The technique of using base isolation system to protect the structure and the contents of the structure has been proven effective in past earthquakes. Modelling of base-isolated structure is crucial to ensure that the response is predicted correctly. In this study, the linear viscous damper was used to represent the energy dissipation mechanism to study the effect of damping in the isolation system to the response of the structure. The peak isolator displacement, peak interstory drift and the roof floor acceleration response spectrum were used to evaluate and compare the effectiveness of the isolation system. Other variables in this study were the period separation and the level of damping in the isolation system. The use of bilinear model to model base-isolated structure is common due to the simplicity of the model and the ease to implement in computing software. However, the simplicity of the bilinear model has also restricted the ability to define the mechanical properties of the model over a wide range of shear strain. This study found that some degree of damping in the isolation system is beneficial to the structure but when the damping is too high, the key objective of isolating the structure-to protect the contents of the structure-is hampered due to the presence of high frequency vibration.
Base isolation technique is one of the methods to mitigate the effect of earthquakes to structures, but this is the only technique among others that can protect both the structure and the contents of the structure from the ground shaking. Base isolation can be achieved through various approaches, such as the use of high damping elastomeric bearings, lead rubber bearings and friction pendulum bearings. A hybrid system can be created by combining two of more isolation mechanisms or by introducing additional damper system to increase the energy dissipation capability. The fundamental of this technique lies in the concept of lengthening of the fundamental period of the structure such that the natural period of the system does not fall in the range of the predominant period of the earthquakes . The base isolation technique is widely used around the world, including the United States, Armenia, Chile, China, Indonesia, New Zealand, and particularly Japan . Damping is the most critical property for a system responding at resonance, such as base-isolated system. Damping plays a role in controlling the dynamic response of the system and reduces the peak displacement demand at resonance. In structural dynamics, the linear viscous damper model is very often used to represent the energy dissipation mechanism. Bilinear model is one of the most well-known mathematical model to model hysteresis or energy dissipation. This model is simple and easy to implement in computer software. Despite the model has energy dissipation feature, the energy dissipation mechanism varies with material. In
addition, due to the simplicity of the model, the mechanical properties of the model are correct at the calibration shear strain but can quickly deviate from the actual properties of the material if the energy dissipation mechanism of the material and the bilinear model is different.
Earthquake protection of structures has been an ongoing effort and over the years, many technologies and techniques have been developed. At the same time, most of these technologies and techniques have been tested by real earthquakes. Base isolation is one of the technologies that can be used not only to protect the structure but the contents of the structure from earthquake shaking. Base isolation can be achieved by introducing an isolation system between the superstructure and the substructure. The isolation system alters the dynamic properties of the structural system. This alteration leads to a fundamental mode of the system whereby the superstructure acts like a rigid body while the deformation is mainly in the isolation system. Some of the commonly used components to isolate a structure are elastomeric bearings and friction pendulum bearings. In addition to installing bearings as an isolation system, additional components are sometimes introduced to the isolation system such as dampers to dissipate energy and to control the response of the superstructure .
Bilinear model is a highly simplified mathematical model to
represent an element with restoring force and energy dissipation
capabilities. This model has been widely used due to the simplicity
both in the concept and the implementation in computer modelling.
When a bilinear model is subjected to sinusoidal input of amplitude
D, the peak displacements are represented by D− and D+
, the minimum and maximum peak displacement respectively, and
the minimum and maximum forces correspond to the minimum
and maximum peak displacements are F− and F+ respectively.
The force value when the hysteresis loop crosses the vertical axis
is called the characteristic force, Q. In addition, the hysteresis loop
of a bilinear model has two stiffnesses, the initial stiffness, and
the second stiffness, K2. The graphical representation of the bilinear
model and the parameters are shown in Figure 1.
Nonetheless, only four basic parameters are required to properly
define a bilinear model, i.e., the initial stiffness, K1, second
stiffness, K2 and the characteristic strength, Q and the yield displacement,
Dy . Therefore, the mechanical properties of the bilinear
model can be defined using these basic parameters.
