INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 1, No 4, 2011

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1 Behavr f hgh frequenc mdal respnses n nn lnear sesmc analss Dhleep. M 1, Trved. A 2, Bse.P.R 3 1 Assstant Prfessr, Department f Cvl Engneerng, SCMS Schl f Engneerng and Technlg, Karukutt, Ernakulam, Kerala 2 Dean, Facult f Technlg, Unverst f Delh, Delh Cllege f Engneerng Campus, Bawana Rad, Delh 42 3 Head, Engneerng and Dsaster Management, DDF cnsultants Pvt. Ltd., Delh kavad@bsnl.n ABSTRACT Practcal dffcultes asscated wth the nn lnear drect numercal ntegratn f the equatns f mtn leads t the use f nn lnear statc pushver analss f structures. Pushver analss s gettng ppular due t ts smplct. Hgh frequenc mdes and nnlnear effects ma pla an mprtant rle n stff and/r rregular structures. The cntrbutn f hgher mdes n pushver analss s nt full develped. The behavr f hgh frequenc mdal respnses n nn lnear sesmc analss f structures s nt knwn. In ths paper an attempt s made t stud the behavr f hgh frequenc mdal respnses n nn lnear sesmc analss f structures. Kewrd: Pushver, Hgh Frequenc Mdal Respnse, Rgd Frequenc, Nn lnear, Sesmc Analss 1. Intrductn In sesmc analss and desgn f structures, a drect ntegratn f the equatn f mtn s cnsdered t eld the true respnse f a structure when subjected t the partcular earthquake tme hstr cnsdered n the analss. Practcal dffcultes asscated wth the analss f large real lfe structural mdels dctate the use f respnse spectrum methd f analss nstead f the drect ntegratn methd. Furthermre, the earthquake nput needed n the desgn f structures subjected t future earthquakes s defned n terms f a respnse spectrum. The respnse spectrum methd f analss utlzes the cncept f mdal superpstn. In tpcal regular buldng structures nl a few lwer rder mdes are suffcent t evaluate the ttal respnse wth reasnable accurac. Unlke regular structures, hgh frequenc mdes and nn lnear effects cntrbutes sgnfcantl n the sesmc analss f rregular structures. Nn lnear sesmc analss f structures nvlves drect numercal ntegratn f the equatns f mtn. Ths methd s mpractcal and cmputatnall tedus fr mst practcal applcatns. Therefre an apprxmate nn lnear statc analss prcedure knwn as pushver analss s used nstead f nn lnear dnamc analss (ATC40, 1996; FEMA273, 1997; FEMA274, 1997; FEMA356, 2000). The gudelnes gven b Federal Emergenc Management Agenc (FEMA) and Appled Technlg Cuncl (ATC) fr pushver analss assumes the lateral frce dstrbutn based n the 723

2 fundamental mde f vbratn. In the case f smple regular structures, the fundamental mde s suffcent t evaluate the ttal respnse wth reasnable accurac. Unlke regular structures, the fundamental mde n an rregular structure s tpcall a lcalzed mde that ma nt have an bearng n the respnse f certan ther parts wthn a structure (Dhleep and Bse, 2009). FEMA356 recmmends that hgher mde effects shall be cnsdered f the shear n an str resultng frm the mdal analss cnsderng mdes requred fr btanng 90% mass partcpatn exceeds 130% f the crrespndng str shear cnsderng nl the frst mde respnse. T satsf ths crtern, structural engneers ma cnsder several hgher mdes whse frequences ma be hgher than the fundamental mde. In rder t take the respnse cntrbutns f hgher mdes nt accunt, a mdal pushver analss s prpsed (Chpra and Gel, 2001; Kalkan and Kunnath, 2004; Barrs and Almeda, 2005). Unlke lnear analss, the cntrbutn f hgh frequenc mdes n pushver analss s nt full develped and s a develpng area f research. Further the effect f hgh frequenc mdal respnses n nn lnear analss f structures s nt knwn. Ths paper dscusses the behavr f hgh frequenc mdal respnses n the nn lnear statc pushver analss f structures. 2. Nn lnear sesmc analss Equatn f mtn fr an dealzed sngle degree f freedm sstem wth a stffness K, mass M and dampng ceffcent C s gven b, { } u & g M U & + C U & + f ( U, sgn U & ) = M 1 (1) where, f ( U, sgn U & ) dentes the hsteretc relatn between the lateral frce f s and the lateral dsplacement U (Chpra 1998; Chpra and Gel, 2001). On ntal ladng, wthn the eld strength f, the sstem s lnearl elastc wth stffness K. Yeldng begns when the frce reaches f and the defrmatn reaches the eld defrmatn U. In the pst eldng regn the stffness f the frame ma be taken as αk, where, α vares between 0 t 10% (FEMA 273, 1997). The eld strength f s related t the strength requred fr the structure t reman elastc f, durng the grund mtn, thrugh the eld strength reductn factr, R, defned b f U R = = (2) f U where U s the eld defrmatn and R s equal t 1 fr lnearl elastc sstems and greater than 1 fr sstems that defrms n t the nelastc range. Perfrmance based desgn requres an estmate f the maxmum dsplacement, a structure wll underg due t a desgn earthquake. In rder t acheve ths, a nn lnear dnamc analss s requred. Hwever due t the cmputatnal effrts and practcal dffcultes f nn lnear dnamc analss, nn lnear statc pushver analss s nw wdel used as an alternatve. 724

