1. Introduction
Liquid-containing tanks are used to store a variety of materials, including water and chemical fluids. As a result, they form an integral part of the infrastructure of our modern society and industries. Water storage tank damage can result in several consequences, including water shortages and economic losses. Liquid storage tanks have experienced severe damage in past earthquakes such as Northridge, Hokkaido, Kobe, Kocaeli, and Chi-Chi, illustrating their vulnerability to seismic activity [
1,
2,
3,
4,
5].
Under seismic excitations, liquid storage tanks exhibit a complex behavior primarily rooted in the interaction between fluid and structure. The fluid–structure interaction is found to influence not only the sloshing response of liquid storage tanks but also the impulsive pressure, amplified by the tank wall’s elasticity [
6]. Additionally, the structural flexibility of tanks’ walls, while increasing this complexity, has been demonstrated to significantly affect the seismic response of these tanks, particularly base shear and dynamic pressure [
7,
8,
9,
10]. In one study, which was conducted by Kolbadi et al. [
10], it was demonstrated that the flexibility of tank walls increases the base shear and base torque by approximately 100% compared to rigid tanks, demonstrating the importance of considering the flexibility of tank walls as part of numerical models. Mitra and Sinhamahapatra studied the effect of considering the fluid–structure interaction on the sloshing response considering the coupled fluid and structural dynamics [
11]. The sloshing motions have been found to be amplified by increasing tank wall flexibility in this paper.
To address the complexity of modeling the fluid–structure interaction, several analytical and numerical models have been proposed in the literature. Tsao and Huang studied the sloshing response of rectangular liquid storage tanks occupied by porous media [
12]. They conducted analytical, numerical, and experimental studies to characterize the fluid–structure interaction using a reduced mechanical sloshing model. The analytical solution used in this paper is based on the linear wave assumption and Darcy’s law in order to model the sloshing motion. The regularized boundary integral method (RBIM) is also applied in this paper for modeling the sloshing behavior numerically. Compared with the experimental results, the analytical method has been determined to be valuable in the linear flow regime, whereas the numerical model has demonstrated success in the nonlinear flow regime.
In general, earthquake activity causes the most severe damage to civil engineering structures in areas closest to faults, often referred to as “near-field/source” regions. As a result of the high input energy and velocity pulses induced by near-field earthquakes in these regions, seismic demands of such structures are shown to substantially increase [
13,
14,
15,
16,
17,
18,
19]. Near-fault ground motions can exhibit velocity pulses that can be more damaging than far-fault motions. Chen et al. performed several studies on the characteristics and effects of pulse-like ground motions on the seismic behavior of various structures [
20,
21,
22,
23]. In one study, the authors examined the seismic behavior of a burial tunnel in soft soil under ordinary and pulse-like ground motions [
22]. The researchers found that the tunnel’s seismic demands significantly increased under pulse-like ground motions. Moreover, multi-pulse ground motions were investigated in another study in order to determine the effects of such motions on the seismic response of frame structures, soil slopes, and concrete dams [
23]. In that study, the researchers used 21 different records of the Chi-Chi earthquake, including 7 non-pulse, 7 single-pulse, and 7 multi-pulse horizontal and vertical ground motions. Multi-pulse ground motions were found to cause more severe damage to the studied structures than non- and single-pulse motions.
It has also been demonstrated that the seismic response of liquid storage tanks under such ground motions increases when they are exposed to near-source earthquakes [
24,
25,
26,
27]. Experimental studies conducted by Zhou et al. showed that cylindrical liquid storage tanks exhibit higher sloshing responses when subjected to long-period and near-field ground motions [
26]. Luo et al. studied the influence of the pulse-like ground motion on the sloshing response of cylindrical LNG tanks [
27]. They found that pulse-like ground motions cause a significant increase in this response. Another study, conducted by Pirsoltan et al., numerically studied nine two-dimensional liquid storage tanks with different dimensions under 10 near-fault earthquakes and 10 far-fault earthquakes [
19]. It was demonstrated that the maximum sloshing response of the tanks significantly increases under near-fault earthquakes.
The strength and characteristics of ground motions generated by earthquakes are qualified using parameters called intensity measures (IMs). Using these measures, it is possible to assess the potential impact of earthquakes on different structures. These IMs can be classified into three main categories: duration-based IMs (like significant durations—Ds5-75 and Ds5-95), amplitude-based IMs (like PGA and PGV), and frequency-based IMs (like mean period Tm). Engineering demand parameters are relatively weakly correlated with duration- and amplitude-based IMs, particularly when there are different fundamental structural periods involved in the system [
28].
