Seismic Performance of Large Underground Water Tank Structures Considering Fluid–Structure Interaction

Abstract

The widespread application of large underground water tank structures in urban areas necessitates reliable design guidelines to ensure their safety as critical infrastructure. This paper investigated the seismic response of large underground water tank structures considering fluid–structure interaction (FSI). Coupled Eulerian–Lagrangian (CEL) was employed to analyze the highly nonlinear FSI caused by intense fluid sloshing during earthquakes. The patterns of fluid sloshing amplitude observed from the finite element model were summarized based on analyses of fluid velocity, hydrodynamic stress components, and overall kinetic energy. In addition, the seismic response of the water tank structure was thoroughly assessed and compared with the simulation results of the empty tank structure. The results indicate that significant fluid sloshing occurs within the structure under seismic excitation. The amplitude of fluid sloshing increases horizontally from the center toward the edges of the structure, corresponding to higher hydrodynamic loads at the side area of the structure. By comparing the analysis results of the water tank structure with and without water, it was concluded that FSI is the primary cause of structural damage during an earthquake. The hydrodynamic loads on the roof, diversion walls, and external walls lead to significant localized damage.

1. Introduction

Underground water tank structures are frequently utilized for rainwater storage, clear water storage, and sewage treatment, depending on their specific purpose [1]. Rainwater storage tank structures collect and store rainwater during the flood season, aiding in the mitigation of urban water scarcity mitigation and flood control [2,3,4]. Clearwater and sewage treatment tank structures are integral components of municipal water supply, sewage and reclaimed water facilities, all critical components of urban water systems. Damage to these systems would severely disrupt essential activities such as firefighting, medical services, and daily life [5]. Earthquake disasters frequently cause damage to urban infrastructure [6,7,8]. The seismic activity subjects tank structures to hydrodynamic and seismic loads, leading to structural damage [9].

Underground water tank structures offer space-saving benefits and flexible site selection, allowing the space above the structure to be utilized in various ways. Depending on demand, underground water tank structures can also be located closer to service areas, hence reducing investment and maintenance costs associated with long-distance transport. These advantages have led to their widespread use globally [10]. Due to the elevated demand for water reclamation and urban flood control in China [11], the development of underground water tanks has accelerated in recent years [12]. However, due to their recent construction, most of these structures have not yet experienced earthquakes, and few studies have analyzed their seismic performance.

Investigations of fluid sloshing in tank structures have shown that dynamic water loads significantly affect the dynamic response of the structure [13,14,15]. According to China’s current code for seismic design of outdoor water supply, sewerage, gas, and heating engineering (GB50032-2003) [16,17], the seismic design of underground water tank structures employs the concentrated mass method to account for the impact of the hydrodynamic pressure on the walls. However, this method does not adequately address the impact of local high hydrodynamic pressure on the structure caused by fluid sloshing during an earthquake [18,19]. It is significant to consider an intense fluid sloshing load in the seismic design of underground water tank structures [20,21].

To accurately simulate the nonlinear fluid sloshing and the impact of fluid on structures during an earthquake, this study employed the CEL method to analyze the dynamic response of fluid–structure interaction under dynamic loads [22,23,24]. The Lagrangian method allows the mesh to deform with the material and therefore is typically used to analyze structures and fluids with minor deformations. It becomes unstable and loses accuracy if deformations are excessive, necessitating frequent re-meshing to prevent excessive mesh distortion. In contrast, the Eulerian method allows the mesh to remain static as fluid flows or deforms within it and therefore is more suitable for complex fluid dynamics and material phase changes [25]. In this study, the Lagrangian method was applied to the structure, while the Eulerian approach was used to simulate the fluid [26].

