Sloshing of Liquid in a Cylindrical Tank with Multiple Baffles and Considering Soil-Structure Interaction

Abstract

In this study, the liquid sloshing in a cylindrical tank considering soil–structure interaction and undergoing horizontal excitation is investigated analytically. Multiple rigid annular baffles are positioned on the rigid wall to mitigate the liquid sloshing. Firstly, combined with the subdomain partition method for sloshing, the complex liquid domain is partitioned into simple subdomains with the single condition for boundary. Based on continuity conditions of velocity and pressure as well as the linear sloshing equation for free surface, the exact solution for convective velocity potential is derived with high accuracy. By yielding the similar hydrodynamic shear and moment as those of the original system, a mechanical model is developed to describe continuous sloshing, and parameters of the model are given in detail. Then, by means of the least squares approach, the Chebyshev polynomials are utilized to fit impedances for the circular surface foundation. A lumped parameter model is employed to represent influences of soil on the superstructure. Finally, by using the substructure method, a coupling model of the soil–tank system is developed to simplify the dynamic analysis. Comparison investigations are carried out to verify the effectiveness of the model. Detailed sloshing characteristics and dynamic responses of sloshing are analyzed with regard to different baffle sizes and positions as well as soil parameters, respectively. The novelty of the present study is that an equivalent analytical model for the soil–foundation–tank–liquid system with multiple baffles is firstly obtained and it allows the dynamic behaviors of the coupling system to be investigated with high computation efficiency and acceptable accuracy.

1. Introduction

The seismic behavior for liquid storage tanks on soil foundation is still an extremely significant engineering design index due to abundant periodic components of earthquake waves. The storage tanks are subjected to hydrodynamic pressures during horizontal seismic excitation which can exert great influences on the superstructure [1]. The hydrodynamic pressures generated by the resonant sloshing of the free surface can induce serious damage to the tank structures, for instance, splashing and overflowing of the liquid may occur, which can even cause explosions, fires and environmental pollution. Therefore, liquid sloshing is a fundamental issue in storage containers undergoing seismic loads [2]. Amabili [3] obtained a formulation of eigenvalue problem for a circular cylindrical shell partially filled with incompressible sloshing fluid. By using an isogeometric scaled boundary finite element technique, Wang et al. [4] conducted parameter analysis on the effects of the shape and size of internal bodies in a circular tank on fundamental frequencies and sloshing forces. Considering fluid–structure interaction, Lee and Lee [5] investigated the dynamic response of a rectangular tank undergoing three-directional seismic ground motion whose directionality can influence the hydrodynamic pressure. Housner [6] proposed a famous mass-spring model to illustrate the continuous liquid in cylindrical tanks. Xue et al. [7] mainly focused on the differences in sloshing wave kinematics and dynamics for the same volume of liquid in storage vessels with various shapes by using the open source code OpenFOAM. A great deal of efforts are devoted to the sloshing dynamics and pressure distribution for liquid storage containers; however, much less attention is paid to the anti-sloshing problem for the storage vessels in the investigation [7].

Actually, internal structures such as annular or vertical baffles can be positioned in storage tanks to diminish the free sloshing [8,9]. Zhang et al. [10] obtained sloshing mitigation numerically in a rectangular container with baffle with satisfactory accuracy. Biswal et al. [11] performed a series of numerical investigations on liquid responses in baffled containers. Unal et al. [12] conducted the numerical analysis to obtain sloshing in a container equipped with a T-shaped baffle, showing that the existence of the annular baffle is highly effective to decrease sloshing forces. However, the applications of numerical approaches may face restrictions as a result of the ineffective grid generation technology for moving boundaries and complicated structure configurations. Fang et al. [13] utilized experiment investigation to analyze the seismic responses of a cylindrical container with ring baffles. Yu et al. [14] performed an experimental analysis for vertical slat screen influences on decreasing sloshing in a storage container. Xue et al. [15] employed experiments to analyze inhibitory impacts of various kinds of vertical baffles on the hydrodynamic pressure. Nevertheless, it is almost impractical to conduct experiment studies on sloshing of liquid in tanks with multiple various baffles when combined with other complex systems that may need large amounts of tests.

Compared with the existing numerical and experimental investigations, the theoretical method is more suitable to obtain parameter analysis for slight liquid sloshing. In terms of the analytical methods for the sloshing problems of storage containers, Sakai et al. [16] obtained an analytical result for the resonant frequency of sloshing in a container with perforated baffles. Wang et al. [17,18] utilized the subdomain partition method for liquid sloshing to perform parameter studies on dynamic characteristics and seismic responses of cylindrical tanks with multiple annular baffles while providing a small calculation and high accuracy. By employing the same approach, Meng et al. [19] obtained the dynamics in a rectangular storage tank with a horizontal baffle. Similarly, Cao and Wang [20] obtained the semi-analytical results for dynamic response in a container with a baffle by solving the Stokes–Joukowski potential function. However, the available analytical and numerical methods are based on the motion equation for continuous fluid. Tedious derivation and a great amount of calculation will be required when it comes to the application to the complicated tank-liquid systems. Sun et al. [21,22] proposed a mass-spring model for replacing continuous liquid analysis in a cylindrical container with an anti-sloshing baffle, which can be simply assembled with other complex systems with acceptable precision and small calculation cost.

Ignoring effects of flexible soil on the superstructure can induce uncontrollable analysis errors in the practical engineering [23]. Kumar and Saha [24] adopted the lumped-mass model for the container and finite element technique for soil to study seismic responses for containers considering soil–structure interaction (SSI). Park et al. [25] simulated the dynamic performance of cylindrical tanks on a surface foundation and pile group foundation under seismic excitation by centrifugal model tests. Hernandez-Hernandez et al. [26] determined influences of SSI and sloshing mitigation on the development of axial stress by shake table experiments. Ormeño et al. [27] analyzed dynamic responses for a cylindrical tank on the surface foundation through shaking table tests, showing that flexible soil amplifies the wall acceleration, however, diminishes the axial compressive stress of wall. For the soil–tank system, the numerical methods can establish the overall analysis model; however, the computational cost is great and at the same time the calculation accuracy is dependent on the discrete model. The restriction of shake table experimental investigations may exist due to the high cost for tests and the limit of the ultimate bearing capacity of the shaking table.

