Optimum Design Method of Viscous Dampers in Building Frames Using Calibration Model

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The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China Optimum Design Method of iscous Dampers in Buiding Frames sing Caibration Mode M. Yamakawa, Y. Nagano, Y. ee 3, K.

1 The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China Optimum Design Method of iscous Dampers in Buiding Frames sing Caibration Mode M. Yamakawa, Y. Nagano, Y. ee 3, K. etani 4 Assistant Professor, Dept. of Architecture and Architectura Engineering, Kyoto niversity, Kyoto, Japan Associate Professor, Dept. of Architecture and Civi Eng., Fukui niversity of Technoogy, Fukui, Japan 3 Researcher, Dept. of Architecture and Architectura Engineering, Kyoto niversity, Kyoto, Japan 4 Professor, Dept. of Architecture and Architectura Engineering, Kyoto niversity, Kyoto, Japan Emai: yamakawa@archi.kyoto-u.ac.jp, nagano@fukui-ut.ac.jp, archi_yujin@yahoo.co.jp, uetani@archi.kyoto-u.ac.jp ABSTRACT : We present a new practica optimum design method of viscous dampers in buiding frames. In the design fied of structura engineering, structura designers aways try to buid mathematica modes to simuate the eact behavior of rea word system which are often very compe for the verification anaysis and are not suited for the optimum design owing to the fact that they come with a good dea of computationa cost. Therefore, we present a method using caibrated response mode, which is simper than the verification mode. The caibration mode is a statistica prediction of the verification mode and with its usage the optimum design method of dampers can be made efficient and numericay stabe. The efficiency of the presented method has been demonstrated by a numerica eampe. KEYWORDS : Optimum design, iscous damper, Caibration, Bayesian inference. INTRODCTION Structura designers, in structura design, aways try to deveop mathematica modes which can accuratey simuate the behavior of the rea word system. However, these modes for the verification anaysis are often compe and are not suited for the optimum design owing to the face that they come with a good dea of computationa cost. Furthermore, it is difficut to understand the characteristics of the optimum soution. For this reason some studies of an optimum design method of viscous dampers [for eampe, Tsuji et a., ] in buiding frames have deat with simper mode than the mode widey used for the verification anaysis. Here, we present a new practica optimum design method of viscous dampers in buiding frames using statistica prediction mode within Bayesian framework. The presented method is based on the caibrated output of the simpe mode and is very efficient in view of computationa cost and numericay stabiity... Muti-eve Modes Kennedy and O Hagan ( modeed the outputs of muti-eve computer codes using a spatia autocorreation structure within Bayesian framework. In this paper, we use this muti-eve mode: verification mode and simpe mode. We define a mode using noninear time history response anaysis as verification mode and a mode using etended CQC method [Igusa et a., 984] as simpe mode. The caibration mode is defined as the scaed simpe mode based on computer eperiments. The response by verification mode is predicted as the scaed simpe mode. The method is very efficient and numericay stabe. sing this method, we formuate the optimum design probem, which finds a set of viscous dampers, the cross sections of coumns and beams that minimize the cost function subject to some constraints.. CAIBRATION MODE From a Bayesian perspective, uncertainty about the output of compe mode can be epressed by a stochastic process. The prediction method is assumed to be a function of a set of inputs denoted by nx = (,, n R, with output of compe mode represented by y R. et us consider a mode: X

2 The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China y ( = η( + ε, (. where ε is a random noise and foows an independent norma distribution: ε N (, ασ /( + α. The parameters σ and α is unknown scae parameter and variance ratio. Such modes typicay incude an unknown smooth response surface, which is knows as nugget effect... Gaussian Process We assume that ( η is represented by Gaussian process [for eampe, Rasmussen et a., 6] with mean m and covariance function (,. Gaussian process using a hierarchica formuation is epressed as ( ( β,, GP ( m (, (, where GP (, denotes Gaussian process distribution, and η σ ψ, (. ( ( T m = h β, (.3 (, = σ R(, ; ψ /( + α, (.4 nx q q where h( : R R is a function of the inputs, which is the simpe mode. The parameter β R is an unknown vector of coefficients, and R (, ; ψ is a given correation function. In this paper, we use Gaussian correation function as foows q (, ; ep ( ( ( R ψ = hk h k / ψ k, (.5 k= where the hyper parameter ψ = { ψ,, ψ q } are caed as correation ength parameters. We have a training dataset { i n X, ( i n D = R yi = y ; i =,, n}. According to (., the distribution of η = ( η(,, η( is mutivariate norma as foows where η β, σ, ψ N ( Hβ, σ /( + α R, (.6 H ( T h =, n T h( R n R(, ; ψ R(, ; ψ =. (.7,(.8 n n n R(, ; ψ R(, ; ψ sing standard techniques for conditioning in mutivariate norma distributions, we get Gaussian process * * prediction with mean m ( and covariance function (, as where * * (, β,, ψ, y GP ( m (, (, η α σ, (.9 ( ( T ( T m = h β+ r R ( y Hβ, (. * ( = σ { R( ( + α r( R r( } * T,, ; ψ /, (.

