Analysis and Design of One Dimensional Periodic Foundations for Seismic Base Isolation of Structures

WitartoWitarto et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 6) January 2016, pp.05-15
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Analysis and Design of One Dimensional Periodic Foundations for
Seismic Base Isolation of Structures
Witarto Witarto1
, S.J. Wang2
, Xin Nie3
, Y.L. Mo1
, Zhifei Shi4
, Yu Tang5
,
Robert P. Kassawara6
1
Department of Civil and Environmental Engineering, University of Houston, Houston, Texas
2
National Center for Research on Earthquake Engineering, Taipei, Taiwan
3
Department of Civil Engineering, Tsinghua University, Beijing, China
4
School of Civil and Environmental Engineering, Beijing Jiaotong University, Beijing, China
5
Argonne National Laboratory, Argonne, Illinois
6
Electric Power Research Institute, Palo Alto, California
ABSTRACT
Periodic foundationis a new type of seismic base isolation system. It is inspired by the periodic material crystal
lattice in the solid state physics. This kind of material has a unique property, which is termed as frequency band
gap that is capable of blocking incoming waves having frequencies falling within the band gap. Consequently,
seismic waves having frequencies falling within the frequency band gap are blocked by the periodic foundation.
The ability to block the seismic waveshas put this kind of foundation as a prosperous next generation of seismic
base isolators. This paper provides analytical study on the one dimensional (1D) type periodic foundations to
investigate their seismic performance. The general idea of basic theory of one dimensional (1D) periodic
foundations is first presented.Then, the parametric studies considering infinite and finite boundary conditions are
discussed. The effect of superstructure on the frequency band gap is investigated as well. Based on the analytical
study, a set of equations is proposed for the design guidelines of 1D periodic foundations for seismic base
isolation of structures.
Keywords–finite element, frequency band gap, one dimensional periodic foundation, phononic crystal, seismic
isolation
I. INTRODUCTION
The research on periodic material has shown that
an infinite series of lattice layers has the ability to
manipulate certain waves travelling through its
medium [1–9]. Periodic material is classified into
photonic crystal, which can manipulate
electromagnetic waves and phononic crystal, which
can manipulate elastic waves. According to the
number of directions where the unit cell is repeated,
periodic material can be classified as: one-
dimensional (1D), two-dimensional (2D) and three-
dimensional (3D) periodic material. This man-made
material has the property of preventing the
propagation of waves having frequencies within
certain frequency bands through the crystal’s
medium. These frequency bands are termed as
frequency band gaps or attenuation zones [10].
Utilizing the unique feature of the
periodicmaterial especially phononic crystal,
researchers in civil engineering field have started to
apply the crystal lattice into the structural element. In
the beginning, mechanical and civil engineering
researchers [11–15] studied the periodic beams and
found the existence of flexural wave band gaps in the
periodic beams. Then, theoretical and experimental
studies were conducted to investigate the application
of periodic rod on offshore platforms to isolate the
sea wave [16–17]. Subsequently, the phononic crystal
was applied to the structural foundations to isolate
seismic waves to protect the superstructure, which
later known as periodic foundation. Although the
idea of periodic foundations is relatively new,
experimental testing on periodic foundations has
already been conducted by several researchers[18–
20]. Due to its uniquewave isolation mechanism,
periodic foundation can isolate superstructure from
the incoming seismic waves without having a large
relative horizontal displacement in the isolation layer
that generally occurs in the conventional seismic
isolation, such as: rubber bearings and friction
pendulum systems[21].
II. BASIC THEORY OF 1D PERIODIC
MATERIAL
In 1D periodic material, the crystal lattice
possesses periodicity in one direction.Consider that
the unit cell is composed of N layersin which the
periodicity is in z direction, as shown in Fig.1.The
equation of motion in each layer n subjected to elastic
wave is shown in equation (1)
RESEARCH ARTICLE OPEN ACCESS

Figure 1.One dimensional periodic material
2 2
2
2 2
n n
n
n
u u
C
t z
 

 
(1)
The Cn constant is expressed in equation (2)and
equation (3)for input wavesof S-Wave and P-Wave,
respectively.
/n n nC   (2)
( 2 ) /n n n nC     (3)
Where 𝜆 𝑛 and𝜇 𝑛 are the Lamé elastic constantsand
𝜌 𝑛 is material density at layer n. Consider a steady
state oscillatory waves of angular frequency:
   , ei t
n n n nu z t u z
 (4)
Substituting equation (4) into equation (1) yields:
2
2 2
2
( )
( ) 0n n
n n n
n
u z
C u z
z