The effect of these IMs on the seismic response of liquid storage tanks has also been studied in the literature. Peak ground acceleration (PGA), as an intensity-based IM, and the first mode’s spectral acceleration Sa(T1), as a frequency-based IM, are widely used to assess the response of building frames. It is not guaranteed, however, that these scalar variables can predict the intensity of different seismic responses of liquid storage tanks. In one study, it was found that an increase in PGA (in specific ranges) increases the conditional probability of failure of an elevated liquid storage tank [
29]. The maximum sloshing response of liquid storage tanks was reported to increase with an increase in PGA [
26,
30,
31,
32,
33]. A study has also shown that increasing the excitation amplitude can affect the resonance frequency of the sloshing response [
34]. In another study, on the other hand, PGA was reported to be less effective than the other scalar IM in capturing the seismic response of a squat cylindrical liquid storage tank [
35].
Fluid-filled tanks can be dynamically analyzed using generalized single-degree-of-freedom (SDOF) systems, representing the impulsive and convective vibration modes of the tank–liquid system. Using this method to analyze the seismic behavior of liquid storage tanks, one study showed that only the first sloshing frequency (Tc1) is important [
36]. Another study, however, showed that, unlike the sloshing response of liquid storage tanks under far-fault earthquakes, the second convective mode’s response is significantly larger than that of the first one (Tc1) for near-fault earthquakes and cylindrical liquid storage tanks [
25]. It was also shown in another study that the first three odd sloshing frequencies of rectangular LNG tanks tend to excite the most obvious sloshing under surge or sway excitations [
37].
As mentioned above, the characteristics of external loads applied to liquid storage tanks and their effects on the tanks’ responses have been studied in the literature. The proximity of the predominant earthquake period to the fundamental sloshing period of liquid storage tanks was reported to increase their maximum sloshing response [
26,
31]. Lijian Zhou et al. studied this relationship in a cylindrical tank under 38 unidirectional earthquake excitations [
26]. This relationship was also studied by Ersan Güray et al. for a two-dimensional rectangular tank using the smoothed-particle hydrodynamics (SPH) method [
31]. It is worth noting that regarding rectangular liquid storage tanks under three-dimensional earthquake excitations, the corner of the tanks was found to be the critical location for sloshing [
38].
Furthermore, it was found that increasing the pulse period can increase the maximum sloshing height of cylindrical liquid storage tanks [
26]. Zhou and their colleagues studied a cylindrical liquid storage tank with a natural convective period of 5.42 s under 38 near-fault pulse-like earthquakes with pulse periods ranging from 1.5 to 8 s [
26]. Dynamic responses of liquid storage tanks have also been shown to be highly sensitive to the characteristics of the applied loads. It has been shown that the pulse characteristics of ground motion increase dynamic pressures on the tanks’ walls as well as dynamic loads [
27,
39]. Ren et al. studied the sloshing response of rectangular liquid storage tanks under earthquakes with different frequency contents [
40]. Accordingly, they modeled six two-dimensional liquid storage tanks, including squat, square, and slender tanks, with different water height to tank length ratios. According to their findings, the maximum sloshing response is highly sensitive to earthquake records with high-frequency contents.
Veletsos et al. demonstrated the importance of considering higher sloshing modes when calculating the maximum linear sloshing [
41]. The higher sloshing modes have, therefore, been considered in more recent studies of the seismic behavior of liquid storage tanks. For example, Merino et al. considered the first three convective modes to calculate the maximum sloshing wave height [
42]. It is worth mentioning that most design codes, except NZSEE [
43], ignore the effect of higher convective modes on the maximum sloshing response.
In the case of rectangular liquid storage tanks under multi-directional earthquake ground motions, little research has been conducted on the effects of seismic wave incidences on the seismic response of liquid storage tanks [
44]. Isaacson et al. studied the effect of incident angle on the maximum force on a rigid rectangular tank under unidirectional harmonic and seismic loads [
45]. Under both types of loads, the direction of motion parallel to the tank walls induced the maximum load in the tank.