Another key parameter of the bilinear model is the effective
stiffness, eff K , defined by Equation (1).
With the mass of the system known, the effective stiffness is
used as a proxy to estimate the effective period of the system. The
effective stiffness of the system, in terms of the basic parameters
of the bilinear model is shown in Equation (2).
In addition, the effective damping ratio of the system, eff β , for
the displacement amplitude D is shown in Equation (3).
In is evident from Equations (2) and (3) that both the effective
stiffness and effective damping ratio of the bilinear model is displacement
In order to evaluate the effectiveness of the isolation system, a
single-degree-of-freedom (SDOF) system was extended to a twodegree-
of-freedom (2DOF) base-isolated system. The SDOF system
consists of one lumped mass, ms , that represents the mass of
the superstructure, and is supported by a massless column with
stiffness coefficient, ks and damping coefficient, cs (Figure 2).
The damping in the superstructure was assumed to be represented
by a linear viscous damper model where the damper force is
proportional to the relative velocity of the system. The fixed-base
SDOF system used in this study is assumed to have a period of
0.4sec. The 2DOF base-isolated model is an extension of the SDOF
model shown in Figure 2. The 2DOF base-isolated system consists
of one additional mass, mb , that represents the base mass and the
restoring force in the isolation system is defined as fb (Figure 3).
The detailed explanation of this approach is available in the book
Earthquake-Resistant Design using Rubber by Kelly .
The absolute displacement of the ground, base mass, b m , and
the superstructure mass, ms , are represented by ug , ub and us
respectively. Two isolator models were considered in this study,
namely the linear spring with linear viscous damper model and
the bilinear isolator model. Although the mechanical properties of
the former are unlikely to be true in practical use since almost all
isolators have some degree of nonlinearity, but the response can
be linearized in some form such that the analysis and computation
can be simplified. In addition, the linear viscous damper model
is commonly used in structural analysis to represent the energy
dissipation mechanism in a structure . For the linear viscous
damper model, the isolator restoring force can be written in the
form shown in Equation (4).
where b c and b k are the linear viscous damper coefficient and
linear stiffness coefficient of the isolator model, and b v and b v are
the relative velocity and relative displacement of the base mass respectively. The relative displacements of the masses are defined
as vb:=ub-ug and vs:=us-ub For the bilinear model, the restoring
force depends on the loading and unloading path on the hysteresis
loop (Figure 1). Another form of energy dissipation mechanism is
the hysteretic damping. This energy dissipation model requires
solving for the response of the structure in the frequency domain.
This will further increase the complexity of the problem [6,7].
By taking the section cuts through the columns and the isolation
system and equating the inertia force experienced by the
masses to the restoring force, the equations of motion can be established.
The relative displacements for the masses in terms of
the absolute displacements are :s
vs:=us-ub and vb:=ub-ug for
the superstructure mass and base mass respectively
The equations of motion obtained from the section cut are
shown in Equations (5) and (6).
where mt is the isolated mass,
mt := ms + mb .
The equations of motion for the system can be further transformed
into the form of matrix equation of motion. The matrix
equation of motion for the system with linear spring and linear
viscous damper isolation system is shown in Equation (7) while
the equation of motion for the system with bilinear isolation model
is shown in Equation (8).
The matrix form equation of motion shown in Equations (7)
and (8) can be solved numerically using mathematical tools such
as Matlab. Before the second order matrix ordinary differential
equation is solved, the system was transformed into a system of
first order differential equations. The built-in toolbox, ode45 in
Matlab, was used to solve for the response.
Response spectrum is a representation of the peak response
of a series of linear SDOF systems with different periods subjected
to the same excitation. The floor response spectrum is a record of
peak response of linear SDOF systems subjected to the acceleration
experienced by the mass where the SDOF systems are placed
on. This approach can be a useful proxy for the performance of a
piece of equipment placed on a certain floor within a superstructure
that does not interact with the dynamic response of the superstructure
during an earthquake.