3 2.1 Pushver analss Pushver analss s a perfrmance based analss prcedure, usuall cnssts f applng a vertcal dstrbutn f lateral lads t a mdel f an exstng r prevusl desgned structure. These lads are ncreased untl the peak respnse f the structure s btaned (Fgure 1). Fgure 1: Pushver Analss The pushver curve s a plt between the base shear and rf dsplacement. Ths pushver curve s then cnverted t acceleratn dsplacement respnse spectrum (ADRS) knwn as capact spectrum (ATC40, 1996; FEMA356, 2000). Further the elastc respnse spectrum s cnverted t the demand spectra n ADRS frmat. The capact spectrum and demand spectrum are then super mpsed t btan the perfrmance pnt. 3. Spectral regns n a Respnse Spectrum The spectral regns n a respnse spectrum can be dvded nt three, 1) a hghfrequenc regn, 2) a md frequenc regn and 3) a lw frequenc regn (Mrante et al, 1999; USNRC, 2006). In hgh frequenc regn, the spectral acceleratn becmes equal t the peak grund acceleratn, ften referred t as the zer perd acceleratn (ZPA). In ths regn f spectrum, the perdc part f the respnse becmes neglgble and nl the rgd part f respnse remans. The respnses n ths regn are n phase t each ther and cmbne algebracall. The mnmum frequenc n the hgh frequenc regn bend whch the spectral acceleratn curves fr varus dampng rats have same values was defned as the rgd frequenc (Gupta et al, 1996). In the lw frequenc regn f the spectrum, the respnses are nt n phase wth the grund acceleratn, and generall are nt n phase wth each ther. The rgd part f respnse n ths regn s equal t zer and the respnse cntans nl damped perdc part. 725

4 The defntn f the rgd frequenc can be ver mprtant n the respnse spectrum methd fr accuratel evaluatng the structural respnse, the varus cdes and standards defne ts value whch s based prmarl n the ease f mplementatn and nt n the accurac f the calculated respnse. The premse fr dng s s t avd the practce f determnng the rgd frequenc frm a vsual nspectn f respnse spectrum curves. The fxng up f rgd frequenc based upn the cnvergence f dfferent spectral curves wth dfferent dampng rats can be prne t ndvdual judgment and can be lead t the use f dfferent rgd frequences b dfferent engneers (Dhleep, 2007; Dhleep, 2008). Recent studes (Dhleep and Spna, 2011) shw that the spectral curves wth dfferent dampng rats cnverges at dfferent frequences. Instead f rgd frequenc, a damped rgd frequenc s prpsed based n the cnvergence f a spectral curve wth the spectral curves havng hgher dampng rats. 4. Behavr f hgh frequenc mdal respnses At hgh frequences, the perdc part f the respnse becmes neglgble and nl the rgd part f respnse remans. Mrever the perd f a hgh frequenc mde s ver shrt, s the respnse n such a mde s essentall statc than dnamc. Cnsder the frst n mdes f a N degrees f freedm sstem, havng frequences less than damped rgd frequenc and let the respnse n these n mdes be U, and the respnse n the remanng rgd mdes be U.Then, n U = U = φ X ; U = U = φ X (3) = 1 n = 1 N = n + 1 N = n + 1 U = U + U (4) Fr hgh frequenc mdes, Eqs. (3) and (4) gves the equatn f mtn as, M U & + C U & + KU = MU (5) b u& & g where, n U = U φ Γ, (8) b b = 1 M= mass matrx, C= dampng matrx and K= Stffness matrx. Snce the resdual respnse n hgh frequenc mdes are pseud statc, we can neglect the terms U & and U & n Eq. (7), therefre KU b u& & g = MU (9) The respnse n a hgh frequenc mde s essentall statc and culd be determned b statc analss usng equatn (9) nstead f dnamc analss. The respnses f the 726

