Fluid pressure on unanchored rigid flat-bottom cylindrical tanks due to uplift motion and its approximation

2009 ◽  
Vol 31 (11) ◽  
pp. 2598-2606 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Yoshinori Ando ◽  
Teruhiro Nakashima
Keyword(s):  
Fluid Pressure ◽  
Flat Bottom ◽  
Cylindrical Tanks

Fluid Pressures on Unanchored Rigid Flat-Bottom Cylindrical Tanks Under Action of Uplifting Acceleration

2009 ◽  
Vol 132 (1) ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Yoshinori Ando
Keyword(s):  
Velocity Potential ◽  
Fluid Velocity ◽  
Fluid Pressure ◽  
Angular Acceleration ◽  
Bottom Plate ◽  
Center Line ◽  
Cylindrical Tank ◽  
Cylindrical Coordinates ◽  
Flat Bottom ◽  
Cylindrical Tanks

To protect flat-bottom cylindrical tanks against severe damage from uplift motion, accurate evaluation of accompanying fluid pressures is indispensable. This paper presents a mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank in the same manner as the procedure outlined and discussed previously by the authors (Taniguchi, T., and Ando, Y., 2010, “Fluid Pressures on Unanchored Rigid Rectangular Tanks Under Action of Uplifting Acceleration,” ASME J. Pressure Vessel Technol., 132(1), p. 011801). With perfect fluid and velocity potential assumed, the Laplace equation in cylindrical coordinates gives a continuity equation, while fluid velocity imparted by the displacement (and its time derivatives) of the shell and bottom plate of the tank defines boundary conditions. The velocity potential is solved with the Fourier–Bessel expansion, and its derivative, with respect to time, gives the fluid pressure at an arbitrary point inside the tank. In practice, designers have to calculate the fluid pressure on the tank whose perimeter of the bottom plate lifts off the ground like a crescent in plan view. However, the asymmetric boundary condition given by the fluid velocity imparted by the deformation of the crescent-like uplift region at the bottom cannot be expressed properly in cylindrical coordinates. This paper examines applicability of a slice model, which is a rigid rectangular tank with a unit depth vertically sliced out of a rigid flat-bottom cylindrical tank with a certain deviation from (in parallel to) the center line of the tank. A mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank accompanying the angular acceleration acting on the pivoting bottom edge of the tank is given by an explicit function of a dimensional variable of the tank, but with Fourier series. It well converges with a few first terms of the Fourier series and accurately calculates the values of the fluid pressure on the tank. In addition, the slice model approximates well the values of the fluid pressure on the shell of a rigid flat-bottom cylindrical tank for any points deviated from the center line. For the designers’ convenience, diagrams that depict the fluid pressures normalized by the maximum tangential acceleration given by the product of the angular acceleration and diagonals of the tank are also presented. The proposed mathematical and graphical methods are cost effective and aid in the design of the flat-bottom cylindrical tanks that allow the uplifting of the bottom plate.

Approximation of Fluid Pressure on the Cylindrical Tanks in Rock With the Crescent-Like Uplift Part in the Bottom Plate by Radially Sliced Tank Model

2013 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Takumi Shirasaki
Keyword(s):  
Fluid Pressure ◽  
Bottom Plate ◽  
Mathematical Treatment ◽  
Cylindrical Tank ◽  
Flat Bottom ◽  
Tank Model ◽  
Rectangular Tank ◽  
Tank Bottom ◽  
Cylindrical Tanks ◽  
Earthquake Effects

Flat-bottom cylindrical shell tanks may rock and have a crescent-like uplift part in the bottom plate at the event of a severe earthquake. Effects of the deformed tank bottom plate on the fluid pressure on the cylindrical tank have not been, however, quantified yet. Since the crescent-like uplift part appears eccentrically on the periphery of the tank bottom plate, its mathematical treatment would be troublesome. Regarding a cylindrical tank as a set of pieces of a thin rectangular tank with a deformed bottom plate that correspond radially sliced parts of the cylindrical tank with the crescent-like uplift part in the bottom plate, this paper defines the fluid pressure on the cylindrical tank by calculating that on the rectangular tank. For designer’s convenience, the fluid pressure computed are normalized and depicted in accordance with the aspect of the cylindrical tank and the uplift ratio of the tank bottom plate.