In order to reasonably reflect the SSI effect and improve the calculation efficiency, the substructure method is applied to dynamics for the soil–tank system. As the key to the substructure method, dynamic impedance with frequency dependence is employed to portray the force–displacement relationship between foundation and soil. Exact data for flexibility coefficients of the circular surface foundation were presented by Veletsos and Wei [28] without assuming the distribution of the contact pressure. A lumped parameter model (LPM) is utilized to eliminate frequency dependence for impedances and can be employed in a time domain [29]. Considering the rocking motion and lateral translation, an analytical model for cylindrical tanks was utilized with a lumped parameter of soil [30]. Wu and Lee [31] established a nested LPM for unbounded soil; however, the precision of LPMs is remarkably influenced by the impedances employed in the fitting process. The numerical stability of rational approximation for the simple polynomial with a high degree cannot be guaranteed. The employment of the Chebyshev polynomials for a nested LPM has the advantage of reducing the numerical oscillation in the modelling while showing excellent convergence [32]. Lyu et al. [33] established a three lumped-mass model for sloshing in a container and a lumped parameter model for soil considering horizontal and rocking motions of foundation. According to the fluid partition approach, Ying et al. [34] developed an analytical model for sloshing in a two-dimensional aqueduct with a baffle. Then, a dynamic analysis model for the soil–aqueduct system is constructed by using the LPM to describe the impedance for surface foundation. By employing the substructure method, Meng et al. [35] analyzed seismic responses for a circular cylindrical tank resting on a circular surface foundation.

Due to the multi-filed interaction for soil, structure and liquid, it is rare to apply the substructure method to theoretical dynamics for a storage container with multiple baffles and resting on soil foundation. In this article, an equivalent model for sloshing in a cylindrical tank with multiple baffles is established, which can simply be assembled with an LPM representing soil with a small computational cost. By yielding the similar shear and moment as exact results [18] undergoing horizontal excitation, the parameters for the model are determined with high accuracy. The impedance of the circular surface foundation is fitted by the complex Chebyshev polynomials, respectively. The nested LPMs independent of excitation frequency are constructed. The complex soil–tank system is replaced by an analytical model with discrete masses and springs. Parametric analysis is investigated in detail with various baffle sizes and positions as well as shear velocities of soil.

2. Subdomain Partition Method for Fluid

A cylindrical tank resting on soil is partially full of incompressible, irrotational and inviscid liquid in Figure 1. Multiple rigid annular baffles are positioned on the tank wall. The soil foundation is considered as the half space. The rigid tank is supported on the circular base whose radius is the same as that of the storage tank. The cylindrical coordinate system with the origin located at the bottom center is employed to describe the movement for the soil–tank system undergoing a horizontal motion. Due to small amplitude for free surface under small earthquake, the linear liquid sloshing is considered. The influences of the baffle thickness on the sloshing can be overlooked. The number of anti-sloshing baffles is considered as $M$. $R_1$ and $R_2$ are radiuses of the baffles and tank, respectively. The baffles are, respectively, positioned at $z=h_1, h_2, \dots , h_M.$ $h_{M+1}$ and h₀ are the fluid height and the tank bottom, respectively. According to the subdomain partition method [17,18], the subdomains of liquid ${\Omega }_i (i=1, 2, \dots , 2M+2)$ are determined with M circular and $M+1$ cylindrical interfaces, as shown in Figure 2. The velocity potential for fluid yields

$$\varphi (r, \theta , z, t)={\varphi }_i(r, \theta , z, t), (r, \theta , z)\in {\Omega }_i, (i=1, 2, \dots , 2M+2)$$

(1)

where ${\Gamma }_{2M+2}$ and ${\Gamma }_{2M+3}$ are free surfaces for subdomains ${\Omega }_{2M+2}$ and ${\Omega }_{2M+1},$ respectively. The M circular artificial interfaces from bottom baffle to top one are identified as ${\Gamma }_{M+2}, {\Gamma }_{M+3}, \dots , {\Gamma }_{2M+1}.$ The M + 1 cylindrical artificial interfaces from bottom to top are identified as ${\Gamma }_1, {\Gamma }_2, \dots , {\Gamma }_{M+1}$.

The liquid velocity potential ${\varphi }_i$ satisfies Laplace equation based on assumptions and definitions:

$$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial {\varphi }_i}{\partial r})+\frac{1}{r^2}\frac{{\partial }^2{\varphi }_i}{\partial {\theta }^2}+\frac{{\partial }^2{\varphi }_i}{\partial z^2}=0, (i=1, 2, \dots , 2M+2)$$

(2)

${\Gamma }_l$ denotes the artificial interface between the adjacent subdomains ${\Omega }_i$ and ${\Omega }_{i^{\prime }} (i<i^{\prime })$ which meet continuity conditions of pressure and velocity at surface ${\Gamma }_l$, respectively:

$$\frac{\partial {\varphi }_i}{\partial t}=\frac{\partial {\varphi }_{i^{\prime }}}{\partial t},\hspace{1em}\frac{\partial {\varphi }_i}{\partial n_l} =\frac{\partial {\varphi }_{i^{\prime }}}{\partial n_l}, (l=1, 2, \dots , 2M+1)$$

(3)

where n_l is the normal vector for ${\Gamma }_l$. The normal vector for the velocity potential function should be consistent with that for the boundary due to no liquid cavitation at interfaces. Considering free surface waves, the movement equation yields

$${\frac{\partial {\varphi }_i}{\partial t}|}_{z=h_{M+1}}+g{f}_i=0, (i=2M+1, 2M+2)$$

(4)

where g is the gravity acceleration. f_i represents the sloshing height for ${\Omega }_i$ and has

$$f_i={{\displaystyle {\int }_0^t\frac{\partial {\varphi }_i}{\partial z}|}}_{z=h_{M+1}}\mathrm{d}t, (i=2M+1, 2M+2)$$

(5)

Regard $\ddot{u}(t)$ as horizontal absolute acceleration of the cylindrical container. The liquid can be regarded as the impulsive component moving with the rigid wall and convective one undergoing the sloshing [6]. The velocity potential is the sum of the impulsive velocity potential ${\varphi }_i^{\mathrm{I}}(r,\theta ,z,t)$ and convective velocity potential ${\varphi }_i^{\mathrm{C}}(r,\theta ,z,t)$, namely, ${\varphi }_i(r,\theta ,z,t)={\varphi }_i^{\mathrm{I}}(r,\theta ,z,t)+{\varphi }_i^{\mathrm{C}}(r,\theta ,z,t)$. Combined with Equations (2)–(5), ${\varphi }_i^{\mathrm{I}}(r,\theta ,z,t)$ and ${\varphi }_i^{\mathrm{C}}(r,\theta ,z,t)$ should meet the movement equation, continuity and boundary conditions, respectively. The impulsive velocity potential yields

$${\varphi }_i^{\mathrm{I}}=\dot{u}(t)r\cos \theta , (i=1, 2, \dots , 2M+2)$$

(6)