In this study, the floor response spectrum is used to study the
effectiveness of the isolation system. One of the objectives of isolating
a structure is to also protect the contents of the structure.
Hence, if the floor response spectra for the floor masses within
the isolated structure indicate that the peak responses are mainly
around the nominal isolation period, then the isolation system has
successfully protected the contents of the structure from high frequency
vibration. However, if peak responses are observed to be
present at shorter period (higher frequency) region relative to the
nominal isolation period, higher modes response might have excited.
Based on conventional modal analysis in structural dynamics,
the modal period decreases with increasing mode; therefore,
when higher modes response is excited, high frequency (short period)
vibration will be present within the superstructure; equipment
that is sensitive to high frequency vibration might be damaged.
The steps in generating a generic floor response spectrum
are shown in Figure 4.
The equipment with mass m is placed on the upper floor of
a two-degree-of-freedom (2DOF) system. The mass of the equipment
is assumed to be very small relative to the floor masses (
ms m and b m m ) and does not have significant contribution to
the dynamic response of the superstructure. The response of the
2DOF system with masses ms and ms was obtained by solving the
equation of motion (Equation (8)) numerically using the ground
motion recording as an input.
The absolute acceleration for the floor mass, s u , where the
equipment is placed is obtained.
The response of the mass m is obtained by solving the equation
of motion using the floor acceleration from 2 as the input.
Step 3 is repeated for the SDOF system of mass m with various periods
and the peak response is plotted against the natural period
to generate the floor response spectrum.
In this study, an arbitrary site located at 37.78 °N, 122.39 °W,
with Class D (stiff soil) soil type was selected. A suite of ground
motions was selected from the Ground Motion Selection Database
on the Pacific Earthquake Engineering Research Center (PEER)
website. The design spectrum for the site was established according
to the procedure outlined in the ASCE 7-16 design document
A suite of seven ground motions that has magnitude, M between
5 and 7, and rupture distance, rup R between 0-20km and matches the above site condition was selected using the Ground
Motion Selection Database by PEER. The details of the suite of
ground motions are summarized in Table 1.
Linear Spring and Linear Viscous Damper Isolation System
The first study on the effect of damping in the isolation system
was carried out using the 2DOF base-isolated structural model
with linear spring and linear viscous damper isolator. The fixedbase
frequency, s ω
, is defined as
while the nominal isolation frequency b ω
, is defined as
These two frequencies are assumed to be well separated. The
ratio of these frequencies, ε, is defined as
For systems with well separated frequencies, is assumed to
be in the order of between 10−1and 10−2 . The damping ratios of the
structure and the isolation system, and , are
Both damping ratios of the structure and the isolation system
are of the magnitude of .
The dynamic properties of the base-isolated structure were
obtained by solving the eigenproblem for Equation (7). The modal
frequencies, expressed in terms of the SDOF structural frequencies
and the nominal isolation frequencies are
For cases where the fixed-base and isolated frequencies are
well separated, the modal frequencies of the base-isolated system
can be approximated to . It can be observed
that the first mode period is the nominal isolation period,
i.e. the superstructure behaves and responds like a rigid body
and the displacement is only taking place in the isolation system;
whereas in the second mode, as the mass ratio, γ, is always less
than unity, the vibration frequency is increased slightly as compared
to the fixed-base frequency.
The mode shapes obtained from the eigen analysis, where the
eigenvector for the base mass is assumed to be unity, are
The graphical representation of the mode shapes is shown in
The mode shape for the first mode response agrees with the
modal frequency interpretation where the deformation is mainly
in the isolation system and deformation in the superstructure is
only of the order ε. However, the mode shape for the second mode
indicated that the deformation is mainly in the superstructure,
which is a less desired response as this defeat one of the objectives
in isolation a structure, i.e., to protect the contents of the structure.
