Engineering Structures 26 (2004) 485–497
www.elsevier.com/locate/engstruct
A parametric study of linear and non-linear passively damped
seismic isolation systems for buildings
Cenk Alhan, Henri Gavin Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708-0287, USA
Received 18 April 2003; received in revised form 21 October 2003; accepted 10 November 2003
Abstract
The effects of near-field ground motions with large velocity pulses have motivated passive damping requirements for the protection of seismically isolated structures. Structures in which the first mode damping exceeds 20% or 30% typically do not exhibit
classical modes, and simulation via a simple superposition of uncoupled second order equations is not possible. When the damping is produced by viscous or linear visco-elastic devices, we can, however, gain insight into the dynamic behavior of these structures using a convenient first-order formulation and frequency domain methods. When the damping effects are created by nonlinear mechanisms such as yielding or friction, the behavior of the structure is amplitude dependent and analyses are commonly
carried out in the time domain. In this paper, frequency domain analysis and earthquake time history analysis are applied to
study the influence of isolation damping on higher-mode effects and inter-story drift ratios. Because higher mode effects, plan irregularities, and bi-directional ground motions are all important attributes of the dynamic behavior of these structural systems, a
simple comparison of isolation damping mechanisms can not be carried out via simple single or two degree of freedom realizations. In order to incorporate these important details in the study of the dynamic behavior of these structures, a set of 8-story
proto-type building models with L-shaped floor plans, different isolation periods, isolation damping characteristics, and levels of
isolation stiffnesses are examined.
# 2004 Elsevier Ltd. All rights reserved.
Keywords: Seismic base isolation; Viscous damping; Hysteretic damping; Vibration control; Earthquake engineering
1. Introduction
Seismic isolation systems for building structures are
designed to preserve structural integrity and to prevent
injury to the occupants and damage to the contents by
reducing the earthquake-induced forces and deformations in the super-structure [6]. The compliant elastomeric bearings and frictional sliding mechanisms
installed in the foundations of seismically isolated
structures protect these structures from strong earthquakes through a reduction of stiffness and an increase
in damping [9]. The reduction of stiffness is intended to
detune the structure’s fundamental period from the
characteristic period of earthquake ground motions,
Corresponding author. Tel.: +1-919-660-5201; fax: +1-919-6605219.
E-mail addresses: [email protected] (H. Gavin), http://www.duke.edu/hpgavin/ (H. Gavin).
0141-0296/$ - see front matter # 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engstruct.2003.11.004
and isolated structures have long fundamental periods
of vibration as compared to the periods of most commonly encountered earthquake ground motions. Base
isolation systems are being implemented in an increasing number of projects in highly seismic areas throughout the world. In full-scale implementation and modelscale investigation, seismic isolation methods performed as desired [1,14]. In addition, during the 1994
Northridge Earthquake, base isolated structures
behaved as designed [17].
Ground motion records from recent earthquakes,
such as the January 17, 1995 Kobe Earthquake, the
August 17, 1999 Kocaeli Earthquake, and the September 21, 1999 JiJi Earthquake show that near-field
ground motions may contain strong, long period velocity pulses. This type of ground motion may be very
damaging to base-isolated structures. Protection of
seismically isolated structures from strong, long-period
velocity pulses can be challenging [4,21]. A study by
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C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
Heaton et al. [4] showed that a base isolated building
subjected to a near-field Mw 7.0 blind thrust earthquake could result in large isolator drifts (in excess of
50 cm) even when damped at 25% critical damping.
Large displacements at the isolation interface during a
strong earthquake can lead to buckling or rupture of
the isolation bearings [11,16,21]. Koh and Balendra
[11] examined the seismic response of base isolated
buildings including the P–D effects of the isolation
bearings and showed that due to large displacements of
the bearings under earthquake excitation, buckling may
become an important issue especially when the buckling safety factor is low. Kelly [10] discussed the need
for high levels of damping in order to limit isolator
drifts within code-mandated limits and used a two
degree of freedom model to demonstrate the potential
for dynamic amplification in isolated systems with high
isolation damping. Although, large levels of damping
reduce isolator displacements in the fundamental
mode, they impart forces into the structure which can
increase structural accelerations and deformations in
higher modes, and can increase inter-story displacements [10]. The potential for increased damping to
increase high-mode response depends on the flexibility
or rigidity of the structure. Feng and Shinozuka [3] and
others have shown that increased damping causes
increased accelerations in flexible isolated bridge structures. Sadek and Mohraz [19] analyzed a set of 6-dof
structures equipped with a damper and reported that
for flexible structures, increased damping decreases the
base displacement but increases the floor accelerations.
