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## Chapter 3 Evaluation of numerical pounding force models

Seismic pounding is a form of impact where large impulsive forces act between the participating bodies resulting in a near-instantaneous change in momentum of the colliding bodies. The impact is called elastic if there is no loss of kinetic energy during the collision. If there is some energy loss, the impact is termed inelastic. Pounding is a complex phenomenon involving heterogeneous materials and different types of contact surfaces, supporting structure and foundation system. There may be several collisions within a single earthquake and the impact velocities for each event will be different. As the current state of the art in pounding research cannot account for these factors, the numerical simulations include many assumptions such as lumped mass models, a constant coefficient of restitution and insensitivity of pounding forces to the contact surface geometry.

The lumped-mass model assumes that the building frame or bridge deck acts as a diaphragm, and that the whole mass of the floor or deck contributes to the impact even though classical physics states that only a small part of the colliding mass participates in the contact (Goldsmith 2001). The model also assumes that the actual structural stiffness and the masses of the non-colliding floors above or below the pounding stories have no effect on the build-up of impulsive forces. During numerical simulations, displacement of the structures are computed considering all structural properties. However, when the displacements time history of the neighbouring buildings intersect the contact forces are calculated only from the colliding floor masses and relative velocity of impact.

The coefficient of restitution, defined as the ratio of the final relative velocity to the initial relative velocity of the colliding bodies, is employed as a measure of the elasticity of an impact. In an elastic impact, the coefficient of restitution equals one while it is zero for a completely plastic impact. Coefficient of restitution signifies the loss of kinetic energy due to several complex processes such as material damping, surface friction, surface yielding, Evaluation of numerical pounding force models residual internal vibrations, generation of sound and heat during collision etc. These processes are not fully understood even for the relatively simple collisions of identical spheres at known velocities. In the absence of large scale experimental results, the numerical simulations have to assume values based on the experiments on collision of small masses. The value is also assumed as invariable throughout the simulations even though past experiments have shown it varies according to the mass and the initial relative velocity of the colliding bodies (Goldsmith 2001; Jankowski 2010).

The constraints and approximations have created a dichotomy in pounding research. Most pounding models are validated based on impact force and the numerical evaluation studies of these models also rely on structural acceleration, velocity and relative impact velocities. However, the researchers studying the effect of pounding on structures focus on amplification of structural displacement, bending moment, base shear and storey shear. Past studies that evaluated these models have mostly been based on single impact experiments which do not provide a measure of performance through a full ground motion. Shake table studies on pounding have usually evaluated only one model and performance have not been compared against other models.

To the authors’ knowledge, no past study has attempted a parametric comparison of the predictions of various pounding models with experimental deformation amplification. This study presents the results of a shake table investigation of floor to floor pounding between two steel portal frames and compares them with the results from elastic numerical analysis. A contact element model is recommended for use based on the comparison of predicted maximum displacement amplification with the experimental displacement amplification values. The contact elements are compared based on their displacement response because structural displacement has been found mostly insensitive to the contact element stiffness which cannot be determined with certainty. Structural drift is also the main kinematic parameter of interest to designers as all the internal forces in structures can be computed from displacement. The contact surface between the two frames was kept flat instead of the commonly employed hemispherical (van Mier *et al.* 1991; Chau *et al.* 2003) or cylindrical interfaces (Filiatrault *et al.* 1995).

**Numerical models considered for evaluation**

Contact element models have been employed frequently to predict pounding forces and structural responses. Such models define a constant “gap” between the buildings and if the 27 Evaluation of numerical pounding force models relative closing displacement between the two buildings is more than the “gap”, the contact force is activated. The most common form of contact model assumes the presence of an elastic spring with or without a viscous damper to model energy loss. These models have an advantage that they can be implemented in most existing numerical time history analysis software without significant programming modification. Other approaches like Lagrange-multiplier and Laplace-domain methods have also been employed in some studies (e.g. Papadrakakis *et al.* 1991; Chouw 2002) but their application is limited because they have not been implemented in commercial FE software and need special programming by the users.

This chapter compares the displacement amplification predicted by various viscoelastic contact-element models with the experimental results and select the best performing model. The linear elastic and nonlinear elastic contact elements have not been included in the comparison because they cannot simulate energy loss during impact. The numerical models considered for the evaluation are described below with their underlying assumptions and their intended performance priorities, such as modelling energy loss or better prediction of pounding force.

The formulations were first derived by Anagnostopoulos (1988; 2004). The expressions for the damping coefficient, *c*, and damping ratio, ξ, were derived so that the energy loss is the same as that in the stereo-mechanical model for the same *e*. The model, with respect to the displacement amplification of the structures, is found to be insensitive to the values of contact element stiffness. The impulsive acceleration calculated during impact varies according to the stiffness adopted, but this variation does not translate into variation in structural displacement. This model produces tensile force near the end of the contact which does not agree with experimental force time histories. Several objections to the model’s uniform damping throughout the contact have also been raised and various amendments to the value of ξ have been proposed.

*Modified linear viscoelastic model (MLVe)*

*Modified linear viscoelastic model (MLVe)*

The modified linear viscoelastic element has an approach only dashpot. A similar model was initially proposed by Valles-Mattox and Reinhorn (1996) as impact Kelvin model. However, the impact Kelvin model has not been used in any numerical simulations perhaps due to its complex formulation. The proposing study also did not provide any numerical investigation to assess its performance. The MLVe model (Equation 3.2) was later proposed by Mahmoud (2008). The relationship between ξ and *e* in LVe model has been modified so that the total viscous damping can be incorporated within the approach period.

**Chapter 1 Introduction **

1.1 Motivation for the study

1.2 Methodology

1.3 Outline

**Chapter 2 Shake table simulation of row building pounding**

2.1 Experimental setup

2.2 Results and discussion

2.3 Summary

**Chapter 3 Evaluation of numerical pounding force models **

3.1 Numerical models considered for evaluation

3.2 Experimental setup

3.3 Numerical results and discussions

3.4 Summary

**Chapter 4 Numerically exact viscoelastic force model: derivation and validation**

4.1 Derivation of exact expression for damping constant ζ

4.2 Selection of existing viscoelastic force models for comparison

4.3 Assessment of pounding models using experimental data

4.4 Numerical simulation of pounding structures

4.4.1 Structural model

4.5 Summary

**Chapter 5 Impact of RC slabs: influence of mass, velocity and contact surface **

5.1 Experimental setup

5.2 Results and discussion

5.3 Summary

**Chapter 6 Limitations in building pounding simulations**

6.1 Limitations in numerical simulations

6.2 Limitations in experimental simulations

6.3 Sears impact model

6.4 Summary

**Chapter 7 Damped Sears impact model: derivation and validation **

7.1 Numerical Derivation

7.2 Experimental setup

7.3 Results and discussion

7.4 Summary

**Chapter 8 Damped Sears impact model: application to building pounding **

8.1 Theoretical derivations

8.2 Experimental setup

8.3 Results and discussion

8.4 Practical example

8.5 Summary

**Chapter 9 Recommended further development of damped Sears model for application to**

**building design against pounding**

9.1 Limitation of the proposed model in its current form

9.2 Recommendations for further development of the proposed model

9.3 Summary

**Chapter 10 Conclusions**

10.1 Conclusions

10.2 Evaluation of the existing force models

10.3 Damped Sears impact model: derivation and verification

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A distributed-mass model with end-compliance effects for simulation of building pounding