# Seismic Fragility Assessment of Base-Isolated Steel Water Storage Tank

Authors: Zhuo Zhao; Xiaowei Lu; Yu Guo; Xiaofeng Zhao
Journal: Shock and Vibration
 (2020)

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## Abstract

Steel water storage tanks (WSTs) are among the important components of water treatment industry facilities that are expected to remain functional and applicable after strong earthquakes. In this study, the seismic vulnerability of base-isolated steel WST is investigated. A three-dimensional finite element stick model of the targeted tank is created using OpenSees. This model is capable of reproducing convective, impulsive, and rigid responses of fluid-tank systems. Time-history responses of convective displacement, bearing displacement, and base shear force for base-isolated tank subjected to a typical ground motion are compared. Furthermore, time-history analysis based on a suite of 80 ground motions is conducted. The seismic demand models for various responses are established and the most efficient intensity measure (IM) is determined based on the dispersion and coefficient of determination. Seismic fragility curves for different responses are derived for all three damage states using cloud analysis. The results from this study reveal that (i) the convective displacement is significantly greater than bearing displacement; (ii) peak ground displacement (PGD) is the most efficient and sufficient IM for the targeted tank; and (iii) the characteristic of isolation bearing significantly influences the seismic fragilities of convective displacement and bearing displacement and has a little impact on base shear force, which makes the selection of the proper characteristic parameters for isolation bearing very essential. The analysis technique and procedure mentioned above as well as derived insights are of significance to general liquid storage tank system configuration.