However, they only considered extremely low (f¼ 0:05)
and extremely high (f¼ 0:40) damping cases, and did
not take the intermediate levels of damping into
account. Studies discussed above did not examine a
broad range of isolation damping and stiffness. Isolation damping and stiffness can also be provided by
non-linear isolation mechanisms and therefore the
effects of non-linear isolation parameters, i.e. yield displacement, yield force, and the post-yield to pre-yield
stiffness ratio, on the structural responses may be investigated.
In this study, we examine the potential drawbacks of
heavy isolation damping in the context of the response
of a realistic eight story base isolated building to nearfield ground motions. The objectives of this paper are
to investigate performance limits of passive linear and
non-linear isolation systems, to identify levels of heavy
isolation damping which increase structural accelerations and inter-story drifts, and to identify appropriate
combinations of isolation parameters which reduce
base displacements without significantly increasing
floor accelerations and inter-story drifts. Section 2
describes the structural model, Sections 3 and 4 focus
on base isolation systems with linear viscous damping
and non-linear hysteretic damping, respectively, and
conclusions are given in Section 5.
2. Structural model
To investigate the effect of isolation system parameters on the dynamic response of buildings, the behaviors of a set of prototypical 8-story isolated structures
with 46 different isolation systems, modeled with linear
stiffness and damping elements and with bi-axial hysteretic elements, are examined via complex frequency
response analysis and transient time history response
analysis. The responses of these 16 linearly isolated
structures are compared to those of 30 non-linearly isolated structures using three historical earthquake
records. In these analyses, floor accelerations, drift
ratios, and isolation deformation are evaluated. The
following simplifying assumptions are made: dynamic
soil-structure interaction, P–D effects, in-plane shear
deformation of the floors, and non-uniform ground
motion are neglected; linear material behavior is
assumed for the super-structure and the linear isolation
system; and the response of the non-linear isolation
bearings is simulated with bi-axial hysteretic models.
The combination of isolation system damping and
superstructure damping results in complex modes; the
center of stiffness and center of mass are not coincident; and bi-directional horizontal ground motion excitation is considered in time history analyses.
The prototypical structure used in this study has an
L-shaped, asymmetric plan, as shown in Fig. 1. The
weight of the structure is 33.4 MN and the inter-story
clear height is 3 m. In structures such as this, where the
center of mass does not coincide with the center of
rigidity, the modes exhibit lateral-torsional coupling
and are not aligned with the framing layout of the
structure. Periods of this structure are associated with
modes in which the motion is primarily (but not
entirely) in the NS0 and EW0 directions, as shown in
Fig. 1. The EW0 axis passes through the center of gravity and the center of stiffness. The NW0 axis is perpendicular to the EW0 axis. Each floor has three degrees of
freedom as shown in Fig. 1: two lateral degrees of freedom in the x and y directions and a rotational degree
of freedom about the vertical axis are placed at the
center of the floor mass which is also the geometric
center and which is the same at every floor. The asymmetric plan results in modes with both lateral and torsional deformation [20]. Although this structural model
has an asymmetric floor plan, the mass and rigidity
centers are relatively close to one another, as shown in
Fig. 1, and the lateral-torsional coupling effects are not
prominent.
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
487
Fig. 1. Plan and elevation of the 8-story base isolated structure. Circles indicate isolator locations. The z-axis passes through the center of mass
of each floor. The center of stiffness is at coordinate (1.03,0.55 m).
The equations of motion for a damped structure
subjected to bi-directional base excitation are:
M€
qðtÞ þ C q_ ðtÞ þ KqðtÞ ¼ MH€zðtÞ
ð1Þ
where M is a positive definite structural mass matrix, C
is a non-negative definite structural damping matrix, K
is a positive definite structural stiffness matrix, €zðtÞ ¼
½€zx ; €zy T contains the two orthogonal components of
horizontal ground acceleration, H is a two-column
boolean earthquake input matrix, and q is a displacement vector which contains the displacement of the
centers of masses of the stories. For this isolated structure, the displacement vector q ¼ ½x1 ; y1 ; h1 ; :::; x9 ; y9 ; h9 T
contains the positions of the centers of masses at the
9 levels of the building including the isolation level
with respect to the ground as shown in Fig. 1. The
matrices Miso, Kiso, and Ciso represent the structural
mass, structural stiffness, and structural damping
matrices of the isolated structure and Mfix, Cfix, and
Kfix model the fixed-base structure which has three
fewer degrees of freedom.
