Conference PaperPDF Available

Stochastic Response of Nonlinear Base Isolation Systems

Abstract and Figures

A hybrid base isolation system was used to retrofit two residential buildings in Solarino, Sicily. Subsequently, five free vibration tests were carried out in one of these buildings to assess its functionality. The hybrid base isolation system combined high damping rubber bearings with low friction sliders. In terms of numerical modeling, a single-degree-of-freedom system is used here with a new five-parameter trilinear hysteretic model for the simulation of the high damping rubber bearing, coupled with a Coulomb friction model for the simulation of the low friction sliders. Next, experimentally obtained data from the five free vibration tests were used for the calibration of this six parameter model. Following up on the model development , the present study employs Monte-Carlo simulations in order to investigate the effect of the unavoidable variation in the values of the six-parameter model on the response of the base isolation system. The calibrated parameters values from all the experiments are used as mean values, while the standard deviation for each parameter is deduced from the identification tests employing best-fit optimization for each experiment separately. The results show that variation in the material parameters of the base isolation system produce a nonstationary effect in the response. In addition, there is a magnification effect, since the coefficient of variation of the response, for most of the parameters, is larger than the coefficient of variation in the parameter values.
Content may be subject to copyright.
COMPDYN 2017
6th ECCOMAS Thematic Conference on
Computational Methods in Structural Dynamics and Earthquake Engineering
M. Papadrakakis, M. Fragiadakis (eds.)
Rhodes Island, Greece, 15–17 June, 2017
STOCHASTIC RESPONSE OF NONLINEAR BASE ISOLATION
SYSTEMS
Athanasios A. Markou1, George Stefanou2and George D. Manolis3
1Postdoctoral Researcher, Norwegian Geotechnical Institute
Sognsveien 72, 0806 Oslo
e-mail: athanasios.markou@ngi.no
2Assistant Professor, Aristotle University of Thessaloniki
Panepistimioupolis, Thessaloniki 54124, Greece
e-mail: gstefanou@civil.auth.gr
3Professor, Aristotle University of Thessaloniki
Panepistimioupolis, Thessaloniki 54124, Greece
e-mail: gdm@civil.auth.gr
Keywords: Stochastic Response, Hybrid Base Isolation System, Trilinear Hysteretic Model,
Monte-Carlo Simulations.
Abstract. A hybrid base isolation system was used to retrofit two residential buildings in So-
larino, Sicily. Subsequently, five free vibration tests were carried out in one of these buildings
to assess its functionality. The hybrid base isolation system combined high damping rubber
bearings with low friction sliders. In terms of numerical modeling, a single-degree-of-freedom
system is used here with a new five-parameter trilinear hysteretic model for the simulation of
the high damping rubber bearing, coupled with a Coulomb friction model for the simulation
of the low friction sliders. Next, experimentally obtained data from the five free vibration tests
were used for the calibration of this six parameter model. Following up on the model develop-
ment, the present study employs Monte-Carlo simulations in order to investigate the effect of
the unavoidable variation in the values of the six-parameter model on the response of the base
isolation system. The calibrated parameters values from all the experiments are used as mean
values, while the standard deviation for each parameter is deduced from the identification tests
employing best-fit optimization for each experiment separately. The results show that variation
in the material parameters of the base isolation system produce a nonstationary effect in the
response. In addition, there is a magnification effect, since the coefficient of variation of the
response, for most of the parameters, is larger than the coefficient of variation in the parameter
values.
1
Athanasios A. Markou, George Stefanou and George D. Manolis
Figure 1: The Solarino building in Via Baden Powell 25, Solarino, Sicily.
1 INTRODUCTION
Base isolation has been extensively used over the last decades for the protection of structures
against earthquakes. The concept behind base isolation is the idea of introducing a flexible
layer between the superstructure and its foundation [1], so to simply reduce the transmission
of energy from the ground to the superstructure [2]. To this end, the mechanics behind an
isolation system are: (i) a flexible support in order to elongate the natural period of the structure,
(ii) energy dissipation in order to control the relative displacements and (iii) sufficient rigidity
under service loads to avoid unnecessary motion [3]. The first mode of an isolated structure
involves only deformations in the superstructure, while the higher modes do not contribute to
the response due to orthogonality conditions [4].
