Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4

3145

ABSTRACT: Seismic excited liquid filled tanks are subjected to extreme loading due to hydrodynamic pressures, which can lead to nonlinear stability failure of the thin-walled cylindrical tanks, as it is known from past earthquakes. The overall seismic behavior of tanks is, however, quite complex, since the dynamic interaction effects between tank shell and liquid must be considered. A significant reduction of the seismically induced loads can be obtained by the application of base isolation systems, which have to be designed carefully with respect to the modified hydrodynamic behavior of the tank in interaction with the liquid. Firstly, this paper presents the state of the art of tank design for anchored tanks with a fix connection to a rigid foundation with special focus on the practicability of the available design rules. Then a highly sophisticated fluid-structure interaction model is introduced, which has to be applied for a realistic simulation of the overall dynamic behavior of an isolated tank.

KEY WORDS: liquid storage tank; impulsive vibration mode; fluid structure interaction model; base isolation; LS-DYNA

1 INTRODUCTION Steel tanks are preferably designed as cylindrical shells, because they are able to carry the hydrostatic pressure from the liquid filling by activating membrane stresses with a minimum of material. In combination with the high strength of steel this leads to thin-walled constructions, which are highly vulnerable to stability failures caused by seismic induced axial and shear forces. However, an earthquake-resistant design of rigid supported tanks especially for high seismic hazard levels requires unrealistic und uneconomic wall thicknesses. Compared to increasing the wall thickness an earthquake protection system can be a much more cost-effective alternative. Especially a base isolation with elastomeric bearings offers advantages in terms of an earthquake-friendly tank design. The calculation capabilities of base-isolated, liquid-filled tanks are quite limited because of the complex interaction of the isolation and the combined modes of vibrations of tank and fluid. For rigidly supported tanks different calculation methods are available for describing the vibration behavior and the seismic induced load components in a more or less simplified way. These generally accepted methods are either quite simple [1] or very accurate but complex [2] – but valid only for anchored tanks. To capture the hydrodynamic loading of isolated tanks, a complete modeling of the fluid-structure interaction including the behavior of the seismic isolation is necessary. Firstly, this contribution presents a comprehensive and feasible calculation method for anchored, rigidly supported liquid storage tanks and then introduces the modelling of a complex fluid-structure-interaction model, which is used to calculate the seismic behavior of isolated tank structures.

2 LOAD CALCULATION OF ANCHORED LIQUID STORAGE TANKS

As a result of seismic excitation hydrodynamic pressure components, produced by the movement of the fluid, appear

and have to be superimposed with the hydrostatic pressure. These seismic loads acting on wall and bottom of a cylindrical tank (Figure 1) can be divided into the following components:

- the convective load component; the fluid vibration in the rigid tank (sloshing),

- the impulsive rigid load component; caused by the inertia of the liquid, if the rigid tank moves together with the foundation,

- the impulsive flexible load component; representing the combined vibration of the flexible tank shell with the liquid.

Since the oscillation periods of those individual seismically induced pressure components are far apart, each mode of vibration can be determined individually.

Figure 1. Cylindrical tank.

2.1 Convective pressure component Figure 2 shows the mode of vibration and the pressure distribution corresponding to the convective pressure component. The pressure distribution is defined as:

Seismic isolation of cylindrical liquid storage tanks

Julia Rosin1, Christoph Butenweg1

1Chair of Structural Analysis and Dynamics, RWTH Aachen University, Mies-van-der-Rohe-Str. 1, 52072 Aachen, Germanyemail: Julia.Rosin@LBB.rwth-aachen.de, butenweg@LBB.rwth-aachen.de

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3146

pk ξ, ζ, θ, t =2·R·ρL

(λn2-1)

J1(λn · ξ)

J1(λn)cosh(λn· γ · ζ)

cosh(λn· γ)

∞

n=1

cos(θ) akn t · Γkn

(1)

with: pk convective pressure component due to horizontal

excitation n summation index; number of considered sloshing

modes (here: n = 1) R inner tank radius ρL liquid density J1 first order Bessel function:

