A MEASURE OF DRIFT DEMAND FOR EARTHQUAKE GROUND MOTIONS BASED ON TIMOSHENKO BEAM MODEL

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1 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia A MEASURE OF DRIFT DEMAND FOR EARTHQUAKE GROUND MOTIONS BASED ON TIMOSHENKO BEAM MODEL Juju XIE 1 ad Zepig WEN 1 PH. D. Studet, Istitute of Geophysics, Chia Earthquake Admiistratio, Beijig Chia Professor, Istitute of Geophysics, Chia Earthquake Admiistratio, Beijig Chia wezp@cea-igp.ac.c, xiejujv5@mails.gucas.ac.c ABSTRACT: A ew measure of demads for earthquake groud motio is preseted. It is base o a cotiuous Timosheko beam model. While cosiderig the effect of ouiform distributio of drift demads alog the buildig height ad takig ito accout the couplig effect of bedig ad shear distortio o the vibratio of catilever beam structures, the drift spectrum leads to more accurate estimatios of drift demad for earthquake groud motios. This cotiuous model is compared with the combied flexural-shear beam model proposed by Mirada. Similar to the model proposed by Mirada, the dyamic properties of the catilever Timosheko beam model are highly iflueced by a parameter called the height-width ratio. A formula is give to idicate the relatio of height-width ratio with lateral stiffess ratio. The effects of height-width ratio, dampig ad higher modes o drift demads are examied. KEYWORDS: earthquake, drift demads, groud motio, Timosheko theory, catilever beam. 1. INTRODUCTION The cotrollig of displacemet durig the desig process of buildig structures has bee give more cocer sice the 1971 Sa Ferado earthquake. It has also bee show that the use of lateral displacemets as demad parameters could permit a more direct way to cotrol the damage i the buildig structures durig the desig process. Amog all of the displacemet related parameter, the lateral story drift (defied as the ratio of story drift betwee two cosecutive floors to story height) are cosidered as a sigificat cause that leads to the damage of buildig structures whe subjected to earthquake groud motio. ( Moehle ; Wallace 1994; Wallace ad Moehle 199). Examiatio of damage patters durig the earthquake showed that buildigs i the ear field might suffer large localized ductile displacemet ad story drifts, it could ot be associated with the peak acceleratios aloe. The ear-field groud motios ofte cotai coherece waveforms that appear as distict velocity or displacemet pulses with high itesity. The coheret pulse-like ature of the ear-field groud motio time history may cause the maximum respose of the structure to occur before a resoat mode-like respose ca build up. These groud pulses may be associated with substatial ielastic respose as well as higher-mode effects that geerally caot be represeted adequately by respose spectrum. I this case, Iwa(1997) itroduced a simple measure of drift demads for earthquake groud motio called the drift spectrum. It is based o the wave aalysis of a cotiuous shear beam model. The drift spectrum has become a icreasigly importat topic sice the. Followig Iwa, may researchers studied the drift spectrum i differet ways. Chopra ad Chitaapakdee (1) show that drift spectra based o a shear beam model could also be computed usig covetioal modal aalysis. Gulka ad Akkar () itroduced a simple replacemet method for the drift spectrum. Kim ad Collis s study () idicated that Iwa s model could result i residual drifts for certai groud motio. Mirada (6) used a combied flexural-shear beam model to obtai drift demads for earthquake groud motios, he itroduced the geeralized iterstory drift spectrum which is applicable for differet kids of buildig structures which deform i the way of a combied flexural ad shear type to estimate the maximum iterstory drift demads. The elastic drift spectrum directly shows the story drift that a groud motio record would cause i a multistory shear buildig. It is a coveiet tool that displays the maximum local drift demads at chose elevatio i buildigs, which is very importat durig the buildig desig process. Despite its revolutioary

