performance of a base-isolated building with system ... · pdf filelrb and rb types of...

Performance of a Base-Isolated Building withSystem Parameter Uncertainty Subjected toa Stochastic EarthquakeSudib K. MishraDepartment of Civil Engineering, Indian Institute of Technology, Kanpur, Utter Pradesh, India-208016

Subrata ChakrabortyDepartment of Civil Engineering, Bengal Engineering and Science University, Shibpur, Howrah, West Bengal,India-71103

(Received 21 May 2011, revised 17 February 2012, accepted 14 September 2012)

Base isolation has long been established as an effective tool for improving the seismic performance of structures.The effect of parameter uncertainty on the performance of base isolated structure is investigated in the presentstudy. With the aid of the matrix perturbation theory and first-order Taylor series expansion, the total probabilityconcept is used to evaluate the unconditional response of the system under parameter uncertainty. To do so,the conditional second-order information of responses are obtained by time domain nonlinear random vibrationanalysis through stochastic linearization. The implications of parametric uncertainty are illustrated in terms ofthe responses of interest in design applications. The lead rubber bearing isolator, isolating a multistoried buildingframe, is considered for numerical elucidation. It is observed that, although the randomness in a seismic eventdominates, the uncertainty in the system parameters also affects the stochastic responses of the system. Particularly,the variance of the stochastic responses due to parameter uncertainty is notable.

1. INTRODUCTION

Vibration control technologies are widely acclaimedamongst researchers and practicing engineers as a viable alter-native to traditional seismic design, which relies on energy dis-sipation through inelastic deformations of structural elementsunder earthquake-induced vibrations. In contrast to the tradi-tional design, passive/active vibration control strategies sub-stantially reduce the structural responses to ensure minimaldamage to structures. A reduction of response is achievedthrough control systems, using base isolation (BI), tuned massdampers, liquid column vibration absorbers, etc. Among these,BI systems have been used and globally accepted as an effec-tive technology to reduce the seismic effects on strategicallyimportant structures as well as in retrofitting. In a BI system,the building rests on a system of isolators uncoupling the build-ing from the horizontal component of the ground motion to ef-fectively reduce the seismic load transmission to the structure.Various BI devices, such as rubber bearings (RB), lead rub-ber bearings (LRB), high-damping rubber bearings (HDRB),friction pendulums (FP), and resilient friction bearing isola-tors (R-FBI), are conventionally adopted for seismic protec-tion of buildings, bridges, and other infrastructural facilities.These devices use different materials and design strategies todisconnect the superstructure motion from the ground. Theeffectiveness of BI systems and their performances has beenextensively studied.1–4 Several studies on stochastic responseof base-isolated structures under random earthquakes are alsonotable.5–7 These studies provide important insight into the

behaviour of structures with BI systems. It is well establishedthat the response of BI systems largely depends on the char-acteristics of the isolator, such as the yield strength for theLRB and RB types of isolator, optimal damping for R-FBI,and so on. Earlier studies have also provided parameters toensure optimal performances.7–9 In fact, studies on the opti-mum design of such systems are well known.8–10 However,most of these works are based on deterministic descriptionsof the parameters, characterizing the mechanical model of thesuperstructure-BI system as well as the stochastic load modelfor the earthquake. A major limitation of the deterministic ap-proach is that the uncertainties in the performance-related de-cision variables cannot be included in the parameters for theprocess of optimization. Yet, the efficiency of such a systemmay be drastically reduced if the parameters are off tuned tothe vibrating mode for which it is designed to suppress be-cause of the unavoidable presence of uncertainty in the systemparameters. Therefore, the passive vibration control of struc-tures using BI system with uncertain parameters has attractedthe interest of the vibration control community.

The developments in the field of passive vibration control byusing various passive devices and considering system parame-ter uncertainty have been improved by many researchers.11–18

However, this is not the case for BI systems. Studies on theperformance of BI systems in connection with passive vibra-tion control strategy are very limited. Benfratello et al. indi-cated that the effect of uncertainty on the response of structurewith regard to base isolators and the ground motion filter pa-rameters cannot be ignored.19 Kawano et al. studied the effect

International Journal of Acoustics and Vibration, Vol. 18, No. 1, 2013 (pp. 7–19) 7

S. K. Mishra, et al.: PERFORMANCE OF A BASE-ISOLATED BUILDING WITH SYSTEM PARAMETER UNCERTAINTY. . .

of uncertain parameters on the nonlinear dynamic response ofthe BI structure in the framework of Monte Carlo simulationmethods.20 It has been demonstrated that uncertain parame-ters play a significant role on the maximum responses of theBI system. Scruggs et al. proposed a probability-based ac-tive control synthesis for seismic isolation of an eight storeybase-isolated benchmark structure using uncertain model pa-rameters.21 Zhou, Wen, and Cai22 and Zhou and Wen23 pre-sented two adaptive back-stepping control algorithms for theactive seismic protection of building structures using an uncer-tain hysteretic system. Though studies on the performance ofBI systems supplemented by the active vibration control strate-gies are extensive, the effect of uncertain parameters on the re-sponses and performance of BI systems with passive vibrationcontrol is limited.

Thus, in the present study, the effect of system parameter un-certainty on the performance of BI systems is evaluated undera stochastic earthquake load. The response evaluation involvesconsideration of uncertainty in the properties of the isolated su-perstructure, isolator, and ground motion characteristics. Withthe aid of the matrix perturbation theory using first-order Tay-lor series expansion, the total probability concept is used toevaluate the unconditional response of structures under param-eter uncertainty.24 For this, the conditional second-order in-formation of responses are obtained using time domain anal-ysis of nonlinear random vibration by stochastic linearization.Subsequently, the root mean square (RMS) of the top-floor dis-placement and acceleration (considered to be the performanceindex) are obtained to study the effect of system parameter un-certainty. Numerical analysis elucidates the effects of param-eter uncertainty on the stochastic responses of interest. Theimplication of the parametric uncertainty is demonstrated interms of the disparity between the conditional and uncondi-tional stochastic responses and their associated variances.

