Dynamic behaviour of double and triple adjacent 2D hills using boundary element methos

Dynamic behavior of double
and triple adjacent 2D hills
using boundary element
method
Mobin Afzalirad a,∗
, Mehran Naghizadehrokni b
, Iman Khosravi c
a
Civil Engineering Department, Islamic Azad University, Iran
b
Chair of Geotechnical Engineering, RWTH Aachen University, Germany
c
Islamic Azad University, Iran
∗
Corresponding author.
E-mail address: m.afzali@qaemiau.ac.ir (M. Afzalirad).
Abstract
A 2D boundary element method with material damping is developed in order to
seismic investigation of constructions with different configurations. The boundary
element algorithm, which utilized the time-convoluted kernels is employed in order
to incorporate proportional damping. In order to investigate the seismic reaction of
two dimensional double and triple semi-sine formed hills subjected to vertical P
and SV waves by using a developed viscoelastic boundary element algorithm and
responses are presented in the form of graphs, amplification pictures and
displacement. In addition, the influences of various damping proportions on site of
semi-sine hills with various shapes are analyzed. It can be concluded from the
results that the crest of homogeneous adjacent hills have similar shape ratio,
potential and larger maximum amplification in comparison with the crest of single
hills. Although this difference is increased by increasing shape ratio, it is negligible
within the proportions of study shape. In addition, multiplicity of hills increased
frequency characteristic of amplification curve and the number of peaks and valleys.
Keywords: Civil engineering, Mechanical engineering, Structural engineering,
Geophysics
Received:
3 August 2018
Revised:
17 October 2018
Accepted:
28 December 2018
Cite as: Mobin Afzalirad,
Mehran
Naghizadehrokni,
Iman Khosravi. Dynamic
behavior of double and triple
adjacent 2D hills using
boundary element method.
Heliyon 5 (2019) e01114.
doi: 10.1016/j.heliyon.2018.
e01114
https://doi.org/10.1016/j.heliyon.2018.e01114
2405-8440/Ó 2018 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction
The effect of local conditions of geology and soil have been investigated on the
severity of ground vibration and earthquake damage in recent years. Earth seismic
behavior is affected by different parameters: influence of source, propagation direc-
tion influence and site influence [1, 2, 3]. One of the well-known site effects is the
topography of the earth’s surface since the incoming waves field can be modified by
this surface topographical irregularity [4]. Previous numerical studies on seismic
behavior of topographic features were mainly limited to their single form. Whereas,
a realistic look at the surrounding nature shows the high diversity and complexity of
the ground surface and topographic features often appear in complex form. Among a
number of numerical studies conducted on the seismic behavior of hills, a consider-
able amount of studies is devoted to single semi-sine hills. The reason is that the
semi-sine hills are the most common form of natural topography of ground surface.
These studies showed that the hills can influence different points of the ground’s sur-
face with amplification of the motion component amplitude as well as the phase dif-
ference between adjacent points of the seismic response.
Observations by Trifunac, Hudson and Boore in San Fernando earthquake 1971 in
Pacoima dam that showed a maximum horizontal acceleration about 1.25g in the two
perpendicular directions which was recorded by seismograph located on the crest of
a rocky hill [5, 6, 7, 8] and Lee et al., after the Northridge earthquake in 1994 in Tar-
zana hills [9, 10], were the starting point of investigation. Then various numerical
and analytical studies were done to calculate the seismic behavior of natural topog-
raphy of the ground’s surface. Researchers found that one of the reasons for
recording high accelerations in Pacoima dam site at the time of the earthquake in
San Fernando 1971 [5, 6] and observations of the earthquake in Chile 1985 [11]
was topographic effects.
Davis and West considered the influence of topography for analyzing site. During
the experiment, which was conducted in the Nevada desert in 1968, accelerographs
installed in and out of Tonopah City showed different measurements, while Tonopah
was in a valley and accelerographs installed near the mountain. However, since the
procedure of placing accelerographs was not suitable for providing spectrum ampli-
fication values, attempts were done to consider the effect of topography for studying
site response [12].
Bard stated that there is a good agreement between seismic movements observed on
the top of mountains and at the bottom of valleys, respectively, with the results of
theoretical and numerical studies. He indicated that the ground vibration amplifica-
tion on the top of mountains is usually more for horizontal components than the ver-
tical ones that the horizontal component and vertical component are caused by S
waves and P waves. The difference between the two horizontal components is
2 https://doi.org/10.1016/j.heliyon.2018.e01114
Article Nowe01114

