UNIVERSITY OF SOUTHERN CALIFORNIA Department of Civil Engineering NON-LINEAR EARTHQUAKE WAVES IN SEVEN-STOREY REINFORCED CONCRETE HOTEL

Transcription

1 UNIVERSITY OF SOUTHERN CALIFORNIA Department of Cvl Engneerng NON-LINEAR EARTHQUAKE WAVES IN SEVEN-STOREY REINFORCED CONCRETE HOTEL by Vlado Gcev and Mhalo D Trfnac Report CE 6-3 November, 6 Los Angeles, Calforna

2 ABSTRACT The seven-story renforced concrete bldng stded n ths work was damaged drng the 994 Northrdge, Calforna earthqake The damage was wdespread bt was mostly concentrated n the colmns on the forth floor, below the spandrel beam and the floor slab on the ffth floor We show that sch a concentraton of damage can be eplaned by nterference of non-lnear waves n the bldng resltng from ncdent, pward-propagatng earthqake plses and downward propagaton of plses reflected from the roof of the bldng We conclde that the analyss and predcton of where the localzed damage may occr and the proper desgn to mnmze and control sch damage can be carred ot best by non-lnear wave propagaton methods

3 INTRODUCTION The seven-story hotel bldng n Van Nys, Calforna (VN7SH) s becomng one of the most stded bldngs n sothern Calforna Many aspects of ts response and damage, frst drng San Fernando, 97, earthqake, and then the Northrdge earthqake n 994, have been eamned The relatve smplcty, nformty, and symmetry of ts geometry make ths bldng deal for testng and for calbraton of dfferent analyss methods So far, t has been dffclt to eplan why the man damage drng the Northrdge earthqake occrred n the forth floor colmns rather than at ts base or throghot the strctre In a stdy of the propagaton of non-lnear waves n a smple, nform shear beam, cased by ncdent strong moton plses, Gcev and Trfnac (6) fond that for large grond dsplacement plses the mamm permanent strans n the beam occr manly at the nterface of the beam wth the sol, whle for smaller ampltdes of plses permanent trans occr closer to the top of the beam They dentfed three zones of the permanently deformed beam: () a permanently deformed zone at the bottom; () an ntermedate zone, whch s not deformed at ts bottom part and s deformed n the top part; and (3) a non-deformed zone at the top of the beam They fond that the occrrence and the development of these zones depends pon the dmensonless ectaton ampltdes and the dmensonless freqency of the ncdent strong moton plses, and n partclar on the condtons that lead to the occrrence of the frst permanent stran For large and long strong-moton plses, only zones and 3 are present n the beam For large ampltdes and short strong-moton plses, all three zones can develop and are present For smaller ectaton ampltdes only zones and 3 est n the beam Gcev and Trfnac (6) frther show that for ectaton by near-feld dsplacement plses, falre can occr anywhere n the bldng, before the ncdent wave has completed ts frst travel from the fondaton to the top of the bldng and back to the fondaton Becase ths travel tme s shorter (by one half) than the natral perod of the strctre on the fed base, t s seen that the common response spectrm method of analyss (based on the vbratonal formlaton of the solton) cannot provde the reqred detals for the desgn of strctres for sch ectaton They noted that to the etent that the locatons of the plastc deformaton zones can be controlled by the desgn process, absorpton of the ncdent wave energy by strctral members may become a new and powerfl tool for performance-based desgn To take advantage of sch possbltes, the governng dfferental eqatons mst be solved by the wave propagaton method

4 The prpose of ths work s to show that f the ectaton s vewed as plane waves enterng the beam (strctre) vertcally, the nterference of the waves propagatng p, and ther reflecton off the stress-free roof, can eplan well the observed concentraton of damage n VN7SH Or approach n ths work belongs to a grop of wave propagaton methods (Kana 965; Todorovska and Trfnac 6a,b,c) and s also stable for strctral health montorng and for real-tme damage detecton To mnmze the conseqences of rockng assocated wth sol-strctre nteracton and the contrbton of torson, n ths work we analyze the EW (longtdnal) response of the VN7SH bldng only The reader shold be aware that gnorng the rockng contrbton to the overall EW response wll tend to make or model and the reslts smlar to many engneerng stdes of ths bldng, whch typcally gnore sol-strctre nteracton that s, they assme that the bldng s spported by rgd sol We wll address the two- and three-dmensonal aspects of ths problem and the role of sol-strctre nteracton n or ftre work THE BUILDING The bldng stded n ths work s a seven-story hotel (VN7SH) located n Van Nys, Calforna It was damaged by the 994 Northrdge, Calforna earthqake (Ivanovć et al 999a,b, Trfnac and Hao, Trfnac et al 999a,b) Its response has been descrbed and analyzed n nmeros papers and reports The VN7SH (Fg ) s located n the central San Fernando Valley of the Los Angeles metropoltan area (at 34 o N and 847 o W) The bldng was desgned n 965, and constrcted n 966 (Blme and Assoc 973, Mlhern and Maley 973, Table ) Fgre shows a plan vew of a typcal floor (a), and a sde vew of the bldng frame (b) The bldng s m n plan The typcal framng conssts of colmns spaced on 6 m centers n the transverse drecton and 58 m centers n the longtdnal drecton Spandrel beams srrond the permeter of the strctre Lateral forces n the longtdnal (EW) drecton are ressted by nteror colmn-slab frames and eteror colmn spandrel beam frames The added stffness n the eteror frames assocated wth the spandrel beams creates eteror frames that are roghly twce as stff as nteror frames The floor system s renforced concrete flat slab, 54 cm thck at the second floor, 6 cm thck at the thrd to seventh floors, and 3 cm thck at the roof (Brownng et al, De La Llera at al, Islam 996, L and Jrsa 998, Trfnac and Ivanovć 3) 3

5 Fg Vew of Van Nys Seven Story Hotel (VN7SH) from North-East The bldng s stated on ndfferentated Holocene allvm, ncemented and nconsoldated, wth a thckness of < 3 m, and an age of <, years (Trfnac and Todorovska 998) The average shear-wave velocty n the top 3 m of sol s 3 m/s, and the sol-borng log shows that the nderlyng sol conssts prmarly of fne sandy slts and slty fne sands The fondaton system conssts of 965-cm deep ple caps, spported by grops of two to for pored-n-place 6-cm-dameter renforced concrete frcton ples These are centered nder the man bldng colmns All of the ple caps are connected by a grd of beams Each ple s roghly m long and has a desgn capacty of over N vertcal load and p to N lateral load The strctre s constrcted of normal-weght renforced concrete (Blme and Assoc 973; Table ) Earthqake Damage The M L = 64 Northrdge earthqake of Janary 7, 994 severely damaged the bldng The strctral damage was etensve n the eteror north (D) and soth (A) frames that were desgned to take most of the lateral load n the longtdnal (EW) drecton Severe shear cracks occrred at the mddle colmns of frame A, near the contact wth the spandrel beam of the 5 th floor (Fgs 3 and 4) Those cracks sgnfcantly decreased the aal, moment, and shear capacty of the colmns The shear cracks that appeared n the north (D) 4