Due to its accuracy in fluid analysis, the CEL method is widely applied in fluid–structure interaction, underwater explosions, metal cutting, aerodynamics, and large deformation geotechnics [22,27]. Studies involving FSI in prestressed LNG structures under seismic effects were conducted employing the CEL method, and the results verified its feasibility and validity for such analyses [28]. Notably, the CEL-based simulations successfully captured complex free-surface sloshing and splash phenomena, demonstrating the method’s ability to accurately represent dynamic FSI behavior under seismic excitation. The CEL method was employed to investigate fluid sloshing behavior in both rigid and flexible tank structures under various water levels, and the results confirmed that structural deformation and the use of internal baffles play a crucial role in reducing sloshing intensity and mitigating hydroelastic effects. Specifically, comparisons between rigid and deformable tanks revealed that wall flexibility can effectively dissipate sloshing energy, leading to reduced wave elevation near the tank wall. Thinner tank walls exhibited greater deformation and energy absorption, whereas thicker walls were less responsive to hydrodynamic loading. In addition, the implementation of internal baffles, particularly those optimized in height and material properties, further attenuated sloshing by acting as passive dampers [29]. Similarly, fluid sloshing and FSI in rigid tank structures subjected to seismic excitations have been investigated using both the CEL and Coupled Acoustic–Structural (CAS) finite element methods. The results demonstrated that the CEL approach can provide highly accurate predictions of FSI behavior, particularly for capturing small-amplitude sloshing. Comparative studies have shown that both methods yield sloshing displacement and hydrodynamic pressure responses in close agreement with experimental observations. While the CAS method offers higher computational efficiency, the CEL method is advantageous in resolving complex flow patterns and localized pressure variations [30]. The accuracy of the CEL method was further validated through comparison with experimental data on sloshing mitigation using spring-mounted baffles in rectangular containers. These studies demonstrated that the incorporation of moving baffles can significantly influence the sloshing dynamics. Specifically, the use of moving spring baffles has been shown to attenuate the kinetic energy of sloshing through energy absorption and the formation of recirculation zones within the fluid domain. Furthermore, the damping performance of such baffles improves with higher water storage ratios, while the influence of spring stiffness on sloshing suppression remains relatively minor. These findings further underscore the capability of the CEL approach in capturing complex fluid–structure interaction phenomena in sloshing-dominated systems [31,32]. The seismic performance of rigid and flexible liquid storage tank structures was extensively investigated in previous studies by considering a range of parameters, including surrounding soil conditions, foundation types, and structural aspect ratios [33,34,35]. These studies revealed that while soil–structure interaction (SSI) tends to have a limited effect on the sloshing wave height, other factors, such as the tank’s geometric configuration, water fill levels, and the frequency content and amplitude of the input ground motion, play a more significant role in influencing the FSI response.

Although previous studies considered various factors in the seismic performance analysis of water tank structures [36,37,38], limited research was focused specifically on the seismic response of large underground water tanks [39,40]. To address this gap, the present study investigated the influence of FSI on the seismic behavior of a large-scale underground reinforced concrete water tank. A high-fidelity CEL approach was implemented in Abaqus to capture the nonlinear fluid sloshing and hydrodynamic pressure during seismic excitation. Unlike prior models with simplified configurations, this study incorporated a realistic tank geometry featuring internal columns and diversion walls, along with an elastoplastic material model. The results provide valuable insights into the coupled seismic response of large tank structures and establish a basis for future seismic design and assessment of underground water storage infrastructure.

2. Numerical Model

The seismic performance of reinforced concrete underground water tank structures with multiple diversion walls and a flat slab roof without beams was investigated by employing finite element analysis. This tank structural form design was widely adopted in recent large underground water tank structure projects. Figure 1 illustrates a schematic diagram of this underground water tank structure (The ceiling part is hidden to show the internal structure clearly).

2.1. Description of the Water Tank Structure

Figure 2 illustrates a front view of the underground water tank structure. The diversion walls are numbered (from 1 to 13), providing specific reference for discussion of the analysis results. The external geometric dimensions of the selected underground water tank structure are 71 m in length, 51 m in width, and 6.7 m in height. The structure comprises 117 columns, with masonry walls between them that form 13 diversion walls. The entire structure was constructed using reinforced concrete. The thickness of the external wall is 500 mm, and the thickness of the bottom and top slabs is 400 mm and 300 mm, respectively. The columns have cross-sectional dimensions of 400 mm by 400 mm. The space between each column is 4600 mm. Each diversion wall is 250 mm thick and 5800 mm high, with a spacing of 4750 mm between them. The distance between the walls and the structure wall is 4800 mm at the diversion wall gaps. The structure body, diversion walls, and columns were made of C30 concrete.

To investigate the potential for collapse or severe failure, the nonlinear behavior of both concrete and reinforcing steel was considered in the simulation. The concrete damage plasticity (CDP) model was adopted to simulate the dynamic response of concrete. The material parameters of concrete are listed in

Table 1. Material parameters of C30 concrete.

Density (kg/m3)	Poisson’s Ratio	Elastic Modulus (MPa)	Dilation Angle (°)	Initial Compressive Yield Stress (MPa)
2500	0.2	30,000	36.5	13
Limited compressive yield stress (MPa)		Initial tensile yield stress (MPa)	Compression stiffness recovery parameter	Tensile stiffness recovery parameter
20.1		2.4	1.0	0.0

.

Table 2. Concrete cracking displacement corresponding to tensile stress and damage factor.

Cracking Displacement (mm)	Tensile Stress (MPa)	Tensile Damage Factor
0	2.4	0
0.066	1.617	0.381
0.123	1.084	0.617
0.173	0.726	0.763
0.22	0.487	0.853
0.308	0.219	0.944
0.351	0.147	0.965
0.394	0.098	0.978
0.438	0.066	0.987
0.482	0.042	0.992

and

Table 3. Concrete plastic strain corresponding to compression stress and damage factor.

