Published by Elsevier Elsevier Science Ltd. All rights reserved 12th E uropean uropean Conference on Earthquake Engineering Paper Reference 5 (quote when citing this paper)
FLUID – SOIL SOIL – STRUCTURE INTERACTION IN THE DYNAMIC ANALYSIS OF GROUND AND UNDERGROUND STORAGE TANKS S. M Tajvidi * 1, A. Noorzad*2, A.A. Moinfar 3 *
Civil Engineering Dept. Faculty of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran
ABSTRACT
A comprehensive study is made about the effects of fluid-soil-structure interaction on the dynamic response of ground and underground cylindrical storage tanks. Considering the interaction between fluid and structure, an analytical simple model is proposed. This is resulted from the solution of the Laplace equation in the fluid domain with appropriate boundary conditions. Furthermore, considering the flexibility of the tank base, the relation between the foundation motion and base excitation is derived which itself modifies the dynamic characteristics of the whole structure. Taking into account the two effects simultaneously, it will be shown that the soil-structure interaction affects higher modes of impulsive component of response, especially in rather slender tanks. Keywords: Earthquake Engineering; Dynamic Dynamic Analysis; Interaction; Ground Storage Tanks
INTRODUCTION
The response of ground liquid storage tanks during has been studied to a vast degree. Several researches have been performed and presented considering different assumptions. The subject of fluid-structure interaction for such structures was for the first time investigated by Housner [6] who proposed a simple model separating the impulsive and convective components of fluid motion in the tank and calculated each component characteristics such as added mass, natural frequency of the full tank etc. Several authors who investigated the different conditions and proposed some modifications in the model modified the subject. The initial concept was maintained. An instructive review of the efforts can be found in the report by ASCE [1]. The effect of base flexibility has been investigated in different conditions. The emphasis has been upon vertical excitation. Most of the recent studies have been numerical studies i.e. a verification of analytical models. The wall flexibility has been taken into account by Haroun [5] and rocking component of motion has been investigated by Veletsos [13]. A straightforward design procedure with some approximations of course can be found in [12]. Considering the common dimensional proportions of tanks, these approximations are quite justifiable. 1
PhD Candidate Assistant Professor 3 Senior Consultant Engineer 2
The objective of this paper is 1) to enlighten the nature of fluid-structure interaction for a liquid upright, circular cylindrical tank subject to horizontal earthquake base excitation; 2) to describe the nature of soil-structure interaction for a circular foundation which is allowed to have horizontal and rocking motion due to horizontal excitation; 3) to elucidate the simultaneous effect of the above mentioned phenomena and describe the results in the response of the structure under dynamic harmonic excitations, 4) to investigate the contribution of higher modes of the convective component of motion in the interaction phenomena. MODEL ASSUMPTIONS
The major assumptions that affect the analysis are as follows: 1) The tank is assumed to be upright, circular cylinder with flexible walls 2) The fluid is assumed to be homogeneous, inviscid, and incompressible, with small amplitude surface waves. 3) There is weak interaction between surface waves and wall motion 4) The tank rests on a foundation supported by homogenous, isotropic semi-infinite medium with hysteretic damping. 5) Partial embedment is allowed i.e. the tank can embedded so that the walls have contact with soil thus considering the soil stiffness to have base and wall components. 6) Short-period pressure due to the wall flexibility is taken into account GOVERNING EQUATIONS IN LIQUID
A simple sketch of the tank is shown in figure 1., the dimensions and geometric properties of the tank are shown as well.
Figure 1- Basic Model for the Tank The governing equation in the liquid, considering the above conditions is: (1) ∇ 2Φ = 0 1. Boundary Condition at the bottom
∂Φ & ( t ) cos θ ( r , θ,0, t ) = − r α ∂z
(2)
2. Boundary Condition at the tank wall
∂Φ & (t) + w & (z, t )] cos θ ( R , θ, z, t ) = −[ x& ( t ) + zα ∂z
(3)
3. Boundary Condition at free surface
∂Φ ( r , θ, H, t ) = 0 ∂z
(4)
An analytical solution has been derived [5], with a Bessel Function form, the resulting base shear and moments are:
&& ( t ) + γ &x&( t ) + γ α && Q( t ) = [ γ 1 W 2 3 ( t )]m
(5)
&& ( t ) + γ &x&( t ) + γ α && M(t ) = [γ 4 W 5 6 ( t )]mH
(6)
γ 1 to γ 6 are the integrals derived from analytical solution. This analysis can be interpreted to a mechanical model approach that considers a mass-spring-dashpot model for the liquid motion.
