ESTIMATION OF SEISMIC ACCELERATION DEMANDS IN BUILDING COMPONENTS

Transcription

1 3 th World Conference on Earthquake Engneerng Vancouver, B.C., Canada August -6, 4 Paper No. 399 ESTIMATION OF SEISMIC ACCELERATION DEMANDS IN BUILDING COMPONENTS Shahram TAGHAVI, Eduardo MIRANDA SUMMARY A method to estmate floor acceleraton demands n mult-story buldngs subjected to earthquakes s presented. In the proposed method, buldngs are modeled as a combnaton of a shear and fleural beams. The model s defned by three parameters: the fundamental perod of structure, dampng rato and lateral stffness rato. The accuracy of the method s then evaluated by comparng acceleratons computed wth the method to those measured n three nstrumented buldngs n Calforna. A parametrc study to evaluate the effects of these three parameters on sesmc acceleraton demands of buldngs ncludng peak floor acceleraton and floor response spectra s also presented. INTRODUCTION Nonstructural components typcally represent a major porton of the total cost of buldngs. Furthermore, nonstructural damage often occurs at response ntenstes that are smaller to those requred to produce structural damage. Therefore, t s not surprsng that when losses due to structural and nonstructural components are separated, losses due to nonstructural components have consstently been reported to be far greater than those resultng from structural damage (Ayers et al. [], Whtman et al. [], Rhal [3]). A large porton of nonstructural components and buldng contents are damaged prmarly as a result of beng subjected to large floor acceleraton demands. Components such as suspended celngs, lght ftures, fre sprnklers and parapets are eamples of acceleraton senstve components. Fgure shows photos of a few acceleraton senstve nonstructural components. The functonalty of many facltes such as hosptals depends on functonalty of these components. In the Olve Vew Medcal Center n Sylmar, Calforna, durng the 994 Northrdge earthquake, the water leakage from broken fre sprnkler and chlled water caused the faclty to shut down and forced patents to be evacuated (OSHPD 995 [4]). Smlarly, the San Francsco Internatonal arport was shut down for thrteen hours as a result of the 989 Loma Preta earthquake because of a power falure and nonstructural damage n the control tower, such as fallng celng tles and several broken wndows. Despte ther sgnfcance to control economc losses and downtme, sesmc behavor and desgn of nonstructural components has receved relatvely small attenton from researchers and practcng Graduate Student, Cvl Engneerng Dept., Stanford Unversty, Emal: shahramt@stanford.edu Assstant Professor, Cvl Engneerng Dep., Stanford Unversty, Emal: emranda@stanford.edu

2 engneers compared to the attenton that has been devoted to understand and mprove the sesmc behavor of structural members. Sesmc provsons provde smplfed procedures to estmate acceleraton demands on nonstructural components. In partcular, current U.S. sesmc provsons recommend the use of a trapezodal dstrbuton of peak floor acceleratons along the heght of the buldng and a floor response spectrum regardless of the number of stores n the buldng or ts lateral resstng system. Some lmted evdence suggests that ths varaton may be nadequate for some structures (Sngh [5]; Soong et al. [6]). Vllaverde [7] noted that these problems are due to the fact that these desgn-orented methods stll do not account for all the factors that sgnfcantly affect the response of nonstructural components. More recently, these provsons have seen severely crtczed by some practcng structural engneers (Kehoe and Freeman [8]; Searer and Freeman [9]) who concluded that the ntensty and dstrbuton of floor acceleratons over the heght of the buldng appears to be nfluenced by the predomnant perod of vbraton of the buldng and the mode shapes. Fgure Eamples of acceleraton senstve nonstructural components In ths paper, a smplfed method to estmate peak floor acceleraton demands and floor spectra ordnates n buldngs that are epected to reman elastc or practcally elastc when subjected to earthquake ground motons s presented. The appromate method s drectly relevant to the estmaton of sesmc demands on acceleraton-senstve nonstructural components attached to conventonal buldngs durng small and moderate earthquakes n whch the structure s epected to reman elastc or practcally elastc, as well as to the estmaton of sesmc demands on acceleraton-senstve nonstructural components n crtcal buldng facltes whch are desgned to reman elastc or practcally elastc even durng severe ground motons. The effcency and accuracy of the method s evaluated by a few eamples. A bref parametrc study to evaluate effects of fundamental perod of vbraton of buldngs and lateral resstng systems as well as stffness reducton on peak floor acceleraton and floor response spectra s also presented. APPROXIMATE ESTIMATION OF ACCELERATION DEMANDS Smplfed model of buldng In the method proposed here, mult-story buldngs are modeled usng an equvalent contnuum model consstng of a fleural cantlever beam and a shear cantlever beam deformng n bendng and shear confguratons, respectvely (Fgure ). It s assumed that lateral deformaton of fleural and shear beams

