teraton procedre for each ncrement: K F R j j j () + () j j where K s the nonsnglar tangent stffness matrx, s the ncremental strctre nodal dsplacement vector, s the ncremental load vector, F j s the strctre nternal force vector, and, j are the nmbers of the load step and teraton. The rght sde of Eq. () s nown as a vector of ot-of-balance (resde) loads de to nonlnearty. Convergence s assmed when R < ε, < ε (7) where R s the ot-of-balance load, and ε R and ε are tolerances. R. NUMERICL RESULTS In the frst example, the two-story one-bay frame as shown n Fg. s analyzed wth rgd and sem-rgd connectons, wth and wthot geometrc nonlnearty. The connectons have an ntal stffness of 7 n-p/rad, and an ltmate moment of 7 n-p wth a shape factor of.. The reslts of lnear and nonlnear analyss are presented n Fgs. -. / 7 8 / 88 n Fg.. Two-story one-bay frame wth nform and concentrated loads n n Colmns : W eams : W (frst floor) W (second floor) (ps) 7 sem-rgd conn.- nonlnear analyss x sem rgd conn.- lnear analyss rgd conn.- lnear analyss rgd conn.- nonlnear analyss Dsplacement (n) (ps) 7 sem-rgd conn.- lnear analyss x sem rgd conn.- nonlnear analyss rgd conn.- lnear analyss rgd conn.- nonlnear analyss Moment (n-ps) Fg.. The load-lateral dsplacement crves of node 8 Fg.. The load-moment crves of node s the second example, the frame wth concentrated loads n Fg. s analyzed for dfferent connecton stffnesses and rgd end lengths. The defnton of the fxty factor s gven as γ, where s the rotatonal stffness of the connecton. + EI L October 7 Iranan Jornal of Scence & Technology, Volme, Nmber

4 7 S. Seren avc n n d d Colmns : Wx9 eams : Wx8 n. ft ft ft (ps) d n d n d n d n d n d n γ Fg.. Two-story one-bay frame wth rgd end sectons Fg.. The crtcal loads for the frame as a fncton of the fxty factor and the length of the rgd end sectons The varaton of the crtcal load of the frame wth the connecton stffness and the rgd end length s gven n Fg.. The varatons of the dsplacement and moment wth the connecton stffness and rgd end length are gven n Fgs. 7-8 for ps. The dsplacements and moments are normalzed by dvdng them by the correspondng reslts for the case of flly pnned jonts. Normalzed dsplacement d n d n d n d n d n d n γ Normalzed moment d n d n d n d n d n d n γ Fg. 7. The varaton of dsplacement wth connecton stffness and rgd end length at node Fg. 8. The varaton of moment wth connecton stffness and rgd end length at node The last example (Fg. 9), whch s selected for comparson, s taen from Selovc and Salatc[7]. The crtcal loads of bfrcaton type stablty loss of a sngle bay frame wth a dfferent nmber of storeys fond by the present method has been compared wth those of [7]. I E G I I 77 cm I cm I cm. cm m m Normalzed Crtcal Load cr a. a b c d. b.. c d. resent Stdy. Selovc and Salatc Fxty Factor p Fg. 9. The sngle bay frame wth one Fg.. Inflence of connecton flexblty on the crtcal load of the frame Iranan Jornal of Scence & Technology, Volme, Nmber October 7

5 Nonlnear analyss of sem-rgd frames wth 7. CONCLUSIONS In the nmercal part of ths stdy, some examples have been solved for observng the nflence of the sem rgd connectons on the bendng response of frames. For frames wth rgd connectons and low levels of loadng, the dfference between the dsplacements and moments of lnear and nonlnear analyses s very slght. owever, for frames wth sem-rgd connectons, the dsplacements and moments tae sgnfcantly fewer vales n the lnear case than n the nonlnear one for the same load, even when t s at a low level. Frthermore, as the load level ncreases, t s observed that, the lnear analyss yelds mch lower crtcal loads compared to the nonlnear one. The reslts of examples show that when the connectons are pnned jonts, the effect of the lengths of the rgd end sectons on the crtcal load of the system s very slght, almost none. owever, when the connectons are sem-rgd, the crtcal load of the system ncreases wth the ncrease n the lengths of the rgd end sectons and the stffness of the sprngs. REFERENCES. anotopolos, C. C. & Ivany, M. (). Sem-rgd jonts n strctral steelwor. CISM Lectre Notes 9, Sprnger-Verlag Venna, ISN Ivany, M. & Salod, M. (99). Stablty problems of steel strctres. CISM Lectre Notes,, Sprnger- Verlag, ISN Lghtfoot, E. and Messrer,.. (979). Elastc analyss of frame-wors wth elastc connectons. J. of Strct. Dv. SCE, Vol., pp ng, K. M. & Morrs, G.. (98). nalyss of three-dmensonal frames wth flexble beam-colmn connectons. Can. J. Cvl Eng., Vol., pp. -.. araat, M. & Chen, W. F.(99). ractcal analyss of sem-rgd frames. Eng. J. ISC, Vol. 7, No., pp Kng, W. S. & Chen, W. F. (99). LRFD analyss for sem-rgd frame desgn. Eng. J. ISC, Vol., No., pp Selovc, M. & Salatc, R. (). Nonlnear analyss of frames wth flexble connectons. Compters and Strctres, Vol. 79, pp October 7 Iranan Jornal of Scence & Technology, Volme, Nmber