J1 λn· ξ = -1 k

k! · Γ(1 + k + 1)·λn · ξ

2

2k+1∞

k=0

λn null derivation of Bessel function: λ1 = 1,841, λ2 = 5,331, λ3 = 8,536

ξ dimensionless radius: ξ = r/R ζ dimensionless height: ζ = z/H θ angle of circumference γ tank slenderness: γ = H/R akn(t) horizontal acceleration-time history as a result of an

equivalent single-degree-of-freedom system with a period Tkn for the nth-eigenmode of the sloshing wave. By using the response spectrum method the spectral accelerations corresponding to the natural periods Tkn should be calculated based on the elastic response spectrum.

Γkn participation factor for the convective pressure component for the nth-eigenmode.

Figure 2. Convective pressure component – mode of vibration

and pressure distribution.

Taking into account the first sloshing eigenmode (n = 1) and the pressure distribution of the tank shell (ξ = 1), equation (1) can be simplified to:

pk ξ = 1, ζ, θ, t = R · ρL 0,837 · cosh(1,841· γ · ζ)

cosh(1,841· γ)cos(θ) ak1 t · Γk1

(2)

The natural period Tkn for the nth-eigenmode of the sloshing wave is calculated with

Tkn= 2πg · λn· tanh (λn· γ)

R

(3)

2.2 Impulsive rigid pressure component due to horizontal seismic excitation

Figure 3 shows the mode of vibration and the pressure distribution corresponding to the rigid impulsive pressure component. The pressure distribution is given by the expression:

pis,h ξ, ζ, θ, t = 2 · R· γ· ρL· -1 n

νn2

I1νnγ · ξ

I νnγ

∞

n=0

cos νn· ζ cos(θ) ais,h(t) · Γis,h

(4)

with: pis,h rigid impulsive pressure component due to horizontal

excitation νn νn= 2n + 1

2π

I1 modified first order Bessel function:

I1νnγ

· ξ = J1 i·νnγ · ξ

in=∑ 1

k! · Γ(1 + k + 1)∞k=0 ·

νnγ · ξ

2

2k+1

I1’ Derivation of the modified Bessel function regarding to

[3]:

I νnγ

·ξ = I0νnγ

·ξ -I1

νnγ ·ξ

νnγ · ξ

I =∑ 1k! · Γ(0 + k + 1)

·νnγ ·ξ

2

2k+0∞k=0 -

∑ 1k! · Γ(1+k+1) · 

νnγ · ξ

2

2k+1∞k=0

νnγ · ξ

ais,h(t) horizontal acceleration-time history. By using the response spectrum method ais,h(t) should be replaced by the spectral acceleration Sa corresponding to T = 0 s.

Γis,h participation factor for the rigid impulsive pressure component: Γis,h = 1.0, because the rigid tank is moving together with the foundation.

Figure 3. Impulsive rigid pressure component – mode of

vibration and pressure distribution.

Taking into account the pressure distribution of the tank shell (ξ = 1), equation (4) is simplified to:

The corresponding natural period is T = 0.

2.3 Impulsive flexible pressure component due to horizontal seismic excitation

Figure 4 shows the mode of vibration and the pressure distribution corresponding to the flexible impulsive pressure component.

The flexible impulsive pressure component is calculated in an iterative procedure using the added-mass-model according to [3], Annex A. Within the framework of the procedure the tank wall is loaded with iterative calculated additional mass

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n=0

cos(θ) ais,h(t)·Γis,h

(5)

portgive

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∞

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h: flexible horizontal

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(9)

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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3149

θ = 0°, circumferential compressive stress at θ = 180° and shear stress at θ = 90°.

