BASE-ISOLATION TECHNOLOGY FOR EARTHQUAKE PROTECTION OF ART OBJECTS

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1 The 4 th World Conference on Earthquake Engineering Octoer -7, 008, Beijing, China BASE-ISOLATION TECHNOLOGY FOR EARTHQUAKE PROTECTION OF ART OBJECTS P.C. Roussis, E.A. Pavlou and E.C. Pisiara 3 Lecturer, Dept. of Civil & Environmental Engineering, University of Cyprus, Nicosia, Cyprus Research Associate, Dept. of Civil & Environmental Engineering, University of Cyprus, Nicosia, Cyprus 3 Graduate Student, Dept. of Civil & Environmental Engineering, University of Cyprus, Nicosia, Cyprus ABSTRACT : roussis@ucy.ac.cy In this paper the dynamic response of ase-isolated lock-like slender ojects, such as statues, sujected to horizontal ground excitation is investigated. The structural model employed consists of a rigid lock supported on a rigid ase, eneath which the isolation system is accommodated. Assuming no sliding of the lock relative to the supporting ase, when sujected to ground excitation the system may exhiit two possile patterns of motion, namely pure translation, in which the system in its entirety oscillates horizontally ( degree-of-freedom response), and rocking, in which the rigid lock pivots on its edges with respect to the horizontally-moving ase ( degree-of-freedom response). The dynamic response of the system is strongly affected y the occurrence of impact etween the lock and the horizontally-moving ase, as impact can modify not only the energy ut also the degrees of freedom of the system y virtue of the discontinuity introduced in the response. Therefore, the critical role of impact in the dynamics of the system necessitates a rigorous formulation of the impact prolem. In this study, a model governing impact from the rocking mode is derived from first principles using classical impact theory. Numerical results are otained via an ad hoc computational scheme developed to determine the response of the system under horizontal ground excitation. KEYWORDS: ase isolation, art ojects, earthquake, rigid lock. INTRODUCTION There is a wealth of elements of cultural heritage worldwide and thus a widespread challenge to preserve this important legacy. From irreplaceale museum artifacts and statues to intricate decorative art ojects, the range and complexity of materials, forms and geometry makes protecting this heritage against earthquakes a formidale task. Yet despite the need for conservation efforts, only recently research attempts have een pursued in this direction. In this paper the seismic performance of ase-isolated lock-like ojects, such as statues, is investigated. The seismic ehavior of such ojects can e analyzed within the context of rigid-ody dynamics. In the literature there is a wealth of research papers on the seismic ehavior of lock-like structures. Housner's landmark study has provided the asic understanding on the rocking response of a slender rigid lock and sparked modern scientific interest. His model is ased on the assumption of perfectly-inelastic impact and sufficient friction to prevent sliding during impact. Following Housner s fundamental work, numerous studies [e.g. Yim et al. (980), Ishiyama (98), Spanos and Koh (984), Shenton and Jones (99), Makris and Roussos (000)] have een reported in the literature dealing with various aspects of the complex dynamics of the single rigid lock. Moreover, the dynamic ehavior of two-lock structures has een studied y Psycharis (990) and Spanos et al. (00), who analyzed the non-linear dynamic response of systems consisting of two locks, one placed on top of the other, free to rock without sliding. Such a configuration can e thought of as a model of ancient statue placed on top of a lock-like ase. Only recently has the seismic protection of ojects of cultural heritage ecome a suject of interest of many researchers. Agaian et al. (988) were perhaps the first to explicitly deal with the seismic protection of art

