Stochastic analysis of base-isolated liquid storage tanks with uncertain isolator parameters under random excitation

Engineering Structures 57 (2013) 465–474

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Stochastic analysis of base-isolated liquid storage tanks with uncertain isolator parameters under random excitation S.K. Saha a, K. Sepahvand b,⇑, V.A. Matsagar a, A.K. Jain a, S. Marburg b a b

Department of Civil Engineering, Indian Institute of Technology (IIT) Delhi, Hauz Khas, New Delhi 110 016, India Institute of Mechanics, Universität der Bundeswehr Munich, Germany

a r t i c l e

i n f o

Article history: Received 3 July 2013 Revised 26 September 2013 Accepted 29 September 2013 Available online 4 November 2013 Keywords: Base isolation Laminated rubber bearing Liquid storage tank Non-intrusive method Polynomial chaos Random excitation Uncertain parameters

a b s t r a c t Stochastic response of base-isolated liquid storage tanks is presented herein, considering uncertainty in the characteristic isolator parameters, under random base excitation. The liquid storage tank is modeled using lumped mass mechanical analog, along with laminated rubber bearing (LRB) as the base isolator. The non-sampling stochastic simulation method based on the generalized polynomial chaos (gPC) expansion technique is used for numerical dynamic simulation of the base-isolated liquid storage tanks. The uncertain isolator parameters, the random base excitation and the system response quantities are represented by the truncated gPC expansions. The stochastic discrete model of the system, involving these expansions, is projected to an equivalent deterministic model by using the stochastic Galerkin method. Non-intrusive solution of the projected system is preformed, at a set of collocation points, for calculation of the gPC coefficients of the system response. The analyses are carried out for two different configurations of the liquid storage tanks, i.e. broad and slender tank configurations. It is observed that the uncertainty in the isolation parameters and in the excitation force affect the response of slender tanks more as compared to broad tanks. The statistics of the response quantities are examined, and the effectiveness of the procedure is compared with the results from the Monte Carlo simulations. It is further observed that uncertainty in the isolation damping has insignificant effect on the distribution of peak response quantities. It is also concluded that for specific uncertainties in the isolator parameters and excitation, the response calculated for the base-isolated liquid storage tank may be similar to that of fixed-base tank. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Seismic analysis of structures in the probabilistic domain has gained popularity over the years among the structural engineering community. Considerable attention is paid towards the stochastic seismic response analysis of base-isolated structures considering the uncertainty involved in the earthquake excitation as well as in the system parameters. Liquid storage tanks are strategically important structures for many industries and public service sectors. Base isolation is an effective technique to protect such important structures against devastating earthquakes as it helps to diminish the energy imparted by the earthquake to the structure. Use of base isolation for liquid storage tanks for seismic performance upgradation have been studied deterministically in details by Malhotra [1], Kim et al. [2], Shrimali and Jangid [3] and several other researchers. Consideration of the randomness of the earthquake excitation on the base-isolated buildings can be traced back in the work by Ahmadi [4] for mass block on sliding base. Studies related to the ⇑ Corresponding author. Tel.: +49 8960044110. E-mail address: [email protected] (K. Sepahvand). 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.09.037

base-isolated buildings and bridges, and comparison of different isolation systems, under stochastic earthquake, have been reported by several researchers [5–11]. In all these studies, only the seismic excitation has been considered random. However, the uncertainty in the base isolation system parameters, along with the random nature of the earthquake, may lead to inadequate estimation of the protective measures. Consideration of system parameter uncertainty can be seen in the limited number of studies. Manohar and Ibrahim [12] have presented a brief review on the consideration of the parametric uncertainties in dynamic analysis of structures. Li and Chen [13] have proposed an approach for dynamic response analysis of structures considering parameter uncertainty using Monte Carlo (MC) simulation. They have considered uncertain structural parameters, such as stiffness, and damping, along with the random input excitation. The proposed MC based simulation is robust and simple to apply. Nevertheless, large number of MC realizations is required for a reasonable accuracy in the simulation, which may be extremely expensive in terms of simulation time. Chaudhuri and Chakraborty [14] have presented a general procedure, using the perturbation based stochastic finite element method, to derive unconditional reliability of an idealized multi degree of freedom system considering