3 are dentcal. Floor masses are assumed to reman constant along the heght of the buldng. As shown n fgure 3, the appromate model used here has the advantage of beng able to consder not only the two etremes of deformaton (pure shear and pure fleure), but n addton t can consder buldngs whose lateral deformatons are a combnaton of fleural and shear deformaton. Fleural beam Shear beam H Lnks aally rgd Fgure - Smplfed model to estmate the dynamc propertes of multstory buldngs The dfferental equaton of the combned shear-fleural model used here was frst developed by Traum and Zalewsk [] and by Hedebrecht and Stafford Smth []. More recently, Mranda [] used the model to estmate mamum nterstory drft demands n buldngs subjected to earthquakes. He derved closed-form solutons for the lateral dsplacements normalzed by the dsplacement at the top of the structure and for the rato of the mamum rotaton demand to the roof drft rato (lateral dsplacement at the top dvded by the total heght) when subjected to a wde varety of statc lateral forces. Hs appromate method was more recently etended to buldngs wth non-unform lateral stffness (Mranda and Reyes [3]). (a) fleural-type deformatons (b) shear - type deformatons (c) combned fleural and shear - type deformatons Fgure 3 - Overall lateral deformatons n multstory buldngs The governng dynamc equaton of moton of the contnuum system wth unform lateral stffness shown n Fgure when subjected to a horzontal base acceleraton of ü g (t) s gven by the followng equaton: ρ u(, t) + EI t c EI u(, t) + t H 4 u(, t) α - H 4 u(, t) = ρ u - EI t g ( t) () where ρ() s the mass per unt length n the model, u(, t) s the lateral dsplacement at non-dmensonal heght (varyng between zero at the base of the buldng and one at roof level) at tme t, H s the total

4 heght of the buldng, c() s the dampng coeffcent per unt length, EI s the fleural rgdty at the base of the structure and α s the lateral stffness rato defned as: EI / GA α = H () where GA s the shear rgdty at the base of the structure. The lateral stffness rato s a dmensonless parameter α that controls the degree of partcpaton of overall fleural and overall shear deformatons n the smplfed model of mult-story buldngs and thus, t controls the lateral deflected shape of the buldng. A value of α equal to zero represents a pure fleural model (Euler-Bernoull beam) and a value equal to corresponds to a pure shear model. Intermedate values of α correspond to mult-story buldngs that combne shear and fleural deformatons. Dynamc characterstcs of smplfed model In the method proposed here, the dynamc propertes of multstory buldngs are appromated by those of the smplfed model dscussed n the prevous secton. For the case of unform lateral stffness, the dynamc characterstcs can be obtaned n closed form. In partcular, the mode shape assocated to the th mode of vbraton s gven by (Mranda and Taghav [4]): sn( γ ) γ ( ) = sn( γ ) γ / ( ) α + γ snh α + γ + η cosh α + γ cos( γ / ( ) α + γ snh α + γ + η cosh α + γ cos( γ ) φ (3) where η s defned as: ) γ sn( γ ) + γ η = γ cos( γ ) + (4) ( α + γ ) cosh( α + γ ) α + γ snh( α + γ ) γ s an egenvalue parameter assocated wth mode and the root of the followng characterstc equaton: 4 cos( ) cosh sn( )snh α + + = γ α + γ + γ α γ ( ) γ + γ α γ α + γ α (5) + Once γ s known for th mode of vbraton, the modal partcpaton factor and the perod rato of th mode are gven by: Γ = φ φ ( ) d ( ) d γ γ + α = γ γ + α T T (6) (7) Eamnaton of equatons 3 to 7 shows that mode shapes, modal partcpaton factors and perod ratos are fully defned by a sngle parameter, the lateral stffness rato, α (see fgures 4 and 5). Mranda and Reyes [3] have ndcated that ths parameter can be estmated based on the type of lateral resstng system n the buldng. Shear wall and braced frame buldngs usually have values of α between and.5; buldngs