6 Table Propertes of the constrcton materals of the VN7SH bldng Concrete (reglar weght, 5 pcf () ) Locaton n the strctre Mnmm specfed compressve strength f c ps () Modls of elastcty E ps () Colmns, st to nd floors 5, 4 6 Colmns, nd to 3 rd floors 4, 37 6 Beams and slabs, nd floor 4, 37 6 All other concrete, 3 rd floor to roof 3, 33 6 Renforcng steel Locaton n the strctre Beams and slabs Colmn bars Grade Intermedate grade deformed bllet bars (ASTM A-5 and A-35) Mnmm specfed yeld strength f y ks (3) Modls of elastcty E ps () Deformed bllet bars (ASTM A-43) () Ponds per cbc foot () Ponds per sqare nch (3) Kps per sqare nch frame on the 3 rd and 4 th floors and the damage to colmns D, D3, and D4 on the st floor cased mnor to moderate changes n the capactes of these strctral elements No major damage to the nteror longtdnal (B and C) frames was observed, and there was no vsble damage to the slabs or arond the fondaton The nonstrctral damage was also sgnfcant The recorded peak acceleratons n the bldng were 46g (L), 4g (T), and 8g (V) at the base, and 59g (L) and 58g (T) at the roof, along the longtdnal (L), transverse (T), and vertcal (V) aes of symmetry (there were no sensors nstalled on the roof to measre vertcal motons) (Trfnac et al 999b) Photographs and detaled descrptons of the damage from the earthqake can be fond n Trfnac et al (999b) and Trfnac and Hao () Analyss of the relatonshp between the observed damage and the changes n eqvalent vertcal shear-wave velocty n the bldng can be fond n Ivanovć et al (999b), and Todorovska and Trfnac (6d) A dscsson of the 5

7 4 m 5 65 m 3 m 64 m 9 m D a) C N B A 8 57 m = 457 m b) N D C B A roof 3 cm slab 7th 6 cm slab 6th 6 cm slab 5th 6 cm slab 4th 6 cm slab 3rd 6 cm slab nd 54 cm slab st cm slab 635 m 635 m 635 m 9 m Fg Typcal Floor Plan (a), and North-Soth secton (b) etent to whch ths damage has contrbted to the changes n the apparent perod of the solstrctre system can be fond n Trfnac et al [a,b] Strong-Moton Records The response of VN7SH was recorded by a 3-channel CR- central recordng system and by one tr-component SMA- accelerograph, wth an ndependent recordng system bt wth common trgger tme wth the CR- recorder (Trfnac et al 999b) 6

8 roof FRAME D (North vew) th 6th " " shear cracks <5 cm 5th " " shear cracks <5 cm cracks throgh the beam 5 - cm 4th 3rd nd " " shear cracks <5 cm dagonal cracks along the colmn, 5 cm " " shear cracks <5 cm dagonal cracks n beams <5 cm cracks along the colmn, 5 cm cracks throgh the beam 5 - cm 'short colmn' cracks, 5 - cm st 6 cracks between brcks cracks between brcks roof FRAME A (Soth vew) th 6th "" - shear cracks ~ 5 cm, bendng of long renforcement " " shear craks ~5 cm "" - shear cracks ~ 5 cm, bendng of long and trans renforcement 5th 4th "" - shear cracks > 5 cm, bendng of long renforcement "" - shear cracks ~ 5 cm, bendng of long renforcement cracks n the beam < cm 3rd nd cracks n the beam < cm st complete frst floor on the soth sde was covered - there may have been some damage there 6 Fg 3 Observed Damage of frames D (top) and A (bottom) 7

9 The locatons of the fve transdcers, whch recorded EW response of the bldng drng earthqake, s shown n Fg 3 Roof Fg 4 Post-earthqake vew of damaged colmns A7 and A8 3 PREVIOUS WORK Snce 994, the VN7SH bldng has emerged as a sefl benchmark for comparson of dfferent analyses Trfnac et al (999b) and Trfnac and Hao () presented photographs of the damage followng the earthqake and collected all of the strong-moton data dgtzed ths far for the perod between 97 and 994 Two fll-scale ambent vbraton tests were performed 8

10 (Ivanovć et al, 999a; ) Drng the second ambent vbraton srvey, measrements of wave moton throgh the bldng fondaton showed that the fondaton s fleble and deforms wth the passage of mcro-tremor waves, whch ndcates that for stdes of solstrctre nteracton the rgd fondaton assmpton may not be approprate (Trfnac et al, 999a) The apparent perod of the sol-strctre system and ts dependence pon the response ampltdes n VN7SH were descrbed by Trfnac et al (a,b), and an applcaton of off-lne and on-lne dentfcaton technqes to the bldng response data n VN7SH was presented by Loh and Ln (996) A contnm mechancs representaton of VN7SH was consdered n terms of sotropc and ansotropc two-dmensonal models and ther response to ncdent wave moton by Todorovska et al (a,b) The feasblty of dentfyng the observed damage throgh wave propagaton stdes sng recorded earthqake responses was eplored n Ivanovć et al (999b) and Trfnac et al (3) The accontng of ncdent-wave energy, ts redstrbton among dfferent response energes, and the power of ncdent-wave moton and ts capacty to damage the VN7SH strctre has been descrbed n Trfnac et al (c) The engneerng stdes of VN7SH have focsed manly on ts longtdnal (EW) response Wthot ecepton, these stdes have neglected the effect of sol-strctre nteracton and have mplctly assmed that all non-lneartes n the observed response are assocated wth the bldng strctre Islam (996) consdered two two-dmensonal models for EW response of the bldng, one wth and one wthot the brck nfll walls n the for bays of the northern permeter frame D (see Fgs and 3) Assmng the bldng to be fed at the grond floor level, he sed the tranglar dstrbted horzontal load to perform a psh-over analyss Fgre 5 shows hs reslts for V/W, the resltng base shear (V), normalzed by the approprate fracton of bldng weght (W), verss roof dsplacement, assmng that the soth permeter frame (A) ressts one thrd of the lateral load Fgre 6 shows the story drfts calclated by Islam, sng elastc tme hstory analyss, at the center of mass (COM), the NE corner, and the SW corner, for the two models, wth and wthot the nfll walls n frame D For comparson, we also show n ths fgre the observed drft ampltdes, evalated on Febrary 4, 994, based on detaled analyss of the cracks, the scratches cased by relatve moton on the partton walls, and the marks on the nteror appearance of the permeter walls (Trfnac and Ivanovć 3) Islam conclded that many of the strctral elements may have eceeded ther elastc lmt state at appromately 4 seconds nto the earthqake However, the most severe damage eg, breakdown of the entre load path n the soth permeter frame colmns mmedately below the 5th floor level may have actally occrred at appromately 9 seconds, whch concdes wth the tme of the peak grond acceleraton n the longtdnal drecton He also notes that a psh-over analyss performed on the longtdnal frame wth a tranglar load pattern was nable to predct the damage observed n the bldng 9