2 The 4 th World Conference on Earthquake Engineering Octoer -7, 008, Beijing, China ojects. Their work aimed at the development of analytical and experimental procedures for the evaluation of the seismic mitigation of various museum contents, at the Jean Paul Getty Museum in Maliu, California. Susequently, Augusti et al. (99) studied the seismic response of art ojects and proposed some simple rules for the design of displays in order to mitigate the seismic risk of valuale exhiits. Aiming at the protection of cultural heritage through the application of ase isolation, Vestroni and Di Cinto (000) performed a parametric study on the response of an isolated statue modeled as a single-degree-of-freedom system, with the isolator characterized y a hysteretic force-displacement law. Moreover, Myslimaj et al. (003) proposed the installation of Tuned Configuration Rail (TCR), a rolling type ase-isolation system, underneath showcases, preservation racks, shelves and statues to control their seismic response. More recently, Caliò and Marletta (003) examined the virations of art ojects modeled as rigid locks isolated with viscoelastic devices and performed numerical investigations under impulsive and seismic excitations. In this paper the dynamic response of ase-isolated lock-like slender ojects, such as statues (Figure ), sujected to seismic excitation is investigated. The structural model employed consists of a rigid lock supported on a rigid ase, eneath which the isolation system is placed. Assuming sufficient friction to prevent sliding of the lock relative to the supporting ase, when sujected to ground excitation, the system may exhiit two possile patterns of motion, namely pure translation, in which the system as a whole realizes only horizontal displacement ut (), and rocking, in which the rigid lock experiences rotation θ ( t) with respect to the horizontally-moving ase. Thus, the active degrees of freedom of the system considered are dependent on the nature of the response, with the system possessing degrees of freedom for motion realized in the rocking mode and degree of freedom for motion realized in the pure-translation mode. The formulation of the prolem involves derivation of the nonlinear equations of motion, transition from one mode to the other, and a rigorous formulation of the impact model ased on classical impact theory. Figure : Schematic of a ase-isolated statue. ANALYTICAL MODEL.. Model Considered The system considered consists of a symmetric rigid lock of mass m and centroid mass moment of inertia I, supported on a rigid ase of mass m (Figure a). The entire system is ase-isolated with a linear isolation system composed of a linear spring k and a linear viscous dashpot c. The rigid lock of height H = h and width B = is assumed to rotate aout the corners O and O '. The distance etween one corner of its ase and the mass centre is denoted y r and the angle measured etween r and the vertical when the ody is at rest is denoted y α, where α = tan ( / h).

3 The 4 th World Conference on Earthquake Engineering Octoer -7, 008, Beijing, China Inertial Frame Inertial Frame Figure : Model at rest and oscillation patterns The horizontal and vertical displacements of the mass center of the lock relative to an inertial frame of reference are denoted y X ( t ) and Z( t ) respectively, while the corresponding displacements relative to the ase are denoted y x( t ) and zt ( ). The angular rotation of the lock is denoted y θ ( t), positive in the clockwise direction, and the horizontal displacement of the ase relative to the foundation is denoted y ut ( )... Initiation of Rocking When sujected to ground acceleration x&& g, the supporting ase will oscillate in the horizontal direction with a displacement ut ( ) relative to the foundation. The rigid lock will e set into rocking on top of the moving ase M = mu&& + && x h, exceeds the availale resisting when the overturning moment due to external loads, over ( g ) moment due to gravity, M res = mg, yielding u&& + && xg > g (.) h.3. Equations of Motion Assuming no sliding of the lock relative to the supporting ase, when sujected to ground excitation the system may exhiit two possile patterns of motion: (a) pure translation, in which the system in its entirety oscillates horizontally with displacement ut ( ) ( degree-of-freedom response), and () rocking, in which the rigid lock pivots on its edges with rotation θ ( t) as the supporting ase translates horizontally with ut ( ) ( degree-of-freedom response). The governing equations for each pattern of motion are herein formulated y means of the Lagrange method..3. Pure-translation mode The equation of motion of the system in the pure-translation mode is ( + ) + + = ( + ) m m u&& cu& ku m m && x (.) g which is the classical linear second-order differential equation governing the response of single-degree-of-freedom system to ground excitation. a

4 The 4 th World Conference on Earthquake Engineering Octoer -7, 008, Beijing, China.3. Rocking mode In the rocking mode the system possesses two degrees of freedom. Using as generalized coordinates q u, the horizontal translation of the ase relative to the ground, and q θ, the rotation angle of the oject aout a ottom corner, Lagrange s equations take the form d T T V Qu and d T T + = + V = Q dt u& u u dt & θ θ θ θ (.3) in which T denotes the kinetic energy of the system, V the potential energy of the system, and generalized nonconservative forces. The kinetic energy of the system is otained as Q k the & & & & & g g θ θ & θ θ & θ θ & θ θ & θ (.4) ( ) ( cos sin ) ( cos sin ) T = m u+ x + m u+ x + h + + h + I in which the first term is associated with pure translation of the ase, and the second and third term are associated with general planar motion of the lock. The potential energy of the system is otained as = + sinθ ( cosθ ) (.5) V ku mg h in which the first term is associated with the potential energy due to elastic deformation of the spring and the second term is associated with the potential energy due to gravity. The generalized forces Q k are derived via the virtual work of the nonconservative forces as Qu = cu&, Q θ = 0 (.6) Sustituting Equations (.4) through (.6) into Equations (.3) yields the governing equations of motion for rotation aout O ( θ > 0). The governing equations of motion for rotation aout O ( θ < 0) can e derived in a similar manner. Comining the equations for rotation aout O and O, leads to a compact set of equations governing the rocking mode of the oject on top of the moving ase: ( + ) cosθ+ sgnθ( sinθ) && θ+ sgnθ( cosθ) sinθ & θ = ( + ) m m u&& cu& ku m h m h m m && xg (.7) ( ) θ θ θ( θ) θ( θ) θ θ θ( θ) mr + I && + mu&& hcos + sgn sin + mg sgn cos hsin =m hcos + sgn sin && xg (.8) where sgnθ denotes the signum function in θ. Note that equations (.7) and (.8) hold only in the asence of impact ( θ 0 ). At that instant, oth corner points O and O are in contact with the ase, rendering the aove formulation invalid. The impact prolem is addressed separately in the following section.