Effective Mass of Fluid for Rocking Motion of Flat-Bottom Cylindrical Tanks

2009 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Toru Segawa
Keyword(s):  
Effective Mass ◽  
Fluid Pressure ◽  
Angular Acceleration ◽  
Inertia Force ◽  
Bottom Edge ◽  
Flat Bottom ◽  
Effective Moment ◽  
Rocking Motion ◽  
Tank Bottom ◽  
Cylindrical Tanks

In analyzing the rocking motion of the flat-bottom cylindrical tanks subjected to severe earthquakes, the effective mass of fluid for the rocking motion and its moment inertia around the pivoting bottom edge of the tank would be indispensable dynamical properties, because they couples the fluid-shell interaction motion, the so-called bulging motion, with the rocking motion. This paper quantifies them based on the equilibrium of the fluid pressure and inertia force accompanying the angular acceleration acting on the pivoting bottom edge of the tank. Employing a general mathematical solution for the fluid pressure that can calculate either fully or partially uplifted tank bottom, this paper presents mathematical formulae of the effective mass of fluid for the rocking motion and its moment inertia. These quantities are given by an explicit function of dimensional variables of the tank but with Fourier series. For designer’s convenience, the effective moment inertia and effective mass of fluid for the rocking motion and its center of gravity from the pivoting bottom edge are normalized accordingly and are depicted on diagrams.

Fluid Pressure on Rectangular Tank Consisting of Rigid Side Walls and Rectilinearly Deforming Bottom Plate Due to Uplift Motion

2008 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Toru Segawa
Keyword(s):  
Velocity Potential ◽  
Fluid Velocity ◽  
Fluid Pressure ◽  
Angular Acceleration ◽  
Parabolic Partial Differential Equation ◽  
Bottom Plate ◽  
Flat Bottom ◽  
Fourier Cosine ◽  
Rectangular Tank ◽  
Rigid Walls

Although the uplift motion of flat bottom cylindrical shell tanks has been considered to contribute toward various damage to tanks, the mechanics were not fully understood. As well as computing uplift displacement of the tanks, accurate prediction of fluid pressure accompanied with uplift motion is indispensable to prevent the tanks from severe damage. This paper mathematically derives the fluid pressure on a rectangular tank with unit depth consisting of rigid walls and rectilinearly deforming bottom plate accompanied with the uplift motion. Assuming perfect fluid and velocity potential, the continuity equation is given by Laplace equation. The fluid velocity imparted by motion of rigid walls, immobile bottom plate and rectilinearly deformed bottom plate of tank accompanied with uplift of the tank constitutes boundary conditions. Since this problem is set as the parabolic partial differential equation of Neumann problem, the velocity potential is solved with Fourier-cosine expansion. Derivatives of the velocity potential with respect to time give the fluid pressure at arbitrary point inside the tank. The proposed mathematical solution well converges with a first few terms of Fourier series. Diagrams that depict the fluid pressure normalized by product of angular acceleration and diagonals of the tank are presented.

Fluid Pressures on Unanchored Rigid Rectangular Tanks Under Action of Uplifting Acceleration

2009 ◽  
Vol 132 (1) ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Yoshinori Ando
Keyword(s):  
Cylindrical Shell ◽  
Fourier Series ◽  
Velocity Potential ◽  
Fluid Pressure ◽  
Angular Acceleration ◽  
Parabolic Partial Differential Equation ◽  
Bottom Edge ◽  
Flat Bottom ◽  
Mathematical Solution ◽  
Rectangular Tank

Although uplift motion of flat-bottom cylindrical shell tanks has been considered to contribute toward various damages to the tanks, the mechanics were not fully understood. As well as uplift displacement of the tanks, fluid pressure accompanying the uplift motion of the tanks may play an important role in the cause of the damage. An accurate estimate of the fluid pressure induced by the uplift motion of the tanks is indispensable in protecting the tanks against destructive earthquakes. As a first step of a series of research, this study mathematically derives the fluid pressure on a rigid rectangular tank with a unit depth accompanying angular acceleration, which acts on a pivoting bottom edge. The rectangular tank employed herein is equivalent to a thin slice of the central vertical cross section of a rigid flat-bottom cylindrical shell tank. Assuming a perfect fluid and velocity potential, a continuity equation is given by the Laplace equation in Cartesian coordinates. The fluid velocities accompanying the motions of the walls and bottom plate constitute the boundary conditions. Since this problem is set as a parabolic partial differential equation of the Neumann problem, the velocity potential is solved with the Fourier-cosine expansion. The derivative of the velocity potential with respect to time gives the fluid pressure at an arbitrary point inside the tank. A mathematical solution for evaluating the fluid pressure accompanying the angular acceleration acting on the pivoting bottom edge of the tank is given by an explicit function of a dimensional variable of the tank, but with the Fourier series. The proposed mathematical solution well converges with a few first terms of the Fourier series. Values of the fluid pressure computed by the explicit finite element (FE) analysis well agrees with those by the proposed mathematical solution. For the designers’ convenience, diagrams that depict the fluid pressures normalized by the maximum tangential acceleration given by the product of the angular acceleration and diagonals of the tank are also presented. Consequently, the mathematical solution given by the Fourier series converges easily and provides accurate evaluation of the fluid pressures on a rigid rectangular tank accompanying the angular acceleration acting on the pivoting bottom edge. Irregularity in the fluid pressure distribution increases as the tank becomes taller.

