Free Vibration Analysis of a Liquid in a Circular Cylindrical Rigid Tank Using the Hierarchical Finite Element Method

Abstract

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 therefore be solved very accurately by using a higher hierarchical mode number in the longer direction than in the shorter one. Furthermore, it is possible to idealize the liquid by using only one element. The solution can therefore be obtained to any desired degree of accuracy simply by increasing the hierarchical mode number. In this method, the liquid behavior is described by the displacements alone. The pressure and velocity potential are not considered as unknowns. The results are compared with other methods and show good agreement.

Pressure and stress analysis of liquid‐filled cylindrical tank

“…These are suitably handled with the Eulerian formulation of equilibrium equations 28 . In the case where the shape of the fluid domain is expected to change significantly, a modified formulation called arbitrary Lagrangian–Eulerian (ALE) formulation was adopted to adequately simulate the physical behavior of the domain of interest 29 . The ALE description was designed to follow the boundary motions rather than the fluid particles.…”

“…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).…”

“…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) 13 …”

“…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. Three different finite element approaches can be used to describe the fluid motion: (1) the Eulerian approach (the potential [or pressure] is used to describe the fluid behavior), [14][15][16][17][18][19][20][21][22][23] (2) the Lagrangian approach (displacement), [24][25][26][27][28][29][30][31][32][33] and the mixed approach (combining displacement and pressure).…”

“…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.…”