2. FORMULATIONS

2.1. Response of a Base-Isolated Structureunder a Random Earthquake

The structure considered in the present study is idealized asa shear frame isolated by an LRB type of isolator. The ideal-ization of the structure and the LRB is shown in Figs. 1(a) and(b), respectively. Since the BI system substantially reduces thestructural response, the isolated structure can reasonably be as-sumed to behave linearly. The damping of the superstructureis assumed to be the viscous type. The energy is dissipatedthrough the hysteresis of LRB by large shear deformation andby yielding of the lead core. Therefore, the force-deformationbehaviour of the LRB is highly nonlinear and is idealized as bi-linear (Fig. 1(c))25–27 with associated parameters such as yielddisplacement (q), yield strength (FY ), pre-yield (kb), and post-yield stiffness (αkb). The structure is considered to be excitedby the horizontal component of the ground motion only.

The equation of motion for an n-storied superstructure canbe expressed as

Mx + Cx + Kx = −Mr(xg + xb); (1)

in which M, K, and C are the mass, stiffness*, and damp-ing matrices of the structure, respectively, of order n × n;x = [x1 x2 . . . xn]T is the displacement vector of thesuperstructure containing the lateral displacement of any floorrelative to the isolator, shown in Fig. 1(a). The symbol r isthe influence coefficient vector implying the pseudo-elastic de-formation of the respective floor due to a unit deformation ofthe ground. The symbol xb is the acceleration of the isolatorwith respect to the ground and xg is the earthquake ground ac-celeration. The details of these matrices are presented in theappendix.

The governing equation of motion for the isolator mass(Fig. 1(b)) can be written as

mbxb + cbxb + Fb − c1x1 − k1x1 = −mbxg; (2)

where mb is the mass of the base, Fb is the restoring force ofthe isolator, cb is the viscous damping in the rubber of the LRB,and k1 and c1 are the stiffness and damping of the first storeyof the superstructure, respectively.

The bilinear force-deformation behaviour is adopted in thisstudy to model the LRB in which the force-deformation be-haviour is expressed by the differential Bouc-Wen model.25, 26

Following this, the isolator restoring force can be expressed as

Fb(xb, xb, Z) = αkbxb + (1−α)FyZ; (3)

where kb is the initial elastic stiffness, xb and xb are the relativedisplacement and velocity, α is the ratio of post- to pre-yieldstiffness (referred as rigidity ratio), and Fy is the yield strengthof the LRB. The variable Z is a variable quantifying the hys-teretic behaviour of the isolator. Substituting Eq. (3) in Eq. (2)and dividing by mb, the equation reduces to

xb+cbmb

xb+αkbmb

xb+(1−α)Fy

mbZ− c1

mbx1−

k1mb

x1 = −xg.(4)

The variable Z is governed by the differential equation

qZ = −γ|xb|Z|Z|η−1 − βxb|Z|η + δxb; (5)

where q is the yield displacement of the isolator. The five pa-rameters β, γ, η, α, and δ in Eq. (5) control the shape of thehysteresis loop. The variable η controls the transition from theelastic to plastic phase; when η →∞ (infinity), the model be-comes elasto-plastic. The nature of the hysteretic behaviour iscontrolled by β; β > 0 implies hardening and β < 0 results insoftening. Presently, the parameters are adopted as α = 0.05,β = γ = 0.5, δ = 1, and η = 1 to correspond the bilinearbehaviour. However, such choices lead to smooth transitionfrom the elastic to plastic state, which is adequately taken tobe close enough to sharp bilinear behaviour.

The post-yield stiffness of the LRB (αkb) is selected to pro-vide the specific time period of isolation (Tb), given by

Tb = 2π

√M

αkb; (6)

where M =∑ni=1mi + mb is the total mass of the isolation-

superstructure system, which is the sum of all the floor mass

8 International Journal of Acoustics and Vibration, Vol. 18, No. 1, 2013


Figure 1. (a) Schematic of the base-isolated structure; (b) Mechanical model of the LRB; and (c) Idealized bilinear hysteresis of the LRB under cyclic loading.

(mi) and the isolator mass (mb). The viscous damping (cb) inthe rubber of the LRB is written as

cb = 2ξbMωb; (7)

where ξb is the damping ratio and ωb is the frequency (ωb =

2π/Tb) of the isolator. The variable q is the displacement cor-responding to the yield strength, conveniently normalized (F0)with respect to the weight (W = Mg) of the isolation-structuresystem:

F0 =FYW

; (8)

g is the gravitational acceleration. The characteristic parame-ters of the isolator are, therefore, the time period of isolation(Tb), viscous damping (ξb), and the normalized yield strength(F0). These parameters dictate the performance of isolation inseismic vibration mitigation. The response quantities of the BIsystem varies monotonically with respect to the isolator timeperiod (i.e., increasing time period reduces the response andvice versa).7, 8 The response also reduces monotonically withincreasing damping for the LRB.7, 8 However, depending onthe type of BI system (e.g., R-FBI), this trend might not be thesame, and instead, the optimal value of damping minimizesthe response.7, 8 In the present study, the isolator time period isfixed at 2 s, and the viscous damping in the isolator is taken as10%.

The nonlinear force-deformation characteristic in Eq. (5)of the LRB is too complicated to be readily incorporated inthe state-space formulation for the evaluation of the stochas-tic responses and the associated sensitivity statistics. This has

been facilitated through stochastic linearization by writing astochastically equivalent linear form of the nonlinear Eq. (5)as27, 28

qZ + Cexb +KeZ = 0; (9)

where Ce and Ke are the equivalent damping and stiff-ness values obtained from the least-square error minimizationamong the nonlinear equation (Eq. (5)) and the linear equation(Eq. (9)). Presently, for η = 1, the closed form expressions ofthe equivalent linear damping and stiffness values are adoptedas27

Ce =

√2

π

{γ

E[xbZ]√E[x2b ]

+ β√

E[Z2]

}− δ; (10a)

Ke =

√2

π

{γ√

E[x2b ] + βE[xbZ]√

E[Z2]

