Effect of Flexible Wall Boundary in Seismic Design of Liquid Storage Tanks with Fluid Structure Interaction

Effect of Flexible Wall Boundary in Seismic Design of Liquid Storage Tanks with Fluid Structure Interaction

Manuel George Akash Rajan

M.Tech Student Research Scholar

Dept. of Mech. Engg. Dept. of Mech. Engg.

Saintgits College of Engg. Kottayam College of Engg. Trivandrum

Abstract In this paper, the effect of fluid structure interaction on the modal characteristics of a cylindrical water tank without free surface effect is considered. Acoustic structure interaction using unsymmetric pressure based formulation is used to solve the coupled system using FEM and the procedure is validated using results from published literature. A 2D axisymmetric model is also presented to evaluate the same for axisymmetric mode shapes of the system and matched with the axisymmetric modes of the 3D tank model. The perturbation caused in the natural frequency of the coupled system is presented by comparing the coupled frequencies with the ordered uncoupled frequencies of the tank and fluid alone. Parametric study on the uncoupled frequencies of different tank configurations are also presented. The effect of flexible wall boundary condition on the convective and impulsive modes of tall and shallow aspect ratio tanks is shown and are compared to uncoupled structural modes. Parametric study is done for different fluid levels to characterize the dynamics of coupled system. Free surface is considered in fluid alone model to predict the slosh frequencies employing rigid wall boundary.

Keywords Perturbation; Slosh; Convective And Impulsive Modes

1. INTRODUCTION

   The dynamic interaction between fluid and structure is a significant concern in many engineering problems. These problems include systems as diverse as off-shore and submerged structures, biomechanical systems, aircraft, suspension bridges, storage tanks. The interaction can drastically change the dynamic characteristics of the structure and consequently its response to transient and cyclic excitation. Therefore, it is desired to accurately model these diverse systems with the inclusion of Fluid Structure Interaction (FSI).

   Fluid Structure Interaction (FSI) is defined as the interaction of some deformable or movable structure with an internal or surrounding flow. The important aspect of FSI is that there must be genuine interaction between the fluid and solid component. This implies that there is a transfer of energy from fluid to the solid and vice versa. Fluid structure interaction becomes particularly important when the liquid is almost incompressible and deformation on the solid cannot be neglected.

   A literature survey has been conducted on seismic design of liquid storage tanks and its simulation. Most of the papers are having experimental evaluation of the structure with changes in shape and height. With the addition of another structure like roof on the tank also contribute significant effects on to it. Dynamic analysis of different shapes of tanks with varying quantities has been studied by various researchers. A. Ergin [1], introduced vibration problems by the interactions that take place between structure and fluid. This is due to the vibration of the structural surface in contact with the fluid medium imparting motion to the fluid, thus altering its pressure, and, hence, inducing reactive forces on its surface. In his investigation, it is assumed that the fluid is ideal, and fluid forces are associated with inertial effects only: namely, the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration.

   M. Moslemi [12], bring up to identify the major parameters affecting the dynamic response of acoustic structures and to address the interaction between these parameters. Examined parameters include sloshing of liquid free surface, tank wall flexibility, vertical ground acceleration, tank aspect ratio, and base fixity. M. Amiri[11] introduced a series of ambient vibration tests on three tall liquid storage tanks with same height and different radius are considered, to determine the natural frequencies and, the modes of the vibration. Juan C. Virella5 introduced the fundamental impulsive modes of vibration of cylindrical tank-liquid systems anchored to the foundation under horizontal motion. The roof and walls are represented with shell elements and the liquid is modeled using two techniques: the added mass formulation and acoustic finite elements. The literature survey gives a deep insight to the effects of various parameters on seismic structures and impact of different shapes on the natural frequency of the system, location of liquid storage tanks and the analysis on them.

   The objective of the project is to evaluate the dynamic interaction between fluid and structure. FE modeling of tank containing the fluid is modeled in MATLAB using Isoparametric 2 D four noded quadrilateral acoustic element with 2-D quadrilateral plane stress element and validated with literature. The main reason for the study is better predictive Finite Element model development for design of structures which are prone to earthquake. By knowing the natural

   frequency and mode shapes, the structure can be better designed which are prone to common frequencies of earthquake. Parametric study is conducted by considering different wall flexibility, fluid damping properties/densities, earthquake frequency content, liquid level, tank dimensions etc. to characterize their significance in seismic response.

2. FINITE ELEMENT FORMULATION

   1. Abbreviations and Acronyms

      o : Mean fluid density

      k : Bulk modulus of fluid

      P : Acoustic pressure

      t : Time

      [M] : Structural mass matrix
      [R] is a "coupling" matrix that represents the effective surface area associated with each node on the fluid-structure interface. It also takes into account the direction of the normal vector defined for each pair of coincident fluid and structural element faces that comprises the interface surface. The positive direction of the normal vector, as the ANSYS program uses it, is defined to be outward from the fluid mesh and in towards the structure.