&& ( t ) Q( t ) = m f &y&( t ) + m r &x&( t ) + m r H r α
(7)
M ( t ) = m f H f &y&( t ) + m r H r &x&( t ) + ( m r H r
2
&& ( t ) + I r )α
(8)
Figure 2- Discrete Mass Model of the Tank Closed–form relations for m f , m r , H f and Hr derived from γ i values are presented in [5]. In figure 3, the changes of these quantities with the tank geometry are shown by a number of curves.
Figure 3.a
Figure 3.b
Figure 3- Rigid And Flexible mass and their Height in the model a ri id mass. b flexible mass EMBEDDED FOUNDATION RESPONSE
Basic assumptions in soil and foundation are: 1- Soil medium is assumed to be infinite halfspace, homogenous, isotropic, viscoelastic with frequency independent (hysteretic) damping. 2- Foundation is assumed to be rigid, circular, massless cylinder. 3- No separation between soil medium and foundation occurs. 4- The soil undergoes small displacements only, no non-linear behaviour is considered. 5- Plane strain case is assumed. In figure 2, the modes of motion of a unit length cylinder are shown. Its possible motions are horizontal and vertical translation, rocking and torsion. For the present case, because of the types of the equations, only horizontal and rocking motions are investigated. The response of tanks under vertical excitation can be found in [7].
Figure 4 - Plane Strain model for a rigid cylinder with infinite length GOVERNING EQUATION IN SOIL
The equations of motion in the soil as viscoelastic medium in the polar coordinates are:
∂∆ 2G * ∂w z ∂ 2u * ∂w θ − + 2G =ρ 2 (λ + G ) ∂r r ∂θ ∂z ∂t *
*
(λ
+ 2G
(λ
+ 2G
*
*
∂∆ ∂2v * ∂w r * ∂w θ ) − 2G + 2G =ρ 2 r ∂θ ∂z ∂r ∂t
(10)
G * ∂w θ ∂∆ 2G * ∂ (rw θ ) ∂2w − +2 =ρ 2 ) r r ∂θ ∂z ∂r ∂t
(11)
*
*
(9)
In the above equations, γ i is the dilatation, wr ; wθ and wz are the components of rotation vector. For the plain strain case (figure 2.) with translational and rocking response, the equations reduce to Bessel ordinary differential equation, thus giving the soil dynamic stiffness as: K x
= πGa 02 T
(12)
In which: 4K 1 ( b *0 )K 1 (a *0 ) + a *0 K 1 ( b *0 ) K 0 (a *0 ) + b *0 K 0 ( b *0 )K 1 (a *0 )
T=− * b 0 K 0 ( b *0 ) K 1 (a *0 ) + a *0 K 1 ( b *0 ) K 0 (a *0 ) + b *0 a *0 K 0 ( b *0 ) K 0 (a *0 ) a *0
= sr 0 =
b *0
=
a
* 0
a0 1 + iD s
a0
η
1 + iD l
= sr 0 =
i
(13)
(14)
i
(15)
a 1
0
+ iD
i s
(16)
Dl
λ ′ + 2G ′ = λ + 2G
Ds
=
(17)
G′
(18)
G
a0 = Non dimensional frequency Vs = Shear wave velocity in the soil R = Cylinder radius G = Shear modulus of soil Manipulating the relation and separation of the real and imaginary parts of impedance will result in the forms: Translational impedance: K x = g [Su1 (a0, Ds) + iSu2 (a0, Ds)] (19) Rocking Impedance: (20) K ψ = Gr 02 [S ψ1 (a 0 , D s ) + iS ψ 2 (a 0 , D s )] Graphical presentation of S values is in Figure 5
Figure 5a.
