NON-LINEAR EFFECTS IN SEISMIC BASE ISOLATION

THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Serie A, OF THE ROMANIAN ACADEMY Volume 5, Numer 3/4, pp. - NON-LINEAR EFFECTS IN SEISMIC BASE ISOLATION Dinu BRATOSIN Intitute of Solid Mechanic - Romanian Academy, Calea Victoriei 15, 711 Bucharet, Romania. E-mail: ratoin@acad.ro The ae-iolated tructural ytem contain an iolatory layer compoed y ruer like material with oviou non-linear mechanical propertie. The ojective of thi paper i to analye the effect of thee propertie on ae-iolation tructure repone. Uing a non-linear one-degree-of-freedom model for upertructure-ae ytem y numerical non-linear imulation one preent ome feature of the non-linear tructural repone. 1. INTRODUCTION The ojective of eimic iolation ytem i to decouple the uilding tructure from the damaging component of the earthquake impute motion. By interpoing a layer with low horizontal tiffne ut with high damping characteritic etween the tructure and the foundation, the upertructure i partially decoupled from the horizontal component of the earthquake ground motion. In the lat decade everal ae iolation ytem have developed for eimic protection of the tructure with the pecial detination a hopital, emergency communication center, firetation, traffic management center, hitorical uilding, a..o [1], [17], [1]. Performance of the ae iolated uilding in different part of the world during earthquake in the recent pat etalihed that the ae iolated technology i a viale alternative to the conventional earthquake reitant deign of a large category of uilding. Some of the commonly ued iolation ytem i the laminated teel-ruer earing (LRB) intalled etween tructure and the foundation [1]. The LRB ytem conit of alternating layer of ruer and teel vulcanized etween them. Thee element upport entire vertical tatically load of the uilding and play a diipative roll for horizontal eimic excitation. The ruer material upported permanent vertical compreion and horizontal hearing loading during earthquake. Another notale ae-iolation ytem i aed on pendulum column, which are introduced etween tructure and loc foundation (PRB ytem) []. The ytem conit of a erie of hort pendulum column, with pherically teel calotte, placed etween the upertructure and the foundation and laterally emedded in a ma of ruer. The gravitational load are tranported only throughout column and ruer material i ujected to compreion only during earthquake, therefore the fatigue rik i diminihed. Thi ytem comine the advantage of kinematic ytem with thoe of the laminated-ruer earing. The main performance of thee iolation ytem i governed y the dynamic ehaviour of the iolatory material a ruer, neoprene or another material, which play the ame role. The dynamic propertie of thee material uch a horizontal tiffne and damping capacity determine the filtering role of the iolation layer and, finally, the tructural dynamic repone. All of material ued in iolation layer ytem exhiit, more or le, a non-linear ehaviour. Thi fact i an induitale experimental reality [1], [6], [16], [4]. But, many time thi non-linearity i jut like that ignored, other time the tructural effect are preume ineential. Depending of loading tip, one can encounter either oftening or hardening non-linearity. Thu, for example, in hear or torional loading the ruer exhiit a oftening non-linearity and in compreion tet the ame material how hardening ehaviour. Thu, it i poile that the tructural repone will depend not only on non-linearity himelf ut non-linearity tip alo. Reccomended y Panaite MAZILU, memer of the Romanian Academy