 

(5)
The general displacement solution of equation (5)is
expressed as:
( ) sin( / ) sin( / )n n n n n n n nu z A z C B z C   (6)
In which An and Bn are the amplitudes of the general
displacement solution on layer n. Subsequently, the
shear stress component on the periodic foundation
can be calculated fromequation (7).
( ) /
[ cos( / ) sin( / )]/
n n n n n
n n n n n n n n
z u z
A z C B z C C
 
   
  
 
(7)
Equation (6) and equation (7) are arranged into
matrix form.
( ) ( )n nz zn n nw H ψ (8)
or
sin( / ) sin( / )
( )
cos( / ) sin( / )(z )
n n n n
n n n
n n
n n n nn n n
n n
z C z C
u z A
z C z C B
C C
 
   
 
 
             
(9)
The left hand side vector of equation (8) at the
bottom of layer n is defined as𝐰 𝐧
𝐛
which gives
information regarding the displacement and the stress
at the bottom of layer n. As for the top of layer n, the
left hand side vector is defined as 𝐰 𝐧
𝐭
which gives the
information of displacement and the stress at the top
of layer n.
(0) (0) b
n n n nw w H ψ (10)
( ) ( )n nz z t
n n n nw w H ψ (11)
Equation (10) can be related to equation (11)
through a transfer matrix 𝐓𝐧.
t b
n n nw T w (12)
Hence, the transfer matrix 𝐓𝐧 for a single layer n is:
 
1
( ) (0)nZ

n n nT H H (13)
Each layer interface of the unit cell is assumed to be
perfectly bonded and hence the displacement and
shear stress need to satisfy continuity. Therefore, the
displacement and shear stress of the top of layer n is
equal to that of the bottom of layer n+1.
b t
n+1 nw w (14)
The relation of displacement and shear stress of the
bottom and top surfaces of the unit cell containing N
layers are:
 ... ...
 
  
t b t
N N N N N-1
b b
N N-1 N-1 N N-1 1 1
w T w T w
T T w T T T w
(15)
The displacement and shear stress vector of top
and bottom surface of the unit cell is 𝐰 𝐭
= 𝐰 𝐍
𝐭
and
𝐰 𝐛
= 𝐰 𝟏
𝐛
. Equation (15)can be shortened into:
(ω)t b
w T w (16)
Based on the Bloch-Floquent theorem the
periodic boundary conditions can be expressed as:
eika
t b
w w (17)
Wherea is the unit cell thickness. Subtraction of
equation (17) by equation (16) yields:
( ) e 0ika
   
b
T I w (18)
The nontrivial solution can be achieved when:
( ) e 0ika
  T I (19)
Equation (19) is the so called Eigenvalue
problem, with e𝑖𝑘𝑎
equal to the Eigenvalue of the
transformation matrix T(ω). Thus, the relation of
wave number k and frequency ω can be obtained by
solving the corresponding Eigenvalue problem. The
relationship between the wave number and frequency
forms the dispersion curve, which provides the
information of the frequency band gaps.
III. PARAMETRICSTUDY OF 1D PERIODIC
FOUNDATIONS WITH INFINITE UNIT
CELLS
The most simple unit cell of 1D periodic
foundation is the two layered unit cell. In this
configuration, a single unit cell consists of 2 different
layers; A and B; in which hA and hBrepresents the
height of each of both layers A and B, respectively.
Consider rubber and concrete as the unit cell’s
component with material properties as shown in
Table 1 and each layer height, hrand hc, equals to 0.2
m as shown in Fig.2.
Figure 2. Benchmark unit cell of 1D periodic
foundation