5 wth dual structural systems consstng of a combnaton of moment-resstng frames and shear walls or a combnaton of moment-resstng frames and braced frames usually have values of α between.5 and 5; whereas moment-resstng frame buldngs usually have values of α between 5 and. Hence, the smplfed model presented n the prevous secton has the mportant advantage of allowng estmaton of the dynamc characterstc of a mult-story buldng based only on ts lateral resstng system and ts fundamental perod of vbraton α α = α = 55 α α = α α = φ φ φ 3 Fgure 4 - Effect of α on mode shapes when the lateral stffness remans constant. In the proposed method, floor acceleraton demands are appromated by only ncludng the frst few modes of vbraton. Therefore, usng modal analyss equatons, the absolute (total) floor acceleraton at non-dmensonal heght can be appromated as: N t u& (, t) u&& & ( t) + Γ φ ( ) D ( t) (8) g = & where (t) s the relatve acceleraton of the th mode SDOF system subjected to ground acceleraton. D In equaton 8, modal partcpaton factors and mode shapes are functons of lateral stffness rato, α and D & (t) s a functon of perod of the th mode whch s a functon of α and T, and modal dampng rato ξ. Therefore total acceleraton at a certan locaton can be computed by knowng fundamental perod of vbraton of the buldng T, lateral stffness rato α, modal dampng rato ξ and ground acceleraton. T /T ΓΓ ΓΓ. T/T T / T Γ3Γ T3/T T 3 / T α α Fgure 5 - Effect of α on modal partcpaton factors and perod ratos. Γ..5.

6 The computatonal effort n the proposed method s very small. In partcular, the computatonal effort s much smaller than the computatonal effort nvolved n the computaton of a lnear elastc response spectrum. Equatons 3 to 8 assume that the lateral stffness of the buldng remans constant along the heght of the buldng. Wth the ecepton of one to three story buldngs such assumpton s not usually realstc. Mranda and Taghav [4] studed the effect of reducton of stffness along the heght on the product of the mode shape and the modal partcpaton factor (product of equatons 3 and 6) and on perod ratos (equaton 7). They consdered lnear and parabolc reductons of stffness along the heght and up to 75% reducton n lateral stffness from the base to the roof. Ther study showed that reductons n lateral stffness along the heght have a relatvely small effect on the product of the mode shape and the modal partcpaton factor (product of equatons 3 and 6) and on perod ratos (equaton 7). Hence, usng Γ φ () and T / T computed from a unform model provdes a relatvely good appromaton to these dynamc propertes n non-unform buldngs. VALIDATION OF THE PROPOSED METHOD Accuracy of the proposed method s evaluated n ths secton by comparng floor acceleraton demands computed wth the contnuum model to those recorded n three nstrumented buldngs n Calforna. The frst buldng s a 3-story renforced concrete buldng n Emeryvlle that recorded the 989 Loma Preta earthquake. The second and thrd buldngs are 3-story renforced concrete buldng and 6-story steel buldng that were shaken by the 994 Northrdge earthquake. When the lateral stffness s assumed to reman constant along the heght of the buldng, the contnuum model used n the method s fully defned wth knowledge of only three parameters: the fundamental perod of the structure, the dampng rato and the lateral stffness rato α. As mentoned before, α can be appromated based on knowledge of the lateral resstng system. The parameters used for each of the buldngs are shown n table for each component. The fundamental perod of vbraton and the dampng rato of these buldngs correspond to those avalable n the lterature and the lateral stffness rato s based on the lateral resstng system of each buldng. Table Informaton of the buldngs used for evaluaton of the method No. of Structural Bldg. Locaton Earthquake Dr T stores (s) ξ (%) α System Reference N-S.59 3 MRF.5 [5] Emeryvlle 3 Loma Preta E-W.69 3 MRF.5 Sherman N-S 3. 5 MRF.5 [6] 3 Northrdge Oaks N-S.8 8 MRF.5 N-S.33 Dual System 3. [7] 3 Sylmar 6 Northrdge E-W.33 8 Dual System 3. Comparson of acceleraton demands Fgure 6 shows a comparson of peak floor acceleraton predcted wth the proposed method usng the parameters lsted n table wth peak recorded acceleratons. It can be seen that for all three buldngs and for both drectons the proposed method produces very good estmates. Also shown n the fgure are the peak floor acceleratons the floor acceleratons computed accordng to the FEMA-368 [8] assumng that the peak ground acceleraton s known. As shown n the fgure, these provsons recommend a lnear varaton of lateral acceleraton demands varyng from an acceleraton equal to peak ground acceleraton at the base to three tmes the peak ground acceleraton at the roof. As shown n the fgures, n many cases these recommendatons can lead to sgnfcant errors.