The lateral inter-story stiffness of a story of the fixed
base structure in the x, y, and h directions is given by
P
2P
3
0
i kxi
P
P i kxi Yi
5
KS ¼ 4 0 P
Pi kyi
Pi kyi Xi 2
2
i kxi Yi
i kyi Xi
i kxi Yi þ kyi Xi
ð2Þ
where kxi and kyi represent the lateral stiffness of the ith
column in the x and y directions, and Xi and Yi denote
the x and y coordinates of the ith column with respect
to the center of the coordinate system. The torsional
stiffness of each individual resisting element is considered negligible [8]. Formulation of the element stiffness matrix corresponding to the isolation level follows
the same procedure. Since the locations of isolation
bearings in this model are the same as the locations of
the columns, the resulting stiffness matrix of the isolation level, KI, will have the same form as Eq. (2) with
the lateral stiffnesses of columns replaced by the lateral
stiffnesses of isolation bearings. The block-tri-diagonal
symmetric structural stiffness matrices of the fixed base
structure, Kfix, and the isolated structure, Kiso, are then
assembled according to the conventions of a shear
building model [15].
The mass center and the origin of the coordinate system are coincident for each floor in this model, and the
mass matrix of each floor is diagonal. The story mass
matrix is:
3
2ð
dm
0
0
7
6
7
6 A
ð
7
6
7
0
dm
0
ð3Þ
MS ¼ 6
7
6
A
7
6
ð
5
4
0
0
r2 dm
A
488
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
where A is the floor area, dm is a differential mass of
the floor, and r is the distance from the origin of the
coordinate system. The diagonal mass matrices Miso
and Mfix are assembled by stringing the story mass
matrices MS along the diagonal.
In this study the damping matrix of the fixed base
structure, Cfix, is determined from the eigenvector
matrix of the general eigenvalue problem
K fix Ufix ¼ M fix Ufix Kfix
fix
ð4Þ
th
where K is a diagonal matrix with its i diagonal
element being the ith eigenvalue corresponding to the
ith mode and Ufix is the mass normalized eigenvector
matrix. For the fixed base structure, the damping
matrix is calculated from
fix T fix fix
U
C U ¼C
ð5Þ
where C is a diagonal matrix with its ith diagonal
element being equal to 2fixni, where fi is the modal
damping ratio of the ith mode. From Eq. (5), Cfix can
be obtained as
T fix 1
C fix ¼ Ufix
C U
:
ð6Þ
Note that in general, Cfix is not mass and stiffnessproportional, and is typically diagonally-dominant and
fully populated.
The fixed-base periods, Tfix, of the superstructure for
the first three modes are 0.90, 0.81, and 0.60 s in NS0 ,
EW0 and torsional directions. The fixed-base damping
ratios, ffix, of the superstructure corresponding to the
first three modes are 2%, 3%, and 2%. The floor plan
dimensions, natural periods and damping ratios for the
fixed-based and nominal-isolated structure are similar
to the USC Hospital Building.
3. Linear isolation system
This section primarily addresses the linear viscous
damping in the isolation system. Approximating the
energy dissipation mechanisms with a linear viscous
damping model results in a simplified linear model
which allows a convenient frequency domain analysis.