Table 1.Material properties for benchmark unit
cell of 1D periodic foundation
Component
Young’s
modulus
(MPa)
Density
(kg/m3
)
Poisson’s
ratio
Concrete 31400 2300 0.2
Rubber 0.58 1300 0.463
Figure 3. Dispersion curve of the periodic
foundation with benchmark unit cell
Fig.3shows the dispersion curves of the periodic
foundation with benchmark unit cell subjected to
transverse wave (S-Wave) and longitudinal wave (P-
Wave). The curves were obtained by solving the
Eigenvalue problem stated in equation (19). The first
two frequency band gaps under S-Wave are observed
at 13.51 Hz– 30.87 Hz and 36.65 Hz to 50 Hz. While
the first frequency band gap under P-Wave are 51.5
Hz – 117.6 Hz.
By changing each of the material and geometric
properties while keeping the rest of the properties the
same as in the benchmark unit cell, the effect of each
property on the first frequency band gap can be
observed.
Figure 4.Effect of rubber material properties on
the first frequency band gap
Fig.4 shows the effect of rubber material
properties on the first frequency band gap. The
starting frequency of the first frequency band gap
represented by the blue curve and the width of the
first band gap (obtained by subtracting the starting
frequency from the end frequency of the first band
gap) represented by the red curve increase rapidly
with the increase of the Young’s modulus of rubber.
The increase of rubber density would slightly reduce
the starting frequency of the first frequency band gap
and would greatly reduce the width of the band gap.
The effect of Poisson’s ratio is different when the
periodic foundation is subjected to S-Wave or P-
Wave. While both the starting and the width of the
frequency band gap are steadily decreasing as the
Poisson’s ratio increases under the S-Wave, the
starting and the width of the frequency band gap
increase tremendously as the Poison’s ratio gets
closer to 0.5 under the P-Wave. This is so because
the volumetric locking occurs in the continuum body
subjected to the P-Wave. As shown in equation (3),
the constant C for P-Wave is a function of the first
Lamé constant λ. As the Poison’s ratio gets closer to
0.5, theλ value goes to infinite causing the volumetric
locking that makes the continuum body to become
very stiff.
Fig.5shows the effect of each of the concrete
material properties on the first frequency band gap. It
is observed that the Young’s modulus and Poisson’s
ratio of concrete does not affect the first frequency
band gap. The increase of concrete density would
allow lower starting frequency and wider attenuation
zone, which is the most desired band gap for periodic
foundation.

Figure 5.Effect of concretematerial properties on
the first frequency band gap
Figure 6.Effect of rubber to concrete thickness
ratio on the first frequency band gap
Figure 7.Effect of unit thickness on the first
frequency band gap
Fig.6shows the effect of rubber to concrete
thickness ratio on the frequency of the band gap. It is
shown that the increase of the rubber to concrete
thickness ratio would reduce the starting frequency of
the first attenuation zone when the thickness ratio is
lower than 1.5. For the thickness ratio greater than
1.5, the starting frequency of the first attenuation
zone would increase along with the increase of the
thickness ratio. Meanwhile, the width of the
attenuation zone keeps decreasing with the increase
of the thickness ratio. Fig.7shows that the increase of
the unit cell thickness would result in the reduction of
the starting frequency and the width of the frequency
band gap.
IV. PARAMETRIC STUDY OF 1D PERIODIC
FOUNDATIONS WITH FINITE NUMBER
OF UNIT CELLS
The parametric studies conducted in Section III
have provided the insight of how the material and
geometric properties affect the frequency band gap.
However, the real structures would have a finite
geometry. Thus, the number of unit cells and the
plane area has to be finite. In order to simulate the
behavior of periodic foundation as close as possible
to the reality, finite element model of 1D periodic
foundations was utilized. The material properties
were assumed to be linear elastic.
4.1. Effect of periodic foundation plane size
This section investigates the plane size effect of
the periodic foundation on the response inside the
frequency band gap. The periodic foundation was
modeled using one unit cell consisting of a rubber
layer and a concrete layer. The unit cell was placed
on top of a concrete base layer. Each layer has a
thickness of 0.2 m. The material properties are listed
in Table 2. Three cases were investigated: Case A1
with plane size 1m x 1m, Case A2 with plane size 2m
x 2m, and Case A3 with plane size 3m x 3m. All
three cases are shown in Fig.8.
Table 2. Material properties
Material
Young’s
modulus
(MPa)
Density
(kg/m3
)
Poisson’s
ratio
Concrete 40000 2300 0.2
Rubber 0.1586 1277 0.463
Figure 8. (a) Case A1 (1m x 1m); (b) Case A2 (2m
x 2m); (c) Case A3 (3m x 3m)