The first efforts for Italian buildings to be retrofitted with base isolation started in 2004, [5].
Among those buildings were two four-story R/C residential buildings in Via Baden Powell 23-
25, Solarino, Eastern Sicily, [6]. The retrofit included a hybrid base isolation system (HBIS),
which combined 12 high damping rubber bearings (HDRB) with 13 low friction sliding bearings
(LFSB), [6]. In July 2004, static and dynamic tests were performed on one of the two Solarino
buildings, [7], see Fig. 1. The static tests were used for the identification of the static friction
force, while the dynamic ones were in the form of free vibration tests following application and
instantaneous release of a displacement close to the design value.
In the years following these experiments, research efforts were made towards dynamic iden-
tification of the Solarino HBIS by using several mechanical models and various identification
techniques [8, 9, 10, 11, 12]. In the present study, a five-parameter trilinear hysteretic model
(THM) developed in [10, 11] will be used for the HDRB response, while a single-parameter
constant Coulomb friction model (CCFM) will be used for the LFSB response.
Uncertainties inherently exist in the loading as well as in the material and geometric param-
eters of engineering systems. Within the framework of safe engineering design, papers in the
literature primarily deal with the effect of stochastic earthquake excitation on the structural re-
sponse. For instance, Ref. [13] studies the stochastic response of secondary systems attached
to a BI structure undergoing random ground motions described by a filtered white noise model.
In Ref. [14], the randomness of earthquake loads is considered, but a parametric investigation
with regard to deterministic structural - isolator parameters is also conducted. In [15], only the
properties of the superstructure are treated as random variables in an optimization procedure.
The effect of uncertain near-field excitations on the reliability-based performance and design of
2
Athanasios A. Markou, George Stefanou and George D. Manolis
Table 1: List of abbreviations.
BHM bilinear hysteretic model
CCFM constant Coulomb friction model
CMA-ES covariance matrix adaptation - evolution strategy
HBIS hybrid base isolation system
HDRB high damping rubber bearing
LCFM linear Coulomb friction model
LFSB low friction sliding bearing
MCS Monte Carlo simulation
SDOF single-degree-of-freedom
THM trilinear hysteretic model
base-isolated systems is explored in [16]. In fact, very few publications consider uncertainty in
the base isolator parameters. For instance, the stochastic response of base isolated liquid stor-
age tanks is computed in [17] using a polynomial chaos expansion to represent the uncertainty
in the characteristic parameters of a laminated rubber bearing isolator. Finally Ref. [18], [19]
perform robust optimum design of BI systems taking into account the uncertainty in the isolator
parameters.
In the above work, assumptions were made regarding the statistical characteristics of the
isolator parameters. In the present paper, the parameters of the adopted HBIS are calibrated
by using experimental evidence [20]. Specifically, the aforementioned five free vibration tests
performed in Solarino will be used to define the mean value and standard deviation of the six-
parameter mechanical model. The effect of parameter variation on the response of the HBIS will
be investigated in the framework of Monte Carlo simulation (MCS), leading to useful conclu-
sions about the probabilistic characteristics of the response. Finally, for a list of abbreviations
used throughout the paper, the reader is referred to Table 1.
2 MECHANICAL MODELS
Two possible THMs based on different mechanical representations exist, but as it was shown
in [11] only one is able to describe the HDRB response satisfactorily. This THM comprises
three elements, a linear elastic spring of stiffness ke(element 1) in series with a parallel system,
namely a plastic slider of characteristic force fs(element 2) connected in parallel with a trilinear
elastic spring with stiffnesses kh1, kh2and characteristic displacement uc(element 3), see Fig. 2.
The compatibility, equilibrium and constitutive equations of the THM are presented in Table 2.
As shown in Fig. 2(e) the THM has three plastic phases (1-3) and one elastic phase. Plastic
phase 1 has stiffness k2(shown in yellow), plastic phase 2 has stiffness k1(shown in green),
plastic phase 3 has stiffness k2(shown in blue) and the elastic phase has stiffness k0(shown
in red), Fig. 2(e). The two characteristic displacements are also shown, namely the first yield
displacement uyand the second yield displacement u3. The force at zero displacement after
yielding (F2) and the force at second yield displacement u3in the loading phase with positive
displacement, (F3) are defined as follows:
F2= (k0k1)uy;F3=k0uy+k1(u3uy)(1)
The resulting THM is a five-parameter system and the relationships between the mechanical
parameters (ke, kh1, kh2, fs, uc) shown in Fig. 2(a) and the mathematical ones (k0, k1, k2, uy, u3)
shown in Fig. 2(e) are listed in Table 3.