considerable. Bard showed that the maximum amplification is associated with a
sharpness of topography and the amplification will increase by growing the slope
of topography [13].
Kamalian et al. used time domain 2D BEM for analyzing response of 2D ridges and
valleys. In addition, different basic rules presented for the seismic design of struc-
tures with various 2D topographic structures. Although, they provide the findings
achieved by an extensive numerical study on seismic behavior of two dimensional
semi-sine formed hills, the formulation adopted was limited to a linear elastic me-
dium except for the effects of material damping on site response [14, 15, 16, 17,
18, 19, 20]. Later, Chaillat studied the seismic behavior of topographic features
by utilizing fast baoundary element method in frequency space and for visco-
elastic environments. They have accelerated the standard BEM by using fast multi-
pole method and have developed the basic elastodynamic responses based on this
method [21].
Sohrabi Bidar et al., presented elastodynamic 3D cores with analytical solutions of
convolution integrals of traction basic responses in two constant and linear changes.
Based on boundary element algorithm and 3D elastodynamic cores, they provided
BEMSA (boundary element method for 36 seismic analyses) software, which is
used for dynamic analysis of 3D topographic effects [22, 23].
Vincenzo assessed the site effect and its relationship with local topographical fea-
tures of a slope. A FEM is employed to calculate the seismic amplification. In this
paper, models are considered horizontally and unlimited at the slope’s crest and
foot. Input waves utilized in this investigation, is a SV seismic wave in different fre-
quency 0.5e32 Hz [24]. Moreover, Cavallaro et al., examined the effect of topog-
raphy on Monte Po hill where is located in an area with high seismic hazard.
They analyzed the hill behavior using one and two-dimensional method, and exam-
ined the results [25].
Review of technical literature, especially the aforementioned works indicates that no
comprehensive numerical studies have been done on seismic behavior of the com-
pound hills. Most researches are restricted to single hills and in the frequency space
and no studies were performed on time space. The principal object of this investiga-
tion is to discuss the findings of BEM on seismic behavior of double and triple adja-
cent 2D homogenous semi-sine hills with material damping.
For this purpose, the simplest combination of adjacent hills including double and tri-
ple semi-sine hills are considered. First, the seismic behavior of the double homog-
enous adjacent hills is examined and is compared with single hills; then the seismic
behavior of triple homogenous adjacent hills as well as the behavior in the valley
between two hills is considered. An attempt is made to analyze the results of numer-
ical sensitivity study and infer the initial answers to three key questions: how and to
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what extent can adjacency of double semi-sine adjacent hills increase the ampliﬁca-
tion potential of each of them? Can the behavior of valley between two adjacent hills
be equivalent to the seismic behavior of a single valley with the same shape? Can the
adjacency of hills easy interpret large amounts of ampliﬁcation ratios observed in
experimental studies for earlier researchers?
2. Methodology
2.1. Governing equations
The numerical research was implemented through developing a two-dimensional
visco-elastodynamics. The fundamental equation for a body may be presented as
Eq. (1):
À
c2
1 À c2
2
Á
$uj;ij þ c2
2$ui;jj þ bi À €ui ¼ 0 ð1Þ
In the above equation ui and bi are the movement vector the body force vector,
respectively and the propagation velocities of the compression and shear waves
can be introduced by c1 and c2, which are given by C2
1 ¼ ðl þ 2mÞ=r; C2
2 ¼
m=r, with l and m being the Lame constants and r the mass density.
The fundamental boundary integral equation for the elastic, isotropic and homoge-
neous half-plane exposed to an incoming in-plane movement (uinc:
j ), may be
achieved utilizing the weighted residual procedure as Eq. (2) [26]:
cijðsÞ:uiðs;tÞ ¼
Z
G

G*
ijðx;s;tÞ*tiðx;tÞ À H*
ijðx;s;tÞ*uiðx;tÞ

$dG þ uinc:
j ðs;tÞ ð2Þ
In which, Gij and Hij can be recognized as the transient movement and traction
fundamental solutions.
Integrating by parts can result in the below superseded boundary integral equation,
which is as follows Eq. (3) [27]:
cijðsÞ:uiðs;tÞ ¼
Z
G