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parametric uncertainty. Gupta and Manohar [15] have proposed Taylor series based method to consider the parametric uncertainty on the stochastic response of pipeline structures in the context of reliability analysis. Nevertheless, in all those works on stochastic response of base-isolated structures, the isolator parameters have been considered deterministic. There are few studies reported, which address the stochastic seismic response of liquid storage tanks. Chatterjee and Basu [16,17] have used wavelet based technique for non-stationary seismic response analysis of liquid storage tanks. Talaslidis et al. [18] have investigated the response of industrial tanks for risk analysis under chemical blast and seismic excitation. They have considered the uncertainty in the seismic excitation only, while other structural input parameters are assumed deterministic. Mishra and Chakraborty [19] have considered the parametric uncertainty for reliability analysis of tower mounted base-isolated liquid storage tanks under random earthquake. They have used perturbation technique to consider the uncertainty involved in the isolator properties. They have concluded that the uncertainty in the earthquake motion dominates the variability in the reliability, however the system parameter uncertainty cannot be neglected for the precise estimation of the overall reliability. Furthermore, the major issue in using the perturbation technique is that the uncertainties cannot be too large for a range of acceptable accuracy. Recently, Curadelli [20] have studied the stochastic seismic response of base-isolated ground supported liquid storage tanks considering two soil conditions. The base excitation has been considered as the stationary random process, and it is characterized by the design spectrum compatible power spectral density function (PSDF). However, uncertainties in the important isolator parameters have not been considered in their study. From the literature review, it is evident that there is a need of extensive work towards stochastic formulation of base-isolated liquid storage tanks under random excitation considering uncertain isolation parameters. The objectives of the present study are to address three major issues: (i) considering the uncertainty involved in the isolator parameters, (ii) to develop an efficient stochastic simulation procedure and (iii) the effect of individual parameter uncertainties and level of uncertainty on the peak response distribution. To address these issues, generalized polynomial chaos (gPC) expansion technique is used. A detailed step-by-step procedure for numerical simulation, of the time variant response quantities, by using non-intrusive technique is explained. The need of considering the parametric uncertainty in the dynamic analysis of liquid storage tanks is also explained by comparing the response of fixed-base and base-isolated liquid storage tanks considering parametric uncertainties. The gPC is an efficient tool to solve stochastic system with less computational involvement. The idea of polynomial chaos expansion was first developed as homogeneous chaos by Wiener [21]. The procedure has been later applied extensively in the physical and engineering applications reported by several researchers [22–25]. The advantages of the gPC over other methods, such as perturbation techniques, MC simulation, Karhunen–Loéve (KL) expansion etc., are evident from the recent works on stochastic finite element approach in uncertainty modeling [26–29]. Herein, the stochastic response of ground supported base-isolated liquid storage tanks, with uncertain isolator parameters, subjected to random excitation, is investigated. The uncertainty in the input parameters and system response quantities are represented using gPC expansion. Lumped mass mechanical analog of the base-isolated liquid storage tank is considered with uncertain isolator parameters, and random base excitation is employed to obtain the stochastic dynamic response of the tank. The randomness in the excitation is considered in the amplitude and frequency parameters with predefined probability distributions. The gPC

expansions of all the parameters are constructed, and used to represent uncertainty in the system model. Accordingly, the system response quantities are represented by the gPC expansions with unknown deterministic functions and stochastic basis orthogonal polynomials. The stochastic Galerkin technique is employed to project the governing equations into a set of deterministic equations. The unknown coefficients of the deterministic equations are determined by using non-intrusive solution method, which uses a set of collocation points of the random basis function. It is shown that the results are in good agreement with the sampling MC simulations even for few number of collocation points. 2. Stochastic modeling of base-isolated liquid storage tank The deterministic model of a base-isolated liquid storage tank, subjected to horizontal base excitation, can be represented by a 3-degrees of freedom (DOF) lumped mass system using the mechanical analog proposed by Haroun and Housner [30], cf. Fig. 1(a). Afterwards, Haroun [31] has validated the dynamic response of liquid storage tanks using this model with experimental results, and the model has been widely used for base-isolated liquid storage tanks in reported research works [3,32]. In this model, the top liquid layer is considered as sloshing mass (mc), middle layer of the liquid is considered as impulsive mass (mi), and the liquid mass moving rigidly with the tank wall at the bottom is considered as rigid mass (mr). They are considered to act at heights Hc, Hi and Hr, respectively, from the base of the tank. The radius and height of the liquid column are denoted by R and H, respectively. The isolator is considered as laminated rubber bearing (LRB) with linear force–deformation behavior as described by Jangid and Datta [33], cf. Fig. 1(b), where Fb is the isolator restoring force. The equations of motion for such a model of the base-isolated liquid storage tank are expressed in the matrix form as

€ þ Cx_ þ Kx ¼ Mru €g Mx

ð1Þ

T

in which x = {xc, xi, xb} is the displacement vector; xc = uc  ub, xi = ui  ub and xb = ub  ug are the relative displacements of the sloshing, impulsive and rigid masses, respectively; and r = {0, 0, 1}T. Here, uc, ui, ub and ug represent the absolute displacements of the sloshing mass, the impulsive mass, the base (top of the isolator) and the ground, respectively. Accordingly, the mass matrix, M, the damping matrix, C and the stiffness matrix, K are expressed as

