INTRODUCTION
The destructive nature of earthquakes has always been important to structural
engineers. The reports on damage after strong earthquakes show that the effect
of structural pounding is often one of the reasons for damage. The earthquake
that struck Alaska (1964), San Fernando (1971), Mexico City (1985), Loma Prieta
(1989) and Kobe (1995) are clear examples of the serious seismic hazard due
to pounding. Bertero (1985) reported that pounding was
present in about 40% of 330 collapsed buildings during the 1985 Mexico City
earthquake and in 15% of all cases it led to collapse.
Seismic pounding occurs between adjacent buildings with different structural
characteristics, which are insufficiently separated. The differences in geometrical
and/or material properties result in the outofphase oscillations and significant
differences in the responses increase the probability of impacts. One of the
key issues is to estimate the required separation distance to avoid pounding
of adjacent structures. Four different expressions (Valles
and Reinhorn, 1996; Penzien, 1997; Lopez
Garcia, 2004) have been used in building codes to calculate the minimum
separation distance required to prevent structural interactions. One of them
considers the equivalent to the absolute sum of adjacent buildings’ maximum
displacements. The second specifies the minimum gap size without considering
the dynamic properties of adjacent structures. Instead, a coefficient multiplied
by the building height is needed to evaluate the critical gap distance. The
third expression specifies a fixed distance for construction considerations
and the last one uses the square root of sum of squares taking into account
the fact that the maximum displacements in the structures will not occur at
the same time.
Because of the complexity of the pounding phenomenon, most of the past studies
on structural pounding incorporated elastic models of buildings assuming that
the ground motion causes deformations that do not exceed the elastic limit.
In addition, the buildings were often idealized as SingleDegreeofFreedom
(SDOF) systems (Maison and Kasai, 1992; Chau
and Wei, 2001; Chau et al., 2003). One of
the studies on the seismic response of adjacent structures, which permitted
the evaluation of the nonlinear behaviour of buildings subjected to pounding,
was conducted by Athanassiadou and Penelis (1985). Anagnostopoulos
(1988) carried out a more detailed study of pounding of adjacent buildings
in a row modelled as SDOF nonlinear systems. Pantelides and
Ma (1998) considered the dynamic behavior of damped SDOF elastic and inelastic
structural systems with onesided pounding during an earthquake using the Hertz
contact model to capture pounding. Muthukumar and Des Roches
(2006) studied the pounding of adjacent structures modelled as elastic and
inelastic SDOF systems using different pounding models. Jankowski
(2006a) proposed the idea of impact force response spectrum for two adjacent
elastic and inelastic structures modelled as SDOF systems.
In order to conduct numerical simulations of the responses of colliding buildings
under earthquake excitation, a number of different impact force models have
been used. Among them, the linear and nonlinear viscoelastic models (Anagnostopoulos,
1988; Jankowski, 2005) are the most attractive since
they allow us to simulate the dissipation of energy during collisions. Mahmoud
et al. (2008) confirmed the accuracy of the nonlinear viscoelastic
model in capturing the pounding force through a comparison based on different
ground motion records with different Peak Ground Accelerations (PGA). On the
other hand, it is known that the use of the linear viscoelastic model (Anagnostopoulos,
1988) results in tension force just before separation, which is considered
as a major shortcoming of the model for pounding simulation. Mahmoud
(2008) proposed a modified version of the linear viscoelastic model to simulate
pounding without inducing the sticky tensile force, yet this modified model
requires further investigations.
Besides the fact that the study on earthquakeinduced structural pounding has been recently much advanced, the comparison between the poundinginvolved responses of buildings modelled as elastic and inelastic systems has not been conducted. Moreover, in the numerical analyses, the elastic pounding force models have often been used. Therefore, the aim of this study is to conduct a comparative study in order to verify the importance of the necessity of nonlinear modelling of structural behavior. In the study, buildings are modelled as multidegreeoffreedom systems and the nonlinear viscoelastic model is used to simulate impact force during collisions. Different gap distances between structures exposed to different ground motion excitations are considered in the study.
EQUATION OF MOTION
Let us consider the models of fourstorey buildings shown in Fig.
1. For i = (1,2,3,4) and j = (5,6,7,8) let m_{i}, c_{i}
and k_{i} be the masses, damping and stiffness coefficients for the
left and the right building, respectively.

