Access to this full-text is provided by MDPI.
Content available from Applied Sciences
This content is subject to copyright.
Citation: Uz, M.E.;
Jakubczyk-Gałczy´nska, A.;
Jankowski, R. Numerical Analysis of
Seismic Pounding between Adjacent
Buildings Accounting for SSI. Appl.
Sci. 2023,13, 3092. https://doi.org/
10.3390/app13053092
Academic Editor: Maria Favvata
Received: 3 December 2022
Revised: 15 February 2023
Accepted: 24 February 2023
Published: 27 February 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
applied
sciences
Article
Numerical Analysis of Seismic Pounding between Adjacent
Buildings Accounting for SSI
Mehmet Eren Uz 1, Anna Jakubczyk-Gałczy ´nska 2and Robert Jankowski 2, *
1Department of Civil Engineering, Faculty of Engineering, Aydin Adnan Menderes University,
Aydin 09010, Turkey
2Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 80-233 Gdansk, Poland
*Correspondence: jankowr@pg.edu.pl
Abstract:
The structural pounding caused by an earthquake may damage structures and lead to their
collapse. This study is focused on the pounding between two adjacent asymmetric structures with
different dynamic properties resting on the surface of an elastic half-space. An exploration of the
relationship between the effects of the seismic analysis with the impact response to the torsional
pounding between adjacent buildings under different SSI effects has been presented. In this paper,
the authors have proposed a procedure for analyzing the response for adjacent buildings subjected to
the pounding effects, considering systems with multiple degrees of freedom and modal equations of
motion with four types of soil. All the calculations have been performed based on the fourth-order
Runge–Kutta method. The novelty of the present study is related to the fact that the rigorous and
approximate methods are used to examine the effects of pounding and SSI simultaneously. As a
result, these two methods have been thoroughly investigated for both effects and the results have
been compared. The results show that the approximate method produces results that are slightly
different from those obtained by the rigorous direct integration method in the case of small SSI
effects due to an increase in the pounding force. The efficiency of the method is also validated using
numerical examples.
Keywords:
seismic analysis; structural pounding; soil–structure interaction; earthquake; torsional response
1. Introduction
Seismic researchers have often observed collisions between adjacent buildings that
are not sufficiently separated. This phenomenon, known as earthquake-induced struc-
tural pounding, may lead to minor damage at the points of interaction during moderate
ground motions [
1
,
2
]. It may also cause serious damage to colliding structures and even
result in their collapse during major earthquakes. Examples include the Loma Prieta earth-
quake in 1989 [
3
] or the Athens earthquake in 1999 [
4
], when many cases of structural
pounding of buildings occurred. Three important aspects of a complex seismic analysis
of structure–foundation systems (see [
5
]) include the frequency-dependent interaction
forces, nonproportional damping of soil–structure interaction (SSI), liquefaction [
6
–
10
],
and pounding responses of adjacent buildings. Earthquake-induced structural pounding
between symmetric buildings has been investigated [
11
–
15
]. Numerous researchers have
studied the effects of structural interactions by applying different structural models and us-
ing different models of collisions [
16
–
20
]. Numerous researchers have also analyzed the SSI
of asymmetric buildings exposed to seismic excitations [
21
–
25
]. In contrast, pounding be-
tween adjacent asymmetric structures with different dynamic properties and incorporating
SSI has not been sufficiently explored.
SSI problems (see [
5
,
26
]) have been addressed in the frequency domain by means
of either a Fourier or a Laplace transform [
27
–
29
] to consider the frequency-dependent
interaction forces. However, in frequency-domain analysis, only linear responses can be
Appl. Sci. 2023,13, 3092. https://doi.org/10.3390/app13053092 https://www.mdpi.com/journal/applsci
Appl. Sci. 2023,13, 3092 2 of 18
considered. The soil springs and dashpots associated with the translational rocking and
torsional modes of vibration can be satisfactorily calculated for typical multistory buildings
in the time domain using frequency-independent expressions [30]. Employing traditional
modal analysis methods to derive equivalent modal damping from the diagonal terms
of the transformed damping matrix without examining the off-diagonal elements is one
of the common approximate approaches for analyzing nonclassically damped systems.
Certain conditions may lead to unacceptable errors in the response when off-diagonal
terms are ignored in the transformed damping matrix [
31
]. Jui-Liang and Keh-Chyuan [
32
]
verified the accuracy of the two degree-of-freedom (DOF) modal equations of motion
for conserving nonproportional damping compared with that of the damped one-way
asymmetric buildings.
A simplified modal response analysis was developed for engineering applications that
do not require complicated calculations of the equivalent modal damping. The outcomes
of a parametric investigation, conducted by varying the structural parameter values, have
also been investigated in the previous studies of the authors [
33
,
34
]. The parametric study
is beyond the scope of this paper, which focuses on considering the SSI and impact effects
for rigorous and approximate methods. According to the results of the response analysis
conducted by Uz and Hadi [
33
], the pounding of buildings during ground motion exci-
tation has a considerable influence on the longitudinal behavior of the lighter structure.
Simplifications of the modal analysis and investigations on pounding-involved structural
responses have been proposed [
31
,
33
,
35
–
37
]. Studies have also been conducted on the
earthquake-induced responses of colliding symmetric structures incorporating SSI. How-
ever, to the best of our knowledge, our study is the first exploration of the effect of SSI
on the response of adjacent asymmetric buildings exposed to pounding, although some
studies [
38
–
41
] investigated the pounding effect or SSI effects individually on adjacent
buildings. Therefore, our main objective in this study was to investigate the relationship
between the effects of the seismic analysis with the response to the torsional pounding
between adjacent buildings under different SSI effects. Both the SSI and pounding effects
have not been adequately researched in the past in terms of rigorous and approximate
methodologies. In most cases, they were evaluated independently. The novelty of the
present study is related to the fact that the rigorous and approximate methods are used to
examine the effects of pounding and SSI simultaneously. As a result, these two methods
have been thoroughly investigated for both effects and the results have been compared.
The response analysis procedure has been developed for adjacent buildings subjected to
pounding effects using multi-DOF modal equations of motion with four types of soil. The
fourth-order Runge–Kutta method has been applied in all the calculations. The numerical
examples have been used to validate the efficiency of the proposed method.
2. Theoretical Model Framework
The basic model for SSI that considers the effects of the pounding force on adjacent
buildings has been used in this study.
2.1. Equation of Motion
Figure 1shows a structural model of two adjacent buildings on a half-space with
elastic properties.
Appl. Sci. 2023,13, 3092 3 of 18
Appl. Sci. 2023, 13, 3092 3 of 19
Figure 1. Modeling of asymmetric adjacent buildings.
Subscripts and in Figure 1 indicate the numbers of stories in the buildings.
These are 1,2,.., for Building and 1,2,.., for Building . In this study, the fol-
lowing values have been determined: the mass, stiffness, damping coefficients, and
moment of inertia of the floor with respect to the axes parallel to the and axes of
Building and in relation to the center of mass. In addition, Buildings and are
represented either by the subscript or superscript and , respectively. A building with
stories has 3+5 DOF; hence, Buildings A and B have 3+5 and 3+5 equa-
tions, respectively. Here, only the equations for Building are highlighted. The equa-
tion of motion noted from previous studies [30,35,42–44] is briefly presented herein for
completeness. The three equations of motion of each floor of the building may be written
in a matrix form as (3 N):
[
]
{
}
+
[
]
{
}
+
[
]
{
}
+
(
)
=
{
0
}
(1)
[
]
{
}
+
{
}
+
{
}
+
(
)
=
{
0
}
[
]
+
[
]
{
}
−
{
}
+
[
]
+
[
]
{
}
−
{
}
+
[
]
{
}
+
(
)
=
{
0
}
The mass, damping, and stiffness of Building A in the submatrices are denoted as
, , , , and , in relation to both directions. These matrices are sized with N
× N dimensions. As described by Equation (2), the parameters
,
, and
are the
displacements of the center of mass of the floors along the longitudinal and transverse
axes, and their twists about the upward axis () in Building A, respectively. The forces
used for pounding in the longitudinal direction
() are obtained from the nonlinear
viscoelastic model developed by Jankowski [45].