}; (10b)

in which E[ ] is the expectation operator. In stochastic lin-earization, the responses (xb, xb) of the system are assumedto be jointly Gaussian. This assumption might not be correct,given that the system is nonlinear in the presence of bilinearhysteresis. However, it is demonstrated that this assumptiondoes not result in serious error so far as the stochastic responseevaluation is concerned.27, 28

In evaluation of the stochastic response of a structure un-der a random earthquake, unlike the deterministic time histo-ries, stochastic models are employed to describe the under-lying stochastic process. Presently, the well-known Kanai-Tajimi model is considered;29, 30 the power spectral density of

International Journal of Acoustics and Vibration, Vol. 18, No. 1, 2013 9


the ground motion, Sxg(ω), in this model is expressed as

Sxg(ω) = S0

(1 + 4ξ2g(ω/ωg)

2

[1− (ω/ωg)2]2

+ 4ξ2g(ω/ωg)2

); (11)

where S0 is the white-noise intensity of the rock bed excitationresponsible for the seismic event; ωg and ξg are the characteris-tic frequency and damping of the soil media over the rock bedand underlying the building. The variable ω is the frequencycomponent of the ground motion. The parameter S0 is relatedto the RMS ground acceleration (ugmax) of the earthquake as31

S0 =2ξgugmax

π(1 + 4ξ2g)ωg. (12)

The state-space formulation can include the white-noise typeof ground excitation directly in the formulation, whereasthe coloured-noise type of excitation (as in the Kanai-Tajimimodel) can be included in the formulation by incorporating theequations for the Kanai-Tajimi filter within the dynamic equa-tions of motion for the superstructure and the isolator. Theseequations convert the rock bed white noise to colour whilepassing through the filter. The equations for the Kanai-Tajimifilter can be expressed as

xg = xf + w; (13a)

xf + 2ξgωgxf + ω2gxf = −w. (13b)

Substituting Eq. (13b) in Eq. (13a),

xg = −2ξgωgxf − ω2gxf ; (14)

in which w is the white-noise intensity at the rock bed withpower spectral density of S0. The variables xf , xf , and xf arethe response of the Kanai-Tajimi filter. The seismic motion,thus, is introduced in the formulation through incorporatingEq. (13b) and substituting the expression xg from Eq. (14) inthe rest of the equations.

The above equations are now rearranged to represent thestate-space form. Multiplying both sides of Eq. (1) with M−1

and substituting the expression with (xg + xb) from Eq. (4),Eq. (1) can be rewritten as

x = −M−1Cx−M−1Kx+

r

(cbmb

xb + αkbmb

xb +(1−α)FY

mbZ − c1

mbx1 −

k1mb

x1

).

(15)

Similarly, Eq. (2) for the base mass/isolator can be rewrittenby dividing both sides by base mass mb and substituting theexpression xg from the filter Eq. (14) as

xb = − cbmb

xb − αkbmb

xb −(1−α)FY

mbZ +

c1mb

x1 +k1mb

x1+

2ξgωgxf + ω2gxf . (16)

Equation (9) for Z can be rewritten as

Z = −Ceqxb −

Ke

qZ. (17)

Also, the filter equations from Eq. (13b) can be expressed as

xf = −2ξgωgxf − ω2gxf − w. (18)

The state variables are introduced in a state vector as

Y =[xT xb Z xf xT xb xf

]T. (19)

Equations (15)–(18) can be expressed in state-space form as

d

dtY = AY + w; (20)

where A is the augmented system matrix and

w =[0 0 0 0 0 0 −w

]T. (21)

In the equations above, the vector x has a dimension equal tothe number of structural degrees of freedom (n), and Y has asize of (2n+5). The structure of the augmented system matrixA is provided in the Appendix.

The response of the system can be evaluated by solvingEq. (20). In stochastic dynamic analysis, the statistics, suchas covariance of responses, are of interest. It can be shown thatthe covariance matrix CYY of the response vector Y (assumedas Markovian) evolves following an equation of the form32

d

dtCYY = ACYY

T+ CYYAT+ Sww. (22)

CYY has dimensions of (2n+ 5, 2n+ 5) with its terms as

CYiYj = E[YiYj ]. (23)

The matrix Sww contains a term that quantifies the intensityof the white-noise excitation at the rock bed, denoted as S0.Following the structure of w, matrix Sww has all terms thatequal zero, except the last diagonal, which is 2πS0.

The covariance of responses can be obtained by solvingEq. (22). It should be noted that even though the stochasticlinearization is adopted for the nonlinear isolator behaviour,the system still shows the nonlinear characteristics because theequivalent linear stiffness and damping are still functions ofthe responses

Ce = f(xb, Z); Ke = g(xb, Z); (24)

where f and g refer to the nonlinear functions of the responsequantities xb and Z, the isolator velocity, and hysteretic dis-placement, respectively (as in Eq. (10a) and (10b)). In solvingEq. (22), these terms are modified in each iteration, followingthe response statistics of the previous steps. Iterations stops(converges) when the response from two successive steps arepractically identical.

The equations for response statistics of their derivative pro-cess (such as acceleration x, xb) are obtained as

CYY = ACYYAT+ Sww. (25)

The RMS responses are obtained from their covariance:

σYi=√CYiYi

. (26)



The absolute top-floor acceleration (un) and the relative top-floor displacement (xn) are two important design parametersfor a BI system. The RMS of the top-floor displacement re-sponse is given by

σxn=√CY Y (n, n); (27)

and the top-floor acceleration (un) is obtained by summing upthe relative top-floor (xn), isolator-base (xb), and the groundacceleration (xg) as

un = xg + xb + xn. (28)

Then, the RMS top-floor acceleration can be written as

σunun=√CY Y (2n+3, 2n+3) + CY Y (2n+4, 2n+4)+

CY Y (2n+5, 2n+5) ; (29)

where σ2xg

= CY Y (2n+3, 2n+3), σ2xb

= CY Y (2n+4, 2n+4),and σ2

xn= CY Y (2n+5, 2n+5) are the variance of ground

acceleration, acceleration of the isolator, and top-floor acceler-ation, respectively.