3.1 Earthquake protection by base isolation To prevent an earthquake damage of an arbitrary building structure two different possibilities exist: One the one hand a stiffening of the structure results in higher resistance to the earthquake impact and on the other hand a decoupling of structure and underground results in reduced seismic loads acting on the structure. For a tank structure the stiffening could be render possible by an increase of the shell thickness, which especially for high seismic hazard levels requires unrealistic und uneconomic wall thicknesses. Compared to increasing the wall thickness an earthquake protection system can be a much more cost-effective alternative. Especially a base isolation with elastomeric bearings offers advantages in terms of an earthquake-friendly tank design.

A base isolation is aimed at a decoupling of the building and the ground motion. Elastomeric bearings are a widely used base isolation and can optionally be installed with or without reinforcement, often in combination with a lead core [9]. It is recommended to use reinforced bearings, since elastomers are subjected to high deformations up to 25% under vertical loads. These deformations cause lateral strains, by which unwanted rocking motion in vertical direction in case of seismic excitation can occur. Reinforced bearings can be considered as quasi-rigid in the vertical direction, so they are suitable to transfer vertical loads. Under cyclic loading, elastomers behave almost like springs. They have – depending on the material properties – a certain stiffness, which causes a reset of the bearing and thus of the entire system after the release. Through the use of high-damping elastomers (addition of oils, resins, extra fine carbon black and other fillers), or a lead core, the damping capacity of the bearings can be increased significantly. In case of a distortion of 100%, high damping elastomers have damping rates from 0.1 to 0.2 while normal elastomers are about 0.04 to 0.06 [9]. The use of elastomeric bearings as earthquake protection systems for tanks has already been realized, e.g. in [10]. So far, to capture the hydrodynamic loading due to seismic excitation simple mass oscillators with simplistic approaches have been used: In this case of calculation the fluid was assumed as rigid, so that the interaction between the tank wall and fluid was neglected. In addition, the determination of the stress distribution in the tank wall is not possible.

4 FLUID STRUCTURE INTERACTION MODEL The software LS-DYNA [11] is used for the simulation of the fluid-structure interaction of a liquid-filled tank. The software provides an explicit solver, which offers advantages especially for the solution of dynamic contact problems. In addition, LS-DYNA provides materials for the modeling of fluids and standardized contact formulations to represent the interaction of the tank shell and the fluid during a seismic excitation. Details of the following material, element and contact formulations can be found at the LS-DYNA manuals [12].

4.1 Material formulation Basically both, elastic and plastic approaches are applicable for the tank shell. Since the focus is set on the reduction of the seismic loading by applying a base isolation, an elastic

behavior of the tank shell is assumed (MAT_ELASTIC). For the base plate the concrete material model MAT_CSCM is used, which is applied with default settings. The fluid is performed by a linear (elastic) material formulation (MAT_ELASTIC_FLUID), which idealizes the fluid as incompressible and friction-free. Here, only the fluid density as well as the bulk modulus of the fluid is needed as input for the calculation. The change in the hydrostatic pressure fraction p is described by the bulk modulus K and the strain rate εii in the main directions of the material:

p = -K εii (13)

The shear modulus is set to zero. The deviatoric stress component Sik

n+1 is given by:

Sikn+1 = VC ΔL C ρL εik

' (14)

Herein, VC is a tensor viscosity coefficient, ΔL is the characteristic element length, C is the fluid bulk sound speed (C = K/ρL), ρL is the fluid density, and εik

' is the deviatoric strain rate. The chosen values for water are given in Table 1.

Table 1. Material properties for the fluid formulation.

