CUMULATIVE DAMAGE ESTIMATION USING WAVELET TRANSFORM OF STRUCTURAL RESPONSE Ryutaro SEGAWA 1, Shizuo YAMAMOTO, Akira SONE 3 Ad Arata MASUDA 4 SUMMARY Durig a strog earthquake, the respose of a structure with biliear restorig force has discotiuity i jerk. I this paper, the method of estimate the ductility factor by the wavelet aalysis of absolute acceleratio respose is proposed. It is foud that the wavelet coefficiet obtaied by wavelet aalysis of acceleratio respose is proportioal to the velocity at yield poit ad the ductility factor is evaluated from the velocity at yield poit by the eergy coservatio law. Usig the proposed method, the ductility factor is estimated from wavelet coefficiet with good accuracy ad its applicability is verified through some simulatios. INTRODUCTION The cumulative damage of structure subjected to a strog seismic motio is the low cycle fatigue with large amplitude ad small umber of cycle, which occurs durig time iterval of 0 or 30 secod. The ductility factor; that is defied as the ratio of peak relative displacemet ad the yield relative displacemet of structure is utilized for estimatig the cumulative damage of low cycle fatigue. For the estimatio of cumulative damage, it is ecessary to kow the followig two factors. (1) Peak displacemet i each cycle oscillatio. () Number of these oscillatios. Whe the structures with hysteretic restorig force behave from the elastic regio to plastic regio, the acceleratio respose chages rapidly ad its time derivative (jerk) has discotiuity at the yield poit. Therefore, for estimatig secod item, the wavelet aalysis seems to be a suitable method. The authors [Soe, et al., 1995] have bee proposig the estimatio method of cumulative damage due to low cycle fatigue of structures subjected to a strog earthquake. It was assumed that the discotiuity might be cotaied i the observed acceleratio records, which should be produced at the yield poit chagig from the elastic regio to the plastic regio ad might be detected by wavelet aalysis. Through the umerical simulatios usig a sigle degree of freedom model, it was prove that this assumptio was correct ad it was cofirmed that the wavelet aalysis was used to estimate the cumulative damage of structures by earthquake, effectively. However, we do ot apply this aalysis to the estimatio of above first item. Cosiderig the low cycle fatigue of structures, it is importat to estimate ot oly the umber of discotiuity but also damage rate due to the low cycle fatigue. I this paper, we cosider sigle degree of freedom model with biliear restorig force subjected to filtered white oise excitatio with domiat frequecy. This paper aims at idetifyig the ductility factor, usig the magitude of wavelet coefficiet obtaied by the wavelet aalysis of acceleratio respose. Especially, the relatio betwee the velocity at the yield poit ad the ductility factor is derived based o the eergy balace techique. The relatio betwee the velocity at the yield poit ad the magitude of wavelet coefficiet for acceleratio respose is also discussed. 1 3 4 Dept of Mechaical ad System Egieerig, Kyoto Istitute of Techology, Japa.Email: segawa@viblab.mech.kit.ac.jp Dept of Mechaical ad System Egieerig, Kyoto Istitute of Techology, Japa.Email: segawa@viblab.mech.kit.ac.jp Dept of Mechaical ad System Egieerig, Kyoto Istitute of Techology, Japa.Email: soe@ipc.kit.ac.jp Dept of Mechaical ad System Egieerig, Kyoto Istitute of Techology, Japa.Email: masuda@ipc.kit.ac.jp