() and
() are derived by the
Coulomb friction model used by Chopra [46]. The impact model (nonlinear viscoelastic
model) is used in this study for the rigorous model as it is. On the other hand, the impact
model for the approximate method is deeply examined. , , and are the dis-
Figure 1. Modeling of asymmetric adjacent buildings.
Subscripts
i
and
j
in Figure 1indicate the numbers of stories in the buildings. These
are 1, 2,
. . .
,
N
for Building
A
and 1, 2,
. . .
,
S
for Building
B
. In this study, the following
values have been determined: the mass, stiffness, damping coefficients, and moment of
inertia of the floor with respect to the axes parallel to the
x
and
y
axes of Building
A
and in
relation to the center of mass. In addition, Buildings
A
and
B
are represented either by the
subscript or superscript
a
and
b
, respectively. A building with
N
stories has
3N+5 DOF
;
hence, Buildings Aand Bhave 3
N+
5 and 3
S+
5 equations, respectively. Here, only the
equations for Building
A
are highlighted. The equation of motion noted from previous
studies [
30
,
35
,
42
–
44
] is briefly presented herein for completeness. The three equations of
motion of each floor of the building may be written in a matrix form as (3 N):
[Ma]n..
xt
ic o+[Cax ].
xi+[Kax ]{xi}+hFp
xij (t)i={0}
[Ma]n..
yt
ic o+Cay .
yi+Kay{yi}+hFp
yij (t)i={0}
r2
a[Ma]..
θt
i+fa[Cax ].
xi−eaCay.
yi+Ca
θRn.
θic o+fa[Kax ]{xi}
−eaKay{yi}+Ka
θR{θic }+hFp
θij (t)i={0}
(1)
The mass, damping, and stiffness of Building Ain the submatrices are denoted as
M
a
,C
ax
,C
ay
,K
ax
, and K
ay
, in relation to both directions. These matrices are sized with
N
×
Ndimensions. As described by Equation (2), the parameters
xt
ic
,
yt
ic
, and
θt
ic
are the
displacements of the center of mass of the floors along the longitudinal and transverse axes,
and their twists about the upward axis (z) in Building A, respectively. The forces used for
pounding in the longitudinal direction
Fp
xij (t)
are obtained from the nonlinear viscoelastic
model developed by Jankowski [
45
].
Fp
yij (t)
and
Fp
θij (t)
are derived by the Coulomb friction
model used by Chopra [
46
]. The impact model (nonlinear viscoelastic model) is used in
Appl. Sci. 2023,13, 3092 4 of 18
this study for the rigorous model as it is. On the other hand, the impact model for the
approximate method is deeply examined. x
i
,y
i
, and
θic
are the displacement vectors of the
center of resistance (CR) in the xand ydirections and the twist of each floor with respect to
the base, respectively. Additionally, two-way asymmetric buildings have been considered
such that the CRs and center of masses (CMs) are not symmetrical along two axes of the
horizontal plane as shown in Figure 2. The center of rigidity is the centroid of stiffness in a
floor-diaphragm layout. The floor diaphragm experiences translational displacement in
two directions and rotation when the center of stiffness is subjected to lateral loading.
Appl. Sci. 2023, 13, 3092 4 of 19
placement vectors of the center of resistance (CR) in the and directions and the twist
of each floor with respect to the base, respectively. Additionally, two-way asymmetric
buildings have been considered such that the CRs and center of masses (CMs) are not
symmetrical along two axes of the horizontal plane as shown in Figure 2. The center of
rigidity is the centroid of stiffness in a floor-diaphragm layout. The floor diaphragm ex-
periences translational displacement in two directions and rotation when the center of
stiffness is subjected to lateral loading.
Figure 2. An asymmetric view of adjacent shear buildings in two directions.
The static eccentricity of the center of rigidity from the center of mass ( and ) for
each story level does not differ, although the CR varies from floor to floor. Hence, the CR
lies at the co-ordinates , of Building A and , of Building B. The radius of gyra-
tion of the floor mass is taken from the center of mass. The torsion stiffness matrix de-
fined about the CR is denoted as
, whereas
in Equation (1) is defined in relation
to CM. Furthermore, the damping matrices of Building A are given as , , and
,
in Equation (1), considered proportionally to the stiffness matrices [47]. The displacement
vectors of Building A in the related directions without the effects of SSI can be calculated
as
{
}
=
{
1
}
+
{
1
}
+
{
ℎ
}
+
{
}
−
{
}
(2)
{
}
=
{
1
}
+
{
1
}
+
{
ℎ
}
+
{
}
+
{
}
{
}
=
{
1
}
+
{
}
where and are vectors of displacement with relation to the CM of the superstruc-
ture;
,
,
, and
represent the DOF at the foundation that is related to transla-
tions and rocking about the and axes; and
is the rotation around the axis. In
each building, the five equations of motion of the foundation are described as transla-
tional and rocking distances in the x and y directions as well as a torsional mode of vi-
bration.
2.2. Pounding Forces
The equation of motion for Building A can be expressed in terms of translation and
rocking along the and axes and twist along the axis, as described by Richart et al.
[30], Sivakumaran and Balendra [23], and Jui-Liang et al. [35]. The static impedance
functions have been used in this study (see [30]). Equation (3) shows the pound-
ing-involved equation of motion for buildings with SSI:
0
0
(
)
(
)
+
0
0
(
)
(
)
+
0
0
(
)
(
)
+
(
)
−
(
)
=
−
()
(
)
(3)
Figure 2. An asymmetric view of adjacent shear buildings in two directions.
The static eccentricity of the center of rigidity from the center of mass (
e
and
f
) for
each story level does not differ, although the CR varies from floor to floor. Hence, the
CR lies at the co-ordinates
ea
,
fa
of Building Aand
eb
,
fb
of Building B. The radius of
gyration of the floor mass is taken from the center of mass. The torsion stiffness matrix
defined about the CR is denoted as
Ka
θR
, whereas
Ka
θM
in Equation (1) is defined in relation
to CM. Furthermore, the damping matrices of Building Aare given as
Cax
,
Cay
, and
Ca
θR
,
in Equation (1), considered proportionally to the stiffness matrices [
47
]. The displacement
vectors of Building Ain the related directions without the effects of SSI can be calculated as
xt
ic =xa
o{1}+xg{1}+ϕa
o{hi}+{xi}−fa{θic }
yt
ic =ya
o{1}+yg{1}+ψa
o{hi}+{yi}+ea{θic }
θt
ic =θa
o{1}+{θic }(2)
where
xi
and
yi
are vectors of displacement with relation to the CM of the superstructure;
xa
o
,
ya
o
,
ψa
o
, and
φa
o
represent the DOF at the foundation that is related to translations and
rocking about the
x
and
y
axes; and
θa
o
is the rotation around the
z
axis. In each building,
the five equations of motion of the foundation are described as translational and rocking
distances in the xand ydirections as well as a torsional mode of vibration.
2.2. Pounding Forces
The equation of motion for Building Acan be expressed in terms of translation and
rocking along the
x
and
y
axes and twist along the
z
axis, as described by Richart et al. [
30
],
Sivakumaran and Balendra [
23
], and Jui-Liang et al. [
35
]. The static impedance functions
have been used in this study (see [
30
]). Equation (3) shows the pounding-involved equation
of motion for buildings with SSI:
Ma0
0Mb(..
Ua(t)
..
Ub(t))+Ca0
0Cb(.
Ua(t)
.
Ub(t))+Ka0
0KbUa(t)
Ub(t)+Fp(t)
−Fp(t)
=−Pa(t)
Pb(t)
(3)
Appl. Sci. 2023,13, 3092 5 of 18
where
Fp(t)
,
Pa(t)
, and
Pb(t)
are the vectors including the impact forces between floor
masses mi,mj.
In this study, the pounding force was simulated based on the nonlinear viscoelastic
model [45,48–50].
