Engineering Structures 75 (2014) 340352

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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct

Improved equivalent viscous damping model for base-isolated

structures with lead rubber bearings
Tobia Zordan a, Tao Liu a,, Bruno Briseghella b, Qilin Zhang a
a
Department of Building Engineering, Tongji University, Shanghai 200092, China
b
College of Civil Engineering, Fuzhou University, Fuzhou, Fujian 350108, China

a r t i c l e i n f o a b s t r a c t

Article history: Nowadays, seismic isolation system has been widely applied in the world to mitigate damage risk of
Received 17 September 2013 structures. Although maximum displacement demand can be obtained through nonlinear time history
Revised 5 April 2014 (NLTH) analysis, many approximate methods are frequently recommended in structural specications
Accepted 27 May 2014
to reduce the required computational time. One of the best-known methods is the equivalent linear
Available online 2 July 2014
(EL) method, in which the nonlinear response of isolator can be adequately modeled using a ctitious vis-
cously damped elastic structure. In this paper, a comparison between existing expressions supplying the
Keywords:
state of research is carried out and then, an improved expression is presented for equivalent linearization
Seismic isolation
Bilinear behavior
of structures supported on lead rubber bearings (LRB). Based on the concept of secant stiffness, the opti-
Peak displacement mal damping ratios, which minimize the errors of maximum displacement between EL analysis and NLTH
NSGA-II optimization analysis, are calculated and averaged over 12 ground motions. Then, a rational model to estimate equiv-
Equivalent damping ratio alent damping ratio is derived through statistic analysis of the optimal damping ratios. To examine the
prediction accuracy of the proposed model, mean ratios of approximate to exact maximum displacement
and root mean square error for different isolated period are calculated as evaluation indicator. Compared
with other EL models, the newly proposed model predicts a displacement that is in better agreement with
the one obtained through NLTH analysis.
2014 Elsevier Ltd. All rights reserved.

1. Introduction simplication, bilinear forcedeformation behavior is generally

assigned to LRB, which can be characterized by the initial elastic
Seismic isolation, decoupling the structure from the ground, stiffness Ki, the yield displacement xy, and the post-to-pre yield
provides a very effective passive method of protecting structures stiffness ratio a, as presented in Fig. 1b.
against severe seismic events. The mitigation of seismic risk is Due to the structural exibility introduced by the isolation sys-
primarily achieved through period shift and modication of mode tem, large deformation often occurs under a given earthquake
shape to focus most of deformation at isolators. Various seismic iso- ground motion. Therefore, predicting the maximum inelastic
lators have been developed and used practically for anti-seismic deformation demands becomes a very important step in the eval-
design of structures during the last twenty years [1], including elas- uation of seismically isolated structures. As well known, maximum
tomeric bearings, frictional/sliding bearings and roller bearings. inelastic deformation demand can be obtained through nonlinear
Compared to other passive devices, the lead rubber bearings time history (NLTH) analysis. However, solving of a system with
(LRB) (Fig. 1a) require minimal initial cost and maintenance [2]. a large number of degrees of freedom may require an exorbitant
The lead core is the crucial element of LRB, which provides the amount of time when time history analysis methods are used. Even
initial rigidity against minor earthquakes and exhibits nonlinear for SDOF systems, the number of different loading cases needed to
behavior to add hysteretic damping in the structure when be solved may be quite large. In addition, in the preliminary stage
subjected to severe earthquakes. Due to its wide applications, the of structural design, structural congurations are not completely
present research study is focused on LRB bearing. For the sake of dened. Thus, there will always be a need for good approximate
methods of analysis of nonlinear systems [3].
Among the approximate methods, the equivalent linear (EL)
Corresponding author. Tel.: +86 021 65980644.
method, which estimates the maximum displacement of an
E-mail addresses: tobia.zordan@gmail.com (T. Zordan), taoliu.liu@gmail.com
inelastic system by the maximum displacement of an EL system,
(T. Liu), Bruno@fzu.edu.cn (B. Briseghella), qilinzhang0@gmail.com (Q. Zhang).

http://dx.doi.org/10.1016/j.engstruct.2014.05.044
0141-0296/ 2014 Elsevier Ltd. All rights reserved.
 T. Zordan et al. / Engineering Structures 75 (2014) 340352 341

Fig. 1. (a) Lead rubber bearing (LRB) and (b) idealized bilinear hysteresis model.