3
Athanasios A. Markou, George Stefanou and George D. Manolis
Figure 2: Trilinear hysteretic model (THM): (a) mechanical model (b) fe1uefor element 1 (c) fe2uhfor
element 2 (d) fe3uhfor element 3 and (e) overall fTugraph.
Table 2: Compatibility, equilibrium and constitutive equations of the THM.
Compatibility u=ue+uh
Equilibrium fT=fe1=fe2+fe3
Constitutive law
fe1=keue
fe2( ˙uh6= 0) = fssgn( ˙uh)
fe2( ˙uh= 0) = fe1fe3
fe3(|uh| ≤ uc) = kh1uh
fe3(|uh|> uc)=(kh1uc+kh2(|uh| − uc))sgn(uh)
Table 3: Relationships between mechanical and mathematical parameters of the THM, see Figs. 2(a),(e).
ke=k0;kh1=k1k0
k0k1;kh2=k2k0
k0k2;fs=k0uy;uc= (u3uy)k0k1
k0
4
Athanasios A. Markou, George Stefanou and George D. Manolis
Figure 3: Constant Coulomb friction model (CCFM): (a) mechanical model (b) overall fFugraph.
Figure 4: Single-degree-of-freedom (SDOF) system representing a base-isolated building.
The constant Coulomb friction model is used for the description of the behavior of the LFSB
component, Fig. 3. This model is defined by the characteristic force ffand its constitutive
equation after initiation of motion ( ˙u6= 0) as follows:
fF=ffsgn( ˙u)(2)
When motion stops ( ˙u= 0) the friction force fFcan take any value between ff< fF< ff.
3 EQUATION OF MOTION UNDER FREE VIBRATIONS
In terms of numerical modeling, a single-degree-of-freedom (SDOF) system is used, see
Fig. 4. The equation of motion of the SDOF system under free vibration excitation is given by:
m¨u+fT+fF= 0 (3)
where fTdenotes the force in the trilinear model of the HDRB component and fFdenotes the
force in the friction model of the LFSB component.
3.1 Constitutive equations for the THM
The restoring force in the THM, fT(u, ˙u)assumes different forms according to whether
the system experiences an elastic phase or a plastic phase of motion. The force-displacement
5
Athanasios A. Markou, George Stefanou and George D. Manolis
relationship for the elastic phases is given by the following expression:
fT(u, ˙u) = Fe
I( ˙u) + k0(uue
I)(4)
where (Fe
I, ue
I) is the starting point of the elastic phase. The three plastic phases are governed
by the following equations:
fT(u, ˙u) = FJsgn( ˙u) + hJ(uuJsgn( ˙u)),(J= 1,2,3) (5)
where (FJ, uJ) are characteristic points of the upper plastic phases. As it may be seen from
Fig. 2(e), u2= 0,h1=h3=k2and h2=k1.
3.2 Constitutive equation for the CCFM
Independently of the phase of motion, the resisting force in the slider, fF(u, ˙u), is always
given by Eq. 2. At times when the system stops, the friction force must satisfy the following
inequality:
|˜
fF(uR,0)| ≤ ff(6)
where uRis the residual displacement.
3.3 Rest conditions
The system will come to rest if the following conditions are satisfied:
sgn(¨u) = sgn( ˙u)(7)
|¨u| ≤ 2fF
m(8)
where ¨uand ˙udenote the acceleration and the velocity just before the stoppage. When the
system stops ( ˙u= 0), it can reach a position of static equilibrium different from the original
unstrained one, as long as the following equation is satisfied:
fT(uR,0) + ˜
fF(uR,0) = 0 (9)
3.4 Analytical solution
The above expressions for the restoring force in the THM, and for the friction force in the
slider, show that each phase of motion, whether elastic or plastic, is governed by linear equa-
tions. The differential equation of motion can then solved analytically, see [20].