2 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia cocept, the drift spectrum itroduced i Iwa s model also has some disadvatages. It s based o the wave aalysis of a cotiuum shear beam, the effect of bedig is ot cosidered i the shear beam model. Furthermore, the wave aalysis is ot familiar with structural egieers ad could result i residual drifts for certai groud motio as idetified by Kim ad Collis. x yxt (, ) b H ug () t Figure 1 Simplified model for catilever structures I this paper, a alterative method to estimate the drift demads for groud motios is developed ad preseted. Cotiuous catilever beam model based o Timosheko beam theory is utilized to estimate drift spectrum which is a measure of demad for earthquake groud motios o structures (see Fig.1). It has bee commoly recogized that the iterstory drift demads are ot uiformly distributed alog the buildig height. While cosiderig this kid of ouiform effect ad takig ito accout the couplig effect of bedig ad shear distortio o the vibratio of catilever beam structures, the drift spectrum leads to more accurate estimatios of drift demad for earthquake groud motios. This study ca also avoid the residual drifts problem by usig the covetioal modal aalysis techiques. The maximum drift spectra ca supply as a rapid evaluatio for the drift demads of earthquake groud motios, which is very importat for the desig of buildigs to resist the seismic load.. CONTINUOUS BEAM MODELS May researchers have used the cotiuous beams to model the behavior of catilever structures whe subjected to static ad dyamic loads sice the last cetury. Westergaard (1933) itroduced a cotiuous shear beam model with uiform-mass ad stiffess to estimate lateral deformatios of buildigs subjected to earthquake groud motio. Roseblueth et al. (1968) used shear beams ad respose spectrum aalysis to estimate story shears ad overturig momets i buildigs structures. Motes ad Roseblueth (1968) used flexural beam model to estimate shear ad overturig momet demads i chimeys. Based o a cotiuum model cosistig of combiatio of a flexural beam ad a shear beam, Mirada (1999, ) evaluated drift ad acceleratio respose for differet kids of buildig structures. The Timosheko beam theory was first itroduced by Timosheko i 191. I this cotiuous beam theory, Timosheko (191; 19) exteded to iclude the effect of trasverse shear deformatio o the basis of Rayleigh beam theory. Followig Timosheko, may researchers studied the dyamics ad vibratios of Timosheko beams usig travelig wave method or model vibratio aalysis (Miklowitz 1953; Aderso 1953). More recetly, Geist ad McLaughli (1997) discussed the pheomea of double eigevalues i Timosheko beams. Kausel () studied the effect of rotatioal motio o the ormal modes of urestraied shear beams.

3 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia Aristizabal-Ochoa (4) ivestigated the vibratio of Timosheko beam-colum with geeralized support coditios ad subjected to a costat axial load alog its spa. Challamel (6) compared Timosheko ad shear models i beam dyamics. Height Flexural beam model Shear beam model Lateral deflectio Figure Deflectio shapes of flexural beam model ad shear beam model May studies have show that the structures may deform i a flexural type or a shear type whe subjected to lateral loads (see Fig.), but more commoly, the structures deform i the form of a combied flexural-shear shape, ad the overall lateral deformatio is deoted by a superpositio of shear deformatio ad flexural deformatio (Kha ad Sbarouis 1964; Heidebrecht ad Stafford Smith 1973; Mirada 1999; Mirada ad Carlos J ). I this paper, the couplig effects of bedig ad shear distortio o the vibratio of catilever beam structures are cosidered accordig to Timosheko theory. It is show that the joit effects of bedig ad shear deformatios i Timosheko beam structures are highly iflueced by the height-width ratio. It is iterestig to ote that the deflectio shapes of catilever Timosheko beam i the fudametal mode exhibit the form of a flexural type or a shear type whe a certai value of beam height-width ratio is give. 3. DYNAMIC PROPERTIES OF TIMOSHENKO BEAMS The goverig equatios for the free vibratio of Timosheko beam structures (see Fig.1 ad Fig.3) are y EI θ θ + k GA( θ) ρi = (3.1) t y θ y kga( ) ρ A = (3.) t The Timosheko beam i the model has the followig material property: desity ρ, Youg modulus E ad shear modulus G. Its trasverse cross sectio is deoted by total area A ad momet of iertia I = Ar ( r = radius of gyratio of the cross sectio). k is the shear correctio coefficiet depedig o the shape of the y cross sectio ad material s Poisso ratio υ ; y is the trasverse deflectio; is the slope of the ceterlie of the beam; θ is the rotatio of the cross sectio. The trasverse deflectio y of Timosheko beam is deoted by a superpositio of shear deformatio ad bedig deformatio. The relatio is deoted by y = γ + θ (3.3)