Fig. 2. Equivalent linearization of bilinear hysteretic behavior.

Fig. 3. R&H model: (a) secant stiffness and (b) equal energy dissipation principle.

Table 1
Recorded earthquake ground motions used in this study.

Date Earthquake Ms Station name Rrup (km) Vs30 (m/s) Com. (deg) PGA (m/s2) PGV (m/s) PGD (m) Duration (s)
1966 Parkeld 6.2 Temblor pre-1969 16.0 527.9 205 3.504 0.215 0.038 30.3
1971 San Fernando 6.6 Castaic-Old Ridge Route 22.6 450.3 021 3.177 0.156 0.024 30.0
1972 Managua-Nicaragua-01 6.2 Managua-ESSO 4.1 288.8 090 4.131 0.214 0.060 26.0
1979 Imperial Valley-06 6.5 Compuertas 15.3 274.5 015 1.826 0.138 0.029 36.0
1980 Mammoth Lakes-01 6.1 Convict Creek 6.6 338.5 090 4.084 0.232 0.047 30.0
1980 Victoria-Mexico 6.3 Cerro Prieto 14.4 659.6 045 6.091 0.316 0.131 24.5
1983 Coalinga-01 6.4 Parkeld-Cholame 2WA 44.7 184.8 000 1.069 0.113 0.026 40.0
1989 Loma Prieta 6.9 Foster City-Menhaden Court 45.6 126.4 270 1.048 0.206 0.080 30.0
1992 Cape Mendocino 7.0 Petrolia 8.2 712.8 000 5.782 0.481 0.219 36.0
1994 Northridge-01 6.7 LA-Wonderland Ave 20.3 1222.5 095 1.101 0.087 0.018 30.0
1995 Kobe-Japan 6.9 Kakogawa 22.5 312.0 000 2.466 0.187 0.058 41.0
1999 Kocaeli-Turkey 7.5 Izmit 7.2 811.0 090 2.153 0.298 0.171 30.0

Note: Ms is the surface-wave magnitude of recorded earthquake; Rrup is the rupture distance to the horizontal projection of the fault; Vs30 is shear-wave velocities in the upper
30 m of the site prole; Com. is the horizontal component of the considered ground motions.

is the best-known. The equivalent stiffness Keq and the equivalent When EL analysis is performed, it is obviously noted that the
damping ratio neq should be determined such that the maximum rational estimation of EL properties is crucial for the prediction
displacement responses of the two systems are approximately accuracy. The main difference among the existing EL methods is
equal, as shown in Fig. 2. the way in which the EL properties are determined. According to
342 T. Zordan et al. / Engineering Structures 75 (2014) 340352

Fig. 4. Estimation accuracy of R&H model based on isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.