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## Body

## 1. Introduction

Liquid storage tanks (LSTs), as lifeline structure, are extensively used to store a variety of liquids, for example, water, petroleum, chemicals, and liquefied natural gas [1, 2]. With the development of economics, the amount and capacity of LST are gradually increasing. At present, the capacity and radius of large-scale LST can reach 20 × 104 m3 and 50 m, respectively. Once the damage of such large-scale LST occurs due to natural disaster, this will cause a tremendous hazard for social and public safety. Earthquake is one of the most common and destructive natural disasters. Especially in recent years, a large number of LSTs are constructed in coastal region, southeast of China. In addition, these coastal regions are significantly influenced by Ring of Fire (i.e., Circum-Pacific Belt). As such, seismic performance of LSTs during earthquake has attracted increased attention in industrial and academic communities. Previous earthquake investigations have demonstrated that the buckling of LST wall and uplift of LST base are the most common types of damage due to strong ground motion. For example, Malhotra [3] presented an approximate method for the analysis of earthquake-induced uplifting in base plate of unanchored liquid storage tanks supported on flexible soil foundations. Malhotra et al. [1] reported that the larger axial compressive stress of tank wall can cause bulking of the tank wall. Alembagheri and Estekanchi [4] investigated the seismic response of unanchored steel LST using the endurance time dynamic analysis procedure. They found that the uplifting of the tank base obviously influenced the seismic response of the tank.Various studies have been performed to explore the dynamic characteristic and failure mechanism of LST during earthquake. In fact, the liquid in the LST demonstrates three distinct motions [2]: (1) convective (or sloshing) motion, that is, motion of top liquid; (2) impulsive motion, that is, motion of intermediate liquid along with tank wall; and (3) rigid motion, that is, rigid motion of lower liquid along with tank wall. Based on the above, Malhotra et al. [1] presented a simplified analysis procedure for ground-supported cylindrical tanks. Such procedure can consider impulsive and convective action of the liquid in the tank. Haroun [5] systematically studied the dynamic response of ground-supported cylindrical LST through theoretical and experimental investigation. Barton and Parker [6] investigated the seismic response of anchored and unanchored cylindrical LST using finite element method. Their results illustrated that the base restraint conditions should be carefully considered in the seismic design of LST. Tedesco et al. [7] developed an analytical procedure for ground-supported cylindrical LST to estimate its seismic response subjected to a horizontal ground motion. Veletsos and Tang [8] investigated the effect of soil-structure interaction on the response of cylindrical tanks subjected to a horizontal ground motion. Cho et al. [9] performed seismic response analysis of LST using refined numerical technique along with added mass method. Virella et al. [10] reported the fundamental impulsive modes of liquid-tank system during horizonal motions through general finite element program. Recently, Shakib and Alemzadeh [11] investigated the effect of earthquake site-source distance on the seismic response of reinforced concrete elevated water tanks with shaft type support. The results showed that the seismic response of fluid-tank system highly depended on site-source distance of the earthquake records. Li et al. [12] conducted the shaking table tests and finite element modeling on the 1/20 scale liquid-tank-foundation system to investigate the dynamic response of the cylindrical oil-storage tank under seismic excitation.However, in order to mitigate the hydrodynamic-induced seismic response of fluid-tank system due to strong ground motion, the base isolation is a very common technology and is extensively employed in the industrial design. In the aspect of base-isolated tank subjected to earthquake excitation, Liang and Tang [13] reported the effect of lead-rubber bearing isolator on the seismic response of flexible LST. They showed that such isolator can greatly reduce the hydrodynamic pressure, shell, and liquid sloshing amplitudes. Malhotra [14] proposed a seismic base isolation method for ground-supported cylindrical LST through using a flexible membrane between tank wall and base plate and found that such isolation method can decrease significantly the hydrodynamic-induced base shear force, overturning moment, and axial compressive stress in the tank wall and does not obviously increase the vertical displacements of the free liquid surface due to convective motion. Shrimali and Jangid [15] studied the seismic response of elevated liquid storage steel tanks isolated by the linear elastomeric bearings under real earthquake ground motion. Furthermore, Jadhav and Jangid [16] investigated the seismic response of the liquid storage tanks isolated by the elastomeric bearings and sliding systems subjected to near-fault earthquake motions. In their paper, it was shown that the seismic response of the isolated tank is mainly governed by fault normal component of earthquake motions. At present, based on an equivalent mechanical model, Hashemi and Aghashiri [17] analyzed the seismic response of base-isolated flexible rectangular fluid containers under horizonal motion. The results demonstrated that the base isolation can efficiently reduce base shear force, wall deformation, and hydrodynamic pressure. Tsipianitis and Tsompanakis [18] focused on the influence of damping modeling approach on the dynamic response of base-isolated LST subjected to strong near-fault ground motion.Seismic vulnerability assessment of engineering structure is carried out by fragility curve. The assessment based on fragility curve is used extensively in building engineering [19], bridge engineering [20], and harbor engineering [21]. Seismic vulnerability assessment of LSTs plays a critical role in uninterrupted operation of an industrial facility. Technically, seismic fragility can present capacity of tank component resistance to failure subjected to different seismic hazard levels. For instance, Phan et al. [22] assessed the seismic vulnerability of steel tank supported by reinforced concrete columns using probabilistic seismic assessment approach. A three-dimensional finite element stick model is established to perform the nonlinear time history analyses using OpenSees. Their results illustrated that a higher damage vulnerability of tank-supported column occurs due to excessive lateral displacement of tank base. Lately, for base-isolated tank, Tsipianitis and Tsompanakis [23] carried out the fragility analysis of tank isolated by sliding-bearing under near-fault earthquakes using a surrogate model. Kildashti et al. [24] explored the influence of base flexibility on the seismic performance of fully anchored LST using fragility curve. In summary, there are a large number of researches regarding seismic response of base-isolated tanks and seismic fragility of LSTs. Nevertheless, until recently, only very limited studies were reported about seismic fragility of base-isolated tanks.Based on the above, in this study, the seismic fragility of base-isolated water storage tank is studied. A three-dimensional (3D) finite element stick model is created through OpenSees computer program. Seismic response of base-isolated tank is analyzed under a typical earthquake ground motion. A suite of 80 earthquake ground motion records are employed to assess the seismic performance of base-isolated tank. The seismic demand models of base-isolated tank are constructed from the obtained intensity measure-seismic demand data pairs. Combining with the obtained seismic demand models and appropriate bound limit for various damage states, seismic fragility curves are derived based on the cloud analysis. The influence of the characteristic of isolation bearing on the seismic fragility curve of the tank is systematically investigated. In short, the novelties of this study are twofold: (1) A simplified 3D FE stick model of fluid-tank system is created considering fluid-structure interaction; (2) the seismic fragility of such a system is investigated under various properties of isolation bearing.