G*
ijðx;s;tÞ*piðx;tÞ À Z*
ijðx;s;tÞ*uiðx;tÞ þ W*
ijðx;s;tÞ*viðx;tÞ

dG
þ uinc:
j ðs;tÞ ð3Þ
In which, vi is the velocity vector and G*
ij; Z*
ij and W*
ij represent the corresponding
solutions for movement, traction and velocity, subsequently.
Assume that a unit point force is applied at a two-dimensional body at time t¼0. If
the body has a liner elastic behavior, the displacement at time t is thought to be u+ðtÞ.
In addition, if the body has viscoelastic behavior, the displacement at time t, equals
Article Nowe01114

to Eq. (4) considering material damping. This has been carried out successfully by
Feng et al., [28]:
uðtÞ ¼ u+ðtÞð1 À hÞ
ðtÀtÞ
2T
ð4Þ
In Eq. (4), T denotes the prominent period of earthquake loading, approximately and
h denotes the coefficient of proportional damping.
According to Eq. (5), the boundary integral equation, Eq. (3) takes the following
modified form for a viscoelastic body:
cijðsÞ:ujðs;tÞ ¼
Zt
0
2
4
Z
G

G*
ijðx;s;t À tÞ:piðx;tÞ

dG
3
5ð1 À hÞðtÀtÞ=2T
dt
þ
Zt
0
2
4
Z
G

W*
ijðx;s;t À tÞ:viðx;tÞ

dG
3
5ð1 À hÞ
ðtÀtÞ=2T
dt
À
Zt
0
2
4
Z
G

Z*
ijðx;s;t À tÞ:uiðx;tÞ

dG
3
5ð1 À hÞ
ðtÀtÞ=2T
dt þ uinc:
jd ðs;tÞ
ð5Þ
Where uinc:
jd ðs; tÞ is damped incident wave.
2.2. Theory
The executed of boundary integral in Eq. (3) requires estimation in both temporal
and spatial changes of the field parameters. For temporal integration, the time axis
from 0 to t is divided into N equal steps of duration Dt, i.e. t ¼ N: Dt. Application
of a linear time variation of the field variables, the displacement, velocity and trac-
tion are represented as follows in Eqs. (6), (7), and (8):
uiðx;tÞ ¼ 41ðtÞ$un
i ðxÞ þ 42ðtÞ$unÀ1
i ðxÞ ð6Þ
_uiðx;tÞ ¼ _41ðtÞ$un
i ðxÞ þ _42ðtÞ$unÀ1
i ðxÞ ð7Þ
piðx;tÞ ¼ 41ðtÞ$pn
i ðxÞ þ 42ðtÞ$pnÀ1
i ðxÞ ð8Þ
Where 41ðtÞ and 42ðtÞ are linear temporal shape functions, which are expressed by
Eq. (9):
41ðtÞ ¼
t À TnÀ1
Dt
42ðtÞ ¼
Tn À t
Dt
TnÀ13t3Tn ð9Þ
Subscripts 1 and 2 mention to the forward and backward temporal nodes, during a
time step, respectively. Therefore, the time integration including only the kernels
and is represented by Eqs. (10), (11), (12), and (13):
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GNÀnþ1
ij1 ¼ ð1 À hÞ
ðNÀnþ1ÞDt=2T
ZnDt
ðnÀ1ÞDt
Gijðx;s;t À tÞ 41ðtÞ$dt ð10Þ
GNÀnþ1
ij2 ¼ ð1 À hÞ
ðNÀnþ1ÞDt=2T
ZnDt
ðnÀ1ÞDt
Gijðx;s;t À tÞ42ðtÞdt ð11Þ
FNÀnþ1
ij1 ¼ ð1 À hÞ
ðNÀnþ1ÞDt=2T
ZnDt
ðnÀ1ÞDt
À
Zijðx;s;t À tÞ41ðtÞ À Wijðx;s;t À tÞ _41ðtÞ
Á
dt
ð12Þ
FNÀnþ1
ij2 ¼ ð1 À hÞðNÀnþ1ÞDt=2T
ZnDt
ðnÀ1ÞDt
À
Zijðx;s;t À tÞ42ðtÞ À Wijðx;s;t À tÞ _42ðtÞ
Á
dt
ð13Þ
After spatial discretization and some rebuilding, the complicated boundary element
method equation for linear temporal variation can be expressed as Eq. (14):
cij$uN
i ðsÞ ¼
XN
n¼1
XQ
q¼1
8