Plastic Strain (%)	Compression Stress (MPa)	Compression Damage Factor
0	14.02	0
0.04	17.33	0.113
0.08	19.44	0.246
0.12	20.1	0.341
0.16	20.18	0.427
0.2	18.72	0.501
0.24	17.25	0.566
0.36	12.86	0.714
0.5	8.66	0.824
0.75	6.25	0.922
1	3.98	0.969

show tensile and compressive damage factors of concrete. The values are consistent with those commonly employed in seismic response simulations utilizing the concrete damage plasticity (CDP) model originally proposed by Lee and Fenves [41]. In the CDP framework implemented in ABAQUS, the damage factors represent the progressive stiffness degradation due to microcracking under tension and crushing under compression. These parameters were calibrated based on experimental uniaxial stress–strain data under cyclic loading conditions. Although the calibration data are mainly obtained under quasi-static conditions, previous studies demonstrated that the CDP model is capable of adequately capturing the essential characteristics of concrete degradation and energy dissipation under dynamic loading scenarios. Longitudinal reinforcement in the structure walls, diversion walls, structure top, structure bottom, and columns is with HRB400 steel, with HPB300 steel used for column stirrups. Material parameters for the structure, including concrete and reinforcement, are provided in

Table 4. Material parameters of reinforcing steel.

Material	Density (kg/m3)	Poisson’s Ratio	Elastic Modulus (GPa)	Yield Strength (MPa)	Ultimate Strength (MPa)
HRB400	7500	0.3	200	416	489
HPB300	7850	0.3	210	309	358

.

2.2. Finite Element Model of Structure

Solid elements were initially employed in the simulation of the tank structure; however, their use in the current CEL framework resulted in convergence difficulties. These difficulties primarily stem from the strong fluid–structure interaction at the Eulerian–Lagrangian interface, particularly under seismic excitation, where large pressure gradients and localized fluid sloshing induce high mesh distortion and instability in the solid mesh. In contrast, shell elements exhibited improved numerical stability and efficiency in this context. This modeling strategy is consistent with established practices in previous CEL-based studies of liquid storage tanks. Consequently, the modeling was adapted to employ composite shell elements. The thickness direction is discretized in the shell elements into different material layers, representing separate longitudinal and transverse rebar and concrete layers. This element can be used to simulate the reinforcement of a structure and the corresponding structural properties. In contrast to solid elements, composite shell elements can decrease computational load while maintaining both convergence and accuracy. Figure 3 presents a finite element schematic of the structure. Both the Eulerian fluid domain and the Lagrangian tank structure were discretized using a uniform element size of 0.5 m. For numerical stability and convergence in CEL simulations, it is essential to select a sufficiently fine mesh and ensure compatible mesh densities between the fluid and the structure. The choice of mesh size draws upon precedent in existing studies. For instance, prior literature demonstrated that a mesh size of 0.14 m applied to a fluid domain with dimensions of 3 m (length) × 1 m (width) × 1 m (height) was sufficient to achieve accurate results in fluid–structure interaction analyses [25]. By comparison, our mesh-depth-to-fluid-depth ratio remains appropriately scaled. Preliminary sensitivity tests were performed using mesh sizes of 0.5 m, 0.75 m, and 1 m. The 0.5 m mesh displayed stable sloshing patterns, smooth wave propagation, and consistent structural damage distributions, demonstrating its validity for the large-scale FSI model.

Soil–structure interaction effects were not considered in this study. This modeling simplification was deliberately made to isolate and better examine the FSI mechanisms in large-scale reinforced concrete water tanks. Including SSI in the analysis would have introduced additional complexity related to soil flexibility and foundation damping, which may have obscured the specific tank structure response behavior by FSI, which was the primary focus of this investigation. In the case of large underground reinforced concrete water tanks with considerable self-weight and significant hydrodynamic mass due to the stored water, both inertial and kinematic effects become important [42]. Neglecting SSI may influence the accuracy of the structural seismic response, and site effects may potentially amplify or attenuate seismic intensity depending on local soil conditions. If the surrounding soil of the underground tank structure were to be considered, the lateral confinement and energy dissipation provided by the soil would likely reduce the hydrodynamic load and seismic intensity of the structure. In that case, the structural deformation and damage response observed in this study might have been lower. For above-ground water tank structures, when stiff soil overlies bedrock, seismic waves may experience attenuation, particularly at higher frequencies. Conversely, in the presence of soft or deep soil layers, site conditions may lead to amplification of seismic waves and structural dynamic response. Future research will incorporate appropriate soil constitutive models and boundary conditions to comprehensively evaluate FSI and SSI effects in a fully coupled framework to address this limitation.

2.3. Fluid–Structure Interaction Simulation

The CEL method is utilized to simulate the fluid–structure interaction system. Compared to traditional equivalent fluid models, the Eulerian analysis provides a high-fidelity simulation of fluid sloshing within the structure and FSI. Liquid is considered a free surface large deformation fluid. The Lagrangian method allows the mesh to deform with the material and therefore is typically used to analyze structures and fluids with minor deformations. It becomes unstable and loses accuracy if deformations are excessive, necessitating frequent re-meshing to prevent excessive mesh distortion. In contrast, the Eulerian method allows the mesh to remain static as fluid flows or deforms within it and therefore is more suitable for complex fluid dynamics and material phase changes, as shown in Figure 4 [25]. The structure and fluid were analyzed separately using the Lagrangian and Eulerian methods [26]. The general algorithm was applied to the contact between the fluid and the tank structure. Hard contact and frictionless were used in normal and tangential directions, respectively. The Lagrangian part can be considered as the displacement boundary of the Eulerian region, while the Eulerian part can be considered as the load boundary of the Lagrangian element.