## 2. Numerical Models of Storage Tank

A targeted cylindrical steel WST is commonly used in the water treatment project (SunRui Marine Environment Engineering Co., Ltd.). Traditionally, such tank is connected to base reinforced concrete slab through eight anchors evenly distributed along the tank base. To alleviate seismic damage of steel WST, the eight anchors at the tank base are replaced by eight isolation bearings (i.e., single concave friction pendulum bearing), shown in Figure1(a), which are installed between tank base and base reinforced concrete slab. The radius of steel WST is 3.6 m and the thickness of tank wall is 16 mm. The thickness of tank bottom is 30 mm. The angle irons are arranged in the bottom of tank to avoid out-of-plane buckling deformation. The total height of steel WST is 9.2 m. The water height is 7.2 m for normal operating conditions. The mechanical properties of steel WST are summarized in Table 1.Figure 1
Base-isolated steel water storage tank: (a) physical model; (b) stick model.
(a)(b)Table 1
Mechanical properties of steel storage tank.
ComponentPropertiesUnitValue16 mm thickness steel tankYoung’s modulusMPa200000Yield strengthMPa205Densitykg/m37900WaterDensitykg/m31000During earthquake, the considered water body is assumed as an incompressible, inviscid, and irrotational flow. Many studies [1, 2, 17, 22] have shown that three distinct vibration modes are presented, that is, convective mode, impulsive mode, and rigid mode. Although there are various vibration modes for WST under earthquake excitation, the first convective mode and first impulsive mode can sufficiently predict the seismic response of water body based on the experimental results of Kim and Lee [25] and numerical results of Malhotra [14]. As such, a three-dimensional (3D) finite element stick model of fluid-tank system, shown in Figure 1(b), is established using open-source OpenSees computational platform [26]. In the numerical modeling, the convective mass (mc) and impulsive mass (mi) are located at the corresponding heights (hc) and (hi), respectively. The convective and impulsive masses, defined by lumped mass, connect to tank base by mean of elastic beam-column element with corresponding stiffness (kc and ki) and Rayleigh damping (cc and ci). The tank base is simulated by eight rigid links with a length equal to the tank radius considering the thickness of tank bottom and the arrangement of angle irons in the bottom of tank. The one end of rigid link connects to elastic beam-column element, and the other end of rigid link connects to isolation bearing element with corresponding horizontal stiffness (kb) and viscous damping (cb). The other end of bearing element is fixed for three translation and rotation degrees of freedom. The displacements of convective, impulsive, and base parts are denoted by xc, xi, and xb, respectively. According to [1], the mass, frequency, stiffness, and height of mass for impulsive and convective parts are obtained. The damping for impulsive and convective parts is set to 2% and 0.5%, respectively, referring to Phan et al. [22]. The dynamic property parameters of steel WST are summarized in Table 2.Figure 2
Various earthquake ground motions: (a) El Centro station record of 1940 Imperial Valley earthquake; (b) Capitola station record of 1989 Loma Prieta earthquake; (c) Rinaldi Receiving station record of 1994 Northridge earthquake.
(a)(b)(c)Table 2
Dynamic properties of steel water storage tank.
ComponentUnitImpulsiveConvectiveMassT223.769.5Damping%20.5Frequencyrad/s132.42.2StiffnesskN/m3923816.5347.3Height of massm3.225.41


## 3. Seismic Response Analysis of Storage Tank

To investigate the seismic response of storage tank, various earthquake ground motions are employed as a base excitation, including El Centro station record of 1940 Imperial Valley Earthquake (El Centro record), Capitola station record of 1989 Loma Prieta earthquake (Capitola record), and Rinaldi Receiving station record of 1994 Northridge earthquake (Rinaldi Receiving record), shown in Figure2. The peak ground acceleration of El Centro, Capitola, and Rinaldi Receiving records is 0.348 g, 0.511 g, and 0.445 g, respectively. As recommended by Shrimali and Jangid [2], the seismic demands/responses of interest are as follows: the convective displacement (xc), the bearing displacement (xb), and the base shear force (Fb). In fact, the base shear force is directly proportional to earthquake loading exerted in the tank and the convective and bearing displacements are very important from the design point of view. To validate the effectiveness of 3D FE stick model created by OpenSees, a similar stick model of broad tank is established based on [2] and seismic response of broad tank is computed. Figure 3 presents the comparison of the computed time history response for broad tank [2] subjected to El Centro record through the finite element method and modal superposition method. From Figure 3, the seismic response computed by the stick model is similar to that obtained by modal superposition method in the reference, especially for the convective displacement. This indicates that the proposed stick model can reproduce the seismic response of WST. Figure 4 demonstrates time-history response of stick model for water storage tank subjected to various ground motions. From Figure 4(a), different seismic demands present similar sinusoidal responses under base excitation. The amplitude of convective displacement is significantly greater than that of bearing displacement. The main reason is that the mass and height of the convective part are larger than those of the impulsive part. In addition, the stiffness of the impulsive part is significantly greater than the counterpart of the convective part. The similar comments are made for Capitola and Rinaldi Receiving records, demonstrated in Figures 4(b) and 4(c).Figure 3
Comparison of the computed time-history response for broad tank [2] through the finite element method and modal superposition method.Figure 4
Time-history response of stick model for water storage tank subjected to various ground motions (isolation periodTb = 2.0 and damping ratio ξb=0.1): (a) El Centro record; (b) Capitola record; (c) Rinaldi record.
(a)(b)(c)