3 The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China R = ( R + αi n / ( + α, (. ( (, ; / ( (, ; / ( T r = R ψ + α R ψ + α. (.3 ( n where I n denotes the identity matri of size n. The mode is aso known as kriging. Kriging is mosty used in two or three dimensiona input spaces for spatia prediction, athough Gaussian process prediction coud be used in a genera regression contet... Gaussian Process within Bayesian Framework sing a weak prior for p( β, σ σ, integrating out β and σ of (.9 by Bayes theorem, the Student s t process predictor with n q degrees of freedom, mean m ( and covariance function (, can be shown as ( y,, TP ( n q, m(, (, where TP (,, denotes Student s t process distribution, and η α ψ (.4 ( ( T ˆ ( T m = h β + r R ( y Hβ ˆ, (.5 ( ( ( ( T, = σˆ C, ; ψ / α ( + r R r T T ( ( T ( T + h r R H( H R H ( h( H R r(, ˆ T ( T = H R H H R y, T T T σˆ = ( / n q (.6 β (.7 ( (. y R R H H R H H R y (.8 According to (. and (.4, conditiona epectation and variance are obtained as foows: E ˆ ˆ (.9 [ ( ] ( T ( T y y, α, ψ = h β+ r R ( y Hβ, T T T T + ( h( H R r( ( H R H h( H R r( [ ( ] ( T y y, α, ψ = σˆ r R r( (. (. 3. OPTIMM DESIGN METHOD We present an optimum design method which enabes to find minimum cost so as to satisfy member-end strain constraints, story drift constraints, and side constraints. Consider a panar stee buiding frame added Mawe-type viscous dampers as shown in Figure. The noda mass is taken into account in every node. Rigid diaphragms are assumed for foor sabs and the dampers connect these rigid diaphragms. 3.. Design ariabe The cross sectiona area of the i th member A i is associated with the design variabes = { ; =,, nx} as Ai = ( i I A with the inde set I A. We simpy write it as Ai (. The cross sectiona area Ai (, second moment area Ii ( and section moduus Zi ( are assumed to satisfy:.5 for beams: I ( 4. ( (, (.5( ( i = Ai Zi = Ai ( i =,, n M, (3.,(3..5 for coumns: I (. ( (, (.8( ( i = Ai Zi = Ai ( i =,, n M, (3.3,(3.4

4 The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China where n M denotes the number of members. Mawe-type biinear viscous eements which depend on veocity such as oi damper are added to the frame shown in Figure. The j th damper has biinear damping coefficient: c Dj and. cdj, reief oad f Rj, imum oad f Dj as shown in Figure and the stiffness kdj (. The f Dj is aso associated with as fdj = ( j I D with the inde set I D, which we simpy write as fdj (. The damping coefficient cdj ( and stiffness kdj ( are given as foows: c Dj ( = ( /4.38 f (, k ( =.f ( ( j =,, n F (3.5,(3.6 Dj Dj where n F denotes the number of stories, and the units of fdj (, cdj ( and kdj ( are kn, kn /(cm / sec and kn / cm, respectivey. Dj damping force f Dj f Rj.c Dj Figure Panar stee frame added Mawe-type viscous dampers c Dj 3. 5 veocity [cm/s] Figure Damping characteristic of damper 3.. erification Mode We perform the noninear time history response anaysis by Newmark- β method as the verification mode. The design earthquake motions and the ampitude of the scaed design earthquakes are as shown in Tabe. Tabe Design earthquakes (by Buiding Center of Japan Maimum ampitude seismic veocity(cm/s seismic acceeration E CENTRO 94 NS 55 5 TAFT 95 EW 48 5 BCJ- (artificia seismic wave 7 9 (cm/s et δ j ( denote the imum story drift of the j th story to the design earthquakes by time history response anaysis. In this mode, the dampers are biinear viscous eements, however, the coumns and the beams are eastic eements for simpicity. We suppose that the noninear easto-pastic eements can be used Simpe Mode We use the etended CQC method as simpe mode. et ω and h denote an eigen frequency and a moda damping ratio. Design response spectrum [Newmark and Ha, 98] is modified for the design earthquakes shown in Tabe. The foowing design dispacement response spectrum is used here. (, min SD ω h = SD ( ω,h, SD ( ω,h (3.7 A SD ( ω, h =6.{ 4.4-.n( h }/ ω ( cm (3.8 S ( ω, h =3.4 {.6-.5n( h }/ ω ( cm (3.9 D