3.1. Modeling of the linear isolation system
Formulation of the damping matrix of the isolated
structure requires a methodology which makes use of
Cfix and the damping of the isolation level, CI. Let
CIxx ; CIyy , and CIhh be the total damping of the isolation system in the x, y and h directions, respectively,
and let CI and CF be defined as
2
3
0
CIxx 0
6
7
CI ¼ 4 0
CIyy 0 5
0
0
CIhh
2P
3
fix
0
0
i;j Cij
6
7
P
fix
7
0
CF ¼ 6
k;l Ckl
40
5
P
fix
0
0
C
m;n mn
ð7Þ
where i;j ¼ 1;4;7;:::;22; k;l ¼ 2;5;8;:::;23; and m;n ¼
3;6;9;:::;24. Here, CF is the viscous damping matrix of
the first story, neglecting the lateral-torsional coupling
of the damping terms. Then, the damping matrix for
the isolated model with linear viscous damping, Ciso is
2
3
0318
CI þ CF CF
6 CF
7
7
C iso ¼ 6
ð8Þ
fix
4
5
C
0318
In order to study the effect of different levels of isolation damping on structures with different levels of
isolation stiffness, two dimensionless parameters, fk and
fc, are selected to represent the stiffness and damping
levels. The stiffness, KI, and damping, CI, of the isolation system are calculated to give the natural isolation periods, Tiso, of the nominal structure to be 2.25,
2.12, and 1.94 s with fiso equal to 0.15 for the first
three modes. Recall that the fixed base periods, Tfix, of
this superstructure are 0.90, 0.81, and 0.60 s for the
first three modes, as described in Section 2. The superstructure remains unchanged for all of the isolation
systems examined in this analysis. The fundamental
isolation period of this proto-type structure falls into
the 2–3 s range where most isolation periods of baseisolated buildings lie, as reported by Makris [13]. Some
examples of fixed base periods, Tfix, and isolated base
periods, Tiso, of existing buildings include the 5-story
New Zealand Parliament Building with T fix ¼ 0:5 and
T iso ¼ 3:5 s, the 6-story West Japan Computer Center
with T fix ¼ 0:7 and T iso ¼ 3:0 s, and the 8-story USC
University Hospital Building with T fix ¼ 0:7 and T iso ¼
2:25 s [2,12,17], which are close to the fixed and nominal isolation periods of our prototype building. The
isolation period and damping ratio of the San Bernardino County Medical Center are 3.0 s and 0.50 [10]. In
this study, structures with different isolation damping
and stiffness levels are obtained by multiplying KI and
CI with scaling factors fk and fc. Note that fk ¼ 1 and
fc ¼ 1 correspond to the nominal case. The fundamental mode period, Tiso, the fundamental mode damping
ratio, fiso , and the corresponding average total lateral
~ ¼ fk ðKIxx þ KIyy Þ=2 and the average total
stiffness k
damping ~c ¼ fc ðCIxx þ CIyy Þ=2 of the isolation level for
16 different combinations of fk and fc are given in
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
489
Table 1
~, and isolation damping rates, ~c, for all cases
Fundamental mode periods, Tiso, damping ratios, fiso , isolation stiffness, k
Case
~ ¼ 144;305
(1) k
~ ¼ 288;610
(2) k
~ ¼ 577;220
(3) k
~ ¼ 865;830
(4) k
kN=m
kN=m
kN=m
kN=m
fk ¼ 0:5
fk ¼ 1:0
fk ¼ 2:0
fk ¼ 3:0
(a) ~c ¼ 19;286 kN=m=s
fc ¼ 0:6
(b) ~c ¼ 32;144 kN=m=s
fc ¼ 1:0
(c) ~c ¼ 64;288 kN=m=s
fc ¼ 2:0
(d) ~c ¼ 96;432 kN=m=s
fc ¼ 3:0
Tiso
fiso
Tiso
fiso
Tiso
fiso
Tiso
fiso
3.08 s
2.25 s
1.70 s
1.47 s
0.14
0.09
0.06
0.04
3.06 s
2.24 s
1.69 s
1.46 s
0.23
0.15
0.09
0.05
3.00 s
2.20 s
1.67 s
1.45 s
0.46
0.30
0.18
0.12
2.87 s
2.12 s
1.63 s
1.42 s
0.72
0.46
0.26
0.18
Table 1. The damping rates are the same in each case,
1–4, and the isolation stiffness is the same in each case,
~ and ~c in
a–d. This is demonstrated by the values k
Table 1. As a result, the damping ratios are larger in
the low-stiffness/long-period cases, even though the
damping rates are the same for all stiffness cases. For
purposes of the parametric study, a broad range of
values are considered, including long periods and high
damping ratios. The largest values are comparable to
those of the San Bernardino County Medical Center.
3.2. Complex frequency response analysis
The linear isolation system is considered first for the
convenience of complex frequency response analyses
including non-proportional damping effects. Transmissibility ratios for roof drift, floor acceleration, base
drift, roof rotation drift, and base rotation are calculated for all cases with the excitation in x and y directions, separately. The transmissibilities are observed to
be small when the excitation direction is perpendicular
to the response direction. In this section, the frequency
response plots for x-direction excitation is presented
and the trends are similar for the y-direction excitation.
Five transmissibility performance indices are defined
as follows:
T1 ¼
T2 ¼
ðx9 x1 Þ=h
€zx
9
X
ð€
xi þ €zx Þ=8
i¼2
€zx
ð9Þ
ð10Þ
T3 ¼
x1
€zx
ð11Þ
T4 ¼
ðh9 h1 Þ
€zx
ð12Þ
T5 ¼
h1
€zx
ð13Þ
Here, €zx and €zy are the two orthogonal components of
horizontal ground acceleration in x and y directions,
and h is the inter-story height.