Substituting Equations (5) and (6) into Equation (4) has

$${\frac{\partial {\varphi }_i^{\mathrm{C}}}{\partial t}|}_{z=h_{M+1}}+g{f}_i^{\mathrm{C}}=-\ddot{u}(t)r\cos \theta , (i=2M+1, 2M+2)$$

(7)

where $f_i^{\mathrm{C}}$ is the surface wave height corresponding to ${\varphi }_i^{\mathrm{C}}(r, \theta , z, t)$ which is expanded according to sloshing modes by introducing generalized coordinates $q_n(t)$. ${\varphi }_i^{\mathrm{C}}(r, \theta , z, t)$ can take the following form based on the mode superposition method:

$${\varphi }_i^{\mathrm{C}}={\displaystyle \sum _{\overline{m}=0}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{\dot{q}}_n(t){\Phi }_{\overline{m}n}^i}(r,z)\cos \overline{m}\theta , (i=1, 2, \dots , 2M+2)}$$

(8)

where $\overline{m}$ represents the circumferential wave number for ${\Phi }_{\overline{m}n}^i(r, z)$ which denotes the mode for Ω_i and meets Laplace equation, impermeability conditions at the boundary and continuity conditions at interfaces. Substituting Equation (8) into Equation (7) and combined with the mode orthogonality [18] and trigonometric function orthogonality, the sloshing mode with $\overline{m}$ equal to 1 is excited under horizontal excitation. The movement equation for the free surface yields

$${\frac{\partial {\Phi }_{1n}^i}{\partial z}|}_{z=h_{M+1}}-{\frac{{\omega }_{1n}^2}{g}{\Phi }_{1n}^i|}_{z=h_{M+1}}=0, (i=2M+1, 2M+2)$$

(9)

The nth order natural frequency ${\omega }_{1n}$ and corresponding mode ${\Phi }_{1n}^i(r, z)$ were given by the subdomain partition method [17]. Four significant digits can be acquired to guarantee the calculation precision when the number of terms is equal to 20 [17]. In the following analysis, 20 terms of series are utilized.

3. Equivalent Model for Sloshing

Introducing Equations (5) and (8) into Equation (7) has

$${{\displaystyle \sum _{n=1}^{\infty }{{\ddot{q}}_n(t){\Phi }_{1n}(r,z)|}_{z=h_{M+1}}+g{\displaystyle \sum _{n=1}^{\infty }{q}_n(t)\frac{\partial {\Phi }_{1n}(r,z)}{\partial z}|}}}_{z=h_{M+1}}=-\ddot{u}(t)r$$

(10)

Multiplying both sides with ${{\Phi }_{1\overline{n}}(r,z)|}_{z=h_{M+1}} (\overline{n}=1, 2, \dots , \infty )$ and integrating rdr on interval [0, R₂], the spatial coordinates r are eliminated. Combined with orthogonality of modes [18], the movement equation with respect to the generalized coordinate $q_n(t)$ is acquired:

$$M_{1n}{\ddot{q}}_n(t)+K_{1n}{q}_n(t)=-\ddot{u}(t)$$

(11)

in which

$$M_{1n}={\displaystyle {\int }_0^{R_2}r{\left({{\Phi }_{1n}(r,z)|}_{z=h_{M+1}}\right)}^2\mathrm{d}r}/{\displaystyle {\int }_0^{R_2}{r^2{\Phi }_{1n}(r,z)|}_{z=h_{M+1}}\mathrm{d}r}$$

(12)

$$K_{1n}=g{\displaystyle {\int }_0^{R_2}{r\left[\left(\partial {\Phi }_{1n}(r,z)/\partial z\right){\Phi }_{1n}(r,z)\right]|}_{z=h_{M+1}}\mathrm{d}r}/{\displaystyle {\int }_0^{R_2}{r^2{\Phi }_{1n}(r,z)|}_{z=h_{M+1}}\mathrm{d}r}$$

(13)

According to ${\varphi }_i={\varphi }_i^{\mathrm{C}}+{\varphi }_i^{\mathrm{I}}$, the surface wave height is expressed as

$$f_i=-\frac{\cos \theta }{g}\left({\displaystyle \sum _{n=1}^{\infty }{{\ddot{q}}_n(t){\Phi }_{1n}^i(r,z)|}_{z=h_{M+1}}}+\ddot{u}(t)r\right), (i=2M+1, 2M+2)$$

(14)

Supposing horizontal excitation along the direction $\theta =0$, the shear force acting on the wall of the tank is acquired by integrating pressure over the wall based on Bernoulli equation:

$$F_{wall}={\displaystyle \sum _{m=1}^{M+1}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{h_{m-1}}^{h_m}{P}_{2m-1}(R_2,\theta ,z,t)\cos \theta R_2\mathrm{d}z\mathrm{d}\theta }}}$$

(15)

where $\rho$ denotes the fluid density. The hydrodynamic moments exerting on the wall, bottom center and the mth baffle with bottom center have, respectively,

$$M_{wall}={\displaystyle \sum _{m=1}^{M+1}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{h_{m-1}}^{h_m}{P}_{2m-1}(R_2,\theta ,z,t)z\cos \theta R_2\mathrm{d}z\mathrm{d}\theta }}}$$

(16)

$$M_{bottom}={\displaystyle {\int }_0^{R_1}{\displaystyle {\int }_0^{2\pi }{P}_2(r,\theta ,0,t)r^2\cos \theta \mathrm{d}\theta \mathrm{d}r}}+{\displaystyle {\int }_{R_1}^{R_2}{\displaystyle {\int }_0^{2\pi }{P}_1(r,\theta ,0,t)r^2\cos \theta \mathrm{d}\theta \mathrm{d}r}}$$

(17)

$$M_{baffle}^m={\displaystyle {\int }_{R_1}^{R_2}{\displaystyle {\int }_0^{2\pi }{P}_{2m+1}(r,\theta ,h_m,t)r^2\cos \theta \mathrm{d}\theta \mathrm{d}r}}-{\displaystyle {\int }_{R_1}^{R_2}{\displaystyle {\int }_0^{2\pi }{P}_{2m-1}(r,\theta ,h_m,t)r^2\cos \theta \mathrm{d}\theta \mathrm{d}r}}$$

(18)