Using the information from the mode shapes, the participation
factors for the system are and the effective
modal masses are
Considering these terms to the zeroth order of ε, i.e., the frequencies
are well separated, the second mode does not contribute to
the overall response of the structure. In other words, the second
mode response is almost orthogonal to the earthquake input and
should there be any energy in the earthquake input at this frequency,
the energy will not be transmitted into the superstructure.
Therefore, these results indicated that the effectiveness of
the isolation system depends on the period separation and more
importantly, the isolation system works by deflecting the energy through orthogonality in the dynamics of the structure and the
earthquake, and not by absorbing the energy of the earthquake.
However, energy dissipation mechanism still has an important
role in an isolation system
Further investigating into the modal damping ratios for each
mode shows that
The damping ratio for the first mode, 1 β
is a slight modification
of the damping ratio of the isolation system by the product
of the mass ratio and the frequency separation. However, the second
mode damping ratio, 2 β is influenced by the damping ratio in
the superstructure and the damping ratio in the isolation system.
If the damping ratio in the superstructure is small relative to the
damping ratio in the isolation system, the product of b β
could cause the second term in Equation (18) to be the governing
term for this expression.
In order to be able to work independently of the geometry of
the isolator, the mechanical properties of the bilinear model were
normalized by the area of the isolator, A. In addition, the pressure
exerts on the isolator, p is assumed to be 1000psi. Hence, the effective
stiffness, eff K and effective damping ratio, eff β of the bilinear
model shown in Equations (2) and (3) are normalized and
modified to the form shown in Equations (19) and (20) for the
normalized effective stiffness, Keff and βeffrespectively.
where K is the normalized second stiffness and Q
is the normalized
characteristic strength. The expression for the effective
stiffness can be further written in terms of the properties of the
isolator and the dynamic properties of the system, i.e., the pressure
on the isolator, p and the period correspond to the maximum
displacement, M T , as shown in Equation (21).
Likewise, the normalized characteristic strength, Q
cast into the form shown in Equation (22).
Prior to the analysis, the suite of ground motions listed in Table
1 was scaled according to the procedure outlined in the ASCE
7-16 design document . The design code requires that each
ground motions should be scaled to match the design spectrum
within the period range of 0.75 TM and 1.25 TM , where TM is the
period at the maximum displacement. In this study, TM was taken
as 3sec. In addition, the average acceleration response spectrum
obtained from the suite of ground motions should not be less than
the design spectrum for any period over the same period range.
These conditions resulted in a typical constrained linear leastsquares
problem that can be solved using computational mathematical
tools such as Matlab. The detailed solution to this problem
is explained in Doctoral Dissertation by Lee . The plot of the
scaled ground motions response spectra, the average acceleration
response spectrum and the target design spectrum is shown in
The case study on the structural model with linear spring and
linear viscous damper model is carried out by varying the period
separation and the damping ratio in the isolation system, as
these two factors were observed to influence the response of the
base-isolated system. The nominal isolation periods were chosen
to be 3sec and 4sec while the damping ratio in the isolation system
varied from 5% to 45%. The 2DOF base-isolated model was
used in this study and the equation of motion for the system was
solved numerically using Matlab with the scaled ground motion as
input. The comparison on the effect of damping ratio and period
separation was carried out. The peak isolator displacement and
peak story drift are summarized in Table 2 and Table 3 respectively.
The peak isolator displacement was observed to reduce with
the increasing damping ratio in the isolation system. In addition,
longer nominal isolator period, Tb also resulted in larger displace ment demand. However, there is no significant difference in the
peak roof drift for these cases. Besides displacement and drift,
the comparison was made on the roof floor acceleration response
spectra for systems with various damping ratios and period separation
values. The plot of the average roof floor spectrum for a
system with 3sec nominal isolation period and various damping
ratios in the isolation system is shown in Figure 7.