2

mc

6 M ¼ 40

mc

0 mi mi

mc

3

2

cc

7 6 mi 5 C ¼ 4 0 0 M

0 ci 0

0

3

2

kc

7 6 0 5 K ¼ 40 cb 0

0 ki 0

0

3

7 0 5 kb ð2Þ

in which the pairs (cc, kc), (ci, ki) and (cb, kb) are damping and stiffness of the sloshing mass, the impulsive mass and the base isolator, respectively. The laminated rubber bearing is characterized by its damping, cb = 4pfbM/Tb and isolation time period, T b ¼ 2p pffiffi ðM=kb Þ, where fb is the isolation damping ratio and M = mc + mi + mr. For the present study, damping and stiffness of the isolator, and amplitude and frequency of the uni-directional base excitation are considered as uncertain parameters. Hence, the overall damping and stiffness matrices of the base-isolated liquid storage tank, as well as the forcing terms of the model become uncertain. Considering these uncertainties Eq. (1) can be written as

€ðt; nÞ þ Cðnc Þxðt; _ nÞ þ Kðnk Þxðt; nÞ ¼ Mru € g ðt; ng Þ Mx

ð3Þ

where x(t, n) is the unknown random displacement vector, and nc, nk and ng are vectors of random variables representing the randomness

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(a)

(b)

Fig. 1. (a) Model of base-isolated liquid storage tank and (b) laminated rubber bearing (LRB) and its force-deformation behavior.

in damping, stiffness and base excitation, respectively. The random vector n represents the random variables involved in the system response and the excitation, which takes the following form

n ¼ fnc ; nk ; ng gT

ð4Þ

In general, the random variable vectors nc, nk and ng belong to various random Hilbert spaces which construct the new random basis function n belonging to the same space [28]. Here, all random variables are assumed to be standard. This allows to preserve the orthogonality properties for the polynomial basis expansion in the random space by using the Hilbert space properties. The uncertain damping and stiffness matrices can be represented by the truncated generalized polynomial chaos (gPC) expansions as follows [26,28].

Cðnc Þ ¼

N1 X Ci1 Wi1 ðnc Þ

ð5Þ

i1 ¼0

Kðnk Þ ¼

N2 X

Ki2 Wi2 ðnk Þ

ð6Þ

i2 ¼0

where Ci1 and Ki2 are the deterministic unknown coefficient matrices and Wi1 ðnc Þ and Wi2 ðnk Þ are the stochastic basis functions for damping and stiffness, respectively. The random base acceleration and response are modeled as random fields, and can be represented by the gPC expansions as

€ g ðt; ng Þ ¼ u

N3 X € gi ðtÞWi3 ðng Þ u i3 ¼0

xðt; nÞ ¼

N4 X

ð7Þ

3

xi4 ðtÞWi4 ðnÞ

ð8Þ

i4 ¼0

€ gi ðtÞ is the deterministic unknown coefficient for in which, again, u 3 the base excitation, and xi4 ðtÞ is the deterministic unknown coefficient vector of the response. Wi3 ðng Þ and Wi4 ðnÞ are the stochastic basis functions for the excitation and the response, respectively. Substitution of these expansions in Eq. (3) yields an approximated stochastic form of the system equations. The stochastic approximation error, denoted by (t, n), is expressed as

ðt; nÞ ¼ M

N4 N1 N4 X X X €xi4 ðtÞWi4 ðnÞ þ Ci1 Wi1 ðnc Þ x_ i4 ðtÞWi4 ðnÞ i4 ¼0

þ

i1 ¼0

i2 ¼0

i4 ¼0

i3 ¼0

3

3. Procedure for numerical simulation In the stochastic variational concept, the non-intrusive method is same as the collocation method which forces the residual error to be deterministically zero at specific points, i.e. in Eq. (9),