According to Equations (14)–(18), Equations (19)–(22) are given in the form of

$$F_{wall}=-{\displaystyle \sum _{n=1}^{\infty }{\ddot{q}}_n(t)A_{1n}-\rho \pi R_2^2{h}_{M+1}\ddot{u}(t)}$$

(19)

$$M_{wall}=-{\displaystyle \sum _{n=1}^{\infty }{\ddot{q}}_n(t)B_{1n}-\frac{1}{2}\rho \pi R_2^2{h}_{M+1}^2\ddot{u}(t)}$$

(20)

$$M_{bottom}=-{\displaystyle \sum _{n=1}^{\infty }{\ddot{q}}_n(t)C_{1n}-\frac{1}{4}\rho \pi R_2^4\ddot{u}(t)}$$

(21)

$$M_{baffle}=-{\displaystyle \sum _{n=1}^{\infty }{\ddot{q}}_n(t)D_{1n}}$$

(22)

in which

$$A_{1n}=\rho \pi R_2{\displaystyle \sum _{m=1}^{M+1}{\displaystyle {\int }_{h_{m-1}}^{h_m}{\Phi }_{1n}^{2m-1}(R_2,z)dz}}$$

(23)

$$B_{1n}=\rho \pi R_2{\displaystyle \sum _{m=1}^{M+1}{\displaystyle {\int }_{h_{m-1}}^{h_m}{\Phi }_{1n}^{2m-1}(R_2,z)zdz}}$$

(24)

$$C_{1n}=\rho \pi \left({\displaystyle {\int }_0^{R_1}{\Phi }_{1n}^2}(r,0)r^2\mathrm{d}r+{\displaystyle {\int }_{R_1}^{R_2}{\Phi }_{1n}^1}(r,0)r^2\mathrm{d}r\right)$$

(25)

$$D_{1n}=\rho \pi \left[{\displaystyle \sum _{m=1}^M{\displaystyle {\int }_{R_1}^{R_2}{\Phi }_{1n}^{2m+1}}(r,h_m)r^2\mathrm{d}r-{\displaystyle \sum _{m=1}^M{\displaystyle {\int }_{R_1}^{R_2}{\Phi }_{1n}^{2m-1}}(r,h_m)r^2\mathrm{d}r}}\right]$$

(26)

Supposing that $q_n^{*}(t)=M_{1n}{q}_n(t)$, Equation (11) has

$$A_{1n}^{*}{\ddot{q}}_n^{*}(t)+A_{1n}^{*}{\omega }_{1n}^2{q}_n^{*}(t)=-A_{1n}^{*}\ddot{u}(t)$$

(27)

in which $A_{1n}^{*}$ ($A_{1n}^{*}=A_{1n}/M_{1n}$) is the nth order convective mass for the LPM. $q_n^{*}(t)$ represents the sloshing displacement relative to the wall for the nth convective mass. Substituting ${\ddot{q}}_n(t)={\ddot{q}}_n^{*}(t)/M_{1n}$ into Equation (14) and truncating series, Equations (28)–(32) are given [22]:

$$f_i=-\frac{1}{g}\left({\displaystyle \sum _{n=1}^N{\frac{{\ddot{q}}_n^{*}(t)}{M_{1n}}{\Phi }_{1n}^i(r,z)|}_{z=h_{M+1}}}+\ddot{u}(t)r\right), (i=2M+1, 2M+2)$$

(28)

$$F_{wall}=-{\displaystyle \sum _{n=1}^N\left[{\ddot{q}}_n^{*}(t)+\ddot{u}(t)\right]A_{1n}^{*}-\left(\rho \pi R_2^2{h}_{M+1}-{\displaystyle \sum _{n=1}^N{A}_{1n}^{*}}\right)\ddot{u}(t)}$$

(29)

$$M_{wall}=-{\displaystyle \sum _{n=1}^N\left[{\ddot{q}}_n^{*}(t)+\ddot{u}(t)\right]B_{1n}^{*}-\left(\frac{1}{2}\rho \pi R_2^2{h}_{M+1}^2-{\displaystyle \sum _{n=1}^N{B}_{1n}^{*}}\right)\ddot{u}(t)}$$

(30)

$$M_{bottom}=-{\displaystyle \sum _{n=1}^N\left[{\ddot{q}}_n^{*}(t)+\ddot{u}(t)\right]C_{1n}^{*}-\left(\frac{1}{4}\rho \pi R_2^4-{\displaystyle \sum _{n=1}^N{C}_{1n}^{*}}\right)\ddot{u}(t)}$$

(31)

$$M_{baffle}=-{\displaystyle \sum _{n=1}^N\left[{\ddot{q}}_n^{*}(t)+\ddot{u}(t)\right]D_{1n}^{*}}-\left(-{\displaystyle \sum _{n=1}^N{D}_{1n}^{*}}\right)\ddot{u}(t)$$

(32)

where $B_{1n}^{*}=B_{1n}/M_{1n}, C_{1n}^{*}=C_{1n}/M_{1n}, D_{1n}^{*}=D_{1n}/M_{1n}.$ Based on Equations (28)–(32), the impulsive mass for the model can be written as

$$A_{10}^{*}=\rho \pi R_2^2{h}_{M+1}-{\displaystyle \sum _{n=1}^N{A}_{1n}^{*}}$$

(33)

The heights for the nth order convective and impulsive masses have, respectively,

$$H_{1n}^{*}=\frac{B_{1n}^{*}+C_{1n}^{*}+D_{1n}^{*}}{A_{1n}^{*}}$$

(34)

$$H_{10}^{*}=\frac{\left(\frac{1}{2}\rho \pi R_2^2{h}_{M+1}^2-{\displaystyle \sum _{n=1}^N{B}_{1n}^{*}}\right)+\left(\frac{1}{4}\rho \pi R_2^4-{\displaystyle \sum _{n=1}^N{C}_{1n}^{*}}\right)+\left(-{\displaystyle \sum _{n=1}^N{D}_{1n}^{*}}\right)}{\rho \pi R_2^2{h}_{M+1}-{\displaystyle \sum _{n=1}^N{A}_{1n}^{*}}}$$

(35)

The spring stiffness for the nth convective mass is $k_{1n}^{*}={\omega }_{1n}^2{A}_{1n}^{*}.$ In Figure 3, an equivalent model for sloshing in a cylindrical storage tank with multiple rigid annular baffles is given [22]. The convective sloshing is considered as the linear combination of modes from the first order to the Nth order.