The volume of fluid (VOF) method, a common free surface modeling technique in computational fluid dynamics (CFD), is used to track the sloshing pattern of a fluid with large deformation. In the CEL approach, the Eulerian Volume Fraction (EVF), a VOF method, computes the fluid volume fraction in each Eulerian element independently. When the element is completely filled and void of fluid, EVF is 1 and 0 respectively. When the fluid partially flows through the Eulerian element, EVF is the fraction of the fluid volume to the total volume of the element. To fully realize fluid motion tracking, it is essential to establish the Eulerian domain wherever the fluid may flow.

The CEL method is sensitive to time increment selection due to the use of an explicit time integration scheme. The stability of the simulation is governed by the Courant–Friedrichs–Lewy (CFL) condition. In this study, automatic time increment control was applied to ensure numerical stability throughout the analysis. The smallest element size in the model was 0.5 m, and the maximum wave speeds in the materials were approximately 4000 m/s for concrete and 1500 m/s for water, resulting in a stable maximum allowable time increment on the order of 10⁻⁴ s. The time step was automatically adjusted based on these parameters during the simulation. Additionally, energy balance and kinematic consistency were closely monitored, and no numerical instabilities or divergence were observed, indicating that the simulation maintained stable and reliable behavior under the adopted conditions.

The CEL method employs the operator splitting method to independently solve the Lagrangian material property update equation and the Eulerian update convection equation [43]. The rate of heat exchange and the rate of mechanical work done by stresses together govern the rate of increase in internal energy per unit mass, $E_m$. When the heat conduction between the environment and the fluid is neglected, the energy conservation equation can be expressed as follows [44]:

$$\mathsf{\rho }{\displaystyle \frac{\partial E_m}{\partial t}}=\left(P-P_{bw}\right){\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial t}}+S : \dot{e}+\rho \dot{Q}$$

(1)

where $\mathsf{\rho }$ is the density, $P$ is the pressure stress, which is positive in compression, $P_{bw}$ is the pressure stress due to the bulk viscosity, $S$ is the partial stress tensor, $\dot{e}$ is the deviatoric component of strain rate, $\dot{Q}$ is the heat rate per unit mass, $E_m$ is the internal energy per unit mass. Mie-Gruneisen equation of statement (EOS) describes the hydrodynamic behavior. EOS assumes that fluid pressure is a linear function of internal energy and has a nonlinear relationship with density. The equation is as follows:

$$P-P_H=\Gamma \rho \left(E_m-E_H\right)$$

(2)

where $P_H$, $E_H$, and $\Gamma$ are Hugoniot pressure, Hugoniot specific energy, and Grüneisen ratio, respectively; $\Gamma$ is calculated by the following equation:

$$\Gamma ={\Gamma }_0{\displaystyle \frac{{\rho }_0}{t}}$$

(3)

where ${\Gamma }_0$ is material constant, ${\rho }_0$ is reference density. The relationship between $E_H$ and $P_H$ is as follows:

$$E_H={\displaystyle \frac{P_H\eta }{2{\rho }_0}}$$

(4)

where $\eta$ is the nominal volume compressive strain. This is calculated by the following equation:

$$\eta =1-{\displaystyle \frac{{\rho }_0}{\rho }}$$

(5)

Substituting $\Gamma$ and $E_H$ into Equation (2), the equation for $P$ can be obtained as follows:

$$P=P_H\left(1-{\displaystyle \frac{{\Gamma }_0\eta }{2}}\right)+{\Gamma }_0{\rho }_0{E}_m$$

(6)

The linear Hugonoit form fit to $P_H$ is given by the following:

$$P_H={\displaystyle \frac{{\rho }_0{c}_0^2\eta }{{(1-s\eta )}^2}}$$

(7)

where $c_0$ is the velocity of sound propagation in the fluid, $s$ is the material constant and also the slope of the $U_s-U_p$ curve, and ${\rho }_0{c}_0^2$ defines the elastic bulk modulus at small nominal strains. The linear relationship between fluid shock velocity ($U_s$) and fluid particle velocity ($U_p$) can be expressed as follows:

$$U_s=c_0+s{U}_p$$

(8)

By substituting Equation (7) into Equation (6), the relationship between fluid pressure and density is established as the following:

$$P={\displaystyle \frac{{\rho }_0{c}_0^2\eta }{{(1-s_{\eta })}^2}}\left(1-{\displaystyle \frac{{\Gamma }_0\eta }{2}}\right)+{\Gamma }_0{\rho }_0{E}_m$$

(9)

The fluid in the model adopts EOS material defined by four parameters ${\rho }_0$, $c_0$, ${\Gamma }_0$, and $s$. The fluid used in this study is water, and the material property parameters are shown in

Table 5. Water material parameters.