7 Floor NS 6 Appro. Recorded FEMA PFA (cm/s ) Floor EW Floor NS PFA (cm/s ) Floor EW 5 3 Floor NS 3 3 PFA (cm/s ) Floor EW PFA (cm/s ) PFA (cm/s ) 5 5 PFA (cm/s ) Fgure 6 Comparson of recorded peak floor acceleratons wth those of smplfed model and NEHRP provsons In addton to estmaton of peak floor acceleraton demands, the proposed method can also be used to estmate floor spectra and floor acceleraton tme hstores. Fgure 7 shows comparson of recorded and estmated acceleraton tme hstores at roof level n the 6-story Sylmar Medcal Center and the 3-story Sherman Oaks buldng. Consderng the smplcty of the method, the results are very promsng. Floor response spectra at roof level for the perpendcular drecton of the same buldngs are shown n fgure 8. As shown n ths fgure, the proposed method s also able to estmate floor spectra relatvely well. Fgure 7 Comparson of recorded acceleraton tme hstores wth those of smplfed model at roof levels of 6-story Sylmar and 3-story Sherman Oaks Buldngs

8 S FA (cm/s ) Appromate From Eact Records Perod (s) S FA (cm/s ) Appromate Eact From Perod (s) Fgure 8 Comparson of floor response spectra computed from recorded acceleratons and those calculated by the method at roof levels of 6-story Sylmar and 3-story Sherman Oaks Buldngs PARAMETRIC STUDY ON PEAK FLOOR ACCELERATIONS Some studes have suggested that the varaton of acceleraton demands along the heght of buldngs and n partcular the rato of the peak floor acceleraton demand to peak ground acceleraton s ndependent of the perod of vbraton of the structure (Bachman and Drake [9] and Drake and Gllengerter []). However, as shown fgure 6, the acceleraton profle can change sgnfcantly from one buldng to another. Kehoe and Freeman [8] have crtczed the NEHRP provsons to estmate floor acceleratons n buldngs and have ndcated that the perod of vbraton may nfluence the dstrbuton of acceleratons along the heght of the buldng, but have not provded specfc recommendatons on how ths parameter should be taken nto account. In the followng paragraph the results of a parametrc study of the effects of fundamental perod of vbraton, lateral stffness rato and stffness reducton along the heght on sesmc peak floor acceleraton demands are summarzed and dscussed. Structural parameters For buldng wth unform stffness along the heght the smplfed model s defned by three parameters: fundamental perod of the structure, modal dampng rato and lateral stffness rato. For buldngs wth non-unform stffness, a fourth parameter correspondng to the rato of the lateral stffness at roof to the lateral stffness at the base. (Mranda and Taghav [4]). In ths study, all models are assumed to have the modal dampng rato equal to 5 percent. The fundamental perod of the structure was vared from.5s to 4.s wth ncrement of.5 s. The lateral stffness rato, α, was vared from (fleural behavor) to (nearly shear behavor) wth ncrements of. Fnally, the stffness reducton parameter was vared from to 75 percent wth ncrement of 5 percent. Ground motons consdered Eghty recorded ground motons were used n ths study. The ground motons were recorded on stes classfed as class D accordng to recent NEHRP provsons. These ground motons were then classfed nto four bns accordng to ther earthquake magntude and epcentral dstance as follows: () SMSR (Small Magntude, Small Dstance); () SMLR (Small Magntude, Large Dstance); (3) LMSR (Large Magntude, Small Dstance); (4) LMLR (Large Magntude, Large Dstance). The earthquakes wth magntude of 5.8 to 6.5 are referred as small magntude and from 6.6 to 6.9 are referred as large magntude. The dstance of recordng staton to epcenter from 3 to 3 km s referred to as small dstance and from 3 to 6 km s referred to as large dstance. The ground motons have PGAs rangng from 3g to.44g. More nformaton regardng the ground motons can be found n Medna [].