4 IDENTIFICATION PROCEDURE
As previously mentioned a static test and six free vibration tests were performed on one of
the two Solarino buildings in 2004. Test numbers 1 and 2 were the static test and a trial test for
the push-and-release device, respectively. The following six tests, numbered from 3 to 8, were
dynamic free vibration tests under imposed initial displacements. The identification procedure
was applied to five of the six dynamic tests, namely tests 3, 5, 6, 7 and 8. Test number 4 was
not considered, since it was performed under a nominal initial displacement of only 4.06cm.
The parameters defining the dynamic system described by the models shown in Fig. 2(e) and
Fig. 3(a) are listed in the following system parameter vector: [m, k0, k1, k2, uy, u3, ff]. Consider
first the following relationships:
ωi=rki
m;fi=ωi
2π;uf=ff
k0
; (i= 0,1,2) (10)
6
Athanasios A. Markou, George Stefanou and George D. Manolis
Next, if we include the imposed initial displacement u0, the system parameter vector to be
optimized becomes:
S= [u0, uy, uf, f0, f1, f2, u3](11)
where f0the elastic frequency, f1the first post-yield frequency, f2the second post-yield fre-
quency.
The identification procedure is based on fitting the acceleration response predicted by the
model to that measured during the experiments. Accelerations are used because they can be
measured reliably. Let A0and t0be the experimental acceleration and time vectors, while A
and tare the acceleration and time vectors of a candidate solution. Then the error, or fitness
function, of the identification procedure, can be defined as:
e2=(A0A, A0A)
(A0, A0)+(t0t, t0t)
(t0, t0)(12)
where
(A, B) =
N
X
i=1
AiBi(13)
is the standard inner product and Nis the length of the vectors considered. The Covariance
Matrix Adaptation-Evolution Strategy (CMA-ES) was used to minimize the error defined by
Equation 12, [21].
Finally, the mass of the system was evaluated as
m=F0FfS
ω2
0uy+ω2
1(u3uy) + ω2
2(u0u3)(14)
when using the THM. In the above equation, F0is the magnitude of the force applied to impose
the initial displacement u0, and FfS is the static friction force measured in the first static test.
5 NUMERICAL RESULTS
The effect of parameter variation on the response of the HBIS is examined here in the frame-
work of MCS. The set of parameters derived by the identification procedure from the previous
section will constitute the mean parameter set for all the experimental tests to be used in the
MCS (see Table 4). The identified mass for each test is presented in Table 5 along with the
static friction force in the first static test FfS and the magnitude of the force applied to generate
the initial displacement u0,F0. The standard deviation (std), for each parameter is deduced
from the identification tests employing best-fit optimization for each experiment separately, see
Table 6, [10].
From Tables 4 and 6 it can be observed that the second yield displacement u3is the parameter
with the largest coefficient of variation (cov =std
mean ), which is equal to 15.4%. A normal
distribution is assumed for all parameters since there is inadequate amount of data to validate a
(more realistic) non-Gaussian assumption. The monitored response quantity is the acceleration
of the HBIS, whose recorded and identified values are plotted in Fig. 5 for each test.
Next, one thousand MCS were performed considering the variation of each parameter sep-
arately. Fig. 6 shows that statistical convergence is achieved in all cases with this number of
samples. In the same figure, it can be observed that the std of the acceleration at three different
time instants is substantially different, particularly when uf,uyand f0are varying, which means
that the response is non-stationary. This is reflected in Fig. 7, where the acceleration versus time
7
Athanasios A. Markou, George Stefanou and George D. Manolis
01234567
t (s)
-1.5
-1
-0.5
0
0.5
1
a (m/s2)
Test 3
(a)
e2 = 0.0122
01234567
t (s)
-1.5
-1
-0.5
0
0.5
1
a (m/s2)
Test 5
(b)
e2 = 0.0095
01234567
t (s)
-1.5
-1
-0.5
0
0.5
1
a (m/s2)
Test 6
e2 = 0.0066
(c)
01234567
t (s)
-1.5
-1
-0.5
0
0.5
1
a (m/s2)
Test 7
e2 = 0.0099
(d)
01234567
t (s)
-1.5
-1
-0.5
0
0.5
1
a (m/s2)
Test 8
e2 = 0.0095
(e)
Figure 5: Identified and recorded accelerations (a) Test 3, (b) Test 5, (c) Test 6, (d) Test 7, (e) Test 8 (the red line
denotes the recorded signal and the blue line the identified one).
