A Hertz contact model with non-linear damping for pounding simulation

Transcription

1 EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2006; 35: Published online 10 February 2006 in Wiley InterScience ( DOI: /eqe.557 A Hertz contact model with non-linear damping for pounding simulation Susendar Muthukumar 1 2; ; and Reginald DesRoches 1 Browder + LeGuizamon & Associates; 174 West Wieuca Road NE; Atlanta; GA 30342; U.S.A. 2 School of Civil and Environmental Engineering; Georgia Institute of Technology; Atlanta; GA 30332; U.S.A. SUMMARY This paper investigates the cogency of various impact models in capturing the seismic pounding response of adjacent structures. The analytical models considered include the contact force-based linear spring, Kelvin and Hertz models, and the restitution-based stereomechanical approach. In addition, a contact model based on the Hertz law and using a non-linear hysteresis damper (Hertzdamp model) is also introduced for pounding simulation. Simple analytical approaches are presented to determine the impact stiness parameters of the various contact models. Parameter studies are performed using two degreeof-freedom linear oscillators to determine the eects of impact modelling strategy, system period ratio, peak ground acceleration (PGA) and energy loss during impact on the system responses. A suite of 27 ground motion records from 13 dierent earthquakes is used in the analysis. The results indicate that the system displacements from the stereomechanical, Kelvin and Hertzdamp models are similar for a given coecient of restitution, despite using dierent impact methodologies. Pounding increases the responses of the stier system, especially for highly out-of-phase systems. Energy loss during impact is more signicant at higher levels of PGA. Based on the ndings, the Hertz model provides adequate results at low PGA levels, and the Hertzdamp model is recommended at moderate and high PGA levels. Copyright? 2006 John Wiley & Sons, Ltd. KEY WORDS: seismic pounding; impact models; Hertz model with non-linear damper; bridges; impact energy loss INTRODUCTION Earthquakes can induce out-of-phase vibrations in adjacent structures due to dierences in dynamic characteristics, which can result in impact if their at-rest separation is insucient to accommodate the relative displacements. This impact, commonly referred to as seismic pounding, generates high magnitude and short duration acceleration pulses that can cause Correspondence to: Reginald DesRoches, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA , U.S.A. reginald.desroches@ce.gatech.edu Received 21 March 2004 Revised 21 November 2005 Copyright? 2006 John Wiley & Sons, Ltd. Accepted 28 November 2005

2 812 S. MUTHUKUMAR AND R. DESROCHES structural damage. Closely spaced buildings can experience inll wall damage, column shear failure and possible collapse due to pounding. In the case of bridge structures, dynamic impact can result in concrete spalling at the joints, damage to column bents, restrainer units, bearings and abutments, and possibly contribute to the collapse of bridge spans. Structural damage due to pounding has been identied in several recent earthquakes. During the 1989 Loma Prieta earthquake, pounding of adjacent unreinforced masonry (URM) buildings resulted in shear failure of the brickwork leading to partial collapse of the wall [1]. Examples of veneer spalling were also reported from buildings in downtown San Francisco [1]. Signicant pounding damage was observed at the expansion hinges and abutments of standing portions of connectors at the I-5/SR-14 Interchange which was located approximately 12 km north northeast of the epicentre during the 1994 Northridge earthquake [2]. Reconnaissance reports from the 1995 Kobe earthquake identify pounding as a major cause of fracture of the bearing supports and potential contributor to the collapse of several bridge decks [3]. Impact between a six-storey building and two-storey building in Golcuk, Turkey during the 1999 Kocaeli earthquake contributed to column failure above the third-oor slab in the taller building, and shear failure of two second-oor piers in the smaller building [4]. Pounding of abutments and deck joints was also observed in several highway bridges during the same earthquake. Reconnaissance visits after the 1999 Chi-Chi earthquake in Taiwan revealed hammering at the expansion joints in some bridges which resulted in damage to shear keys, bearings and anchor bolts [5]. Pounding of adjacent simply supported spans was observed in the Old Surajbadi highway bridge, India Bridge and several other bridges on National Highway 8A during the 2001 Bhuj (Gujarat, India) earthquake [6]. Structural damage included the failure of girder ends, superstructure dislocation and bearing damage. Pounding is a highly non-linear phenomenon, which results in several uncertainties in its mathematical modelling. Primarily, two modelling techniques have been used the stereomechanical approach and the contact element approach. The stereomechanical approach assumes that impact is instantaneous and uses the principle of momentum and the coecient of restitution (e) to modify the velocities of the colliding bodies after impact. The theory assumes a direct, central impact and does not consider transient stresses and deformations in the impacting bodies. This method has been used by several researchers to analyse seismic pounding [7 10]. However, the stereomechanical approach is no longer valid if the impact duration is large enough so that signicant changes occur in the conguration of the system. Furthermore, it cannot be easily implemented into existing software for the analysis of multi-degree-offreedom structures. The contact element approach is a force-based approach, where a contact element is activated once impact occurs. Typically, a linear spring with stiness proportional to the axial stiness of the colliding structures is used to represent the force during impact [11 13]. The linear spring model cannot account for the energy loss during impact. Hence, the Kelvin model represented by a sti linear spring in combination with a damper has been used in some studies [14 16]. The damping coecient can be related to the coecient of restitution (e) that describes the energy dissipation during impact [15]. However, it has been shown that the Kelvin model is untenable as it results in tensile forces acting on the bodies just before separation [17, 18]. Alternatively, a non-linear spring based on the Hertz contact law can be used to model impact [19 22]. The Hertz contact law is representative of elastic impact but fails to include the energy dissipation during impact. A Hertz contact model with a non-linear hysteresis