Z fni g

i4 ¼0

N2 N4 N3 X X X €gi ðtÞWi3 ðng Þ Ki2 Wi2 ðnk Þ xi4 ðtÞWi4 ðnÞ þ Mr u

tion of the error, (t, n). Consequently, any optimization process to minimize stochastic error must be performed with respect to the random space discretization by the gPC. The gPC coefficients of the system matrices and the excitation terms are to be calculated in pre-processing step using suitable projection scheme. One can calculate the deterministic gPC coefficients of the system response by minimizing the error, (t, n). The selection and the form of the optimization process depend on the available information. For instance, one can attempt to minimize the error between the statistical moments calculated from the gPC expansions of the response and the corresponding values from theoretical or experimental results. Another possibility is to use the deterministic responses of the system at specific points or roots of the orthogonal polynomials, and minimizing the least squares error between these responses and the responses approximated by the gPC expansion. Generally, there are two broad classes of methods that can be used to solve the above stochastic model: (i) intrusive and (ii) non-intrusive methods. Implementation of the intrusive method requires projection of the stochastic model into an equivalent deterministic model by using stochastic Galerkin projection, whereas in the non-intrusive method, the model is employed as third party solver or black-box, and the solution is investigated at specific collocation points of the stochastic basis function. The choice between these methods depends on the problem under investigation, and the information available on the system model. For instance, if the governing equations of the system are readily available and linear in nature, the intrusive method can be used to generate the deterministic equivalent model. However, the order of the projected model depends on the order of the gPC expansion representing the uncertainty in the system response quantities. In this study, the non-intrusive method is used for its simplicity. The step-by-step procedure used for the numerical simulation is discussed in the following section.

ð9Þ

Once the stochastic basis functions, W(n)s, are chosen, the solution process reduces to computation of the unknown xi4 ðtÞ by minimiza-

ðt; nÞdðni  pki Þf ðni Þdni ¼ 0;

for k ¼ 0; 1; . . . ; N5

ð10Þ

in which d is the delta function, pki denotes the set of specific collocation points and f(ni) is probability density function (PDF) of random variable ni. The method provides this major facility to use the deterministic model as black-box to get the system response associated to each realization of the random vector. These

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characteristics make the method very attractive for parametric quantification in complex models where the deterministic model has been already developed. The procedure for the numerical simulation, considering uncertain system parameters, using non-intrusive approach can be summarized in following steps. All the model parameters are assumed independent and normally distributed. (i) Representation of uncertain input parameters: To consider the parameter uncertainty in the stochastic formulation, the first step is to select the uncertain input parameters. The selected input parameters are then required to be represented using a set of standard random variables {nj} with the help of the gPC expansion. In the present study, the isolator stiffness, kb is considered in terms of the isolation time period, Tb. The isolation damping, cb is considered in terms of the damping ratio, fb. Tb and fb are represented by gPC, which are included in Eqs. (5) and (6). The base excitation is assumed as sinusoidal acceleration with random amplitude, Aðna Þ and the frequency, x(nx) as given by

€ g ðt; ng Þ ¼ Aðna Þ sin½xðnx Þt u

ð11Þ

(ii) Determination of the coefficients for the uncertain parameters: Assuming the uncertain parameters (P j ) are normally distributed, they can be best represented by Nth order Hermite polynomial, Hk(n) as the following equation.

P j ðnj Þ ¼

N X

pjk fHk ðnj Þg

ð12Þ

k¼0

where j stands for cb ; kb ; A or x. To calculate the deterministic gPC coefficients (pjk ), Galerkin projection technique is applied for each uncertain parameter as given by the following equation [28].

pjk ¼

1

Z

hH2k i

(iv) Selection of collocation points: To determine the unknown coefficient, yi(t), the response of the system is to be evaluated for some set of input parameters. The input parameters are obtained for specific values of the uncertain random variable (n), known as collocation points. The minimum number of the collocation points should be at least equal to the number of unknown coefficients (Nr + 1) in the approximating gPC expansion. These points can be selected as zero and the roots of the one order higher polynomial that is used to approximate the response. If more points are required, they should be selected in such a way that they lay symmetrically about the origin. (v) Determination of the unknown polynomial coefficients for the response: Using the selected collocation points, a set of the input random variables is generated through the idealization in step (ii). Now, these values are to be given as inputs to the already developed deterministic model of the baseisolated liquid storage tank, Eq. (1). For each set of the input parameters, the dynamic analysis has to be carried out, and the response time histories are to be obtained independently. At each time step, the system of the simultaneous Eq. (16) is to be solved to determine the unknown coefficient yi(t). When the number of collocation points are higher than the number of unknown coefficients (Nr + 1), regression analysis based on the least squares method may be applied to solve the simultaneous equations. (vi) Calculation of the response statistics: Once the polynomial coefficients are determined, they are substituted back into Eq. (16). Now, the response of the base-isolated liquid storage tank is expressed in terms of the uncertain input random variables, and at each time step, the response statistics can be obtained. The mean (lr) and variance (r2r ) of the response can be calculated by the following equation [28].