All three cases were subjected to the vibration of
a scanning frequency ranging from 0 to 50 Hz which
is a typical main frequency content of seismic waves.
The responses of these foundations are presented in
the form of frequency response function (FRF)
defined as 20log 𝛿 𝑜/𝛿𝑖 , where 𝛿 𝑜 is the
instantaneus displacement amplitude recorded at the
top of the periodic foundation and 𝛿𝑖 is the amplitude
of instantaneous displacement input at the base of the
periodic foundation.
The FRF of all three foundations are presented in
Fig.9. The negative value shows that the output
response is smaller than the input. The yellow gaps
are the theoretical frequency band gap obtained from
solving the wave equation as described in Section II.
It is observed that inside the theoretical frequency
band gap, all three foundations show response
reduction. Occasional spikes, due to rocking modes,
inside the frequency band gap that goes to positive
value were observed on the curves. However, the
larger the plane size, which is the closer it is to the
theoretical infinite boundary condition, the less
spikes occur inside the frequency band gap because
the rocking modes are eliminated.
Figure 9. Frequency response function of periodic
foundation with different plane size
4.2. Effect of number of unit cells
This section investigates the effect of number of
unit cells on the response inside the frequency band
gap. Three cases of periodic foundation were
investigated. The first case (designated as Case B1) is
a periodic foundation with one unit cell and plane
size of 3m x 3m, similar to Case A3 studied in
Section 4.1. The second case (designated as Case B2)
is a periodic foundation with two unit cells. The third
case (designated as Case B3) is a periodic foundation
with three unit cells. The illustration of each of the
three cases is shown inFig.10.
Figure 10.(a) Case B1 (one unit cell); (b) Case B2
(two unit cells); (c) Case B3 (three unit cells)
As observed inFig.11, the FRF value inside the
theoretical frequency band gap gets lower as the
number of unit cells increases. The lower FRF value
represents more response reduction. The more
number of unit cells is, the greater the response
reduction because it is closer to the theoretical
infinite number of unit cells condition.

Figure 11.Frequency response function of periodic
foundation with different number of unit cells
4.3. Effect of multilayered unit cells
In this section, the effect of multilayered unit cell
as a combination of two different unit cells is studied.
The first periodic foundation (designated as unit cell-
1) is composed of Layer A (made of rubber material)
with a layer thickness of 0.2 m and Layer B (made of
concrete material) with a layer thickness of 0.2 m.
The second foundation (designated as unit cell-2) is
composed of Layer C (made of rubber material) with
a layer thickness of 0.15 m and Layer D (made of
concrete material) with a layer thickness of 0.25 m.
Both periodic foundations have plane size of 3 m by
3 m and seat on a concrete base (Fig.12(a) and
Fig.12(b)). The third periodic foundation (designated
as unit cell-3) is composed of all four layers
composing the two periodic foundations mentioned
above (Fig.12(c)). The material properties follow
Table 2.
(a) (b)
(c)
Figure 12. (a) Unit cell-1, (b) Unit cell-2, and (c)
Unit cell-3
The FRF of each of the three foundations
subjected to transverse wave can be seen inFig.13.
For both the periodic foundation made of unit cell-1
and the periodic foundation made of unit cell-2, the
FRF values are found to be negative inside the
theoretical frequency band gap. The frequency band
gap of the unit cell-1 and unit cell-2 can be seen
overlapping one another. Therefore, the unit cell-3,
which is the combination of the two unit cells, is
presumed to have frequency band gap coming from
the union of the two frequency band gaps (7 Hz to 50
Hz). However, Fig.13(c) shows that the FRF has a
positive peak around frequency 17 Hz which
represents amplification inside the combined
frequency band gap. This result invalidates the
previous presumption.
The contradicting result indicates that the
frequency band gap of four-layer unit cell is not the
union of the frequency band gaps of two separate
two-layer unit cells. Therefore the frequency band
gap of the four-layer unit cells is calculated.
Using the transfer matrix method, the theoretical
frequency band gap of the four-layer unit cell is
obtained, as shown in Fig.14. It is observed that
multiple frequency band gaps are located very close
to each other in the region of 7.37 Hz to 50 Hz. The
pass bands that separate the frequency band gaps in
this region are 17.36 Hz –17.85 Hz, 23.14 Hz –23.45
Hz, 33.32 Hz–33.44 Hz, 44.16 Hz–44.22Hz, and
49.38Hz–49.48Hz. These very thin pass bands
responsible for the amplification of the response at 17
Hz and peak FRFs at 23 Hz, 33 Hz, 44.5 Hz, and 50
Hz. Since the pass band is very small, some peaks
still have negative FRF values representing the
response reduction.
(a)
(b)