Parameters	Value
Density (ρ)	1000 kg/m3
Viscosity (η)	0.001 MPa
Velocity of sound through water (c0)	1500 m/s
Material constant (s)	0
Material constant (Γ0)	0

. In the present analysis, the dynamic viscosity of water is assumed to be constant, consistent with conventional practice in seismic FSI simulations of tank structures. While local turbulence-induced variations in viscosity may influence detailed flow patterns, previous studies showed that such effects have limited impact on the seismic response of tank structures and the global fluid sloshing pattern, which constitutes the primary focus of this study. Therefore, the assumption of constant viscosity is considered justified. Moreover, the CEL method employed in Abaqus does not incorporate turbulence modeling or spatially variable viscosity by default, yet remains effective in capturing the dominant features of fluid sloshing and transient hydrodynamic loading under seismic excitation. The water depth inside the tank structure is set to 3600 mm, corresponding to 60% of the water storage capacity.

2.4. Ground Motion Input

The El Centro 1940 ground motion, with a peak ground acceleration (PGA) of 0.2 g, was selected as the seismic input and applied at the base of the structure. Figure 5 shows the acceleration and velocity time histories of this ground motion. This input represents a typical moderate-intensity earthquake and corresponds to the seismic design requirements of many urban regions worldwide, such as areas with seismic intensity VII in China and parts of the central and eastern United States and Europe.

The present study serves as a preliminary investigation to evaluate the applicability of the CEL method in capturing fluid sloshing behavior and FSI effects in large reinforced concrete water tanks. In this initial phase, only the aforementioned ground motion was considered. While using a single input facilitates controlled analysis and model verification, it inherently limits the evaluation of the variability in structural response [45,46]. Specifically, vertical ground motions, particularly those near fault zones or with high-frequency content, may alter the fluid sloshing behavior. In addition, ground motions with varying intensities or frequency characteristics may directly influence the dynamic response of the water tank structure [47]. Future research will incorporate multiple ground motions with varying directions, frequency contents, and intensity levels to enable a more comprehensive seismic response analysis.

2.5. Model Validation Analysis

In order to validate the accuracy of the CEL method, the free surface sloshing analysis of fluid under dynamic load excitation conducted by Tippmann et al. [48] was studied. Tippmann et al. compared the study results with those obtained from the CFD approach to ensure reliability. The tank structure was modeled with rigid shell elements with a length (L) of 610 mm, as shown in Figure 6. The initial static fluid height (h) was 200 mm. To accommodate fluid sloshing, the water tank height was set to 500 mm. The fluid material was also water, so the same material parameters as in this study were used.

The water tank structure is subjected to unidirectional harmonic ground motion. The equation of motion is as follows:

$$\mathrm{X}\left(t\right)={\mathrm{X}}_0\mathrm{s}\mathrm{i}\mathrm{n}\left(2\mathsf{\pi }{\mathsf{\omega }}_{d}t\right)$$

(10)

where $\mathrm{X}\left(t\right)$ is the displacement at time $t$. ${\mathrm{X}}_0$ represents the amplitude of the ground motion. ${\mathsf{\omega }}_d$ is the forcing frequency for excitation. The driving frequency used in the verification model was 0.9 Hz and the displacement amplitude of the tank structure was 5.4 mm. Figure 7 illustrates the variation in free surface sloshing height (${\mathrm{h}}_{\mathrm{s}})$ over time on the left side of the tank structure. A time-history line graph of free-surface elevation indicates that the amplitude of fluid sloshing progressively increases over time. The results obtained through the CEL method in this study aligned closely with those of Tippmann et al. [48]. While they did not provide CFD results for the 0.9 Hz case, their CFD and numerical simulation data at 1.09 Hz, which display a sloshing pattern closely resembling that at 0.9 Hz, show excellent agreement. The CFD results are slightly lower at five critical moments, specifically by 12.8%, 10.7%, 20.6%, 14.3%, and 28.1% at 2.88 s, 3.66 s, 3.84 s, 4.31 s, and 4.78 s, respectively. Sloshing wave elevations from CEL resulting at five corresponding extreme value time instants under 0.9 Hz excitation (2.62 s, 3.24 s, 3.70 s, 4.19 s, and 4.75 s) were extracted. The deviations relative to Tippmann et al.’s simulation results at these instants were 14.7%, −7.1%, 1.6%, 8.6%, and 0.9%, respectively. These findings demonstrate the CEL approach can accurately analyze free surface sloshing behavior and provides a credible alternative to traditional CFD methods for large-scale FSI problems.

3. Fluid Sloshing Analysis

3.1. Fluid Sloshing in Structures During Earthquakes

In large underground water tanks, fluid sloshing and wave propagation are influenced by the interaction between external walls and partition walls, unlike in simple structures. This interaction exerts hydrodynamic pressure on the diversion walls, which, in turn, alters the fluid’s sloshing behavior. Figure 8 illustrates the sloshing of the fluid at various moments during seismic excitation. Figure 8a displays minor sloshing that occurs 0.2 s after the seismic input. With continued seismic excitation, fluid sloshing intensity and height vary over time. Certain regions experience significant sloshing, causing the fluid height to increase until it contacts the structure roof. Figure 8b shows intense sloshing at 3.05 s. Despite the complex seismic inputs and structural environment, localized sloshing tendencies can still be analyzed. Figure 8c shows a cross-sectional view of the fluid sloshing near the right side of diversion wall No. 13, where the fluid level reaches 6 m, coming into contact with the structure roof.