9 ....8 α o =.8 α o = 4.8 α o = T =.5 s T =. s T =. s T = 4. s T =.5 s T =. s T =. s T = 4. s T =.5 s T =. s T =. s T = 4. s Fgure 9 Effect of T of peak floor acceleraton profle Effects of fundamental perod of vbraton and lateral stffness rato on PFA profle Fgure 9 shows the effect of the fundamental perod of vbraton on the varaton of peak floor acceleratons along the heght of the buldng. Results shown n ths fgure correspond to mean ratos (average of 8 records) of peak floor acceleraton demands to peak ground acceleraton. It can be seem that floor acceleratons are amplfed as the perod of vbraton decreases. In partcular, short perod structures ehbt large amplfcaton of acceleraton demands as heght ncreases. For buldngs wth small values of α mean amplfcatons at roof level can be larger than those currently recommended n NEHRP provsons. It can also be observed that the effect of the fundamental perod of vbraton of the structure s larger n buldngs that deflect laterally lke shear beams than those that deflect laterally lke fleural beams. However, the latter buldngs are more lkely to eperence sharp local amplfcatons near the top of the buldng as a result of hgher modes. Fgure shows the effects of the lateral stffness rato α on the varaton of peak acceleraton demands a long the heght of buldngs. It can be seen that for short perod structures, floor acceleraton ncrease as heght ncreases regardless of the lateral stffness rato. It can be seem that long perod buldngs that deflect laterally lke shear beams on average wll have acceleraton demands that are smaller than those occurrng at the base.. T =.5 s. T =. s. T = 4. s α = α = α = 6 α = α = α = α = 6 α = α = α = α = 6 α = Fgure Effect of α on the varaton of peak floor acceleratons along the heght of the buldng

10 4. 3. =.5 α = α = 4 α = =.... z / H =.5. z / H = Perod (s) Perod (s) Fgure Varaton of peak floor acceleraton at md heght and roof level wth changes n the fundamental perod of vbraton T for dfferent lateral stffness ratos. Fgure shows changes n peak floor acceleratons normalzed by peak ground acceleratons at mdheght and roof levels wth changes n the fundamental perod of vbraton. It can be seen that mean PFA to PGA ratos tend to decrease as the fundamental perod of vbraton ncreases. However, reductons are more mportant for buldngs wth large values of α and are more pronounced at roof level than those at md-heght. In some cases the mean reductons are substantal. For eample for buldngs wth large values of α the PFA to PGA rato decreases from appromately 3. for a perod of.5s to appromately. for perod of vbraton of 4.s. Effect of stffness reducton on PFA profle Fgures 9, and correspond to buldngs n whch the lateral stffness was assumed to reman constant along the heght. Mranda and Taghav [4] studed the effect of the reducton of lateral stffness on the dynamc propertes requred to estmate lateral acceleraton n buldngs. They consdered varatons n lateral stffness defned by two parameters, δ that controls the lateral stffness at roof to that at the base and λ that controls the shape of the stffness profle (see fgure ). They showed that the effect of λ s neglgble so only the effect of δ was consdered n the parametrc study. Fgure 3 shows the effect of stffness reducton for moment frame buldngs. PFA s plotted for δ =. (unform stffness),.75,.5 and.5. It s seen that regardless of fundamental perod of the structure, stffness reducton does not has a sgnfcant effect on the varaton of acceleraton demands along the heght of the buldng for most of the heght. A small effect s observed near the top of the structure λ = λ = δ=. δ=.75 δ=.5 δ= Stffness/Base Stffness Fgure Lateral stffness profle T =.5 s δ =. δ =.75 δ =.5 δ = T =. s δ =. δ =.75 δ =.5 δ = T = 4. s δ =. δ =.75 δ =.5 δ = Fgure 3 Effect of stffness reducton on peak floor acceleraton profle