Subsequently, whenever they are discussed, the displace-ment and accelerations represent the motion of the top floor inthe superstructure. The stochastic dynamic analysis, presentedherein, is based on an earthquake load modelled as a stationarystochastic process. Extending this analysis to a non-stationaryearthquake model will be straight forward. However, doing sowill involve evaluating the time-dependent response statistics.

2.2. Sensitivity of Stochastic Responseunder Parametric Uncertainty

The stochastic response evaluation presented above assumesthat the system parameters are deterministic.33 However, un-certainties in the system parameters may lead to large and un-expected excursion of responses, causing drastic reduction inaccuracy and precision of safety evaluation.34 In design ofsuch a system, apart from the stochastic nature of earthquakeloading, the uncertainties with regard to these parameters areexpected to be influential. The sensitivities of the stochasticresponses with respect to the uncertain parameters are essen-tial in considering the effects of parametric uncertainty. Theformulation to obtain the sensitivities is presented.

The random variability is reasonably assigned to the param-eters of the isolator, the structure, and in the earthquake load,denoted by

θ =[k c kb cb FY ξg ωg S0

]T; (30)

where θ is the vector of random design parameters, k is the uni-form storey stiffness, c is the uniform damping of each storey,and the other parameters are defined earlier. For simplicityof presentation, uniform storey stiffness and damping are con-sidered herein. However, the proposed formulation is not re-stricted to such an assumption and can easily be applied forvarying values of k and c along different stories. First-ordersensitivity of the base of Eq. (22) with respect to the i-th pa-rameter θi is written as

d

dt

∂CYY

∂θi= A

∂CYY

∂θi

T

+∂CYY

∂θiAT+ B; (31)

in which ∂CYY

∂θiis the sensitivity of response covariance

(CYY) with respect to the parameter θi, and B is defined by

B =∂A

∂θiCYY

T+ CYY∂A

∂θi

T

+∂Sww

∂θi. (32)

It should be noted that Eq. (31) has the same form as Eq. (22)and can be solved similarly. The sensitivity of the time deriva-tive processes (i.e., acceleration) can be obtained with

∂CYY

∂θi= A

∂CYY

∂θiAT+ B1; (33)

where

B1 = ACYY∂A

∂θi

T

+∂A

∂θiCYYAT+

∂Sww

∂θi. (34)

The second-order sensitivity is obtained by further differenti-ating Eq. (33) with respect to the parameter θj . After rear-ranging the terms, the equation for the second-order sensitivitybecomes

d

dt

∂2CYY

∂θi∂θj= A

∂2CYY

∂θi∂θj

T

+∂2CYY

∂θi∂θjAT+ C; (35)

where C is given as

C =∂A

∂θj

∂CYY

∂θi

T

+∂CYY

∂θi

∂A

∂θj

T

+∂2A

∂θi∂θjCYY

T+

∂A

∂θi

∂CYY

∂θj

T

+ CYY∂2A

∂θi∂θj

T

+∂CYY

∂θj

∂A

∂θi

T

+∂2Sww

∂θi∂θj.

(36)

The statistics for the respective time derivative processes arealso expressed as

∂2CYY

∂θi∂θj= A

∂2CYY

∂θi∂θjAT + C1; (37)

where C1 is given by

C1 = A∂CYY

∂θi

∂A

∂θj

T

+∂A

∂θj

∂CYY

∂θiAT+

∂A

∂θjCYY

∂A

∂θi

T

+

A∂CYY

∂θj

∂A

∂θi

T

+ ACYY∂2A

∂θi∂θj

T

+∂2A

∂θi∂θjCYYAT+

∂A

∂θiCYY

∂A

∂θj

T

+∂A

∂θi

∂CYY

∂θjAT+

∂2Sww

∂θi∂θj. (38)

Equations (31) and (35) can be solved similarly for first- andsecond-order sensitivity, respectively. It should be mentionedthat some of the matrices involved in Eq. (35) for evaluating C

are null. This is because A and Sww have zero-th/first-orderterms, which are a function of random variables θi, which van-ishes after first-/second-order differentiation with respect to θi.

The sensitivities of the RMS responses are obtained by dif-ferentiating Eq. (26) with respect to the i-th random parameteras

∂σYm

∂θi=

1

2

1√CYmYm

∂CYmYm

∂θi;

∂2σYm

∂θi∂θj=

1

2

1√CYmYm

[∂2CYmYm

∂θi∂θj− 1

2

1

CYmYm

(∂CYmYm

∂θi

)2].

(39)



In the above equation, Ym is the response; σYmis the RMS re-

sponse of Ym. The terms ∂σYm

∂θiand ∂2σYm

∂θi∂θjrepresent the first-

and second-order sensitivity with respect to the parameters θiand θj , respectively.

It should be mentioned here that the system parameter ma-trix A, defined in the Appendix, is an explicit function of un-certain model parameters, θ. Thus, the derivatives can be di-rectly obtained by differentiating with respect to each parame-ter. However, the formulation does not impose any limit to thenumber of elements or degrees of freedom used in the anal-ysis. But with an increasing number of elements, the matri-ces will be bigger to accommodate the computational demand.For more complex super-structural systems, involving finite el-ement modelling for response evaluation, the matrix A cannotbe obtained explicitly. For implicitly generated element mass,stiffness, and the damping matrix of the system, the differenti-ation can be carried out through a sequence of calculations or,alternatively, by finite difference approximation and can sub-sequently be used to obtain the partial derivative of A. Thereare important practical considerations for computing deriva-tives that are required in structural sensitivity analysis.34, 35

2.3. Parameter Uncertainty andUnconditional Stochastic Response

The stochastic response of a structure under earthquakeloading depends on the system parameters and can be ex-panded around the mean value of the uncertain parameters(with the assumption that the random variability is small) us-ing the Taylor series. The random system parameter (θi) can beviewed as the superposition of the deterministic mean compo-nent (θi) with a zero mean deviatoric component (∆θi). Thus,the Taylor series expansion of the RMS response at the meanvalue of the random parameters can be written as