2.3. Approximate Normal Modes of Adjacent Buildings
Equation (4) can be expressed in the following form, as proposed by Chopra and
Goel [51]:
Pa(t)=
3N+5
∑
n=1
sa
nΓa
xn
..
xg+Γa
yn
..
yg;Pb(t)=
3S+5
∑
n=1
sb
nΓb
xn
..
xg+Γb
yn
..
yg(4)
The
nth
-mode modal inertia force distribution can be calculated as
Maϕa
n
for vector
sa
n
and
Mbϕb
n
for vector
sb
n
. The
ϕa
n
calculated from
Ka
and
Ma
is the
nth
-mode shape
without damping.
ϕb
n
is also obtained in the same way as
ϕa
n
.
Γa
xn
,
Γa
yn
,
Γb
xn
, and
Γb
yn
in
the longitudinal and perpendicular directions for the
nth
modal contribution values for
both buildings. The output values of
U(t)
contain translations in the
x
and
y
directions,
rotations for each floor of the superstructure and the foundation, and the rocking angles
for the foundation only in both directions. The
nth
mode of the contribution factor in the
modal analysis is calculated as
Γa
xn =ϕa
nT×Ma×[1T0T0T10000]T
ϕa
nT×Ma×ϕa
n;
Γa
yn =ϕa
nT×Ma×[0T1T0T01000]T
ϕa
nT×Ma×ϕa
n
(5)
where
1
and
0
are the unit and zero vectors, respectively, sized as
N×
1. Equation (5)
proves that the
nth
modal participation factors rely on the path of the horizontal ground
motion excitations. The 1940 El Centro (117 El Centro Array-9 station) and the 1995
Kobe (KJMA station) ground motions have been used for the analysis of both buildings,
which is shown in Figure 3. The top accelerations of the related excitations were scaled
to 0.3
g
and 0.2
g
for the 1940 El Centro NS-EW and 0.8
g
and 0.6
g
for the 1995 Kobe
NS-EW, respectively.
Appl. Sci. 2023, 13, 3092 5 of 19
where (), (), and () are the vectors including the impact forces between floor
masses ,.
In this study, the pounding force was simulated based on the nonlinear viscoelastic
model [45,48–50].
2.3. Approximate Normal Modes of Adjacent Buildings
Equation (4) can be expressed in the following form, as proposed by Chopra and
Goel [51]:
(
)
=
Γ
+
Γ
;
(
)
=
Γ
+
Γ
(4)
The -mode modal inertia force distribution can be calculated as
for vector
and
for vector
. The
calculated from and is the -mode shape
without damping.
is also obtained in the same way as
. Γ
, Γ
, Γ
, and Γ
in
the longitudinal and perpendicular directions for the modal contribution values for
both buildings. The output values of () contain translations in the and direc-
tions, rotations for each floor of the superstructure and the foundation, and the rocking
angles for the foundation only in both directions. The mode of the contribution fac-
tor in the modal analysis is calculated as
Γ
=
×
×
[
1
0
0
0
0
]
××
;
Γ
=
×
×
[
0
1
0
0
0
]
×
×
(5)
where and are the unit and zero vectors, respectively, sized as ×1. Equation (5)
proves that the modal participation factors rely on the path of the horizontal ground
motion excitations. The 1940 El Centro (117 El Centro Array-9 station) and the 1995 Kobe
(KJMA station) ground motions have been used for the analysis of both buildings, which
is shown in Figure 3. The top accelerations of the related excitations were scaled to 0.3 g
and 0.2 g for the 1940 El Centro NS-EW and 0.8 g and 0.6 g for the 1995 Kobe NS-EW,
respectively.
(
a
)
(
b
)
Figure 3. (a) 1940 El Centro and (b) 1995 Kobe earthquakes with maximum ground acceleration
scaled to 0.3 g and 0.8 g, respectively.
and
are defined in Equation (4) by the time variations in () and (). In
the analysis, the vertical motion of the ground was not in use. The force distribution can
be expressed with the sum of the modal inertia forces, as given in Equation (4).
and
are (3+5)×8 and (3+5)×8 diagonal matrices for both buildings, respective-
ly.
,
, and
in
are subvectors of the natural vibration mode in the
Figure 3.
(
a
) 1940 El Centro and (
b
) 1995 Kobe earthquakes with maximum ground acceleration
scaled to 0.3 g and 0.8 g, respectively.
Ua
n
and
Ub
n
are defined in Equation (4) by the time variations in
..
xg(t)
and
..
yg(t)
. In the
analysis, the vertical motion of the ground was not in use. The force distribution can be
expressed with the sum of the modal inertia forces, as given in Equation (4).
Ta
n
and
Tb
n
Appl. Sci. 2023,13, 3092 6 of 18
are
(3N+5)×
8 and
(3S+5)×
8 diagonal matrices for both buildings, respectively.
ϕa
xn
,
ϕa
yn
, and
ϕa
θn
in
Ta
n
are subvectors of the
nth
natural vibration mode in the superstructure,
sized as
N×
1. They are related to the displacement in both directions and the rotating
DOF.
φa
xon
,
φa
yon
,
φa
θon
,
φa
ψon
, and
φa
φon
in
Ta
n
are five subvectors of the mode shapes of the
SSI system for Building A. With generalized modal co-ordinates for both buildings, the
nth
undamped modal displacement responses, Ua
nand Ub
n, can be redefined as
Ua
n(t)
Ub
n(t)=Ta
n0
0Tb
nDa
n(t)
Db
n(t)(6)
where
Da
n
and
Db
n
are the
nth
generalized modal co-ordinate of both buildings. By substi-
tuting Equation (6) and rearranging each side of the equation of motion by
Ta
n0
0Tb
nT
,
we obtain
Ma
n0
0Mb
n"..
Da
n(t)
..
Db
n(t)#+Ca
n0
0Cb
n".
Da
n(t)
.
Db
n(t)#+Ka
n0
0Kb
nDa
n(t)
Db
n(t)+"Fap
n(t)
−Fbp
n(t)#
=−
Ma
nιΓa
xn
..
xg+Γa
yn
..
yg
Mb
nιΓb
xn
..
xg+Γb
yn
..
yg
(7)
where
Ma
n=Ta
nTMaTa
n
,
Ca
n=Ta
nTCaTa
n
,
Ka
n=Ta
nTKaTa
n
,
Mb
n=Tb
nTMbTb
n
,
Cb
n=Tb
nTCbTb
n
,
and
Kb
n=Tb
nTKbTb
n
are sized 8
×
8;
ι
is a vector sized as an 8
×
1 column with all elements
the same as unity; the 3
N+
5 and 3
S+
5 multi-DOF modal equations of motion, as given
in Equation (7) for both buildings, comprise a nonproportionally damped system.
Ca=Ta
nT(αMa+βKa)Ta
n=αMa
n+βKa
n
Cb=Tb
nTαMb+βKbTb
n=αMb
n+βKb
n
(8)
Cruz and Miranda [
52
] found that the Rayleigh damping model underestimates the
damping of higher modes that contribute to the seismic response, resulting in an over-
estimation of the seismic response. This manuscript does not address the validity of
the mass-proportional and stiffness-proportional assumptions in Rayleigh damping. The
approximation approach and rigorous methods are contrasted in the first three modes
of this study. If
Ca
and
Cb
are proportionally damped, i.e.,
Ca=(αMa+βKa)
and
Cb=αMb+βKb
, the
nth
modal damping matrices for each building can be described as
Equation (8). In here,
α
and
β
are coefficients found by the damping ratios of the two specific
modes, since the original SSI system is not proportionally damped, i.e.,
Ca6=(αMa+βKa)
,
Cb6=αMb+βKb
. In approximate method, Equation (8) converts
Ca6=αMa
n+βKa
n
and
Cb6=αMb
n+βKb
n
based on [
35
]. The nonlinear viscoelastic model is only modified for the
approximate method used in this study, as given in Equation (9). Notably, the elements of
Da
nand Db
nare not the same, even in an elastic state.
Appl. Sci. 2023,13, 3092 7 of 18
Fp
xijn(t)=0 for δijxn(t)≤0;
Fp
xijn(t)=βδijxn(t)3/2 +cijn (t).
δijxn(t)for δijx n(t)>0 and .