3 HERTZ CONTACT MODEL WITH NON-LINEAR DAMPING 813 damper has been used in other areas such as robotics and multi-body systems (slider-crank mechanisms) to analyse the impact [17, 18, 23]. This model accounts for non-linearity associated with impact as well as the energy dissipated during collision. More recently, a non-linear viscoelastic model based on the Hertz contact law in conjunction with a damper that is active only during the approach period of impact has been used to analyse structural pounding [24]. The parameters of the impact model are obtained numerically through iterative simulations. The contact element approach has its limitations, with the exact value of spring stiness to be used being unclear. Moreover, using a very sti spring can result in unrealistically high impact forces and also lead to numerical convergence diculties. With several analytical models available for the investigation of dynamic impact, there is a need to perform an evaluation and comparison of these models to determine their applicability and ecacy in capturing the pounding phenomenon during earthquakes. In particular, it is important to compare the stereomechanical and contact force-based approaches, to ascertain the eect of impact modelling methodology on the response of participating systems. Furthermore, the eect of energy loss during pounding on the responses of participating systems needs to be examined. Although, several analytical studies have been performed, few experiments have been conducted to investigate the eects of pounding. van Mier et al. [25] studied concrete-to-concrete collisions between breakwater armour elements through a series of dynamic impacts between a prestressed concrete pile and a concrete striker of variable mass. The load-time histories during contact were determined from the experiments and an impact stiness parameter was calculated using the Hertz law. Papadrakakis and Mouzakis [26] performed shaking table experiments on pounding between two-storey reinforced concrete buildings with zero gap separation subject to sinusoidal excitation. A comparison of the experimental results with analytical predictions using the Lagrange multiplier method (corresponding to the stereomechanical method with e = 1) showed good agreement. This, in fact is to date the best validation for a pounding model, but it is not force based. Filiatrault et al. [27] conducted shaking table tests on dynamic impact between adjacent three- and eight-storey single-bay steel frames (1=8 scale model), with 0 and 15 mm gap separations, subject to the 1940 El Centro earthquake. The experimental results were compared to analytical predictions based on a linear, elastic spring, with a stiness of 12.8 kn=mm (73 kips=in). The amplitude and phase of the displacement, and impact forces obtained from the experiment were well predicted by analytical models. However, the accelerations at the contact locations were not well predicted, since the analytical models could not reproduce all the high-frequency components from experiment. Chau et al. [28] performed shake table tests on pounding between adjacent two-storey steel towers (2 m tall) subjected to both harmonic excitation and ground motions from the 1940 El Centro earthquake. The experimental ndings were compared with the results from an analytical model using the Hertz contact law, with an impact stiness parameter, e =2: N=m 3=2 ( kip in 3=2 ). The estimated relative impact velocity and the maximum stand-o distance to prevent pounding agreed qualitatively with the experiments. However, discrepancies did exist in many cases. Results from experimental tests on collisions between spheres, impact of spheres on plates, central longitudinal impact of elastic bars, and transverse impact tests on elastic beams are presented in Reference [29]. Materials used in the experiments included steel, brass, lead, aluminium, zinc, tin and cast iron. The coecient of restitution is presented as a function of the

4 814 S. MUTHUKUMAR AND R. DESROCHES impact velocity for collisions involving spheres. For the case of impact between longitudinal bars, the contact force is presented as a function of the time after contact. Past research on pounding has attempted to compare experimental ndings with analytical predictions from one impact model. The various analytical models have not been compared with one another, and thus it is not clear which analytical model (either contact force-based or restitution-based) is most appropriate to simulate pounding. Furthermore, experimental testing has yet to identify an appropriate value of the coecient of restitution (e) to be used in the analysis. Test results indicate that the impact stiness parameter for contact force-based models is a function of material, contact surface geometry and magnitude of forces during collision. Most of the experiments were performed on relatively small-scale specimens involving small values of the impact spring stiness and translation to large-scale situations, such as impact between bridge decks or oor diaphragms is dicult. Thus, it is important to determine appropriate values for the impact stiness parameter, before comparing the various contact force-based models. The primary objective of this study is to investigate the ecacy of various analytical models in representing the pounding response of closely spaced structures. The adjacent structures are represented by a simplied two degree-of-freedom (DOF) model. Elastic system response is considered, with three values for the system period ratio, T 1 =T 2 =0:3, 0.5 and 0.7 (where T 1 and T 2 refer to the fundamental periods of the adjacent structures). The existing impact models implemented include the stereomechanical approach, and the contact force-based linear spring, Kelvin and Hertz models. A Hertz contact model with a non-linear damper is introduced to study seismic pounding. Two values of the coecient of restitution, e =1:0 (no energy loss) and e =0:6 (some energy loss) are selected. Appropriate values of impact stiness parameters are chosen and parameter studies are then conducted with the two-dof system subject to a suite of ground motion records to assess the eect of ground motion characteristics, system period ratio, and impact energy loss on the performance of the various pounding models. Finally, the performance of the various models in simulating pounding in bridges is evaluated through a case study using an inelastic, simplied multiple-frame bridge model. SIMPLIFIED MODEL FOR POUNDING INVESTIGATIONS A simplied two-dof model is developed, as shown in Figure 1(c), to investigate seismic pounding between adjacent structures. The adjacent structures can be closely spaced buildings or bridge frames as shown in Figure 1. Each DOF is characterized by mass m i, initial stiness k i and viscous damping coecient c i and is assumed to behave elastically. Using a forcebased approach to model impact, the equations of motion for the two-dof system subjected to horizontal ground motion u g can be written as [ ]{ } [ } { } { } m1 0 u1 c1 0 ]{ u1 R1 (u 1 ) Fc (u 1 u 2 g p ) m 2 u 2 0 c 2 u 2 R 2 (u 2 ) F c (u 1 u 2 g p ) [ ]{ } m1 0 1 = u g (1) 0 m 2 1