      Both the structural and fluid load quantities that are produced at the fluid-structure interface are functions of unknown nodal degrees of freedom. Placing these unknown "load" quantities on the left hand side of the equations and combining the two equations into a single equation produces the following:

      [C] : Structural damping matrix

      Ms

      0 ..

      Ks

      R Fs

      [K] : Structural stiffness matrix

      U U

      (2)

      RT

      M .. 0

      K P F

      {u} : Nodal displacement vector c : Speed of sound

      0 f P

      f

      f

      {Fa} : Applied load vector

      {n} : Unit normal to the interface S

      {N} : Element shape function for pressure

      {N'} : Element shape function for displacements

      {Pe} : Nodal pressure vector

      {ue} : Nodal displacement component vectors

      [R] : Coupling matrix

      FSI : Fluid Structure Interaction FEM : Finite Element Method

      The foregoing equation implies that nodes on a fluid-structure

      interface have both displacement and pressure degrees of freedom.

      Where

      M 1 NNT dV

      f V c2

      f V

      K BT BdV

      R NnT N 'T dS

      (3)

   2. Governing Equations and assumptions

      In acoustical fluid-structure interaction problems, the

      e

      M fs

      S

      o e

      R T

      structural dynamics equation needs to be onsidered along with the Navier-Stokes equations of fluid momentum and the flow continuity equation. The fluid momentum (Navier- Stokes) and continuity equations are simplified to get the acoustic wave equation using the following assumptions (Kinsler):

      1. The fluid is compressible (density changes due to pressure variations).

      2. The fluid is inviscid (no viscous dissipation).

      3. There is no mean flow of the fluid.

      4. The mean density and pressure are uniform throughout the fluid. Note that the acoustic pressure is the excess pressure from the mean pressure.

      5. Analyses are limited to relatively small acoustic pressures so that the changes in density are small compared with the mean density.

         ..

         The interaction of the fluid and the structure at a mesh interface causes the acoustic pressure to exert a force applied to the structure and the structural motions produce an effective "fluid load." The governing finite element matrix equations then become:

         M s U Ks U Fs RP

         K fs R

         e

3. PROBLEM FORMULATION

   1. Case A

   In this study, two cylindrical concrete tank models with different aspect ratios, representative of two classes of tanks namely Shallow and Tall are investigated in both 2D axisymmetric and 3D spaces. The terms Shallow and Tall used in the current study have only relative meaning. The aspect ratio of the Tall tank model (H/D) is about three times higher than that of the Shallow tank model. The simplified geometries of the tanks are shown in fig.1.Dimensions of the tanks are selected such that the volume of stored water remains unchanged for both tank models. The tanks are assumed to be anchored to the rigid ground such that no sliding or uplift may occur. Therefore, all base nodes located along the floor perimeter are fully restrained in all directions. As a result of this perfect anchorage assumption, no bending moment can be transferred from the wall to the floor and vice versa and therefore the tank floor may not be included

   f P f f

   0 U

   M .. K P F

   RT ..

   (1)

   in FE modelling of such containers. Since only anchored tanks are considered in this study, the tank floor is not modelled in FE simulation. Properties of the concrete are elastic Modulus=24.86GPa, Poisons ratio=0.16 and density=2400 kg/m3 and for water density =1000 kg/m3 and sonic velocity-1533m/s.

   Figure 2:

   Mode No:	Natural frequency in Hertz
   	Fluid alone	Structure alone	Ordered uncoupleda	Coupled (FSI)	Deviation
   1	39.04	21.78	21.78 (s)	9.8	11.98
   2	71.51	34.86	39.04 (f)	23.33	15.71
   3	103.7	82.58	34.8 (s)	56.53	-21.73
   4	136.06	136.02	71.51 (f)	73.62	-2.11
   5	139.7	164.79	82.5 (s)	88.389	-5.889
   6	145.1	286.66	103.7 (f)	108.05	-4.35
   7	157	411.35	136.02 (s)	112.82	23.2

   Values in brackets (s) denotes structure mode and (f) denote fluid mode

   Figure 1: a) Shallow tank model b)Tall tank model Error! Reference source not found.

   Finite element models of tanks are constructed using the general purpose finite element package ANSYS and their dynamic parameters are derived from free vibration analyses. Modal analysis of 2D axisymmetric tank liquid systems is done in three stages. 1) Modal analysis of tank structure alone, 2) Modal analysis of Fluid alone and 3) Modal analysis of tank structure with fluid. SHELL208 is a 2-node axisymmetric shell

   1. Case B

      Figure 3:

      element with three degrees of freedom at each node: translations in the x and y directions and rotation about the z-axis. FLUID29 has four corner nodes with three degrees of freedom per node: translations in the nodal x and y directions and pressure at the interface for the structure present option. Only pressure DOF is present for structure absent option.

4. RESULTS AND DISCUSSIONS

1. Case A

2D axisymmetric analysis of tank model to find the effect of coupling. Figure below shows mode shape of tank structure and fluid motion. Table shows effect of coupling modes 1,2,7 shows effective coupling reduced frequency considerably. Graphical interpretation is also provided to show the deviation between coupled and uncoupled system.

