# Development of pounding model for adjacent structures in earthquakes

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2 Figure 1. Pounding induced damage of buildings in Christchurch CBD a single column system and representing the lumped mass by nodes on the column. The pounding effects are simulated by the laws of stereo-mechanics (Goldsmith, 2001) or by a collision element introduced between the lumped masses that are at the same level in the adjoining buildings (Fig. 2). There have also been studies that employ wave theory (Cole, 2010) to calculate the pounding forces or that combine both buildings as a single unit during collisions (Chouw, 2002). However, the collision element approach is the one most commonly employed by researchers. Figure 2. Lumped Mass model with a gap and an elastic spring in series The collision element can be defined as a link between the two masses that is activated when the gap between the two lumped masses becomes zero or less. All types of collision element have at least two properties: a gap or an opening which is the distance between the surfaces of colliding masses, and a stiffness value (k) which is the spring constant of a linear spring. It will impart equal and opposite forces on both structures when they collide. This pounding model will result in an unrealistic, perfectly elastic contact and to avoid this drawback Anagnostopoulos (1988 and 2004) proposed a linear link that has both stiffness as well as a damping value to simulate the energy lost during collisions. This model has been further revised to include the nonlinear effects inherent in the Hertz contact law (Muthukumar and Desroches, 2004 and 2006; Jankowski, 2005 and 2006)). Jankowski (2005) and Muthukumar and Desroches (2006) provided a succinct analysis of these different collision elements (not repeated here). There have also been various experimental studies on pounding (van Mier et al., 1991; Zhu et al., 2002; Chau et al., 2003). Van Mier et al. (1991) proposed an elastoplastic impact element model based on the experimental results. 2

4 parameters for general application for all structures. Van Mier et al. (1991) employed no viscous parameter in their elastoplastc impact model, and this seems to be a weakness, as is apparent from all the loading curves. The loading curves produced by the first part of Equation 1 will be concave shaped but the loading curves seen in the actual experiments are all convex in shape. 2.2 Nonlinear viscoelastic impact element (Jankowski, 2005) Jankowski (2005) proposed a nonlinear viscoelastic impact element, composed of a nonlinear elastic element stiffness β in parallel with a nonlinear damper that only becomes active during the approach period. The approach-only dashpot tries to emulate the experimental observations that most of the energy loss during impact occurs during approach (Goldsmith, 2001). The pounding force is given by: ( ) = / ( ) + ( ) ( ) for ( ) >0 ( ) = / ( ) for ( ) 0 (2) Where β is the impact stiffness parameter, δ (t) is the velocity of the left hand mass relative to the right hand mass, and c is the instantaneous damping of the impact element, given by =2 ( ) (2a) where m 1 and m 2 are the masses of the colliding bodies, and the damping ratio, according to Jankowski (2006), is given by: = (2b) ( where e is the coefficient of restitution. The parameter differs from the Hertz s coefficient k h, due to the inclusion of the viscous parameter and has to be derived iteratively. Jankowski (2005) proved that, a posteriori, the iterative procedure can produce a close fit to the experimental time histories, but it suffers from two drawbacks. First, the value of parameter can only be derived iteratively if the collision force time history is available after the event, and second, the relation can explain the results when the pounding force is in the elastic range proposed by van Mier, but not when it is in the elastoplastic range, i.e. it cannot replicate the plateau in the elastoplastic curve. The proposed model will address this second limitation of the nonlinear viscoelastic model. 2.3 Viscous elastoplastic impact element This study proposes a viscous elastoplastic impact model. The model combines the elastoplastic and nonlinear viscoelastic impact elements and describes the pounding force as: ( ) = / ( ) + ( ) ( ) for / ( ) + ( ) ( ) < ( ) >0 = / ( ) + ( ) ( ) ( ) >0 = / ( ) / ( ) < ( ) 0 = / ( ) ( ) 0 (3) Where F is the yield strength of the material or structural element at the contact point. The parameter should be deduced judiciously from the study of the model. For instance, if there is mid-column pounding, the value of F E is the force that will cause the failure in column. Similarly, if the impact is on a corner of the slab, F E is the force that will crush that particular corner, and if the collision is on the face of a beam, F E should be able to cause flexural failure or material crushing failure at that particular surface of contact. The parameters and are the same as those of the nonlinear viscoelastic impact element and are given by: c =2ξ β δ(t) m 1 m 2 m 1 + m 2 (3a) 4

6 Figure 5. Adjacent buildings under investigation The restitution coefficient has been calculated for each impact, from the relationship of the concrete vs. concrete impact, as given by Jankowski (Jankowski 2010), = (4) where v is the relative velocity at the initiation of impact. Figure 6. Simplified models of the adjacent buildings The model is analyzed with 6 s of the 1940 North-South El Centro ground acceleration time history (Fig. 7) downloaded from Pacific Earthquake Engineering Research Center and linearly scaled to the maximum ground acceleration of 0.5 g. The analysis is performed using both the nonlinear viscoelastic and viscous elastoplastic models. ground acceleration (g) Time (s) Figure 7. El-Centro ground motion The analysis includes the effect of the local plasticity on the global structural response only. The gap widening due to permanent compression of participating surfaces is ignored. This will introduce some errors, as the gap size widens as the contact area yields, an effect which is not incorporated in this analysis. Similarly, the impact of strain rates on the local yielding may be significant for concrete. 6