5 HERTZ CONTACT MODEL WITH NON-LINEAR DAMPING 815 g p (a) u u 1 2 g p Impact element m m1 2 k 1 k 2 c c 1 2 (b) (c).. U g Figure 1. The pounding problem in: (a) bridge structures; (b) closely spaced buildings; and (c) model idealization of adjacent structures. where u i, u i, u i are the acceleration, velocity and displacement relative to the ground, and dot denotes dierentiation with respect to time; R i is the system restoring force and F c is contact force due to pounding. Impact occurs when the gap between the two bodies closes, i.e. u 1 u 2 g p 0. Several studies have shown that the damping of participating systems is not a signicant factor aecting the pounding response [8, 11, 15]. Hence, modal damping of 5% is assigned to each DOF. The solution of Equation (1) is obtained numerically using the 4th-order Runge Kutta method [30]. ANALYTICAL MODELS FOR IMPACT Seismic pounding is essentially a problem of dynamic impact. The forces created by collision act over a short period of time, where energy is dissipated as heat due to random molecular vibrations and the internal friction of the colliding bodies. Usually, contact is modelled using

6 816 S. MUTHUKUMAR AND R. DESROCHES either a continuous force model or via a stereomechanical (coecient of restitution) approach, as described earlier. Several of the existing impact models are considered in this study. In addition, a contact model based on the Hertz law and using a non-linear hysteresis damper for energy dissipation is also introduced. The analytical formulations of the various impact models are outlined below. (1) Stereomechanical model. This approach uses the momentum conservation principle and the coecient of restitution to model impact. The duration of impact is neglected. The coecient of restitution (e) is dened as the ratio of separation velocities of the bodies after impact to their approaching velocities before impact [29]. Since this is not a force-based approach, the eect of impact is accounted by adjusting the velocities of the colliding bodies, as shown in Equation (2). v 1 = v 1 (1 + e) m 2(v 1 v 2 ) (2a) m 1 + m 2 v 2 = v 2 +(1+e) m 1(v 1 v 2 ) (2b) m 1 + m 2 where v 1, v 2 are the velocities of the colliding masses (m 1;m 2 ) after impact, v 1 ;v 2 are the velocities before impact and e is the coecient of restitution. (2) Linear spring model. A linear spring of high stiness (k l ) can be used to simulate impact once the gap between adjacent bodies closes. The contact force during impact is taken as F c = k l (u 1 u 2 g p ); u 1 u 2 g p 0 (3a) F c =0; u 1 u 2 g p 60 (3b) Figure 2(a) presents the contact force displacement relationship. This approach is relatively straight forward, and can be easily implemented in commercial software. However, energy loss during impact cannot be modelled. (3) Kelvin model. A linear spring of stiness, k k, is used in conjunction with a damper element (c k ) that accounts for energy dissipation during impact. The impact force displacement relation, as shown in Figure 2(b) can be represented as F c = k k (u 1 u 2 g p )+c k ( u 1 u 2 ); u 1 u 2 g p 0 (4a) F c =0; u 1 u 2 g p 0 (4b) The damping coecient c k can be related to the coecient of restitution (e), by equating the energy losses during impact. ( ) m1 m 2 ln e c k =2 k k ; = (5) m 1 + m (ln e) 2 where m 1, m 2 are the masses of the colliding bodies. (4) Hertz model. Another popular model for representing pounding is the Hertz model, which uses a non-linear spring of stiness (k h ), as illustrated in Figure 2(c). The impact force

