CONSISTENT EARTHQUAKE ACCELERATION AND DISPLACEMENT RECORDS
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1 APPENDX J CONSSTENT EARTHQUAKE ACCEERATON AND DSPACEMENT RECORDS Earhqake Acceleraons can be Measred. However, Srcres are Sbjeced o Earhqake Dsplacemens J. NTRODUCTON { XE "Acceleraon Records" }A he presen me mos earhqake moons are approxmaely recorded by acceleromeers a eqal me nervals. Afer correcng he acceleraon record, as a resl of he dynamc properes of he nsrmen, he record may sll conan recordng errors. Assmng a lnear acceleraon whn each me nerval, a drec negraon of he acceleraons generally prodces a velocy record wh a non-zero velocy a he end of he record ha shold be zero. And an exac negraon of he velocy record does no prodce a zero dsplacemen a he end of he record. One mehod crrenly sed o mahemacally prodce a zero dsplacemen a he end of he record s o nrodce a small nal velocy so ha he dsplacemen a he end of he record s zero. However, hs nal condon s no aken no accon n he dynamc analyss of he comper model of he srcre. n addon, hose dsplacemen records canno be sed drecly n ml-sppor earhqake response analyss. The prpose of hs appendx s o smmarze he fndamenal eqaons assocaed wh me hsory records. wll be demonsraed ha he recovery of acceleraons from dsplacemens s an nsable nmercal operaon. A new
2 APPENDX J- STATC AND DYNAMC ANAYSS nmercal mehod s presened for he modfcaon of an acceleraon record, or par of an acceleraon record, so ha sasfes he fndamenal laws of physcs n whch he dsplacemen, velocy and acceleraon records are conssen. J. GROUND ACCEERATON RECORDS Normally, 00 pons per second are sed o defne an acceleraon record, and s assmed ha he acceleraon fncon s lnear whn each me ncremen, as shown n Fgre J.. TME Fgre J. Typcal Earhqake Acceleraon Record Grond veloces and dsplacemens can hen be calclaed from he negraon of he acceleraons and veloces whn each me sep. Or: ( ( ( ( The evalaon of hose eqaons a recrsve eqaons: (J. prodces he followng se of
3 CONSSTENT EARTHQUAKE RECORDS APPENDX J- (,, (J. The negraon of grond acceleraon records shold prodce zero velocy a he end of he record. n addon, excep for near fal earhqake records, zero dsplacemens shold be obaned a he end of he record. Real earhqake acceleraons are normally correced o sasfy hose reqremens. { XE "Cbc Dsplacemen Fncons" }Noe ha he dsplacemens are cbc fncons whn each me ncremen. Therefore, f dsplacemens are sed as he specfed sesmc loadng, smaller me seps or a hgher order solon mehod, based on cbc dsplacemens, ms be sed for he dynamc srcral analyss. On he oher hand, f acceleraons are sed as he basc loadng, a lower order solon mehod, based on lnear fncons, may be sed o solve he dynamc response problem. J. CACUATON OF ACCEERATON RECORD FROM DSPACEMENT RECORD Rewrng Eqaon (J., shold be possble, gven he dsplacemen record, o calclae he velocy and acceleraon records from he followng eqaons: [ ] (J. On he bass of lnear acceleraon whn each me sep, Eqaons (J. and (J. are heorecally exac, gven he same nal condons. However, comper rond off nrodces errors n he veloces and acceleraons and he recrrence
4 APPENDX J-4 STATC AND DYNAMC ANAYSS Eqaon (J. s nsable and canno be sed o recover he np acceleraon record. Ths nsably can be llsraed by rewrng he eqaons n he followng form: ( ( (J.4 f he dsplacemens are consan, he recrrence eqaon wren n marx form s: (J.5 Or, f a small rond-off error, ε, s nrodced as an nal condon, he followng resls are prodced: ε 0 0 0, ε /, ε 4/ 7 (J. s apparen from Eqaon (J. ha he nrodcon of a small rond-off error n he velocy or acceleraon a any sep wll have an oppose sgn and be amplfed n sbseqen me seps. Therefore, s necessary o se an alernae approach o calclae he veloces and acceleraons drecly from he dsplacemen record. { XE "Splne Fncons" } s possble o se cbc splne fncons o f he dsplacemen daa and o recover he velocy and acceleraon daa. The applcaon of Taylor s seres a pon prodces he followng eqaons for he dsplacemen and velocy: ( ( (J.7