5 The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China S et δj ( denote the imum story drift to the design response spectrum by etended CQC method. Note that, in this simpe mode, the j th damper is inear viscous eement which has the initia damping coefficient of the verification mode c Dj. Additionay, we aso use output of CQC method with no damper mode, that is, S c Dj = as simpe mode. et δj ( denote the imum story drift to the design response spectrum by CQC method in the no damper mode. δ ( S wi be in between δ ( S and δ (. j 3.4. Origina Optimum Design Probem The cost function of the frame mode is given as j j nm nf f = w ρ A + w f (3. ( ( ( F i i D Dj i= j= wehre i, ρ, w F and w D are the ength of the i th member, the density of stee, the cost factor of frame and damper, respectivey. We define the Origina Optimum Design Probem (OODP as foows: OODP min nm nf ( = ( ( Fρ i i + D Dj i= j= f w A w f (3. subject to δ ( j δj δj ( j =,, nf (3. ε ( εi ε ( j =,, nf (3.3 ( =,, nx (3.4 where εi ( denotes the imum member-end strain of the i th member. δ j, ε, and denote the imum story drift, the imum member-end strain, ower and upper bound of the design variabe, respectivey. For simpicity, we use etended CQC method for εi ( Soution Agorithm with Bayesian Inference Time history response anaysis needs much computationa cost than etended CQC method. For this reason, the directy soving OODP is epensive in terms of the computer time. Besides, the time history response anaysis is strongy noninear and non-conve, hence soving OODP is not numericay stabe. Therefore, we present an optimization method with the statistica prediction mode instead of soving OODP directy. The soution agorithm is shown as foows. Step. Soving the Simpe Optimum Design Probem Firsty, we sove the foowing Simpe Optimum Design Probem (SODP. SODP nm nf ( ( ( f = wfρa i i + wdfdj i= j= min (3.5 subject to S δ ( j δj δj ( j =,, nf (3.6 ε ( εi ε ( i =,, nm (3.7 ( =,, nx (3.8 The soution of the SODP is denoted by ˆ = { ˆ ; =,, nx }. Etended CQC method is moda anaysis for inear mode, as a resut soving SODP needs ow computationa cost.

6 The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China Step. Carrying out computer eperiments We carry out the computer anayses using both the simpe mode and the verification mode at the set i of the points { ; i =,, n}. The points are chosen to be satisfied i ( i =,, n, =,, n. (3.9,(3. ˆ r ˆ r ( i,, n,,, n. (3.,(3. X i Δ Δ = = X i where and Δ r denote the th eement of i and change of the th design variabe, respectivey. Here, we set the equations presented in section as foows: ( j y ( = δ ( ( j =,, n, (3.3 j h. (3.4 S S S ( ( S = δ (,, δn (, δ (,, δn ( F In equation (3.3, the method in section is appied for each j =,, nf. Thus, we get the training i ( j dataset of j th story Dj = {, yi ; i =,, n}. arious methods of choosing points are presented [for eampe, Santner, et a., 3]. Here, the points are generated randomy for simpicity. Step 3. Finding the hyper parameters by cross vaidation ( j We find the optimum hyper parameters of each y ( by minimizing eave-one-out cross-vaidated sum of squared error as foows: { } n ( j ˆ( j ( j ( j ( j ( j αˆ, ψ argmin y (,, ( j ( j y yi α ψ k α, ψ k= { Ik } F = E, (3.5 F T where Ik = {,, n} \ { k} and ( j ( j k k ( y ; k k y I = I. Step 4. Finding the optimum soution with Bayesian inference We sove the foowing Optimum Design Probem with Bayesian Inference (ODPBI. ODPBI min nm nf f = w ρa + w f (3.6 ( ( ( F i i D Dj i= j= subject to ( j ( j ( ( ( ( ( ˆ j j j ( ˆ j E δj, αˆ, ψ δj, αˆ, y + y ψ δ j ( j =,, nf (3.7 ε ( εi ε ( i =,, nm (3.8 ( =,, nx (3.9 ˆ Δr ˆ +Δ r ( =,, nx (3.3 The soution of ODPBI is denoted by ˆ = { ˆ ; =,, n X }. ODPBI is based on the caibrated simpe mode, besides, the output of the caibration mode is smoothed out, as a resut soving ODPBI needs ow computationa cost and is numericay stabe. Step 5. erification of the soution ( We anayze the soution ˆ by verification mode. If the output ˆ j { y ( ˆ ; j =,, n F } doesn t satisfy ( the design criteria with sufficient accuracy, ˆ ˆ j {, y ( ˆ} is added to the training data set D j as a new data. Moreover, we repeat from step3 unti the soution satisfies the design criteria.