The effects of incident angle have been shown to affect both convective and impulsive responses. Lee and his colleague studied the effects of incident angle on the behavior of a rectangular liquid storage tank with a planar aspect ratio of 3 under a far-fault three-directional ground motion by rotating the horizontal earthquake components from 0 to 170 degrees with 10-degree increments [
46]. They found that the impulsive and convective responses of the tank are greatly affected by changing the incident angle. The effects of incident angle on the convective and impulsive responses of a concrete liquid storage tank were studied by [
47]. In the presence of seven different incident angles, both the convective and impulsive responses of the liquid storage tank were found to be sensitive to angle variations. It has been demonstrated by [
47] that the angle of incidence of input ground motions greatly affects the seismic response of rectangular liquid storage tanks. According to that study of rectangular liquid storage tanks with three different planar aspect ratios, structural displacement is related to the sloshing response, such that the sloshing height increases when the structural displacement decreases with a change in the angle of incidence.
Various methods have been employed in the literature to study the impact of the parameters mentioned above on liquid storage tanks’ dynamic behavior. Among these methods, numerical methods are widely used to study this behavior. As a part of the research conducted by Daalen, the Lagrangian principle and the Hamiltonian formulation for water waves were extended to three-dimensional problems of nonlinear water waves in hydrodynamic interaction with floating bodies [
48]. It was shown that the extended formulations can clearly describe wave–body dynamics, including momentum and energy transfer. Chen et al. proposed a method for numerically modeling 2D and 3D nonlinear sloshing responses using a new boundary integral method (BIM) [
49]. Using an artificial linear damping coefficient proportional to the fluid particle velocity, they simulated liquid motion energy dissipation. They conducted several small-scaled shaking table tests to show the efficiency and reliability of the new BIM model. Wu et al. extended a two-dimensional numerical scheme to a three-dimensional model to simulate the free-surface waves in liquid storage tanks with different shapes. The extended scheme enabled the precise prediction of the trajectory of each free-surface node based on the accurate estimation of the partial derivatives of the velocity potentials, shown by comparing with experimental results [
50].
The FE method developed by Tezduyar et al. was shown to be highly effective in modeling the fluid–structure interaction and tracking free-surface fluid [
51]. This method updates the mesh every time step to handle fluid changes in the spatial domain. The Lagrangian–Lagrangian method is another numerical method that enables the precise modeling of water-tank systems and the tracking of free-surface water nodes’ movement over time. Poorakarparast et al. used this method to study the seismic behavior of a liquid storage tank considering the fluid–soil–structure interaction [
52]. They validated their model by comparing the first frequency of their numerical model with the one calculated using ACI regulation.
In this study, the sloshing response of a real-scale rectangular liquid storage tank under seven three-dimensional near-field earthquakes was numerically studied. In order to numerically model the liquid storage tank, the Lagrangian–Lagrangian method was applied. This method was validated using the experimental results provided by [
53]. In the subsequent step, a real-scale rectangular liquid storage tank was modeled using the same numerical method, and its dynamic properties, including the first five sloshing periods of vibration, were calculated. The tank was then subjected to seven near-field earthquakes under two different conditions. In the first condition, the major and minor horizontal earthquake components were applied perpendicularly to the shorter and longer tank walls, respectively. These two components were then rotated 90 degrees around the vertical earthquake component and applied to the tank. The sloshing response of the tank to the seven earthquakes was recorded at its corners, in the middle of the shorter tank wall, and in the middle of the longer tank wall at the water’s surface. Finally, the relationship between the maximum sloshing response and the frequency content of the earthquakes was studied.
2. Numerical Model and Verification
This study uses the Lagrangian–Lagrangian approach to model the dynamic behavior of liquid storage tanks, which is based on the total energy and stain energy of liquids. It has been demonstrated that this method is capable of modeling fluid behavior in rigid tank systems [
52,
54,
55]. This method is applied in this study by taking into account the flexibility of the tank’s wall. The fluid potential energy (
), in this approach, is equal to the sum of strain energy and increasing potential energy (
), as shown in Equation (1).
where
and
are the strain and potential energies, respectively.
The strain energy equation is defined as follows:
In Equation (2),
is the velocity vector,
ρ is the mass density of the fluid, and
T is the strain energy. The Lagrange equation can be therefore written as follows:
where
and
are the displacement and force vectors of the
jth component.