4 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia y θ M Q d θ M+dM dx Q+dQ γ θ y Figure 3 Differetial aalysis of Timosheko beams x The solutios to Eq. 3.1 ad 3. are of the form y ( xt, ) = φ( x)si( ωt) (3.4) θ ( x, t) = ψ( x)si( ωt) (3.5) Substitutig Eq. 3.4 ad Eq. 3.5 ito Eq. 3.1 ad Eq. 3., ad elimiatig all fuctios of time, the followig pair equatios i terms of the shape fuctios φ ( x) ad ψ ( x) are obtaied: d φ dφ + ψ + ρ ω ψ = EI k GA( ) I (3.6) dx dx d φ dψ kga( ) + ρaω φ = (3.7) dx dx Eq. 3.1 ad 3. describe coupled rotatio ad lateral displacemet of the uiform beam, where θ ca be elimiated to form a sigular equatio y y E y I y EI ρ ρa + ρi = (3.8) 4 t kg t kg t For a cotiuous catilever beam with total height of H, itroducig the followig odimesioal variables: x x = H ad φ φ = H Usig the odimesioal variables just listed above, substitutig Eq. 3.4 ito Eq. 3.8, ad elimiatig the fuctios of time, the the followig differetial equatio is obtaied: 4 4 d φ 1 1 d φ ρaωh ρr ω + ρω 1 4 H + φ dx E k G dx = EI k G (3.9) Itroducig the followig parameters: E μ = kg (3.1) α = H ρ m = H kg kga (3.11)

5 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia ρ A m β = H = H (3.1) EI EI where m = ρ A is the uiform mass per uit legth of the cotiuum beam, the Eq. 3.9 ca be writte as follows: (4) φ + α ω 1+ φ β ω α ω φ = (3.13) μ μ Similar to the dimesioless parameter i the combied flexural-shear beam model proposed by Mirada, the lateral stiffess ratio η of the Timosheko beam is defied as β kga η = = H (3.14) α EI E Substitutig I = Ar ad μ = ito Eq. 3.14, we have kg H kg H 1 η = = (3.15) r E r μ The solutio of Eq is of the form φ = Ce λx (3.16) The followig polyomial is obtaied after substitutig the above equatio ito Eq λ + α ω 1+ λ β ω α ω = (3.17) μ μ The solutio of the above equatio is of the form αω 1 1 4η λ1 = μ μ αω (3.18) αω 1 1 4η λ = μ μ αω The sig of λ 1 is clearly egative. However, the sig of λ is idefiite, it is cotrolled by the sig of Δ = 1 αω 1 μ η (3.19) Case 1: whe Δ<, the sig of λ is positive. The four roots of Eq are λ1 = ± j, λ = ± ε where αω 1 1 4η = μ μ α ω (3.) αω 1 1 4η ε = μ μ α ω The above relatio of Eq. 3. ca also be expressed as follows: 1 ε = (1 + ) α ω μ (3.1) ε = β ω α ω μ The fuctio of the lateral deflectio φ = φ/ H is expressed as follows φ = C si( x) + C cos( x) + C si h( ε x) + C cos h( ε x) (3.) 1 3 4

6 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia Substitutig of where ad φ = Hφ ad x = Hx ito Eq. 3.7, the shape fuctio of the rotatio ψ is obtaied i the form [ C cos( x) C si( x) ] [ C cosh( x) C sih( x) ] ψ = χ + ϑ ε + ε (3.3) α ω χ = α ω ϑ = ε + ε C 1, C, C 3, ad C 4 are coefficiets that deped o the followig boudary coditios: At the base where x = : φ () = ; ψ () = At the roof where x = 1 : dψ dφ (1) (1) = ; ψ (1) = dx dx Four homogeeous equatios i terms of coefficiets C 1, C, C 3 ad C 4 is obtaied accordig to the boudary coditios. Settig the determiat of this set of equatios equal to zero, ad elimiatig the parameters α ad β, the characteristic equatio ca be expressed as follows: [( + με ) + ( μ + ε) ]coscosh ε ( + με)( ε + μ ) ε ( + sisih ε) = (3.4) ε β Usig the relatioη =, elimiatig α, β ad ω from Eq. 3.1, the followig equatio i terms of α eigevalues ad ε is obtaied: 1 1 ( ε ) η ( ε ) 1 ε μ 1 + = (3.5) + μ The eigevalues ad ε of the th mode ca be easily obtaied from Eq. 3.4 ad 3.5. As it is show above, double eigevalues ad ε deped o the lateral stiffess ratio η ad the dimesioless ratio μ. The mode shape of lateral deflectio ormalized by C = 1 is expressed as follows. = + + μ ε + ε φ ζ si( x) cos( x) ζ sih( ε x) cosh( ε x) με Where ζ is odimesioal parameter for the th mode of vibratio ad give by ( ε + μ ) cos + cosh ε ( + με ) ζ = ε si + sihε Takig the derivative of the shape fuctio of lateral deflectioφ, we obtai με + dφ = ζcos( x) si( x) ζ cosh( ε ) sih( ) x ε εx dx μ 1+ ε (3.6) (3.7) (3.8)