δijxn(t)>0;
Fp
xijn(t)=βδijxn(t)3/2 for δijxn(t)>0 and .
δijxn(t)≤0;
δijxn(t)=δijyn (t)=Da
xn ϕa
ixn (t)−Db
xn ϕb
jxn (t)−D;δijθn(t)=Da
θnϕa
iθn(t)fa−
Db
θnϕb
jθn(t)fb−D;
.
δijxn(t)=.
Da
xn ϕa
ixn (t)−.
Db
xn ϕb
jxn (t);.
δijyn(t)=.
Da
yn ϕa
iyn (t)−.
Db
yn ϕb
jyn(t);
.
δijθn(t)=.
Da
θnϕa
iθn(t)fa−.
Db
θnϕb
jθn(t)fb;
Fap
ijn =hFa p
xijn Fap
yijn Fap
θijn 00000iT
(3N+5)×1
(9)
Here
,
δij (t)
is the total displacement between the buildings with respect to the foun-
dation;
.
δij (t)
is their velocity;
β
is the damping; and
cij (t)
is the stiffness of the impact
element. The damping ratio
ξ
in relation to the coefficient of restitution
(e)
provides the
dissipation of energy during impact [
53
].
D
is the distance between the buildings.
Fap
θijn
can
be calculated as
Fap
xijn
using the related
δijθn(t)
and
.
δijθn(t)
in Equation (9).
Fap
n(t)=Ta
nTFap
ijn
in Equation (7) is an 8
×
1 vector with pounding forces. As a result, the approximate
method substantially reduces the size of the matrices from 3 N+ 5 to 8 for each building.
The modal responses with regard to the displacement of each building,
Da
n(t)
and
Db
n(t)
,
are derived using the method of direct integration in Equation (7). The following equation
is derived to obtain all the responses of the two nonproportionally damped asymmetric
buildings on the top of an elastic half-space:
Ua(t)=
3N+5
∑
n=1
Ua
n(t)≈
3N+5
∑
n=1
Ta
nDa
n(t);Ub(t)=
3S+5
∑
n=1
Ub
n(t)≈
3S+5
∑
n=1
Tb
nDb
n(t)(10)
The first few modal responses in Equation (10), to acquire a satisfying result, should
be summed similarly to the displacement analysis.
2.4. Equation of Motion for SSI System
Based on Figure 1, the equation of motion for the entire foundation system for Building
A can be derived from Equation (11) for the translation along the xand yaxes, rotation
around the zaxis, and rocking along the xand yaxes.
ma
o..
xg+..
xa
o+{1}T[Ma]n..
xt
ic o+Pxa(t)=0
ma
o..
yg+..
ya
o+{1}T[Ma]n..
yt
ic o+Pya(t)=0
r2
ama
o
..
θa
o+r2
a{1}T[Ma]..
θt
i+Ta(t)=0
N
∑
i=0Ixi
..
ψa
o+{hi}T[Ma]n..
yt
ic o+Qxa (t)=0
N
∑
i=0Iyi
..
φa
o+{hi}T[Ma]n..
xt
ic o+Qya (t)=0
(11)
The soil–structure interface has been modelled using a parallel set of frequency-
independent springs and dashpots considered in the study of Richart et al. [
30
].
Pxa
,
Pya
,
Appl. Sci. 2023,13, 3092 8 of 18
Ta
,
Qxa
, and
Qxa
in Equation (11) are the interaction forces of Building Aas given in
Equation (12).
Pxa(t)=CT
.
xa
o+KTxa
o
Pya (t)=CT
.
ya
o+KTya
o
Ta(t)=Cθ
.
θa
o+Kθθa
o
Qxa (t)=Cψ
.
.
ψ
a
o+Kψψa
o
Qya (t)=Cφ
.
.
φ
a
o+Kφφa
o
(12)
where
KT,θ,ψ,∅
and
CT,θ,ψ,∅
are the spring and dashpot coefficients of translations along
both x and y axes, the torsion and rocking movements along both xand yaxes, respectively.
These constants of the static impedance functions are shown in Table 1with many subscripts,
as seen in Equation (12) [30].
Table 1. Spring and dashpot constants used by Richart et al. [30].
Sliding Torsion Rocking
Spring KT=32(1−υ)Gro
7−8υKθ=16Gr3
o
3Kψ,φ=8Gr3
o
3(1−υ)
Mass Ratio BT=(7−8υ)MT
32(1−υ)ρr3
oBθ=Iθ
ρr5
oBψ,φ=3(1−υ)Iψ,φ
8ρr5
o
Damping
Ratio DT=0.288
√BTDθ=0.5
1+2BθDψ,φ=0.15
(1+Bψ,φ)√Bψ,φ
Coefficient CT=2DT√KTMTCθ=2Dθ√KθIθCψ,φ=2Dψ,φpKψ,φIψ,φ
Where
MT
,
Iθ
,
and Iψ,φ
are the total mass, polar moment of inertia, and moment of
inertia of the rigid body for rocking, respectively.
G
,
ρ
,
υ
, and
νs
are the shear modulus, mass
density of half-space, Poisson’s ratio, and shear velocity of the elastic medium, respectively.
rois the radius of the massless disc on the surface of an elastic homogeneous half-space.
3. Numerical Study
In this study, five- and four-story asymmetric buildings are placed on an elastic half-
space as Buildings Aand B, respectively.
3.1. Structure Properties
Buildings Aand Bhad the dimensions of 20
m
by 15
m
and 25
m
by 20
m
, respectively,
with the longer lengths in the longitudinal direction (
x
) for each building. The ratio of the
foundation mass to the floor mass was 3 for each building. Each story in each building was
2.85
m
high. Table 2provides the basic values describing the structural characteristics that
have been used for this study.
Table 2. Building details [31,51].
Story No.
Story Height
hi,hj
(m)
Building ABuilding B
mi×106
(kg)
ki×108
(N/m)
mj×106
(m)
kj×108
(N/m)
1F 2.85 0.30 3.46 0.4065 5.06
2F 5.7 0.30 3.46 0.4065 3.86
3F 8.55 0.30 3.46 0.4065 3.86
4F 11.4 0.30 3.46 0.4065 3.86
5F 14.25 0.30 3.46 - -
Based on the reported results [
34
,
54
,
55
], we used
β
= 2.75
×
10
9N/m3/2
and
ξ
= 0.35
for the pounding force parameters in the nonlinear viscoelastic model, with an established
coefficient of friction
µf
of 0.5. In Equation (13), the translational and torsional stiffness
Appl. Sci. 2023,13, 3092 9 of 18
at the center of mass for each story of a building are longitudinally in proportion to the
stiffness of the same story [56].
βy=kyi
kxi
=kyj
kxj
;βt=kθi
r2
akxi
=kθj
r2
bkxj
(13)
The values of
βy
and
βt
were 1.32 and 1.69 for both buildings, respectively. Both
buildings were characterized by 2% of the critical damping as a constant of propor-
tionality (
α
) in the first vibration mode. Poisson’s ratio (
υ
) was 0.333, and the density
of the soil (
ρ
) was 1922
kg/m3
. Specifically, four types of soil at shear wave velocities
ranging from 65, 130, 200, to 300
m/s
as Case I to IV have been examined, respectively.
These ranges are described as soft to hard soil, based on Abdel Raheem, et al. [
57
] and
Sulistiawan et al. [
58
]. Soft soil (Case I) indicates the large SSI effect on the building’s
response, while hard soil (Case IV) indicates the small SSI effect. The original distance,
D
, between the buildings was 0.04
m
. The selection of this value has been based on the
previous studies [
48
,
59
,
60
]. A rigorous method using the direct integration method has
been applied to compute the equation of motion for the responses of the SSI system of each
building shown in Equation (1).
3.2. Response Analysis
The results have been compared with those reported by Balendra et al. [
42
], Sivaku-
maran and Balendra [
23
], and Jui-Liang et al. [
35
], who did not address the effect of
pounding. As the reference building (RB) in this study, another four-story building without
considering SSI has been chosen.