11 PARAMETRIC STUDY OF FLOOR SPECTRA Floor response spectra are useful to estmate sesmc demand of fleble acceleraton senstve components mounted on floors of buldngs whose mass s sgnfcantly smaller to that of the buldng. Varous studes have shown that acceleraton demands can be greatly amplfed for buldng components whose perod of vbraton concde wth those of the prmary structure. A parametrc study was conducted to study the effects of the fundamental perod of vbraton, the lateral stffness rato and the reducton of stffness along the heght on floor spectra. Effects of fundamental perod of vbraton and lateral stffness rato on floor spectra ordnates Fgure 4 shows mean floor response spectra at roof level for buldngs wth fundamental perods T equal to.,. and 3. s and wth lateral stffness rato of α = and. All floor spectra are normalzed by peak floor acceleraton (sometmes also referred to as zero perod acceleraton). It can be seen that the ampltude and locaton of the peaks n the floor spectra change wth changes n the fundamental perod of vbraton. The amplfcaton at a perod equal to the fundamental perod decreases as the fundamental perod of the buldng ncrease. Ths amplfcaton s appromately 3. for T =.s,.5 for T = s and. for T = 3 s. Ths trend does not hold for hgher modes. It can be seen that the amplfcaton for a perod equal to the second mode of vbraton of the buldng s 4. when T =., 4.4 when T =. and 3.8 when T = 3. s. In all three plots, t can be observed that normalzed spectral ordnates ncrease around T when lateral stffness rato ncreases. In other word, there s a slght ncrease of floor spectra around the frst mode of structure n buldngs deflectng laterally as shear beams (e.g. moment frame buldngs) compared to that n buldngs that deflect laterally lke fleural beams (e.g. shear wall buldngs). However, around the second mode, ths trend s reversed. Fgure 5 shows the varaton of spectral amplfcatons along the heght for fleble nonstructural components whose perods concde to those of the frst and second perods of vbraton of the buldng. As shown n ths fgure the amplfcaton n spectral ordnates changes not only wth the fundamental perod of vbraton of the structure but also wth the heght level. In general, spectral amplfcatons for perods around T are larger n the upper part of the buldng and can be on average as large as fve for short perod structures at two thrd of the heght. For long perod structures, ths value reduces to.. Also t s clear that for buldngs wth longer fundamental perod, the demand s lower around ther fundamental perod. The spectral amplfcaton around T also vares sgnfcantly along the heght of the buldng. Mamum amplfcaton n ths case are epected to occur at one thrd of the heght. The mean spectral acceleraton ordnate can be as low as PFA and ncrease to values as large at 4 tmes PFA wth changes n heght locaton wthn the buldng. The lowest amplfcatons occur, as epected, at the buldng heght where the mode shape of the second mode has a node. S fa / PFA α = α = S fa / PFA α = α = S fa / PFA α = α = 3.. T =. s 3.. T =. s 3.. T = 3. s Ts (s) Ts (s) Ts (s) Fgure 4 General observatons of effects of T and α on floor response spectra