:
Tn
ik$
Z
Gq
h
GNþ1Àn
ij ðrÞ
i
$NkðhÞ$jJj$dh À Un
ik$
Z
Gq
h
FNþ1Àn
ij ðrÞ
i
$NkðhÞ$jJj$dh
9
=
;
þ uinc:
jd ðs;tÞ ð14Þ
The following description can be called linear time-convoluted visco-elastodynamic
displacement and traction kernels, subsequently through Eqs. (15) and (16):
GNÀnþ1
ij ðrÞ ¼
h
GNÀnþ1
ij1 ðrÞ þ GNÀn
ij2 ðrÞ
i
ð1 À hÞðNÀnþ1ÞDt=2T
ð15Þ
FNÀnþ1
ij ðrÞ ¼
h
FNÀnþ1
ij1 ðrÞ þ FNÀn
ij2 ðrÞ
i
ð1 À hÞ
ðNÀnþ1ÞDt=2T
ð16Þ
In the above equations, N and Q can show the last time step and the total number of
boundary elements, respectively and L demonstrates the incorporation function of
an isoperimetric quadratic boundary element. The two-dimensional full space
visco-elastodynamic kernels, which have been derived analytically [27].
By sequentially writing Eq. (14) for each boundary node at time tþDt, and with
some rearrangement, the assembled system of equation takes the following matrix
form in Eqs. (17) and (18):
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F1
$UN
¼ G1
$TN
þ ZN
ð17Þ
In which:
ZN
¼
XNÀ1
n¼1
À
GNÀnþ1
$Tn
À FNÀnþ1
$Un
Á
þ Ud
inc:N
ð18Þ
In the above equations, UN
and TN
demonstrate the nodal movement and traction
vectors at the current time node, subsequently. ZN
involves both the effects of the
past dynamic history and the incident waves on the current time node [29, 30, 31, 32].
2.3. Model
The presented viscoelastic boundary element algorithm and derived time convoluted
kernels are applied in a program code, which is named VBEMSA (Viscoelastic
BEM for seismic analyses) to conduct investigation of topographic structures
exposed to incoming waves including SV and P. Fig. 1 represents the shape and dis-
cretization of a semi-circular canyon exposed to vertically propagating incoming
waves including SV and P by Eq. (19):
f ðtÞ ¼ Amax
h
1 À 2
À
pfpðt À t0Þ
Á2
i
eÀðp fpðtÀt0ÞÞ
2
ð19Þ
In which, fp and t0 demonstrate the dominant frequency and a proper time shift fac-
tor, subsequently. f (t) determines the horizontal part of the incoming displacement
whenever the vertical one is equal to zero for SV, and it would be vice versa for P
waves. Fig. 2 illustrates a layer, which is soft wit following parameters: height ¼ 10
m, SV ¼ 70.5 m/s, overlying a stiffer half-space with SV¼141 m/s.
Different material damping ratios of 0.03, 0.06 and 0.10 is considered for soft layer.
The amounts of 2.0 ton/m3
and 0.33 are selected for densities and the Poisson’s ra-
tios, subsequently.
Fig. 1. Schematic geometry and discretization of a semi-circular canyon subjected to vertically propa-
gating incident SV and P waves of the Ricker type.
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The same examples are also analyzed with the SHAKE program, which is em-
ployed for 1-D analysis of site response [33]. A comparison of the horizontal ac-
celeration time histories calculated by mentioned programs (SHAKE and
VBEMSA)for the top (O) and bottom (I) of the soft layer can be seen in Figs.
3 and 4, subsequently. It may be observed that there is a good compatibility be-
tween the acquired results.
A semi-elliptical hill, as represented in Fig. 5, is exposed to the vertically propa-
gating Ricker type P wave. This issue was investigated in a dimensionless form
by Alvarez-Rubio et al. [34] for viscoelastic media. The Ricker wave has a predom-
inant frequency of 2 Hz, time shift parameter of 0.8 s and maximum amplitude of
0.001 m. The hill has a radius of 200 m and height of 400m. The shear wave velocity,
damping ratio, Poisson’s ratio and mass density are 800 m/s, 0.25, 1% and 2.23 ton/
m3
, subsequently.
Fig. 6 shows the ampliﬁcations acquired by the oﬀered VBEMSA program
along the canyon with those acquired by the previous study for a frequency
of 1.5 (U ¼ 1.5).
Fig. 2. Geometry and discretization of the layered half-plane problem.
Fig. 3. Horizontal acceleration time histories at the base of the viscoelastic soft layer (point I) for ma-
terial damping ratio 0.06.
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3. Results discussion
In this research, we investigate the behavior of single hills and adjacent double and