3.2. Analysis of Fluid Sloshing via Fluid Velocity Field and Dynamic Pressure

The variation of fluid velocity over time offers insights into violent sloshing behavior. Figure 9 shows the velocity field in the structure at 2.80 s and 3.05 s. In Figure 9a, localized fluid velocity increases significantly due to seismic input and fluid–structure interaction. Figure 9b demonstrates upward fluid motion, with the fluid sloshing violently and reaching the roof. Figure 10 shows sectional views of the fluid velocity distribution from 2.8 s to 3.8 s. During this period, fluid between the diversion walls No. 4 and No. 8 exhibits intense sloshing due to the combined effects of seismic excitation and fluid–structure interaction. In Figure 10a, seismic loads increase the fluid velocity, causing it to flow towards the nearby diversion wall. The confined space leads to fluid accumulation and a rising liquid level. The fluid velocity then shifts upward in Figure 10b. In Figure 10c, the fluid level rises until it contacts the tank roof, where the direction shifts horizontally. Figure 10d,e show that the fluid velocity decreases and turns downward after impacting the roof. Finally, the fluid level subsides, and the intense sloshing ends in Figure 10f.

The fluid velocity is dependent on both position and time. Figure 11 presents the velocity–time-history and von Mises stress–time-history curves for three points from different positions. Locations of P1, P2, and P3 are shown in Figure 9a. The curves indicate that seismic loads and fluid–structure interaction initially create significant horizontal fluid velocities, which subsequently turn upward and decrease upon contact with the roof. Hydrodynamic pressure similarly fluctuates with changes in fluid speed, showing a positive correlation. The peak hydrodynamic pressures were found to vary temporally with the progression of sloshing, reaching approximately 0.83 MPa at 2.8 s and 0.31 MPa at 3.15 s. The design pressure estimates prescribed by Eurocode 8 Part 4 typically range from 0.2 to 0.5 MPa for large concrete tanks under moderate seismic excitation. The values are reasonably consistent with, or slightly exceed the estimated values. The elevated peak pressures observed can be attributed to the fully nonlinear fluid–structure interaction captured by the CEL method, which accounts for dynamic amplification, localized wave impacts, and internal structural discontinuities. Moreover, the pressure outputs from the simulation represent instantaneous localized peaks, while code-based estimates reflect smoothed envelope values intended for global design. These findings indicate that code provisions are generally conservative but may underestimate localized pressure amplification in tanks with complex internal configurations. Figure 12 shows the hydrodynamic von Mises stress distribution. The highest local hydrodynamic pressure is at position P3, which corresponds to the peaks of both the fluid velocity curve and the hydrodynamic pressure curve shown in Figure 11. The significant differences in the velocity–time-history curves of the three points are due to their different positions. To study the relationship between fluid velocity trend and position, velocity–time-history curves were obtained from different positions and are shown in Figure 13. Figure 13a illustrates the velocity data collected at seven locations perpendicular to the diversion wall’s direction. Each curve is named after the number of the diversion wall on the right side of the corresponding position. The curves reveal more significant velocity fluctuations at the edges than at the structure center, where the fluid velocity is relatively small. This indicates that both the fluid sloshing and the seismic response of the structure are more pronounced near the edges. Figure 13b illustrates the fluid velocity at various locations along the diversion wall for comparison. Each curve is named based on the left diversion wall number and the specific position near the wall (a for gaps, b for mid-wall positions) where the data is collected. Figure 13b shows that the fluid velocity near gaps is typically higher than at points near the middle of the diversion wall. The fluid velocity gradually increases from the center of the structure to the surrounding area.

The fluid velocity distribution is assessed at three vertical positions: up, middle, and down. Figure 13c displays the fluid velocity distribution in the vertical direction at the diversion wall gap, with the upper fluid regions showing higher velocities and variability than the lower regions. This trend aligns with Haroun–Housner’s flexible wall model, but due to intense sloshing, areas of the structure bottom are intermittently exposed, excluding impulse components [49]. Figure 13d shows the vertical velocity distribution at the middle position of the diversion wall, with similar trends across the three curves over time.