σYm = σYm(θi)+

nv∑i=1

∂σYm

∂θi∆θi+

1

2

nv∑i=1

nv∑i=1

∂2σYm

∂θi∂θj∆θi∆θj ;

(40)in which nv is the total number of random variables involved,and the derivative of the respective response quantities are thesensitivity terms addressed in the previous section. Assum-ing that the uncertain random variables are uncorrelated, thequadratic approximation provides the expected value of the un-conditional RMS response as

σYm = σYm(θi) +1

2

nv∑i=1

∂2CYmYm

∂θ2iσ2θi ; (41)

where σ2θi

is the standard deviation of the i-th random param-eter. The linear approximation of the Taylor series expansionfurnishes the variance of the RMS response:

var [σYm ] = σ2σYm

=

nv∑i=1

(∂σYm

∂θi

)2

σ2θi . (42)

Such responses are referred to as unconditional because thecondition that the structural parameters to be deterministic hasbeen relaxed while estimating such responses.33

Table 1. Statistical properties of the random system parameters.

Parameters MeanCoefficient Distribu-of variation tion

storey stiffness (k) 5830 kN/m normalstorey damping (c) 264499.5 kNs/m normal

isolator stiffness (kb) 56791296 kN/m normalisolator damping (cb) 10% normal

isolator yield- 5% of total 5%–15%normal

strength (FY ) weight (W )ground damping (ξg) 60% normal

ground frequency (ωg) 5 rad/s normalseismic intensity (S0) 0.05 m/s2 normal

Figure 2. The normalized (a) second-order and (b) first-order terms dictatingthe contribution of individual stochastic parameters in affecting the uncondi-tional response and their variances.

3. NUMERICAL ILLUSTRATIONS

A five-storied shear building (n = 5) is studied to investigatethe effects of parametric uncertainty on the performance of aBI system subject to stochastic earthquake. The mass, stiff-ness, and damping (mi, ki, ξi) of each floor are assumed to beidentical. The values of stiffness and mass for each storey areassigned to provide the desired value of time period (T ) to thesuperstructure (varying from 0.1 s to 1 s) employed for para-metric study. The uniform viscous damping ratio (ξ) is takenas 2%, unless specifically mentioned. The LRB is character-ized with a mass ratio (mb/m) of 1, the viscous damping ratio(ξb) of 10%, and the normalized yield strength (F0) of 0.05,unless otherwise specified. The parameters characterizing therandom earthquakes are ωg , ξg , and S0. All random param-eters in the study are assumed to be statistically independent,normally distributed, and the properties are listed in Table 1.

In prior to present the response of the BI system under pa-rameter uncertainty, the relative importance of the random sys-tem parameters are studied through a bar chart sensitivity anal-ysis (see Figs. 2(a) and 2(b)) in order to assess the contributionof the individual random variable in affecting the responses(shown in Eqs. (41) and (42)). In these figures, the first- andsecond-order sensitivity terms are multiplied by the varianceof the respective random variables. While plotting, the termsare normalized with respect to the square root of sum of allsquared terms. For first-order terms (governing the variance of



Figure 3. Conditional and unconditional RMS top-floor (a) displacement and (b) acceleration, and coefficient of variation of top-floor (c) displacement and(d) acceleration with varied superstructure flexibilities and different degrees of uncertainty.

Figure 4. Conditional and unconditional RMS top-floor (a) displacement and (b) acceleration, and coefficient of variation of top-floor (c) displacement and(d) acceleration with varied superstructure damping for different degrees of uncertainty.



Figure 5. Conditional and unconditional RMS top-floor (a) displacement and (b) acceleration, and coefficient of variation of top-floor (c) displacement and(d) acceleration with varied isolation time period and different degrees of uncertainty.

Figure 6. Conditional and unconditional RMS top-floor (a) displacement and (b) acceleration, and coefficient of variation of top-floor (c) displacement and(d) acceleration with varied isolation damping and different degrees of uncertainty.



Figure 7. Conditional and unconditional RMS top-floor (a) displacement and (b) acceleration, and coefficient of variation of top-floor (c) displacement and(d) acceleration with varied yield strength of isolator and different degrees of uncertainty.

Figure 8. Conditional and unconditional RMS top-floor (a) displacement and (b) acceleration, and coefficient of variation of top-floor (c) displacement and(d) acceleration with varied seismic intensity and different degrees of uncertainty.



unconditional response), this is expressed as

(∂f/∂xi)σxi√∑nvi=1(∂f/∂xi)2σ2

xi

; (43)

and a similar expression is used for contribution of the second-order terms (governing the unconditional response). Figure 2clearly indicates that the superstructure stiffness is the mostimportant parameter as far as the top-floor displacement re-sponses are concerned, whereas the top-floor acceleration re-sponse is primarily affected by the uncertainty in the groundfrequency (ωg) and intensity of the earthquake (S0). Thesecond-order sensitivity of both the responses with respect tothe seismic intensity is negative. It is noteworthy that the first-order sensitivity of top-floor acceleration with respect to theground frequency is much higher than the ground damping,whereas for the second-order sensitivity, the trend is the oppo-site. These observations conform to the fact that the displace-ment response is governed by the system stiffness, whereasthe floor acceleration, being a measure of the amount of seis-mic force transmitted to the structure, is dictated by the seismicground motion.

Though the uncertainty in the excitation process dominatesthe performance of the BI system, the uncertainties in thesystem parameters, in particular, the superstructure stiffness,damping, and yield strength of the LRB have noticeable in-fluence and cannot be neglected. With this nature of the para-metric randomness and the relative importance, the conditionaland unconditional response of the system and the associatedcoefficient of variation are studied subsequently.

The conditional and the unconditional responses of the BIsystem are presented for varied superstructure flexibilities inFigs. 3(a) and 3(b). The superstructure flexibilities are ex-pressed in terms of the time period of the superstructure. Theparametric uncertainty increases the RMS displacement andacceleration, shown in Figs. 3(a) and 3(b). The increase in theRMS acceleration is, however, lesser than the RMS displace-ment. This difference might be the result of the absolute ac-celeration including ground acceleration, which is irrespectiveof the parametric uncertainty. The coefficient of variation forthe respective unconditional responses are shown in Figs. 3(c)and 3(d). It should be noted that the coefficient of variation in-creases with increasing randomness for the range of consideredsuperstructure flexibilities. The coefficient of variation for theRMS displacement is almost three orders of magnitude highercompared to that of RMS acceleration.

