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A hierarchical finite element is developed for the free vibration analysis of a liquid in a rigid cylindrical tank with or without a free surface. It is a hierarchical quadrilateral element and has the advantage that the hierarchical mode number is allowed to vary independently of direction. Liquid behavior in tanks with large aspect ratios can the...

## Contexts in source publication

**Context 1**

... order to verify its effectiveness, the hierarchical liquid element is used for the free vibration of a liquid with respectively a volumetric mass and a bulk modulus equals to 10 3 Kg/m3 and 2.068 10 9 KPa contained firstly in a rigid circular cylindrical tank of a radius equal to 6.48m for a height equal to 6.24m and secondly in a rigid circular cylindrical tank of a radius equal to 1.88m for a height equal to 4.75m (see figure 5). The results are compared with those obtained by a standard FEM program where the axisym- metric liquid element with four nodes is used and with the theoretical frequencies of a liquid in a circular cylindrical storage tanks given by (Blevins, 1980) ...

**Context 2**

... g is the gravity acceleration, R is the radius of the cylinder, H is the liquid height and εn is the nth root of J(ε) Bessel function. For the first example ( Fig. 5.a), the comparison is carried out for one and two hierarchical ele- ments with radial and axial hierarchical mode numbers varying from 2 to 8. The number of the axisymmetric liquid finite elements used in the standard FEM program is equal to 4212. The rigid tank of the second example ( Fig. 5.b) being long, the comparison is carried out ...

**Context 3**

... root of J(ε) Bessel function. For the first example ( Fig. 5.a), the comparison is carried out for one and two hierarchical ele- ments with radial and axial hierarchical mode numbers varying from 2 to 8. The number of the axisymmetric liquid finite elements used in the standard FEM program is equal to 4212. The rigid tank of the second example ( Fig. 5.b) being long, the comparison is carried out for only one hierar- chical element with radial hierarchical mode number varying from 2 to 4 and axial hierarchical mode number varying from 2 to 8. The number of the axisymmetric liquid finite elements used in the standard FEM is equal to 2408. Table 2 shows the convergence study for the first ...

**Context 4**

... being long, the comparison is carried out for only one hierar- chical element with radial hierarchical mode number varying from 2 to 4 and axial hierarchical mode number varying from 2 to 8. The number of the axisymmetric liquid finite elements used in the standard FEM is equal to 2408. Table 2 shows the convergence study for the first example ( Fig. 5.a) of the first six modes with an increasing of the two hierarchical mode numbers p and q following respectively the radius and the axis directions for one and two elements. For the two idealization (Table 2), one and two ele- ments, an accuracy of two digits after the comma is reached for the first mode with p=q=4. For the second mode, ...

**Context 5**

... for the first mode with p=q=4. For the second mode, an accuracy of two digits is reached for p=q=8 with one element and for p=q=6 with two elements. For the modes 3, 4, 5; the convergence is reached for p=q=8 for the two idealization, but the matrices size is smaller if one increases p and q rather than the element number. For the second example, (Fig. 5.b) where the rigid tank is long, the convergence of the first six modes is given in table 3. The liquid is idealized by only one element. The height of the liquid being greater than the tank radius, the increasing of the two hierarchical mode numbers p and q isn't the same. More the size is greater; more the increase of the hierarchical ...

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... The use of FEM 12,13 was proposed as a solution to the problems of simulation of the ground-supported liquid tanks exposed to earthquakes. The fluid-structure interaction (FSI), soil-structure interaction (SSI), and sloshing of fluid are fundamental factors in the dynamic analysis of this kind of structure. ...

... The impulsive flexible pressure component emerges only in the flexible tank shell (e.g., steel tank). 13 The cylindrical coordinate system r, z, θ is used with origin at the center of the tank bottom and with z vertical axis. R is the radius of liquid filling, and H is the original height of the free surface of liquid filling (see Figure 3). ...

During earthquakes, the liquid-filled storage tank generates hydrodynamic pressures, in addition to hydrostatic pressure, on the solid domain of the tank. The theoretical background of hydrodynamic pressure analysis, as well as the numerical simulation of the liquid-filled cylindrical steel reinforced concrete tank, is the focus of this paper. The finite element method (FEM) modeling, along with arbitrary Lagrangian–Eulerian and fluid–structure interactions formulation, is used for simulating the seismic response of cylindrical steel reinforced concrete liquid-filled tank, fixed to the rigid foundation. The Loma Prieta accelerogram is utilized for recording the seismic ground motion. In the numerical study, two states are observed: (1) static condition where only hydrostatic pressure acts and (2) seismic excitation where hydrodynamic pressure occurs. When exposed to an earthquake situation, the tank liquid gives the total pressure of the liquid domain. The dynamic analysis considers the pressure response of the liquid domain, as well as the stress response of the solid domain of the coupled system, that is, liquid-filled cylindrical steel reinforced concrete tank.

... A possible damages of storage facilities and potential leakage of liquids have financial and especially environmental far-reaching consequences [3,4]. The knowledge of the behaviour of the contained liquid, the knowledge of dynamic interaction problems between contained fluid and storage structure, the knowledge hydrodynamic effect of fluid domain on solid domain of storage structure, the knowledge of effect of fluid storage tank on sub-soil, the knowledge of frequency properties of fluid-storage structures are very important, and they play a decisive role by designing and building of the storage facilities/devices which are resistant to earthquakes [5][6][7][8][9]. ...