Dinu BRATOSIN The ojective of thi paper i to evaluate the effect of the non-linear ehaviour of iolatory material on dynamic uilding repone. Uing a implified method a non-linear one-degree-of-freedom model for ae-iolation tructure are otained and with the aid of thi model a numerical comparative linear veru non-linear tudy are performed. A can ee in the next, the non-linear ehaviour of the iolatory material lead to the qualitative and quantitative different tructural repone y comparion with linear hypothei. More than, the kind of non-linearity, the oftening or hardening non-linearity ha a coniderale influence.. SIMPLIFIED METHOD FOR STRUCTURAL RESPONSE EVALUATION Let a tructure iolated from thi ae y a certain iolation ytem. Due to large difference etween mechanical characteritic of the tructural material and iolatory material two degree of freedom (dof) implified model can e ued to predict the dynamic repone of a uch ae iolated tructure [], [3]. The upertructure i aimilated to ingle degree of freedom (dof) ytem (characterized y it generalized ma m, it generalized damping c and it generalized tiffne k ) mounted on the ae aimilated to another dof ytem, characterized y it generalized ma m, it generalized damping c and it generalized tiffne k (fig..1). If xg, x and x are ground, ae and upertructure aolute diplacement, the ae and upertructure diplacement relative to the ground i (fig..): u = x x ; u = x x (.1) g g u c m x u c m k k x x g x g x x Fig..1 Fig.. mx!! m ( ) k x x (!! ) c x x mx!! mx!! m + m ( ) k x x g (!! ) c x x g Fig..3 reult: In order to determine the motion equation one can diconnect the two mae a in fig..3 and thu

3 Non-linear effect in eimic ae iolation Becaue: mx!! + mx!! + c x! x! g + k x xg = mx!! + c( x! x! ) + k( x x) = (.) x = u + x ; x = u + u + x (.3) g g the ytem (.) ecome: mu!! + m ( u!! + u!! ) + cu! + ku = ( m + m)!! x m( u!! + u!! ) + cu! + ku = m!! xg or, in matricial form: g (.4) Mx!! + Cx! + Kx = Mδx!! g (.5) where the ma, damping and tiffne matrice are, repectively: M m m c k = t ; ; m m C = = c K k (.6) mt eing the total ma: mt m m δ= 1 T. = + and δ a poition vector: { } Auming the tructure together with it ae a perfectly rigid ody mounted on iolator, a ingle degree of freedom model reult (fig..4), with the circular frequency: k = (.7) mt Alo, auming the tructure with fixed ae another dof ytem reult (fig..5), with the circular frequency: k = (.8) m m + m x m x c c k k x g x By introducing the frequential ratio: Fig..4 Fig..5 and the ma ratio: ε= (.9)

Dinu BRATOSIN 4 the dynamic matrix D ecome: m µ= (.1) m 1+µ 1 ε µ µ D = (.11) 1+ µ 1+ µ ε µ µ The fundamental frequencie of dof ytem reult now a the eigenvalue of dynamic matrix in the form: ( µ+ )( ε+ ) ( µ+ )( ε ) 1 1 1 1 4ε " (.1) µ µ ( µ+ 1)( ε 1) 1, = 1+ p By developing the term under the radical in inomial erie the fundamental frequencie of the dof ytem ecome: ( µ + 1)( ε+ 1) ε 1 p1 = ; p ε+ 1 ε+ 1 µ and the normalized form of the fundamental mode reult: T µ 1 = { 1 ε+ 1 } ; = 1 T (.13) X X (.14) ε+ 1 Becaue # the frequential ratio ε ha mall value. In thee condition, the firt fundamental frequency ecome: p1. Thi mean that the firt fundamental frequency of the ae iolated uilding i cloe to the frequency of the dof ytem contituted y rigid upertructure mounted on flexile ae iolator and the iolated uilding tend to ehave gloally a a dof ytem. Alo, the expreion for the econd fundamental frequency denote that the addition of the ae iolator increae the tructural frequency, in particular for mall value of the ma ratio. The modal participation factor are [14]: T i T i L i = XMδ XMX thu, y replacement of the appropriate fundamental mode value (.15) reult: i 1 1 L = µ+ ε+ ; L = ε ε+ µ+ ε+ 1 µ+ ε+ 1 1 (.15) (.16) and y erie development ecome: ε ε L1 = 1 ; L = (.17) µ + 1 µ + 1 Thee relation how that the participation factor for the firt mode approache one, which denote the dof ehaviour. The participation factor for the econd mode i very mall, therefore even if the econd mode frequency p fall in the range of high pectral acceleration, the mallne of the participation factor enure that the econd mode i not highly excited y the ground motion. In order to exemplify the aove-implified method and the ehaviour of the ae iolated tructure in the next will e preented a comparative tudy of a tructure with and without ae iolation.