different treatments in estimating the equivalent period and the xd

l 2
equivalent damping ratio, several analytical or empirical formulas xy
used in EL elastic analysis were developed in the past decades.
Employing the equal energy dissipation principle, rst proposed by
Generally, they can be classied into two main groups based on
Jacobsen [21], the hysteretic damping ratio of the linear elastic
the denition of equivalent period of vibration (or equivalent stiff-
system can be derived, as shown in Fig. 3(b),
ness). The rst group [410] includes methods with equivalent
period dened using the concept of secant stiffness at the design EH 4Q y xd  xy
nhyst 3
displacement of systems. In the second group [1117] of the exist- 4pES 2pK eq x2d
ing EL methods, equivalent stiffness of the EL system is determined
where EH is hysteretic energy dissipated per cycle of motion
using empirical formulas. As known, the estimation accuracy of EL
through inelastic deformation, ES is the elastic strain energy.
analysis is strongly related to different assumptions when various
Considering the post-to-pre yield stiffness ratio a and the duc-
EL methods are derived or tted, such as the hysteretic model and
tility ratio l, the equivalent damping ratio can be expressed as:
the range of displacement ductility. In the present study, only three
EL models are considered due to the limitation of space, as follows. 21  al  1
neq n0 nhyst n0 4
As the representative of the rst group, Rosenblueth and Herre- pl1 al  1
ra [4] rstly proposed the secant stiffness at maximum deforma-
tion as the basis for selecting the period shift, as presented in where n0 is the inherent viscous damping ratio of the structure.
Fig. 3(a). This method is also referred to as geometric stiffness In order to acquire a more accurate estimation of seismic
method, which has been adopted by AASHTO [18], Eurocode 8 response quantities, Dicleli and Buddaram [9] concluded that the
[19] and the new Italian code (NTC 2008) [20]. Here, for brevity, equation of equivalent damping ratio used in the design of seismi-
this model is abbreviated to R&H model. cally isolated structures must incorporate the equivalent period of
In this model, period shift of the EL system is given by: the structure and the frequency characteristics of ground motion. A
new equation was proposed by modifying R&H model, which is
q p
abbreviated to D&B model and expressed as:
T eq =T 0 K i =K eq l=1 al  1 1
s
 
p 21  al  1 T eq
where T0 is the initial period of isolator, dened as 2p M=K i (M is neq n0 0:41 1 5
pl1 al  1 T0
the weight supported by isolator), Keq is the secant stiffness at
maximum deformation, l is the displacement ductility ratio dened In the research study of Guyader and Iwan [17], the optimal pair
as the ratio of the maximum inelastic displacement to the yield of Teq and neq were taken as the values that maximize the probabil-
displacement, namely: ity that the percentage error between the actual nonlinear system
 T. Zordan et al. / Engineering Structures 75 (2014) 340352 343

Fig. 5. Standard deviation of measured ratios based on R&H model and isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.

and its EL counterpart was within the range 10% to +20%. Several equivalent linear models are evaluated in Section 2 through com-
hysteretic systems including bilinear, stiffness degrading, strength parison of results between EL analysis and NLTH analysis. Section 3
degrading and pinching models were considered. The proposed presents the optimal damping ratios obtained using the genetic
empirical equations were abbreviated to G&I model and shown algorithm. Then, in Section 4, an improved equivalent damping
in Eq. (6). model is proposed based on statistic analysis of optimal damping
For l < 4.0: ratios. Conclusions are given in Section 5.

T eq =T 0 1 0:1262l  12  0:0224l  13 6a
2. Evaluation of the considered EL models

neq n0 0:05073l  12  0:01083l  13 6b 2.1. Earthquake records

For 4:0 6 l 6 6:5:
A set of 12 earthquake time histories selected from the Pacic
T eq =T 0 1:1713 0:1194l  1 6c Earthquake Engineering Research Center (PEER) [22] is used as
seismic input in this study. The EL model derived by Iwan [12]
neq n0 0:1169 0:01579l  1 6d using 12 recorded earthquake ground motions were investigated
by Liu et al. [23] and satised accuracy could be obtained within
For l > 6.5: the limited conditions, which indicates that the use of 12 ground
s ! motions could capture the majority variation of equivalent damp-
l1 ing ratio.
T eq =T 0 1 0:87 1 6e
1 0:10l  1  1 All the ground motions are selected based on the following cri-
terions: (1) recorded on accelerograph stations where enough
 2 information exists on the geological conditions; (2) recorded on
0:36l  1  1 T eq
neq n0 0:24383 6f free eld stations or in the rst oor of low-rise buildings with
0:36l  12 T 0
negligible soil-structure interaction effects; (3) recorded in earth-
The main objectives of this study are: (i) to study the estimation quakes with surface-wave magnitudes between 6.1 and 7.5; and
accuracy of equivalent linearization models described above, (ii) to (4) the records have the peak ground acceleration greater than
investigate the variation of optimal damping ratio of the EL system 0.10 g. A complete list of all used ground motions is given in
under different parameters, and (iii) to improve the prediction Table 1, including date, earthquake name, magnitude, recorded
accuracy of EL analysis of structures with LRB by proposing a station, rupture distance to the horizontal projection of the fault,
new expression to estimate the equivalent damping ratio. Three shear-wave velocity in the upper 30 m of the site prole, selected
344 T. Zordan et al. / Engineering Structures 75 (2014) 340352

Fig. 6. Estimation accuracy of D&B model based on isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.