7 The numerical analysis was conducted with an in-house program. The accuracy of the program was verified by comparing the results of a linear visco-elastic analysis with that from SAP2000. The results, including the effect of stiffness, damping and local plasticity are presented in Figure 8. It is apparent that the relative displacement between the structures is reduced as the plasticity effects are considered. The two graphs are identical until 2.4 seconds at which instant the pounding force is large enough to induce plasticity. The graphs then diverge and the largest relative displacement in the viscous elastoplastic model is, as expected, less than that obtained using the nonlinear viscoelastic model. Since the right hand side masonry structure in this numerical study is relatively rigid, the relative displacement time history is nearly identical to the roof displacement time history for the flexible frame structure and so the latter is not reproduced here. Relative Displacement (mm) Time (s) Figure 8. Relative displacement time histories Viscous Elastoplastic Model Non-linear Viscoelastic model 4 CONCLUSIONS This paper studied two impact elements, namely, an elastoplastic impact element proposed by van Mier et al. (1991) and a nonlinear viscoelastic impact element proposed by Jankowski (2005). A new impact element is proposed, combining the parameters and ideas of the two earlier models into one coherent model. A sample numerical study has been presented to illustrate the difference in the non-linear viscoelastic model and the proposed viscous elastoplastic model. The effect of pounding on the relative displacement of a reinforced concrete building adjacent to a relatively rigid masonry building with no separation gap, and a common wall between them is considered. The proposed impact element highlights the effects of local yielding, and shows that the relative displacement between the two floors, and as a result, the floor displacement response of the flexible frame structure is reduced if the plasticity effects at the contact location are considered. Further studies may include the effect of strain rate on concrete deformation and the effect of the gap widening on the pounding force development. This may help in explaining the damage propagation in the participating structures. ACKNOWLEDGEMENT The authors would like to thank the anonymous reviewer for the constructive comments and the University of Auckland for providing the first author with the International Doctoral Scholarship. REFERENCES Anagnostopoulos, S. A. (1988). Pounding of buildings in series during earthquakes. Eathquake Engineering and Structural Dynamics 16:

8 Anagnostopoulos, S.A. (2004). Equivalent viscous damping for modeling inelastic impacts in earthquake pounding problems. Earthquake Engineering and Structural Dynamics 33(8): Anagnostopoulos, S.A. and Karamaneas C.E. (2008). Collision shear walls to mitigate seismic pounding of adjacent buildings, Proceedings.of the 14 th World Conference on Earthquake Engineering, October 12-17, 2008, Beijing, China Bothara, J.K., Jury R.D., Wheelers K. And Stevens C. (2008). Seismic Assessment Of Buildings In Wellington: Experiences And Challenges. Proceedings of the 14 th World Conference on Earthquake Engineering, October 12-17, 2008, Beijing, China Chau, K. T., Wei, X. X., Guo, X., And Shen, C. Y. (2003). Experimental and theoretical simulations of seismic poundings between two adjacent structures. Earthquake Engineering and Structural Dynamics 32(4): Chouw, N. (2002). Influence of soil-structure Interaction on pounding response of adjacent buildings due to near-source earthquakes. JSCE journal of Applied Mechanics 5: Goldsmith, W. (2001). Impact: the theory and physical behaviour of colliding solids, Dover Pubns. Cole, G., Dhakal, R., Carr, A., and Bull, D. (2010). An investigation of the effects of mass distribution on pounding structures. Eathquake Engineering and Structural Dynamics. DOI: /eqe.1052 Ip, F. (1999). Compressive Strength and Modulus of Elasticity of Masonry Prisms, ME Thesis, Carleton University. Jankowski, R. (2005). Non linear viscoelastic modelling of earthquake induced structural pounding. Earthquake Engineering and Structural Dynamics 34(6): Jankowski, R. (2006). Analytical expression between the impact damping ratio and the coefficient of restitution in the non linear viscoelastic model of structural pounding. Earthquake Engineering and Structural Dynamics 35(4): Jankowski, R. (2010). Experimental study on earthquake-induced pounding between structural elements made of different building materials. Earthquake Engineering and Structural Dynamics 39(3): Jeng, V. and Tzeng, W. (2000). Assessment of seismic pounding hazard for Taipei City. Engineering Structures 22(5): Muthukumar, S. and Desroches, R. (2004). Evaluation of impact models for seismic pounding. Proceedings of the 13 th world Conference on Earthquake Engineering, August 1-6, 2004 Vancouver, B. C., Canada. Muthukumar, S. and DesRoches, R. (2006). A Hertz contact model with non linear damping for pounding simulation. Earthquake Engineering and Structural Dynamics 35(7): Rosenblueth, E. and Meli, R. (1986). The 1985 earthquake: causes and effects in Mexico City. Concrete International 8(5): van Mier, J.G., Pruijssers A., Reinhardt, H.W. and Monnier, T. (1991). Load Time Response of Colliding Concrete Bodies. Journal of Structural Engineering 117(2): Zhu, P., Abe, M. and Fujino, Y. (2002). Modelling three dimensional non linear seismic performance of elevated bridges with emphasis on pounding of girders. Earthquake Engineering and Structural Dynamics 31(11):

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