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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

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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= (k0−k1)uy;F3=k0uy+k1(u3−uy)(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.

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Athanasios A. Markou, George Stefanou and George D. Manolis

Figure 2: Trilinear hysteretic model (THM): (a) mechanical model (b) fe1−uefor element 1 (c) fe2−uhfor

element 2 (d) fe3−uhfor element 3 and (e) overall fT−ugraph.

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) = fe1−fe3

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

k0−k1;kh2=k2k0

k0−k2;fs=k0uy;uc= (u3−uy)k0−k1

k0

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Athanasios A. Markou, George Stefanou and George D. Manolis

Figure 3: Constant Coulomb friction model (CCFM): (a) mechanical model (b) overall fF−ugraph.

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(u−ue

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(u−uJsgn( ˙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=(A0−A, A0−A)

(A0, A0)+(t0−t, t0−t)

(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=F0−FfS

ω2

0uy+ω2

1(u3−uy) + ω2

2(u0−u3)(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.

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