8
Athanasios A. Markou, George Stefanou and George D. Manolis
200 400 600 800 1000
n
0
0.05
0.1
0.15
std
uf
(a)t1 = 1.25 s
t2 = 3.45 s
t3 = 4.45 s
200 400 600 800 1000
n
0
0.05
0.1
0.15
std
uy
(b)t1 = 1.25 s
t2 = 3.45 s
t3 = 4.45 s
200 400 600 800 1000
n
0
0.05
0.1
0.15
std
u3
(c)t1 = 1.25 s
t2 = 3.45 s
t3 = 4.45 s
200 400 600 800 1000
n
0
0.05
0.1
0.15
std
f0
(d)t1 = 1.25 s
t2 = 3.45 s
t3 = 4.45 s
200 400 600 800 1000
n
0
0.05
0.1
0.15
std
f1
(e)t1 = 1.25 s
t2 = 3.45 s
t3 = 4.45 s
200 400 600 800 1000
n
0
0.05
0.1
0.15
std
f2
(f)t1 = 1.25 s
t2 = 3.45 s
t3 = 4.45 s
Figure 6: Convergence of the acceleration standard deviation vs Monte-Carlo simulations at three different time
intervals for test 5 with varying parameters: (a) uf, (b) uy, (c) u3, (d) f0, (e) f1, (f) f2.
9
Athanasios A. Markou, George Stefanou and George D. Manolis
Table 4: Set of identified parameters for all tests.
Test 3
u0(m)
0.1041
Test 5 0.1132
Test 6 0.1097
Test 7 0.0859
Test 8 0.0893
LFSB uf(m)0.0032
HDRB
uy(m)0.0138
u3(m)0.0799
f0(Hz)0.5400
f1(Hz)0.4159
f2(Hz)0.3227
Error P5
i=1 e2
i(%) 4.77
Table 5: Identified mass, F0and Ff S measures.
Test 3 5 6 7 8
F0(kN )1027 1140 1177 828 927
m(kN s2
m)1306 1392 1470 1147 1274
Ff S (kN )100
graphs are given for a separate variation of the six model parameters. Based on this figure, it
is concluded that uy,u3and f1are the most critical parameters in terms of response variability.
The non-stationary effect is verified in Fig. 8, where the complete temporal evolution of the
std is shown. The above effects can be attributed to the high level of nonlinearity in the base
isolation system for large initial displacements.
6 CONCLUSIONS
MCS have been employed here in order to investigate the effect of the uncertainty in the
values of a six-parameter mechanical model used to simulate the response of base isolation sys-
tems. The parameters of the hybrid base isolation system examined herein was used in practice
in the Solarino 2004 retrofit project, and were calibrated from experimental data. The results
have shown that variation in the material parameters of the isolation system produce a non-
stationary effect in the response, which can be traced by the time evolution of the standard
deviation computed from the response at different time intervals. The first and second yield
displacements and the first post-yield frequency have been identified as the most critical param-
eters in terms of response variability. In addition, there was a magnification effect, due to the
fact that the coefficient of variation of the response was larger than the coefficient of variation
Table 6: Standard deviation (std) of the six system parameters.
Parameter uf(m)uy(m)u3(m)f0(Hz)f1(Hz)f2(Hz)
std 0.00025 0.00174 0.01233 0.00935 0.01319 0.04094
10
Athanasios A. Markou, George Stefanou and George D. Manolis
Figure 7: Acceleration time histories for test 5 with varying parameters: (a) uf, (b) uy, (c) u3, (d) f0, (e) f1, (f) f2
(the red line denotes the recorded signal and the yellow line denotes the average computed acceleration).