Symbol Property Water value K Bulk modulus 2100 [N/mm²] ρL Density 1000 [kg/m³] C Speed of sound 1450 [m/s] VC Tensor viscosity coefficient 0.1

4.2 Element formulation The tank wall and the tank bottom are idealized by Belytschko-Lin-Tsay shell elements with reduced integration, which are characterized by a high efficiency in terms of computing power required for explicit analysis [13]. To avoid hourglassing, which is particulary a problem by application of reduced integrated elements, LS-DYNA offers an hourglass control. The foundation plate is idealized by 8-node solid elements and the fluid is represented by an Arbitrary-Lagrangian-Eulerian finite element formulation (ALE). When using the ALE formulation extra volume elements are generated within the region of the freeboard up to the top edge of the tank wall, so the fluid surface can move freely (sloshing). Also, the ALE mesh must enclose the Lagrange mesh. For this reason a series of elements at the top and bottom of the tank, below the base plate and outside of the tank wall are generated. The elements are assigned to the vacuum material MAT_VACUUM, which has no physical meaning, but merely represents a region within the ALE mesh in which the fluid can move. Due to the very moderate deformation of the actual FE-mesh by using the ALE element formulation, no hourglass problems for the fluid result.

4.3 Contact formulation The interaction between the tank shell and the fluid represents an important aspect of modeling. If a contact of the two parts appears, compression stresses are transferred while the transfer of tensile and shear stresses is disregarded. LS-DYNA provides essentially two different contact formulations for coupling the tank shell (Lagrange) and the fluid (ALE) with

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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3151

value relating to the range occurs at the peripheral angle of θ = 0°.

Figure 13. FSI-model: Qualitative hydrodynamic pressure

distribution over tank height and range.

To validate the fluid-structure interaction model the hydrodynamic pressure and the stress results of a rigidly supported tank are compared to those from an analysis with equivalent static pressure distributions (ESP) presented in chapter 2. For the analysis with equivalent static pressure distributions the seismic induced pressure components are calculated separately for each vibration mode using the response spectrum method. Afterwards they are superimposed to the resulting hydrodynamic pressure using the SRSS-rule. Finally the resulting hydrodynamic pressure is applied as an equivalent static load to the dry tank wall to calculate the stress distribution of the tank shell. For this calculation the hydrodynamic pressure is combined with dead load and hydrostatic pressure.

The models are validated by means of parametric studies with different tank geometries. The varied calculation parameters are the tank slenderness (H/R-ratio), the wall slenderness (R/t-ratio) and the seismic hazard level (different subsoil conditions according to Eurocode 8 [4]. The investigated H/R and thickness ranges are 0.3 ≤ H/R ≤ 3 and 500 ≤ R/t ≤ 1000, which contains the calculation example presented in chapter 5.1 (H/R = 2 and R/t = 833).

Figure 14. Hydrodynamic pressure distribution for two

different tank geometries.

Figure 14 shows the hydrodynamic pressure distribution due to horizontal seismic action at the peripheral angle of 0° over the wetted tank height for two selected tank geometries: the geometry of the calculation example and, in addition, a very squat geometry. According to [3] damping values of 2% for the tank shell and 0.5% for the fluid are applied for the

rigidly-supported FSI-model. By applying these damping values the numerical simulation results of the rigidly-supported tank considering fluid-structure interaction effects show a very good agreement with results according to the method with equivalent static pressure distribution (ESP).

5.3 Base-isolated support The calculations of the base isolated tank are carried out for an elastic behavior of the tank itself and a damping of the fluid of 0.5%. Figure 15 shows the resulting displacement and the acceleration time history of two different nodes: the one is a supporting node, which means a node directly affected from the seismic excitation. So the displacement and the acceleration are exactly the same than the ground movement raised by the earthquake. As the ground movement is filtered by the isolation system the tank structure itself moves different. The acceleration affecting a node of the tank bottom is reduced significantly. Of course the reduced acceleration is at the expense of an increase of the displacement. The maximum horizontal displacement of the isolated tank structure is about 3 cm.

Figure 15. Resulting displacement and acceleration time history of the isolated tank above and under the isolation

support.

Figure 16 shows a significant decrease of the resulting hydrodynamic pressure, which results from the frequency shift and the increasing damping rate caused by the isolated support.

Figure 16. Hydrodynamic pressure distribution for the isolated

tank in comparison to a rigid support.

Figure 17 to Figure 19 show the circumferential (Figure 17), axial (Figure 18) and shear stress distribution (Figure 19) over

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