4. Soil–Tank Coupling Model

The dynamic impedance $K(\omega )$ describes the force displacement relation between foundation and soil. The dynamic flexibility $F(\omega )$ is normalized associated with the static flexibility $F_{\mathrm{s}}$ in the form of

$$F(\omega )=\frac{1}{K(\omega )}=F_{\mathrm{s}}{F}_{\mathrm{d}}(a_0)$$

(36)

in which $F_{\mathrm{d}}(a_0)$ denotes the dynamic flexibility coefficient. $a_0=\omega r_{\mathrm{s}}/V_{\mathrm{s}}$ denotes the normalized frequency. r_s is the radius of the circular surface foundation. $V_{\mathrm{s}}$ represents the soil shear wave velocity. The dynamic flexibility coefficient for soil foundation is fitted in the form of a ratio of two Chebyshev polynomials [32]:

$$F_{\mathrm{d}}(a_0)=F_{\mathrm{d}}(\lambda )\approx \frac{Q^{(0)}(\lambda )}{P^{(0)}(\lambda )}=\frac{1+q_1^{(0)}\lambda +q_2^{(0)}{\lambda }^2+\dots +q_{N_{\mathrm{s}}}^{(0)}{\lambda }^{N_{\mathrm{s}}}}{1+p_1^{(0)}\lambda +p_2^{(0)}{\lambda }^2+\dots +p_{N_{\mathrm{s}}}^{(0)}{\lambda }^{N_{\mathrm{s}}}+p_{N+1}^{(0)}{\lambda }^{N_{\mathrm{s}}+1}}$$

(37)

in which $\lambda ={i{a}_0/a_0}_{\max },$ $a_0{}_{\max }$ denotes the maximum in the approximate frequency range. N_s denotes the degree for the Chebyshev polynomials. The unknown coefficients in Equation (37) are obtained by the least squares technique [31]. The dynamic flexibility function $F(\omega )$ is written in the nested form of

$$\begin{array}{l}F(\omega )=F_{\mathrm{s}}{F}_{\mathrm{d}}(a_0)=\frac{F_{\mathrm{s}}}{\frac{1+p_1^{(0)}\lambda +p_2^{(0)}{\lambda }^2+\dots +p_{N_{\mathrm{s}}}^{(0)}{\lambda }^{N_{\mathrm{s}}}+p_{N+1}^{(0)}{\lambda }^{N_{\mathrm{s}}+1}}{1+q_1^{(0)}\lambda +q_2^{(0)}{\lambda }^2+\dots +q_{N_{\mathrm{s}}}^{(0)}{\lambda }^{N_{\mathrm{s}}}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{\frac{1}{F_{\mathrm{s}}}+i\omega \frac{{\delta }_0{r}_{\mathrm{s}}}{V_{\mathrm{s}}{F}_{\mathrm{s}}}+\frac{1}{\frac{F_{\mathrm{s}}}{{\chi }_1}+\frac{1}{i\omega \frac{{\delta }_1{r}_{\mathrm{s}}}{V_{\mathrm{s}}{F}_{\mathrm{s}}}+\frac{1}{\frac{F_{\mathrm{s}}}{{\chi }_2}+\frac{1}{i\omega \frac{{\delta }_2{r}_{\mathrm{s}}}{V_{\mathrm{s}}{F}_{\mathrm{s}}}+\frac{1}{\begin{array}{l}\\ \ddots \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{\frac{F_{\mathrm{s}}}{{\chi }_{N_{\mathrm{s}}}}+\frac{1}{i\omega \frac{{\delta }_{N_{\mathrm{s}}}{r}_{\mathrm{s}}}{V_{\mathrm{s}}{F}_{\mathrm{s}}}}}\end{array}}}}}}}\end{array}$$

(38)

in which the dimensionless coefficients of the dashpot δ_j and spring χ_j for the nested LPM in Figure 4 are calculated from

$${\delta }_0=\frac{p_{N_{\mathrm{s}}+1}^{(0)}}{q_{N_{\mathrm{s}}}^{(0)}{a}_0{}_{\max }},{\chi }_j=\frac{p_{N_{\mathrm{s}}-j+1}^{(j)}}{q_{N_{\mathrm{s}}-j+1}^{(j-1)}},{\delta }_j=\frac{p_{N_{\mathrm{s}}-j+1}^{(j)}}{q_{N_{\mathrm{s}}-j}^{(j)}{a}_0{}_{\max }} \mathrm{for} j=1,\cdots , N_{\mathrm{s}}$$

(39)

where $q_n^{(j)}=q_n^{(j-1)}-\frac{q_{N_{\mathrm{s}}-j+1}^{(j-1)}}{p_{N_{\mathrm{s}}-j+1}^{(j)}}{p}_n^{(j)},$ $p_n^{(j+1)}=p_n^{(j)}-\frac{p_{N_{\mathrm{s}}-j+1}^{(j)}}{q_{N_{\mathrm{s}}-j}^{(j)}}{q}_{n-1}^{(j)}$ for $n=1,\cdots , N_{\mathrm{s}}-j.$ In the following analysis, the Chebyshev polynomials of the degree N_s = 5 are considered.

Based on the substructure approach, the model for the sloshing of liquid is conveniently assembled with the LPM for soils, as shown in Figure 5. The governing equation for motion of the coupling model can be formulated based on the Hamilton’s principle:

$${\displaystyle {\int }_0^t\left[\delta (E_K-E_P)+\delta W_c\right]} dt=0$$

(40)

in which the kinematic energy $E_K$ and potential energy $E_P$ for the coupling system have

$$\begin{array}{l}{E}_K=\frac{1}{2}{\displaystyle \sum _{n=1}^N{A}_{1n}^{*}{({\dot{q}}_n^{*}+{\dot{u}}_0+H_{1n}^{*}{\dot{\phi }}_0+{\dot{u}}_g)}^2}+\frac{1}{2}{A}_{10}^{*}({\dot{u}}_0+H_{10}^{*}{\dot{\phi }}_0+{\dot{u}}_g{)}^2\\ +\frac{1}{2}{M}_{\mathrm{t}}({\dot{u}}_0+y_{\mathrm{t}}{\dot{\phi }}_0+{\dot{u}}_g)2+\frac{1}{2}{J}_{\mathrm{t}}({\dot{\phi }}_0{)}^2\end{array}$$

(41)

$$E_P=\frac{1}{2}{\displaystyle \sum _{n=1}^N{k}_{1n}^{*}{(q_n^{*})}^2}+\frac{1}{2}{k}_0^h{(u_0)}^2+\frac{1}{2}{\displaystyle \sum _{j_1=1}^{N_h}{k}_{j_1}^h{(u_{j_1-1}-u_{j_1})}^2}+\frac{1}{2}{k}_0^r{({\phi }_0)}^2+\frac{1}{2}{\displaystyle \sum _{j_2=1}^{N_r}{k}_{j_2}^r{({\phi }_{j_2-1}-{\phi }_{j_2})}^2}$$