triple homogenous hills with shape ratios 0.1, 0.3, 0.5 and 0.7 with damping mate-
rials, 0.00, 0.03 and 0.06 under the effect of P and SV waves with the modeled
dimension ranges from Æ 5000 to Æ 7240 m for semi-sine hills. The geometric char-
acteristics of adjacent double semi-sine hills are provided in Fig. 7. The reason for
choosing this amounts is that they are most used in previous studies in addition,
the extensive studies to ensure hills behavior with different specifications seems
sufficient.
Fig. 4. Horizontal acceleration time histories at the top of the viscoelastic soft layer (point O) for material
damping ratio 0.06.
Fig. 5. Schematic geometry and discretization of a semi-elliptical hill subjected to vertically propagating
incident SV and P waves of the Ricker type.
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Fig. 6. Ampliﬁcation of surface displacements for a semi-elliptical hill in the case of an incident P wave
and U ¼ 1.5. (The symbols “Hrz” and “Vrt” represent the horizontal and vertical components of ampli-
ﬁcation, respectively.)
Fig. 7. Geometric properties of homogeneous adjacent double semi-sine hills.
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3.1. Displacement pattern (time-place-displacement) for the hill
Fig. 8 shows graphs of vertical displacement component of ranges around double
adjacent semi-sine hill for P and SV waves, Poisson’s ratio 0.33, shape ratio 0.5
and all damping ratios including 0, 0.03 and 0.06 which are represented in
Fig. 8a, b and c, respectively.
Observations from Fig. 8 can be stated that in the semi-sine hills, each point on the
topography surface can create wave propagation. Wave propagation begins from the
first moments of the incident wave hit to topography from two points in domain. By
increasing distance from the hill, the propagated waves from different parts of the
topography will interact and create a wave train per each phase. Increasing the ratio
of hill shape can lead to rising the domain ranges of propagated waves; as a result,
the motion time of points will be increased. The displacement pattern in the range of
hills due to the adjacent hills has many complexities. In addition, to the propagated
waves inside each of the hills, other propagated waves influence on the displacement
pattern because of the surrounding hills.
Careful consideration of Fig. 8 (a), (b) and (c) represents that increasing damping can
lead to reducing displacement along the hill and the motion time of points will be
decreased. In addition, the range of error waves that are emitted is smaller and
this issue can create a condition for surveying results in a greater range. Increased
number of hills increases the complexity of displacement pattern and increases mo-
tion time of points.
Time history of vertical displacement of hill’s crest, adjacent point of the reference hill
with different shape ratio and affected by P waves in triple hills are presented in Figs. 9
and 10. As it can be seen in the central hill’s crest, we will have longer time and an
increase in the peak and valley points by increasing the shape ratio. However, a similar
behavior can be seen in the hills crest by drawing time history in different shape ratio.
As it can be observed, as damping increases, displacement along the hills decreases,
as a result the motion time will be decreased. Also, amplitude of error waves that are
emitted from trimming range are smaller and this issue can create a condition for
surveying the results in a greater range of the surrounding. Increasing the number
of hills adds the complexity of the displacement pattern and increases motion
time. Time history of vertical displacement of triple hill’s crest are presented in
various shape proportions and under the effect of SV waves in Fig. 11.
3.2. Amplification pattern (frequency-placeeamplification) for
the hill
Fig. 12 depicts amplification pattern (frequency-place-amplification) for a single,
double and triple adjacent semi-sine hill for damping ratios of 0.0, 0.03 and 0.06
Article Nowe01114