7 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia Accordig to Eq. 3.1, the circular frequecy ω of the th mode ca be computed by ( ) μ ε ω = (3.9) ( μ+ 1) α where ad ε are eigevalues of the th mode of vibratio correspodig to the th roots of the characteristic equatios 3.4 ad 3.5. Case : whe Δ>, the sig of λ is egative. For higher frequecy modes with larger value of ω, the four roots of Eq are where λ1 = ± j, λ = ± jσ ad αω 1 1 4η = μ μ α ω αω 1 1 4η σ = μ μ α ω Usig the boudary coditio as case 1, the trasverse deflectio φ i the mode shape are obtaied as follows: μσ φ = ζsi( x) + cos( x) ζ si( σ ) cos( ) x σx (3.3) μ σ σ where ( σ μ ) cos cosσ ( ) μσ ζ = (3.31) σ si siσ The circular frequecy ω of the th mode at this case is computed by ( ) μ + σ ω = (3.3) ( μ + 1) α 4. INFLUENCE OF HEIGHT-WIDTH RATIO ON DYNAMIC PROPERTIES OF CONTINUOUS TIMOSHENKO MODEL Cosiderig a uiform beam with rectagular cross sectio, ad that the beam is costructed of a homogeeous isotropic material with Poisso s ratioυ. Commo values of Poisso s ratio υ are.5 to.3 for steel, approximately.33 for most other metals, ad. for cocrete. The shear correctio coefficiet k ca be calculated usig the followig equatio(cowper, 1966) 1( 1+ υ ) k = (4.1) υ The relatio of Youg modulus E ad shear modulus G ca also be writte i terms of υ usig the well-kow formula (see for istace Popov 1968) E = 1 ( + υ ) (4.) G Substitutig Eq. 4.1 ad 4. ito Eq. 3.15, the dimesioless ratio μ ca be expressed as follows: υ μ = (4.3) 5

8 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia 3 Let υ =, the μ = 3, ad accordig to Eq the lateral stiffess ratio 11 3 H η = (4.4) 3 r 3b For a uiform beam with rectagular cross sectio, the radius of gyratio r =, where b =width of the cross 6 sectio alog the vibratio directio (see Fig.1); Itroducig the parameter called the height-width ratio H R = (4.5) b The the lateral stiffess ratio η is computed by: η = R (4.6) Accordig to Eq ad Eq , whe the dimesioless ratio μ has a costat value of 3, the eigevalues ad ε is determied by the height-width ratio R. That is to say, the mode shapes of lateral deflectio, rotatio ad shear distortio i the cotiuous model deped oly o a sigle parameter, the height-width ratio R. Figure 4 Effects of height- width ratio o deflectio of the fudametal mode Figure 4 shows lateral deflectio shape of cotiuous Timosheko beam i the fudametal mode. Products correspodig to five values of height-width ratio are show for to examie the ifluece of this ratio o the deflectio shape. It ca be see that the dyamic properties of the cotiuous Timosheko beam model are highly iflueced by the height-width ratio, the deflectio shows a shape of a flexural type for bigger values of height-width ratio ad it chages to a shear type whe this ratio becomes smaller. 5. EARTHQUAKE RESPONSE HISTORY MODAL ANALYSIS OF THE CONTINUOUS MODEL The respose of the uiform Timosheko beam model show i Fig.1 whe subjected to a groud motio at the.. base ug () t is give by the followig partial differetial equatio: EI y y ρi E y ρ I y ug () t + ρa 1 ρa = (5.1) 4 H t H kg t kg t t The total lateral displacemet respose of the system ca be calculated by superpositio of all modes. y xt, = y xt, (5.) ( ) ( ) where yi (, ) i( ) i i( ) i ( ) Where Γ is modal participatio factor of the ith mode of vibratio; φ ( x ) i i= 1 xt =cotributio of the ith mode with classic dampig ratio, ad it is give by y xt, =Γ φ x D t (5.3) i i =amplitude of the ith deflectio