Fig. 7. Standard deviation of measured ratios based on D&B model and isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.
 T. Zordan et al. / Engineering Structures 75 (2014) 340352 345

horizontal component, peak ground properties and duration. The motion on seismic isolation system may be up to 20 and higher.
results from NLTH analyses are averaged over the selected ground Although many research works [8,10,14,23,25] contain ductility
motions and used in this study. ratios smaller than 5, it is unrealistic that the LRB system has a
maximum displacement of 0.05 m under design earthquakes. So,
2.2. Parameter variation and assessment procedure the maximum displacement of LRB system is considered as param-
eter in the present paper, which varies from 0.1 m to 0.4 m with
Since the superstructure in general remains linear elastic and increment equal to 0.05 m. To differentiate the concept of viscous
moves like a rigid body when subjected to an earthquake, seismi- damping and hysteretic damping, the inherent damping ratio of
cally isolated structures can be simplied to be a single-degree-of- the bilinear SDOF system is assigned to be zero, i.e., n = 0. Herein
freedom (SDOF) system in many structural specications, lies the assumption that the seismic input energy in nonlinear sys-
particularly for low-to-medium rise buildings. Considering a SDOF tem is dissipated only through hysteretic behavior.
system, the characteristics of bilinear hysteresis model can be The accuracy of approximate method is examined through com-
selected as varied parameters. For a typical LRB, the yield displace- parison of results between NLTH and EL analysis. Detailed proce-
ment is only a function of the lead core height since lead has a pre- dures to perform both NLTH and EL analysis of seismically
dened yield stress and shear modulus, which could be considered isolated structure are described below.
as a constant (0.01 m). It is not consistent to x both the yield dis- Exact maximum inelastic displacement can be considered as
placement and the pre-to-post yield stiffness ratio since the post- the specied isolator displacement. However, an iterative scaling
yield stiffness is based on the rubber stiffness and the pre-yield process of the selected ground motions should be performed until
stiffness is based mostly on the lead core strength, both of which the computed displacement is, within a tolerance, equal to the
can be varied independent on one another. Therefore, the normal- desired isolator displacement. The scaling factor is considered
ized yield strength f (yield of lead divided by supported weight) satisfactory if the relative error between computed and specied
p

and isolated period Ti (i.e., 2p M=K p ) are selected as the two isolator displacement is within 1%. NLTH analysis of the isolated
parameters to be varied, which is also consistent with those used structure is conducted using the Open System for Earthquake Engi-
by designers of LRB systems. Here, the normalized yield strength neering Simulation (OpenSees) [26] due to its modularity and high
f varies from 0.04 to 0.22 with increment equal to 0.02. Fundamen- execution speed. Bilinear forcedeformation behavior is modeled
tal period of the isolated structure is generally considered to range by elastomeric Bearing element in OpenSees. To solve the nonlin-
between 1.5 s and 3.0 s [24]. Thus, a total of 16 isolated periods of ear equations of motion of the bilinear system, the Newmark-b
vibration between 1.5 s and 3.0 s are also assigned with period method [27] is used with b = 0.25 and c = 0.5.
increment equal to 0.1 s. Furthermore, as mentioned by Hwang For EL analysis, the EL properties can be determined based on
and Chiou [14], ductility ratio demanded by an earthquake ground displacement ductility ratios and hysteretic characteristics of the