Figure 1: Model of structure with biliear restorig force BILINEAR RESTORING FORCE AND DISCONTINUITY IN JERK AT YIELD DISPLACEMENT Biliear Restorig Force The equatio of motio of structure show i Fig.1 is give as follows: mx&& + cx& + q( x) = mu&& g ( 1) Where m, c ad q(x) are mass, dampig coefficiet, restorig force, respectively. &&u g ad x are seismic iput acceleratio ad relative displacemet. Dividig the both sides of Eq.(1) by mx y (x y : yield relative displacemet), the followig odimesioal equatio is give by X&& + ζω X& + Q( X ) = U&& ( ) g Where, the restorig force Q(X) is show as the liear combiatio of a liear compoet ad a hysteretic compoet [Suzuki ad Miai, 1994]. Q( X ) = γω X + ( 1 γ ) ω Z ( 3) Where γ is the ratio of postyield stiffess ad preyield stiffess. X ad Z are odimesioal parameters as follows. x z c k X Z x x mk m U ug =, =, =, =, && && ζ ω g = ( 4) y y xy ζ ad ω are the dampig ratio ad atural circular frequecy of structure. U&& g is the ormalized acceleratio. Time derivative & Z of odimesioal restorig force Z is give by usig the step fuctio U Z & = X 1 U ( X & ) U ( Z 1 ) U ( X & ) U ( Z 1 ) ( 5) The physical meaig for Eq.(3) is explaied by takig the examples γ = 1 ad γ = 0 as show i Fig.. That is, whe γ = 1, the secod term i Eq.(3) becomes zero ad the first term shows the elastic restorig force as show i Fig.(a). O the other had, whe γ = 0, the first term i Eq.(3) becomes zero ad the secod term shows the restorig force for γ = 0 as show i Fig.(b). Whe γ = 05., the value of vertical axis i Fig.(a) ad (b) reduces by half ad the combiatio of two figures is show i Fig.(c). bg. Discotiuity i Jerk Respose at Yield Poit Rearragig Eq.(), the absolute acceleratio X&& + && is give by the followig equatio. U g Figure : Explaatio of biliear restorig force for example of γ = 0. 5
&& && & X + U = ζω X γω X ( 1 γ ) ω Z ( 6) g I the right had of this equatio, the displacemet X, the velocity & X ad the restorig force Z are cotiuous because the equatio of motio is secod order. Differetiatig the both sides of Eq.(6) with respective to time, the followig equatio is give by &&& &&& && & X + U = ζω X γω X ( 1 γ ) ω Z& ( 7) g I this equatio, the acceleratio X && is still cotiuous, but the time derivative of restorig force Z is ot cotiuous. The time derivative of restorig force is show i Fig.3 ad it is foud that Z & is equal to X & o the iclied lies ad is equal to zero o the flat lies. The, the value of Z at the yield poit chagig from the iclied lie to the flat lie has discotiuity. Cosequetly, the jerk; that is the time derivative of acceleratio ( X&& + && ), as show i Eq.(7) has discotiuity at the yield poits ad the magitude of the discotiuity is U g b g X &. related to 1 γ ω ESTIMATION OF DUCTILITY FACTOR Figure 3: Time derivative of restorig force Wavelet Aalysis The absolute acceleratio show i Eq.(6) is expaded by the wavelet series ψ j, k as follows [Soe, et al, 1995 ad Ashio ad Yamamoto, 1997]. X&& + U&& g = α ψ j, k j, k ( t) ( 8) j k Where ψ j, k btg is a aalysig wavelet. j, k are the dilatio ad traslatio parameters, respectively. It is kow that wavelet coefficiets α j, k are chaged dramatically whe the absolute acceleratio respose is aalyzed by the wavelet trasform [Ashio ad Yamamoto, 1997]. The, the wavelet coefficiet α j, k obtaied by Eq.(8) ca detect discotiuity cotaied i the jerk at the yield poit as show i Fig.3. Besides, because of the relatioship betwee the magitude of discotiuity ad b1 γgω X &, the wavelet coefficiet α j, k is assumed by the followig equatio. α j, k = 1 γ ω vy a b g ( 9) Where a is the costat value which is obtaied whe the kid of wavelet ad the level j used i the wavelet aalysis are fixed. v y is the relative velocity at the yield poit. The physical meaig for Eq.(9) is that the sudde discotiuity at the yield poit causes a jerk, which registers as high frequecy ad high amplitude acceleratios, ad the jerk depeds o the velocity. So, as the velocity at the yield poit becomes larger, the chage i acceleratio becomes remarkable. The, we assumed that the discotiuity at the yield poit could be detected by the wavelet coefficiet α j, k for the acceleratio respose. Moreover, it is expected that as the velocity at the yield poit icreases, the peak displacemet (ductility factor) becomes large. Eergy Balace 3