5 Non-linear effect in eimic ae iolation Mechanical characteritic of the upertructure are chooe a in [3]: m = 9485 kg ; c = 371 N/m ; k = 1191 N/m Auming the tructure with fixed ae a dof ytem reult (a in fig..5), with the pulation frequency f and the natural period T : k 1191 1 1 = = = rad/ ; f = = 3. Hz ; T = = =.313 m 9485 π π f 3., the natural We notice that thee dynamic characteritic of upertructure are cloe to predominant dynamic characteritic for a uual ite (conolidated aluvionary depoit). Therefore, thi tructure i a proper candidate for iolated ae technology. Mechanical characteritic of the ae with the iolated layer (uppoed linear in [3]) are: m = 68 kg ; c = 374 N/m ; k = 3 N/m Auming the tructure together with it ae a perfectly rigid ody mounted on iolator, a dof model reult (a in fig..5), with the following dynamic characteritic: k 3.53 1 1 = = =.53 rad/ ; f = =.4 Hz ; T = = =.5 m + m 3685 π π f.4 The maic rapport i in thi cae: and pulation rapport: m 68 µ= = =.3 m 9485 6.394 ε= = =.16 44 are mall enough to make acceptale the approximation ued in eq. (.16) and from thee relation, the natural pulation of the dof-ytem reult: 1 1 p1 = = =.99 =.99.53 =.51 rad/ ε+ 1.16 + 1 p ( µ+ 1)( ε+ 1) (.3 + 1)(.16 + 1) = = =.331 =.331 = 46.6 rad/ µ.3 Thu, the natural frequency and natural period of the dof ytem are: and from eq. (.) the participation factor are: p1.51 1 1 f1 = =.4 Hz ; T1 = = =.5 π π f1.4 p 46.6 1 1 f = = 7.44 Hz ; T = = =.135 π π f 7.44 ε.16 L1 = 1 = 1 = 1.13 =.987 µ+ 1.3 + 1 ε.16 L = = =.13 µ+ 1.3 + 1 One can oerved that due to the iolator layer the dynamic characteritic undergo a jump:

Dinu BRATOSIN 6 f f1 =.4 Hz T1 =.5 = 3. Hz ; T =.314 f = 7.44 Hz T =.135 which for firt viration mode take out the tructure from dangerou zone (fig..7). For econd viration mode the tructure remain in the dangerou zone ut due to mall participation factor ( L =.13 ) the tructural effect are negligile. Fig..7 Magnification function in term of frequency and period The aove example prove that the quai-rigid ehaviour of the uilding from ae iolated tructure allow to ue a implified ingle degree of freedom modell to predict the dynamic ehaviour. In order to implify thi demontration in thi example we preerve the linear mechanical characteritic for upertructure and iolated layer ued in [3]. For the tructural material thi hypothei i acceptale ut for iolatory material i at leat moot. Experimental tudie performed upon either material ample or whole iolator prove evident train dependence ehaviour [1], [6], [7], [9], [16], [4]. For thi reaon the implified dof ytem mut e nonlinear. Finally, we notice that: The iolated layer give the tructure a fundamental frequency that i much lower than it fixed-ae frequency and alo much lower than the uual predominant frequencie of the ground motion. The firt dynamic mode of the iolated tructure involve deformation only in the iolation ytem, the tructure aove eing to all intent and purpoe rigid. The higher mode have high frequencie ut do not participate in the motion due to it mall participation factor. The quai-rigid ehaviour of the uilding from ae iolated tructure allow uing a implified ingle degree of freedom modell. But, due to nonlinear characteritic of the iolated layer material thi dof ytem mut e nonlinear. 3. NONLINEAR SINGLE-DEGREE-OF-FREEDOM MODEL A can ee in the previou chapter, for implified evaluation of the tructure-ae iolated ytem a non-linear dof model are needful. Alo, all device ued for experimental determination of the dynamic characteritic of the iolated layer material are in fact dof ytem ujected to teady-harmonic excitation at top ide (e. g. reonant column apparatu [7], [15], [5]) or y ae diplacement (e. g. electrodynamic haker)[4 ]. For thi reaon, in thi chapter will e preent a nonlinear ingle-degree-offreedom otained a an extenion in non-linear domain of the linear Kelvin-Voigt model.