Fig. 8. Estimation accuracy of G&I model based on isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.
346 T. Zordan et al. / Engineering Structures 75 (2014) 340352

system. Here are the steps to estimate the maximum inelastic Denote Dap;l T eq ; neq and Dex;l T 0 ; a; l; n0 to be the approximate
displacement using EL analysis: and the exact maximum inelastic displacement when subjected
to the lth ground motion, respectively, and mean ratio R can be
(1) Calculate the period of vibration of the equivalent system expressed as:
using the equations for period shift corresponding to differ-
ent EL models. 1X N
Dap;l T eq ; neq
R 7
(2) Compute the equivalent damping ratio of EL system using N l1 Dex;l T 0 ; a; l; n0
the equations corresponding to different EL models.
where N is the number of earthquake records.
(3) Compute the response time history of EL system with a
In the following illustrations, ratios smaller than one indicate
linear time history analysis based on the EL properties
that the approximate method underestimates on average the exact
computed in steps 1 and 2. Note that the seismic input
maximum displacement in inelastic system and ratios larger than
should be the scaled earthquake record used in NLTH analy-
one mean that the approximate method overestimates on average
sis to produce the specied isolator displacement.
the exact maximum inelastic displacement. Furthermore, in order
(4) Calculate the approximate maximum inelastic displacement
to assess the dispersion of ratios computed using different earth-
as the maximum absolute value of displacement response
quake ground motions, the standard deviation of the ratios is also
computed in step 3.
investigated, as given by:
s
2.3. Results of parametric analysis  2
1 XN Dap;l T eq ; neq
rR  R 8
N1 ex;l T 0 ; a; l; n0
l1 D
According to the parameters specied in Section 2.2 (i.e., 10
normalized yield strengths, 16 isolated periods, 12 ground motions Due to the limitation of paper space, the following results are only
and 7 isolator displacements), a large number of numerical simula- based on isolator displacement equal to 0.1 m, 0.2 m, 0.3 m and
tions are performed, including both nonlinear and linear dynamic 0.4 m. In addition, for the sake of clarity, the results are plotted
analysis. A total of 10  16  12  7 = 13,440 inelastic displace- against less parameter than specied. Fig. 4 shows mean approxi-
ment demands are computed by iteration. Meanwhile, the scaled mate to exact ratios corresponding to R&H model. When the isola-
ground motions are used as the seismic input of linear dynamic tor displacement is 0.1 m, the maximum inelastic displacement is
analysis. Accordingly, 13,440  3 = 40,320 linear time history anal- accurately predicted for normalized yield strength f less than 0.08.
yses are performed based on three EL models described previously. However, the exact displacement is signicantly underestimated
Ratios of approximate to exact maximum inelastic displace- for f larger than 0.12. It can be seen that the mean ratios in general
ment are computed and averaged over the 12 ground motions. increase with increasing the isolator displacement from 0.1 m to

Fig. 9. Standard deviation of measured ratios based on G&I model and isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.
 T. Zordan et al. / Engineering Structures 75 (2014) 340352 347

0.3 m. For isolator displacement larger than 0.3 m, the maximum f the inuence of isolated period on the standard deviation is not
displacement is overestimated by R&H model and the estimation clear. However, for large f the standard deviation in general
accuracy could be considered to be independent of various decreases with increasing the isolated period.
parameters. Mean and standard deviation of approximate to exact displace-
Regarding standard deviation of the measured ratios computed ment ratios produced by G&I model are illustrated in Figs. 8 and 9,
by R&H model, as shown in Fig. 5, larger dispersions will be respectively. It can be noted that the maximum inelastic displace-
obtained for relatively higher f except for cases with short isolated ment is generally underestimated, particularly when the normal-
period when small isolator displacement is considered. In general, ized yield strength is equal to 0.04. With increasing the isolated
the standard deviation decreases with increasing the isolator period and the isolator displacement, the estimation accuracy
displacement. becomes better. Although for large f most of the measured ratio
Fig. 6 presents the mean approximate to exact displacement is in between 0.9 and 1.0, the standard deviation computed based
ratios calculated by D&B model. In general, it can be observed that on G&I model is much larger than those obtained by R&H model or
the exact isolator displacement is signicantly overestimated. The D&B model. In addition, the standard deviation in general increases
estimation accuracy increases with increasing the isolated period. with increasing the isolator displacement.
Furthermore, it is interesting to see that for small normalized yield
strength the measured ratio R decreases with increasing the isola- 3. The optimal damping ratio required in equivalent
tor displacement while for large f the measured ratio R generally linearization
increases with increasing the isolator displacement.
Compared with R&H model, larger standard deviation of the In order to derive a rational expression of equivalent damping
measured ratios could be obtained by D&B model, as presented ratio, the optimal damping ratio nopt, which minimizes the error
in Fig. 7. Similar to the results of mean ratio R, for relatively small of maximum displacement between EL analysis and NLTH analysis,