The eergy balace of i-th step is show i Fig.4. If the effect of seismic excitatio is ot cosidered, the kietic eergy at the poit A is equal to the potetial eergy show by the shaded area. This potetial eergy W i is give by c b gh b gb g ( 10) 1 W i = ω 1 γ η i 1 1 + γ η i 1 η i 1 Where, η i is the ductility factor at i-th step (= xmi x ). x y mi ad x y are maximum relative displacemet of i-th step ad yield relative displacemet. The kietic eergy T i at the poit A is give by Ti = 1 v yi ( 11) The, after equatig Eqs.(10) with (11), the relatio betwee velocities v yi at the yield poit ad ductility factors η i is obtaied as follows. c b gh b gb g ( 1) v yi ω γ η i 1 γ η i η i = 1 1 + 1 1 Figure 4: Eergy balace Equatio(1) meas that the velocity v y at the yield poit depeds o the curret ductility factor η i ad the prior ductility factor η i 1. The, if a iitial ductility factor is zero, it will be able to evaluate ductility factors from the velocities at the yield poit successively. Therefore, usig Eq.(9) ad Eq.(1), it is possible to estimate the ductility factor from the wavelet coefficiet, which is obtaied by wavelet aalysis of absolute acceleratio respose. VERIFICATION OF PROPOSED METHOD BY SIMULATION Respose Aalysis Sice the horizotal seismic motio is ideally assumed to be a motio with a domiat frequecy ad is stochastic i ature, stochastic modelig of groud motio seems appropriate [Tajimi, 1960]. Therefore, i this paper, a seismic acceleratio &&u g to a structural model is the filtered white oise which is passed through the secod order system (ω f = 4 π, ζ f = 05. ). It stimulates the respose of structure i coditio of resoace. The mea-square amplitude of iput acceleratio ad its time duratio are set o referrig to post earthquake motio. The mea-square amplitude of &&u g is 1549(cm /s 4 ). The structural parameter of the model as show i Eq.() is take as ω = π, ζ = 0. 01, x = 5( cm), γ = variable(0-1) ( 13) y I order to obtai the relatioship betwee the wavelet coefficiet ad the ductility factor, respose aalyasis is performed from 0 to 1000 secods. I this sectio, the case of system parameter γ = 0. 05 is examied The samplig frequecy is 100Hz. The iput acceleratio ad absolute acceleratio respose from 0 to 5 secods are show i Fig.5 ad Fig.6, the time history of restorig force is show i Fig.7 ad the absolute jerk respose evaluated by umerical differetiatio of absolute acceleratio is show i Fig.8. Though it ca be see from Fig.7 that the displacemet respose gets ito the plastic regio five times, the discotiuity at the yield poit ca ot be foud oly by observig the acceleratio respose i Fig.6. While, the absolute jerk respose ought to cotai some discotiuous iformatios. But uless kowledge of the time eterig to plastic regio ad observatio of the sigal ear the 4
Figure 5: Iput acceleratio Figure 6: Respose of system (g =0.05) Figure 7: History of restorig force (g =0.05) Figure 8: Absolute jerk respose (g =0.05) time carefully, it is impossible to detect the discotiuity. I proportio as the rigidity ratio icreases, the detectio of discotiiuity becomes more difficult. Detectio of Discotiuity by Wavelet Aalysis The absolute acceleratio respose simulated i previous sectio is aalyzed by the wavelet aalysis ad the wavelet coefficiet α j, k is obtaied as show i Fig.9. The ceter frequecy of each aalyzig wavelet (level j =1-4) are 9Hz, 14.5Hz, 7.5Hz ad 3.65Hz respectively. The kid of wavelet is Daubechies wavelets (N=6) i this study. I the case of level j = 4, the sigal detectig the discotiuity is buried i oise, while i the other cases (level j =1,,3), the wavelet coefficiet detects discotiuity clearly as a pulse at the yield poit. As metioed above, this pheomeo meas that sudde discotiuity at the yield poit causes the chage i acceleratio; that is the jerk. Thus, rearragig Eq.