1

P j ðnj ÞHl ðnj Þf ðnj Þ dnj ;

l ¼ 0; 1; . . . ; N

ð13Þ

lr ¼ y0 ;

1

r2r ¼

P j ¼ lj þ rj n;

where n  N ð0; 1Þ

ð14Þ

(iii) Approximate polynomial model of the response: The time variation of the output response quantity of interest, Y r ðt; nÞ is to be represented using suitable polynomial expansion as given by

Y r ðt; nÞ ¼

Nr X

yi ðtÞwi ðnÞ

ð15Þ

i¼0

where yi(t) is the unknown deterministic coefficient at each time instant, and wi(n) is the stochastic basis function. This polynomial consists of all the uncertain input random variables. The number of unknown coefficients increases rapidly with the number of uncertain input random variables and the order of the approximating polynomial. As all the input random variables are assumed to be normally distributed, using 3rd order Hermite polynomial the response, as a function of the time, Yr(t, n) can be represented as

Y r ðt; nÞ ¼

3 X

yi ðtÞHi ðnÞ

ð16Þ

i¼0

where H0 = 1, H1 = n, H2 = (n2  1) and H3 = (n3  3n) are the Hermite polynomials of the order 0, 1, 2 and 3, respectively; yi(t) is the deterministic coefficient at each time step corresponding to each term of the gPC expansion.

2

y2i hi

ð17Þ

i¼1

hH2k i

where denotes the inner product in the Hilbert space spanned by the Hk. Considering the uncertain parameter P j is normally distributed with the mean lj and the standard deviation rj, Eq. (12) takes the following simple form.

Nr X

2

where hi is the norm of the polynomial. For 1-D Hermite polyno2 mial with normally distributed uncertain parameters, h1 ¼ 1, 2 2 h2 ¼ 2 and h3 ¼ 6. The convergence and stability of the stochastic non-intrusive method follow the same as the deterministic collocation due to the fact that the formulation of the method and deterministic collocation are the same if the probability density function (PDF) of random variable basis is not zero for all the collocation points [34]. With the non-intrusive method, the uncertainty analysis of a given model is performed as an extension of the deterministic analysis of the model. 4. Numerical study Circular steel tank, with laminated rubber bearing (LRB) as base isolator, is considered for the present study, and the contained liquid is considered as the water. The input base excitation is considered as uni-directional sinusoidal sine wave acceleration as given in Eq. (11). Table 1 summarizes the geometrical properties of the liquid storage tanks with two different configurations, such

Table 1 Properties of the cylindrical tanks, cf. Fig. 1(a). Type of tank

H/R

H (m)

ts/R

Broad tank Slender tank

0.6 1.85

14.6 11.3

0.004 0.004

S.K. Saha et al. / Engineering Structures 57 (2013) 465–474

as broad and slender. The thickness of the tank wall is denoted by ts. The density and modulus of elasticity for the tank wall material are considered as 7800 kg/m3 and 2  105 MPa, respectively. The density of the contained liquid is considered as 1000 kg/m3. Small damping of 0.5% and 2% are assumed corresponding to the sloshing and impulsive masses, respectively [31]. The mean and standard deviation of the normally distributed uncertain input parameters are shown in Table 2. Base shear (Vb), base displacement (xb), sloshing displacement (xc) and the overturning moment (Mb) of the base-isolated liquid storage tank are considered as the important response quantities. The base shear and the overturning moment are presented in terms of the total weight of the liquid storage tank (W = Mg), where g is the gravitational acceleration. The deterministic model of the base-isolated liquid storage tank is analyzed for nine sets of input variables, and the coefficients of the gPC expansion for the response quantities are determined using least squares regression. The dynamic equations of motion in Eq. (1) are solved incrementally at each time step using Newmark’s-b method to obtain the deterministic response time histories. Total time of the excitation is kept constant at 15 s, and the time increment is considered as 0.02 s. Incorporating the Hermite polynomials, Eq. (16) is rewritten as

Y r ðn; tÞ ¼ y0 ðtÞ þ y1 ðtÞn þ y2 ðtÞðn2  1Þ þ y3 ðtÞðn3  3nÞ

ð18Þ

The stochastic model in Eq. (10) is solved for the sets of nine collocation points generated from the roots of the 4th order Hermite polynomial to derive the coefficients yi as shown in Figs. 2 and 3. It is observed from these figures that the first coefficient (y0), which represents the mean response, dominates in all the cases. However, the second coefficient (y1), which influences the variance of the response, is comparable with the y0 in case of base shear, base displacement and overturning moment. This indicates that these response quantities are more sensitive to the uncertainty in the isolator parameters and base excitation. The influence of y1 is more in case of the slender configuration of the tank. Moreover, the higher order coefficients, y2 and y3, are very small, which indicates the effectiveness of the gPC expansion with respect to the convergence of the response. Table 3 shows the mean (l) and standard deviation (r) of the response quantities at different time instants. The deterministic response, taking the mean values of the input parameters, of the base-isolated liquid storage tanks are also given for comparison purpose. From the tabulated values of the response quantities, as shown in Table 3, it is observed that deterministic analysis may underpredict the response of base-isolated liquid storage tanks. Hence, the uncertainties of the input parameters must be considered to accurately estimate the design forces. It is also observed that the mean base shear, base displacement and overturning moment, considering the uncertain parameters, are deviating more from the deterministic values, whereas the deviation in the mean sloshing displacement, from its deterministic value, is less. Further, it can be noted that slender tank configuration is more sensitive to the parametric uncertainty. 4.1. Probability distribution of peak response quantities Structural safety is defined more precisely in probabilistic way with reliability analysis. Keeping this in mind, the peak values of Table 2 Mean and standard deviation of the input parameters. Parameters