Fig. 10. Flowchart of the specically developed program.

348 T. Zordan et al. / Engineering Structures 75 (2014) 340352

Fig. 11. The optimal damping ratios computed based on isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.

4. Derivation of improved model

Table 2
Determination of parameters A, B and C. If the equivalent damping ratio is not capable to represent the
Isolator A B C Coefcient of energy dissipated by the hysteretic behavior of the bearings, the
displacement (m) determination maximum displacement demand will not be accurately predicted.
0.1 0.3790 0.09899 0.2268 0.9770 According to the evaluation of other EL models and the optimal
0.15 0.3952 0.1412 0.3170 0.9804 damping ratios obtained in previous sections, an improved formula
0.2 0.3948 0.1730 0.3936 0.9847 is derived here for estimating the accurate equivalent damping
0.25 0.4312 0.2648 0.5721 0.9895 ratio of LRB systems.
0.3 0.4603 0.3682 0.7599 0.9921
Considering the selected parameters, the proposed formula for
0.35 0.4675 0.4410 0.9042 0.9917
0.4 0.4994 0.5881 1.141 0.9913 equivalent damping ratio is assumed to have the following form:
B
neq A 9
f  Td C
has to be calculated and saved as a database. The optimization
analysis is performed using the NSGA-II genetic algorithm [28]. A For a given isolator displacement, surface tting toolbox provided
total of 13,440 optimal damping ratios can be obtained based on by MATLAB is used to determine the parameters A, B and C through
the parameter variations. For given normalized yield strength minimizing the sum of squares of the differences between the
and isolated period, the optimal damping ratio nopt is averaged over optimal damping ratios and those predicted by Eq. (9). The values
the selected ground motions. To perform these optimization anal- of parameters A, B and C are presented in Table 2 as well as the coef-
yses expediently and systematically, a program is developed using cients of determination.
MATLAB in conjunction with Opensees, as presented in Fig. 10. In For a perfect t, the coefcient of determination is equal to 1.0.
this program, the rule of Opensees is to compute the time history Excellent ts generally have a coefcient of determination greater
response of linear or nonlinear systems, while MATLAB is used to than 0.95. To visualize the situation of surface ts, the optimal
control the global operations, such as parameter input, Opensees damping ratios (scatter data) and those predicted by Eq. (9) (gray
calling and result processing. grids) are plotted together in Fig. 12. For other isolator displace-
Fig. 11 presents the optimal damping ratio nopt versus the iso- ments, analogous results could also be obtained.
lated period and the normalized yield strength. It can be observed To consider the effect of isolator displacement on the determi-
from this gure, nopt increases with increasing the isolated period nation of the optimal damping ratio, a simple nonlinear regression
of isolation system. As the normalized yield strength increases, nopt procedure is performed using the curve tting toolbox of MATLAB,
also increases. However, it is found that nopt decreases with as shown in Fig. 13. Therefore, parameters A, B and C could be com-
increasing the isolator displacement. puted using the following equations:
 T. Zordan et al. / Engineering Structures 75 (2014) 340352 349

Fig. 12. Surface tting of optimal damping ratios based on isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.

8
< A 0:3395  exp0:9499  x
>
B 0:0622  exp5:644  x 10
>
:
C 0:1549  exp5:043  x

where x is the isolator displacement caused by earthquakes.