11 V/W L and Jrsa (998) performed a non-lnear tme hstory analyss of VN7SH n the longtdnal (EW) drecton only becase most of the damage occrred n ths drecton Acceleraton tme hstores recorded at grond level were sed as npt grond moton, and colmns were assmed to be fed at the base Sol-strctre nteracton was not nclded n 6 Ths stdy (nform) UBC-94, V/W=54 Frst colmn yeld Eteror colmn falre Model A Frst colmn yeldng Soth permeter frame (Islam, 996) (tranglar load; V s assmed to resst W/3) Frst beam yeld Frst beam yeldng Model B 7 Ths stdy (tranglar) Formaton of mechansm Mamm recorded dsplacement drng the 994 Northrdge earthqake 6 L and Jrsa (998) Frst colmn shear falre Tranglar Unform Islam (996) Brownng () Model A Model B 8 Roof Dsplacement (n) Fg 5 Base shear (V) coeffcent normalzed by total bldng weght, W, verss EW roof dsplacement of VN7SH (Ths stdy; Islam, 996; L and Jrsa, 998; and Brownng et al, : Models A and B) the models Effectve stffness and resdal lateral capacty were chosen so that the perod of calclated response-tme hstory wold match the recorded tme hstory, and 35 EI g was chosen as an effectve stffness for all beams and colmns Ther analyss was two-dmensonal, and therefore no torsonal effects of ectaton and of response cold be nclded Calclated mamm drft ratos from L and Jrsa (998) are reprodced n Fgre 6, where they can be compared wth other drft estmates The athors nterpreted ther reslts to ndcate that the bldng was very close to collapse Therefore, the drft lmt recommended by NEHRP-94 appears to be reasonable as a lfe safety lmt for ths strctre Whle descrbng ther psh-

12 Story Level over analyss (see Fg 5), L and Jrsa stated that Psh-over analyss sccessflly predcted that the strctre almost lost ts lateral load-resstng capacty, and the shear falres of colmns occrred pror to reachng the mamm roof dsplacement the bldng eperenced drng the earthqake Brownng et al () at tme of mamm roof dsplacement 8 Approach A: absolte mamm Drft lmt set by NEHRP THIS STUDY Measred Low-pass L and Jrsa (998) COM (wth nfll), Islam (996) EW De la Llera et al () Approach A: COM (no nfll), Islam (996) SW corner (wth nfll), Islam (996) L and Jrsa (998) 8 Drft Rato (%) Fg 6 Comparson of dfferent estmates of EW drft n VN7SH drng 994 Northrdge earthqake Brownng et al () compared three ndependent analyses, ncldng ther own reslts, wth regard to the response of VN7SH to the Northrdge earthqake: Approach A (by Lynn and Moehle); Approach B (by Brownng and Sozen); and Approach C (by L and Jrsa) Becase approach C has already been smmarzed, we wll descrbe brefly only the reslts of the analyses based on approaches A and B Approach A dealzes the bldng as a two-dmensonal frame and consders only longtdnal (nteror and eteror) framng lnes A smple b-lnear relaton wthot stffness or strength degradaton s sed to descrbe load deformaton propertes of the frames The fondaton s assmed to be rgd that s, no sol-strctre nteracton s consdered, and the athors sed tranglar load dstrbton wth monotoncally ncreasng ampltde n ther psh-over analyss (Fg 5) Dynamc non-lnear response hstores were compted for the moton measred at the base of the bldng, and drft ampltdes were

13 Floor compted () at the tme of mamm roof response, and () sng absolte mama of drft over tme The range between these two estmates s shown as the gray zone n Fg 6 Approach B sed a model geometry smlar to that of model A, bt the n-fll was assmed not to contrbte to resstance to the lateral forces A Takeda non-lnear model wth nloadng stffness redcton eqal to 4 was adopted, and non-lnear statc and dynamc response analyses were condcted The reslts of the psh-over analyss are shown n Fg 5 Approach A gves smaller drft between the frst and second floors relatve to all other drft predctons shown n Fg 6 Between the thrd and ffth floors, t s n ecellent agreement wth the drfts predcted by L and Jrsa (993) and Islam (996) Brownng et al () Model A Model B Model C THIS STUDY 4 Islam (996) 3 Recorded De la Llera et al () Dsplacement (n) Fg 7 Comparson of recorded and calclated mamm floor dsplacements n VN7SH drng the Northrdge 994 earthqake De la Llera et al () noted that planar analyses of the bldng reported prevosly are obvosly not capable of predctng torsonal moton They developed an dealzaton of the bldng consstng of a sngle colmn-lke element (SEM) connectng two consectve floors