Mean (l)

Standard deviation (r)

Damping (%) Time Period (s) Amplitude (m/s2) Frequency (Hz)

0.1 2.5 3.6 10

0.02 0.50 0.72 2

469

the response quantities obtained from deterministic analysis are insufficient to accurately define the probability of failure. To address this aspect, the probability distributions of the response quantities are obtained. A comparison is shown in Fig. 4 between the gPC and the MC simulation to obtain the probability distribution of the response quantities. The response of the base-isolated liquid storage tanks are obtained deterministically using Eq. (1) for 50,000 MC realizations of the input parameters. The peak response quantities for broad and slender tanks are determined from the time history, for each set of input parameters, to plot the probability distribution. The deterministic values of the peak response quantities are also plotted for the broad and slender tank configurations. It is evident that the gPC expansion predicts the probability distribution of the peak response quantities by employing nine collocation points with sufficient accuracy as compared to computationally involved MC simulation. Moreover, gPC expansion provides the simple relationship between the selected response quantity and the uncertain input parameters in the form of the polynomial expansion. Hence, the uncertainty of input parameters for analysis of base-isolated liquid storage tanks can be accurately handled with the gPC expansion, which is computationally faster than MC simulation. Further, the deterministically obtained peak response quantities are observed near the maximum probability density, more evidently for broad tank configuration. However, the deterministic peak sloshing displacement is away from the maximum probability density in case of slender tank configuration. 4.2. Effect of individual parameter uncertainties on the peak response The effect of uncertainty in each parameter on the distribution of peak response of base-isolated liquid storage tank is investigated. The probability distribution of each response quantity is obtained considering uncertainty in one parameter only at a time, while the other parameters are kept deterministic. The values of the uncertain parameters are taken as given in Table 2. The standard deviation is taken zero for the deterministic parameters. The gPC expansion models of the response quantities are suitably modified to consider the uncertainty of a single input parameter only, keeping the other input parameters as deterministic. Fig. 5 shows the distribution of peak response quantities of base-isolated liquid storage tank for four cases. Each case represents the effect of individual parameter uncertainties on the distribution of the peak response. The gPC expansion is used to obtain the distribution of the peak response quantities for broad and slender base-isolated liquid storage tanks. The deterministic peak response quantities are also shown for comparison purpose. It is observed that the isolation damping (nb) has the least effect on the distribution of the peak response quantities, as the dispersion of the distribution is very less from the deterministic peak value. While the uncertainties in the isolation time period (Tb), amplitude ðAÞ and frequency (x) of the excitation force have major influence on the distribution of the peak response quantities. The distributions of the peak base shear, base displacement and overturning moment are predominantly distributed symmetrically about the deterministic peak response. Moreover, when the uncertainty is considered only in amplitude of the excitation force, the peak response is following normal distribution about the deterministic peak response. This implies that the effect of the amplitude is closely related to the peak response, and the variation in the excitation force amplitude affects the peak response significantly as compared to the other uncertain parameters. The effect of uncertainty is also observed in sloshing displacement, however the range of variation is less as compared to the other response quantities, specifically in the broad tanks. Fig. 6 shows the similar plot using 50,000 Monte Carlo simulation to compare the effectiveness of the gPC expansion to model individual parameter uncertainty. Comparison of the peak

S.K. Saha et al. / Engineering Structures 57 (2013) 465–474

xb (cm)

0.70 0.35 0.00 -0.35 -0.70 60 30 0 -30 -60 14 7 0 -7 -14

Vb (W)

xc (cm)

Mb (W-m)

470

0.12 0.06 0.00 -0.06 -0.12

Broad ( S = 0.6)

y0

0

3

6

9

y1

y2

y3

12

15

Time (sec)

V b ( W)

xb (cm)

xc (cm)

Mb (W-m)

Fig. 2. Time history of the gPC expansion coefficients for broad tank (S = 0.6).

