Earthquake frequency content has a significant effect on sloshing wave amplitude and height in liquid storage tanks. In this paper, the finite element method had been used to obtain the three dimensional fluid-structure interaction response of the Rectangular tanks to access the sloshing interference effects at the tank corners under various seismic input motions with different frequency contents. The flexibility of the tank wall as well as the structural and fluid damping have been taken into account to obtain more reliable and realistic results. It has also been shown that the 3D sloshing interference may increase the total wave height significantly at the corners of the tanks compared to the values presented in the design codes, which shows the maximum sloshing wave with much lower values and at a different location. It has been finally shown that the 3D sloshing effects relates to the ratio of the width and the length of the tank.

Sloshing is a well-known phenomenon in liquid storage tanks subjected to base or body motions. Up to now the use of multiple vertical baffles for reducing the sloshing effects in tanks subjected to earthquake has not been taken into consideration so much. On the other hand, although some of the existing computer programs are able to model sloshing phenomenon with acceptable accuracy, the full dynamic analysis subjected to random excitations, such as earthquake induced motions, is very time consuming. In this paper a method is presented for reducing the analysis duration based on first, conducting several dynamic analysis cases by using ANSYS-CFX for Rectangular tanks of various dimensions, subjected to seismic excitations, and then, using neural network to create simple relationships between the dominant frequency and amplitude of the base excitations and the maximum level of liquid in the tank during the sloshing. The numerical modeling has been verified by using some existing experimental data, and several cases of time history analysis have been conducted to obtain the required numerical results for training a neural network. Finally, the predicted results of the neural network have been compared to those obtained by some other cases of analyses as control values.

Many liquid storage tanks around the world have affected by earthquakes. This structure can store dangerous chemical liquids. Hence dynamic behavior of ground supported Rectangular storage tanks is very important due to their applications in industrial facilities. In current research, the seismic behavior of two water storage Rectangular concrete tanks is examined. For this purpose, these tanks are modeled in FEM software for analyzing. These tanks are analyzed under four type of analysis: static, modal, response-spectrum and time-history analysis. Time history analysis can take all the nonlinear factors into the analysis, so it is used to estimate the exact amount of structural response. In time history analysis, earthquake accelerograms of Tabas, Kobe and Cape Mendocino have been applied to tanks. Finally, it is resulted that Displacement, base shear and wave height obtained from time history analysis are more than those of response spectrum analysis, indicating insufficiency of response spectrum analysis. In time history analysis, the maximum displacement is achieved in highest part of the tanks. It is due to the wave height which created in earthquake. By increasing in dimension, the wave height is also increased.

This paper presents an in-depth analytical approach to understanding the free vibration dynamics of compressible fluids within rigid-walled Rectangular tanks, emphasizing the significance of fluid-structure interaction (FSI). The study derives the governing partial differential equations and boundary conditions, providing exact solutions for two-dimensional and three-dimensional tank geometries. Through the calculation of natural frequencies using derived analytical expressions, the research examines the impact of varying tank dimensions on these frequencies via comprehensive sensitivity analyses.The introduction underscores the importance of understanding FSI in engineering fields like storage tanks, pipelines, and offshore platforms. It traces the historical development of research in this domain, highlighting key contributions from pioneers such as Westergaard, Housner, Jacobsen, Lemm, and others. The paper acknowledges the significant advancements made through analytical and numerical modeling techniques, as well as the growing role of machine learning in enhancing FSI simulation accuracy.The paper then presents the mathematical framework governing the behavior of compressible fluids within Rectangular tanks. The core principles of mass and momentum conservation are expressed through the continuity and Navier-Stokes equations, respectively. The relationship between stress and strain rate is defined by the constitutive equation, while the Reynolds Transport Theorem relates the rate of change of a quantity within a control volume to the flux across the control surface and the rate of change within the volume itself.For the three-dimensional tank analysis, the paper derives the governing Helmholtz equation in the frequency domain, representing the spatial variation of the fluid velocity field within the tank. The boundary conditions are specified, ensuring no flow normal to the tank boundaries, mass conservation within the tank, and no normal gradient of velocity at the boundaries. The method of separation of variables is employed to solve the partial differential equation, leading to a general solution for the velocity field.The sensitivity analysis section explores the effects of varying tank dimensions on the first five natural frequencies. Each subsection focuses on a specific dimension (length, height, or width), systematically investigating how changes in that dimension influence the natural frequencies while keeping the other two dimensions constant. The results are presented in tabular and graphical forms, revealing a consistent decrease in natural frequencies across all modes as the length, height, or width of the tank increases.The paper attributes this phenomenon to the fascinating interplay between the fluid's kinetic and potential energies. As the tank dimensions increase, the fluid has more space for deformation, reducing the restoring force acting on the fluid elements when compressed. This decrease in potential energy is identified as the primary contributor to the observed lowering of natural frequencies with increasing reservoir size.Furthermore, the paper introduces the Rayleigh-Ritz method, a work and energy-based approach to analyzing vibrating systems. By utilizing approximate shape functions to represent the displacement of the vibrating body, the method calculates the potential and kinetic energies, leading to the derivation of fluid frequencies. The results obtained through the Rayleigh-Ritz method align well with the accurate method, validating its effectiveness in calculating fluid frequencies in reservoirs.The discussion section delves deeper into the theoretical underpinnings of the observed behavior. Analogies are drawn to familiar concepts, such as the sound produced by flutes of varying lengths and the behavior of waves in the ocean, to provide intuitive explanations for the observed trends. The paper emphasizes the significance of understanding this relationship between tank dimensions and resonance frequencies, as it offers engineers the ability to optimize tank designs for safe and efficient performance under various loading conditions, including seismic events.In conclusion, this study provides a comprehensive analytical framework for understanding the free vibration dynamics of compressible fluids in rigid-walled Rectangular tanks, considering the crucial effects of fluid-structure interaction. The derived analytical expressions, combined with the Rayleigh-Ritz method, offer a powerful tool for calculating natural frequencies and analyzing the impact of tank dimensions on these frequencies. The results not only validate the analytical model but also provide valuable insights for engineers working on seismic-resistant storage facilities, acoustically tuned containers, and other applications involving fluid-structure interaction phenomena.