For a system with very low damping in the isolation system,
i.e., 5%, the roof floor response spectrum only peaked at the
nominal isolation period. However, as the damping ratio in the
isolation system increases, the peak at the nominal isolation period
becomes less apparent and another peak near the period of
0.3sec began to become significant. When the damping ratio in the
isolation system exceeded 35%, the peak near the period of 0.3sec
became the dominant peak in the roof floor response spectrum.
When the peak at the shorter period appears in the roof floor response
spectrum, this indicates that the vibration of the mass consists
of high frequency element. In other words, there is also high
frequency (low period) vibration components in the response.
This type of respond is less favorable as any vibration sensitive
equipment placed on this mass might be affected. The mean roof
floor response spectrum for a system with nominal isolation period
of 4sec and various damping ratios in the isolation system is
shown in Figure 8.
Similar observation can be made on the roof floor response
spectra in terms of the effect of increasing damping ratio in the
isolation system for the system with 4sec nominal isolation period.
For an isolation with low damping, the isolation system acts almost
like a filter, allowing only energy near the first mode period
to enter the structure. This shows that the isolation system is effective
when the damping is low. When the damping ratio increases,
the presence of high frequency vibration in the superstructure
becomes more significant thus reducing the effectiveness of the
isolation system. On the other hand, the larger the period separation,
the lower the response spectrum ordinate of the roof floor
mass. Hence, the acceleration experienced by the mass is reduced.
However, the price to pay for the reduced acceleration is the increased
displacement demand in the isolation system.
As mentioned before, the presence of high frequency component
is more significant as the damping in the isolation system
increases. From Figure 7 and Figure 8, it can also be observed that
when the damping in the isolation system is low, the dominant
peak in the roof floor acceleration response spectrum is near the
nominal isolation period. As the damping increases, the peak near
the second mode period, i.e. around 0.3sec, become more obvious
and when the damping ratio is beyond 45%, the dominant peak
in the floor acceleration response spectrum is at the second mode
The case study using bilinear model is a typical design scenario
where the design shear strain was chosen as 100%, which corresponds
to 40in. displacement, the damping ratio at the design
shear strain was set to 10% and the target period at the design
shear strain was 4sec. In addition, the design code also requires
the following set of equations (corresponds to Equations 17.5-1,
17.5-2 and 17.2-3 in ASCE 7-16) that is stipulated in the code to
be satisfied .
This set of equations are coupled; thus, these parameters will
have to be solved iteratively. Besides, three ground motion inten sities are considered in this study, namely the service level earthquake
(SLE), design basis earthquake (DBE) and maximum considered
earthquake (MCER). These ground motion intensities are
related where SLE is 50% of DBE and MCER is 150% of DBE. At
the MCER ground motion intensity, the ground motion parameter
at 1-second period, SM1 was chosen to be 1.2. By using the above
correlations for different ground motion intensities, the ground
motion parameter at 1-second period for 0.5DBE and DBE, SS1 and
SD1 were 0.4 and 0.8 respectively.
Using Equations (21) and (22), the normalized effective stiffness,
eff k and normalized characteristic strength, Q
ground motion intensity are 6.4psi/in. and 41.4psi respectively.
The normalized initial stiffness, k1 and normalized second stiffness,
k2 are 93psi/in. and 5.4psi/in. The ratio between the second
and initial stiffness, k1 / k 2 is approximately 18. This bilinear
model resembles an isolation system using lead plug bearings .
The set of equations stipulated in the design document shown
above was used to determine the targeted displacement, effective
period and effective stiffness of the isolation system at the SLE
and DBE intensities. The maximum displacements at the SLE
and DBE ground motion intensities for the system are 7.1in. and
21in. respectively. The normalized effective stiffness of the system
at the SLE and DBE ground motion intensities are 11psi/in. and
7.3psi/in. while the effective damping ratios at the SLE and DBE
ground motion intensities are 28% and 16% respectively. The effective
periods are 3sec and 3.7sec for the SLE and DBE ground
motion intensities respectively. The distribution of the effective
stiffness and effective damping ratio of the bilinear model up to
the displacement correspond to the design shear strain is shown
in Figure 9 and Figure 10 respectively.