Fig. 2 shows the transmissibility performances in xdirection for roof drift ratio, T1, floor acceleration, T2,
and base drift, T3, transmissibilities. Fig. 3 shows the
transmissibility performances for roof rotation drift, T4
and base rotation, T5. These plots correspond to the
extreme cases of levels of stiffness: case 1, where fk ¼
0:5 (lowest stiffness case), and case 4, where fk ¼ 4
(highest stiffness case), and all values of damping are
presented. It can be seen from these figures that
increasing damping decreases the first mode responses.
However, this trend is reversed in frequency regions
between higher mode resonances. Increasing damping
increases the response significantly in-between the first
and second modes for roof drift and floor accelerations, especially for the low stiffness case (Fig. 2(i) and
(iii)). A similar trend can be observed for the roof
rotation. However, base drift transmissibility does not
increase with high damping for both low and high stiffness cases (Fig. 2(v) and (vi)). For base rotation, one
observes some negative effect of high damping,
especially for the low stiffness case, case 1, (Fig. 3).
These trends vary monotonically through cases 2 and 3.
Note that the logarithmic scale indicates that the
increase in response in the higher modes due to larger
levels of isolation damping are at most only 10% of the
reduction in the fundamental mode. The significance of
these relatively minor higher mode effects are examined
further via time history analyses.
3.3. Time history analysis
Time history analyses of the structures are carried
out for bi-directional ground motions, for 10 different
historical earthquake records: the 1940 Imperial Valley
El Centro record, the 1979 Imperial Valley Meloland
record, the 1994 Northridge Rinaldi record, the 1994
Northridge Sylmar record, the 1989 Loma Prieta Los
Gatos record, the 1995 Kobe JMA record, the 2000
Kocaeli YPT record, the 1999 JiJi Shikhkang record,
the 1999 JiJi TCU078 record, and the 1999 JiJi TCU129
record. Representative results obtained for three earthquake records, namely 1940 El Centro, 1995 Kobe
JMA, and 1999 JiJi TCU129, which account for low,
moderate and large level ground motions, respectively,
are presented here. The bi-directional acceleration and
displacement response spectra of the three earthquake
records for 5% and 10% damping are shown in Fig. 4.
In time history analyses, the HHT-a method [7] is used
490
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
Fig. 2. Transmissibility performances for excitation in the x-direction, (i), (ii), roof drift ratio, T1, (iii), (iv) floor acceleration, T2, (v), (vi), base
drift, T3.
to compute the responses. The HHT-a method is well
suited for numerical integration of large second order
systems which may have very high frequency modes
which do not contribute significantly to the response,
but which may require very short time steps for
numerical stability. The HHT-a method enables accu-
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
491
Fig. 3. Transmissibility performances for excitation in the x-direction, (i), (ii), roof rotation drift, T4, (iii), (iv) base rotation, T5.
rate simulation of these systems with a larger time step.
In the software developed, the algorithm constants are
a ¼ 0:05, b ¼ 0:276, and c ¼ 0:55, which ensures
stability [7]. The time step used is 0.005 s. Three criteria, base drift, J1, floor acceleration, J2, and story
drift ratio, J3 are considered for the evaluation of performance of the analyzed structures are as follows:
J1 ¼ maxt ðd1 Þ
ð14Þ
J2 ¼ maxt;i ½ðdi di1 Þ=h
"
!, # 12
X
2
2
axi þ ayi
J3 ¼ maxi
t
ð15Þ
ð16Þ
t
where t is time, h is the inter-story height, i is the floor
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
number, di ¼ x2i þ y2i is the displacement of the ith
Fig. 4. Acceleration and displacement bi-directional response spectra for Imperial Valley-El Centro, JiJi-129 and Kobe-JMA earthquake records.
floor, xi is the displacement in the x-direction, yi is the
displacement in the y-direction, as shown in Fig. 1,
€i þ €zx is the total acceleration in the x-direction,
axi ¼ x
and ayi ¼ €yi þ €zy is the total acceleration in the y-direction. Recall that €zx and €zy are the two orthogonal components of horizontal ground acceleration in x and y
492
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
Fig. 5. Performance of structures with different combinations of levels of stiffness and damping, (i), (iv), (vii) base drift, J1 (ii), (v), (viii) story
drift ratio, J2 (iii), (vi), (ix) floor acceleration, J3.
directions. A threshold is set to discard the values of
acceleration lower than 0.2 m/s2, to avoid including
the portions of the records in the average where insignificant or zero ground accelerations are recorded for a
long time period.