1.2 0.6 0.0 -0.6 -1.2 60 30 0 -30 -60 18 9 0 -9 -18

Slender (S = 1.85)

0.26 0.13 0.00 -0.13 -0.26

y0

0

3

6

9

y1

y2

y3

12

15

Time (sec) Fig. 3. Time history of the gPC expansion coefficients for slender tank (S = 1.85).

Table 3 Statistics of the response quantities. Response

Time instant (s)

Broad (S = 0.6)

Slender(S = 1.85)

l

r

Deterministic

l

r

Deterministic

Mb (W m)

0.5 5 10

0.472 0.015 0.042

0.0963 0.0848 0.0652

0.456 0.025 0.052

0.544 0.068 0.370

0.6628 0.1223 0.9956

0.509 0.139 0.106

xc (cm)

0.5 5 10

7.8 28.2 43.6

0.75 3.91 1.62

7.9 26.8 44.1

5.3 26.47 15

0.56 2.21 14.52

5.43 21.8 14.7

xb (cm)

0.5 5 10

12.3 1.3 2.3

1.22 3.96 2.74

13.4 1.6 3.6

14.6 3.1 2.4

1.41 1.18 6.55

15.8 6.5 1.6

Vb (W)

0.5 5 10

0.086 0.019 0.022

0.0417 0.0598 0.0404

.075 0.032 0.053

0.106 0.004 0.07

0.0521 0.0145 0.1392

0.967 0.0215 0.0166

471

S.K. Saha et al. / Engineering Structures 57 (2013) 465–474

20

80

28

3.2

gPC MC simulation Deterministic

Probability density

Broad (S = 0.6) 15

60

21

2.4

10

40

14

1.6

5

20

7

0.8

0 0.00

0 0.13

0.26

0.39

8

Vb (W)

16

20

0 42

xb (cm)

0.0 48

54

60

0

xc (cm) 60

2.8

9

90

45

2.1

6

60

30

1.4

3

30

15

0.7

0 0.0

1

2

3

Mb (W-m)

120

12

Probability density

12

Slender (S = 1.85)

0.2

0.4

Vb (W)

0.6

0 13

17

21

xb (cm)

25

0 20

30

40

50

0.0 0.0

xc (cm)

0.8

Fig. 4. Comparison of probability distributions of response quantities obtained using gPC and MC simulation.

Fig. 5. Effect of individual parameter uncertainties on the peak response using gPC expansion.

1.6

Mb (W-m)

2.4

472

S.K. Saha et al. / Engineering Structures 57 (2013) 465–474

Probability density

25

30

Broad (S = 0.6)

20

10

24

8

4

15

18

6

3

10

12

4

2

5

6

2

1

Deterministic Uncertain Uncertain Uncertain

0 0.0

0.1

0.2

0

0.3

5

12

Vb (W)

19

26

38

61

84

0 0.0

30

4

12

36

18

3

8

24

12

2

4

12

6

1

0.24

1.4

2.1

MC simulation

24

0.12

0.7

5

48

0 0.00

Uncertain

Mb (W-m)

xc (cm)

60

Slender (S = 1.85)

16

0 15

xb (cm)

20

Probability density

5

0

0.36

7

Vb (W)

13

19

0 20

25

32

44

56

0 0.0

0.7

1.4

2.1

Mb (W-m)

xc (cm)

xb (cm)

Fig. 6. Effect of individual parameter uncertainties on the peak response using MC simulation.

35

120

60

5

28

96

48

4

21

72

36

3

14

48

24

2

7

24

12

1

Standard deviation ( ) 0% 5% 10 % 15 % 20 %

Probability density

Broad (S = 0.6)

0 0.00

0.08

0.16

0 0.24 10.0

11.5

13.0

14.5

0 45

xb (cm)

Vb (W) 20

49

53

57

0 0.0

xc (cm)

0.6

1.2

200

30

5

16

160

24

4

12

120

18

3

8

80

12

2

4

40

6

1

Probability density

Slender (S = 1.85)

0 0.00

0.12

0.24

Vb (W)

1.8

Mb (W-m)

gPC

0.36

0 14

15

16

xb (cm)

17

0 27

33

39

45

xc (cm)

Fig. 7. Peak response distribution for different levels of uncertainty.