The first six natural frequencies of the eight-story building-foundation system used
by Balendra et al. [
42
] are shown in Table 3. As can be seen from the table, the x-direction
displacement component is dominant in the first and fourth modes, while the y-direction
displacement component is dominant in the second and fifth modes. The major component
in the third and sixth modes corresponds to the rotation around the vertical axis. Building B
has a modal response that is less than that of Building A, but with the same trend; therefore,
the results of Building B are not provided here. Figure 4shows the mode shapes of Building
A placed on rigid ground as well as the mode shapes of each case using the approximate
method. The thin lines in Figure 4represent the translations and rotations of the base, as
shown by the offsets and slopes. In Figure 4, the first to third mode shapes of the dominant
motions are the
x
- and
y
-translation and rotation around the vertical axis for both the RB
without the effects of SSI and for all cases.
Table 3.
First six natural frequencies of an eight-story building-foundation system considered in [
38
]
compared with those in the current study.
Cases
Frequency Case I Case I * Case IV Case IV * RB RB * Case I *
RB * Case IV *
RB *
W1(Hz) 0.698 0.685 0.788 0.787
0.794
0.792 0.865 0.994
W2(Hz) 0.783 0.787 0.932 0.931
0.941
0.940 0.837 0.990
W3(Hz) 1.120 1.107 1.224 1.222
1.233
1.231 0.899 0.993
W4(Hz) 1.896 1.874 1.942 1.937
1.943
1.941 0.965 0.998
W5(Hz) 2.227 2.175 2.301 2.297
2.306
2.303 0.944 0.997
W6(Hz) 2.905 2.795 3.013 3.007
3.019
3.015 0.927 0.997
* Obtained by current studies; RB: reference building.
Appl. Sci. 2023,13, 3092 10 of 18
Appl. Sci. 2023, 13, 3092 10 of 19
Figure 4. Building A in the first to third modes without considering SSI effects and for all cases.
However, the third mode shape in Case I is characterized by a rocking movement in
the -direction. In Cases I to IV, the first to third modes of shapes are consistent with
those of the reference building. Figure 5 shows that the modal dislocation–time interac-
tions for each of the eight DOFs for Case I are different.
Figure 5. First to third modes during soft and hard soil earthquakes used in this study.
Figure 4. Building Ain the first to third modes without considering SSI effects and for all cases.
However, the third mode shape in Case I is characterized by a rocking movement in
the
y
-direction. In Cases I to IV, the first to third modes of shapes are consistent with those
of the reference building. Figure 5shows that the modal dislocation–time interactions for
each of the eight DOFs for Case I are different.
Appl. Sci. 2023, 13, 3092 10 of 19
Figure 4. Building A in the first to third modes without considering SSI effects and for all cases.
However, the third mode shape in Case I is characterized by a rocking movement in
the -direction. In Cases I to IV, the first to third modes of shapes are consistent with
those of the reference building. Figure 5 shows that the modal dislocation–time interac-
tions for each of the eight DOFs for Case I are different.
Figure 5. First to third modes during soft and hard soil earthquakes used in this study.
Figure 5. First to third modes during soft and hard soil earthquakes used in this study.
In Case IV, the modal responses for the eight DOFs display a similar pattern to those
of the second and third modes. Case I shows that the modal responses for the first three
modes are mainly affected by SSI effects, and they are out of phase. Figure 6shows the
response of the first mode in Case IV without any pounding.
Appl. Sci. 2023,13, 3092 11 of 18
Appl. Sci. 2023, 13, 3092 11 of 19
In Case IV, the modal responses for the eight DOFs display a similar pattern to those
of the second and third modes. Case I shows that the modal responses for the first three
modes are mainly affected by SSI effects, and they are out of phase. Figure 6 shows the
response of the first mode in Case IV without any pounding.
Figure 6. First-mode responses for El Centro earthquake in 1940 and Kobe earthquake in 1995
without pounding.
As shown in Figure 6, when the necessary distance is present between the buildings
to avoid pounding, the modal responses of the eight DOFs for each vibration mode are
similar to those in Case IV. Figure 7 illustrates the response histories of Cases I and IV
obtained with the proposed method under the 1995 Kobe earthquake. The dashed line in
Figure 7 denotes the approximate method (App.) given by Equation (7), whereas the
solid line denotes the rigorous method (Rig.) obtained by Equation (3).
Figure 6.
First-mode responses for El Centro earthquake in 1940 and Kobe earthquake in 1995
without pounding.
As shown in Figure 6, when the necessary distance is present between the buildings
to avoid pounding, the modal responses of the eight DOFs for each vibration mode are
similar to those in Case IV. Figure 7illustrates the response histories of Cases I and IV
obtained with the proposed method under the 1995 Kobe earthquake. The dashed line in
Figure 7denotes the approximate method (App.) given by Equation (7), whereas the solid
line denotes the rigorous method (Rig.) obtained by Equation (3).
Appl. Sci. 2023, 13, 3092 12 of 19
Figure 7. Results for both approximate and rigorous solutions on the fourth floor for the 1995 Kobe
earthquake.
For Cases I and II, excellent agreement between the peak and phase responses for
both methods has been found for the selected earthquakes. Notably, both methods pro-
duced slightly different response histories for the fourth story due to the larger forces
under the small SSI effects, as shown in Figure 7. As a result of the high pounding forces
in the first through third modes, the responses at the foundation obtained by the rigorous
method in Figure 8 are not fully in agreement with those achieved by the approximate
method in Case IV. Based on the comparison between the conventional rigorous method
and the approximate method, it could be concluded that the approximate method
markedly improves the accuracy of the analytical results without increasing the compu-
tational effort and by reducing the dimensions of the matrix.
Figure 7.
Results for both approximate and rigorous solutions on the fourth floor for the 1995
Kobe earthquake.
Appl. Sci. 2023,13, 3092 12 of 18
For Cases I and II, excellent agreement between the peak and phase responses for both
methods has been found for the selected earthquakes. Notably, both methods produced
slightly different response histories for the fourth story due to the larger forces under
the small SSI effects, as shown in Figure 7. As a result of the high pounding forces in
the first through third modes, the responses at the foundation obtained by the rigorous
method in Figure 8are not fully in agreement with those achieved by the approximate
method in Case IV. Based on the comparison between the conventional rigorous method
and the approximate method, it could be concluded that the approximate method markedly
improves the accuracy of the analytical results without increasing the computational effort
and by reducing the dimensions of the matrix.
Appl. Sci. 2023, 13, 3092 13 of 19
Figure 8. Time histories of Cases I and IV for the foundation during the 1995 Kobe earthquake.
Regarding the large SSI effect (Case I), both methods produced the same results for
the 1995 Kobe earthquake in terms of translation, twisting, and rocking. Figure 9 com-
pares both methods based on the story shears and torque with and without pounding.
Figure 9 shows the maximum story shears in the x- and y-directions, together with the
maximum story torque in relation to the upright direction for shear waves traveling at 65
and 300 m s
⁄. This is based on simulations by Balendra et al. [42,43]. For validation pur-
poses, these two methods used in this study are compared with the findings of Balendra
et al. [42,43] without considering the pounding effect.
Figure 8. Time histories of Cases I and IV for the foundation during the 1995 Kobe earthquake.
Regarding the large SSI effect (Case I), both methods produced the same results for
the 1995 Kobe earthquake in terms of translation, twisting, and rocking. Figure 9compares
both methods based on the story shears and torque with and without pounding. Figure 9
shows the maximum story shears in the x- and y-directions, together with the maximum
Appl. Sci. 2023,13, 3092 13 of 18
story torque in relation to the upright direction for shear waves traveling at 65 and
300 m/s
.
This is based on simulations by Balendra et al. [
42
,
43
]. For validation purposes, these
two methods used in this study are compared with the findings of
Balendra et al. [42,43]
without considering the pounding effect.
Appl. Sci. 2023, 13, 3092 14 of 19
Figure 9. Maximum shear and torque of an eight-story building in Cases I and IV for the effects of
pounding during the 1940 El Centro earthquake [42,43].
Based on the results for the large SSI case, the approximate and rigorous methods
provided in Figure 9 for adjacent buildings are completely in agreement. Figure 9 also
shows that, in Case IV, the maximum responses to the story shear and torque are reduced
due to reduced pounding.