## 4. Seismic Fragility Assessment

### 4.1. Selection of Ground Motion

In order to perform the seismic fragility analysis of WST, the existing ground motion set is selected. This ground motion set comprised 80 ground motions extracted from the Pacific Earthquake Engineering Research Center’s (PEER) Strong Motion Database by Medina and Krawinkler [27]. These 80 ground motions are classified into four magnitude-distance bins according to various moment magnitudes (5.8 < Mw < 7.0) and epicentral distances (13 km < R < 60 km) and their peak ground displacement (PGD) distribution is illustrated in Figure 5.Figure 5
PGD distribution of 80 ground motions.


### 4.2. Seismic Demand Models

Based on the above-mentioned seismic demands (i.e., responses of interest), the seismic demand models for the convective displacement, the bearing displacement, and the base shear force are established to map the relationship between intensity measure (IM) and seismic demand (D). Reference [28] recommended that the estimate for the median of the seismic demand be predicted by a power function expressed in the following equation:(1)SD=aIMb,where a and b are the power-law model regression coefficients based on the collection of the peak demand and IM quantity from time-history analyses of the targeted tank using a suite of 80 selected ground motions. SD is the median estimate of the demand as a function of IM. By expressing the median seismic demand in logarithmic space, the expression of SD with IM takes a linear form presented in the following equation:(2)lnSD=lna+blnIM.The dispersion (βD|IM) of the demand conditioned on the IM can be estimated by statistical analysis of lnIM−lnD pairs, as given in the following equation:(3)βD|IM=∑i=1nlndi−lnSD2n−2,where di is the ith computed value of the seismic demand; and n is the number of performed time-history analyses. Usually, the efficiency (i.e., goodness of fit) of linear regression is evaluated by computing the coefficient of determination (R2), which is the proportion of the variance in the dependent variable that is predictable from the independent variable in statistics, illustrated in the following equation:(4)R2=1−SSresSStol=1−∑i=1nlndi−lnSD2∑i=1nlndi−meanlndi2,where SSres is defined as the sum of residual distances from model with IM and SStol is also defined as the sum of residual distances from intercept model with IM. If R2 gets closer to 1.0, the linear fit is better. The model can perfectly fit all the points if R2 is 1.0, and it cannot be useful if R2 is 0. Usually, the value of R2 is between 0 and 1.0.Based on the above-mentioned 3D finite element stick model, the time-history analysis is carried out for each of the selected 80 ground motions to obtain the responses of interest. Finally, 80 input-output data pairs are collected for each IM and response of interest. As such, combining with the collected data pairs, the seismic demand models for various responses of interest can be obtained based on equations (1) and (2). The dispersion (βD|IM) and coefficient of determination (R2) are calculated through equations (3) and (4), respectively.In the vulnerability assessment of LSTs, the selection of the proper IM is a fundamental issue, which will influence the evaluation of seismic fragility. To obtain an efficient IM, a comparative analysis for various IMs is performed for different seismic demands with dispersion (βD|IM) and coefficient of determination (R2). Figure 6 illustrates the seismic demand model of convective displacement in the logarithm scale for various IMs. From this figure, compared with other IMs (i.e., PGA, PGV, CAV, Ia, and Sa(T2)), the log-linear fitting is the best for convective displacement with respect to PGD, indicating that the adopted log-linear model is effective for determining the seismic demands. Similarly, Buratti and Tavano [29] also found that the PGD is the most efficient and sufficient IM. The dispersion (βD|IM) and coefficient of determination (R2) are 0.55 and 0.72 for convective displacement with respect to PGD, respectively, presented in Table 3. Likewise, the dispersion (βD|IM) and coefficient of determination (R2) for bearing displacement and base shear force with respect to various IMs are summarized in Table 3. The same comment can be made for PGD. As such, Figure 7 demonstrates the seismic demand models of bearing displacement and base shear force with PGD. As seen from Figure 7, the log-linear model can fit well the bearing displacement and base shear force with PGD.Figure 6
Linear fit demand model of convective displacement with different intensity measures.
(a)(b)(c)(d)(e)(f)Table 3
Linear regression analysis results.
Intensity measureConvective displacementBearing displacementBase shear forceβR2βR2βR2PGA1.030.030.860.070.860.07PGD0.570.710.520.660.520.66PGV0.610.660.460.740.460.74CAV0.680.580.540.640.540.64Ia0.860.330.670.430.670.43Sa(T2)1.050.010.890.020.890.02Note:β is standard deviation; R2 is coefficient of determination; CAV is Cumulative Absolute Velocity; Ia is Arias Intensity.Figure 7
Linear fit demand model with PGD: (a) bearing displacement; (b) base shear force.
(a)(b)