7 HERTZ CONTACT MODEL WITH NON-LINEAR DAMPING 817 F c F c k l k k (a) g p u u 2 1 (b) g p u u1 2 F c F c k h k h (c) g p u u 2 1 (d) g p u 1 u 2 Figure 2. Contact force displacement relationship for various impact models: (a) Linear spring element; (b) Kelvin model; (c) Hertz non-linear spring; and (d) Hertzdamp model. representation is: F c = k h (u 1 u 2 g p ) n ; u 1 u 2 g p 0 (6a) F c =0; u 1 u 2 g p 60 (6b) The use of the Hertz contact law has an intuitive appeal in modelling pounding, since one would expect the contact area between the colliding structures to increase as the contact force increases, leading to a non-linear stiness described by the Hertz coecient (n). The impact stiness parameter, k h, depends on the material properties of the colliding structures and the contact surface geometry. The Hertz coecient, n, is typically taken as 3=2. Studies have shown that the system displacement response is relatively insensitive to the exponent, n, in the contact law [20, 22]. (5) Hertz model with non-linear damper (Hertzdamp model). The Hertz model suers from the limitation that it cannot represent the energy dissipated during contact. Hence, an improved version of the Hertz model is considered herein, whereby a non-linear damper is used in conjunction with the Hertz spring. Similar models have been used in other areas such as robotics, and multi-body systems [17, 18, 23]. However, its ecacy in structural engineering has not been considered. The contact force can be expressed as F c = k h (u 1 u 2 g p ) n + c h ( u 1 u 2 ); u 1 u 2 g p 0 (7a) F c =0; u 1 u 2 g p 60 (7b)

8 818 S. MUTHUKUMAR AND R. DESROCHES where c h is the damping coecient, u 1 u 2 g p is the relative penetration and u 1 u 2 is the penetration velocity. A non-linear damping coecient is proposed so that the expected hysteresis loop during impact matches the one shown in Figure 2(d). The damping coecient is taken as follows [23]: c h = n (8) where is the damping constant, and is the relative penetration (u 1 u 2 g p ). Equating the energy loss during stereomechanical impact to the energy dissipated by the damper, an expression for the damping constant () can be found in terms of the spring stiness (k h ), the coecient of restitution (e) and the relative approaching velocity (v 1 v 2 ), as follows [23]: = 3k h(1 e 2 ) (9) 4(v 1 v 2 ) Hence, the force during contact in Equation (7a) can be expressed as [ F c = k h (u 1 u 2 g p ) n 1+ 3(1 ] e2 ) 4(v 1 v 2 ) ( u 1 u 2 ) ; u 1 u 2 g p 0 (10) Now, all the parameters of the model are known and the Hertz model with non-linear damper (n =3=2) can now be used in impact analysis. The Hertz model with non-linear damper shall be referred to as the Hertzdamp model throughout the rest of this paper. SELECTION OF IMPACT STIFFNESS PARAMETER In order to compare the various contact force-based models, their impact stiness parameters need to be selected appropriately. Among the various experimental studies, only van Mier et al. [25] and Papadrakakis and Mouzakis [26] involve concrete as the material. The study by Papadrakakis and Mouzakis [27] attempted to validate the elastic stereomechanical model. Thus, it is possible to select the stiness parameter of the Hertz model from the range of values (2 kn=mm 3=2 (60 kip in 3=2 )to80kn=mm 3=2 (2300 kip in 3=2 )) described by van Mier et al. [25]. However, the experiments were performed on relatively small-scale specimens and translation to large-scale situations is dicult. Furthermore, the relatively small values of impact stiness would cause a greater penetration between the two bodies (for the larger masses involved in collisions between oor diaphragms or bridge decks), which would be unacceptable. In the absence of relevant large-scale experimental data involving concrete-to-concrete collisions, the rule of thumb is to select the stiness of the linear contact spring (k l ) proportional to the axial stiness of the colliding structure (EA=L) [11 13]. Typical values of EA=L range from 5000 to kips=in for bridge decks and to kips=in for building diaphragms. Several studies have shown the system response to be insensitive to changes in the impact spring stiness by one order of magnitude [12, 15]. Hence, a value of kips=in is chosen for the linear spring stiness, k l. The stiness parameter of the Hertz model (k h ) is a function of the elastic properties and geometry of the two colliding bodies. For elastic contact between two isotropic spheres of