Fig. 1: 
Model of colliding fourstorey buildings 
Let us first assume that the two
buildings remain in the linear elastic range and hence they do not yield under
earthquake excitation. In such a case, the coupling equation of motion for the two buildings shown
in Fig. 1 can be written as Eq. 1:
where, M^{l}, C^{l}, K^{l} and M^{r}, C^{r},
K^{r} are the mass, damping and stiffness matrices of the left and the
right building, respectively; U^{l}, ,
Ü^{l} and U^{r}, ,
Ü^{r} denote the displacement, velocity and acceleration vectors
of the left and the right structure, respectively; F is a vector containing
the forces due to impact; I is a vector with all its elements equal to unity
and Ü_{g} is the earthquake acceleration.
If the two buildings are assumed to be inelastic under the considered earthquake excitation, the coupling equation of motion can be expressed in Eq. 2 as:
where, R^{l} and R^{r} are vectors consisting of the system inelastic storey restoring forces for the left and the right building, respectively; R_{i}(t) = k_{i}(U_{i}(t)  U_{i1}(t)), R_{j}(t) = k_{j}(U_{j}(t)  U_{j1}(t)) for the elastic range and R_{i}(t) = F_{yi}, R_{j}(t) = F_{yj} for the plastic range, where F_{yi} and F_{yj} are the storey yield strengths.
In the study, the nonlinear viscoelastic model is used to simulate the pounding
force during impact F_{ij}(t) between the storeys of the two adjacent
buildings, which is based on the following formula in Eq. 3
(Jankowski, 2005, 2008):
where, δ_{ij}(t) is the relative displacement: δ_{ij}(t)
= (U_{j}(t)U_{i}(t)d), is
the relative velocity, is
the impact stiffness parameter and is
the impact element’s damping, which can be calculated as in Eq.
4 (Jankowski, 2005):
where, is
an impact damping ratio related to a coefficient of restitution e, which can
be defined as in Eq. 5 (Jankowski, 2006b):
NUMERICAL STUDY
A comprehensive study is conducted in order to investigate the influence of modelling structural behaviour using either elastic or inelastic systems on the response of buildings with different separation distances under different ground motions. The El Centro (1940), Cape Mendocino (1992) and Kobe (1995) earthquake records (Fig. 2) are considered to examine the seismic response of the two buildings. The PGA levels of the above earthquakes are 0.34, 0.59 and 0.82 g, respectively (g is the acceleration of gravity). Different separation distances are considered to examine the influence of the gap distance on the peak structural displacements, accelerations and pounding forces. The study is carried out for two cases: (1) two fourstorey buildings are considered to be elastic and (2) two fourstorey buildings are considered to be inelastic.
In the study, we compare the Maximum Elastic Responses (MER) (i.e., displacements,
accelerations and pounding forces) with the responses obtained considering the
buildings to behave inelastically (MIR). The difference between the results
of elastic behavior and the results of inelastic one is assessed by calculating
the Normalized Error (NE) according to the formula in Eq. 6
(Jankowski, 2005):
Properties of the structures: In the present study, two adjacent fourstorey
buildings are considered (Fig. 1). The mass values as well
as the stiffness and damping parameters are assumed to be the same for all storeys
of each structure. The mass of the storey of the left building is equal to 25×10^{3}
kg, whereas each storey of the right building has the mass of 10^{6}
kg. The storey stiffness is equal to 3.46×10^{6} N m^{1} and
2.215×10^{9} N m^{1} for the left and the right building, respectively.
The damping ratio for both buildings is taken as 5%. For the above values of
structural parameters the fundamental natural elastic periods of the structures
have been calculated as equal to 1.2 and 0.3 sec for left and the right building,
respectively. In the case of the inelastic response, the storey yield force
for each storey of the left building is set to be 1.369×10^{5} N, whereas
each storey of the right building yields when the force is equal to 1.589×10^{7}
N.