(42)

where M_t, y_t and J_t are the mass, barycenter and moment of inertia for the cylindrical tank with the base, respectively. N_h and N_r denote orders for horizontal and rocking nested LPMs, respectively. The variation of the energy dissipation $\delta W_c$ is expressed as

$$\delta W_c=-c_0^h{\dot{u}}_0\delta u_0-{\displaystyle \sum _{j_1=1}^{N_h}{c}_{j_1}^h{\dot{u}}_{j_1}^{}\delta u_{j_1}^{}}-c_0^r{\dot{\phi }}_0\delta {\phi }_0-{\displaystyle \sum _{j_2=1}^{N_r}{c}_{j_2}^r{\dot{\phi }}_{j_2}^{}\delta {\phi }_{j_2}^{}}$$

(43)

Introducing Equations (41)–(43) into Equation (40) can obtain the movement equation for the soil–tank–liquid system

$$M\left\{\ddot{U}\right\}+C\left\{\dot{U}\right\}+K\left\{U\right\}=-M\left\{\xi \right\}{\ddot{u}}_g$$

(44)

in which $\left\{U\right\}={\left\{\left\{q_n^{*}\right\} u_0 \left\{u_{j_1}\right\} {\phi }_0 \left\{{\phi }_{j_2}\right\}\right\}}^{\mathrm{T}}$ represents the displacement vector. $u_{j_1}$ and ${\phi }_{j_2}$ are the horizontal displacement and rotation angle of each degree of freedom relative to those at bedrock in LPMs, respectively. $u_0$ and ${\phi }_0$ represent the horizontal displacement and rotation angle at the circular base relative to those at bedrock, respectively. M, C and K are the mass, damping and stiffness matrices, respectively. Detailed expressions are presented in Appendix A. The coefficient vector $\left\{\xi \right\}={\left\{\underset{N}{\underbrace{0, \cdots 0,}} 1, \underset{N_h+N_r+1}{\underbrace{0, \cdots 0}}\right\}}^{\mathrm{T}}$ describes impacts of the excitation direction on loads. ${\ddot{u}}_g(t)$ is the horizontal acceleration at bedrock along the direction θ = 0. The Newmark-β approach with the constant average acceleration can be employed to obtain sloshing responses of the soil–tank coupling system. The control parameters for the Newmark-β approach are considered as $\gamma =1/2$ and $\beta =1/4$ which can unconditionally guarantee the convergence and stability for the method.

5. Comparison Studies

Assuming the displacement condition at the contact surface between foundation and soil, the horizontal and rocking coefficients of dynamic flexibility for the normalized frequencies a₀ $(a_0\in [0, 8])$ and soil Poisson ratio $\nu$ = 1/3 are obtained by Veletsos and Wei [28] based on the elastic half space theory. The horizontal and rocking dynamic flexibility coefficients in Reference [28] are fitted by using the complex Chebyshev polynomials [32]. Figure 6 shows horizontal and rocking dynamic impedances for the circular surface foundation with regard to the normalized frequencies a₀. Two cases with regard to Chebyshev polynomials of degrees N_s = 3 and N_s = 5 are illustrated for various degrees of precision compared with the elastic half space results [28]. It can be seen from Figure 6 that good agreement is achieved for N_s = 5. In the following analysis, the Chebyshev polynomials of the degree N_s = 5 are considered. Then, the selected dynamic flexibility can be expressed as the nested form depicted in Equation (38) according to the recursive function theory. In

Table 1. The coefficients for the springs and dashpots for the nested LPMs with the degree N_s = 5.

Coefficient	Stiffness		Coefficient	Damping
	Horizontal	Rocking		Horizontal	Rocking
${\chi }_1$	−0.1400	−0.4977	${\delta }_0$	0.6545	0.3927
${\chi }_2$	0.6235	−2.7993	${\delta }_1$	−0.1187	−0.4663
${\chi }_3$	−0.1846	167.2036	${\delta }_2$	0.0726	2.5569
${\chi }_4$	0.1653	−0.0038	${\delta }_3$	−0.0392	−2.4834
${\chi }_5$	−0.4609	0.0038	${\delta }_4$	0.0655	−0.0001
-	-	-	${\delta }_5$	−0.0694	0.0188

, the coefficients of mechanical elements of the nested LPM including the springs χ_j and dashpots δ_j in Equation (39) are presented with N_s = 5.

Haroun and Abou-Izzeddine [30] utilized two groups of spring–dashpot systems to analyze soil–tank interaction under seismic excitation. The tank is rigid and without baffle. The ratio of maximum shear F_max under flexible foundation to that under rigid foundation is defined as α₁. The ratio of maximum moment M_max under flexible foundation to that under rigid foundation is defined as α₂. The maximum of the horizontal absolute acceleration for the base is normalized and associated with the maximum of the horizontal acceleration at the bedrock, that is to say, α₃ = |(ü₀ + ü_g)|_max/|ü_g|_max. The densities of the circular base and tank are fixed at 2500 kg/m³ and 7800 kg/m³, respectively. The density of soil is fixed at 2000 kg/m³. The thickness of the circular base is equal to 0.05H. The thicknesses of the tank and the baffles are obtained as 0.003R₂. The base radius is the same as that of the cylindrical tank. The seismic record is El Centro earthquake (S 90 W component) in 1940.

Table 2. Comparison of responses between present results and those obtained from an analytical model [30].