### 4.3. Determination of Seismic Fragility Curve

Typically, the fragility curve can be determined through many approaches, for example, observation of earthquake damage, static structural analyses, and analytical fragility function [22]. The analytical fragility function is extensively used and is derived from a proper seismic demand model [30, 31]. The cloud analysis approach is one of the most common procedures for deriving the fragility curve. In this approach, firstly, the seismic response analysis for the targeted tank is performed subjected to a suite of ground motions; secondly, the linear regression-based seismic demand model is built based on the numerical results; finally, the fragility curve is obtained. The seismic fragility curve is defined as the conditional probability that a seismic demand required for a specific structure exceeds its bound limit state for a specific IM in the form of the following equation:(5)PD≥C|IM=ΦlnSD−lnSLSβD|IM2+βLS2,where SLS is the median estimate of the bound limit state of seismic demand; βLS is the dispersion of the limit state; Φ• denotes the cumulative standard normal distribution function. As such, the fragility curve of seismic demands for the targeted tank will be obtained if the associated parameters SD, SLS, βD|IM, and βLS are first determined. Herein, SD is given based on the above obtained seismic demand models, depicted in Figures 6 and 7. SLS of different bound limit states is extremely important for fragility assessment. In fact, the determination of the value of SLS is rather difficult. A variety of approaches are used to determine SLS for LSTs, for example, earthquake investigation, experimental data, and analytical method. In particular, SLS of different bound limit states for various seismic demands is assumed according to the computed results of the targeted tank, summarized in Table 4. In fact, SLS of different bound limit states is determined by the following steps: (1) for every hazard level (i.e., damage state), the seismic response analysis is conducted using seven input earthquake motions, which have similar spectra accelerations; (2) the seismic response of tank is obtained under different hazard levels; (3) the average of maximum response for different hazard levels is used as SLS of different bound limit states. βLS is assumed to be 15%.Table 4
Limit states for fragility assessment of tank.
Response of interestSlightModerateExtensiveConvective displacement,xc (m)0.050.10.15Bearing displacement,xb (m)0.0060.0120.018Base shear force,Fb (kN)154575


### 4.4. Fragility Analysis

Based on equation (5), the fragility curves of different seismic demands for the targeted tank are obtained and are illustrated in Figure 8. The fragility curves associated with convective displacement for different damage states are compared in Figure 8(a), which provides a clear picture of damage exceedance probability for different damage responses with respect to the different PGD levels. For example, the fragilities of the convective displacement are 95.2%, 67.1%, and 48.7% for the light, moderate, and extensive damage states, respectively, while the PGD is specified as 1.0 m. Obviously, the fragility of the convective displacement for the extensive state is greater than that for slight and moderate states. The fragilities of the convective displacement for all three damage states reach 100% at the PGD of 6 m. Figure 8(b) presents the fragility curves of bearing displacement. The same characteristics are observed for bearing displacement. Figure 8(c) shows the fragility curves of base shear force. It is clear that the fragilities of base shear force for all three damage states are similar, when the PGD is under 0.1 m. The fragility of base shear force for the slight state is obviously greater than that for the moderate and extensive states, while the PGD is above 0.1 m. The fragilities of base shear force for moderate and extensive states reach 60.9% and 54.8% at the PGD of 6 m, respectively.Figure 8
Seismic fragility curves: (a) convective displacement; (b) bearing displacement; (c) base shear force.
(a)(b)(c)To explore the influence of isolation bearing characteristics on the seismic fragility curves, the parametric study is performed through changing the period and damping ratio of isolation bearing. Figure9 displays the influence of isolation bearing period on fragility curves of different seismic demands through changing isolation period from 1.0 to 2.0, according to [18]. As seen from Figure 9(a), for convective displacement, the fragilities for all three damage states present an increasing trend with increasing of isolation period, which indicates that the fragility of convective displacement can be increased by increasing the isolation bearing period. In Figure 9(b), for bearing displacement, the fragilities for all three damage states significantly increase with increasing of isolation period. This demonstrates that the isolation period of bearing influences the fragility of bearing displacement. From Figure 9(c), the fragilities of base shear force for all three damage states slightly increased with increasing of isolation period. Based on the above, this shows that the fragilities of different seismic demands are influenced by the isolation period of bearing.Figure 9
Influence of bearing isolation period on fragility curve: (a) convective displacement; (b) bearing displacement; (c) base shear force.
(a)(b)(c)Figure10 presents the influence of bearing damping ratio on fragility curves of different seismic demands through changing damping ratio from 0.02 to 0.1, referring to Shrimali and Jangid [2]. In Figure 10(a), the fragilities of the convective displacement for all three damage states increase with increasing of damping ratio. A similar characteristic is also found in the fragilities of the base shear force, shown in Figure 10(c). This signifies that the smaller damping ratio of isolation bearing can effectively decrease the fragilities of convective displacement and base shear force. As illustrated in Figure 10(b), the increasing of damping ratio can decrease the fragilities of base shear force.Figure 10
Influence of bearing damping ratio on fragility curve: (a) convective displacement; (b) bearing displacement; (c) base shear force.
(a)(b)(c)In summary, the isolation period and damping ratio of bearing appreciably influence the fragilities of convective displacement and bearing displacement and have a little influence on base shear force. The decreasing of isolation period can decrease the fragilities of various seismic demands for different damage states. The decreasing of damping ratio can reduce the fragilities of the convective displacement and base shear force and increase the fragility of the bearing displacement for all three damage states. For the above reason, to decrease the fragilities of convective displacement and base shear force, the damping ratio and period of isolation bearing are designed as small as possible. A proper damping ratio for bearing should be designed to obtain the appropriate fragilities for bearing displacement.