Fluid Pressure on Rectangular Rigid Tanks Due to Uplift Motion

2007 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Yoshinori Ando
Keyword(s):  
Cylindrical Shell ◽  
Velocity Potential ◽  
Fluid Velocity ◽  
Fluid Pressure ◽  
Design Procedure ◽  
Parabolic Partial Differential Equation ◽  
Bottom Plate ◽  
Flat Bottom ◽  
Side Walls ◽  
Unit Depth

This paper mathematically derives fluid pressure on a rectangular rigid tank with unit depth, which models a section of a center part of a flat-bottom cylindrical shell tank, accompanied with uplift motion. Employing boundary conditions consisting of fluid velocity imparted by motion of the side walls and bottom plate of the tank along with uplifting, the equation of continuity of fluid given by the Laplace equation is solved as the parabolic partial differential equation of Neumann problem. The fluid pressure is given by a function of the velocity potential. Comparison of mathematical results with numerical ones based on explicit FE analysis corroborates its accuracy and applicability on design procedure of flat-bottom cylindrical shell tanks.

scholarly journals Modeling Problem of Equivalent Mechanical Models of a Sloshing Fluid

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Yu-Chun Li ◽  
Hong-Liang Gou
Keyword(s):  
Traditional Method ◽  
Equivalent Model ◽  
Flat Bottom ◽  
Modeling Methods ◽  
Equivalent Models ◽  
Modified Method ◽  
Equivalent Mechanical Model ◽  
Mechanical Models ◽  
Correct Location ◽  
Cylindrical Tanks

Because the defects in the existing modeling methods for the equivalent mechanical model of a sloshing fluid have led to incorrect or inaccurate results in the existing equivalent models, this paper discusses three different modeling methods for the equivalent models: the traditional method, Housner’s method, and the modified method. The equivalent models obtained by the three methods are, respectively, presented and compared with each other for a liquid in rectangular and upright cylindrical tanks. The results show that the traditional method cannot provide the correct location expressions of the equivalent masses because the two types of different excitations are simultaneously used in one equivalent model. An equivalent model is exclusively applicable to a certain excitation (a translational excitation in a certain direction or a rotational excitation about a certain axis). Housner’s method is based on physical intuition, instead of fluid dynamics theory, therefore the calculation precision of Housner’s solution is not satisfactory. Housner’s method is only suitable for vertical tanks with a flat bottom subjected to a horizontal excitation. Based on a conceptual mistake in the traditional method, the concept of the equivalent model is reclarified, and the modified equivalence method is therefore suggested. A supplementary solution for the equivalent model in a cylindrical tank is presented. The correct models can be acquired using the modified equivalence method, which is applicable to tanks of arbitrary shape.

Mathematical Solution for Evaluating Fluid Pressure on Rigid Flat-Bottom Cylindrical Shell Tanks Due to Uplift Motion

2007 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Yoshinori Ando
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Fluid Pressure ◽  
Cylindrical Coordinates ◽  
Case Scenario ◽  
Flat Bottom ◽  
Actual Case ◽  
Mathematical Solution ◽  
Unit Depth ◽  
Rigid Shell

Solving the equation of continuity given by the Laplace equation on the cylindrical coordinates, the mathematical solution for evaluating fluid pressure on rigid flat-bottom cylindrical shell tanks is derived. However, since difficulty of applying asymmetric boundary conditions on the cylindrical coordinates which trace realistic uplift motion of tanks restricts application of the preceding mathematical solution to the actual case scenario, the slice model consisting of a rigid rectangular tank with unit depth on a parallel plane to the uplift motion and offsetting from a center of tanks is developed. Comparison reveals applicability of the slice model for evaluating the fluid pressure on the rigid flat-bottom cylindrical rigid shell tanks.

Approximation of Uplift of Flat-Bottom Cylindrical Tanks and Effects of Out-of-Plane Deformation of Cylindrical Shell on it

2015 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Teruhiro Nakashima ◽  
Daisuke Okui
Keyword(s):  
Cylindrical Shell ◽  
Plane Deformation ◽  
Angular Acceleration ◽  
Bottom Plate ◽  
Flat Bottom ◽  
Out Of Plane ◽  
Prone Area ◽  
Analytical Accuracy ◽  
Tank Bottom ◽  
Cylindrical Tanks

For the unanchored flat-bottom cylindrical tanks located in the seismic prone area, uplift of the tank bottom plate is inevitable. Besides the work of Nakashima, effects of out-of-plane deformation of the cylindrical shell on uplift of the tank bottom plate have been paid little attention. In analyzing uplift of the tank bottom plate, for design purpose in particular, its effects should be included. First, employing a cylindrical shell tanks with multistage rigid or elastic stiffeners, their uplift responses to the horizontal sinusoidal base acceleration are compared to highlight effects of out-of-plane deformation on uplift of the tank bottom plate. Next, employing the numerical results of the cylindrical shell tank with multistage rigid stiffeners, analytical accuracy of the simplified calculation for evaluating the angular acceleration accompanying the tank rock motion is examined.

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