Liquid in a rigid tank

Liquid in a rigid tank

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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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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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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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... 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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