5 The 4 th World Conference on Earthquake Engineering Octoer -7, 008, Beijing, China 3. IMPACT MODEL The dynamic response of the system is strongly affected y the occurrence of impact(s) etween the lock and the horizontally-moving ase. In fact, impact affects the system response on many different levels. On one level, it renders the prolem nonlinear (aside from the nonlinear nature of the equations themselves) y virtue of the discontinuity introduced in the response (i.e. the governing equations of motion are not valid for θ = 0 ). That is, impact causes the system to switch from one oscillation pattern to another (potentially modifying the degrees of freedom), each one governed y a different set of differential equations. Further, the integration of equations of motion governing the post-impact pattern must account for the ensuing instantaneous change of the system s velocity regime. On another level, the effect of impact on the dynamic response is also evident in the energy loss of the system manifested through the reduction of post-impact velocities. Therefore, the critical role of impact in the dynamics of the system necessitates a rigorous formulation of the impact prolem. In this paper a model governing impact is derived from first principles using classical impact theory. According to the principle of impulse and momentum, the duration of impact is assumed short and the impulsive forces are assumed large relative to other forces in the system. Changes in position and orientation are neglected, and changes in velocity are considered instantaneous. Moreover, this model assumes a point-impact, zero coefficient of restitution (perfectly inelastic impact), impulses acting only at the impacting corner (impulses at the rotating corner are small compared to those at the impacting corner and are neglected), and sufficient friction to prevent sliding of the lock during impact. Under the assumption of perfectly inelastic impact, there are only two possile response mechanisms following impact: (a) rocking aout the impacting corner when the lock re-uplifts (no ouncing), or () pure translation when the lock s rocking motion ceases after impact. The formulation of impact is divided into three phases: pre-impact, impact, and post-impact as illustrated schematically in Figure 3. In the following, a superscript - refers to a pre-impact quantity and a superscript + to a post-impact quantity. Fdt x x&& g θ = 0, & θ, u& u Fdt z Fdt z Fdt x θ = θ = 0, u = u θ = θ = 0, u = u & θ = βθ&, u& = u& + βθ& & + + θ = 0, u& = u& + βθ& Figure 3: Impact from rocking aout O followed y (a) re-uplift aout O and () termination of rocking 3.. Rocking continues after impact Consider the system at the instant when the lock hits the moving ase from rocking aout O and re-uplifts pivoting aout the impacting corner, O (Figure 3a). As mentioned efore, impact is accompanied y an instantaneous change in velocities, with the system displacements eing unchanged. Therefore, the impact analysis is reduced to the computation of the initial conditions for the post-impact motion, u & + and & θ +, given the position and the pre-impact velocities, u& and & θ.

6 The 4 th World Conference on Earthquake Engineering Octoer -7, 008, Beijing, China With regard to the lock, the principle of linear impulse and momentum in the x and z direction states that F dt = mx& mx& = mu& + mx& mu& mx& (3.) x + + F dt = mz& mz& = mz& mz& (3.) z in which Fx dt and Fz dt are the horizontal and vertical impulses (assumed to act at O ); X& = u& + x& g + x&, X& = u& + x& g + x& and Z & = z &, Z & + z + = & are the asolute pre- and post-impact horizontal and vertical velocities of the mass center of the lock, respectively. In addition, the principle of angular impulse and momentum states that ( z ) ( x ) F dt h F dt = I & θ + I & θ (3.3) In Equations (3.) and (3.), the pre- and post-impact horizontal and vertical components of the relative translational velocity of the mass center can e expressed in terms of the angular velocity of the lock as Sustituting Equations (3.4) into Equations (3.) and (3.) yields x& = h& θ, z& = & θ, x& = h& θ, z& =& θ (3.4) F dt = mu& + + mh & θ + mu& mh & θ (3.5) x F dt =m & θ + m& θ (3.6) z Equations (3.3), (3.5) and (3.6) constitute a set of three equations in four unknowns. Equivalently, these equations can e comined in one equation (y eliminating the two impulses) in two unknowns (θ& +, u + & ): ( ) & θ ( ) h + 3hu& = 4h & θ + 3hu& (3.7) in which for rectangular lock the centroid mass moment of inertia was taken as I m( h ) = + /3. One additional equation is therefore required to uniquely determine the post-impact velocities & θ + and u & +. By considering the system in its entirety during the impact, it can e stated that the horizontal impulse on the system is zero, resulting in the conservation of the system s linear momentum in the horizontal direction. That is, ( ) ( ) m + m u& = m + m u& mh & θ + mh & θ (3.8) + + Comining Equations (3.7) and (3.8) gives the post-impact velocities as and in which ( m+ 4) ( m+ ) ( m+ 4) + 4( m+ ) & + λ θ = & θ βθ & λ (3.9) + 6mh u& = u& + & θ u& + βθ& (3.0) λ ( m+ 4) + 4( m+ ) λ = h is the geometric aspect ratio and m= m m is the mass ratio.