(c)
Figure 13. Frequency response function (a) Unit
cell-1, (b) Unit cell-2, and (c) Unit cell-3
(a)
(b)
Figure 14. (a) Theoretical band gap of unit cell-3
(b) FRF of unit cell-3
4.4. Effect of superstructure
As periodic foundation is designed to support a
superstructure, the presence of superstructure in 1D
periodic foundation may affect the frequency
response function curve. Fig.15(a) illustrates a
superstructure seating on a 1D periodic foundation.
The mass and stiffness of the superstructure are
tuned, so that the natural frequency of the
superstructure alone is 10 Hz (typical natural
frequency of nuclear reactor building or low rise
building). Fig.15(b) shows that the FRF curves of the
1D periodic foundation with superstructure (red and
blue curves) are quite different especially for the
attenuation zone in the first band as compared to that
of 1D periodic foundation without superstructure
(black curve).
Performing finite element analysis of the full
superstructure with periodic foundation can be time
consuming. Since the presence of superstructure will
alter the FRF curve especially in the first frequency
band gap, it is more convenient if one can predict the
altered frequency band gap without having to model
the entire superstructure. This is important especially
during the preliminary design phase.
(a)
(b)
Figure 15. (a) 1D periodic foundation with
superstructure (b) FRF of 1D periodic foundation
with superstructure
When the superstructure is stiff enough (such as
nuclear reactor building), the superstructure can be
transformed into an equivalent additional layer of a
unit cell, as shown inFig.16. The superstructure is
assumed as an additional layer with thickness h*s,
which can be set the same as the upmost concrete
layer hc. The total weight of the superstructure is then
transferred into an equivalent density by dividing the
total weight with the multiplication of the designed

cross sectional area (horizontal area) of the periodic
foundation and the additional layer thickness.
Figure 16.Unit cell with equivalent superstructure
layer
The corresponding frequency band gaps of the
unit cell with equivalent superstructure layer are
shown in Fig.17. It is observed that the negative
values in the FRF curves of 1D periodic foundation
with superstructure (blue and red curves in Fig.15(b))
coincide with the theoretical frequency band gaps of
unit cell with equivalent superstructure layer.
Therefore, to predict the altered frequency band gap
due to the presence of superstructure, it is much more
convenient to solve for the dispersion curve of the
multi-layer unit cell with equivalent layer of
superstructure.
Figure 17.Theoretical band gap of unit cell with
equivalent superstructure layer
4.5.Damping in 1D periodic foundation
It is explained in structural dynamics that the
damping effect will reduce the structural
response[22]. In 2009, Hussein [23] introduced the
theory of damping in phononic crystal.This section
will discuss the effect of damping in 1D periodic
foundation. For under-damped condition, the wave
propagation will experience amplitude decay and
shortening of frequency. The damped frequency can
be obtained using equation (20). Therefore, each
wave number k will have corresponding damped
natural frequencies and subsequently the damped
dispersion curve can be obtained [51].
2
dω ( ) ω( ) 1 ζ( )k k k  (20)
Assuming the damping in the whole body of 1D
periodic foundation is 10%. The damped and
undamped dispersion curvesare shown inFig.18. It
can be observed that the difference in the
undampedand the damped dispersion curves is
negligible. Therefore, in periodic foundation, the
frequency band gap of the damped unit cell can be
considered the same as the undamped unit cell.
Figure 18. Dispersion curve of damped unit cell
Figure 19. FRF of damped unit cell
Consider the Case B1 periodic foundation in
Section 4.2. The damping ratio of 4% and 10% are
assigned to the concrete and rubber layers,
respectively. The FRF curves of the periodic
foundation with and without damping are shown
inFig.19. It can be seen that inside the frequency
band gap, the FRF values of the damped and
undamped periodic foundations are the same since
the waves are not propagated through the periodic
foundations. Outside the frequency band gap, the
response are reduced tremendously especially in the
pass band after the first frequency band gap.
Therefore, damping contribution is very significant to
reduce the response outside the frequency band gap.
V. DESIGN GUIDELINES OF 1D PERIODIC
FOUNDATION
Based on the parametric study shown in Section
III, it is very clear that the change in the starting
frequency band gap and the band width due to the
change of material and geometric properties follow
certain pattern. If the pattern can be quantified, the
frequency band gap can be calculated without solving
the wave equation.