The change in the kinetic energy of the tank structure system reflects the overall sloshing amplitude of the fluid within the structure. Figure 14 shows the overall kinetic energy–time-history curve of the tank structure at 60% water storage capacity and the empty tank structure, and the seismic acceleration–time-history and velocity–time-history curves are added for comparison. Despite being subjected to the same seismic input, the tank structure filled with 60% water exhibited significantly higher overall kinetic energy than the empty tank structure. For the empty tank structure, the overall kinetic energy remains stable despite minor fluctuations, while the overall kinetic energy of the tank structure containing water tends to increase over time. The changes in the kinetic energy of the tank structure without water closely resemble the ground motion velocity curve. While some fluctuations in the kinetic energy of the tank structure containing water resemble seismic velocity waves, their overall trends differ. This deviation results from the fluid’s continued movement within the structure, which sustains the overall kinetic energy even when the seismic velocity subsides. During the time interval of 4.2 s to 5.8 s in Figure 14a, the earthquake velocity experiences significant changes; however, the overall kinetic energy curve of the structure remains stable. Each kinetic energy peak corresponds to a significant rise in fluid velocity within some areas of the tank, with the highest peak at 2.7 s corresponding to the first peak of fluid velocity at position P3 in Figure 11a. Each kinetic energy peak follows intense local fluid motion, and the peak heights reflect the severity of sloshing. Based on Figure 14 and the analysis of intense fluid sloshing, it can be inferred that the rise in fluid level lags the kinetic energy peaks by approximately 0.1 s to 0.3 s. In Figure 14c,d, the kinetic energy curve of the tank structure with water show phase lag and decoupled dynamic behavior compared with the ground motion acceleration curve.

4. Fluid–Structure Interaction Analysis

4.1. Interaction of Tank Structure and Fluid During Earthquakes

During an earthquake, underground water tank structures are subjected to seismic and hydrodynamic loads. The fluid’s sloshing imposes uneven hydrodynamic loads on the roof, floor, columns, external walls, and diversion walls of the tank structure. Intense fluid sloshing can significantly increase hydrodynamic loads in specific regions, potentially resulting in structural overload and damage. The increased hydrodynamic pressures in local areas, as shown in Figure 12, lead to higher loads on nearby structures. Based on the hydrodynamic pressure curves in Figure 10 and the fluid velocity field in Figure 10, the seismic response of the structure at 2.9 s was analyzed to assess the impact of fluid on the seismic performance of the structure. At 2.9 s, two areas exhibit violent fluid sloshing within the structure, making this moment representative. Figure 15a,b show the stress distribution in the tank structure roof and interior due to intense local fluid sloshing.

In Figure 15a, the upward impact of the fluid near the gap of the diversion wall no. 13 causes the ceiling stress to reach 19.13 MPa, which exceeds the structural limit load and results in deformation and damage to the structure. Figure 16 illustrates the distribution of relative deformation across different regions of the tank structure. This fluid impact process results in severe damage to the ceiling of the structure, with maximum deformation reaching 515.6 mm in a localized area, as illustrated in Figure 17. In this study, the CDP model was employed to simulate the cyclic degradation of concrete, and the plasticity parameters for the reinforcing steel bars were incorporated to consider the yielding and strain hardening effects. However, due to the use of shell elements for modeling the tank structure, certain limitations exist. A shell element possesses only surface displacement degrees of freedom and lacks a volume dimension, which prevents the structure from fully capturing actual crack openings or localized collapse mechanisms. Despite this limitation, the structural deformation observed in this study effectively reflects the seismic responses of various areas of the water tank structure. After substantial damage occurred at the ceiling of the tank at 2.8 s, the surrounding region, identified as a weak area, also experienced progressive damage propagation due to continued seismic excitation.

There are deformations and damages to the walls and columns of the water tank structure. Longitudinal members experience continuous hydrodynamic pressure, unlike the ceiling, which faces intermittent fluid impacts. Additionally, components below the waterline are subject to hydrostatic pressure. The excitation of longitudinal components primarily causes fluid sloshing during earthquakes. These components push the fluid, which in turn causes the components also to bear the hydrodynamic and hydrostatic load. At 2.9 s, intense fluid motion occurs between diversion walls no. 12 and no. 13. Figure 15a illustrates the stress distribution along the longitudinal members, with the stress on diversion wall no. 13 peaking at 21.78 MPa, which resulted in damage. Figure 16b illustrates that the maximum relative deformation of this wall is 432.3 mm.

Additionally, the longitudinal members connected to the damaged ceiling area experienced deformation and damage, as illustrated in Figure 18, with maximum deformations measuring 116 mm and 278 mm, respectively. Earthquakes and fluid–structure interactions can lead to uneven and unpredictable loading, causing damage to the tank structure. Once such damage occurs, the tank structure may lose its functionality, which can impair the operations of the facility.

This section investigates damage distribution to the tank structure after the earthquake. Figure 19 presents the tensile damage distribution of the structure, with a maximum tensile damage value of approximately 0.967. The longitudinal side of the tank structure, including the roof, walls, and columns, sustained severe damage. As the fluid flow space near the gaps in the diversion wall increases, violent fluid sloshing occurs near these areas, causing severe and large-scale damage to the ceiling and surrounding longitudinal components. In contrast, the central region of the tank structure shows less severe damage, limited to column–roof joints and specific diversion wall and column areas. Figure 20 displays compressive damage distribution, with a maximum value of approximately 0.996. The distribution pattern resembles tensile damage; however, the areas affected by compressive damage are smaller.