7 Non-linear effect in eimic ae iolation In linear dynamic a uual decription of a olid ingle-degree-of-freedom ehaviour i given y the Kelvin-Voigt model coniting of a pring (with a tiffne k) and a dahpot (with a vicoity c) connected in parallel. The governing equation of thi ytem for harmonic viration i [7], [1], [13]: M!! x+ c x! + k x= A in t, (3.1) where x i the ytem' diplacement (linear diplacement x or rotation θ), M i the ma characteritic (ma m or maic moment of inertia J), A i external force amplitude (F or M ) and i the pulation of thi force. Dynamic characteritic c and k have for linear elatic material known expreion. Thu, for a rod ujected to longitudinal excitation we have: S k = E, c= mζ long, (3.) h where S i the cro ectional area, h i the high, E i the Young modulu, m i the ma, i the natural pulation and ζlong i the damping ratio. Alo, for the ame circular rod ut ujected to torional excitation we have: I p k = G, c= Jζ tor, (3.3) h where I p i the rotational moment of inertia, G i the hear modulu, J i the maic moment of inertia and ζ tor i the damping ratio for thi type of loading. Uing the method that decrie the non-linearity y train or diplacement dependence of the material moduli: E = E() ε or E = E( x) and G = G( γ ) or G = G( θ ) [], [3], [4], [7] we hall aume that the pring tiffne k and the damper vicoity c are function in term of diplacement: S k( x) = E( x), c( x) = mζlong ( x), h I k G c J tor h p ( θ ) = ( θ) ; ( θ ) = ζ ( θ) We mention that a in thi non-linear cae the pring tiffne i a function, we hall define the undamped natural pulation in term of initial value of tiffne function k() : = k ( )/ M. Therefore the mot expected form of the governing equation for non-linear ehaviour of a ingledegree-of-freedom ytem i: k = k(x) A= A in t m ; J Fig. 3.1 NKV model c = c(x) in t (3.4) M!! x+ c x x! + k x x= A, (3.5) with the analogic non-linear Kelvin-Voigt (NKV) model from fig. 3.1 [11], [19]. For complete determination of the eq. (3.5) the material function c( x) and k( x) from eq. (3.4) mut e evaluation from experimental data. In order to exemplify thi nonlinear extenion in the next will ue the experimental data otained from torional reonant column tet performed upon ruer ample. The modulu and damping function in term of rotation θ wa otained with a reaonale accuracy in exponential form (fig.3.).

Dinu BRATOSIN 8 Fig. 3. Modulu G = G() θ and damping ζ = ζ() θ function for teted ruer In thee figure one can ee that material function have imilar form with the relaxation and creep function of the analogic tandard vicoelatic olid [11], [19]. Thi can denote that a olid tandard model i more adequate then NKV model. But, changing the parameter of the linear Kelvin-Voigt model with the non-linear material function with initial value lead in fact to a nonlinear tandard model. Becaue in the next will not ue analogic model, the denomination non-linear Kelvin-Voigt model (NKV model), a an extenion of the linear Kelvin-Voigt model, wa preerve. Take into account the ample dimenion (height h =7.4 cm and diameter φ = 3. cm) and ma 3 characteritic ( m= 147g, J =.8 1 Nm ) from eq. (3.9) the dynamic nonlinear function of the NKV model reult a in fig.3.3. Fig. 3.3 NKV function

