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.

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.

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.

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.

Effective Mass of Fluid for Rocking-Bulging Interaction of Rigid Rectangular Tank Whose Bottom Plate Rectilinearly Uplifts

2010 ◽  
Author(s):  
Tomoyo Taniguchi
Keyword(s):  
Effective Mass ◽  
Apparent Density ◽  
Total Mass ◽  
Fluid Pressure ◽  
Bottom Plate ◽  
Rotational Inertia ◽  
Cylindrical Tank ◽  
Effective Masses ◽  
Rocking Motion ◽  
Tank Bottom

In experimental and analytical studies of the rocking response of a circular cylindrical tank under the action of the purely horizontal and translational ground motion, the author analogically quantified the mass of fluid contributing to both bulging and rocking motion of the tank. It was called “the effective mass of fluid for the rocking-bulging interaction.” Its dynamical role in the rocking motion of the tank was thoroughly investigated. However, applying it to design process requires us to use its rigorous definition. To date, the fluid pressure on the tank induced by the impulsive (= bulging) motion and the rocking motion and their effective masses of fluid for each motion were mathematically defined, respectively. Therefore, this paper tries to define the effective mass of fluid for the rocking-bulging interaction based on the fluid pressure on the tank mathematically. The effective mass of fluid for the rocking-bulging interaction is understood as a part of the effective mass of fluid for the bulging motion that is also under the action of the rotational inertia. The influence of the rotational inertia on the effective mass of fluid for the bulging motion is measured by a ratio of the apparent density of fluid contributing to the rocking motion to the original density of fluid. The distribution of the apparent density of fluid contributing to the rocking-bulging interaction is drawn for the various aspects of tanks. The ratio of the effective mass of fluid for the rocking-bulging interaction to the total mass of fluid of the tank is given as the function of the aspect ratio of the tank and the ratio of the uplift width of the tank bottom.

Masses of Fluid Contributing to Rocking Motion of Cylindrical Tanks With Partial Uplift of Bottom Plate

2014 ◽  
Author(s):  
Tomoyo Taniguchi ◽  
Yukihiro Katayama
Keyword(s):  
Effective Mass ◽  
Bottom Plate ◽  
Rotational Axis ◽  
Cylindrical Tank ◽  
Effective Moment ◽  
Easy Calculation ◽  
Rocking Motion ◽  
Tank Bottom ◽  
Cylindrical Tanks ◽  
The Masses

Accurate and easy calculation of the mass of fluid contributing to the rocking motion of cylindrical tanks with partial uplift of bottom plate, which is the effective mass of fluid for rocking motion, that for rocking-bulging interaction, effective moment inertia of fluid for rocking motion and their centroid, is proposed. Asymmetric deformation of the tank bottom plate due to crescent-like uplift is used to put quantification of the masses away from rigorous treatments. This study considers the cylindrical tank as a set of thin rectangular tanks, so-called a slice model, and puts them perpendicular to the rotational axis of the tank rock motion. Then solve a boundary-value problem of each slice model specified by uplift of the tank bottom plate and its location, the mass of fluid contributing to rocking of cylindrical tanks is quantified as the sum of that of each slice model. Values of the effective mass of fluid for rocking motion, that for rocking-bulging interaction, effective moment inertia of fluid for rocking motion and their centroid are tabulated and depicted as a function of the aspect of tanks for different values of the ratio of the uplift width of the tank bottom plate to the diameter of tank.

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.

A Study of Applicability of Finite Displacement Analyses With Semi-Analytical Finite Elements for Analyzing Uplift Displacement of Flat-Bottom Cylindrical Shell Tanks Statically

2013 ◽  
Author(s):  
Teruhiro Nakashima ◽  
Tomoyo Taniguchi
Keyword(s):  
Finite Elements ◽  
Dynamic Pressure ◽  
Displacement Function ◽  
Bottom Plate ◽  
Inertia Force ◽  
Flat Bottom ◽  
Ring Element ◽  
Spring Element ◽  
Rocking Motion ◽  
Tank Bottom