... The use of FEM [11,12] was proposed as a solution to the problems of simulation of the ground-supported liquid tanks exposed to earthquakes. The fluid-structure interaction (FSI), soil-structure interaction (SSI), and sloshing of fluid are fundamental factors in the dynamic analysis of this kind of structure. ...

... The convective pressure component satisfies the boundary conditions and the correct equilibrium condition at the free surface. The impulsive flexible pressure component emerges only in the flexible tank shell (e.g., steel tank) [12]. The cylindrical coordinate system: r, z, is used with origin at the centre of the tank bottom and with z vertical axis. ...

During earthquakes, the liquid-filled storage tank generates hydrodynamic pressures, in addition to hydrostatic pressure, on the solid domain of the tank. The theoretical background of hydrodynamic pressure analysis, as well as the numerical simulation of the liquid-filled cylindrical concrete tank, is the focus of this paper. The Finite Element Method (FEM) modeling, along with Arbitrary Lagrangian-Eulerian and Fluid-Structure Interactions formulation, are used for simulating the seismic response of cylindrical concrete liquid-filled tank, fixed to the rigid foundation. The Loma Prieta accelerogram is utilized for recording the seismic ground motion. In the numerical study, two states are observed: 1) static condition where only hydrostatic pressure acts, and 2) seismic excitation where hydrodynamic pressure occurs. When exposed to an earthquake situation, the tank liquid gives the total pressure of the liquid domain. The dynamic analysis considers the pressure response of the liquid domain, as well as the stress response of the solid domain of the coupled system, i.e., liquid-filled cylindrical concrete tank.

The response of liquid-storage tanks under seismic loading is a key research area. The study of free vibration characteristics is a prerequisite for understanding the dynamic behaviour of liquid tanks under seismic loading. Hence, the present study focuses on the analysis of different parameters that may influence the frequency characteristics of cylindrical liquid tanks. For this, a database of 560 three-dimensional cylindrical liquid tank models was developed by carrying out modal analysis using ANSYS Mechanical APDL software. The database was analysed with the aid of artificial neural networks and nonlinear regression analysis. Average deviations of 30% and 32% were observed for the impulsive frequency values estimated based on IS1893 (Part2):2014 and Eurocode-8, respectively, compared to the finite element results. Hence, modification coefficients were suggested with aspect ratio as the demarcating parameter and obtained a Pearson correlation coefficient of rXY > 0.9, between the predicted values of frequency and actual values. The predicted formulae reduced the deviations observed between the frequency estimated based on the codal expressions and those obtained from finite element analysis to 13% and 15%, corresponding to IS1893 (Part2):2014 and Eurocode-8, respectively.

Optimum safe design through numerically investigation and simulation of FSI due to seismic loading on acid tank with piping attachment is presented. A nonlinear FSI based on the FEM is performed on a full-scale 3D model. Investigations are supplemented by a CFD to simulate the fluid motion inside the tank using acceleration time history of Kocaeli earthquake, the response of the maximum stress, deformation, and displacement of rigidly restrained fixed and flexible tanks at different fill levels and thickness are evaluated. The results are compared and analyzed with design codes and the difference observed in hydrostatic pressure is less than 0.08%, and in maximum values of hydrodynamic pressure are less than 4.3%, 0.8%, and 1.5% at three fill level while the average difference in transient time history total pressure is less than 0.4%. Finally, the provision given in the design codes and response of parameters is computed and polynomial correlation is proposed with an accuracy of above 0.99 and average difference less than 5% in fixed tank and less than 2% in the flexible tank for designing a safe tank by analysis.

Assuming that an ideal liquid has irrotational, incompressible, and inviscid flows, a mathematical model is presented to efficiently and simply study liquid sloshing problems under longitudinal excitation in horizontal cylindrical containers with complex baffles. A semianalytical scaled boundary finite-element method (SBFEM) is combined with the zoning technique to solve the liquid sloshing problem. This method can significantly increase the efficiency and accuracy of the calculation using few nodes. Using scaled boundary coordinates with both radial and circumferential directions, the analytical solution in the radial direction can be obtained through approximation in the circumferential direction via a discretization technique similar to that used in the FEM. Thus, the entire calculation domain can be analyzed based on the problem boundary. Continued-fraction expansion is applied to build the eigenvalue problem, and the interior eigenvectors are solved by using asymptotic expansion in detail. Based on the previously mentioned decomposition and eigenvalue problem, the corresponding sloshing mass and motion equations are proposed by an efficient methodology. The simplicity and efficiency of SBFEM applied to sloshing problems with different baffles are obtained through numerical examples. This paper investigates the effects of the arrangement and length of different baffles and liquid fill levels on the sloshing frequencies, modes, and response. The conclusions illustrate that SBFEMcan easily and semianalytically achieve good results for complex sloshing problems with singularity and complex geometry by placing the scaling centers at the tip of the baffles with very few degrees of freedom.