9 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia shape of vibratio at odimesioal height x ; ad Di ( t ) =deformatio respose of a sigle-degree-of-freedom (SDOF) system correspodig to the ith mode, whose respose is computed with the followig equatio:..... () ξω () ω () Di t + i i Di t + i Di t = ug ( t) (5.4) For a cotiuum model with uiform mass, the modal participatio factor of the ith mode is give by 1 φ i x dx Γ i = (5.5) 1 φ i x dx φ cosists of shear deformatio γ ad bedig deformatio ψ. Lateral drift ratio The lateral deflectio ( x ) of the system is defied as the derivative of total deflectio y ( xt, ) with respect to x as follows: From Eq. 5. ad 5.3, we have yxt (, ) IDR = yxt (, ) 1 (5.6) = Γiφi ( x ) Di( t) (5.7) H i= 1 The ordiates of drift demad spectrum are defied as maximum lateral drift ratio over the height of the buildig structures ad are give by yxt (, ) IDRmax = max (5.8) tx, 6. INFLUENCE OF HEIGHT-WIDTH RATIO ON INTERSTORY DRIFT DEMANDS 3 Givig the material properties as Youg s modulus E =3Gpa, mass desity ρ =5 kg/m, the circular frequecy ω of catilever structures with height H ca be computed usig Eq. 3.9 ad 3.3 accordig to π solutio case 1 ad respectively, ad the periods of the vibratio modes ca be computed by T =. The ω maximum iterstory drifts are computed accordig to Eq. 5.7 ad 5.8, ad oly the first three modes are cosidered. Figure 5 Ifluece of height-width ratio o drift demad spectra Figure 5 presets iterstory drift demad spectrum for udamped models subjected to the NS compoet of the groud motio recorded at Rialdi Receivig Statio durig the 1994 Northridge earthquake whe the height-width ratio is equal to 3, 5 ad 8 respectively. It ca be see that the ordiates of the drift spectra augmet with icreasig height-width ratio. That is to say, the catilever structures with bigger height-width ratio will suffer larger drifts ad ted to be more fragile whe subjected to the earthquake groud motios.

10 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia 7. INFLUENCE OF DAMPING ON INTERSTORY DRIFT DEMANDS Figure 6 Ifluece of dampig o iterstory drift demads Figure 6 shows chages i iterstory drift demads owig to differece i dampig whe subjected to the Rialdi Receivig Statio groud motio durig the 1994 Northridge earthquake. The drift spectrum is obtaied by givig height-width ratio a costat value of 5. The iterstory drift demads are computed usig the first three modes ad the dampig ratio is assumed to be the same i all modes. For three differet levels of dampig, % ad 5%, it ca be see that the iterstory drift demads decrease with icreasig dampig ad the ifluece varies i differet vibratio period rage. 8. INFLUENCE OF HIGHER MODES ON INTERSTORY DRIFT DEMANDS Figure 7 Ifluece of higher modes o iterstory drift demads Figure 7 presets drift spectra of the NS compoet of the Rialdi Receivig Statio record usig 1,, 3 ad 4 modes respectively. I all cases, the dampig ratio is assumed to be zero. It ca be see that demads computed with 3 modes are practically the same as those computed with 4modes. It is cocluded that oly a small umber of modes is eough to obtai maximum iterstory drift demads. For this record, oly 3 modes ca provide adequate iterstory drift estimates. Usig oly the fudametal mode is eough to obtai drift estimates whe the fudametal periods are smaller tha s. However, whe the periods are loger tha s, usig oly the first mode will result i uderestimatio of maximum iterstory drift ratio. 9. COMPARISON WITH PREVIOUS STUDY The vibratio properties of this cotiuous Timosheko beam model are compared with the combied flexural-shear beam model proposed by Mirada i Fig. 8. I the combied flexural-shear beam model proposed