${\mathit{V}}_{\mathbf{s}}$$(\mathbf{m}/\mathbf{s})$	${\mathit{\alpha }}_1$			${\mathit{\alpha }}_2$			${\mathit{\alpha }}_3$
	Present	Literature [30]	Errors	Present	Literature [30]	Errors	Present	Literature [30]	Errors
150	1.9442	1.8412	5.59%	2.0398	1.9148	6.53%	1.1008	1.0525	4.59%
200	1.8109	1.6652	8.75%	1.8394	1.7410	5.65%	1.2835	1.2030	6.69%
250	1.5388	1.4484	6.24%	1.5690	1.5172	3.41%	1.2270	1.1619	5.60%
600	1.2051	1.3190	−8.63%	1.2416	1.3602	−8.72%	0.9780	1.0012	−2.32%
800	1.0508	1.1324	−7.21%	1.0933	1.1587	−5.64%	1.0202	1.0024	1.78%
1200	1.0204	1.0430	−2.17%	1.0210	1.0882	−6.18%	1.0085	1.0012	0.73%

depicts comparison results between the present model and those by using an analytical model [30] under different $V_{\mathrm{s}}$ with R₂ = 10 m. The liquid height is 20 m. It is seen that maximum values of relative errors for ${\alpha }_1,$ ${\alpha }_2$ and ${\alpha }_3$ are 8.75%, −8.72% and 6.69%, respectively. The reason for the discrepancy between the present study and the literature results is that the 20 order convective masses for liquid sloshing are computed in the present analysis; however, only the first order sloshing mode for liquid is utilized in Reference [30]. The difference may also originate from the horizontal and rocking nested LPMs with degrees of freedom N_s = 5 respectively employed in the present model, whereas soil is only simulated by two springs and two dashpots in the analytical model [30]. Therefore, the proposed soil–tank coupling model can be used to investigate dynamic responses effectively while providing acceptable accuracy.

6. Parameter Analysis

6.1. The Effect of Baffle

Consider nondimensional baffle heights, the fluid height and the baffle inner radius as ${\beta }_i=h_i/h_{M+1} (i=1, 2,\cdots , M),$ ${\beta }_{M+1}=h_{M+1}/R_2$ and $\gamma =R_1/R_2$, respectively. The near-fault (NF) and far-fault (FF) seismic records are given in detail in

Table 3. Details of near-fault and far-fault earthquake records.

Earthquake	RSN	Event	Year	Station	Tp (s)	Record	PGA (g)
NF	1106	Kobe	1995	KJMA	1.092	KJM-000	0.834
FF	154	Coyote Lake	1979	San Juan Bautista	-	SJB-303	0.106

. Figure 7 depicts the variations of the first-order convective frequency ${\omega }_{11}^{\mathrm{C}}$ with regard to different baffle heights for M = 2, ${\beta }_3=1.0$ and R₂ = 10 m. The soil shear wave velocity vs. is fixed at 150 m/s. The height of the lower baffle exerts little impact on the first order convective frequency in Figure 7a, however, the first order convective frequency declines by increasing the upper baffle location in Figure 7b. Figure 8 illustrates the variation of the rotational impulsive frequency ${\omega }_{\mathrm{r}}^{\mathrm{I}}$ for the baffle height. It is seen that the rotational impulsive frequency increases by increasing the height of the lower and/or upper baffle. In Figure 9, the impacts of the baffle inner radius on the frequency are presented. It is observed that the first-order convective sloshing frequency and horizontal impulsive frequency both increase by increasing the baffle inner radius.

The amplitudes of the surface height at the wall f_wall associated with the location and size of the upper baffle are presented in Figure 10 for M = 2 and R₂ = 0.508 m. The motion amplitude for the tank is X₀ = 0.001 m and horizontal harmonic excitation frequency is $\varpi$ = 5.811 rad/s. The liquid height is 0.508 m. It is seen that as the baffle height increases, f_wall monotonically declines, however, f_wall monotonically increases as the inner radius increases. Result indicates that the existence of the annular baffles can significantly mitigate the surface height. Furthermore, the computation time of the present reduced model (V_s = 1500 m/s) simulated by the MATLAB procedure is compared with that of the finite element model simulated by ADINA under rigid soil [22] for M = 2, ${\beta }_1=0.5,$ ${\beta }_2=0.75,$ ${\beta }_3=1.0,$ $\gamma =0.5$ and R₂ = 1 m. For each response solution, it only takes approximately 18 s by the present method; however, the ADINA model takes about 508 s [22] on the same personal laptop with Intel Core i7-5500U CPU. Therefore, the present analytical model is 25–30 times faster than the ADINA model.

The impacts of the upper baffle height on the maximum convective shear $F_{\max }^{\mathrm{C}}$ and maximum impulsive shear $F_{\max }^{\mathrm{I}}$ under NF and FF seismic excitations are illustrated for M = 2, ${\beta }_1=0.2, 0.3, 0.4,$ ${\beta }_3=1.0,$ $\gamma =0.7, 0.8, 0.9$ and R₂ = 10 m in Figure 11. It is seen that by increasing the baffle height, $F_{\max }^{\mathrm{C}}$ monotonically decreases; however, $F_{\max }^{\mathrm{I}}$ monotonically increases, which is attributed to the fact that the liquid mass corresponding to the impulsive sloshing response is increased with increase in the baffle height [22]. In Figure 12, as the baffle height increases, the maximum of the hydrodynamic shear $F_{\max }^{}$ increases. The maximum of the hydrodynamic moment $M_{\max }^{}$ first decreases and then increases. The effects of dimensionless baffle radius on $F_{\max }^{\mathrm{C}}$ and $F_{\max }^{\mathrm{I}}$ under NF and FF seismic excitations are investigated for M = 2, ${\beta }_1=0.3,$ ${\beta }_2=0.5, 0.6, 0.7,$ ${\beta }_3=1.0$ and R₂ = 10 m in Figure 13. It is observed that as the baffle radius increases, $F_{\max }^{\mathrm{C}}$ monotonically increases; however, $F_{\max }^{\mathrm{I}}$ monotonically diminishes due to the declined impulsive mass by increasing the inner radius of baffles [22]. In Figure 14, $F_{\max }^{}$ declines with increase in the baffle inner radius; however, $M_{\max }^{}$ monotonically increases.

Additionally, the maximum hydrodynamic sloshing responses of the tank undergoing NF earthquake motion are larger than those undergoing FF earthquake motion in Figure 11, Figure 12, Figure 13 and Figure 14, which is attributed to the large amplitude in the range of the lower frequency for power spectral density of NF seismic records. The natural frequencies for the liquid sloshing of the soil–tank system are in proximity to the predominant low frequency of the NF earthquake records. The hydrodynamic responses of the tank are more sensitive to NF seismic action in comparison with FF seismic action. Consequently, attention should be paid to earthquake responses in proximity of active faults.

6.2. The Effect of Soil

The first five convective frequencies, horizontal impulsive frequency and rotational impulsive frequency are calculated for V_s in

Table 4. Convective frequency ${\omega }_{1n}^{\mathrm{C}} (n=1, 2, 3, 4, 5)$, the horizontal impulsive frequency ${\omega }_{\mathrm{h}}^{\mathrm{I}}$ and the rotational impulsive frequency ${\omega }_{\mathrm{r}}^{\mathrm{I}} (\mathrm{rad}/\mathrm{s})$ of the soil–tank system for V_s.