One should have in mind that these tted functions are only
valid for isolator displacement varies from 0.1 m to 0.4 m. For iso-
lator displacement beyond this range, the suitability of these tted
functions needs to be validated.
In accordance with Eqs. (9) and (10), mean of approximate to
exact displacement ratios are computed again and presented in
Fig. 14. Compared with the results obtained by other EL models,
the prediction accuracy of EL analysis is improved by the newly
proposed expression and most of mean approximate to exact dis-
placement ratios range from 1.00 to 1.05, which indicates that
the relative error between NTHA and EL is in general less than 5%.
Fig. 13. Nonlinear curve tting of A, B and C as functions of isolator displacement.
To directly present the accuracy improvement leaded by the
proposed model, root mean square error (RMSE) for different iso-
lated period is plotted in Fig. 15. The RMSE of different EL models model and R&H model. Lowest RSME is obtained by the proposed
is dened as the square root of the mean squared error between model. Therefore, within the considered parameter range, the pre-
computed ratio R and 1.0. As can be observed in this gure, highest diction accuracy of EL analysis is improved by the newly proposed
values of RSME are produced by D&B model, followed by G&I model.
350 T. Zordan et al. / Engineering Structures 75 (2014) 340352

Fig. 14. Estimation accuracy of the proposed model based on isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.

Fig. 15. RMSE of different EL models based on isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.
 T. Zordan et al. / Engineering Structures 75 (2014) 340352 351

Fig. 16. Estimation accuracy of the proposed model based on isolator displacement equal to (a) 0.1 m, (b) 0.2 m, (c) 0.3 m and (d) 0.4 m.

Regarding standard deviation of the measured ratios, the dis- (2) The optimal damping ratio nopt computed using generic algo-
persion obtained by the proposed model is comparable to those rithm increases with increasing the isolated period of bilin-
from R&H model, as shown in Fig. 16. ear isolation system. As the normalized yield strength
In a word, the newly proposed EL model leads to accurate and increases, nopt also increases. However, nopt decreases with
conservative results, and may be used in the practical analysis increasing the isolator displacement.
and design of seismically-isolated structures with LRB systems. (3) The improved expression of equivalent damping ratio, as
Since the research in this paper are based on bilinear SDOF system, functions of normalized yield strength, isolated period and
the proposed model could be applicable as long as the investigated isolator displacement, can accurately predict the optimal
systems (building or bridge) can be simplied into bilinear SDOF damping ratios.
systems. (4) Compared to other EL models, the proposed EL model could
accurately predict the isolator displacement. Most of mean
5. Conclusions approximate to exact displacement ratios range from 1.00
to 1.05, which indicates that the relative error is in general
In this paper, an improved model is presented for equivalent less than 5% and the estimated results are conservative.
linearization of structures isolated by LRB. Bilinear hysteresis (5) In many codes, the use of equivalent linearization is limited
model with a yield displacement of 1 cm is assumed to LRB. Three by several requirements, which are usually involved in
equivalent linear models are evaluated through comparison of equivalent stiffness, equivalent damping ratio and restoring
results between EL analysis and NLTH analysis. The optimal damp- force and so on. However, the improved model is proposed
ing ratios obtained by the genetic algorithm are used in derivation through a smart methodology, which seems to overtake
of an improved equivalent viscous damping model. some of these limitations.
Based on the results obtained from this study, the following
conclusions could be drawn: In spite of the good agreement obtained in this study, a larger
set of data are needed to let the proposed model be considered
(1) Among the investigated EL models, R&H model yields better for practical purposes. In addition, more research is needed to
estimation accuracy than D&B model and G&I model. In assess the accuracy of the expression proposed when it is intended
addition, D&B model overestimates the exact maximum dis- for multi-degree-of-freedom systems and also to determine the
placement of LRB systems while G&I model in general parameters to be used for other types of hysteretic devices
underestimates the peak isolator displacement. commonly adopted in the engineering practice.
352 T. Zordan et al. / Engineering Structures 75 (2014) 340352

References [16] Kwan WP, Billington SL. Inuence of hysteretic behavior on equivalent period
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