Fig. 2: 
Acceleration time histories of the El Centro (1940), Cape
Mendocino (1992) and Kobe (1995) ground motion records (PEER Strong Motion
Database: http://peer.berkeley.edu/smcat/) 
The parameters considered in the study make the left building to be lighter
and more flexible when compared with the right structure, which is heavier and
stiffer.
In the numerical analysis, the following values of parameters of the nonlinear
viscoelastic pounding force model are used: =
2.75×10^{9} N m^{3/2}, =
0.35 (e = 0.65) (Jankowski, 2008).
Solution of equation of motion: The coupling equation of motion (1)
is solved in the incremental form using Implicit RungeKutta (IRK) methods (Chen
and Mahmoud, 2008; Mahmoud and Chen, 2008) with two
stage Burrage coefficients and step size of 0.001 sec. Moreover, we use the
slanting Newton method (Chen et al., 2000) to
solve the system of nonsmooth equations in each iteration of the IRK method.
On the other hand, Newmark’s stepbystep method is used to solve the coupling
equation of motion (2) with a constant time step size of 0.001 sec. In addition,
we use the constant average acceleration approach (i.e., γ = 0.5, β
= 0.25) to ensure high degree of numerical stability. The numerical simulations
are performed using MATLAB 7.0 software on a Dell PC with 2 MB memory and 800
MHz.
Response analysis: Here, pounding of two adjacent buildings modelled
as discrete systems shown in Fig. 1, with either elastic or
inelastic structural behavior, is studied. First, the numerical analysis is
conducted for the value of the gap size between the structures equal to 0.04
m. The results of this analysis in the form of the displacement and pounding
force time histories for both buildings under the El Centro (1940), Cape Mendocino
(1992) and Kobe (1995) earthquake records are shown in Fig. 35.
The responses of the elastic systems are significantly different comparing to
the responses of the inelastic ones (Fig. 3a,b,
5a,b). This concerns especially the left building, which
is lighter and more flexible. In the case of this structure modelled as inelastic
system, the first collision with the heavier and stiffer right building results
in substantial movement into the opposite direction including entering into
the yielding range. This movement is so big that the structures do not come
into contact for the second time and the permanent deformation of the left structure
can be observed (Fig. 3b, 4b, 5b).
It can also be seen from the figures that the values of peak pounding forces
and the number of impacts are larger in the elastic case as compared with the
inelastic one. Moreover, in some cases (Fig. 5), the lower
storeys of the elastic systems come into contact with each other, while in the
case of the inelastic systems collisions between lower storeys do not take place.
Further analysis is carried out for different values of the gap size between
colliding structures in order to investigate the influence of the separation
distance on the poundinginvolved structural response. The results of this analysis
showing the peak displacements, accelerations and pounding forces for elastic
and inelastic systems under the El Centro (1940), Cape Mendocino (1992) and
Kobe (1995) earthquake records are presented in Fig. 68.

Fig. 3: 
Displacement and pounding force time histories under the El
Centro earthquake for: (a) two buildings modelled as elastic systems; (b)
two buildings modelled as inelastic systems 

Fig. 4: 
Displacement and pounding force time histories under the Cape
Mendocino earthquake for: (a) two buildings modelled as elastic systems;
(b) two buildings modelled as inelastic systems 

Fig. 5: 
Displacement and pounding force time histories under the Kobe
earthquake for: (a) two buildings modelled as elastic systems; (b) two buildings
modelled as inelastic systems 

Fig. 6: 
Peak displacements, accelerations and pounding forces with
respect to the inbetween gap distance under the El Centro earthquake for:
(a) two buildings modelled as elastic systems; (b) two buildings modelled
as inelastic systems 

Fig. 7: 
Peak displacements, accelerations and pounding forces with
respect to the inbetween gap distance under the Cape Mendocino earthquake
for: (a) two buildings modelled as elastic systems; (b) two buildings modelled
as inelastic systems 

Fig. 8: 
Peak displacements, accelerations and pounding forces with
respect to the inbetween gap distance under the Kobe earthquake for: (a)
two buildings modelled as elastic systems; (b) two buildings modelled as
inelastic system 
Table 1: 
Normalized errors obtained for the El Centro, Cape Mendocino
and Kobe earthquakes 