${\mathit{V}}_{\mathbf{s}} (\mathbf{m}/\mathbf{s})$	150	200	250	600	800	1200	Rigid
${\omega }_{11}^{\mathrm{C}}$	1.2673	1.2678	1.2681	1.2684	1.2685	1.2685	1.2685
${\omega }_{12}^{\mathrm{C}}$	2.2772	2.2773	2.2773	2.2774	2.2774	2.2774	2.2774
${\omega }_{13}^{\mathrm{C}}$	2.8903	2.8904	2.8904	2.8904	2.8904	2.8904	2.8904
${\omega }_{14}^{\mathrm{C}}$	3.3859	3.3859	3.3859	3.3860	3.3860	3.3860	3.3860
${\omega }_{15}^{\mathrm{C}}$	3.8164	3.8165	3.8165	3.8165	3.8165	3.8165	3.8165
${\omega }_{\mathrm{h}}^{\mathrm{I}}$	25.4219	33.8798	42.3405	101.5847	135.4423	203.1592	-
${\omega }_{\mathrm{r}}^{\mathrm{I}}$	81.1651	108.2175	135.2704	324.6435	432.8573	649.2853	-

. The parameters are selected from the utilized parameters in Figure 11, Figure 12, Figure 13 and Figure 14 which depict the effects of baffles to further analyze the effects of soil for the same soil–tank coupling system: M = 2, ${\beta }_1=0.3,$ ${\beta }_2=0.6,$ ${\beta }_3=1.0,$ $\gamma =0.8$ and R₂ = 10 m. It can be observed that the shear wave velocity of soil V_s has little effect on convective frequencies. The greater the shear wave velocity the soil is, the closer the results under flexible soil are to those under rigid soil, implying that the convective characteristics of the sloshing in tanks resting on flexible soil are similar to those on rigid soil. Nevertheless, the impulsive frequencies remain almost the linearized growth with increase in V_s.

For the NF and FF seismic excitation, the ratios ${\alpha }_1,$ ${\alpha }_2$ and ${\alpha }_3$ are investigated in Figure 15, Figure 16 and Figure 17 with different vs. for M = 2 and R₂ = 10 m. It is observed that the amplitudes of the seismic responses do not vary monotonically with the increase in V_s. The values of ${\alpha }_1,$ ${\alpha }_2$ and ${\alpha }_3$ at lower shear wave velocities (representative of more flexible soils) are relatively larger than those at greater shear wave velocities (stiffer soils), showing that soil–tank interaction magnifies the tank response. For Kobe wave excitation, the maximum values of ${\alpha }_1$ and ${\alpha }_2$ occur at V_s equal to 320 m/s and ${\beta }_3=3.0$; for Coyote Lake wave excitation, the maximum values of ${\alpha }_1$ and ${\alpha }_2$ appear at $V_{\mathrm{s}}$ equal to 400 m/s and ${\beta }_3=3.0$. The phenomenon that the peak values pertaining to the shear wave velocity of the soil deviate under the two excitations may be attributed to the fact that the predominant frequencies of NF and FF seismic motions are different. Then, by further increasing the shear wave velocity of the soil, the values of ${\alpha }_1$ and ${\alpha }_2$ for ${\beta }_3=3.0$ both decline typically. The peak values of ${\alpha }_1$ and ${\alpha }_2$ both increase as the fluid height increases from ${\beta }_3=2.0$ to ${\beta }_3=3.0$. The magnification is a factor of both the shear wave velocity of soils (greater magnification for flexible soils) and the geometry characteristic of tanks (greater magnification for tall tanks). With an increase in soil stiffness, the values of ${\alpha }_1$ and ${\alpha }_2$ approach 1.0, indicating that the responses with respect to soils vanish.

In Figure 18, the impulsive displacement characteristically declines with the increase in $V_{\mathrm{s}}$. The greater the fluid height is, the greater the horizontal impulsive displacement becomes typically if the shear velocity of the soil is held constant. It can be also seen from Figure 15, Figure 16, Figure 17 and Figure 18 that with increase in the soil stiffness, ${\alpha }_1,$ ${\alpha }_2,$ ${\alpha }_3$ approach 1.0 and the impulsive displacement reaches zero gradually, indicating that corresponding seismic responses with respect to soil vanish.

6.3. The Effect of Liquid Height

In this section, the parameters are given as follows: M = 2, ${\beta }_1=0.3,$ ${\beta }_2=0.6,$ $\gamma =0.8$ and R₂ = 10 m. Figure 19 illustrates the effects of the liquid height ${\beta }_3$ on hydrodynamic responses for $V_{\mathrm{s}}$ equal to $150 \mathrm{m}/\mathrm{s}, 200 \mathrm{m}/\mathrm{s}, \mathrm{and} 250 \mathrm{m}/\mathrm{s}$ undergoing NF and FF earthquake actions. It is seen that the greater liquid height induces the larger shear and moment. Moreover, discrepancies of hydrodynamic responses for $V_{\mathrm{s}}$ are obtained by increasing the liquid height.

7. Conclusions

An analytical model for the soil–foundation–tank–liquid coupling system with multiple baffles has first been developed in the present study. The continuous liquid sloshing under horizontal excitation in a cylindrical tank with multiple rigid annular baffles was substituted by an equivalent model with discrete masses and springs. A nested lumped parameter model was utilized to replace soils by fitting the dynamic impedances combined with Chebyshev polynomials. According to the substructure approach, the model of the superstructure can be simply assembled with the lumped parameter model to establish the soil–tank coupling model and then obtain dynamics for the whole system with multiple baffles while providing satisfactory precision and high computation efficiency, which is the significance of the present work.

It has been newly found that the height of the lower baffle exerts little impact on the convective sloshing frequency which diminishes with an increase in the upper baffle location and/or with the decline of the baffle inner radius. The rotational impulsive frequency increases by increasing the height of the annular baffles. Furthermore, by using the present coupling model, a good acquaintance with convective and impulsive behaviors for liquid sloshing can be acquired. When the upper baffle has a greater dimensionless height and/or smaller inner radius, the maximum convective component of shear can be reduced significantly; however, the maximum impulsive one shows the contrary tendency.

The fact that soil–tank interaction has remarkable influence on seismic responses of baffled tanks has been explicated in the present study. Flexible soil and/or greater dimensionless liquid height can contribute to magnifying the dynamic responses. As soil stiffness increases, the amplitude of sloshing response does not change monotonically and the response in the tank on flexible foundation approaches that on rigid foundation gradually.