9 Non-linear effect in eimic ae iolation 4. EVALUATION OF THE NON-LINEAR SDOF RESPONS After the qualitative and quantitative determination of the dynamic material function c( x ) and k( x) the differential equation of the non-linear dof ytem (3.5): in!!! x+ c x x! + k x x= " t (4.1) can e numerical olved for ytem diplacement x. Becaue the dimenionle equation aure a etter numerical accuracy olving it i preferale to tranform the eq. (4.1) into a dimenionle equation. Thu, y uing the change of variale τ = t and y introducing a new "time" function [5]: ϕτ () = xt () = x τ (4.) one otain for eq. (4.1) another form: ϕ + C( ϕ) ϕ + K( ϕ) ϕ = µ inυτ (4.3) where the upercript accent denote the time derivative with repect to τ: and: ϕ 1 ϕ 1 ϕ ( τ ) = = x! ; ϕ ( τ )= =!! x (4.4) τ τ t k() cx kx k x C( ϕ) C( x) = = ζ( x) ; K( ϕ) K( x) = = = kn x M M k " " µ= = = x ; υ=! (4.5) For torional excitation M = M in t when the diplacement i the rotation θ the (4.3) form i a dimenionle one. But, if the excitation i of longitudinal tip F = F in t the tranformed equation (4.3) have length dimenion. In thi cae one can otain a dimenionle form y introducing a new τ function: ψ() τ =ϕ()/ τ µ (4.6) and the dimenionle longitudinal equation reult in the form: ψ + ζµψ ψ + ( µψ) ψ = inυτ (4.7) where: k n () () ; () () ; () () ϕ τ =µψ τ ϕ τ =µψ τ ϕ τ =µψ τ (4.8) For a given normalized amplitude µ and relative pulation υ, the non-linear dimenionle equation (4.3) or (4.7) can e numerically olved and a olution of the form: ( υτ+ψ) ϕ(; τ µ, υ ) =µψ υ ; µ $ = µ Φ υ ; µ in, (4.9) can e otained in n dicreet point [5]. In eq. (4.9) Φ( υ ; µ ) are the maximum magnification function (one function Φ( υ ) for each normalized amplitude µ) and Ψ i the phae difference. Finally, from invere tranformation the queted olution x= x( t; A, ) i otained. When the non-linear ytem are ujected to a harmonic acceleration applied at the ytem ae (a in eimic cae):

Dinu BRATOSIN 1 the movement equation ha the form: or y uing eq. (3.4): ut!!() = u!! in t (4.1) in mx!! + c x x! + k x x = mu!! t (4.11)!! x+ ζ x x! + k x x= u!! int (4.1) n By the ame variale change (4.) a new tranformed equation like eq. (4.3) ut with length dimenion, are otained: ϕ + ζϕ ϕ + ( ϕ) ϕ = µ inυτ (4.13) k n where the normalized amplitude µ i in thi cae: thu: u!! µ= Becaue the excitation ha harmonic form the ae diplacement may e in the form: u!! ut () = uin t= in t u!! µ = = = υ u u (4.14) (4.15) (4.16) The dimenionle form of the eq. (4.13) can e otained y introducing in eq. (4.13) the τ function: that lead to: where µ $ = µ /u =υ. ψ() τ =ϕ() τ /u (4.17) ψ + ζ( u ψ) ψ + k ( u ψ) ψ = µ $ inυτ (4.18) n Now, the dimenionle equation (4.18) can e numerical olved and olution ψ = ψ( τ ; µ$ ) can e ued for otained the olution of the eq. (4.13): ϕ(; τ µ, υ ) = uψ( υ ; µ $ ) = u Φ( υ ; µ $ ) in ( υτ+ψ), (4.19) and then x= x( t; u!!, ) 5. NON-LINEARITY INFLUENCE The modelling of the ae iolated tructure a nonlinear ingle-degree-of-freedom and numerical olving poiility of thi ytem allow a to qualitative evaluate the influence of the iolatory material nonlinearity on tructural repone. All of material ued in iolation layer ytem of the ae iolated tructure exhiit, more or le, a non-linear ehaviour. Thi fact i an induitale experimental reality. Many time thi non-linearity i jut like that ignored, other time are neglected. Depending of loading tip, one can encounter either oftening or hardening non-linearity. Thu, for example, in hear or torional loading the ruer exhiit a oftening nonlinearity and in compreion tet the ame material how hardening ehaviour. Thu, it i poile that the tructural repone depend not only on non-linearity himelf ut non-linearity tip alo.