11
Athanasios A. Markou, George Stefanou and George D. Manolis
0 5 10
t (s)
0
0.05
0.1
std a(m/s2)
uf
(a)
0 5 10
t (s)
0
0.05
0.1
std a(m/s2)
uy
(b)
0 5 10
t (s)
0
0.05
0.1
std a(m/s2)
u3
(c)
0 5 10
t (s)
0
0.05
0.1
std a(m/s2)
f0
(d)
0 5 10
t (s)
0
0.05
0.1
std a(m/s2)
f1
(e)
0 5 10
t (s)
0
0.05
0.1
std a(m/s2)
f2
(f)
Figure 8: Standard deviation of the acceleration evolution over time for test 5 with varying parameters: (a) uf, (b)
uy, (c) u3, (d) f0, (e) f1, (f) f2.
12
Athanasios A. Markou, George Stefanou and George D. Manolis
of the parameter itself. The high level of nonlinearity in the base isolation system amplitude
of vibration brought about by large initial displacement helps explain the previously described
effects. The above observations can serve as guidelines and indicators in the design of new base
isolation systems.
ACKNOWLEDGEMENT
The authors wish to acknowledge financial support from the Horizon 2020 MSCA-RISE-
2015 project No. 691213 entitled ‘Exchange-Risk’, Prof. A. Sextos, Principal Investigator.
REFERENCES
[1] F. Naeim and J. Kelly, Design of Seismic Isolated Structures: From Theory to Practice.
John Wiley & Sons, New York, 1999.
[2] R. Skinner, W. Robinson, and G. McVerry, An Introduction to Seismic Isolation. John
Wiley & Sons, New York, 1993.
[3] G. Oliveto, Innovative Approaches to Earthquake Engineering. WIT Press, Ashurst
Lodge, Southampton, UK, 2002.
[4] Y. Bozorgnia and V. Bertero, Earthquake Engineering from Engineering Seismology to
Performance-Based Engineering. CRC Press, New York, 2004.
[5] A. Martelli and M. Forni, “Seismic retrofit of existing buildings by means of seismic isola-
tion: some remarks on the italian experience and new projects,” in Proceedings of the 3rd
ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and
Earthquake Engineering (M. Papadrakakis, M. Fragiadakis, and V. Plevris, eds.), Corfu,
Greece, 2011.
[6] G. Oliveto and M. Marletta, “Seismic retrofitting of reinforced concrete buildings using
traditional and innovative techniques,ISET Journal of Earthquake Technology, vol. 42,
pp. 21–46, 2005.
[7] G. Oliveto, M. Granata, G. Buda, and P. Sciacca, “Preliminary results from full-scale
free vibration tests on a four story reinforced concrete building after seismic rehabilita-
tion by base isolation,” in JSSI 10th anniversary symposium on performance of response
controlled buildings, Yokohama, Japan, 2004.
[8] N. D. Oliveto, G. Scalia, and G. Oliveto, “Dynamic identification of structural systems
with viscous and friction damping,Journal of Sound and Vibration, vol. 318, pp. 911–
926, 2008.
[9] N. D. Oliveto, G. Scalia, and G. Oliveto, “Time domain identification of hybrid base isola-
tion systems using free vibration tests,Earthquake Engineering and Structural Dynamics,
vol. 39, pp. 1015–1038, 2010.
[10] A. A. Markou, G. Oliveto, and A. Athanasiou, “Recent advances in dynamic identification
and response simulation of hybrid base isolation systems,” in Proceedings of the 15th
World Conference on Earthquake Engineering, (Lisbon, Portugal), 2012. Paper No. 3023.
13
Athanasios A. Markou, George Stefanou and George D. Manolis
[11] A. A. Markou and G. D. Manolis, “Mechanical formulations for bilinear and trilinear
hysteretic models used in base isolators,Bulletin of Earthquake Engineering, vol. 14,
pp. 3591–3611, 2016.
[12] A. A. Markou and G. D. Manolis, “Response simulation of hybrid base isolation sys-
tems under earthquake excitation,Soil Dynamics and Earthquake Engineering, vol. 90,
pp. 221–226, 2016.
[13] G. Juhn, G. D. Manolis, and M. C. Constantinou, “Stochastic response of secondary sys-
tems in base-isolated structures,Probabilistic Engineering Mechanics, vol. 7, pp. 91–
102, 1992.
[14] R. S. Jangid and T. K. Datta, “Stochastic response of asymmetric base isolated buildings,
Journal of Sound and Vibration, vol. 179, pp. 463–473, 1995.