Plots including these performance criteria versus different combinations of fk and fc are shown in Fig. 5. It
can be seen from these figures that the base drift is suppressed as the damping is increased for all stiffness
levels with the level of suppression being much higher
for lower levels of stiffness. However, increasing damping does not always decrease the story drift ratio. On
the contrary, increasing damping increases the storydrift ratio for certain combinations of damping and
stiffness. This undesirable effect is more pronounced
especially for cases with long period isolation systems
ðT iso > 2:5 sÞ and for pulse-like earthquakes (Kobe
and JiJi). This effect is not significant for the El Centro
earthquake record. A similar trend is observed for floor
accelerations. Floor accelerations increase with
increased damping for long period isolation systems in
the pulse-like earthquake records. This behavior is, in
fact, predicted by the frequency domain analysis.
4. Non-linear hysteretic damping
In this section, the structural system is considered to
be isolated by elements which exhibit hysteretic
material behavior. When damping effects are created
by non-linear mechanisms such as yielding or friction,
the behavior of the structure is amplitude dependent
and analyses are commonly carried out in the time
domain.
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
4.1. Modeling of yielding hysteretic isolation system
study, the values of A, c, and b are selected as 1, 0.9,
and 0.1, respectively.
A damped structural system subjected to bi-directional base excitation and possibly experiencing hysteretic material behavior is described by:
M€
qðtÞ þ C q_ ðtÞ þ KqðtÞ þ RðqðtÞ; q_ ðtÞÞ ¼ MH€z
493
ð17Þ
The term RðqðtÞ; q_ ðtÞÞ accounts for the non-linear
restoring forces. Typical hysteretic material behavior is
shown in Fig. 6 where fy is the yield force, Dy is the
yield displacement, and a is the post-yield to pre-yield
stiffness ratio. When the damping effects are due to
material yielding, the restoring forces RðqðtÞ; q_ ðtÞÞ may
be described by the equations for biaxial hysteresis [18]:
Fx ¼ aKhi x1 þ ð1 aÞfy Zx
ð18Þ
Fy ¼ aKhi y1 þ ð1 aÞfy Zy
ð19Þ
where Fx and Fy are the hysteretic forces in x and ydirections, respectively. Here, Zx and Zy account for
the bi-direction and biaxial interaction of hysteretic
forces. The differential equation used for biaxial hysteretic behavior is given by:
Z_ x Dy ¼ Ax_1 Zx2 ðcsignðx_ 1 Zx Þ þbÞ x_ 1
Zx Zy csignðy_ 1 Zy Þ þ b y_ 1
ð20Þ
h i
Z_ y Dy ¼ Ay_ 1 Zy2 csignðy_ 1 Zy Þ þ b y_ 1
Zx Zy ðcsignðx_ 1 Zx Þ þ bÞ x_ 1
ð21Þ
in which, A, c, and b are dimensionless quantities to be
selected such that A ¼ b þ c for Zx and Zy to be bounded between +1 and 1 when yielding occurs [15].
Here, x_ 1 and y_ 1 are the relative velocities at the bearing
locations in x and y-directions, respectively. In this
4.2. Time history analysis
In order to solve for the non-linear restoring forces,
Eqs. (20) and (21) are solved via fourth order RungeKutta method to obtain the hysteretic quantities Zx
and Zy at each time step. These are then inserted in
Eqs. (18) and (19) to find the non-linear restoring
forces in x and y directions, respectively.
The hysteretic behavior, which in turn creates damping, depends primarily on three parameters: yield force,
fy, yield displacement, Dy, and post-yield to pre-yield
stiffness ratio, a. Combinations of different values of
these three parameters are used to study the effect of
different isolation characteristics.
The ‘‘nominal’’ structure with hysteretic yielding isolation system is defined to have a yielding force of
fy ¼ 490 kN, a yielding displacement of Dy ¼ 0:007 m
and a post-yield to pre-yield stiffness ratio of a ¼ 0:15.
With these values, the response of the structure is very
close to the ‘‘nominal’’ structure with the linear isolation system, which corresponds to the case with fk ¼
1:0 and fc ¼ 1:0 in Table 1. Now, to study the effects of
fy and Dy, two sets of time history analyses, set A and
set B, are carried out.