14 and sed ths model to nterpret the three-dmensonal response of the VN7SH bldng to the earthqake Ther estmates of the envelopes of mamm floor drfts are shown n Fg 6, and ther estmates of mamm floor dsplacements sng the sngle-element model are compared n Fg 7 wth recorded and other calclated relatve dsplacements at the tme of mamm response Ther estmates of EW (longtdnal) drfts are n far agreement wth the correspondng estmates of Islam (996) As n all prevos nvestgatons of the response of ths bldng, De la Llera et al () gnored sol-strctre nteracton effects n ther analyses of translatonal and torsonal responses They nterpreted the recorded motons to show that the mamm acceleraton along plane A (or D) de to torson s g, whch s abot 4% of that de to translaton Becase the bldng s symmetrc, De la Llera et al () conclded that these sgnfcant torsonal motons mst be attrbted prmarly to the yeldng of the strctre, predomnantly n one resstng plane (plane A) That s, they epressed the belef that these sgnfcant torsonal motons began at abot 4 s nto the recorded strong moton drng the earthqake and that they were followed by another epsode of large rotatons startng at abot 75 s, the tme when most of the damage occrred Persal of the analyss of drfts n ths bldng (Trfnac and Ivanovć 3) wll show, however, that the relatve torsonal response was promnent drng all prevos recorded responses n ths bldng, as was the case drng the Whtter 987, Landers 99, and Bg Bear 99 earthqakes, when the VN7SH bldng eperenced no damage Ths, whle the eccentrcty cased by damage mst have contrbted to the coplng of translatonal and torsonal responses drng the Northrdge earthqake (Trfnac et al, 999b), ths ncreasng the relatve torsonal response, the blk of the observed torsonal response appears to have reslted from the wave-passage effects alone (Todorovska et al, a,b) 4 ONE-DIMENSIONAL WAVE-PROPAGATION MODEL The locaton of the nstrments that recorded the EW response s shown n Fg 3 To search for the model parameters, we calclate the EW response of the bldng by assmng that the drvng moton can be appromated by strong moton recorded at the basement (channel 6) and then comparng the reslts wth the recorded motons at hgher floors n the bldng (channel at the frst floor, channel at the second floor, channel at the ffth floor, and channel 9 at the roof) The assmpton that the fondaton moves accordng to the recorded dsplacement at the locaton of channel 6 s eqvalent to assmng that () grond s rgd, () there s no sol-strctre nteracton, and (3) all wave energy created by the moton of the base (grond floor) s trapped nsde the bldng These assmptons are analogos to what many prevos nvestgators assmed n the analyses of the response of ths bldng (Islam 996, L and Jrsa 993, 3

15 Brownng et al, De la Llera et al ), bt they are physcally not correct Under deal condtons, f one cold select a bldng model that wold delay and modfy the waves propagatng p and down the bldng, eactly as wold the real bldng, the phase of the downpropagatng waves wold be correct, and the moton wold nterfere wth the drvng dsplacement of the grond floor, eactly as ths wold occr n the presence of fleble sol, wth sol-strctre nteracton, and wth the transparent bondary at the base, allowng the downpropagatng waves to be transmtted nto the half space Or model wll not be able to acheve all of ths, however, and therefore or analyss s only appromate The one-dmensonal model we chose for ths analyss conssts of forteen layers representng seven floors, wth dmensons (thckness) of the layers h(), veloctes of shear waves β (), and materal denstes ρ (), for 4, where = k (k =,,,7) represent nter-story space ( soft layers), and = k (k =,,,7) represent floor slabs ( hard layers) (see Table 4) The velocty of shear waves and the densty of the layers we adopted for ths work are based on the analyss of the mplse response for EW recorded motons n VN7SH (Todorovska and Trfnac 6d) As can be seen from Table 4, the stffness of the floor slabs s mch larger than the stffness of the nter-story space, so t can be epected that the floors wll move wth lttle elastc deformaton and neglgble contrbton to drft We assme the b-lnear constttve law σ = σ(), where σ s shear stress and s shear stran at a grd pont (Fg 8) The frst slope, µ b, represents the lnear (ntal) shear modls, whle the second slope, µ b = γµ b, represents the shear modls after yeldng The yeldng stran n the bldng s yb The ponts at the contacts between the layers are assmed to be n a lnear state To mantan the contnty of the dsplacements and stresses at the contacts between the layers, we ntrodce the eqvalent shear modl at the contact pont as b( ) ( ) µ b( ) µ b() µ c =, (4) µ µ b() where µ ob ( ) and µ ob() are ntal shear modl of the layers above and below the contact pont, and are the spatal ntervals above and below, and µ c s the ntal eqvalent shear modls of the contact pont 4

16 σ (KPa) µ b =γµ b µ b µ c µ b µ b yb µ b Fg 8 The constttve laws, σ, for the bldng (sold lne) and for the contacts (dashed lne) The eqaton of moton s v t = ( σ ) ρ, (4a) and the relaton between the dervatves of the stran and of the velocty s t = v, (4b) 5

17 where v, ρ, σ, and are partcle velocty, densty, shear stress, and shear stran, respectvely, and the sbscrpts t and represent dervatves wth respect to tme and space v = and t = are the velocty and the stran of a partcle, and s ot-of-plane dsplacement of a partcle along the propagaton ray Table 4 One-dmensonal bldng model Inte-rstory Heght h nterstory (m) Slab Thckness h slab (m) βnter-story βslab ρnter-story ρslab (m/s) (m/s) (kg/m3) (kg/m3) Roof slab Seventh story Seventh floor slab Sth story Sth floor slab Ffth story Ffth floor slab Forth story Forth floor slab Thrd story Thrd floor slab Second story Second floor slab Frst story The La-Wendroff O( t, ) fnte dfference method (Gcev 5, La and Wendroff 964) for a set of smltaneos eqatons s then sed to solve the problem The tme ncrement s obtaned from the mnmm rato: l t = β l β 4 =, (43) mn 4 where the sbscrpts stand for the layer nmbers conted from the base 6

18 In the nmercal model, the bldng s dscretzed as follows The floor layers are dscretzed wth three spatal ntervals, whle the nmber of the eqal spatal ntervals of the nterstory layers s n l 3 h l / h4 =, where sbscrpt stands for the layer nmber In ths way, we obtan an eqdstant grd n the nter-story layers, so that for layer l, for eample, the spatal nterval s h l h4 l = = (44a) n 3 l At the floor levels, the spatal nterval s h l l =, (44b) 3 and at the contact ponts the spatal nterval s =, (44c) where stands for the ordered nmber of the contact ponts The nmercal tests have shown that ths dscretzaton leads to good and stable reslts In ths way, the heght of the bldng s dscretzed by a grd havng 95 ponts and 94 ntervals Above the top pont ( N = 95), an addtonal dmmy pont N s ntrodced, at dstance 4 For a stress-free pont N, for all tme, the veloctes and the stress at the pont N are pdated as v (45a) N ' = v N σ (45b) N ' = σ N Eqatons (4) can be wrtten n vector form as U F = t, (46) where v U = and σ F = ρ (47) v 7