The variations of the conditional and unconditional re-sponses of the BI system are shown with respect to the varieddamping of the superstructure in Figs. 4(a) and 4(b). It is im-portant to note that the deterministic system underestimates thesuperstructure RMS displacement and acceleration. The dif-ference increases with an increasing level of uncertainty. Ob-serving the parametric variations of responses with the super-structure damping, it can be inferred that the RMS accelerationresponse is more sensitive to the variation of damping. How-ever, the change of coefficient of variation of the accelerationwith an increasing level of randomness in system parameters

is less than that of the displacement, implying less sensitivityof the acceleration to uncertainty.

The conditional and unconditional responses are presentedwith varied time period of isolation in Figs. 5(a)–(d). Thesefigures also show that the unconditional responses are greaterthan the respective conditional responses. The coefficient ofvariation of the acceleration is seen to be a few orders of mag-nitude less than that of displacement.

The parametric variation of the responses and their coeffi-cients of variation are shown with respect to isolator dampingin Figs. 6(a)–(d). As described earlier, for different level ofrandomness in the system parameters, the unconditional re-sponses and associated coefficients of variation are affecteddifferently. The displacement is more affected by the uncer-tainty compared to the acceleration, as shown in terms of theunconditional responses and associated coefficients of varia-tion. The effect of uncertainty uniformly affects the disparitiesamong the conditional and unconditional responses. The co-efficient of variation of the unconditional responses does notchange significantly with the variations of isolator damping,confirmed by Figs. 6(c) and (d).

The responses of the BI system are further studied with re-gard to varied yield strength of the isolator in Figs. 7(a)–(d).As described earlier, the conditional responses are lower com-pared to the respective unconditional responses. It is notewor-thy that the isolator yield strength presents the optimal valuefor which acceleration/displacement is the minimum. Obvi-ously, the optimal yield strength values for minimum displace-ment and acceleration are not necessarily identical. It shouldbe noted that the associated coefficient of variation of the dis-placement is almost irrespective of the parametric variations ofthe yield strength (Fig. 7(c)), whereas the coefficient of varia-tion for the acceleration shows a definite trend with respect tovarying isolator yield strength (Fig. 7(d)). The coefficient ofvariation of the unconditional acceleration response decreasesfirst and then starts increasing with increasing yield strengthof the isolator, which is unlikely for the variations of the re-sponses presented so far. Comparing Figs. 7(b) and 7(d), itis observed that the optimal values of the yield strength cor-responding to the minimum RMS acceleration might not en-sure that the respective coefficient of variation is also at theminimum. Thus, designers must deal with two conflicting ob-jective functions, i.e., the minimization of the unconditionalRMS response and its variance. This problem can formally beaddressed by formulating it as bi-objective optimization in or-der to minimize the unconditional RMS response as well as itsdispersion (coefficient of variation) simultaneously—referredto as robust design optimization (RDO). The optimum LRBparameters in RDO are achieved from a set of Pareto optimalsolutions by ensuring the desired level of robustness in the de-sign. The issue is emphasized, though not studied, in this pa-per.

It is noteworthy that the trend in Figs. 7(a) and (b) might notnecessarily be the same as that observed in an isolated structuresubject to a particular ground motion time history, even thoughfor large numbers of ground motion time histories, the averagetrend should match that of the stochastic response in an ensem-



Table 2. Disparities among the conditional and unconditional responses forranges of parameter variations.

Discrepancy DiscrepancyLower Uncer- among cond. among cond.

System and tainty and uncond. and uncond.parameters upper (cov, top-floor dis- top-floor ac-

value %) placement (%) celeration (%)Lower Upper Lower Upper

Structural 0.15 0.27 0.41 0.020 0.019

period 1.010 1.10 1.43 0.08 0.0715 2.48 3.08 0.19 0.17

Structural 0.015 0.00 0.00 0.03 0.03

damping 0.0910 0.90 0.90 0.90 0.1115 2.70 1.81 0.27 0.25

Isolator 1.55 0.00 0.00 0.044 0.02

period 2.510 1.40 1.25 0.17 0.0815 2.80 2.50 0.39 0.18

Isolator0.01

5 0.0 0.0 0.015 0.017yield

0.2010 0.60 0.78 0.055 0.066

strength 15 1.80 2.34 0.12 0.15

Isolator 0.0255 2.5 0.0 0.15 0.02

damping 0.1010 3.33 1.06 0.25 0.0815 5.00 2.13 0.42 0.20

Seismic 0.015 0.00 0.00 0.018 0.030

intensity 0.0910 0.00 0.60 0.025 0.1215 2.32 2.42 0.20 0.27

ble sense. This is due to the wide disparities which exist in thepower spectrum of a particular ground motion time history andthe idealized Kanai-Tajimi spectra, employed in this stochasticanalysis.

The behaviour is further studied for earthquake scenario ofdifferent intensities, shown in Figs. 8(a)–(d). With increas-ing seismic intensity, the RMS response increases for obviousreasons. The coefficient of variation of the unconditional dis-placement response is almost three times higher than the cor-responding acceleration response. However, the coefficient ofvariation for both is insensitive to the variations in the seismicintensities (Figs. 8(c) and (d)).