For each set of analyses, the post-yield stiffness, aKhi
is kept constant. Thus, a values are changed such that
all isolation systems would have the same post-yield
stiffness of the ‘‘nominal’’ structure which is Klo ¼
0:15 490=0:007 ¼ 10; 496 kN=m for analysis set A,
and Klo ¼ 0:20 490=0:007 ¼ 14; 004 kN=m for set B.
The a values for other isolation systems are changed
such that the post-yield stiffness is 10,496 kN/m for set
A and 14,004 kN/m for set B. The different values of
the parameters, fy ; Dy ; a used for these two sets of
analyses, A and B, are shown in Table 2. The yield
forces and the yield displacements considered in this
study are in the typical range that is reported for elastomeric bearing characteristics [5].
Plots of the performance criteria J1, J2, and J3, versus different combinations of fy and Dy are given in
Figs. 7 and 8. Fig. 7(i), (iv) and (vii) show that for all
Table 2
Post-yield to pre-yield stiffness ratios for set A (aA) and for set B (aB)
(a) Dy ¼
0:0035 m
Case
Fig. 6.
A typical hysteretic material behavior.
(1) fy
(2) fy
(3) fy
(4) fy
(5) fy
¼ 368
¼ 490
¼ 613
¼ 735
¼ 980
kN
kN
kN
kN
kN
(b) Dy ¼
0:0070 m
(c) Dy ¼
0:0140 m
aA
aB
aA
aB
aA
aB
0.10
0.075
0.06
0.05
0.0375
0.133
0.10
0.08
0.067
0.05
0.20
0.15
0.12
0.10
0.075
0.266
0.20
0.16
0.133
0.10
0.40
0.30
0.24
0.20
0.15
0.532
0.40
0.32
0.266
0.20
494
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
Fig. 7. Performance of structures (set A) with a post-yield stiffness of 10,496 kN/m (i), (iv), (vii) base drift, J1 (ii), (v), (viii) story drift ratio, J2
(iii), (vi), (ix) floor acceleration, J3.
types of earthquake excitations, there is a clear monotonic decrease in base drift as the yield force is
increased (cases 1 to 5). As the yield displacement is
decreased (cases a to c), there is a decrease in base
drift. However, it is important to note that the rate
of decrease in base drift diminishes with smaller
yield displacements. This effect is particularly clear for
the Kobe earthquake where there is almost no difference between case b ðDy ¼ 0:007 mÞ and case c ðDy ¼
0:014 mÞ.
On the other hand, story drift ratios do not decrease
monotonically as the yield force of the isolation system
increases (cases 1 to 5). For the lower levels of yield
force, the story drift ratio decreases with increasing
yield force; for higher levels of yield force, the story
drift ratio increases with increasing yield force, depend-
ing on the characteristics of the ground motion, also,
increasing the yield displacement may increase or
decrease the story drift ratio depending upon the level
of the yield force. Therefore, we see that there is an
appropriate combination of yield force and yield displacement required to obtain a minimum level of story
drift. This isolation system corresponds approximately
to case 3b.
The floor acceleration response follows the same
trends as the story drift ratios. Again, for lower values
of yield force, increasing the yield force decreases the
floor accelerations. For higher values of yield force,
increasing fy increases floor accelerations. This effect is
clearly evident for the El Centro and Kobe earthquakes. For the JiJi earthquake, the floor acceleration
seem to monotonically increase with increasing yield
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
495
Fig. 8. Performance of structures (set B) with a post-yield stiffness of 14,004 kN/m (i), (iv), (vii) base drift, J1 (ii), (v), (viii) story drift ratio, J2
(iii), (vi), (ix) floor acceleration, J3.
force. Increasing yield displacement increases floor
accelerations for low levels of yield force and vice-versa
for high levels of yield force. Thus, there is an appropriate combination of yield force and yield displacement required to obtain a minimum level of floor
accelerations, and these values appear to depend upon
the details of the ground motion record.
Fig. 8 shows the results of the analyses of set B with
a post-yield stiffness of 14,004 kN/m (as compared to
the post-yield stiffness of 10,496 kN/m in set A). It is
seen here that increasing post-yield stiffness decreases
the base drift while it both increases the story drifts
and floor accelerations. This is particularly noted for
the Kobe-JMA record, which is a high-level, pulse-like
earthquake. The changes are minor for the low-level,
moderate earthquake record, El Centro. For the JiJi
record, we see a significant increase in inter-story drift
ratios, a small decrease in base-drift and a small
increase in floor accelerations. On the other hand, the
general trend of the curves are very similar to the general trend of curves obtained for set A. Thus, the discussions above for set A are qualitatively the same for
the set B simulations.