7 The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China 4. NMERICA EXAMPE Consider -story and 3-span panar stee frames as shown in Figure 3. Cross sections of coumns and beams are denoted by C-C8 and G-G, respectivey. The dampers of the -th story are denoted by D-D. 4m 4m C C C C G G G G9 C8 G9 C8 G9 G8 C8 G8 C8 G8 G7 C8 G7 C8 G7 G6 C7 G6 C7 G6 G5 C7 G5 C7 G5 G4 C7 G4 C7 G4 G3 C6 G3 C6 G3 G C6 G C6 G G C6 G C6 G C5 C5 C C C C m m Figure 3 -stories and 3-span mode The structura damping ratio is assumed as h = %. A amped mass of 36.5kg and 63.3kg are paced on every interior node and eterior node, respectivey. The parameters of side constrains are as shown: for beams: for coumns: for dampers: = cm, = cm, kn = 4cm, Δ r = 9cm, = 8cm, Δ r = 8cm, r =, = kn, Δ = 3kN. The other parameters are as foows: ε =.57, δ j = cm, w F = 5 / ton, w D =.5 / kn. Here, practica constraint conditions are added to SODP and ODPBI. The condition is that a cross section of a coumn is smaer than cross sections of the ower coumns ( C> C> >, C5> C6> C7 > C8. Firsty, we obtain the initia soution ˆ by soving SODP with sequentia quadratic programming (step. Net, computer eperiments, of which the number is given by n =, are carried out (step. asty, we obtain the optimum soution ˆ by iterations of soving and updating ODPBI with sequentia quadratic programming (from step3 to step 5. The soutions are shown in Tabe. The costs and imum story drift anges by time history response anaysis of the optimum soutions ˆ and ˆ are shown in Tabe 3, where H = 4cm denotes the story height. It can be observed that the costs of ˆ and ˆ are neary the same. The imum story drift anges of the soution ˆ vioate the criteria δ j / H = 5/ rad argey, by contrast, the imum story drift anges of the soution ˆ approimatey satisfy the criteria 5rad / because the predicted mean and variance have good accuracy. 5. CONCSION The concusion may be summarized as foows: ( We present a new optimum design method of a buiding frame with viscous dampers using the caibration mode, which is based on the statistica muti-eve anaysis. The method has simiar accuracy as the verification anaysis and the computationa cost is much smaer than the verification anaysis. ( The efficiency of the presented method is demonstrated by numerica eampe. In the eampe, the predicted mean and variance have good accuracy. Consequenty, the soution approimatey satisfies the constraints.

8 The 4 th Word Conference on Earthquake Engineering October -7, 8, Beijing, China Tabe Optimum soutions Cross sectiona area of Coum (cm Cross sectiona area of Beam (cm ˆ ˆ ˆ ˆ Maimum oad of Damper (kn C G D. 7. C G D G D3.... G D4.. C G D5.. C G D C G D C G D8.. G D9. 7. G. 8. D. 7. ˆ ˆ Tabe 3 Cost and imum story drift ange by time history response anaysis of the optimum soution Maimum story drift anges [/ rad] Cost F F 3F 4F 5F 6F 7F 8F 9F F ˆ 38. δ ( ˆ / H δ ( ˆ/ H E δ ( ˆ / y H ˆ δ ( ˆ / y H δ ( ˆ E δ ( ˆ y / H Acknowedgments This work was partiay supported by the grant from the Kyoto niversity Foundation and Grant-in-Aid for Scientific Research (No from Japan Society for the Promotion of Science. The authors are gratefu for advice from Shiresta K. C. (Research Student, Kyoto niversity. REFERENCES Igusa, T., Der Kiureghian, A. and Sackman, J.. (984. Moda decomposition method for stationary response of non-cassicay damped systems. Earthquake Engng. Struct. Dynamics,, -36 Kennedy, M. C., and O hagan, A. (. A Bayesian caibration of computer modes (with discussion. Journa of the Roya Statistica Society, B, 63, Newmark, N. M., and Ha, W. J. (98. Earthquake spectra and design, Earthquake Engineering Research Institute, Berkeey, CA. Rasmussen, C. E., and Wiiams, C. K. (6. Gaussian Processes for Machine earning, The MIT Press. Santner, T. J., Wiiams, B. J., and Notz, W. I. (3. The design and Anaysis of Computer Eperiments, Springer-erag. Tsuji, M., Nagano, Y., Ohsaki, M. and etani, K. ( Optimum Design Method for High-Rise Buiding

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