19 8 The vector U at pont n tme (j) t epanded n Taylor seres s ) ( 3,,,, t O t t t t j j j j = U U U U, and from Eq(46) ) ( 3,,,, t O t t t j j j j = F F U U ) ( ) ( 3,,,, t O t t j j j j = F A U F U U (48) A(U) s the Jacoban matr = = = ) ( σ ρ ρ σ ρ σ d d v v v v U F U A (49) Becase the tme hstory of the ncdent-wave dsplacements at the base s specfed a pror and s assmed to be gven by the recorded moton n channel 6 (Fg 3), and becase the top of the bldng s a stress-free bondary, or problem s a bondary-vale problem wth med bondary condtons, prescrbed fncton (dsplacement) at the bottom, and prescrbed zero dervatve (of the dsplacement) at the top Assmng that the stffness and denstes of each floor are known, or goal s to fnd the best vales of the parameters yb and γ (Fg 8) of the b-lnear constttve law of the -D model, so that the compted response agrees wth the recorded one 5 ERROR ANALYSIS The natre of the problem we are stdyng s three-dmensonal, and ts constttve relatons are not b-lnear Therefore, we cannot epect that the -D model wll be able to replcate the recorded dsplacements eactly Wth chosen constrants, we se the recorded dsplacements to fnd the best vales of the parameters yb = and γ, whch wll be assmed to be same for all soft layers (Fg 8) The total dsplacements (lnear non-lnear) obtaned by or -D model verss tme are shown n Fg 9a The correspondng plastc stran s shown n Fg 9b Both plots are presented for the fll draton of the record (t = 6 s) It can be seen that the largest response occrs drng the frst s

20 We search for the best vales of and γ by a tral-and-error procedre n the regons 45, wth ncrements = 5, and 5 γ 5, wth ncrements γ = 5 For ths range, we obtan the model responses for ths coarse grd contanng the data for 8 dfferent models Becase the blk of strong moton energy s concentrated n the frst s of the recorded dsplacements (Fg 9), n the followng we lmt or search to those frst s only For each of the mesh ponts, we obtan the synthetc response m (), t t s at the locatons of the and then compte the error vectors recorded responses rm (), t t s () t m () t r m () t e m = To qantfy the for error vectors and to estmate the error, at the second, thrd, and sth floors, and at the roof (the respectve locatons of the recordng channels Ch, Ch, Ch, and Ch9), we compte ther norm m ( e m, e m ) ( ), err = The total error for each pont γ s then chosen to be the sm of the errors n the for specfed channels, err = err m The dependence of err = err ( γ, ) on γ and error for ths coarse grd s at ( γ, ) = (4, 5) 4 m= s llstrated n Fga The mnmm of the Net, we vary and γ n the regons 3, wth ncrement =, and 38 γ 47, wth ncrement γ = Ths gves a fne grd wth ponts ( γ, ) As for the coarse grd, we obtan the err err ( γ, ) mnmm of ths error appears at ( γ, ) = (43, ), and ts vale s err mn = 49 The reslts for err ( ) =, whch s llstrated n Fg b The m γ, at the locatons of the recordng nstrments for the fne grd are presented n Fg The trend s that for the pper floors (Ch9 and Ch) the mnmm error γ = (43, ), whle for Ch the mnmm error appears at γ = 4 at smaller, and at Ch at ( γ, ) = (45, ) Usng the mnmm cmlatve error, we appears at ( ), choose the yeldng stran = and γ = 43 as an appromaton for the constttve law σ = σ n all layers of the materal ( ) 9

21 a) t (s) (m) b) t (s) (m) Fg 9 a) Dsplacement along the beam (n meters) verss tme b) Permanent stran along the beam verss tme

22 err a) err b) 35 3 γ (err ) mn = γ Fg -norm of the cmlatve error vector as a fncton of γ, and the mamm elastc stran, : a) wth coarse grd ( γ, ), and b) fne grd ( γ, )

23 err 35 Ch 9 3 err 5 38 err 6 Ch 39 4 Ch 4 4 err Ch Fg Errors at the recordng channels 9,, and, verss γ and

24 Dsplacement - m - Roof - Ch9-6-th floor Ch - 3-rd floor Ch recorded compted - -nd floor Ch recorded - Grond floor Ch6 5 5 Tme - s Fg Comparson of recorded (fne lne) and compted (wde lne) dsplacements 3

25 In Fg, the tme hstores of the recorded (sold lne) and compted (wde lne) dsplacements are shown for all recordng channels, for = and γ = 43 We note that the error can be redced frther by varyng and γ n dfferent layers, and by searchng for tme dependence of and γ We wll present sch refnements n or ftre stdes, whch wll be based on more detaled models 6 RESULTS In Fg3, the feld of the total dsplacements s shown for all beam ponts Fg 4 shows the feld of the total dsplacements at the ponts where yeldng occrred Both plots descrbe the frst (m) (m) t (s) Fg 3 Total dsplacement along the beam drng the frst s of response 4

26 s of moton, and t can be seen that p to abot t = 4 s the response s lnear, whle startng at abot t = 4 s all the ponts n the nter-story layers yeld The floor slabs reman lnear (m) (m) t (s) Fg 4 Dsplacement along the beam where permanent stran occrred, drng the frst s of response The largest absolte dsplacement at the top of the bldng occrs at t = 938 s, and the dsplacement along the beam, () at ths nstant s shown n Fg 5a Also, n ths fgre, the resdal dsplacement along the beam after the end of the ectaton s plotted sng a dashed 5

27 (m) lne The largest permanent (nonreversble) strans occr n the forth- and ffth-story colmns (seventh and nnth layers), and ther vales are p (Fg 5b) - - (m) Fg 5a Dsplacement along the the beat at t = 938 s when the mamm occrs at the roof (sold lne), and s after the end of ectaton, at t = 6 s (dashed lne) 6

28 -5 p (m) Fg 5b Permanent stran along the beam at t =938 s, when mamm dsplacement occrs at the roof 7

29 Fg 6 shows the permanent stran, p, along the beam verss tme Up to appromately 4 s, there s no permanent stran At abot 4 s, all soft layers yeld From 4 to abot 85 s, the permanent strans are relatvely low, and from 85 to s permanent strans ncrease frther After s, the strans n the bldng are small agan (Fg6) perm (m) t (s) Fg 6 Permanent stran perm along the bldng drng the frst s of ectaton Fg 7 shows the permanent dctlty, defned as the rato of the permanent stran and the yeldng stran, p µ =, (6) y 8

30 as a fncton of tme Ths type of plot s sefl from the damage detecton pont of vew becase t shows when and where the falre n strctral members occrred In Fg 7a, the postve vales, and n Fg 7b the negatve vales of permanent dctlty are shown µ = perm / (m) 5 5 t (s) Fg 7a The rato µ = perm / along the bldng drng the frst s of ectaton Postve vales 9

