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Research Papers: Seismic Engineering

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

[+] Author and Article Information
Tomoyo Taniguchi

Department of Civil Engineering, Tottori University, 4-101 Koyama-Minami, Tottori 680-8552, Japant_tomoyo@cv.tottori-u.ac.jp

Yoshinori Ando

Aisawa Industries Ltd., 1-5-1 Omotemachi, Okayama 700-0822, Japanyoshi-ando-23-@aisawa.co.jp

J. Pressure Vessel Technol 132(1), 011802 (Nov 30, 2009) (8 pages) doi:10.1115/1.4000374 History: Received November 12, 2007; Revised September 16, 2009; Published November 30, 2009

## Abstract

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.

###### FIGURES IN THIS ARTICLE
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Copyright © 2010 by American Society of Mechanical Engineers
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## Figures

Figure 1

(a) Uplift crescent appeared on bottom plate and (b) the analytical model

Figure 2

Asymmetric boundary conditions for flat-bottom cylindrical shell tank due to uplift motion pivoting at left bottom edge

Figure 3

(a) Symmetric boundary conditions for flat-bottom cylindrical shell tank due to rotation at origin of cylindrical coordinate system, and (b) symmetric boundary conditions for flat-bottom cylindrical shell tank due to uniform upward excitation

Figure 4

(a) Fluid pressure on shell (θ=0 deg), (b) shell (θ=180 deg), and (c) bottom plate (θ=0 deg to θ=180 deg)

Figure 5

Rectangular rigid tank model parallel to x-axis and offsetting distance y′ apart from x-axis

Figure 6

(a) Comparison of fluid pressure on the right side wall (x′=0) and (b) on the left side wall (x′=2ξ)

Figure 7

(a) Fluid pressure on shell (θ=0 deg) and (b) bottom plate (θ=0 deg)

Figure 8

(a) Fluid pressure on shell (θ=30 deg) and (b) bottom plate (θ=30 deg)

Figure 9

(a) Fluid pressure on shell (θ=60 deg) and (b) bottom plate (θ=60 deg)

Figure 10

(a) Fluid pressure on shell (θ=120 deg) and (b) bottom plate (θ=120 deg)

Figure 11

(a) Fluid pressure on shell (θ=150 deg) and (b) bottom plate (θ=150 deg)

Figure 12

(a) Fluid pressure on shell (θ=180 deg) and (b) bottom plate (θ=180 deg)

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