Abdel Raheem [61] has studied the bi-directional excitation and biaxial interaction
of the base isolation system, although there is still debate on the base isolation system’s
behavior under the SSI and hammering impacts. Two approaches utilized in this study
will be applied to base-isolated buildings on various soil types, taking impact effects into
account.
4. Conclusions
In this study, the seismic behavior of multistory asymmetric adjacent buildings by
considering SSI and structural pounding has been examined. The effects of the pounding
force on the adjacent structures in a simplified model have been considered. The parallel
set of frequency-independent springs and dashpots to simulate the interaction forces at
the SSI has been applied. The fourth-order Runge–Kutta differential equations for
two-way asymmetric shear buildings have been derived, which have been solved both
with and without impacts. The approximate method using multi-DOF modal equations
of motion and the rigorous method using direct integration have been compared.
The SSI has been incorporated to generate a set of modal equations of motion for the
pounding responses for each building based on the frequency-independent equation of
motion. The results of this study indicated that pounding detrimentally affects the dy-
namic properties of a building. Because the shear wave velocity is very low, the ap-
proximate method produces the same impact force as the rigorous method. However,
these findings are not valid for small SSI effects due to the increased number of collisions.
The response to small SSI effects at the foundation is less than that to large SSI effects. The
top floor deformations of adjacent buildings are somewhat conservative at a high shear
wave velocity. Finally, buildings are considerably affected by increased shear wave ve-
locity. All the vibration modes, rather than only the first few vibration modes, should be
considered to achieve satisfactory results.
The simultaneous effects of SSI and the pounding of adjacent buildings under seis-
mic loading have not been assessed so far by the use of the approximate and rigorous
method. Both methods only evaluated the cases showing the SSI effect in the past. By
comparing the SSI and pounding effects in these two techniques, the originality of this
work has been demonstrated. It is vital to simulate colliding structures as inelastically as
feasible in order to limit the consequences of pounding between buildings. By increasing
the shear wave velocity, the responses, based on the deformation vectors for each struc-
Figure 9.
Maximum shear and torque of an eight-story building in Cases I and IV for the effects of
pounding during the 1940 El Centro earthquake [42,43].
Based on the results for the large SSI case, the approximate and rigorous methods
provided in Figure 9for adjacent buildings are completely in agreement. Figure 9also
shows that, in Case IV, the maximum responses to the story shear and torque are reduced
due to reduced pounding.
Abdel Raheem [
61
] has studied the bi-directional excitation and biaxial interaction
of the base isolation system, although there is still debate on the base isolation system’s
behavior under the SSI and hammering impacts. Two approaches utilized in this study
will be applied to base-isolated buildings on various soil types, taking impact effects
into account.
4. Conclusions
In this study, the seismic behavior of multistory asymmetric adjacent buildings by
considering SSI and structural pounding has been examined. The effects of the pounding
force on the adjacent structures in a simplified model have been considered. The parallel
set of frequency-independent springs and dashpots to simulate the interaction forces at the
SSI has been applied. The fourth-order Runge–Kutta differential equations for two-way
asymmetric shear buildings have been derived, which have been solved both with and
without impacts. The approximate method using multi-DOF modal equations of motion
and the rigorous method using direct integration have been compared.
The SSI has been incorporated to generate a set of modal equations of motion for the
pounding responses for each building based on the frequency-independent equation of
motion. The results of this study indicated that pounding detrimentally affects the dynamic
properties of a building. Because the shear wave velocity is very low, the approximate
method produces the same impact force as the rigorous method. However, these findings
are not valid for small SSI effects due to the increased number of collisions. The response
to small SSI effects at the foundation is less than that to large SSI effects. The top floor
deformations of adjacent buildings are somewhat conservative at a high shear wave velocity.
Finally, buildings are considerably affected by increased shear wave velocity. All the
vibration modes, rather than only the first few vibration modes, should be considered to
achieve satisfactory results.
The simultaneous effects of SSI and the pounding of adjacent buildings under seismic
loading have not been assessed so far by the use of the approximate and rigorous method.
Appl. Sci. 2023,13, 3092 14 of 18
Both methods only evaluated the cases showing the SSI effect in the past. By comparing
the SSI and pounding effects in these two techniques, the originality of this work has
been demonstrated. It is vital to simulate colliding structures as inelastically as feasible in
order to limit the consequences of pounding between buildings. By increasing the shear
wave velocity, the responses, based on the deformation vectors for each structure, are
drastically decreased while the SSI forces at the building’s foundation are enhanced. In
order to determine the efficacy of these approaches in different circumstances, further
studies should compare them to standard isolated buildings using a sensitivity analysis.
Author Contributions:
Conceptualization, M.E.U.; methodology, M.E.U., A.J.-G. and R.J.; software,
M.E.U.; validation, M.E.U.; formal analysis, M.E.U.; investigation, M.E.U., A.J.-G. and R.J.;
writing—original draft preparation, M.E.U.; writing—review and editing, A.J.-G. and R.J. All authors
have read and agreed to the published version of the manuscript.
Funding:
The authors declare that no funds, grants, or other support was received during the
preparation of this manuscript.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
The datasets generated and analyzed during the current study are
available upon request.
Acknowledgments:
The authors would like to thank Muhammad N.S. Hadi for help in the interpreta-
tion of the databases and for providing a working space at the University of Wollongong, Australia.
Conflicts of Interest: The authors have no relevant financial or nonfinancial interest to disclose.
Notations
Asymbolized Building A
a0dimensionless frequency
Bsymbolized Building B
BT,Bψ,φ,Bθmass ratios for sliding, rocking, and torsion
Ca,Cbdamping matrices of the SSI system for each building
Cax ,Cay N×Nsubmatrices of damping in x- and y-axis for Building A
cxi
,
cyi ith floor damping coefficient in the longitudinal and transverse directions for
Building A
cxj
,
cyj jth floor damping coefficient in the longitudinal and transverse directions for
Building B
cij damping of impact element
CM, CR center of mass and resistance
Ca
θR,Ca
θMtorsional damping matrices of Building Ain relation to CR and CM
CT,Cψ,φ,Cθdamping of soil dashpots
Ddistance between buildings
DT,Dψ,φ,Dθdamping ratio of soil dashpots
ecoefficient of restitution
ea,ebeccentricity in x-direction of Buildings Aand B
{ea},{eb}N×1 column vector with all elements equal to ea,eb
Fp(t)pounding force vector
Fp
xij ,Fp
yij ,Fp
θij
pounding force influence coefficient vectors in longitudinal, transverse, and
vertical directions
Fa(t),Fb(t)shear force matrices of SSI system for each building
Fxi,Fyi,Fθishear force of ith floor
Fy
xi,Fy
yi yield strength of ith floor
fa,fbeccentricity in y-direction of Buildings Aand B
{fa},{fb}N×1 column vector with all elements equal to fa,fb
Appl. Sci. 2023,13, 3092 15 of 18
fmax maximum wave frequency
Gshear modulus of soil
hi,hjheight of ith and jth floor level
{hi},nhjocolumn vector composed of story heights
Ixi,Iyi,Ix j ,Iyj moments of inertia of ith and jth floors
Ia
x0,Ia
y0,Ib
x0,Ib
y0moments of inertia of the base for each building
Iθ,Iψ,φpolar moment of inertia and moment of inertia for rocking of each building
Ka,Kbstiffness matrices of SSI system for each building
Kax ,Kay N×Nsubmatrices of stiffness in x- and y-axis for Building A
kxi
,
kyi ith floor stiffness coefficient in longitudinal and transverse directions for
Building A
kxj
,
kyj jth floor stiffness coefficient in longitudinal and transverse directions for
Building B
Ka
θR,Ka
θMtorsional stiffness matrices of Building Ain relation to CR and CM
KT,Kψ,φ,Kθstiffness of soil springs
Ma,Mbgeneralized mass matrix of SSI systems of each building
MTmass of related building
Mamass matrix of superstructure
mimass at ith floor of Building A
mjmass at jth floor of Building B
ma
0,mb
0foundation masse
Nnumber of stories of Building A
Pa(t),Pb(t)loading vector of SSI system
Pax,Pay,Pbx,Pby SSI forces
Qxa ,Qya ,Qxb ,Qyb SSI moments
ra,rbradius of gyration in relation to mass center
r0radius of a circle having same area as building plan
Snumber of stories of Building B
Ta,TbSSI torques of buildings
ttime variable
Ua,Ubdeformation vector of Buildings Aand B
Vsshear wave velocity of soil
x0
ic,y0
ic,x0
jc,y0
jc x- and y-directional displacement vectors of buildings with SSI effects
xic,yic,xj c,yjc x- and y-directional displacement vectors of buildings without SSI effects
xt
ic,yt
ic,xt
jc,yt
jc total displacement vectors of center mass of floors
xa
0,ya
0,xb
0,yb
0x- and y-directional displacements of foundations of buildings
..
xg,..
ygground acceleration records
αconstant for determining classical damping
βimpact stiffness parameter
θt
i,θic rotational vector of building with and without SSI effects
δij relative displacement influence coefficient with respect to ground
.