The parametric study was conducted by first
assuming a reference or benchmark unit cell. The unit
cell size of the periodic foundation is set to be 0.4 m
with a rubber to concrete thickness ratio of one. The
dispersion curve of the benchmark unit cell under S-
Wave and P-Wave are shown inFig.3. The first
frequency band gap under S-Wave starts at 13.51 Hz
with a band width of 17.36 Hz. While the first
frequency band gap under P-Wave starts at 51.5 Hz
with a band width of 66.1 Hz. Since the damping will
reduce the responses at the pass bands after the first
frequency band gap, therefore only the first
frequency band gap is considered in the development
of design guidelines.
Figure 20. Regression for the effect of Young’s
modulus of rubber on the starting of frequency
band gap under S-Wave
Each of the material and geometric properties
was changed while keeping the rest of the properties
constant. The change in the frequency band gaps due
to a certain parameter is observed to follow a certain
pattern. The results are then fitted with a regression
curve that provides the simplest and the closest
equation to the data points. The regression curve for
the starting of frequency band gap under S-Wave as a
function of Young’s modulus of rubber is shown in
Fig.20. The chosen power equation can accurately fit
the data points.
The Young’s modulus and Poisson’s ratio of
concrete were not included since they do not have
any effect on the frequency band gap. The obtained
equations were normalized with the starting of
frequency band gap obtained from the unit cell with a
set of reference properties, which is 13.51 Hz. The
normalization turns the regression equations into a
modification factor. These modification factors will
modify the frequency band gap of the unit cell with
respect to the set of reference properties. The final
equations to find the starting of frequency band gap
and the band width under both S-Wave and P-Wave
are shown in equations(21)to (24).
Starting of frequency band gap (S-Wave)=
   1 2 3 4 5 613.51F F F( ) ( ) ( ( F F)F )r r r cE T r   (21)
Band width (S-Wave) =
   1 2 3 4 5 617.36G G G( ) ( ) ( ( G G)G )r r r cE T r   (22)
Starting of frequency band gap (P-Wave) =
   1 2 3 4 5 6( ) ( ) (51.5H H H H )H) ( Hr r r cE T r   (23)
Band width (P-Wave) =
   1 2 3 4 5 6( ) ( ) (66.1I I I I )I) ( Ir r r cE T r   (24)
with the modification factors as follow:
3 0.5003
1F ( ) 1.3094 10r rE E
 
5
2 4
2.814 1.627 10
F ( )
13.51 13.6451 10
r
r
r



 

 
0.6263
3F ( ) 0.4139 1.2561r r   
0.03885
4F ( ) 14.937 10.0518c c  
 
5F ( ) 0.4/T T
2.878 0.01594
6F ( ) 0.6403e 0.9489er r
r 
 
3 0.4996
1G ( ) 1.3185 10r rE E
 
0.5964
2G ( ) 98.0991 0.3632r r  
 
0.6325
3G ( ) 0.4112 1.2523r r   
0.03885
4G ( ) 11.6244 9.6025c c  
  
5G ( ) 0.4 /T T
0.8319
6G ( )r r

3 0.5
1H ( ) 1.3122 10r rE E
 
5
2 5
8.978 6.493 10
H ( )
51.5 5.9431 10
r
r
r



 

 
1.945 71.9916
3H ( ) 0.3076e 5.9728 10 er r
r
 
 
  
0.04212
4H ( ) 14.0932 9.1689c c  
 
5H ( ) 0.4 /T T
2.838 0.01595
6H ( ) 0.62738e 0.94796er r
r 
 
3 0.4999
1I ( ) 1.3145 10r rE E
 
0.5954
2I ( ) 97.6097 0.3657r r  
 
1.947 71.9916
3I ( ) 0.3074e 5.9592 10 er r
r
 
 
  