9 HERTZ CONTACT MODEL WITH NON-LINEAR DAMPING 819 radii R 1 and R 2, k h can be expressed as follows [29]: [ ] 1=2 4 R1 R 2 k h = (11) 3(h 1 + h 2 ) R 1 + R 2 where h 1, h 2 are material parameters dened as h i = 1 v2 i ; i =1; 2 (12) E i where v i and E i are the Poisson s ratio and modulus of elasticity, respectively, of sphere i. To calculate the value of k h to be used in the analysis, the colliding bodies can be approximated by equivalent spheres, with radii R 1 and R 2. From previous research into pounding between buildings, limited information is available on the exact mass values used for the impacting (large-scale) oor diaphragms. In the case of pounding between bridge decks, Malhotra [9] used mass values of 1: kg ( kn) for the shorter bridge segment and 2: kg ( kn) for the longer bridge segment. DesRoches and Fenves [30] utilized 5000 kips ( kn) for the weight of adjacent bridge frames. Assuming the colliding bodies to be spherical, the radius of an equivalent colliding sphere can be estimated as 3m R i = 3 i ; i =1; 2 (13) 4 where m i is colliding mass and is the density of concrete (150 lb=ft 3 ). Using the mass values from Malhotra s study, R 1 and R 2 can be calculated as 4.92 and 6.2 m, for the shorter bridge segment, and longer bridge segment, respectively. Typical values for Poisson s ratio of concrete are between 0.15 and 0.20 [32]. Using a concrete strength of 4000 psi and a Poisson ratio (v i ) of 0.15 along with Equations (11) and (12) results in a value of k h = 888 kn=mm 3=2 ( kip in 3=2 ). Thus, the stiness parameter of the Hertz model (k h )is taken as kip in 3=2 in this study. Subsequent comparisons of the impact performance of the various pounding models will be made relative to the assumptions described in this section. PARAMETER STUDY TO ASSESS PERFORMANCE OF IMPACT MODELS Having determined the impact stiness parameters for the contact force-based models, it is important to investigate the eects of the various pounding models on the structural response of the colliding structures. A preliminary study comparing the performance of the Hertzdamp model with other impact models has been conducted by the authors [33]. In this section, a parameter study investigating the eects of ground motion characteristics and system period ratio on the pounding response of the various impact models is presented. The two DOF model shown in Figure 1 is considered with equal masses of 7:8 kips 2 =in. The system period ratio (T 1 =T 2 ) and the ground motion eective period ratio (T 2e =T g ) were recognized as critical parameters aecting the pounding response [10]. Three period ratios, T 1 =T 2 =0:3, 0.5 and 0.7 are considered, herein. A suite of 27 ground motion records from 13 dierent earthquakes is selected, as listed in Table I. All the records are taken from the PEER Strong Motion Database ( The ground motion records

10 820 S. MUTHUKUMAR AND R. DESROCHES Table I. Suite of 27 earthquake ground motion records used in parameter study. PGA PGV PGD Tg Dt level PGA (g) Earthquake Mw Station EPD (cm=s) (cm) Soil class (s) (s) Low 0.11 Northridge, Wonderland Ave A,A Imperial Valley, Bonds Corner D,C San Fernando, Pasadena D,B Loma Prieta, Fremont B, Morgan Hill, Gilroy Array # D,C N. Palm Springs, Morongo Valley C,B Whittier Narrows, E Grand Ave A,A Landers, Joshua Tree C,B Morgan Hill, Gilroy Array # B,B Moderate 0.37 Loma Prieta, WAHO D, Northridge, Mulhol C,B Cape Mendocino, Rio Dell Overpass C,B Northridge, Old Ridge Route B,B Loma Prieta, Coyote Lake Dam A, Northridge, W Lost Cany D,C Loma Prieta, Saratoga Aloha Ave D,B N Palm Springs, N Palm Springs D,B Cape Mendocino, Petrolia D,C High 0.61 Loma Prieta, LGPC A, Coalinga, Pleasant Valley P.P D, Northridge, Old Ridge Route B,B Cape Mendocino, Petrolia D,C Duzce, Bolu D,C Coalinga, Transmitter Hill A, Northridge, Rinaldi C,C Superstition Hills, Superstition Mtn A,B Cape Mendocino, Cape Mendocino A,A PGA peak ground acceleration; Mw moment magnitude; component; EPD epicentral distance; PGV peak ground velocity; PGD peak ground displacement; Soil class geomatrix soil class, USGS; Tg characteristic period; Dt time step of record.

11 HERTZ CONTACT MODEL WITH NON-LINEAR DAMPING 821 are grouped into three levels depending on the peak ground acceleration (PGA) as, low (0:1g6PGA60:3g), moderate (0:4g6PGA60:6g) and high (0:7g6PGA60:9g). The records are chosen such that the ground motion eective period ratio (T 2e =T g = eective exible system period over the ground motion characteristic period) is less than one (Zone I response) [10]. Elastic systems are considered and pounding is analysed using all the impact models specied earlier. To study the eect of energy loss during impact, two values of the coecient of restitution are selected, e =1:0 (no energy loss) and e =0:6 (some energy loss). It should be noted that at e =1:0, the Kelvin model reduces to the linear spring and the Hertzdamp model reduces to the Hertz model. The eect of pounding is expressed in terms of response amplication, which is the ratio of the maximum response when pounding occurs to the maximum response when there is no pounding. The gap between the structures is set very large for the no-pounding analysis, and assumed as 1=2 inch for the pounding analysis. Figures 3 and 4 present the mean values of displacement and acceleration amplication due to pounding for the various impact models. Since pounding amplies the sti system response in Zone I, only the sti system amplications are presented. The eects of impact modelling strategy, system period ratio, PGA, and the coecient of restitution (e) are discussed in the following subsections. Eect of the impact model, system period ratio and PGA The stereomechanical and contact force-based models (Kelvin, Hertzdamp) predict similar displacement responses for a given coecient of restitution, despite using dierent impact methodologies. The dierences in displacement amplication between the various models are within 12% of each other, at all levels of PGA, system period ratio (T 1 =T 2 ) and coecient of restitution. However, the contact force-based models predict higher accelerations due to pounding. The system acceleration responses from the stereomechanical model are much smaller than those from the contact models, and follow the corresponding displacement trends. The Hertzdamp model predicts lower acceleration amplications than the Kelvin model, especially for low to moderate level ground motions. This trend is observed at the both values of e. However, at higher PGA levels, the Kelvin model provides the least acceleration amplications among all contact force-based models. Pounding amplies the displacement response of DOF 1 (stier system) since the system has a Zone I response (T 2 =T g 1). The largest amplications in the sti system displacement response due to pounding are observed when the system is highly out-of-phase (T 1 =T 2 =0:3). The displacement amplications get closer to unity, as the system gets more in-phase (T 1 =T 2 =0:7). The dierences in displacement amplication between the various models remain less than 12% at all period ratios, for a given value of e. The displacement and acceleration amplications of the sti system increase with increasing PGA, especially for moderate to highly out-of-phase systems (T 1 =T 2 =0:3; 0:5). This trend is observed at both levels of the coecient of restitution (e), for all models. Dierences in displacement amplication between the various models remain the same ( 12%) at all levels of PGA, for a particular value of the coecient of restitution. Eect of the coecient of restitution (e) Experimental research has not been very conclusive in identifying an accurate impact model. Pounding causes damage resulting in some energy loss during impact, implying that the value