12 X. X..8 α = δ =. ξ = 5%.8 α = δ =. ξ = 5% T =.5 s.4 T =.5 s. T =. s T =. s. T =. s T =. s T = 4. s S fa (T ) / PFA T = 4. s S fa (T ) / PFA Fgure 5 Varaton of floor response spectra peaks around the frst and second modes of the man structure Fgure 6 shows the varaton of floor spectral amplfcatons at roof level wth changes n the lateral stffness rato. As shown n ths fgure, an ncrease n lateral stffness rato ncreases the floor spectral acceleraton around the fundamental perod and decreases t around the second mode. The effect of lateral stffness rato s smaller for buldngs wth short perods of vbraton than for buldngs wth long fundamental perods of vbraton. Effects of reducton of lateral stffness along the heght of the buldng on floor spectra Fgure 7 shows the effect of the reducton of lateral stffness along the heght of the buldng on floor spectra ordnates at the roof level n buldngs wth a fundamental perod of vbraton of 4.s. The fgure compares floor spectra computed for buldngs wth unform stffness along the heght to those computed n buldngs where the lateral stffness at the top of the buldng s one fourth of the lateral stffness at the base of the buldng (δ =.5). can ncrease the perod of hgher modes up to percent and therefore the peaks n floor response spectra move slghtly. It s seen that the floor spectra are not affected sgnfcantly due to stffness reducton. The peak on the frst perod of the structure s reduced by about percent for δ =.5 n a buldng wth fleural behavor and fundamental perod of 4. seconds. Changes n spectral ordnates are prmarly due to changes n buldng perod ratos. A behavor smlar to that shown n ths fgure was observed for buldngs wth other perods of vbraton. S fa (T, α ) / S fa (T, ). S fa (T, α ) / S fa (T, ) T=. s T=. s T=3. s T=4. s 5 5 α α Fgure 6 Effects of lateral stffness rato on amplfcaton of floor spectra ordnates at the roof.

13 S fa / PFA δ=. δ=.5 S fa / PFA δ=. δ=.5. T = 4. s α =. T = 4. s α = T s (s) T s (s) Fgure 7 Effect of stffness reducton on floor response spectra ordnates at roof level. SUMMARY AND CONCLUSIONS A method to estmate floor acceleraton demands n buldngs subjected to earthquakes was presented. The method uses a smplfed model consstng on two contnuous beams. A close form soluton of the dynamc characterstcs of the model was presented when the lateral stffness of the model s unform. The model s fully defned wth only three parameters: the fundamental perod of the structure, a modal dampng rato and the lateral stffness rato. The accuracy of the method was evaluated by comparng the peak floor acceleraton demands, tme hstores and floor spectra computed wth the method to those obtaned from acceleraton records n three nstrumented buldngs. It was shown that the method s able to capture acceleraton demands wth reasonable accuracy wth a very small computatonal effort. A parametrc study was performed to study the effects of varous parameters on acceleraton demands n buldngs. The parameters that were studed are: the fundamental perod of the structure, the lateral stffness rato and a parameter descrbng the amount of reducton of lateral stffness along the heght. Varaton of these parameters was studed together wth 8 ground motons recorded on frm stes n varous earthquakes n Calforna. It was observed that both the fundamental perod of the structure and the lateral stffness rato can sgnfcantly change acceleraton demands n buldngs. On the other hand, results ndcate the reducton n lateral stffness along the heght of the buldng do not have a sgnfcant effect on acceleraton demands. A smlar parametrc study was performed to nvestgate the effect of these parameters on floor spectra ordnates. Results ndcate that spectral amplfcatons around the perods of the man structure can change sgnfcantly wth change n fundamental perod of the structure, lateral stffness rato as well as floor level. Spectral amplfcatons around the frst mode of the structure decrease as the fundamental perod of vbraton ncreases and ncrease as the lateral stffness rato ncreases. Effects of structural nonlnearty are currently beng nvestgated. REFERENCES. Ayers, J.M., Sun T.Y., (973a). Nonstructural damages, San Fernando, Calforna, earthquake of February 9, 97, U.S. Dept. of Commerce, Natonal Oceanc and Atmospherc Admn., Vol.. Whtman, R. V., Hong, S.-T. and Reed, J. (973). Damage statstcs for hgh-rse buldngs n the vcnty of the San Fernando Earthquake, Report No. 7, Massachusetts Insttute of Technology, 4 pages. 3. Rhal, S. S. (99). Performance and behavor of non-structural buldng components durng the Whtter Narrows, Calforna (987) and Loma Preta, Calforna (989) earthquakes: selected case studes, Report No. ATC-9, Proc. Semnar and Workshop on Sesmc Desgn and Performance of

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