0.04212
4I ( ) 10.9803 8.9228c c  
  
5I ( ) 0.4 /T T
0.8321
6I ( )r r

WhereE is the Young’s modulus, ρ is the density, and
ν is the Poisson’s ratio. Each of the subscriptsr and
cshowsthat material properties belongs to rubber or
concrete, respectively. The unit cell thickness and the
rubber to concrete thickness ratio are denoted by T
and r, respectively.
This set of equations ((21)to (24)) is very
convenient to be used for design. Suppose that the
frequency band gap is the design objective, while the
R
2
= 1

materials are given from the product in the market.
Hence, the unit cell size and rubber to concrete
thickness ratio can be tuned to obtain the desired
frequency band gap.For example, the materials that
will be used for the design is a high density concrete
and soft rubber with properties shown in Table 3. The
periodic foundation is expected to have starting
frequency of band gap at around 6 Hz. The unit cell
size are then designed as 2 m with a rubber to
concrete thickness ratio of 1. The frequency band
gaps under S-Wave and P-Wave calculated using
equations(21) to (24)are 6.44–14.51 Hz and 21.18–
50.01 Hz, respectively. Detail calculations are shown
below:
 Starting of frequency band gap (S-Wave)
   1 2 3 4 5 613.51F F F( ) ( ) ( F F) ( )Fr r r cE T r  
13.51 2.278 1.08 1 0.968 0.2 1 6.44Hz       
 Band width (S-Wave)
   1 2 3 4 5 617.36G G G( ) ( ) ( G G) ( )Gr r r cE T r  
17.36 2.27 1 1 1.025 0.2 1 8.07Hz       
 Starting of frequency band gap (P-Wave)
   1 2 3 4 5 651.5H H H( ) ( ) ( H H) ( )Hr r r cE T r  
51.5 2.273 1 0.936 0.968 0.2 1 21.18Hz       
 Band width (P-Wave)
   1 2 3 4 5 666.1I I I( ) ( ) ( I I) ( )Ir r r cE T r  
66.1 2.273 1 0.935 1.025 0.2 1 28.83Hz       
Table 3. Material properties for designed 1D
periodic foundation
Material
Young’s
modulus
(MPa)
Density
(kg/m3
)
Poisson’s
ratio
Concrete 40000 2500 0.2
Rubber 3 1300 0.463
The dispersion curves obtained from solving the
wave equation are shown in Fig.21. The first
frequency band gap under S-Wave is observed to be
at 6–14 Hz. While under P-Wave, the first frequency
band gap is observed to be at 22.7–53.48 Hz. The
proposed method is proven that can accurately
predict the theoretical frequency band gap.
Figure 21. Dispersion curve of designed 1D unit
cell under (a) S-Wave (b) P-Wave
VI. CONCLUSIONS
Theoretical studies have been conducted to
investigate the behavior of 1D periodic foundations
as seismic isolators. In order to get lower and wider
frequency band gaps, the unit cell must consists of at
least two contrasting components, i.e. stiff and dense
component as well as light and soft component. In
the real application one unit cell is capable of
isolating the waves having frequencies inside the
theoretical frequency band gap. The more unit cells,
the better the waves attenuation. Large plane size to
the total thickness of periodic foundation is necessary
in order to eliminate the undesirable rocking mode.
The present of damping is certainly beneficial to
reduce the amplification on the pass bands. The
presence of superstructure can be beneficial as it
alters the first frequency band gap to become lower
and wider. A set of simple equations to predict the
first frequency band gap is developed for the design
of 1D periodic foundations. The equation is straight
forward and can be applied directly in the design
without solving the wave equation.
The analytical studies have shown that 1D
periodic foundation is very promising for seismic
base isolation system. Experimental test needs to be
conducted to prove the obtained analytical results.
Therefore, an experimental program on 1D periodic
foundation is currently under preparation and will be
conducted in the coming years.
VII. Acknowledgements
This work is supported by the US Department of
Energy NEUP program (Proj. No. CFA-14-6446),
National Center for Research on Earthquake
Engineering, Taiwan and Tsinghua University,
Beijing, China. The opinions expressed in this study
are those of the authors and do not necessarily reflect
the views of the sponsors.

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