5 mdes n the md frequenc regn cnssts f a rgd (statc) part and a damped perdc part. Fgure 2: Elastc respnse spectrum f El Centr 1940 earthquake n nrmal scale Fgure 2 shws the elastc respnse spectrum f El Centr 1940 earthquake. The spectral acceleratn n the hgh frequenc zne s equal t the zer perd acceleratn. The lnear respnse n hgh frequenc mdes f ths regn can be btaned b cnductng a statc analss usng zer perd acceleratn n equatn (9). Fgure 3(a) shws the demand spectrum crrespndng t El Centr 1940 grund mtn n ADRS frmat. Fgure 3(b) shws the enlarged prtn crrespndng t the rgd zne f the demand spectrum. In ths zne t s bserved that the spectral acceleratn remans cnstant and equal t zer perd acceleratn (Fgure 3b). In mdal pushver analss, the structure s subjected t an ncremental frce pattern f Γ n MΦ n S A, where Γ n s the mdal partcpatn factr, Φ n s the mde shape and S A s the spectral acceleratn. In hgh frequenc mdes the behavr f rgd mdes are statc and ther mdal respnse can be btaned b a statc analss. In hgh frequenc mdes the structure s subjected t frce pattern gven b the rght hand sde f equatn (9). Therefre pushver analss can be cnducted n hgh frequenc mdes b pushng the structure usng an ncremental lad pattern gven b equatn (9). Pushver analss s a nn lnear statc apprxmatn t the nn lnear tme hstr analss. Therefre hgher mdes are expected t gve mre accurate results n a nn lnear statc analss cmpared t the mdes n the md frequenc and lw frequenc regns f a respnse spectrum. 727

6 (a) Fgure 3: El Centr 1940 grund mtn (a) Demand spectrum (b) Demand spectrum crrespndng t rgd zne 5. Numercal Verfcatn T stud the behavr f hgh frequenc mdal respnses n nn lnear statc pushver analss a 2D frame mdel as shwn n Fgure 4 s cnsdered. The frequences are changed b changng the gemetrcal parameters f the mdal and pushver analss s cnducted usng the frst mde. (b) Fgure 4: Mdel f 2D frame 728

7 The analss s cnducted usng SAP 2000 fr dfferent frequences (CSI, 2003). Fr pushver analss, M3 hnges are assgned t beams and P M2 M3 hnges are assgned t clumns accrdng t FEMA 356 dcument usng the ptns gven n SAP Lad cases are defned and then analses are perfrmed. Frm the pushver curve the perfrmance pnt n terms f spectral acceleratn and spectral dsplacement, and the crrespndng base shear and rf dsplacement were btaned fr 5% damped El Centr, 1940 respnse spectrum. The results are cmpared wth the peak dsplacement btaned frm the nn lnear mdal tme hstr analss (THA) perfrmed usng El Centr, 1940 grund mtn wth 5% dampng. The results btaned are tabulated n Table 1. Table 1: Dsplacements fr dfferent frequences Sl.N. Frequenc (Hz) Dsplacement (m) % Errr Pushver analss Nn lnear THA X X X X X X X X X X X X Frm Table 1 t s bserved that the percentage errr n pushver analss at a lw frequenc f 9.7 Hz s 53.22% when cmpared t tme hstr analss. Fr hgh frequences the pushver analss gves results wth less than 1% errr. Ths shws that the behavr f hgh frequenc nn lnear mdal respnses s essentall statc. Therefre the nn lnear respnses f structures n hgh frequenc rgd mdes can be btaned b cnductng a nnlnear statc analss, nstead f a nn lnear dnamc analss. Further t s bserved that there s a decrease n errr as the frequenc ncreases frm lwfrequenc zne t hgh frequenc zne. Ths s due t the presence f rgd part f respnse n the md frequenc zne f spectrum. 6. Cnclusns Nn lnear statc pushver analss used as an apprxmatn t nn lnear tme hstr analss s becmng a standard tl amng the engneers, researchers and prfessnals wrldwde. Hgh frequenc mdes ma cntrbute sgnfcantl n the sesmc analss f rregular and/r stff structures. In rder t take the cntrbutns f hgher mdes structural engneers ma nclude hgh frequenc mdes n the nn lnear statc pushver analss. The behavr f hgh frequenc mdes n nn lnear statc pushver analss f rregular structures s studed. At hgh frequences, the respnses f nn lnear dnamc analss cnverge t the nn lnear statc pushver analss. Therefre nn lnear respnse f hgh frequenc mdes can be evaluated usng a nn lnear statc push ver analss wth an ncremental frce pattern gven b ther mdal mass cntrbutn tmes zer perd acceleratn. The hgher mdes wth rgd cntent as a majr cntrbutng 729