11 Non-linear effect in eimic ae iolation In order to evaluate the non-linear ehaviour and linear - non-linear difference, a numerical imulation tudy wa performed. For thi, the ame ae-iolated tructure wa uing, firt neglecting the iolatory layer non-linearity and then dicard thi implification. Alo, the tructural repone wa determinate uing iolator with oftening non-linearity a well a with hardening non-linearity. For detect the oftening - hardening difference the ame tructure mounted upon two different iolator layer wa ued. In the firt cae wa provided the LRB (Laminated Ruer Bearing) iolator where the ruer i ujected to hear and exhiit oftening non-linearity and in the econd cae, the PRB (Pendulum Ruer Bearing) iolator where the ruer i ujected to compreion and exhiit hardening non-linearity. In oth cae a comparion with linear approximation will e made. The tructure ued in thee imulation wa the ame tructure ued in chapter for exemplify the difference etween fixed and iolated ae [3]. Recall that in the chapter' example for iolated layer wa aumed a linear ehaviour with the dynamic characteritic: The non-linear tiffne k k( x) m = 68 kg ; c = 374 N/m ; k = 3 N/m = wa uilder y extenion tarting to own tet [6] and experimental data given in [17] for oftening ehaviour and [4] for hardening ehaviour: koftening = 186 + 46exp( 45 x) [N/m] k = 3 + 76exp 45 x [N/m] hardening Alo, for detect the tructural difference etween oftening and hardening ehaviour in oth cae the ζ=ζ x wa ued: ame damping function ( x) ζ ( x) =.5.3exp 5 In order to compare the non-linear reult with the linear calculu from chapter the initial value of thee non-linear material function wa caled for coincide with contant value of the linear ehaviour hypothei. We notice that material function k = k( x) and ζ=ζ ( x) wa chooe with middle non-linearity a oerved in tet performed upon whole iolator [16] though in laoratory tet performed upon ruer ample a more pronounced train dependence wa otained [6]. The dimenionle material function (ee eq.4.5) are given in fig.5.1. Fig. 5.1 Non-linear dimenionle material function

Dinu BRATOSIN 1 The autment excitation ued in the imulation proce wa of harmonic type:!! x g =!! xgin t with the amplitude value x!! g correponding to peak ground acceleration oerved during ome eartqkuake [3]. The tructural non-linear repone wa otained uing a computer program aed on Newmark algorithm [18] and olving method from chapter 3. The imulation reult i ummarized in fig. 5.. From thi figure, reult ome oervation: Both non-linearity type take out the tructure from dangerou zone. But, the jump toward high period i different from linear one. Wherea the linear calculu lead to the unique reonance value the real nonlinear olving lead to the multiple reonant value (in term of excitation amplitude) ituated efore and after linear reonant value. The non-linear magnification function are different hape in comparion with the linear one. At reonance, the amplitude of the non-linear magnification function are inferior vi-a-vi the maximum amplitude of linear magnification function, thu non-linear calculu make apparently the damping capacity neglected y linear calculu. For period value until reonance the linear calculu may underetimate the dynamic magnification. Fig. 5. Non-linear magnification function AKNOWLEDGMENT Thi work i upported y Romanian Academy Grant No.94/3-4.

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