Eight independent displacement modes of a four node liquid element 

Eight independent displacement modes of a four node liquid element 

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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...

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... matrices size is smaller if one increases p and q rather than the element number. Figure 3 shows the eight independent modes of a four node liquid element. The first six modes are constant-strain modes and the other two are bending modes. ...

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Citations

... 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). ...
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... 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. ...
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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.
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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.