Prior to conducting the dynamic analysis, the suite of ground
motions was scaled such that the response spectral ordinate corresponds
to 1-second period matched the ordinate of the design
spectrum at 1-second period, SM1, as shown in Figure 11. The
equation of motion shown in Equation (8) was solved using Matlab
with the scaled ground motions as the exciting force.
The matrix equation of motion for the base-isolated system
was solved numerically to obtain the response of the superstructure.
The absolute acceleration time series of the superstructure
mass was established to be used as the input to generate the roof
floor response spectrum. To further improve the efficiency of the
solution, the scalar equation of motion, i.e., the equation of motion
for generating the response spectrum, was solved using the
closed-form solution instead of the numerical time-stepping procedure.
The roof floor response spectra for the system and the average
roof floor response spectrum when subjected to the suite of
ground motions scaled to the design spectrum is shown in Figure 12. The average roof floor response spectrum shows two distinct
peaks, one at the period of around 4sec while the other peak is at
the period of about 0.3sec. The period of 4sec corresponds to the
nominal isolation period of the structure. The presence of another
peak at shorter period region suggested that higher modes response
has been excited or high frequency vibration has entered
the superstructure. Therefore, the response of the structure could
be detrimental to vibration sensitive equipment housed within
the structure. In other words, the isolation system was not effective
in protecting the contents of the structure from high frequency
Similar procedure was carried out using the suite of ground
motions scaled to the SLE and DBE intensities. The roof floor response
spectra and the average roof floor response spectrum was
generated and shown in Figure 13 and Figure 14 for the SLE and
DBE intensity respectively.
The roof floor response spectra obtained at the SLE and DBE
ground motion intensities showed similar trend as the roof floor
response spectrum obtained from the system subjected to the
MCER ground motions. Two peaks are observed at the period
around 4sec and 0.3sec. The comparison of the average roof floor
response spectrum for different intensities is shown in Figure 15.
The presence of peaks at the short period region (around
0.3sec) indicated that the structural model with bilinear model,
regardless of the ground motion intensity is experiencing high
frequency vibration within the superstructure. In other words,
the isolation system is not effective in protecting the contents of
the structure, particularly vibration sensitive equipment. A comparison
was made on the mean peak isolator displacement and
the maximum design displacement obtained using the equation
stipulated in the ASCE 7-16 design code. The observation is summarized
in Table 4.
The mean peak isolator displacement obtained from a suite of
seven ground motions did not deviate too much from the values
estimated using the equation stipulated in the design code for various
ground motion intensities.
Base isolation is an effective technique to mitigate earthquake
damage on structures. As shown through analytical and numerical
examples, the base isolation is effective when the damping ratio in
the isolation system is low. Nonetheless, some level of damping in
the isolation system is needed so that the design of the isolation
system can be more economical and for better control of the dynamic
response. Incorporating high level of damping to the isolation
system may help to reduce the displacement demand but at
the same time could excite high frequency response. As a result,
the intention of protecting the contents of the structure might be
jeopardize, particularly when the structure houses vibration sensitive
equipment. The bilinear model is a popular mathematical
model for structural analysis due to the simplicity and ease of implementation.
However, an engineer or designer should consider the variation and deviation of the mechanical properties of the
model to the actual isolation system that will be used to ensure
that the superstructure and the isolation system are properly design
Part of the work in this study was carried out as the research
work for a Doctoral Degree awarded at the University of California,
Berkeley. The doctoral study was made possible with the
scholarship provided by the Malaysian Rubber Board and summer
fellowship provided by the Department of Civil and Environmental
Engineering, University of California, Berkeley.