If
S is the rigidity matrix for the surface elements, and
K and
M are the stiffness and mass matrixes of the fluid elements, Equations (1) and (2) can be rewritten as Equations (4) and (5):
The rigidity matrix (
S), stiffness matrix (
K), and mass matrix (
M) are given below:
where
and
are the interpolation functions for three-dimensional elements and two-dimensional surface elements, respectively. In fluid meshes, the interpolation matrix maps degrees of freedom to degrees of freedom in structural meshes. In FSI simulations, information is transferred between fluid and structural domains by the interpolation matrix.
If Equations (4) and (5) are substituted in the Lagrangian equation (Equation (3)), the governing equation of motion can be written as follows (Equation (9)):
where
a is the acceleration vector of the nodes at the structural domain’s boundary elements, and
R is the load vector, which generally varies in each time interval.
To calculate the pressure and displacement of the system and their derivatives, the time-stepping Newmark method is employed, as shown in Equations (10) and (11).
In Equations (10) and (11),
β and
γ are the Newmark parameters allowing engineers to control the trade-off between numerical stability and accuracy. The Newmark formulation is implemented in the ANSYS Mechanical Release 13.0. The typical value for
γ is 0.5, and for
β it is between 0.17 and 0.25 [
56].
2.1. Model Specifications: Fluid–Structure Interaction and Boundary Conditions
We used FEM to approximate fluid–structure interactions using displacement-based fluid approximations. Accordingly, the ANSYS software was used to simulate the water tank system. An analysis of the interaction between the fluid structure and internal fluid flow in a flexible tank domain was carried out in this study. The fluid–structure interaction was investigated regarding a rectangular concrete above-ground liquid storage tank with a fixed base.
Figure 1 shows a cube element, which is used to discretize both the fluid and tank domains. In order to incorporate the effects of wall flexibility in the tank response, the tank was meshed using three-degree-of-freedom eight-node SOLID65 elements, which have successfully been used to simulate concrete’s nonlinear behavior [
57,
58].
The liquid is represented by 8-node fluid elements, known as FLUID80 in ANSYS [
59]. This element type uses a displacement-based formulation, in which the fluid is characterized by its bulk modulus (K). Studies have demonstrated that this element can be used to simulate fluid–structure interactions, fluid sloshing, and hydrostatic pressure [
60]. In addition, unique surface effects were incorporated into the fluid element, represented by gravity springs. To accomplish this, springs were added to each node and positioned at the top of the element with positive constants. The springs have positive stiffness at the top nodes, while the bottom nodes have negative stiffness. Thus, they balance one another’s positive and negative impact on an intermediate node.
Fluid elements were attached to the tank elements by coupling them in the normal direction, normal to the interface, which permits them to move vertically and tangentially. The dynamic behavior of the tank was evaluated using the direct integration method. In order to solve equations, the Newmark time integration technique was implemented using ANSYS software [
59].
In order to prevent settling, the spring closes off the nodes of the water and the wall of the container at the base of the water, assuming that the bottom of the container is rigid. This spring should be neutralized by setting the liquid’s free surface to the coordinate z = 0 in this case, ensuring that only positive springs are used on the free surface of the water.
Compared with experimental results, this modeling approach has been proven to be capable of accurately modeling the sloshing response [
61]. Studies such as [
62,
63] also employed this approach in order to examine the sloshing and other seismic responses of liquid storage tanks.
2.2. Model Verification
The validity of the finite element method was demonstrated by comparing the numerical results with both analytical and experimental results available in the literature. First, a modal analysis was performed on a FE model of a liquid container that has been experimentally studied in the literature. The first three convective frequencies of the filled water were compared with an analytical solution proposed by [
64].
where
and
are the tank’s length and the height of the water, respectively.
A liquid container modeled to verify the numerical method had a length, width, and height of 1, 0.4, and 0.96 m, respectively. The water depth inside the tank (
HL) was 0.624 m. This tank was numerically and experimentally studied by [
53].
Figure 2 shows the schematic of the tank.
This water–tank system was numerically modeled and meshed using Fluid80 elements with different dimensional properties to determine the optimal size for the elements. A modal analysis was conducted for each element size, and the first five frequencies of the water were compared with those of the analytical solution (Equation (12)). The frequency responses obtained from the FE analysis of the tank, discretized with various element sizes, were compared with those obtained from the analytical solution, and are shown in
Table 1.