The peak responses are considerably reduced for all the ground motion excitations
when inelastic modelling of adjacent structures is considered in the analysis
(Fig. 6a,b, 8a,b). For both elastic and inelastic systems, the peak displacements and accelerations
of the storeys of the lighter and more flexible left building increase up to
a certain value of the gap distance and with further increase in the gap distance
a decrease trend can be observed. On the other hand, the storeys of the heavier
and stiffer right building show almost identical values of peak displacements
and accelerations for all considered gap distances (Fig. 68).
Moreover, higher storeys induce higher peak responses than the lower ones for
both buildings under all the ground motion records and the separation distances
considered in the analysis.
The normalized errors in simulating the peak responses of the elastic systems with respect to the peak responses of the inelastic systems, calculated according to Eq. 6, are presented for the three ground motion records in Table 1. It can be seen from the Table 1 that the normalized errors are relatively high, especially for the peak accelerations and pounding forces. It is worth noting that the normalized errors increase with the increase of the levels of PGA of the earthquake records.
CONCLUSIONS
The comparison between the earthquakeinduced poundinginvolved behaviour of two adjacent buildings modelled as elastic and inelastic systems has been investigated in this paper. In the analysis, both structures have been modelled as fourstorey discrete systems and the nonlinear viscoelastic model has been employed to simulate pounding force during collisions at different storey levels. Three different ground motion records with different PGA have been applied to conduct the numerical simulations. The influence of the gap distance on the structural response (peak displacement, acceleration and pounding force) has also been investigated.
The results of the study indicate that the responses of the elastic systems are significantly different comparing to the responses of the inelastic ones. This concerns especially the lighter and more flexible building, which easily enters into the yielding range as the result of pounding. Moreover, the values of maximum impact forces and the number of impacts are larger in the elastic case.
The results of further investigation show that the peak responses are considerably reduced for all the ground motion histories when inelastic modelling of adjacent structures is considered in the analysis. For both elastic and inelastic systems, the peak displacements and accelerations of the storeys of the lighter and more flexible left building increase up to a certain value of the gap distance and with further increase in the gap distance a decrease trend is observed. On the other hand, the storeys of the heavier and stiffer right building show almost identical values of peak displacements and accelerations for all considered gap distances.
The results of this study clearly show that modelling the colliding buildings to behave inelastically is really essential in order to obtain accurate structural poundinginvolved response under earthquake excitation.
IRK methods with slanting Newton iterations have been used efficiently to solve
structural dynamic impact problems which are considered as system of nonsmooth
ordinary differential equations (ODEs) (
Chen and Mahmoud, 2008;
Mahmoud and Chen, 2008;
Mahmoud et
al., 2008).
Chen and Mahmoud (2008) have analyzed
the slanting Newton as an iteration method for solving the nonlinear system of
equations in the IRK. The superlinear convergence of the slanting Newton method
and the convergence order of IRK methods have also been proved. Moreover, the
effect of step sizes on the obtained error using both Explicit RungeKutta (ERK)
and IRK methods has shown that IRK methods are more efficient than the ERK of
fourth order for solving structural dynamic impact problems (
Chen
and Mahmoud, 2008;
Mahmoud et al., 2008). In
addition, a verified inexact IRK method for solving such kind of problems has
been proposed to give a global error bound for the inexact solution (
Mahmoud
and Chen, 2008).
ABBREVIATIONS
c_{i}, c_{j} 
: 
Damping coefficients at storey level i and j 

: 
Impact element’s damping 
C^{l}, C^{r} 
: 
Damping matrices for the left and the right building 
d 
: 
Gap distance 
e 
: 
Coefficient of restitution 
F 
: 
Impact force vector 
F_{ij} 
: 
Impact force between ith and jth storeys 
F_{yi}, F_{yi} 
: 
Yield strengths at storey level i and j 
g 
: 
Acceleration of gravity 
I 
: 
Vector with all its elements equal to unity 
k_{i}, k_{j} 
: 
Stiffness coefficients at storey level i and j 
K^{l}, K^{r} 
: 
Stiffness matrices for the left and the right building 
m_{i}, m_{j} 
: 
Masses at storey level i and j 
M^{l}, M^{r} 
: 
Mass matrices for the left and the right building 
R_{i}, R_{j} 
: 
Inelastic storey restoring forces at storey level i and j 
R^{l}, R^{r} 
: 
Inelastic storey restoring force vectors for the left and the right building 
U_{i}, U_{j} 
: 
Displacements at storey levels i and j 
U^{l}, U^{r} 
: 
Displacement vectors for the left and the right building 

: 
Velocities at storey levels i and j 

: 
Velocity vectors for the left and the right building 
Ü_{i}, Ü_{j}: 
: 
Accelerations at storey levels i and j 
Ü^{l}, Ü^{r} 
: 
Acceleration vectors for the left and the right building 
Ü_{g} 
: 
Earthquake acceleration 
· 
: 
Euclidean norm 

: 
Impact stiffness parameter 
δ_{ij} 
: 
Relative displacement 

: 
Relative velocity 

: 
Impact damping ratio 
ERK 
: 
Explicit RungeKutta 
IRK 
: 
Implicit RungeKutta 
MER 
: 
Maximum elastic response 
MIR 
: 
Maximum inelastic response 
NE 
: 
Normalized error 
ODE 
: 
Ordinary differential equation 
PGA 
: 
Peak ground acceleration 
SDOF 
: 
Singledegreeoffreedom 