The maximum tensile and compressive damage indices obtained from the CDP model reach values of approximately 0.97 and 0.99, respectively. These high damage indices reflect severe material degradation at certain integration points. In the CDP formulation, such values indicate significant stiffness degradation in localized areas caused by microcracking during tension or crushing under compression; however, they do not necessarily imply complete structural failure or loss of load-carrying capacity of the component or system. In reinforced concrete structures, the residual load resistance is often maintained through reinforcement action and redistribution of internal forces. The overall deformation patterns and stress distributions shown in Figure 15 and Figure 16 confirm that there is severe local damage to the tank structure, while the overall structural integrity remains preserved. Figure 21 shows tensile and compressive damage distribution curves for external walls perpendicular to the diversion walls, with maximum damage near the edges and minimum in the center. This finding is consistent with the results in Figure 13a,b. Curves also show localized fluctuations near the diversion wall connections due to differing stiffness in these areas.

4.2. Role of Fluid in Structural Damage

A comparative analysis of the tank structure without water, subjected to the same seismic input, was performed to assess the impact of fluid on structural damage. Figure 22 shows stress and relative deformation for the empty tank structure. Without fluid, stress and deformation distributions are more uniform, in contrast to the extreme localized stresses observed with fluid. Stress concentration mainly occurs at the ceiling–column connection area and columns. Maximum relative deformation occurs on external walls perpendicular to the seismic direction, where stiffness is lower, leading to higher vulnerability. Without hydrodynamic pressure, the peak stress and deformation in the empty tank structure are significantly lower than in the tank structure with water. To quantitatively assess the influence of fluid–structure interaction on the seismic response of the large water tank structure, dynamic amplification factors (DAFs) were calculated for both structural deformations and von Mises stresses. The DAF is defined as the ratio of the peak structural response under water-storage conditions to that under empty conditions. Results show that the presence of fluid significantly amplifies structural responses. The peak relative deformations increased from 0.21 mm (empty tank) to 515.60 mm (60% water-filled), yielding a DAF of approximately 2455.23. For von Mises stress, the peak value increased from 1.45 MPa to 21.78 MPa, corresponding to a DAF of 15.02. These results indicate that FSI markedly enhances both global deformation and stress demand, fundamentally altering the dynamic behavior and damage evolution of the tank structure under seismic excitation.

Figure 23 presents tensile and compressive damage for the empty tank structure. Due to the absence of dynamic water pressure, the compressive damage to the pool structure is relatively minor, with the maximum value being only about 0.0051. Tensile damage peaks at approximately 0.28, concentrated at the ceiling–column connections. In summary, the hydrodynamic load is a primary cause of damage to the tank structure with 60% water storage capacity. When the geometry of the water tank changes (e.g., in terms of aspect ratio or overall shape) or the storage level increases (e.g., to 80–100%), the amplification of structural response induced by fluid–structure interaction is expected to remain qualitatively valid. However, such changes may lead to significant alterations in sloshing modes, hydrodynamic pressure distributions, and local damage patterns. Higher water depth is likely to enhance hydrodynamic pressures, thereby imposing increased demands on the structural load-bearing capacity.

5. Conclusions

This study employed the CEL method to investigate the seismic response of a large underground water tank structure with fluid–structure interaction. Fluid sloshing within the structure was analyzed in detail using fluid velocity, stress components, and total kinetic energy. Structural stress, deformation, and damage were studied to gain insights into fluid–structure interactions during seismic events. By comparing the differences in the seismic responses of the tank structure with and without water, the roles of fluid load and seismic load in the damage to the pool structure were investigated. The following conclusions were derived:

(1) Under seismic load, the fluid sloshing height can reach the full height of the tank structure, impacting the ceiling. Although seismic input excites fluid sloshing, the fluid motion amplitude does not strictly align with seismic fluctuations, indicating substantial fluid motion even during weak seismic intervals.

(2) The fluid sloshing amplitude along the longitudinal sides of the tank structure exceeds that in the central regions. The fluid sloshing amplitude is higher near diversion wall gaps due to the larger sloshing space, subjecting the surrounding components to greater hydrodynamic loads than those near the diversion wall centers.

(3) During earthquakes, significant local damage mainly occurs at the ceiling corners of the tank structure. Significant damage to longitudinal components (i.e., columns, external walls, and diversion walls) appears along the longitudinal sides and near diversion wall gaps.

(4) The stress and damage to the empty tank structure are uniformly distributed and relatively minor, with maximum damage occurring at the connections between the ceiling and columns. Therefore, fluid–structure interaction is the primary cause of damage to the water tank structure.

(5) This study has some limitations that remain to be investigated. First, the analysis was conducted under a fixed water storage condition (60% capacity), which may not fully capture the variability in fluid–structure interaction and sloshing behavior at different water levels. Second, the seismic excitation was limited to a single unidirectional ground motion record, without consideration of multi-directional loading or variations in ground motion characteristics. These factors may influence the structural response and damage mechanisms in practical scenarios. Future work is warranted to explore a broader range of water storage scenarios and seismic inputs to further generalize the findings.