According to
Table 1, the element size for M4 is fine enough to calculate the first five frequencies of the water with 0, 1, 2, 2, and 3.5 percent compared to the analytical solution. Furthermore, the larger element sizes, i.e., larger than M1, have not been able to calculate higher sloshing modes.
This work then compared the time-history sloshing response of the same tank with the experimental results published by [
53]. The tank was subjected to a harmonic load, as shown in Equation (13).
where
D and ω are the displacement and frequency, set to 5 mm and 1.12 times the water’s first natural frequency (0.87 Hz), respectively.
Figure 3 shows the time history of the sloshing response of the tank at the left wall of the tank.
The comparison between the numerical and the experimental results, as shown in
Figure 3, demonstrates that the numerical model can accurately capture the time-history sloshing response. The difference between the maximum sloshing response of the numerical model, which is equal to 60.3 mm, with that of the experimental model, 62.3 mm, is 3.2 percent. It is important to acknowledge that while the maximum sloshing response is accurately captured using this numerical method, increasing the nonlinearity of this response increased the difference between the time-history sloshing response of these two models.
Based on
Figure 3 and
Table 1, the numerical method can accurately model the water–tank system’s maximum sloshing response.
2.3. Geometrical Data and Finite Element Modeling (the Case Study Tank)
A rectangular concrete liquid storage tank was modeled with the finite element method and subjected to several near-fault pulse-like and non-pulse-like ground motions to investigate the effects of ground motion characteristics on the dynamic response of rectangular liquid storage tanks. The tank was modeled with all walls being flexible in order to take into account the effects of wall flexibility on the seismic behavior of the tank.
The maximum pressure in rectangular concrete water storage tanks occurs at the bottom of the tank wall, where the hydrostatic pressure, a function of the liquid’s density and the liquid’s height, is the highest [
65]. Therefore, the tank walls were modeled as tapered, such that the thickness of the walls varied linearly from 0.4 m at the base to 0.2 m at the top of the tank wall. Some liquid storage tanks were constructed with tapered walls, and some numerical models of liquid storage tanks studied in the literature were modeled with tapered walls like the rectangular liquid storage tank studied by [
52].
Figure 4 shows the geometrical parameters of the water–tank system.
3. Record Selection
Seven near-fault records from the Pacific Earthquake Engineering Research Center–Next Generation Attenuation (PEER–NGA) [
66] were selected in this study. It has been shown in the literature that the vertical earthquake component significantly affects the seismic response of liquid storage tanks, particularly the impulsive response [
8,
67,
68]. Therefore, all three earthquake components were considered in the analysis of the three-dimensional liquid storage tank. According to [
69], the site-to-source distance boundary between near-fault and far-fault earthquakes is 10 km. Therefore, the distance between the recording site and the rupture plane of all selected earthquakes was less than 10 km. All records used in this study had a moment magnitude greater than 6.5 (
). ASCE7 classifies records as near-fault if the moment magnitude is greater than 6 and the site-to-source distance is less than 10 km [
70]. According to [
71], the average PGV of near-fault earthquakes is considerably higher than that of far-fault earthquakes. As a result, the records were selected so that they had a PGV greater than 30 cm/s.
Table 4 summarizes these earthquakes.
Among the earthquakes shown in
Table 4. 5 are pulse-like earthquakes (#3, #4, #5, #6, and #7), and the remaining are non-pulse earthquakes (#1 and #2). The pulse period of the pulse-like earthquakes ranges from 3.53 to 6.27. According to [
72], when the pulse period approaches the fundamental sloshing period (
), the sloshing response considerably increases. In selecting the records for this study, the effect of approaching the pulse period to the fundamental sloshing period on the sloshing response of the tank along its length and width was considered.
The spectral accelerations of the selected records at the tank’s first five sloshing modes, along the tank’s length and width, differed from one another.
The major horizontal component of the selected earthquakes, shown in
Table 4, was scaled to 0.4 g. Using the same scaling factor used to scale the major horizontal component of the earthquake, the second (minor) horizontal component and the vertical component were also scaled.
Table 5 shows the PGA and PGV of the horizontal and vertical components of the selected earthquakes after being scaled.
The tank was subjected to the earthquakes in such a way that the main horizontal component of the ground motions, the one scaled to 0.4 g, acted once parallel to the longer wall and once parallel to the shorter wall. The minor horizontal component acted perpendicular to the major component in both loading conditions. The vertical earthquake component was also considered in the analysis of the tank.