[15] H. A. Jensen, M. Valdebenito, and J. Sepulveda, Optimal Design of Base-Isolated Systems
Under Stochastic Earthquake Excitation. Springer, Berlin, chapter 10 ed., 2013. Compu-
tational Methods in Stochastic Dynamics, M. Papadrakakis et al. (eds.).
[16] H. A. Jensen and D. S. Kusanovic, “On the effect of near-field excitations on the reliability-
based performance and design of base-isolated structures,Probabilistic Engineering Me-
chanics, vol. 36, pp. 28–44, 2014.
[17] S. K. Saha, K. Sepahvand, V. A. Matsagar, A. K. Jain, and S. Marburg, “Stochastic analy-
sis of base-isolated liquid storage tanks with uncertain isolator parameters under random
excitation,Engineering Structures, vol. 57, pp. 465–474, 2013.
[18] B. K. Roy and S. Chakraborty, “Robust optimum design of base isolation system in seis-
mic vibration control of structures under random system parameters,Structural Safety,
vol. 55, pp. 49–59, 2015.
[19] R. Greco and G. C. Marano, “Robust optimization of base isolation devices under uncer-
tain parameters,Journal of Vibration and Control, vol. 22, pp. 853–868, 2016.
[20] A. A. Markou, G. Oliveto, and A. Athanasiou, “Response simulation of hybrid base iso-
lation systems under earthquake excitation,Soil Dynamics and Earthquake Engineering,
vol. 84, pp. 120–133, 2016.
[21] N. Hansen, “The cma-evolution strategy: a tutorial.” ”https://www.lri.fr/
˜hansen/cmatutorial.pdf”, 2011.
14
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
The best known model for numerically simulating the hysteretic behavior of various structural components is the bilinear hysteretic system. There are two possible mechanical formulations that correspond to the same bilinear model from a mathematical viewpoint. The first one consists of a linear elastic spring connected in series with a parallel system comprising a plastic slider and a linear elastic spring, while the second one comprises a linear elastic spring connected in parallel with an elastic-perfectly plastic system. However, the bilinear hysteretic model is unable to describe either softening or hardening effects in these components. In order to account for this, the bilinear model is extended to a trilinear one. Thus, two trilinear hysteretic models are developed and numerically tested, and the analysis shows that both exhibit three plastic phases. More specifically, the first system exhibits one elastic phase, while the second one exhibits two elastic phases according to the level of strain amplitude. Next, the change of slope between the plastic phases in unloading does not occur at the same displacement level in the two models. Furthermore, the dissipated energy per cycle in the first trilinear model, as proven mathematically and explained physically, decreases in the case of hardening and increases in the case of softening, while in the second trilinear model the dissipated energy per cycle remains unchanged, as is the case with the bilinear model. Numerical examples are presented to quantify the aforementioned observations made in reference to the mechanical behavior of the two trilinear hysteretic models. Finally, a set of cyclic shear tests over a wide range of strain amplitudes on a high damping rubber bearing is used in the parameter identification of the two different systems, namely (a) trilinear hysteretic models of the first type connected in parallel, and (b) trilinear hysteretic models of the second type also connected in parallel. The results show that the complex nonlinear shear behavior of high damping rubber bearings can be correctly simulated by a parallel system which consists of only one component, namely the trilinear hysteretic system of the first type. The second parallel system was not able to describe the enlargement of the dissipated hysteresis area for large strain amplitudes.