(9), the proportioal relatioship betwee wavelet coefficiet ad velocity at yield poit is give by followig equatio. Figure 9: Wavelet coefficiet α j, k of absolute acceleratio respose (samplig: 100Hz) Figure 10: Relatioship betwee wavelet coefficiet ad velocity at yield poit (samplig: 100Hz) 5
v y = A α, (14) Where, A = b 1 g j k 1 γ ω a It is atural to cosider that the magitude of wavelet coefficiet detectig discotiuity is proportioal to the velocity at the yield poit. Because, as the velocity at the yield poit icreases, the discotiuity i acceleratio becomes large. I order to ivestigate this proportioal relatioship, the relatio betwee the wavelet coefficiet ad the velocity at the yield poits is obtaied from the respose aalysis ad it is show i Fig.10. I the case of level j =3, it is clear that α j, k correlates with v y. Ad applyig this relatioship to Eq.(14), the value of costat a is obtaied as 0.0154. Next, the wavelet aalysis that is similar to above is performed i the case of chagig samplig frequecy (00Hz ad 400Hz). The results of these aalyses are showed i Figs.11-14. As samplig frequecy becomes double, the resolutio of a aalyzig wavelet becomes double, too. So, i the case of 00Hz samplig, the ceter frequecy of each aalyzig wavelet (level j =1-4) are 58Hz, 9Hz, 14.5Hz ad 7.5Hz respectively. Also, i the case of 400Hz samplig, the ceter frequecy of each aalyzig wavelet (level j =1-4) are 116Hz, 58Hz, 14.5Hz ad 7.5Hz respectively. Thus, the clearest case of the pulse detectig discotiuity is reasoably that the samplig frequecy is 400Hz ad level j = 1, but the proportioal relatioship betwee wavelet coefficiet ad velocity at yield poit is approved oly the case that the level j 3. (15) Figure 11: Wavelet coefficiet α j, k of absolute acceleratio respose (samplig: 00Hz) Figure 1: Relatioship betwee wavelet coefficiet ad velocity at yield poit (samplig: 00Hz) Figure 13: Wavelet coefficiet α j, k of absolute acceleratio respose (samplig: 400Hz) Figure 14: Relatioship betwee wavelet coefficiet ad velocity at yield poit (samplig: 400Hz) 6
Table 1 shows the result of some umerical simulatios i which structural parameter is chaged (rigidity ratio γ = 01., 03., 05., 0. 7, 09. ). The coditio of this simulatios is that the samplig frequecy is 400Hz ad level j =4. It is foud that the value of costat a is almost the same value as 0.01056. Thus, keepig the kid of wavelet ad its level j, the value of a is determied as a costat value. Velocity at Yield Poit ad Ductility Factor The relatio betwee the velocity v yi at the yield poit ad the ductility factor η i is give by Eq.(1). This equatio is derived by the assumptio that the effect of seismic motio is ot cosidered from the yield displacemet to the maximum displacemet. Also, it is assumed that the kietic eergy at the yield poit is equal Table 1: Costat a evaluated from some simulatios Rigidity ratio ã Costat a 0.1 0.01055 0.3 0.01055 0.5 0.01056 0.7 0.01056 0.9 0.01058 to the strai eergy from the yield poit to the maximum displacemet. These assumptios may be justified whe the structure is resoat. I this coditio, the structure is dramatically stimulated by the small seismic motio, because the dampig ratio of structure is quite small. The estimated value by Eq.(1) is show i Fig.15 as the area eclosig by two solid lie obtaied by η i 1 = 1 ad η i 1 =, ad the simulated values are show by dots i the same figure. Though the simulated values are effected by the seismic motio applied from the yield poit to the maximum displacemet, these values are closely distributed aroud the two solid lies give by Eq.