The discrepancies among the conditional and unconditionalresponses and the associated coefficients of variation have beenpresented, along with the parametric variations of responseswith the structural and isolator parameters. Though the varia-tions of the response coefficients of variation are quite obviousfrom the plots, the response values and associated disparities(in %) are presented in Table 2 to provide more precise esti-mates of the disparities. The particular cases presented herecorrespond to the minimum and maximum value of the re-spective parameters considered in the parametric study. Ta-ble 2 shows that the differences among the conditional and un-conditional RMS displacement responses could be as high as5%, whereas for acceleration response, the difference is around0.5%. The associated coefficients of variation are around 20%for displacement and 8% for acceleration (Figs. 7(c), 7(d), 8(c),and 8(d)). These differences (particularly for displacement) aresubstantial, considering the fact that the peak value of the re-sponses could be as high as three to four times (peak factor)of the respective RMS values.31, 32 Thus, system parameter un-certainty is a critical issue in response evaluation and designof BI systems in order to ensure the desired level of seismicsafety.35

4. CONCLUSIONS

The effects of system parameter uncertainty on the perfor-mance of BI systems subject to a stochastic earthquake loadhas been examined. This study incorporates the effects of sys-tem parameter uncertainty in the response evaluation throughperturbation-based analysis of the dynamic equations of mo-tion. The degree of parameter uncertainty is assumed to besmall so that the linear first-order perturbation analysis is valid.The responses are found to be in parity with the results ob-tained from the usual random vibration analysis assuming de-terministic system parameters. However, a definite change inthe responses occurs when the effects of system parameter un-certainty are included. In general, the efficiency of a BI sys-tem tends to decrease as the level of uncertainty increases,though efficiency is not completely eliminated. Therefore,even though the randomness in the seismic events dominates,the random variations of the system parameters play a signif-icant role. The randomness in the structural stiffness governsthe unconditional RMS displacements and associated coeffi-cient of variation, whereas the randomness in the characteris-tic ground frequency and intensity governs the unconditionalRMS acceleration response and coefficient of variation. Theunconditional RMS responses are always greater than (floordisplacement, acceleration) the conditional responses. The dis-crepancies among the conditional and unconditional RMS re-sponses are observed to be 5% for the superstructure displace-ment, with the maximum possible value of isolator damping.The respective coefficient of variation of the unconditional re-sponses for the top-floor superstructure displacement and ac-celeration are around 19% and 6%, respectively. With theobserved discrepancies amongst the conditional and uncondi-tional responses and coefficient of variation, the peak value ofthe respective responses could be significantly different fromthe conditional one. Thus, disregarding the system uncertaintymight lead to an unsafe design. From the parametric variationsof the responses with respect to the isolator yield strength, itis observed that the optimal yield strength exists to ensure thebest performance, which might be affected by the uncertainty.The optimum value of the yield strength might be based on twomutually conflicting objective function containing the mini-mization of both the unconditional responses as well as theassociated standard deviations, thus making the problem idealfor reliability-based robust optimization. However, this aspectrequires further study.

REFERENCES1 Spencer, B. F. and Nagarajaiah S. State of the art of struc-

tural control, Journal of the Art of Structural Control,129 (7), 845–856, (2003).

2 Buckle, I. and Mayes, R. Seismic isolation: History, appli-cation, and performance—A world view, Earthquake Spec-tra, 6 (2), 161–201, (1990).

3 Symans, M. D. and Constantinou, M. C. Semi-active con-trol systems for seismic protection of structures: A state-



of-the-art review, Engineering Structures, 21, 469–487,(1999).

4 Jangid, R. S. and Datta, T. K. Seismic behaviour of base-isolated buildings: A state-of-the-art review, Structures andBuildings, 110 (2), 186–202, (1995).

5 Ribakov, Y. Selective controlled base isolation system withmagnetorheological dampers, Earthquake Engineering andStructural Dynamics, 31 (6), 1301–1324, (2002).

6 Taflandis, A. A. and Jia, G. A simulation-based frame-work for risk assessment and probabilistic sensitivity analy-sis of base-isolated structures, Earthquake Engineering andStructural Dynamics, 40 (14), 1629–1651, (2011).

7 Jangid, R. S. Stochastic response of building frames iso-lated by lead-rubber bearings, Structural Control HealthMonitoring, 17 (1), 1–22, (2010).

8 Jangid, R. S. Optimum lead-rubber isolation bearings fornear-fault motions, Engineering Structures, 29 (10), 2503–2513, (2007).

9 Baratta, A. and Corbi, L. Optimal design of base-isolators in multi-storey buildings, Computers and Struc-tures, 82 (23–26), 2199–2209, (2004).

10 Yen, Y. J., Huang, P. C., and Wan S. Modifications onbase isolation design ranges through entropy-based classi-fication, Expert Systems with Applications, 36 (3:1), 4915–4922, (2009).

11 Li, H. N. and Ni, X. L. Optimization of non-uniformly dis-tributed multiple tuned mass damper, Journal of Sound andVibration, 308 (1–2), 80–97, (2007).

12 Marano, C., Trentadue, F., and Greco, R. Stochastic opti-mum design criteria for linear damper devices for seismicprotection of buildings, Structural and MultidisciplinaryOptimization, 33 (6), 441–455, (2007).

13 Taflanidis, A. A., Beck, J. L., and Angelides, D. C. Robustreliability-based design of liquid column mass dampers un-der earthquake excitation using an analytical reliability ap-proximation, Engineering Structure, 29 (12), 3525–3537,(2007).

14 Chakraborty, S. and Roy, B. K. Reliability based optimumdesign of Tuned Mass Damper in seismic vibration con-trol of structures with bounded uncertain parameters, Prob-abilistic Engineering Mechanics, 26 (2), 215–221, (2011).

15 Debbarma, R., Chakraborty, S., and Ghosh, S. Optimum de-sign of tuned liquid column dampers under stochastic earth-quake load considering uncertain bounded system parame-ters, International Journal of Mechanical Sciences, 52 (10),1385–1393, (2010).

16 Debbarma, R., Chakraborty, S., and Ghosh, S. Uncon-ditional reliability based design of tuned liquid columndampers under stochastic earthquake load considering sys-tem parameter uncertainties, Journal of Earthquake Engi-neering, 14 (7), 970–988, (2010).

17 Marano, G. C., Sgobba, S., Greco, R., and Mezzina, M.Robust optimum design of tuned mass dampers devices inrandom vibrations mitigation, Journal of Sound Vibration,313 (3–5), 472–492, (2008).