0 0.0

0.6

1.2

Mb (W-m)

1.8

473

S.K. Saha et al. / Engineering Structures 57 (2013) 465–474

Probability density

24

81

gPC

16

54

3.0

8

27

1.5

0 0.00

0.15

0.30

0.45

15

Probability density

4.5

Broad (S = 0.6)

0 44

0.0 48

52

0

56

60

1

3

3 Fixed-base (FB)

Slender (S = 1.85) 10

2

Base-isolated (BI)

40

Deterministic (FB)

2

Deterministic (BI) Base-isolated (Only

5

0 0.0

20

0.2

0.4

0.6

0 20

uncertain excitation)

1

30

Vb (W)

40

50

xc (cm)

0 0.0

1.6

3.2

4.8

Mb (W-m)

Fig. 8. Effect of uncertainties on peak response of fixed-base and base-isolated liquid storage tanks.

response distributions in Figs. 5 and 6 further show that the gPC expansion technique is capable to model the parameter uncertainty as good as Monte Carlo simulation. 4.3. Variation of peak response distribution with different uncertainty levels Further studies are carried out to investigate the effect of the level of uncertainty on the distributions of peak response quantities of the base-isolated liquid storage tanks. The levels of uncertainties are quantified by the standard deviation of the uncertain parameters. The standard deviation of all the four uncertain parameters is varied from 5% to 20%. The distribution of the peak response quantities are plotted in Fig. 7 using the gPC expansion. Here, the gPC expansion model, considering all the four uncertain parameters together, is used. It is observed that the distributions of the peak response quantities are dispersing more from respective deterministic peak values, as the uncertainty level is increased. For very low level of uncertainty, the distributions are symmetrical about the corresponding deterministic peak response, however for higher level of uncertainty the distributions no more remain symmetrical. Hence, the level of uncertainty plays a crucial role when the probability of failure is under consideration. 4.4. Effect of uncertainties on peak response of fixed-base and baseisolated liquid storage tanks To demonstrate the importance of the uncertainty analysis for liquid storage tanks, herein the behavior of fixed-base and baseisolated liquid storage tanks are compared. Two different types of the tank configurations, broad and slender, are considered for the present study. The distributions of the peak base shear, sloshing displacement and overturning moment are plotted for both fixed-base and base-isolated liquid storage tanks in Fig. 8 using the gPC expansion. The deterministic peak values of the response quantities are also plotted for comparison purpose. For the fixedbase tank, the uncertainties are considered only in the excitation, whereas for the base-isolated case the uncertainties are considered in two different ways. Firstly, the uncertainties involved in the excitation ðA; xÞ are considered only, secondly all the input parameters are considered uncertain as given in Table 2. It is

observed that the deterministic peak base shear and overturning moment are well separated. However, there are considerable overlapping between the distributions of peak base shear and overturning moment, obtained for the fixed-base and base-isolated tanks, when uncertainties are considered, more specifically for slender configuration. It implies that for specific uncertainties in the isolator parameters and excitation, the response calculated for the base-isolated liquid storage tank may be similar to that of fixedbase tank. Moreover, when only the uncertainties in the excitation are considered, the overlapping is less as compared to the case when all the input parameters are considered uncertain for baseisolated liquid storage tanks. Hence, appropriate quantification of the uncertainty is necessary for assessing the safety of base-isolated liquid storage tanks over fixed-base tanks.

5. Conclusions Stochastic response of base-isolated liquid storage tanks under random base excitation, considering uncertainty in the characteristic isolator parameters is presented. The non-sampling stochastic simulation based on the generalized polynomial chaos (gPC) expansion is used to obtain the response under parametric uncertainty. A step-by-step procedure for numerical simulation, of the time variant response quantities, by using non-intrusive technique is explained in details. The time history of the important response quantities, such as base shear, base displacement, sloshing displacement and overturning moment, are expressed in the form of polynomials. The mean and the standard deviation of the response quantities are computed from the gPC expansion at particular time instants. The efficiency of the method is compared with the Monte Carlo simulation. The effect of the individual parameter uncertainty and the level of uncertainty on the distributions of the peak response quantities are studied. The response of fixed-base and base-isolated liquid storage tanks are compared considering the uncertainty. From the numerical study, following conclusions are drawn. 1.

The gPC expansion is an efficient alternative to MC simulation to model the parametric uncertainty in dynamic analysis of base-isolated liquid storage tank.

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2.

3.

4.

5.

6.

S.K. Saha et al. / Engineering Structures 57 (2013) 465–474

Deviation in the base shear, base displacement and overturning moment is more as compared to the sloshing displacement under parametric uncertainty, more specifically for the slender tank configuration. Maximum probability density, of the base-isolated liquid storage tank response quantities, is observed around the deterministic estimation of the response, more evidently in the broad configuration. Uncertainty in isolation damping has less influence on the response, of the base-isolated liquid storage tanks, as compared to the uncertainties in isolation time period, amplitude and frequency of the excitation. The effect of the excitation amplitude on the response quantities is predominant. The distribution of the response for low level of uncertainty is symmetrical about the deterministic peak response, whereas high level of uncertainty disturb the symmetry. For specific uncertainties in the isolator parameters and excitation parameters, the response calculated for the base-isolated liquid storage tank may be similar to that of fixedbase tank.

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