11 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia by Mirada, the vibratio modes i the cotiuous model deped o the parameter called lateral stiffess ratio α. As show i Fig. 8, a value of α = correspods to a flexural beam while a value of α = 65 correspods approximately to a shear beam. Itermediate values of α, represet structural behaviors that combie flexural ad shear deformatio. Similar to the model proposed by Mirada, the mode shapes i the catilever Timosheko beam model are cotrolled by the height-width ratio. Fig.8 also shows lateral deflectio shape of cotiuous Timosheko beam i the fudametal mode. Products correspodig to five values of height-width ratio are show for to examie the ifluece of this ratio o the deflectio shape i the fudametal mode. It ca be see that the deflectio shows a shape of a flexural type for bigger values of height-width ratio ad it chages to a shear type whe this ratio becomes smaller. Similar to the geeralized iterstory drift spectrum proposed by Mirada, the drift spectrum i this study is associated with a parameter called the height-width ratio. I this study, it is showed that the lateral stiffess ratio is determied by the height-width ratio based o the cotiuum model composed of a sigle Timosheko beam, ad this study idicated that the drift spectrum is highly iflueced by the height-width ratio. Figure 8 Compariso of Timosheko beam model i this study with the combied flexural-shear beam model proposed by Mirada I former study of Iwa(1997) ad Mirada (6), the fudametal periods of the cotiuous model were computed usig experimetal relatioship suggested for steel-momet-resistig frames i the 1997 UBC code,.75 amely, T1 =.853H. As poited out by Mirada (6) i his study, there is a sigificat ucertaity i the estimatio of fudametal periods for buildigs whe usig the experimetal period relatioship, ad this ucertaity will cause ucertaity i the estimatio of iterstory drift demads. I this study, the periods of the cotiuous model are computed usig the theoretical formula based o Timosheko beam theory, so the model i this study ca avoid the ucertaity i the estimatio of iterstory drift demads due to period ucertaity. However, the author thiks that this theoretical equatio could oly be accurate eough whe used to evaluate the vibratio periods of RC shear wall structures, but it should ot be used to compute the periods for all kids of buildig structures.

12 The 14 th World Coferece o Earthquake Egieerig October 1-17, 8, Beijig, Chia Figure 9 Compariso of iterstory estimates usig Timosheko beam model i this study with the model proposed by Mirada subjected to Rialdi record Figure 9 shows the Compariso of iterstory drift estimates usig Timosheko beam model i this study with the model proposed by Mirada whe subjected to Rialdi record. As is show by this figure, the iterstory drift demads obtaied i this study based o the Timosheko beam model are smaller tha the results obtaied usig Mirada s model. This is accordat with the kowledge that the lateral resistig system such as the shear walls has the effects of reducig iterstory drift demads, which has bee cofirmed by Mirada i his study. It is cocluded that the drift spectrum based o the cotiuous Timosheko beam model i this study is restricted for the evaluatio of lateral drifts for shear wall structures. It might ot be adequate to estimate lateral drift demads for other buildig structures whe usig the simplified catilever beam model. 1. SUMMARY AND CONCLUSIONS A ew method to estimate the seismic demad of groud motios is developed ad preseted i this paper. Providig a estimate of maximum iterstory drift demads for earthquake groud motios based o the cotiuous Timosheko beam model, the iterstory drift spectrum i this study ca directly show us damage potetial for the catilever structures with differet fudametal periods whe subjected to earthquake groud motios. The drift demads of this catilever Timosheko beam model are highly iflueced by the height-width ratio. It was show that the catilever beam structures with bigger height-width ratio will suffer larger drifts. It was also coclude that drift spectrum is much correlated with dampig, ad the ifluece of higher modes ca ot be eglected i the estimatio of drift demads for catilever structures whe the fudametal periods are loger tha s. REFERENCES Aderso, R. A. (1953). Flexural vibratio i uiform beams accordig to the Timosheko theory. ASME, J. Appl. Mech., 8, Akkar, S., Yazga, U., ad Gülka, P. (5). Drift estimates i frame buildigs subjected to ear-fault groud motios. J. Struct. Eg., 131:7, Aristizabal-Ochoa, J. Dario. (4). Timosheko beam-colum with geeralized ed coditios ad oclassical modes of vibratio of shear beams. J. Eg. Mech., 13:1, Challamel, N. (6). O the compariso of Timosheko ad shear models i beam dyamics. J. Eg. Mech., 13:1, Chopra,A.K., ad Chitaapakdee,C. (1). Drift spectrum vs. modal aalysis of structural respose to ear-fault groud motios. Earthquake Spectra, 17:, Cowper, G. R. (1966). The shear coefficiet i Timosheko s beam theory. ASME, J. Appl. Mech., 33, Geist, B., ad McLaugli, J. R. (1997). Double eigevalues for the uiform Timosheko beam. Appl. Math. Lett., 1:3, Gulka,P., ad Akkar, S. (). A simple replacemet for the drift spectrum. Egieerig Structures, 4:11,

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