31 34 µ = perm / (m) t (s) Fg 7b The rato µ = perm / along the bldng drng the frst s of ectaton Negatve vales As can be seen, at t = 938 s, when the mama of the permanent stran occrred, the dctlty was the largest and had the vale µ = 99 n the colmns of the forth floor, jst below the spandrel beam, and the ffth floor slab (n the seventh layer of the model, Table 4) The dctlty n the colmns of the ffth floor (nnth layer of the model), was µ = 3, close to the largest dctlty jst below The mamm of the permanent stran occrred at t = 858 s Earler, at t = 49 s, at the same locaton, the dctlty reached a postve vale of µ = 7, whle at t = 359 s the largest negatve dctlty, µ = 5 99, occrred n the forth-story colmns 3

32 (seventh layer n the model) These were large dctltes, and as we cold see after the earthqake, the forth-story colmns were severely damaged, startng at t = 359 s and t = 49 s and then throgh t = 938 s (drft()) ma 5 (m) 5 t ma (s) Fg 8 The nterstory drfts along the bldng, at tmes when the mama occr Fg 8 shows the mama of the drfts along the bldng heght, (m), where drft(l) l l =, and where l stands for the layer nmber, at the tme when those occrred h l It can be seen that wthn abot s (from t = 85 s to t = 95 s) all colmns eperenced ther mamm drfts In Fg 9, these drfts are plotted verss the heght of the bldng, and the tmes when the mama occrred are shown The resdal drfts, after the shakng stopped, are shown by a dotted lne It can be seen that the largest drfts, eqal to 3, occrred n the seventh layer (colmns of the forth story) at tme t 4 = 864 s 3

33 drft t 4 = 864 s t5 = 9 s t 3 = 856 s t 6 = 93 s t = 846 s t = 85 s t 7 = 936 s (m) Fg 9 Mama of the nterstory drfts n the Holday Inn bldng drng Northrdge earthqake, showng tme when those occr (sold lne), and at two seconds after the end of ectaton, at t = 6 s (dashed lne) Fgres 8 and 9 show that the peak nter-story drfts propagated p, apparently as a strongmoton plse entered and propagated p the bldng wth velocty of abot 55 m/s, reachng the mamm vales between 846 s and 884 s The largest drft occrred between the forth and ffth floors, precsely at the locaton where the damage cold be seen after the earthqake (Fg 3

34 4) Upon reflecton from the roof, the dsplacements (and drft) ampltdes dobled, as ths brst of strong-moton energy contned on ts path downward Ths doblng of the ampltdes n trn ncreased the drfts at the ffth, sth, and seventh floors (Fgs 8 and 9) As ths downpropagatng energy encontered the broken colmns (jst below the ffth floor slab) t lost enogh energy, and n ts propagaton down t dd not frther ncrease the peak drfts at the ffth throgh frst floors, whch were created drng the precedng to 3 s The above apparent velocty of 55 m/s, assocated wth pward propagaton of peak stran, s n agreement wth the average speed of shear waves, 548 m/s, drng the tme wndow of 7 < t < s (Todorovska and Trfnac 6d), estmated va mplse-response travel-tme analyss Ths shows that the occrrence of peak strans s assocated wth propagaton of promnent grond moton plses and the waves those plses prodce n the bldngs It shows that the peak drfts can occr drng one travel tme, p or down the bldng, or drng one qarter of the fndamental, fed-base perod of the bldng, T /4 Nether the vertcal dstrbton of the peak drft ampltdes, nor the tme of ther occrrence, can be descrbed drectly by the Response Spectrm Method of analyss In engneerng desgn, the drfts can be sed to calclate the nter-story forces Assmng fedfed colmn ends, these forces are EI h 3 l ( l l ) = Fl, (6a) or, sng the drft, EI h 3 l F h l drft = (6b) l In the above eqatons, E s the modls of elastcty, I s the moment of nerta of the colmn cross secton, s dsplacement of the rgd floor, h s the heght of the colmn, and F s the nter-story force The sbscrpt l stands for the order nmber of the colmn Havng E and I, all of the nter-story forces can be evalated, and the eqvalent statc analyss of the bldng can be performed Smmarzng the above reslts, the mamm dctltes (Fg7a,b) and the mamm nterstory drfts (Fg9), as well as the ratos between the mamm strans ( µ ma ) y ma y p = = (see the defnton of the permanent dctlty n Eq 6) and the mamm nter-story drfts at dfferent layers are as shown n Table 6 33

35 Layer µ ma (permanent dctlty Fg 7a,b) Table 6 drft ( ) ma µ drft ma ma y As wold be epected, t can be seen from the last colmn n ths table that the mamm strans are appromately eqal to the mamm drfts 7 PUSH-OVER ANALYSIS For the statc psh-over analyss, we se the propertes of the bldng (Table 4) represented by one-dmensonal shear-beam model For the wave-propagaton analyss, the bondary condton at the bottom was the prescrbed dsplacement For the statc analyss, there s zero dsplacement at ths bondary At the top of the beam, we se the same bondary condton as that sed for the wave propagaton analyss zero dervatve of the dsplacement wth respect to the vertcal coordnate We descrbe the statc load as follows We defne f ( Hb) = f, and f () =, where f ( ) s the horzontal force per nt volme For F = fr W, where F s horzontal force per nt sqare (wdth =, length = ) of the bldng, fr s the fracton < fr <, and weght of the bldng per nt sqare (wdth =, length = ), we have F f = H b From eqlbrm at an arbtrary pont and for tranglar dstrbton of statc load σ σ d σ dy dz = f () d dy dz (7) b 4 W = ρ h s the b f = H b = F and 34

36 35 or ( ) f = σ, (7) where ( ) b H f f =, and for a cell,, appromatng (7) by Fnte Dfferences (FD): ( ) / / f = σ σ (73) Assmng lnear stress and appromatng (73), we have: b / / H f = µ µ (74) By solvng the system F A =, where A s the tr-dagonal (n -,n - ) matr = N N N N,,,, 3,3, 3,3 A µ µ µ µ µ µ µ µ µ and

37 36 = N 3 b H f F s the load vector, we can fnd the dsplacement vector = N 3 and = (75) Ths s the lnear solton The net step s to compare the strans, = ; 3 3,3 = ; ;, = ; ; N N N N,N N = wth the yeldng stran y = (see the descrpton of the tral-and-error procedre n secton 5, whch gves the best vales for (, ) (43,) y γ = If y, >, the dsplacements n Eq (75) shold be recalclated by rearrangng the coeffcents of the matr A (whch are related to the dsplacements and ) by addng a term n the vector F at the locaton of f and f, as n the followng The non-lnear statc stress between the ponts and for the b-lnear constttve law can be epressed as