## 4.1. Selection of Ground Motion

In order to perform the seismic fragility analysis of WST, the existing ground motion set is selected. This ground motion set comprised 80 ground motions extracted from the Pacific Earthquake Engineering Research Center’s (PEER) Strong Motion Database by Medina and Krawinkler [27]. These 80 ground motions are classified into four magnitude-distance bins according to various moment magnitudes (5.8 < Mw < 7.0) and epicentral distances (13 km < R < 60 km) and their peak ground displacement (PGD) distribution is illustrated in Figure 5.Figure 5
PGD distribution of 80 ground motions.


## 4.2. Seismic Demand Models

Based on the above-mentioned seismic demands (i.e., responses of interest), the seismic demand models for the convective displacement, the bearing displacement, and the base shear force are established to map the relationship between intensity measure (IM) and seismic demand (D). Reference [28] recommended that the estimate for the median of the seismic demand be predicted by a power function expressed in the following equation:(1)SD=aIMb,where a and b are the power-law model regression coefficients based on the collection of the peak demand and IM quantity from time-history analyses of the targeted tank using a suite of 80 selected ground motions. SD is the median estimate of the demand as a function of IM. By expressing the median seismic demand in logarithmic space, the expression of SD with IM takes a linear form presented in the following equation:(2)lnSD=lna+blnIM.The dispersion (βD|IM) of the demand conditioned on the IM can be estimated by statistical analysis of lnIM−lnD pairs, as given in the following equation:(3)βD|IM=∑i=1nlndi−lnSD2n−2,where di is the ith computed value of the seismic demand; and n is the number of performed time-history analyses. Usually, the efficiency (i.e., goodness of fit) of linear regression is evaluated by computing the coefficient of determination (R2), which is the proportion of the variance in the dependent variable that is predictable from the independent variable in statistics, illustrated in the following equation:(4)R2=1−SSresSStol=1−∑i=1nlndi−lnSD2∑i=1nlndi−meanlndi2,where SSres is defined as the sum of residual distances from model with IM and SStol is also defined as the sum of residual distances from intercept model with IM. If R2 gets closer to 1.0, the linear fit is better. The model can perfectly fit all the points if R2 is 1.0, and it cannot be useful if R2 is 0. Usually, the value of R2 is between 0 and 1.0.Based on the above-mentioned 3D finite element stick model, the time-history analysis is carried out for each of the selected 80 ground motions to obtain the responses of interest. Finally, 80 input-output data pairs are collected for each IM and response of interest. As such, combining with the collected data pairs, the seismic demand models for various responses of interest can be obtained based on equations (1) and (2). The dispersion (βD|IM) and coefficient of determination (R2) are calculated through equations (3) and (4), respectively.In the vulnerability assessment of LSTs, the selection of the proper IM is a fundamental issue, which will influence the evaluation of seismic fragility. To obtain an efficient IM, a comparative analysis for various IMs is performed for different seismic demands with dispersion (βD|IM) and coefficient of determination (R2). Figure 6 illustrates the seismic demand model of convective displacement in the logarithm scale for various IMs. From this figure, compared with other IMs (i.e., PGA, PGV, CAV, Ia, and Sa(T2)), the log-linear fitting is the best for convective displacement with respect to PGD, indicating that the adopted log-linear model is effective for determining the seismic demands. Similarly, Buratti and Tavano [29] also found that the PGD is the most efficient and sufficient IM. The dispersion (βD|IM) and coefficient of determination (R2) are 0.55 and 0.72 for convective displacement with respect to PGD, respectively, presented in Table 3. Likewise, the dispersion (βD|IM) and coefficient of determination (R2) for bearing displacement and base shear force with respect to various IMs are summarized in Table 3. The same comment can be made for PGD. As such, Figure 7 demonstrates the seismic demand models of bearing displacement and base shear force with PGD. As seen from Figure 7, the log-linear model can fit well the bearing displacement and base shear force with PGD.Figure 6
Linear fit demand model of convective displacement with different intensity measures.
(a)(b)(c)(d)(e)(f)Table 3
Linear regression analysis results.
Intensity measureConvective displacementBearing displacementBase shear forceβR2βR2βR2PGA1.030.030.860.070.860.07PGD0.570.710.520.660.520.66PGV0.610.660.460.740.460.74CAV0.680.580.540.640.540.64Ia0.860.330.670.430.670.43Sa(T2)1.050.010.890.020.890.02Note:β is standard deviation; R2 is coefficient of determination; CAV is Cumulative Absolute Velocity; Ia is Arias Intensity.Figure 7
Linear fit demand model with PGD: (a) bearing displacement; (b) base shear force.
(a)(b)