7 The 4 th World Conference on Earthquake Engineering Octoer -7, 008, Beijing, China Equations (3.9) and (3.0) give the post-impact velocities for impact from rocking aout O (realized when & θ < 0 ). Identical expressions are derived for the case of impact from rocking aout O ' (realized when & θ > 0 ). It is worth noting that the coefficient of restitution e as defined in classical impact theory, relates pre- to post-impact translational velocities normal to the impact surface, and hence it must not e confused with the coefficient of angular restitution β defined in Equation (3.9), which relates the pre- to post-impact angular velocities of the ody. In the derivation presented herein, the coefficient of restitution e enters in the expression z& + O =ez& O which relates pre- to post-impact vertical relative velocities of the impacting corner ( O ' ). The assumption of perfectly inelastic impact is then justified y considering e = 0. From Equation (3.9), it can e seen that the coefficient of angular restitution, β, depends oth on the geometric aspect ratio λ and the mass ratio m. The variation of coefficient β with the slenderness ratio λ is shown in Figure 4a for different values of the mass ratio m. The dependency of coefficient β on the mass ratio m is seen to e weak, and practically diminishes for very slender locks (e.g. for λ > 6 ). The value β =, implying preservation of the magnitude of the angular velocity after impact, presents an upper ound for the coefficient of angular restitution. Evidently, the more slender a lock, the larger the associated coefficient β is. For the assumption of no-ouncing to e satisfied, the coefficient of angular restitution β should have a positive value. In such a case, the angular velocity of the lock will maintain sign upon impact, implying switching pole of rotation from one corner to the other. This requires that λ > ( m+ ) ( m+ 4). The coefficient associated with the reduction of the linear velocity of ase, β, depends not only on the parameters λ and m, ut also on the asolute size of the lock (in terms of its height). The normalized coefficient β β h is plotted against the slenderness ratio λ for different values of the mass ratio m in Figure 4. Oserve that the value of the coefficient β decays rapidly with the slenderness ratio λ. Moreover, the influence of the mass ratio m on the coefficient β is much greater than that on the coefficient β..0.0 Coefficient β Rocking Bouncing m/m = 0.5 m/m =.0 m/m =.5 m/m =.0 Coefficient β /h m/m = 0.5 m/m =.0 m/m =.5 m/m = Slenderness ratio λ (a) Slenderness ratio λ () Figure 4: Variation of coefficients β and β with slenderness ratio 3.. Rocking ceases after impact When rocking of the lock on top of the moving ase ceases, the system will attain a pure-translation mode (Figure 3). In this case, the impact analysis is reduced to the computation of the post-impact translational velocity of the system, u & +, given the position and the pre-impact velocities, u& and & θ.