38 37 ( ) p y /, γ = µ σ, (76) where n or eample y = s the yeldng stran that s assmed to be the same everywhere n the bldng Becase y y p = =, the eqaton (76) can be rewrtten as ( ) y / /, γ µ γ = µ σ, whch means that the system of eqatons correspondng to the th and ( ) st term wll be modfed as = µ γµ µ γµ γµ µ γµ µ N,,,,, =- ( ) ( ) γ µ γ µ H f y, y, b (77)

39 V / W Solvng Eq (77), we obtan the corrected statc solton for tranglar load vector The par (dsplacement N, fracton fr) then gves one pont on the psh-over crve for the tranglar load 6 Unform 7 Tranglar Roof Dsplacement (cm) Fg Psh-over crves for one-dmensonal model descrbed n Table 4 and for tranglar and nform load dstrbtons 38

40 For nform load vector, the system of eqatons s the same, ecept that for the load vector, all of the components become the same and eqal to W b f = H b The compted psh-over crves for horzontal nform load V f = and for tranglar load H V f = are shown n Fg The coordnates of the frst yeld ponts (crcles n Fg ) n H b dfferent nter-story layers are shown n Table 7 (for nform load) and Table 7 (for tranglar load) sorted by the ampltde of V / W b Table 7 Occrrence of yeld ponts n dfferent layers for nform load Inter-story layer V/W roof (cm) As can be seen from the tables, the frst yeldng occrs n the forth-story colmns for appromately the same roof dsplacement and for both the tranglar and nform load cases The yeldng then occrs net at the thrd story Table 7 Occrrence of yeld ponts n dfferent layers for tranglar load Inter-story layer V/W roof (cm)

41 The yeldng at the bottom (the frst story) occrs for the same horzontal shear force, and the top colmn yelds last n both load cases After the yeldng n the colmns of the forth and thrd stores has occrred, for tranglar load the pper stores (fve and s) yeld earler than the lower stores (two and one), whle for the nform load the lower stores yeld earler than the pper stores (floor one yelds earler than fve, and two yelds earler than s) 8 CONCLUSIONS Usng the smple one-dmensonal fnte-dfference representaton of a non-lnear model, we showed that t s possble to nterpret detals of the response of a real seven-story hotel bldng n Van Nys, Calforna, whch was ected by strong grond moton drng the 994 Northrdge earthqake The characterstc damage to ths bldng occrred n the forth-story colmns, jst below the spandrel beam along the ffth-story slab drng the tme nterval between 4 and 9 s from the begnnng of the shakng Wth or smple model, we were able to confrm these observatons and to show why ths falre occrred at ths locaton Ths stdy shows that the fnte dfference calclatons can be sed sccessflly even for the smplest representaton of a bldng and wth non-lnear wave propagaton throgh an eqvalent contnos model Ths also shows that even wth a smple wave-propagaton approach one can predct the response characterstcs that are obvosly present n bldngs drng earthqakes bt that cannot be decphered by a response analyss based on the concept of the response spectrm (Bot 93; 934; 94; 94) or by psh-over analyss, whch favors that part of the response that s close to the frst lnear mode shape The statc psh-over analyss of the layered shear-beam model of the VN7SH bldng predcts the frst occrrence of non-lnear response at the forth and thrd floors, whch s followed by the ntaton of the non-lnear response at the frst, ffth, second, sth, and fnally the seventh floors, for the assmed nform load dstrbton (and smlarly at the forth, thrd, ffth, sth, second, frst, and seventh floors for the tranglar load), for the roof dsplacements between 6 cm and 8 cm Or wave propagaton analyss shows, however, that the occrrence of non-lnear drfts s governed by the passage of large strong-moton plses that propagate p and down the bldng drng tme ntervals that are on the order of T /4, where T s the perod of the fedbase model of the bldng, n ts crrent lnear or non-lnear state of response In VN7SH, drng the 994 Northrdge earthqake, T ncreased from abot s, drng the frst 3 s of response, to 5 s, from 5 < t < 6 s of shakng Drng the tme wndow from 7 to s, when the largest response and most severe damage occrred, the average vale of T was 47 s, and T /4 was abot 37 s (Todorovska and Trfnac 6d) 4

42 Fg 5 shows that n terms of the psh-over analyss, the stffness of or FD model of VN7SH s n ecellent agreement wth all prevos stdes for relatve roof dsplacements p to abot 6 n Beyond ths ampltde, or model contnes to deform wth constant slope, determned by γ = 43 and the tranglar or nform dstrbton load, whle the dscrete models lead to the colmn falres and to formaton of mechansms The agreement of these approaches s to be epected, becase all prevos stdes as well as the present stdy mst satsfy the observed perod of vbraton drng the recorded earthqake response The prncpal dfferences between the present and the prevos stdes are the conseqence of the contnos (ths stdy) verss dscrete (all prevos stdes) representatons of the bldng model The contnos FD model can represent realstcally short waves n the bldng, whch makes t possble to consder, and to analyze, the conseqences of abrpt hgh-freqency plses assocated wth near-feld grond moton, resltng n lnear and non-lnear response The dscrete models, wth lmped mass at the floor levels, present a low-pass fltered vew of the short strong-moton plses, and ths cannot descrbe accrately for tme and space evolton of the non-lnear response Strong grond moton can be vewed as a seqence of plses that propagate p and down the bldng, reflect off the roof, and create zones of non-lnear response, whch depend pon the ampltdes and the draton of those plses Sch representatons not only broaden or nderstandng of the tre natre of the non-lnear response bt also allow s to predct and then to control certan nwanted featres of non-lnear response that are assocated wth stran localzaton, nterference, and amplfcaton of non-lnear deformatons The applcaton of the method we employed n ths paper s not lmted to eqvalent, homogeneos, layered representaton of the bldng We chose sch a smple model only to llstrate the method and to test ts potental for ectaton wth relatvely long ncdent sesmc waves Geometrcally far more detaled fnte-dfference models, n two and three dmensons, can easly be constrcted and mplemented REFERENCES Bot, MA (93) Vbratons of Bldngs Drng Earthqakes, Chapter II n PhD Thess No 59, enttled Transent Oscllatons n Elastc Systems, Aeronatcs Department, Calf Inst of Tech, Pasadena, Calforna Bot, MA (934) Theory of vbraton of bldngs drng earthqakes, Zetschrft für Angewandte Matematk nd Mechank, 4(4), Bot, MA (94) A mechancal analyzer for the predcton of earthqake stresses, Bll Sesm Soc Amer, 3, 5 7 4