## 4.3. Determination of Seismic Fragility Curve

Typically, the fragility curve can be determined through many approaches, for example, observation of earthquake damage, static structural analyses, and analytical fragility function [22]. The analytical fragility function is extensively used and is derived from a proper seismic demand model [30, 31]. The cloud analysis approach is one of the most common procedures for deriving the fragility curve. In this approach, firstly, the seismic response analysis for the targeted tank is performed subjected to a suite of ground motions; secondly, the linear regression-based seismic demand model is built based on the numerical results; finally, the fragility curve is obtained. The seismic fragility curve is defined as the conditional probability that a seismic demand required for a specific structure exceeds its bound limit state for a specific IM in the form of the following equation:(5)PD≥C|IM=ΦlnSD−lnSLSβD|IM2+βLS2,where SLS is the median estimate of the bound limit state of seismic demand; βLS is the dispersion of the limit state; Φ• denotes the cumulative standard normal distribution function. As such, the fragility curve of seismic demands for the targeted tank will be obtained if the associated parameters SD, SLS, βD|IM, and βLS are first determined. Herein, SD is given based on the above obtained seismic demand models, depicted in Figures 6 and 7. SLS of different bound limit states is extremely important for fragility assessment. In fact, the determination of the value of SLS is rather difficult. A variety of approaches are used to determine SLS for LSTs, for example, earthquake investigation, experimental data, and analytical method. In particular, SLS of different bound limit states for various seismic demands is assumed according to the computed results of the targeted tank, summarized in Table 4. In fact, SLS of different bound limit states is determined by the following steps: (1) for every hazard level (i.e., damage state), the seismic response analysis is conducted using seven input earthquake motions, which have similar spectra accelerations; (2) the seismic response of tank is obtained under different hazard levels; (3) the average of maximum response for different hazard levels is used as SLS of different bound limit states. βLS is assumed to be 15%.Table 4
Limit states for fragility assessment of tank.
Response of interestSlightModerateExtensiveConvective displacement,xc (m)0.050.10.15Bearing displacement,xb (m)0.0060.0120.018Base shear force,Fb (kN)154575