Fig. 8. Vertical and horizontal displacement pattern of double adjacent semi-sine hills for Poisson’s ratio
0.33 and for shape ratio 0.5 and diﬀerent damping factors in the vertical incident SV waves, (a) Damping
ratio: 0.0, (b): Damping ratio: 0.03 and (c) Damping ratio: 0.06, respectively.
Article Nowe01114

in Fig. 12 (a), (b) and (c). The trend of amplification function is slow and gradual due
to the homogeneous geometry of semi-sine hill. Increasing the hill shape ratio can
lead to growing the effect of seismic response. Surveying the behavior of the points
on the hill indicates that by increasing incident wave period in comparison with char-
acterized period of the topography; their motion will be uniform and closer to the
free field movement. Increasing the distance of the points from the hill’s center
can lead to increasing the amplification and de-amplification sequence. Results of
Fig. 12 (a), (b) and (c) shows that by increasing the material-damping ratio, the
amplification potential is reduced on hill’s crest and such effect is considerable in
the site specified period. Moreover, the effect of material damping ratio on amplifi-
cation potential is more tangible with increasing hill’s shape ratio.
Fig. 9. Comparing the time history curves in central hill’s crest in triple homogeneous hills in different
shape ratios with damping ratio 3%.
Fig. 10. Comparing the time history curves of the central hill’s crest in triple homogeneous hills in
different shape ratios with damping ratio 6%.
Fig. 11. Comparing the time history curves in the central hill’s crest in triple homogeneous hills in
different shape ratios with damping ratio 6%.
Article Nowe01114

Fig. 12. Vertical and horizontal component of ampliﬁcation of double adjacent semi-sine hills for shape
ratio (0.5) and diﬀerent damping factors for incident SV wave, (a) Damping ratio: 0.0, (b): Damping ra-
tio: 0.03 and (c) Damping ratio: 0.06, respectively.
Article Nowe01114

3.3. Amplification curves on hill’s crest
Crest point behavior in the single, double and triple homogeneous adjacent hills is rep-
resented in Fig. 13. In this Fig, the amplification curves at the hill’s crest are compared
for different shape ratios and damping. Examination of these curves shows that:
Crest of double adjacent hills in comparison with single hills has similar shape ratio,
and greater potential and amplification maximization. Although this difference in-
creases with increasing shape ratio, it is negligible within the studied shape ratio. How-
ever, by increasing the number of hills, the growth has an ascending trend and we even
Fig. 13. Compares crest amplification curve for single, double and triple adjacent hills in the shape ratio
and different damping factors - the vertical axis is amplification and the horizontal axis is the dimension-
less period.
Article Nowe01114

observe different behavior in higher shape ratio. In addition, by increasing the damp-
ing ratio of material, the difference between the top and bottom points will be reduced.
- Behavior of single and double adjacent hills is somewhat similar to each other but by
increasing the shape ratio in triple adjacent hills, we have a different behavior.
Although, by increasing the shape ratio, a different behavior in triple adjacent hills
than the other two modes can be observed, this difference occurs in short periods and
it is equal in long periods. The value of this similarity is more tangible by increasing
damping ratio. Moreover, in shorter periods, number of hills, frequency character-
istics of the amplification curves and the number of peaks and valleys is increased.
- The similarity in long periods is because the crest response in both adjacent and single
hills tends to seismic response of free field. However, this similarity cannot be seen in
short periods. Most changes at the amplification curves is observed in the crest of
semi-sine double hills in both states of placement at intermediate periods. Increasing
number of hills and examining the amplification curves in hill’s crest shows that by
increasing number of hills, maximum amplification occurs earlier and at lower pe-
riods, so that the maximum occurred very quickly in triple hills in high shape ratios.
- By increasing damping ratio, although complication period has no significant
change, but it reduces the amplification potential in the hill’s crest and the effect
is more in the specified period and has significantly reduced the amplification po-
tential. The effect of amplification amounts is more tangible than the damping ra-
tio of materials by increasing the hill shape ration.
3.4. Amplification of edge and central points at semi-sine hills
with damping ratio
Fig. 14 illustrates the single, double and triple adjacent semi-sine hills in Fig. 14 (A),
(B) and (C), respectively. Fig. 15 compares the amplification of edges of single,
Fig. 14. Homogenous semi-sine hills (A): Single, (B): Double and (C): Triple.
Article Nowe01114