43 4 Bot, MA (94) Analytcal and epermental methods n engneerng sesmology, ASCE Transactons, 8, Blme, JA, and Assoc (973) Holday Inn, Chapter 9 n San Fernando, Calforna Earthqake of Febrary 9, 97, Volme I, Part A, US Dept of Commerce, Natonal Oceanc and Atmospherc Admnstraton, Washngton, DC 6 Brownng, JA, L, RY, Lynn, A, and Moehle, JP () Performance assessment for a renforced concrete frame bldng Earthqake Spectra, 6(3), De la Llera, JC, Chopra, AK, and Almazan, JL () Three-dmensonal nelastc response of an RC bldng drng the Northrdge earthqake J of Strctral Eng, ASCE, 7(5), Gcev, V (5) Investgaton of Sol-Fleble Fondaton-Strctre Interacton for Incdent Plane SH Waves, PhD Dssertaton, Dept of Cvl Engneerng, Unv Sothern Calforna, Los Angeles, Calforna 9 Gcev, V, and Trfnac, MD (6) Permanent deformatons and strans n a shear bldng ected by a strong moton plse, Sol Dynamcs and Earthqake Engneerng, 6(), 49-6 Islam, MS (996) Analyss of the response of an nstrmented 7-story non-dctle concrete frame bldng damaged drng the Northrdge earthqake Professonal Paper 96-9, Los Angeles Tall Bldng Strctral Desgn Concl, 996 Annal Meetng Ivanovć, S, Trfnac, MD, Novkova, EI, Gladkov, AA, and Todorovska, MI (999a) Instrmented 7-Story Renforced Concrete Bldng n Van Nys, Calforna: Ambent Vbraton Srvey Followng the Damage from the 994 Northrdge earthqake, Dept of Cvl Eng, Rep No CE 99-3, Unv of Sothern Calforna, Los Angeles, Calforna Ivanovć, S, Trfnac, MD, and Todorovska, MI (999b) On dentfcaton of damage n strctres va wave travel tmes Proc NATO Workshop on Strong Moton Instrmentaton for Cvl Engneerng Strctres, Jne 5, Istanbl, Trkey, Klwer Academc Pb, Dordrecht, (), Ivanovć, S, Trfnac, MD, Novkova, EI, Gladkov, AA, and Todorovska, MI () Ambent vbraton tests of a seven story renforced concrete bldng n Van Nys, Calforna, damaged by the 994 Northrdge earthqake, Sol Dynamcs and Earthqake Eng, 9(6), Kana, K (965) Some new problems of sesmc vbratons of a strctre, Proc Thrd World Conf on Earthqake Engneerng, Ackland and Wellngton, New Zealand, Janary Febrary, 965, pp II-6 to II-75 5 La, PD, and Wendroff, B (964) Dfference Schemes for Hyperbolc Eqatons wth Hgh Order of Accracy, Comm on Pre and Appled Mathematcs, XVII, L, YR, and Jrsa, JO (998) Nonlnear analyses of an nstrmented strctre damaged n the 994 Northrdge earthqake, Earthqake Spectra, 4(), Loh, S-H, and Ln, H-M (996) Applcaton of off-lne and on-lne dentfcaton technqes to bldng sesmc response data, Earthqake Eng and Strctral Dynamcs, 5,

44 8 Mlhern, MR, and Maley, RP (973) Bldng Perod Measrements Before, Drng And After the San Fernando, Calforna, Earthqake of Febrary 9, 97, US Depart Of Commerce, Natonal Oceanc and Atmospherc Admnstraton, Washngton DC, Vol I, Part B, Todorovska, MI, and Trfnac, MD (6a) Damage detecton n the Imperal Conty Servces Bldng I: The data and tme-freqency analyss, Sol Dynamcs and Earthqake Eng, (n press) Todorovska, MI, and Trfnac, MD (6b) Damage detecton n the Imperal Conty Servces Bldng II: Analyss of noveltes va wavelets (sbmtted for pblcaton) Todorovska, MI, and Trfnac, MD (6c) Earthqake damage detecton n the mperal conty servces bldng III: Analyss of wave travel tmes va mplse response fnctons (sbmtted for pblcaton) Todorovska, MI, and Trfnac, MD (6d) Implse response analyss of the Van Nys 7-story hotel drng earthqakes (97-994): one-dmensonal wave propagaton and nferences on global and local redcton of stffness de to earthqake damage, Report CE 6-, Dept of Cvl Eng, Unversty of Sothern Calforna, Los Angeles, Calforna 3 Todorovska, MI, Ivanovć, SS, and Trfnac, MD (a) Wave propagaton n a sevenstory renforced concrete bldng: I Theoretcal models, Sol Dynam and Earthqake Engrg, (3), 3 4 Todorovska, ML, Ivanovć, SS, and Trfnac, MD (b) Wave propagaton n a sevenstory renforced concrete bldng: II Observed wavenmbers, Sol Dynam and Earthqake Engrg, (3), Trfnac, MD, and Hao, TY (c) 7-Story Renforced Concrete Bldng n Van Nys, Calforna: Photographs of the Damage from the 994 Northrdge Earthqake, Dept of Cvl Eng Report No CE -5 Unv of Sothern Calforna, Los Angeles, Calforna 6 Trfnac, MD, and Ivanovc, SS (3) Analyss of Drfts n a Seven-Story Renforced Concrete Strctre, Dept of Cvl Eng Report No CE 3-, Unv of Sothern Calforna, Los Angeles, Calforna 7 Trfnac, MD, and Todorovska, MI (998) Damage dstrbton drng the 994 Northrdge, Calforna, earthqake n relaton to generalzed categores of srface geology, Sol Dynam and Earthqake Engrg, 7(4), Trfnac, MD, Ivanovć, SS, Todorovska, MI, Novkova, EI, and Gladkov, AA (999a) Epermental evdence for fleblty of a bldng fondaton spported by concrete frcton ples, Sol Dynam and Earthqake Eng, 8, Trfnac, MD, Ivanovć, SS, and Todorovska, MI (999b) Seven-Story Renforced Concrete Bldng n Van Nys, Calforna: strong Moton Data Recorded Between 7 Feb 97 and 9 Dec, 994, and Descrpton of Damage Followng Northrdge 7 Janary 994 Earthqake, Dept of Cvl Eng Rep CE 99-, Unv of Sothern Calforna, Los Angeles, Calforna 3 Trfnac, MD, Ivanovć, SS, and Todorovska, MI (a) Apparent perods of a bldng, Part I: Forer analyss, J Strct Engrg, ASCE, 7(5),