8 factr exhbt a better accurac n nn lnear pushver analss f structures when cmpared t the damped perdc mdes. Acknwledgements The authrs are thankful t Hartha P.H., Lecturer, KMCT Engneerng Cllege, Kzhkde fr her assstance n slvng numercal prblems. 7. References 1. Appled Technlg Cuncl (ATC), 1996, Sesmc Evaluatn and Retrft f Cncrete Buldngs, Reprt ATC 40, Appled Technlg cuncl, Redwd Ct, Calfrna Sesmc Safet Cmmssn. 2. Chpra, A.K.(1998). Dnamcs f Structures: Ther and applcatns t earthquake engneerng, 3 rd Ed., Prentce Hall, New Delh. 3. Chpra, A. K., and Gel, R. K., 2001, A Mdal Pushver Analss Prcedure t Estmatng Sesmc Demands fr Buldngs: Ther and Prelmnar Evaluatn, PEER Reprt 2001/03, Pacfc Earthquake Engneerng Research Center, Unverst f Calfrna, Berkele, Calfrna. 4. Cmputers & Structures, Inc., 2003, SAP 2000 Analss Reference Manual, Berkele, Calfrna: CSI. 5. Dhleep M and Bse P.R., 2007, Effect f Hgh Frequenc Mdes n Sesmc Analss f Buldngs, Prceedngs f the Internatnal Cnference n Cvl Engneerng n the New Mllennum: Opprtuntes and Challenges, Hwrah, Inda, Dhleep M. and Bse P. R., 2008, A Cmparatve stud f Mssng Mass Crrectn Methds fr Respnse Spectrum methd f Sesmc Analss, Cmputers & Structures, 86(21 22), Dhleep M. and Bse P.R., 2009, Sesmc Analss f Irregular Buldngs: Mssng Mass Effect, Jurnal f Structural Engneerng, SERC, Vl. 35, N. 5, Dhleep M. and Spna S.N., 2011, Effect f Rgd Cntent n Mdal Respnse Cmbnatn, Structures and Buldngs (Accepted fr publcatn) 9. FEMA 356, 2000, Pre standard and cmmentar fr the Sesmc Rehabltatn f Buldngs, Buldng Sesmc Safet Cuncl fr the Federal Emergenc Management Agenc, Washngtn, D.C. 730

9 10. FEMA273, 1997 NEHRP Gudelnes fr the Sesmc Rehabltatn f Buldngs, Buldng Sesmc Safet Cuncl fr the Federal Emergenc Management Agenc, Washngtn, D.C. 11. FEMA 274, 1997 NEHRP Cmmentar n the Gudelnes fr the Sesmc Rehabltatn f Buldngs, Buldng Sesmc Safet Cuncl fr the Federal Emergenc Management Agenc, Washngtn, D.C. 12. Gupta, A.K., Hassan, T. and Gupta, A. (1996), Crrelatn ceffcents fr mdal respnse cmbnatn f nn classcall damped sstems. Nuclear Engneerng and Desgn 165, Kalkan, E. and Kunnath, S.K.,2004 Methd f mdal cmbnatn fr pushver analss f the buldng, 13 th Wrld Cnference f Earthquake Engneerng, Vancuver, B.C., Canada., Paper N Mrante, R., Wang, Y., Chksh, N., Kenneall, R. and Nrrs W., 1999, Evaluatn f Mdal Cmbnatn Methds fr Sesmc Respnse Spectrum Analss, 15th Internatnal Cnference n Structural Mechancs n Reactr Technlg, Seul (KR), BNL NUREG 66410, paper ID K4 A4 US Ru Carner Barrs and Rcard Almeda, 2005, Pushver Analss f Asmmetrc Three Dmensnal Buldng Frames, Jurnal f Cvl Engneerng and Management, 11(1), pp USNRC, 2006, Cmbnng mdal respnses and spatal cmpnents n sesmc respnse analss, Regulatr gude 1.92, Offce f nuclear regulatr research, US Nuc. Reg. Cmmssn R2. 731