12 822 S. MUTHUKUMAR AND R. DESROCHES Figure 3. Mean displacement amplications due to pounding (DOF1) elastic systems; T1=T2 =0:3, 0.5, 0.7; e =1:0, 0.6; Nine ground motion records used in each PGA level.

13 HERTZ CONTACT MODEL WITH NON-LINEAR DAMPING 823 Figure 4. Mean acceleration amplications due to pounding (DOF1) elastic systems; T1=T2 =0:5; e =1:0, 0.6; Nine ground motion records used at each PGA level. Figure 5. Mean displacement amplications due to pounding (DOF1) inelastic systems (Q-Hyst -Ry = 3); T1=T2 = 0:7; Nine ground motion records used at each PGA level.

14 824 S. MUTHUKUMAR AND R. DESROCHES of e should be less than 1. The authors have assumed that the energy dissipating models are better suited to model impact and discussed the results of the other models in relation to the energy dissipating models. Neglecting energy dissipation due to impact (e =1:0) overestimates both the displacement and acceleration responses of the sti system. Energy loss is more signicant at high PGA levels. On the average, energy loss during impact reduces the sti system displacements by 28, 16 and 10% for the stereomechanical, Kelvin and Hertzdamp models, respectively, when subjected to high levels of PGA, at T 1 =T 2 =0:3. The corresponding reductions in the sti system accelerations are 27, 32 and 18%. Among all the impact models, the Hertzdamp model exhibits the least variation in system response with respect to changes in e. The parameter study was then repeated for inelastic systems, with the Q-Hyst model being used for the system response [34]. A constant reduction factor, R y = 3, was used for both systems. Due to space constraints, only the results corresponding to T 1 =T 2 =0:7 are presented. From Figure 5, it can be observed that the system displacement amplications are very close to one and independent of the model type, as expected. It should be noted that the use of a constant reduction factor results in a higher ductility for system 1( 1 ) than system 2 ( 2 ) due to the dierence in the system periods. The period ratio for the yielding system is actually, T 1 1 =T 2 2, implying that the yielding system is more in-phase than its linear counterpart. The eects of the impact model type, system period ratio, PGA and coecient of restitution for other period ratios showed trends similar to those observed in the elastic case, and can be found in Reference [35]. EFFECT OF POUNDING IN BRIDGES CASE STUDY COMPARING VARIOUS IMPACT MODELS The previous section performed a comparison of various pounding models using a simplied two DOF model. However, the presence of bearings, restrainers and abutments in a bridge can induce greater dierences in the pounding response, through interaction of the various components. Hence, a case study is performed using a simplied multiple-frame bridge model, consisting of four bridge frames connected at three intermediate hinges, as shown in Figure 6. Weights of 2880, 7080, 7080 and 2880 kips are selected for frames 1 4 and 5% modal damping is assigned to the individual frames. The outer frames (F 1 ;F 4 ) have an individual period of 0.47 s and the stand-alone period of the inner frames (F 2 ;F 3 ) is 1.12 s. The Q-Hyst model is selected as the frame-force deformation relation, with yield strengths of 774 kips for F 1 ;F 4, and 1824 kips for F 2 ;F 3. Cable restrainers at the intermediate hinges are modelled using tension-only bilinear elements (strain hardening ratio = 5%), with a slack. The initial stiness and yield strength of restrainers R 1 ;R 3 are 200 kips=in and 840 kips, respectively. The properties of restrainer R 2 are one-half the properties of R 1 ;R 3. The restrainers are designed according to the procedure suggested by DesRoches and Fenves [32]. The elastomeric bearings at the intermediate hinges (B 1 ;B 2 ;B 3 ) have an initial stiness, yield strength and strain hardening ratio of 6 kips/in, 2.4 kips, and 33%, respectively. The bearings at the abutment locations are designed to have a stiness proportional to the passive stiness of the abutment (k p = 2600 kips=in). The active stiness of the abutment is taken proportional to the typical hinge bearing stiness. The hinge gap is taken as 1=2 inch at all intermediate hinge locations. The restrainer slack is assumed