18 Marano, G. C., Greco, R., and Sgobba, S. A. A comparisonbetween different robust optimum design approaches: Ap-plication to tuned mass dampers, Probabilistic EngineeringMechanics, 25 (1), 108–118, (2010).

19 Benfratello, S., Caddemi, S. and Muscolino G. Gaussianand non-Gaussian stochastic sensitivity analysis of discretestructural system, Computers and Structures, 78 (1–3),425–434, (2000).

20 Kawano, K., Arakawa, K., Thwe, M., and Venkastaramana,K. Seismic response evaluations of base-isolated structureswith uncertainties, structural stability and dynamics, Proc.International Conference on Structural Stability and Dy-namics, Singapore, 16–18, 889–894, (2002).

21 Scruggs, J. T., Taflanidis, A. A., and Beck, J. L. Reliability-based control optimization for active base isolation sys-tems, Structural Control and Health Monitoring, 13 (2–3),705–723, (2006).

22 Zhou, J., Wen, C., and Cai, W. Adaptive control of a baseisolated system for protection of building structures, Jour-nal of Vibration and Acoustics, 128 (2), 261–268, (2006).

23 Zhou, J. and Wen, C. Control of a hysteretic structural sys-tem in base isolation scheme, adaptive backstepping controlof uncertain systems, Lecture Notes in Control and Infor-mation Sciences, 372, 199–213, (2008).

24 Li, J. and Chen, J. B. First passage of uncertain sin-gle degree-of-freedom nonlinear oscillations, InternationalJournal for Numerical Methods in Engineering, 65 (6),882–903, (2006).

25 Ismail, M., Ikhouane, F., and Rodellar, J. The hystere-sis Bouc-Wen model, a survey, Archive of ComputationalMethods in Engineering, 16 (2), 161–188, (2009).

26 Wen, Y. K. Method of random vibration of hysteretic sys-tems, Journal of Engineering Mechanics, 102 (2), 249–263,(1976).

27 Atalik, T. S. and Utku, S. Stochastic linearization of mul-tidegree of freedom nonlinear systems, Earthquake Engi-neering and Structural Dynamics, 4 (4), 411–420, (1976).

28 Roberts, J. B. and Spanos, P. D. Random Vibrations andStatistical Linearization, Wiley, New York, (2003).

29 Kanai, K. Semi-empirical formula for the seismic charac-teristics of the ground, Bulletin of Earthquake Research In-stitute, 35, 309–325, (1957).

30 Tajimi, H. A statistical method of determining the maxi-mum response of a building structure during an earthquake,Proc. World Conference on Earthquake Engineering, 11,781–798, (1960).



0 · · · 0 0 0 0 1 · · · 0 0 0... n× n

...... n× 3

...... n× n

......

...0 · · · 0 0 0 0 0 · · · 1 0 0

0 · · · 0 0 0 0 0 · · · 0 1 0

0 3× n 0 0 −keq 0 0 3× n 0 − ceq 0

0 · · · 0 0 0 0 0 · · · 0 0 1

M –1K11− k1mb

· · · M –1K1n α kbmb

(1−α)FY

mb0 M –1C11− c1

mb· · · M –1C1n

cbmb

0... M –1Kij−δi1 k1mb

......

......

... M –1Cij−δi1 c1mb

......

...M –1Kn1− k1

mb· · · M –1Knn α kb

mb

(1−α)FY

mb0 M –1Cn1− c1

mb· · · M –1Cnn

cbmb

0k1mb

· · · 0 −α kbmb− (1−α)FY

mbω2g

c1mb

· · · 0 − cbmb

2ξgωg0 · · · 0 0 0 −ω2

g 0 · · · 0 0 −2ξgωg

(A.1)

31 Sun, J. Q. Stochastic Dynamics and Control, Elsevier, Am-sterdam, (2006).

32 Lutes, D. L. and Sarkani, S. Random Vibrations, Analysis ofStructural and Mechanical Systems, Elsevier, Burlington,Massachusetts, (2004).

33 Jensen, H. A. Design and sensitivity analysis of dynami-cal systems subjected to stochastic loading, Computers andStructures, 83 (14), 1062–1075, (2005).

34 Jensen, H. A. Structural optimization of non-linear systemsunder stochastic excitation, Probabilistic Engineering Me-chanics, 21 (4), 397–409, (2006).

35 Chaudhuri, A. and Chakraborty, S. Sensitivity evaluation inseismic reliability analysis of structures, Computer Meth-ods in Applied Mechanics and Engineering, 193 (1–2), 59–68, (2004).

APPENDIX

The structure of the augmented system matrix (Eqs. (20)and (22)) A for a n-storied shear building is given in Eq. A.1All the parameters in the matrices have already been defined inthe main text. The variable δij is the Kronecker’s delta definedas δij = 1 when i = j and δij = 0 when i 6= j. The aug-mented stiffness (M−1K) and damping (M−1C) matrices areindicated in the respective block of dimension n× n.

The mass matrix is diagonal containing the storey mass ineach diagonal term. The stiffness and damping matrices takethe following form:

K =

k1 + k2 −k2 · · · · · · · · ·−k2 k2 + k3 · · · · · · · · ·· · · −ki ki + ki+1 −ki+1 · · ·· · · · · · · · · kn + kn−1 −kn· · · · · · · · · −kn kn

(A.2)

C =

c1 + c2 −c2 · · · · · · · · ·−c2 c2 + c3 · · · · · · · · ·· · · −ci ci + ci+1 −ci+1 · · ·· · · · · · · · · cn + cn−1 −cn· · · · · · · · · −cn cn

(A.3)

in which ki and ci are the stiffness and damping of the i-thstorey of the building. The damping for the i-th storey is ex-pressed as

ci = 2ξ√kimi; (A.4)

where ξ is the viscous damping ratio of the superstructure.The matrix for the rock bed seismic motion, characterized

by the white noise of intensity S0 is expressed as

Sww =

0 0 · · · 0 0

0 0 · · · 0 0...

.... . .

......

0 0 · · · 0 0

0 0 · · · 0 2πS0

(A.5)

which is a square matrix of dimension (2n+ 5).


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