The tank structure is modelled using four noded isoparametric quadrilateral SHELL 181 element. The element has four nodes with six degrees of freedom at each node: translations in the x, y and z axes and corresponding three rotations and the fluid domain is modelled using Fluid 30 element having eight corner nodes with four degrees of freedom per node: translations in the nodal x, y and z directions and pressure at the interface for the structure present option. Only pressure DOF is present for structure absent option.. Three dimensional finite element models of Shallow and Tall tanks are constructed using the finite element package ANSYS and their fundamental natural frequencies are derived from modal or free vibration analyses. The results of the analyses are presented in this section. There are two types of modes: impulsive and convective modes.

FREQ=12.902 USUM (AVG) RSYS=0

DMX =.0032

SMX =.0032

FREQ=21.213 USUM (AVG) RSYS=0

DMX =.003482

SMX =.003482

MX

MX

IMPULSIVE VS CONVECTIVE MODES OF A SAHLLOW TANK WITH FSI

RFRQ=8.68258 IFRQ=0

MODE Real part USUM (AVG) RSYS=0

DMX =.113E-05 SMX =.113E-05

Y

Z X MN

Figure 4:

RFRQ=11.9541 IFRQ=0

MODE Real part USUM (AVG) RSYS=0

DMX =.627E-06 SMX =.627E-06

Y

Z X MN

MX

FREQ=9.74296 USUM (AVG) RSYS=0

DMX =.001982

SMX =.001982

Z

Y XMX

MX

MN

Figure 5:

FREQ=23.2699 USUM (AVG) RSYS=0

DMX =.001683

SMX =.001683

Z

Y X MN

Values in bold letters denote the actual water tank under test

Structure alone impulsive mode frequency for shallow tank is 21.31Hz and for flexible (FSI) case it is 21.463

IMPULSIVE VS CONVECTIVE MODES OF A TALL TANK WITHOUT FLUID

Y

X

Z

RFRQ=6.77166 IFRQ=0

MODE Real part USUM (AVG) RSYS=0

DMX =.913E-06 SMX =.913E-06

MX

MX

MN

Y MN

X

Z

Figure 6:

RFRQ=12.2594 IFRQ=0

MODE Real part USUM (AVG) RSYS=0

DMX =.250E-06 SMX =.250E-06

MX

Y

Z Z

X

Y X

MN MN

Figure 7:

Of these free surface sloshing is in convective modes and tank wall shear happen in impulsive mode and a comparison between tall tank and shallow tank is made. The 2 modes are found for different water heights and tabulated

Figure 4 show first convective and impulsive modes of shallow tank without fluid and figure 5 shows that of shallow tank with fluid. When fluid is loaded to shallow tank impulsive mode frequency got reduced from 21.21 to 11.96 Hz and convective mode frequency got reduced from 12.83 to

8.68 Hz

Figure 6 shows first convective and impulsive modes of tall tank without fluid and figure 7 shows that of tall tank with fluid. When fluid is loaded to tall tank impulsive mode frequency got reduced from 23.26 to 12.26 Hz and convective mode frequency got reduced from 9.74 to 6.77 Hz

IMPULSIVE VS CONVECTIVE MODES OF A SAHLLOW TANK WITHOUT FLUID

Values in bold letters denote the actual water tank under test

Values in bold letters denote the actual water tank under test

IMPULSIVE VS CONVECTIVE MODES OF A TALL TANK WITH FSI

Values in bold letters denote the actual water tank under test

It is found that at lower fluid levels impulsive frequency increase is observed which is due to stiffening of tank wall structure. When the water height is gradually increased this impulsive frequency is reduced. At higher levels of fluid frequency reduction is observed which is due to added mass effect. Thus at lower levels of fluid impulsive modes are above the seismic region and it become less dangerous. It is also dangerous to maintain fluid at an intermediate level (hw=4 to hw=6 for shallow tank hw=6 to hw=9 for tall tanks without free surface) especially for critical situation and maintain fluid beyond this or below this intermediate level.

1. Case C

   Free surface is considered in fluid alone model to predict the slosh frequencies employing rigid wall boundary. Fluid mode shapes considered are shown here figure 8 shows first mode and figure 9 second mode shape. When considering different heights same mode shape with frequency difference is studied. Graph is plotted with first mode frequency with free surface(slosh).

   the beaker is full of water it is easy to handle the beaker without sloshing than when it is at lower level. Hence it is seen that fluid at lower levels slosh at very low frequency.

   ACKNOWLEDGMENT

   This work was supported by Centre for Engineering Research and Development, Trivandrum, Kerala and All India Council for Technical Education [RPS grant No: 8023/RID/RPS-24/2011-12].

   REFERENCES

   1st slosh mode frequency (Hz)

   0.14

   0.12

   0.10

   0.08

   0.06

   0.04

   Figure 8:

   Figure 9:

   1 2 3 4 5

   height (m)

   Figure 10

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