δij relative velocity influence coefficient with respect to ground
∆ttime step
βdamping ratio related to e
µffriction coefficient during collision
ωfcircular frequency of applied excitation
{1}N×1 or S×1 column vector with all elements equal to 1
0vector of zeros
References
1.
Zembaty, Z.; Cholewicki, A.; Jankowski, R.; Szulc, J. Trz˛esienia ziemi 21 wrze´snia 2004 r. w Polsce północno-wschodniej oraz
ich wpływ na obiekty budowlane (Earthquakes of September 21, 2004 in north-eastern Poland and their effects on structures).
In˙
zynieria I Bud. 2005,61, 3–9. (In Polish)
2.
Zembaty, Z.; Jankowski, R.; Cholewicki, A.; Szulc, J. Trz˛esienie ziemi 30 listopada 2004 r. na Podhalu oraz jego wpływ na obiekty
budowlane (Earthquake of November 30, 2004 in Podhale and its effects on structures). In˙
zynieria I Bud.
2005
,61, 507–511.
(In Polish)
Appl. Sci. 2023,13, 3092 16 of 18
3.
Kasai, K.; Maison, B.F. Building pounding damage during the 1989 Loma Prieta earthquake. Eng. Struct.
1997
,19, 195–207.
[CrossRef]
4. Vasiliadis, L.; Elenas, A. Performance of school buildings during the Athens earthquake of 7 September 1999. In Proceedings of
the 12th European Conference on Earthquake Engineering, London, UK, 9–13 September 2002.
5.
Kamgar, R.; Tavakoli, R.; Rahgozar, P.; Jankowski, R. Application of discrete wavelet transform in seismic nonlinear analysis of
soil–structure interaction problems. Earthq. Spectra 2021,37, 1980–2012. [CrossRef]
6.
Karafagka, S.; Fotopoulou, S.; Pitilakis, D. Fragility assessment of non-ductile RC frame buildings exposed to combined ground
shaking and soil liquefaction considering SSI. Eng. Struct. 2021,229, 111629. [CrossRef]
7.
Duan, W.; Congress, S.S.C.; Cai, G.; Liu, S.; Dong, X.; Chen, R.; Liu, X. A hybrid GMDH neural network and logistic regression
framework for state parameter–based liquefaction evaluation. Can. Geotech. J. 2021,99, 1801–1811. [CrossRef]
8.
Özcebe, A.G.; Giretti, D.; Bozzoni, F.; Fioravante, V.; Lai, C.G. Centrifuge and numerical modelling of earthquake-induced soil
liquefaction under free-field conditions and by considering soil-structure interaction. Bull. Earthq. Eng.
2021
,19, 47–75. [CrossRef]
9.
Duan, W.; Zhao, Z.; Cai, G.; Pu, S.; Liu, S.; Dong, X. Evaluating model uncertainty of an in situ state parameter-based simplified
method for reliability analysis of liquefaction potential. Comput. Geotech. 2022,151, 104957. [CrossRef]
10.
Madabhushi, G.S.P.; Garcia-Torres, S. Sustainable measures for protection of structures against earthquake induced liquefaction.
Indian Geotech. J. 2021,51, 467–481. [CrossRef]
11.
Anagnostopoulos, S.A. Pounding of buildings in series during earthquakes. Earthq. Eng. Struct. Dyn.
1988
,16, 443–456. [CrossRef]
12.
Flenga, M.G.; Favvata, M.J. Fragility Curves and Probabilistic Seismic Demand Models on the Seismic Assessment of RC Frames
Subjected to Structural Pounding. Appl. Sci. 2021,11, 8253. [CrossRef]
13.
Kazemi, F.; Mohebi, B.; Jankowski, R. Predicting the seismic collapse capacity of adjacent SMRFs retrofitted with fluid viscous
dampers in pounding condition. Mech. Syst. Signal Process. 2021,161, 107939. [CrossRef]
14.
Miari, M.; Jankowski, R. Incremental dynamic analysis and fragility assessment of buildings founded on different soil types
experiencing structural pounding during earthquakes. Eng. Struct. 2022,252, 113118. [CrossRef]
15.
Polycarpou, P.C.; Papaloizou, L.; Komodromos, P. An efficient methodology for simulating earthquake-induced 3D pounding of
buildings. Earthq. Eng. Struct. Dyn. 2014,43, 985–1003. [CrossRef]
16.
Chau, K.T.; Wei, X.X. Pounding of structures modelled as non-linear impacts of two oscillators. Earthq. Eng. Struct. Dyn.
2001
,
30, 633–651. [CrossRef]
17.
Favvata Maria, J.; Karayannis Chris, G.; Liolios Asterios, A. Influence of exterior joint effect on the inter-story pounding interaction
of structures. Struct. Eng. Mech. 2009,33, 113–136. [CrossRef]
18.
Jing, H.-S.; Young, M. Impact interactions between two vibration systems under random excitation. Earthq. Eng. Struct. Dyn.
1991,20, 667–681. [CrossRef]
19.
Kazemi, F.; Miari, M.; Jankowski, R. Investigating the effects of structural pounding on the seismic performance of adjacent RC
and steel MRFs. Bull. Earthq. Eng. 2021,19, 317–343. [CrossRef]
20.
Khatami, S.M.; Naderpour, H.; Barros, R.C.; Jakubczyk-Gałczy´nska, A.; Jankowski, R. Determination of Peak Impact Force for
Buildings Exposed to Structural Pounding during Earthquakes. Geosciences 2020,10, 18. [CrossRef]
21. Balendra, T. A simplified model for lateral load analysis of asymmetrical buildings. Eng. Struct. 1983,5, 154–162. [CrossRef]
22.
Novak, M.; Hifnawy, L.E. Effect of soil-structure interaction on damping of structures. Earthq. Eng. Struct. Dyn.
1983
,11, 595–621.
[CrossRef]
23.
Sivakumaran, K.S.; Balendra, T. Seismic analysis of asymmetric multistorey buildings including foundation interaction and P-
∆
effects. Eng. Struct. 1994,16, 609–624. [CrossRef]
24.
Uz, M.E.; Hadi, M.N.S. Optimal design of semi active control for adjacent buildings connected by MR damper based on integrated
fuzzy logic and multi-objective genetic algorithm. Eng. Struct. 2014,69, 135–148. [CrossRef]
25.
Hadi, M.N.S.; Uz, M.E. Investigating the optimal passive and active vibration controls of adjacent buildings based on performance
indices using genetic algorithms. Eng. Optim. 2015,47, 265–286. [CrossRef]
26.
Miari, M.; Choong, K.K.; Jankowski, R. Seismic Pounding Between Bridge Segments: A State-of-the-Art Review. Arch. Comput.
Methods Eng. 2021,28, 495–504. [CrossRef]
27.
Chopra, A.K.; Gutierrez, J.A. Earthquake response analysis of multistorey buildings including foundation interaction. Earthq.
Eng. Struct. Dyn. 1974,3, 65–77. [CrossRef]
28.