This study also enables a comparison of the two different energy dissipation mechanisms, namely, the linear isolation system and the non-linear hysteretic
isolation system, in terms of the three performance criteria that we have defined. Fig. 5(i), (iv), (vii) and
Fig. 7(i), (iv), (vii) show that the base drift values
obtained for the linear and non-linear isolation systems
496
C. Alhan, H. Gavin / Engineering Structures 26 (2004) 485–497
are in the same range, which serve as a basis for comparison. This is particularly true for case 2b which is
defined as the ‘‘nominal’’ system as discussed before.
When we consider the two other performance criteria,
for the El Centro record, story-drift ratios and floor
accelerations are also about the same for the linear and
non-linear isolation systems as can be seen from
Fig. 5(i), (ii), (iii) and Fig. 7(i), (ii), (iii). However,
when a higher level earthquake, i.e. the Kobe JMA record, is considered, we see that there is a significant
decrease, about 100% to 200% change in story-drift
ratio and floor acceleration ranges when we change the
energy dissipation mechanism from linear to non-linear, especially for high stiffness systems. This can be
seen from Fig. 5(v), (vi) and Fig. 7(v), (vi). A similar
trend is observed in story-drift ratio, between linear
and non-linear systems for the JiJi record.
5. Conclusions
An eight story, L-shape planned, structural model is
used to study the influence of isolation damping and
isolation stiffness on higher-mode effects, inter-story
drifts and floor accelerations. Two types of isolations
systems, namely, linear viscously damped and non-linear yielding hysteretic type isolation systems are studied. The linear systems are examined with both time
and frequency domain analyses. The non-linear systems
are examined only in the time domain.
For the linear isolation systems, a complex frequency
response analysis shows that increasing damping
decreases the first mode response but increases the
response at frequencies between the higher mode resonances, especially in terms of the roof drift and floor
accelerations. This effect is more pronounced for structures with lower isolation stiffnesses (longer periods).
Time history analyses show that increasing damping in
the isolation system increases story drift ratios and
floor accelerations for low stiffness isolation systems
(T iso > 2:5 s). These effects are more pronounced in
pulse-like ground motions, such as Kobe-JMA. Base
drift, however, always decreases with increased isolation damping and increased isolation stiffness for all
earthquake records. It is important to note that the fast
and convenient complex frequency response analysis is
predictive of the effects of damping on base drift and
floor acceleration response in damped base-isolated
buildings. This is confirmed via detailed, computationally intensive time history analyses.
For non-linear systems, time history analyses show
that increasing yield force and decreasing the yield
displacement always decreases the base drift. However, inter-story drifts and floor accelerations are not
always reduced by increasing the yield force and
decreasing the yield displacement. Increasing the yield
force, decreases inter-story drifts up to a point.
Further increases in the yield force cause an increase
in inter-story drifts. Thus, there exists an appropriate
value for the yield force to obtain a minimum interstory drift. Similarly, decreasing the yield displacement
can increase or decrease inter-story drifts, depending
on the value of the yield force. Therefore, in general,
for minimum inter-story drift, an appropriate combination of yield force and yield displacement may be
determined. Similar behavior is observed in terms of
the floor accelerations. Increasing the post-yield stiffness has a similar effect to increasing the pre-yield
stiffness: increasing the post-yield stiffness results in a
decrease in base displacements but increases interstory drifts and floor accelerations. The appropriate
combination of yield displacement and yield force is
dependent upon the characteristics of the earthquake
ground motion.
For low-level ground motions, dissipation mechanisms do not significantly affect the performance in
terms of base drift, story-drift ratio, or floor acceleration. However, for high-level earthquakes, structures
with isolation systems with yielding hysteretic mechanisms perform significantly better than structures with
linear isolation systems, particularly in terms of storydrift ratios and floor accelerations.
These results show that increasing damping decreases the isolator displacements. For the structures analyzed in this study, increasing damping eventually
causes an increase in inter-story drifts and floor accelerations. On the other hand, appropriate levels of isolation stiffness and isolation damping can be combined
to limit the base drift without significantly increasing
floor accelerations and inter-story drift ratios, however, the appropriate values of stiffness and damping is
dependent on the type of earthquake record.
6. Acknowledgments
This material is based on work supported by the
National Science Foundation under Award No. CMS9900193. Any opinions, findings, and conclusions or
recommendations expressed in this publication are
those of the authors and do not necessarily reflect the
views of the sponsors.
The authors thank the reviewers for their helpful
comments.
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