15 HERTZ CONTACT MODEL WITH NON-LINEAR DAMPING 825 Figure 6. Four-frame bridge used in case study comparing various impact models. the same as the hinge gap. The Saratoga record from the 1989 Loma Prieta earthquake, having a PGA of 0:5g is chosen for analysis. The time history of displacements and accelerations for the stier frame, F 1 are presented in Figure 7, when pounding is represented using the various models. The models without energy dissipation such as the stereomechanical (e = 1), linear (Kelvin, with e = 1) and Hertz (Hertzdamp, with e = 1) models overestimate the sti frame displacement, as observed in the earlier parameter study. Dierences can be as large as 25% for Frame 1 when the Hertzdamp model is used. The contact force-based models Kelvin and Hertzdamp predict high frame accelerations due to pounding, with larger accelerations for e = 1. The Hertzdamp model predicts the least impact accelerations among the contact force-based models. The above results show good agreement with the ndings from the earlier parameter study. CONCLUSIONS Closely spaced buildings, adjacent frames and girder ends in bridges are vulnerable to seismic pounding damage, as observed in several recent earthquakes. Analytical models for pounding simulation include the contact force-based linear spring, Kelvin, and Hertz models and the coecient of restitution-based stereomechanical approach. The cogency of various impact models in representing the pounding response of closely spaced structures is investigated in this study. In addition to the existing impact models, a contact model based on the Hertz law and using a non-linear hysteresis damper (Hertzdamp model) is also introduced for pounding simulation. In the absence of relevant large-scale experimental data, the stiness of the linear contact spring is taken proportional to the axial stiness of the colliding structure (EA=L). A simple approach for estimating the stiness of the Hertz model is presented based on the theory of elasticity by approximating the colliding masses with equivalent isotropic spheres. Parameter studies are conducted using two degree-of-freedom linear oscillators with varying system period ratios (T 1 =T 2 =0:3, 0.5, 0.7), and coecients of restitution (e =0:6, 1.0) being subjected to dierent levels of ground motion (low, moderate and high).

16 826 S. MUTHUKUMAR AND R. DESROCHES Figure 7. Frame 1 responses from various impact models Saratoga record (PGA = 0.5g), 1989 Loma Prieta earthquake.

17 HERTZ CONTACT MODEL WITH NON-LINEAR DAMPING 827 The results indicate that the displacement responses from the stereomechanical, Kelvin and Hertzdamp models are similar for a given restitution coecient, although they use dierent methodologies to represent impact. Pounding is critical on the responses of the sti system, especially when the system is highly out-of-phase (T 1 =T 2 =0:3). Essentially, in-phase systems (T 1 =T 2 0:7) exhibit displacement amplications that are much closer to one, independent of model type. For a given value of e, the dierences in sti system displacements between various impact models remain small ( 12%) at all system period ratios. Moderate (T 1 =T 2 =0:5) and highly out-of-phase (T 1 =T 2 =0:3) systems show greater amplications in the sti system responses when subjected to higher levels of PGA (0:7g6PGA60:9g). These trends are observed for all the impact models. Neglecting energy loss during impact overestimates the displacement and acceleration responses of the sti system, for all impact models. The energy loss eect is more signicant at higher PGA levels, especially for moderate to highly out-of-phase systems. At low levels of ground excitation, the eects of energy loss are not signicant for the Hertz model. Hence, the Hertz model, with e = 1 is recommended for impact simulation at low PGA levels, for all period ratios. At moderate PGA levels, the Hertz model is no longer appropriate due to the signicance of energy loss eects during pounding. The Hertzdamp model, with e =0:6 appears well suited to represent pounding, as it provides the least acceleration amplications among the contact force-based models. For high PGA levels, the Kelvin model (e =0:6) provides smaller acceleration amplications. However, the Kelvin model is known to result in sticky tensile forces during separation of the colliding bodies. Hence, the Hertzdamp model (e =0:6) is recommended at higher PGAs as well. Finally, a case study conducted with a multiple-frame bridge model to investigate the dierences in bridge response for the various impact models shows good agreement with ndings from the parameter study. All conclusions as to the ecacy of the various contact models are drawn relative to the assumptions made in the study regarding the selection of impact stiness parameters. Experimental tests on large-scale specimens are required to determine accurate values of the Hertz stiness parameter, and to correlate it with the linear spring stiness more precisely. It should also be noted that the energy dissipating models are dicult to implement in commercial software programs. The authors have proposed representative, simplied contact force-based models using truss and link elements for pounding simulation [35]. REFERENCES 1. Earthquake Engineering Research Institute. Loma Prieta Earthquake Reconnaissance Report, Supplement to vol. 6. May Earthquake Engineering Research Institute. Northridge Earthquake Reconnaissance Report, vol. 1, Supplement C to vol. 11. April Otsuka H, Unjoh S, Terayama T, Hoshikuma J, Kosa K. Damage to highway bridges by the 1995 Hyogoken Nanbu earthquake and the retrot of highway bridges in Japan. 3rd U.S.-Japan Workshop on Seismic Retrot of Bridges, Osaka, Japan, December Earthquake Engineering Research Institute Kocaeli, Turkey, Earthquake Reconnaissance Report, Supplement A to vol. 16. December Earthquake Engineering Research Institute Chi-Chi, Taiwan, Earthquake Reconnaissance Report, Supplement A to vol. 17. April Earthquake Engineering Research Institute Bhuj, India Earthquake Reconnaissance Report, Supplement A to vol. 18. July Papadrakakis M, Mouzakis H, Plevris N, Bitzarakis S. A Lagrange multiplier solution for pounding of buildings during earthquakes. Earthquake Engineering and Structural Dynamics 1991; 20: Athanassiadou CJ, Penelis GG, Kappos AJ. Seismic response of adjacent buildings with similar or dierent dynamic characteristics. Earthquake Spectra 1994; 10:

18 828 S. MUTHUKUMAR AND R. DESROCHES 9. Malhotra PK. Dynamics of seismic pounding at expansion joints of concrete bridges. Journal of Engineering Mechanics (ASCE) 1998; 124: DesRoches R, Muthukumar S. Eect of pounding and restrainers on seismic response of multiple-frame bridges. Journal of Structural Engineering (ASCE) 2002; 128: Maison BF, Kasai K. Analysis for type of structural pounding. Journal of Structural Engineering (ASCE) 1990; 116: Maison BF, Kasai K. Dynamics of pounding when two buildings collide. Earthquake Engineering and Structural Dynamics 1992; 21: Kasai K, Maison BF, Patel DJ. An earthquake analysis for buildings subjected to a type of pounding. Proceedings of Fourth U.S. National Conference on Earthquake Engineering, vol. 2, Palm Springs, CA, 1990; Wolf JP, Skrikerud PE. Mutual pounding of adjacent structures during earthquakes. Nuclear Engineering and Design 1979; 57: Anagnostopoulos SA. Pounding of buildings in series during earthquakes. Earthquake Engineering and Structural Dynamics 1988; 16: Jankowski R, Wilde K, Fuzino Y. Pounding of superstructure segments in isolated elevated bridge during earthquakes. Earthquake Engineering and Structural Dynamics 1998; 27: Hunt KH, Crossley FRE. Coecient of restitution interpreted as damping in vibroimpact. ASME Journal of Applied Mechanics 1975; 42: Marhefka DW, Orin DE. A compliant contact model with nonlinear damping for simulation of robotic systems. IEEE Transactions on Systems, Man, and Cybernetics Part A: Systems and Humans 1999; 29: Jing H-S, Young M. Impact interactions between two vibration systems under random excitation. Earthquake Engineering and Structural Dynamics 1991; 20: Davis RO. Pounding of buildings modelled by an impact oscillator. Earthquake Engineering and Structural Dynamics 1992; 21: Ma X, Pantelides CP. Linear and nonlinear pounding of structural systems. Computers and Structures 1998; 66: Chau KT, Wei XX. Pounding of structures modelled as non-linear impacts of two oscillators. Earthquake Engineering and Structural Dynamics 2001; 30: Lankarani HM, Nikravesh PE. A contact force model with hysteresis damping for impact analysis of multibody systems. Journal of Mechanical Design (ASME) 1990; 112: Jankowski R. Non-linear viscoelastic modelling of earthquake-induced structural pounding. Earthquake Engineering and Structural Dynamics 2005; 34: Van Mier JGM, Pruijssers AF, Reinhardt HW, Monnier T. Load-time response of colliding concrete bodies. Journal of Structural Engineering (ASCE) 1991; 117: Papadrakakis M, Mouzakis HP. Earthquake simulator testing of pounding between adjacent buildings. Earthquake Engineering and Structural Dynamics 1995; 24: Filiatrault A, Wagner P, and Cherry S. Analytical prediction of experimental building pounding. Earthquake Engineering and Structural Dynamics 1995; 24: Chau KT, Wei XX, Guo X, Shen CY. Experimental and theoretical simulations of seismic poundings between two adjacent structures. Earthquake Engineering and Structural Dynamics 2003; 32: Goldsmith W. Impact: the Theory and Physical Behaviour of Colliding Solids. Edward Arnold: London, England, Kreyzig E. Advanced Engineering Mathematics (8th edn). Wiley: New York, 1999; DesRoches R, Fenves GL. Simplied restrainer design procedure for multiple-frame bridges. Earthquake Spectra 2001; 17: Park R, Paulay T. Reinforced Concrete Structures. Wiley: New York, Muthukumar S, DesRoches R. Evaluation of impact models for seismic pounding. Proceedings of the 13th World Conference on Earthquake Engineering, Vancouver, Canada, 2004; Paper No Saiidi M, Sozen MA. Simple and complex models for nonlinear seismic response of reinforced concrete structures. Report No. UILU-ENG Structural Research Series No. 465, University of Illinois, Urbana, Muthukumar S. A contact element approach with hysteresis damping for the analysis and design of pounding in bridges. Ph.D. Dissertation, Department of Civil and Environmental Engineering, Georgia Institute of Technology, Fall 2003.