8 The 4 th World Conference on Earthquake Engineering Octoer -7, 008, Beijing, China By considering the system as a whole during impact, it can e stated that the horizontal impulse on the system is zero, resulting in the conservation of the system s linear momentum in the horizontal direction. That is, which upon rearranging terms ecomes ( + ) + ( + + & θ ) = ( + + ) + ( + + ) m u& x& m u& x& h m u& x& m u& x& (3.) g g g g + mh u u u 3 m θ βθ & = & + & & + & (3.) ( + ) 4. NUMERICAL EXAMPLE Numerical results are otained through an ad hoc computational scheme developed to determine the response of the system under horizontal ground excitation. The numerical integration of the equations of motion is pursued in MATLAB through a state-space formulation (MATLAB 006). In each time step, close attention is paid to the eventuality of transition from one pattern of motion to another and to the accurate evaluation of the initial conditions for the next pattern of oscillation, on the asis of the developed impact model. As an example, the response of the system to the N-S component of 995 Koe, Japan earthquake was computed. Results are presented here for a system of a homogeneous lock with height H =.0 m, ase width B = 0.7 m and mass m = 3800 kg, and a rigid ase of mass m = 3800 kg (i.e. m = ). The linear isolation system considered has period T =.5 s and viscous damping ratio ξ =. For the sake of comparison, results are also generated for the case of a non-isolated rigid lock. Figure 5 depicts comparison of the computed response, in terms of the rotation and angular-velocity histories of the rigid lock and the horizontal-displacement history of the ase. It can e oserved that the rocking response of the isolated lock, in terms of the rotation amplitude, the numer of impacts and the overall duration of oscillation, is reduced. In comparison with the non-isolated lock, which is on the verge of overturning ( θ / α max = 0.95), the isolated lock exhiits a maximum rotation amplitude θ / α max = 0.64and as many as 5 times less impacts. Ground acceleration (g) Rotation θ/α Non-isolated Isolated Base displacement (m) Angular velocity (rad/s) Non-isolated Isolated Time (s) Time (s) Figure 5: Response of the system to the N-S component of 995 Koe, Japan earthquake

9 The 4 th World Conference on Earthquake Engineering Octoer -7, 008, Beijing, China 5. CONCLUDING REMARKS The dynamic response of ase-isolated lock-like slender ojects, such as statues, sujected to horizontal ground excitation is investigated. The model considered consists of a rigid lock supported on a rigid ase, eneath which the isolation system is placed. Under the assumption of sufficient friction to prevent sliding of the lock relative to the supporting ase, when sujected to ase excitation the system may exhiit two possile patterns of motion, each eing governed y highly nonlinear differential equation(s). The system can e set in pure translation, in which the system as a whole oscillates horizontally (-DOF response), or rocking, in which the rigid lock pivots on its edges with respect to the horizontally-moving ase (-DOF response). The dynamic response of the system is strongly affected y the occurrence of impact etween the lock and the moving ase, as it can modify not only the energy ut also the degrees of freedom of the system y virtue of the discontinuity introduced in the response. Therefore, the critical role of impact in the dynamics of the system necessitates a rigorous formulation of the impact prolem. In this paper, a model governing impact from the rocking mode is derived from first principles using classical impact theory. Numerical results are otained via an ad hoc computational scheme developed to determine the response of the system under horizontal ground excitation. ACKNOWLEDGEMENTS The project has received research funding from the European Community s Sixth Framework Programme under the Marie Curie Action (IRG ). REFERENCES Agaian, M.S., Masri, S.F., Nigor, R.L. and Ginell, W.S. (988). Seismic damage mitigation concepts for art ojects in museums. Ninth World Conference on Earthquake Engineering, Tokyo-Kyoto, Japan. Augusti, G., Ciampoli, M. and Airoldi, L. (99). Mitigation of seismic risk for museum contents: An introductory investigation. Proceedings of the Tenth World Conference on Earthquake Engineering, Balkema, Rotterdam. Caliò, I. and Marletta, M. (003). Passive control of the seismic rocking response of art ojects. Engineering Structures 5: Ishiyama, Y. (98). Motions of rigid odies and criteria for overturning y earthquake excitations. Earthquake Engineering and Structural Dynamics 0: Makris, N. and Roussos, Y.S. (000). Rocking response of rigid locks under near-source ground motions. Géotechnique 50: 3, MATLAB 7.3 (006). The Language of Technical Computing The Mathworks, Inc.: Natick, MA. Myslimaj, B., Gamle, S., Chin-Quee, D., Davies, A. and Breukelman, B. (003). Base isolation technologies for seismic protection of museum artifacts. The 003 IAMFA Annual Conference, San Francisco, California. Psycharis, I.N. (990). Dynamic ehaviour of rocking two-lock assemlies. Earthquake Engineering and Structural Dynamics 9: Shenton III, H.W. and Jones, N.P. (99). Base excitation of rigid odies. I: Formulation. Journal of Engineering Mechanics (ASCE) 7: 0, Spanos, P.D. and Koh, A.-S. (984). Rocking of rigid locks due to harmonic shaking. Journal of Engineering Mechanics (ASCE) 0:, Spanos, P.D., Roussis, P.C. and Politis, N.P.A. (00). Dynamic analysis of stacked rigid locks. Soil Dynamics and Earthquake Engineering, Vestroni, F. and Di Cintio, S. (000). Base isolation for seismic protection of statues. th World Conference on Earthquake Engineering, Auckland, New Zealand. Yim, C.-S., Chopra, A.K. and Penzien, J. (980). Rocking response of rigid locks to earthquakes. Earthquake Engineering and Structural Dynamics 8: 6,