5 CONSSTENT EARTHQUAKE RECORDS APPENDX J-5 Elmnaon of from hese eqaons prodces an eqaon for he acceleraon a me. Or: ( ( ( ( (J.8 Evalaon of Eqaon (.0 a ± (a and - prodces he followng eqaons: ( ( ( ( (J.9 Reqrng ha be connos, he followng eqaon ms be sasfed a each pon: 4 ( (J.0 Therefore, here s one nknown velocy per pon. Ths well-condoned rdagonal se of eqaons can be solved drecly or by eraon. Those eqaons are dencal o he momen eqlbrm eqaons for a connos beam ha s sbjeced o normal dsplacemens. Afer veloces (slopes are calclaed, acceleraons (crvares and dervaves (shears are calclaed from: ( ( (J. Ths splne fncon approach elmnaes he nmercal nsably problems assocaed wh he drec applcaon of Eqaons (J.4. However, s dffcl o physcally jsfy how he dsplacemens a a fre me pon can affec he veloces and acceleraons a me pon. J.4 CREATNG CONSSTENT ACCEERATON RECORD { XE "Algorhms for:correcon of Acceleraon Records" }Earhqake compresson, shear and srface waves propagae from a fal rpre a dfferen
6 APPENDX J- STATC AND DYNAMC ANAYSS speeds wh he small amplde compresson waves arrvng frs. For example, acceleraon records recorded near he San Francsco-Oakland Bay Brdge from he 989 oma Prea earhqake ndcae hgh freqency, small acceleraon moons for he frs en seconds. The large acceleraon phase of he record s beween 0 and 5 seconds only. However, he offcal record released covers approxmaely a 40-second me span. Sch a long record s no sable for a nonlnear, me-hsory response analyss of a srcral model becase of he large comper sorage and execon me reqred. s possble o selec he large acceleraon par of he record and se as he basc np for he comper model. To sasfy he fndamenal laws of physcs, he rncaed acceleraon record ms prodce zero velocy and dsplacemen a he begnnng and end of he earhqake. Ths can be accomplshed by applyng a correcon o he rncaed acceleraon record ha s based on he fac ha any earhqake acceleraon record s a sm of acceleraon plses, as shown n Fgre J.. Area A TME Fgre J. Typcal Earhqake Acceleraon Plse From splne heory, he exac dsplacemen a he end of he record s gven by he followng eqaon: ( U (J.
7 CONSSTENT EARTHQUAKE RECORDS APPENDX J-7 A correcon o he acceleraon record may now be calclaed so ha he dsplacemen a he end of he record, Eqaon (J., s dencally eqal o zero. Raher hen apply an nal velocy, he frs second or wo of he acceleraon record can be modfed o oban zero dsplacemen a he end of he record. e s assme ha all of he correcon s o be appled o he frs vales of he acceleraon record. To avod a dsconny n he acceleraon record, he correcon wll be weghed by a lnear fncon, from α a me zero o zero a me. Therefore, he dsplacemen reslng from he correcon fncon a he end of he record s of he followng form: α ( α pu pos α nu neg U (J. For Eqaon (J. he posve and negave erms are calclaed separaely. f s assmed ha he correcon s eqal for he posve and negave erms, he ampldes of he correcon consans are gven by: α p U U pos U neg and α n (J.4a and J.4b U Therefore, he correcon fncon can be added o he frs vales of he acceleraon record o oban zero dsplacemen a he end of he record. Ths smple correcon algorhm s smmarzed n Table J.. f he correcon perod s less ha one second, hs very smple algorhm, presened n Table J., prodces almos dencal maxmm and mnmm dsplacemens and veloces as he mahemacal mehod of selecng an nal velocy. However, hs smple one-sep mehod prodces physcally conssen dsplacemen, velocy and acceleraon records. Ths mehod does no fler mporan freqences from he record and he maxmm peak acceleraon s mananed. The velocy a he end of he record can be se o zero f a smlar correcon s appled o he fnal few seconds of he acceleraon record. eraon wold be reqred o sasfy boh he zero dsplacemen and velocy a he end of he record.
8 APPENDX J-8 STATC AND DYNAMC ANAYSS Table J. Algorhm o Se Dsplacemen a End of Records o Zero. GVEN UNCORRECTED ACCEERATON RECORD,,,,,...,0 and 0 4. COMPUTE CORRECTON FUNCTON U 0 α p ( ( U U pos U and α n pos U neg U U neg. CORRECT ACCEERATON RECORD f f > 0 hen < 0 hen ( α ( α p n,... J.5 SUMMARY Acceleraon records can be accraely defned by 00 pons per second and wh he assmpon ha he acceleraon s a lnear fncon whn each me sep. However, he reslng dsplacemens are cbc fncons whn each me sep and smaller me seps ms be ser-defne dsplacemen records. The drec calclaon of an acceleraon record from a dsplacemen record s a nmercally nsable problem, and specal nmercal procedres ms be sed o solve hs problem. The mahemacal mehod of sng an nal velocy o force he dsplacemen a he end of he record o zero prodces an nconssen dsplacemen record ha shold no be drecly sed n a dynamc analyss. A smple algorhm for he correcon of he acceleraon record has been proposed ha prodces physcally accepable dsplacemen, velocy and acceleraon records.
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