(1). Estimatio of ductility factor I this sectio, it is tried to estimate the ductility factor from the wavelet coefficiet for the acceleratio respose of a structural model stimulated by the filtered white oise which is passed through the secod order system (ω f = 4 π, ζ f = 05. ). The structural parameter of the system show i Eq.() is take as ω = 4π, ζ = 0. 01, γ = 0. 5, x = 5( cm ) ( 16) Figure 15: Relatio betwee velocity v y ad ductility factor η y The velocity at the yield poit is estimated from wavelet coefficiet i Eq.(9) with costat a=0.01056 ad the relatio betwee the estimated velocities $v y at the yield poit ad true values v y is show i Fig.16. It is show from this figure that the relatioship is almost liear ad the velocity at the yield poit is estimated with good accuracy. Next, the ductility factor is estimated from estimated velocity at the yield poit i Eq.(1) ad the relatio betwee the estimated values $η ad the true values η is show i Fig.17. It ca be see from this figure that the relatioship is almost liear ad the estimatio of the ductility factor is successful with good accuracy. Where, the maximum value of error i this estimatio is.9%. 7
Figure 16: Estimatio of velocity at the yield poit Figure 17: Estimatio of ductility factor Figure 18: Histogram of ductility factor As the result of these estimatios, from the wavelet coefficiet for acceleratio respose, the ductility factor, amely damage rate ca be easily estimated ad it ca be possible to obtai the histogram of ductility factor as show i Fig.18. CONCLUSIONS It is ecessary to kow the ductility factor ad the umber eterig the plastic regio i order to estimate the cumulative damage of structure. I this paper, the method is proposed to estimate the ductility factor by the wavelet aalysis of absolute acceleratio respose. Wavelet has the ability to detect discotiuity i a acceleratio respose record, furthermore it is foud i this study that the wavelet coefficiet is proportioal to the velocity at the yield poit. Besides, the ductility factor is estimated from the velocity at the yield poit by usig the eergy balace betwee kietic eergy at the yield poit ad strai eergy from the yield poit to the maximum displacemet. Thus, usig these relatioships, the ductility factor is estimated from the wavelet coefficiet obtaied by the wavelet aalysis of acceleratio respose with good accuracy. The applicability of this method is demostrated through some simulatios. REFERENCES 1. Ashio R. ad Yamamoto S. (1997), Wavelet Aalysis (i Japaese), Kyoritsu Syuppa, Tokyo. Soe A., Yamamoto S., Nakaoka A. ad Masuda A. (1994), Health Moitorig System of High-rise Structures by usig Wavelet Trasformatio of Moitorig Sigal i Service. Proceedigs of First World Coferece of Structural Cotrol, Vol., pp.ta3, 43-5. 3. Soe A., Yamamoto S. ad Masuda A., Nakaoka A. ad Ashio R. (1995) Estimatio of Cumulative Damage of Buildig with Hysteretic Restorig Force by Usig Wavelet Aalysis of Strog Respose Records (i Japaese). J. Struct. Costr. Eg., AIJ, No.476: pp.67-74. 4. Suzuki Y., ad Miai R. (1988) Applicatio of Stochastic Differetial Equatios to Seismic Reliability Aalysis of Hysteretic Structures, Probabilistic Egieerig Mechaics, 3-1: pp.43-5. 5. Tajimi, H. (196), A Statistical Method of Determiig the Maximum Respose of a Buildig Structure durig a Earthquake, Proceedigs of Secod WCEE, Vol., pp.781-786. 6. Tamai H., Kodoh K. ad Haai M. (1994), O Low-cycle Fatigue Characteristics of Hysteretic Damper ad Its Fatigue Life Predictio Uder Sever Earthquake (i Japaese), J. Struct. Costr. Eg., AIJ, No.46: pp.141-150. 8