The rocking motion of tanks due to earthquakes causes the large uplift deformation of the tank bottom plate that has been considered to contribute to the various damages of the tanks. For analyzing the uplift displacement of the tank bottom plate statically and precisely, this paper develops a shell element, ring element and spring element partially attached to the ring element. These elements are defined as a semi-analytical finite element. Fourier series give its circumferential displacement function, while the polynomial gives its radial displacement function. In addition, the ring element can deal with effects of the large deformation, while the spring element enables to express the partial contact between the tank bottom plate and foundation. On the other hand, the loads considered are dead load, hydro-pressure and inertia force due to earthquakes acceleration as well as dynamic pressure of fluid induced by bulging and rocking motion of the tank. The numerical analyses model of the LNG Storage Tank was created using the semi-analytical finite elements shown here, and the uplift displacement of the tank bottom plate accompanying the tank rocking motion was calculated with the static analyses. For evaluating analytical accuracy of the proposed method, numerical results of the proposed method are compared with that of the explicit FE Analysis.

A Static Analysis of Uplift During Strong Earthquakes of Unanchored Flat-Bottom Cylindrical Shell Tanks

2016 ◽  
Author(s):  
Teruhiro Nakashima ◽  
Tomoyo Taniguchi
Keyword(s):  
Dynamic Pressure ◽  
Shell Element ◽  
Bottom Plate ◽  
Inertia Force ◽  
Flat Bottom ◽  
Physical Conditions ◽  
Ring Element ◽  
Spring Element ◽  
Rocking Motion ◽  
Tank Bottom

For analyzing the uplift displacement of the tank bottom plate statically and precisely, this paper develops a shell element, ring element and spring element partially attached to the ring element. These elements are defined as a semi-analytical finite element. Moreover in analyzing uplift of the tank bottom plate precisely, the ring element can deal with effects of the large deformation, while the spring element enables to express the partial contact between the tank bottom plate and foundation. Dead load, hydro-pressure and inertia force due to earthquakes acceleration as well as dynamic pressure of fluid induced by bulging and rocking motions of the tank are applied statically. Comparison of results by the proposed method and that computed by the explicit FE Analysis reveals that the accurate uplift displacement is not obtained until all physical conditions involved in the tank rocking motion and the inward deformation of the tank shell is properly considered.

Contributions of Fluid to Rocking–Bulging Interaction of Rectangular Tanks Whose Walls Are Rigid and Bottom Plate Rectilinearly Uplifts

2012 ◽  
Vol 135 (1) ◽  
Author(s):  
Tomoyo Taniguchi
Keyword(s):  
Effective Mass ◽  
Apparent Density ◽  
Total Mass ◽  
Fluid Pressure ◽  
Bottom Plate ◽  
Rotational Inertia ◽  
Combined Model ◽  
Tank Bottom ◽  
Tank Geometry ◽  
Rocking Response

At the event of severe earthquakes, the tank rocking response appears subsequent to the tank bulging response. A spring-mass-rigid body combined model was proposed for analyzing the tank response and the required quantities for the model have been rigorously defined based on the mathematical solution of fluid pressure on the tank accompanying the tank response of interest. To date, the apparent density of fluid for the rocking and the bulging responses and the effective mass of those are available for this purpose. However, observations revealed that the interaction between the rocking response and the bulging response is stimulated by the effective mass of fluid for the rocking–bulging interaction that is understood as a part of the effective mass of fluid for the tank bulging response that is also under the action of the rotational inertia. Therefore, regarding a ratio of the apparent density of fluid for the tank rocking response to the original density of fluid as the intensity of a contribution of fluid to the tank rocking response, the apparent density of fluid for the rocking–bulging interaction is defined intuitively and conveniently. The effective mass of fluid for the rocking–bulging interaction is subsequently defined. The distribution of the apparent density of fluid for the rocking–bulging interaction inside the tank is drawn for a combination of the various aspects of tank and the ratios of the uplift width of the tank bottom. The ratios of the effective mass of fluid for the rocking–bulging interaction to the total mass of fluid of the tank as well as the ratios of the arm length to the centroid of the effective mass of that to the tank geometry are also depicted by the same manner.

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.

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