double and triple homogenous adjacent hills (point E of Fig. 14A, B and C), in
different damping ratios. In addition, Fig. 16 compared amplification of the midpoint
of the double and triple homogenous hills (point C of Fig. 14B and C) with different
damping ratios. Examining amplification curves at these points show that:
Edge of adjacent hills compared with edges of single hills with similar shape ratio,
has greater potential and larger microzonation maximum. The difference between
amplification values of single and adjacent hills will be reduced by increasing ma-
terial damping ratio. Increasing the maximum amplification by enhancing shape ra-
tio has an increasing trend and direct relationship with each other. Moreover,
Fig. 15. Shows the comparison of amplification curve of the hill edge (point E) for single, double and
triple adjacent homogenous hills in shape ration and different damping factor of dimensionless horizontal
axis and vertical axis of corresponding amplification.
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multiplicity of hills can lead to increasing the frequency characteristics of the ampli-
fication curve and the number of peaks and valleys.
The external edge of double hills has a seismic behavior similar to single hills at the
incident of long and short waves.
Maximumamplificationat edge of double hills occurs in longer periods compared with
single hills, but in triple hills, maximum amplification occurs at shorter intervals even
in comparison the single hills. A relatively similar behavior is observed in the edge of
adjacent or single hills such as long periods but the similarity cannot be seen in shorter
Fig. 16. Shows the comparison between amplification curve of midpoint (point C) for homogenous dou-
ble and triple adjacent hills in shape ratio and different damping factors in dimensionless horizontal axis
and vertical axis of corresponding amplification.
Article Nowe01114

periods, and even a dramatic difference can be seen in triple hills. In addition, by
increasing the damping ratio, adjacent and single hills’ behavior will be very similar.
The points between triple homogenous hills compared with double homogenous
hills have similar shape ratio, and greater potential and maximum amplification
and microzonation. Points among the triple homogenous hills compared to double
homogenous hills have more peaks and this peak will be highlighted by increasing
the shape ratio. In addition, the adjacent hills will have similar behavior to each other
by increasing the damping ratio of the midpoints.
The maximum amplification in the points between triple homogenous hills will
occur at shorter periods than the double hills (like Point E). By increasing material
damping ratio, the difference between amplification of adjacent double and triple
hills in points among hills will be less (like Point E).
4. Conclusion
The study represents an overview of amplification model of 2D homogenous semi-
sine shaped hills with material damping exposed to vertical incoming waves
including SV and P, acquired by numerical analyses using the time domain BEM.
The main conclusions of the analysis are as follows:
1- Looking at the trends of amplification function changes along the topography,
a slow and gradual process is observed and the effect of its seismic response
will be increased by increasing the hill shape ratio according to the geometry
of homogenous semi-sine hill.
2- As it can be seen in the displacement patterns, the motion time of the points
and displacement along the hill will be reduced by increasing damping.
3- Material damping changes does not affect the hill amplification pattern, but
potential amplification in and out will be decreased by increasing damping
ratio.
4- At the time of completion of the analysis in triple adjacent hills without damp-
ing of materials with high shape ratio, there is still amplification in and out,
while this cannot be observed in similar analyzes with damping materials.
This issue can show the effect of wave interference and criticality of this effect
in the adjacent hills.
5- Triple hill’s crest compared with double hill’s crest with the same shape pro-
portion, has the greater amplification potential, which this difference is reduced
by increasing damping materials.
6- In amplification curves, in the double homogenous semi-sine hill’s crest in the
long periods, a relatively similar behavior can be seen in the crest of double
homogenous adjacent hills and single hills. The value of this similarity is
Article Nowe01114

more tangible by increasing material damping, but there are no such similar-
ities in short periods.
7- Most changes in amplification curves are observed in the semi-sine hill’s crest
in intermediate periods.
8- Edges of the adjacent hills compared with the edges of single hills have similar
shape ratio and greater potential.
9- Growth of maximum amplification ratio has an upward trend by increasing the
number of hills and shape ratio and has a direct relationship with them.
10- Frequency specifications of amplification curve and the number of peaks and
valleys will be increased by increasing the number of hills.
11- The maximum amplification in points between triple homogenous hills
compared with double homogenous hills occurs in shorter periods (e.g.,
point E).
12- Hills with any shape ratio, regardless of damping, have special response
spectrum.
13- The number of hills has no effect on the response spectrum in a particular shape
proportion and the influence of the number of hills in drawing the response
spectrum can be ignored.
14- Material damping like amplification and displacement affects the response
spectrum.
Declarations
Author contribution statement
Mobin Afzalirad, Mehran Naghizadehrokni, Iman Khosravi: Conceived and de-
signed the analysis; Analyzed and interpreted the data; Wrote the paper.
Funding statement
This research did not receive any specific grant from funding agencies in the public,
commercial, or not-for-profit sectors.
Competing interest statement
The authors declare no conflict of interest.
Additional information
No additional information is available for this paper.
Article Nowe01114

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