## 4.4. Fragility Analysis

Based on equation (5), the fragility curves of different seismic demands for the targeted tank are obtained and are illustrated in Figure 8. The fragility curves associated with convective displacement for different damage states are compared in Figure 8(a), which provides a clear picture of damage exceedance probability for different damage responses with respect to the different PGD levels. For example, the fragilities of the convective displacement are 95.2%, 67.1%, and 48.7% for the light, moderate, and extensive damage states, respectively, while the PGD is specified as 1.0 m. Obviously, the fragility of the convective displacement for the extensive state is greater than that for slight and moderate states. The fragilities of the convective displacement for all three damage states reach 100% at the PGD of 6 m. Figure 8(b) presents the fragility curves of bearing displacement. The same characteristics are observed for bearing displacement. Figure 8(c) shows the fragility curves of base shear force. It is clear that the fragilities of base shear force for all three damage states are similar, when the PGD is under 0.1 m. The fragility of base shear force for the slight state is obviously greater than that for the moderate and extensive states, while the PGD is above 0.1 m. The fragilities of base shear force for moderate and extensive states reach 60.9% and 54.8% at the PGD of 6 m, respectively.Figure 8
Seismic fragility curves: (a) convective displacement; (b) bearing displacement; (c) base shear force.
(a)(b)(c)To explore the influence of isolation bearing characteristics on the seismic fragility curves, the parametric study is performed through changing the period and damping ratio of isolation bearing. Figure9 displays the influence of isolation bearing period on fragility curves of different seismic demands through changing isolation period from 1.0 to 2.0, according to [18]. As seen from Figure 9(a), for convective displacement, the fragilities for all three damage states present an increasing trend with increasing of isolation period, which indicates that the fragility of convective displacement can be increased by increasing the isolation bearing period. In Figure 9(b), for bearing displacement, the fragilities for all three damage states significantly increase with increasing of isolation period. This demonstrates that the isolation period of bearing influences the fragility of bearing displacement. From Figure 9(c), the fragilities of base shear force for all three damage states slightly increased with increasing of isolation period. Based on the above, this shows that the fragilities of different seismic demands are influenced by the isolation period of bearing.Figure 9
Influence of bearing isolation period on fragility curve: (a) convective displacement; (b) bearing displacement; (c) base shear force.
(a)(b)(c)Figure10 presents the influence of bearing damping ratio on fragility curves of different seismic demands through changing damping ratio from 0.02 to 0.1, referring to Shrimali and Jangid [2]. In Figure 10(a), the fragilities of the convective displacement for all three damage states increase with increasing of damping ratio. A similar characteristic is also found in the fragilities of the base shear force, shown in Figure 10(c). This signifies that the smaller damping ratio of isolation bearing can effectively decrease the fragilities of convective displacement and base shear force. As illustrated in Figure 10(b), the increasing of damping ratio can decrease the fragilities of base shear force.Figure 10
Influence of bearing damping ratio on fragility curve: (a) convective displacement; (b) bearing displacement; (c) base shear force.
(a)(b)(c)In summary, the isolation period and damping ratio of bearing appreciably influence the fragilities of convective displacement and bearing displacement and have a little influence on base shear force. The decreasing of isolation period can decrease the fragilities of various seismic demands for different damage states. The decreasing of damping ratio can reduce the fragilities of the convective displacement and base shear force and increase the fragility of the bearing displacement for all three damage states. For the above reason, to decrease the fragilities of convective displacement and base shear force, the damping ratio and period of isolation bearing are designed as small as possible. A proper damping ratio for bearing should be designed to obtain the appropriate fragilities for bearing displacement.


## 5. Summary and Conclusions

Seismic fragility curve, which represents the probability of exceeding a specified damage limit state for a given IM, is a very powerful tool for seismic fragility assessment. The characteristic of isolation bearing has been found to have a significant impact on seismic response of liquid storage tanks. Seismic response of liquid-tank system with base isolation bearing is a complex fluid-structure process involving convective vibration, impulsive vibration, and rigid vibration. As such, this study performs the seismic fragility assessment of base-isolated steel water storage tank. A three-dimensional finite element stick model with the lumped mass is established using the open-source OpenSees computational platform. The modeling in details is provided. The seismic vulnerability of the tank has been assessed according to fragility curves. The seismic demands for the targeted tank are selected. The PGD, the most efficient IM for various seismic demands, is determined based on the dispersion and coefficient of determination. The seismic demand models and fragility curves in terms of PGD have been derived through the cloud analysis. The influence of isolation period and damping ratio of bearing on seismic fragility curves has been investigated. The related key findings are obtained as follows:(1)
The time-history responses of convective displacement, bearing displacement, and base shear force present the similar sinusoidal trend, and the amplitude of convective displacement is obviously larger than the counterpart of bearing displacement.(2)
The fragilities of various seismic demands present an increasing trend from slight damage state to extensive damage state. The fragilities of base shear force for slight damage state are substantially greater than those for moderate and extensive damage states.(3)
The isolation period and damping ratio of bearing have a significant influence on convective displacement and bearing displacement and have a little impact on base shear force, which tell designer that more close attention should be paid to the selection of parameters for bearing of base-isolated tank to decrease seismic fragilities of the tank during earthquake.



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*Source: 8835943-2020-10-28.xml*