Article
Full-text available
Base isolator devices are widely used for mitigation of vibrations induced in structures by seismic actions. In order to achieve high performances in the mitigation of seismic effects, base isolator mechanical properties should be designed by an optimum criterion. In common approaches, the nature of dynamic loads is assumed as the only source of uncertainty. In the present paper a robust optimization criterion for base isolator devices design is proposed, considering the unavoidable effects of uncertainty in structural properties and seismic action. Uncertain parameters are modeled as random variables and are represented by bounded independent probability density function, with uniform law. The structure is described by a single-degree-of-freedom model and is protected by a linear base isolator in order to reduce vibration levels induced by base acceleration, here modeled by the stationary Kanai-Tajimi stochastic process. The optimal design is formulated as a constrained minimization problem, assuming as an objective function a suitable measure of the isolator efficiency and imposing a constraint on the maximum isolator displacement. A sensitivity analysis is carried out on the robust solution in order to assess characteristics and differences with respect to the conventional deterministic solution
Conference Paper
A physical model composed of a tri-linear spring, a friction slider and a viscous damper is proposed for the simulation of the dynamic behaviour of hybrid base isolation systems (HBIS) composed of high damping rubber bearings (HDRB) and low friction sliding bearings (LFSB). After the introduction of the constitutive equations for each device composing the overall system, it is shown that the motion of the system consists of alternating linear phases. An analytical solution is provided in compact form for all possible phases of motion. The end conditions for one phase provide the initial conditions for the next one. The solution is applied to the dynamic identification of the HBIS of the Solarino buildings. A well established evolution strategy (CMA-ES) is used as the dynamic identification algorithm. The estimated values of the physical parameters, together with simulated test responses, contribute to a better understanding of the behaviour of HBIS.
Chapter
The development of a general framework for reliability-based design of base-isolated structural systems under uncertain conditions is presented. The uncertainties about the structural parameters as well as the variability of future excitations are characterized in a probabilistic manner. Nonlinear elements composed by hysteretic devices are used for the isolation system. The optimal design problem is formulated as a constrained minimization problem which is solved by a sequential approximate optimization scheme. First excursion probabilities that account for the uncertainties in the system parameters as well as in the excitation are used to characterize the system reliability. The approach explicitly takes into account all non-linear characteristics of the combined structural system (superstructure-isolation system) during the design process. Numerical results highlight the beneficial effects of isolation systems in reducing the superstructure response.
Article
In the present work, we investigate the response of a hybrid base isolation system under earthquake excitation. The physical parameters of the hybrid base isolation system are identified from dynamic tests performed during a parallel project involving two residential buildings in the town of Solarino, Sicily, using the well-established optimization procedure 'covariance matrix adaptation-evolution strategy' as dynamic identification algorithm in the time domain. The base isolation system consists of high damping rubber bearings and low friction sliding bearings. Two separate models are employed for the numerical simulation of the high damping rubber bearing component, namely a bilinear system and a trilinear system, both in parallel with a linear viscous damper. In addition, a linear Coulomb friction model is used to describe the behavior of the low friction sliding bearing system. Analytical solutions are provided, in compact form, for all possible phases of motion of the hybrid base isolation system under earthquake excitation. A series of numerical simulations are carried out to highlight the behavior of the considered hybrid base isolation system under different excitation and site conditions.
Article
The optimum design of base isolation system to control seismic vibration considering uncertain system parameters are usually performed by minimizing the unconditional expected value of mean square response of a structure without any consideration to the variance of such responses due to system parameter uncertainty. However, the unconditional mean square response based designed may have larger variance of responses due to uncertainty in system parameters and the overall system performance may be sensitive. But, it is desirable that the optimum design should reduce both the mean and variance of dynamic performance measure under system parameter uncertainty. The present study deals with robust design optimization (RDO) of base isolation system considering random system parameters characterizing the structure, isolator and ground motion model. The RDO is performed by minimizing the weighted sum of the expected value of the maximum root mean square acceleration of the structure as well its standard deviation. A numerical study elucidates the importance of the RDO procedure for design of base isolation system by comparing the proposed RDO results with the results obtained by the conventional stochastic structural optimization procedure and the unconditional response based optimization.
Article
In this work, the stochastic response of secondary systems attached to a base-isolated structure undergoing random ground motions is examined. It is assumed that the properties of this combined structural system are deterministic, while the ground motions are described by a filtered white noise model. The only nonlinear component of this structural system is its base isolation mechanism, which is linearized by using equivalent linearization. Also, a substructuring algorithm is developed which requires the dynamic properties of the individual, fixed-base components of the structural system. Both stationary as well as nonstationary cases are considered and comparisons are made with the results of Monte-Carlo simulations to ascertain the validity of this methodology. The example studied herein is a six-storey steel building frame with a base isolation system consisting of sliding bearings and restoring force springs. For this example, spectra are constructed that account for primary-secondary system interaction and depict the effect of variations in the base isolator's structural parameters and in the mass and location of the secondary system on the latter's root-mean-square (RMS) accelerations.