In this paper, genetic algorithm has been used for liquid sloshing modeling in the Rectangular storage tanks. For this purpose, a numerical model was developed to model the sloshing phenomenon in the Rectangular storage tank with different widths and water depths exposed to the horizontal periodic sway motion with different amplitudes and angular frequencies. In the numerical model, coupled finite element and boundary element methods were used to solve the governing equations. The governing equations are the Laplace equation and the dynamic free surface boundary condition. In order to validate the model, a Rectangular tank with 0. 9 m width and 0. 6 m water depth was exposed to a horizontal periodic sway motion with 0. 002m amplitude and 5. 5 rad/sec angular frequency. Finally, the results of the numerical model were compared with those of Nakayama and Washizu. Then, the numerical model was used to model the sloshing phenomenon in the Rectangular storage tanks with different widths and water depths exposed to horizontal periodic sway motions with different amplitudes and angular frequencies. In the next step, a genetic algorithm method was used to model the sloshing phenomenon using the results of numerical model, and some relationships are presented to estimate the maximum free surface and the horizontal force is exerted on the tank perimeter. The correlation coefficient is used to evaluate the accuracy of the presented relations. The results show that genetic algorithm has good accuracy in predicting these two parameters. However, the accuracy of the model in the prediction of the maximum horizontal force exerted on the tank perimeter is more than that of prediction of the maximum free surface displacement. Because the correlation coefficient in the training and testing steps are 82 and 93 percent and 99 and 5 percent for these parameters, respectively. Furthermore, the percent relative errors of prediction of maximum free surface and horizontal force exerted on the tank perimeter are 11 and 5 percent, respectively. Therefore, the present model can be a power tool to model the sloshing phenomenon in the storage tanks.

One of the most important components of water supply systems is the liquid storage tanks. During an earthquake, the interaction of fluid and structure in the liquid storage tanks and the sloshing phenomenon has a significant effect on the response values of the structure. Regarding the importance of the effects of near-fault earthquakes and their effect on seismic behavior and structures loads, in this study, the sloshing height and vibration of 2D concrete Rectangular tanks under near-and far-field earthquakes was investigated using numerical methods. The effect of tank's dimensions, depth of water and ground motion characteristics on the maximum sloshing height was taken into account. Therefore, 9 tank models and 10 near-and far-field ground motion records were considered. The results indicated that the median values of maximum sloshing height in the near-field records are significantly higher than those related to far-field records. The average of increase of sloshing heights in tanks with lengths 20, 40 and 60 meters is 65, 77 and 100 percentage respectively. Also, with increase of fluid depth and tank width, the median of the maximum sloshing height increases and decreases respectively. In tanks, subjected to far-and near-field earthquakes, sloshing height had the highest correlation with Arias Intensity and PGV respectively. According to the results of this research, correction coefficients for the relations presented in the codes can be proposed to consider the effects of near-field earthquakes in calculating the maximum sloshing height.

1. Introduction: The dynamic response of liquid containers to the underground excitation has been studied intensively in recent years. In early investigations, the fluid response in Rectangular liquid storage tanks was represented by impulsive and convective components [1]. The fluid was assumed to be incompressible and the container was assumed to have rigid walls. The Housner’s model [1] has been adopted in most of the current codes and standards for calculating the hydrodynamic pressures in concrete tanks. Very strong earthquakes in the United States and Japan caused heavy damage to many liquid storage tanks.

Sloshing phenomenon is one of the complex problems in free surface flow phenomena. Numerical meshless methods as a new method can be used to solve this problem. In these methods, the lack of a mesh and complex elements for the domain of problems due to the change in geometry of the solution over time provides a lot of flexibility in solving numerical problems. In the previous researches, the sloshing problem in reservoirs was solved, using the Laplace equation with respect to the velocity potential, but the solution to this problem with pressure equations has not much considered; therefore, using the pressure equations and a suitable lagrangian time algorithm, generalized exponential basis function method has been developed for dynamic stimulation reservoirs. The approximation is solved, using a meshless method of generalized exponential basis functions and the entire domain of problem will discrete to a number of nodes and then with appropriate boundary conditions, the unknowns are approximated. In this study, linear and nonlinear examples have been solved under harmonic stimulation, in two-dimensional form of Rectangular cube tanks, and the results of them have been compared with the analysis solving methods, other numerical methods, and experimental data. The results show that the present method in two-dimensional mode is very noticeable compared with other available lagrangian methods because of accuracy in solving problem and spending time.