Gupta, V.K.; Trifunac, M.D. Seismic response of multistoried buildings including the effects of soil-structure interaction. Soil Dyn.
Earthq. Eng. 1991,10, 414–422. [CrossRef]
29.
Veletsos, A.S.; Meek, J.W. Dynamic behaviour of building-foundation systems. Earthq. Eng. Struct. Dyn.
1974
,3, 121–138.
[CrossRef]
30. Richart, F.E.; Hall, J.R.; Woods, R.D. Vibrations of Soils and Foundations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1970.
31.
Goel, R.K. Simplified analysis of asymmetric structures with supplemental damping. Earthq. Eng. Struct. Dyn.
2001
,30, 1399–1416.
[CrossRef]
32.
Lin, J.-L.; Tsai, K.-C. Simplified seismic analysis of one-way asymmetric elastic systems with supplemental damping. Earthquake
Eng. Struct. Dyn. 2007,36, 783–800. [CrossRef]
Appl. Sci. 2023,13, 3092 17 of 18
33.
Uz, M.; Hadi, M.N. Investigating the effects of pouding for inelastic base isolated adjacent buildings under earthquake excitations.
In Incorporating Sustainable Practice in Mechanics of Structures and Materials; Fragomeni, S., Venkatesan, S., Lam, N., Setunge, S.,
Eds.; CRC Press: Leiden, The Netherlands, 2011; pp. 329–334.
34. Jankowski, R.; Mahmoud, S. Earthquake-Induced Structural Pounding; Springer: Cham, Switzerland, 2015.
35.
Lin, J.-L.; Tsai, K.-C.; Miranda, E. Seismic History Analysis of Asymmetric Buildings with Soil–Structure Interaction. J. Struct.
Eng. 2009,135, 101–112. [CrossRef]
36.
Migda, W.; Szczepa´nski, M.; Lasowicz, N.; Jakubczyk-Gałczy ´nska, A.; Jankowski, R. Non-Linear Analysis of Inter-Story Pounding
between Wood-Framed Buildings during Ground Motion. Geosciences 2019,9, 488. [CrossRef]
37.
Favvata, M.J. Minimum required separation gap for adjacent RC frames with potential inter-story seismic pounding. Eng. Struct.
2017,152, 643–659. [CrossRef]
38.
Abdel Raheem, S.E. Mitigation measures for earthquake induced pounding effects on seismic performance of adjacent buildings.
Bull. Earthq. Eng. 2014,12, 1705–1724. [CrossRef]
39.
Raheem, S.E.A.; Alazrak, T.M.A.; Shafy, A.G.A.A.; Ahmed, M.M.; Gamal, Y.A.S. Seismic pounding between adjacent buildings
considering soil-structure interaction. Earthq. Struct. 2021,20, 55–70. [CrossRef]
40.
Abdel Raheem, S.; Fooly, M.; Shafy, A.; Abbas, Y.; Omar, M.; Latif, M.; Mahmoud, S. Seismic Pounding Effects on Adjacent
Buildings in Series with Different Alignment Configurations. Steel Compos. Struct. 2018,28, 289–308. [CrossRef]
41.
Abdel Raheem, S.; Fooly, M.; Omar, M.; Abdel Zaher, A. Seismic pounding effects on the adjacent symmetric buildings with
eccentric alignment. Earthq. Struct. 2019,16, 715–726. [CrossRef]
42.
Balendra, T.; Tat, C.W.; Lee, S.-L. Modal damping for torsionally coupled buildings on elastic foundation. Earthq. Eng. Struct.
Dyn. 1982,10, 735–756. [CrossRef]
43.
Balendra, T.; Tat, C.W.; Lee, S.L. Vibration of Asymmetrical Building‐Foundation Systems. J. Eng. Mech.
1983
,
109, 430–449. [CrossRef]
44.
Flenga, M.G.; Favvata, M.J. Probabilistic seismic assessment of the pounding risk based on the local demands of a multistory RC
frame structure. Eng. Struct. 2021,245, 112789. [CrossRef]
45.
Jankowski, R. Impact Force Spectrum for Damage Assessment of Earthquake-Induced Structural Pounding. Key Eng. Mater.
2005
,
293–294, 711–718. [CrossRef]
46. Chopra, A.K. Dynamics of Structures; Prentice-Hall: Englewood Cliffs, NJ, USA, 1995.
47. Clough, R.W.; Penzien, J. Dynamics of Structures, 2nd ed.; McGraw-Hill Book Co., Inc.: New York, NY, USA, 1993.
48.
Jankowski, R. Theoretical and experimental assessment of parameters for the non-linear viscoelastic model of structural pounding.
J. Theor. Appl. Mech. 2007,45, 931–942.
49.
Hosseini, S.H.; Naderpour, H.; Vahdani, R.; Jankowski, R. Evaluation of pounding effects between reinforced concrete frames
subjected to far-field earthquakes in terms of damage index. Bull. Earthq. Eng. 2022,20, 1219–1245. [CrossRef]
50.
Hadi, M.N.S.; Uz, M.E. Inelastic Base Isolated Adjacent Buildings under Earthquake Excitation with the Effect of Pounding. In
Proceedings of the 5th Civil Engineering Conference in the Asian Region and Australasian Structural Engineering Conference
CECAR 5/ASEC 2010, Sydney, Australia, 8–12 August 2010; pp. 155–201.
51.
Chopra, A.K.; Goel, R.K. A modal pushover analysis procedure to estimate seismic demands for unsymmetric-plan buildings.
Earthq. Eng. Struct. Dyn. 2004,33, 903–927. [CrossRef]
52. Cruz, C.; Miranda, E. Evaluation of the Rayleigh damping model for buildings. Eng. Struct. 2017,138, 324–336. [CrossRef]
53. Goldsmith, W. Impact: The Theory and Physical Behaviour of Colliding Solids; Edward Arnold: London, UK, 1960.
54.
Uz, M.E.; Hadi, M.N.S. Seismic history analysis of asymmetrical adjacent buildings with soil-structure interaction con-sideration.
In Proceedings of the 8th World Conference on Earthquake Resistant Engineering Structures, Chianciano, Italy, 7–9 September
2011; pp. 225–236.
55. Uz, M.E.; Hadi, M.N.S. Earthquake Resistant Design of Buildings, 1st ed.; CRC Press: Boca Raton, FL, USA, 2017. [CrossRef]
56.
Kan, C.L.; Chopra, A.K. Coupled Lateral Torsional Response of Buildings to Ground Motion; University of California: Berkeley,
CA, USA, 1976.
57.
Abdel Raheem, S.E.; Ahmed, M.M.; Alazrak, T.M.A. Evaluation of soil–foundation–structure interaction effects on seismic
response demands of multi-story MRF buildings on raft foundations. Int. J. Adv. Struct. Eng. 2015,7, 11–30. [CrossRef]
58.
Sulistiawan, H.; Supriyadi; Yulianti, I. Shear wave velocity profiling analysis for site classification using microtremor single
station method. AIP Conf. Proc. 2018,2014, 020003. [CrossRef]
59.
Khatami, S.M.; Naderpour, H.; Razavi, S.M.N.; Barros, R.C.; Sołtysik, B.; Jankowski, R. An ANN-Based Approach for Prediction of
Sufficient Seismic Gap between Adjacent Buildings Prone to Earthquake-Induced Pounding. Appl. Sci.
2020
,10, 3591. [CrossRef]
Appl. Sci. 2023,13, 3092 18 of 18
60.
Khatami, S.M.; Naderpour, H.; Barros, R.C.; Jankowski, R. Verification of Formulas for Periods of Adjacent Buildings Used to
Assess Minimum Separation Gap Preventing Structural Pounding during Earthquakes. Adv. Civ. Eng.
2019
,2019, 9714939.
[CrossRef]
61.
Abdel Raheem, S.E. Exploring Seismic Response of Bridges with Bidirectional Coupled Modelling of Base Isolation Bearings
System. Arab. J. Sci. Eng. 2014,39, 8669–8679. [CrossRef]
Disclaimer/Publisher’s Note:
The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
Available via license: CC BY 4.0
Content may be subject to copyright.