Figure 5b. Figure 5- Real and imaginary parts of the foundation impedance in a) rocking and b) translation
DESCRIPTION OF THE INTERACTION MODEL
In order to account the soil-structure and liquid-structure interaction effects a flexible base model (as shown in figure 6.) is introduced. The mechanical characteristics of the model are obtained from the first model in the paper. It can be a SDOF or a MDOF model. Though, increasing the degrees of freedom in the tank does not affect the soil interaction very much, as will be shown later. At present a SDOF model is considered. The motion function of the foundation is calculated from the base excitation function. The foundation motion will then be related to the structure motion Dynamic equilibrium equations of foundation are: Shear
m fon &x&( t ) + Q( t ) + Q s ( t ) = 0
(21)
Moment
M = mr H r A′(t )
(22)
Figure 6 – Basic MDOF interaction model considering the coupling between impulsive and convective modes of motion Qs and Ms are in terms of the relative motion of foundation to soil. If the excitation is assumed to be harmonic: x g ( t ) = X g e iΩt
(23)
Thus the foundation motions in translation and rocking are: x ( t ) = Xe iΩt
(24)
ψ( t ) = Ψe iΩt
(25)
Ignoring coupling effects between x and ψ, Ms and Qs will be
∂∆ 2G* ∂wz ∂ 2u * ∂wθ (λ + G ) − + 2G =ρ 2 ∂r r ∂θ ∂z ∂t
(26)
M s ( t ) = K ψ Ψe iΩt
(27)
*
*
K x and K ψ are the impedance functions of foundation derived in Eqs. 16 and 17. If embedment is to be considered, each K value consists of two parts: one for the bottom stiffness and one for the wall stiffness. The characteristics of soil under the foundation and the ambient soil (in the case of embedded tank) are assumed to be the same.
&x& ( t ) + Q ( t ) + Q s ( t ) = 0
m
fon
M
&& ( t ) && ( t ) + (m r H 2r + I b )ψ = m r H r &x&( t ) + m f H f w
(28) (29)
In the frequency domain, the equations of equilibrium take the form Shear: m r (−Ω 2 ) Xe iΩt
+ m f (−Ω 2 )Tw Xe iΩt + m r H r (−Ω 2 )Ψe iΩt + K x (X − X g )e iΩt = 0
(30)
Moment: m r H r ( −Ω 2 ) Xe iΩt
+ m f H f (−Ω 2 )Tw Xe iΩt + (m r H 2r + I b + I fon )(−Ω 2 )Ψe iΩt + K ψ (X − X g )e iΩt = 0 (31)
In the above Eqs, T w is the frequency transform function of the fundamental mode of vibration of the tank: Tw
1
=
(32)
Ω Ω + 2iξ ω f ωf
1 −
The matrix form for the interaction equations for the impulsive components:
A11 A 21
A 12 X
K x = A 22 Ψ 0
(33)
The impulsive components will be:
φ = m r A l ( t )
Translation: Rocking:
(34)
M = m r H r A ′l ( t )
A i ( t ) = −Ω 2 ( X +
m f
A ′i ( t ) = −Ω 2 [ X +
m f H f
m r
(35)
Tw X + H r Ψ )e iΩt
m r H r
Tw X + (H r +
(36)
I b m r H r
Ψ )]e iΩt
(37)
The A values are the pseudo-acceleration functions which comprise the response spectra. The variations of these quantities with the dimensionless frequency are shown in figure 8. Similar relations can be derived for the convective mode response. The only difference is that there might be more than one mode, in which case the shear and moment at the base are calculated by summing the values for each mode.
SAMPLE PROBLEM
Different tank geometries (H to R ratios) and soil conditions (shear wave velocities) are investigated. The tanks are analysed in two conditions; ground tanks and underground tanks. In ground tanks the Veletsos model is verified (see Figure 7). In underground tanks different embedment ratios i.e. 20%, 50% and 100% of the height is assumed to be embedded. The resulting curves of tank pseudo-acceleration vs. dimensionless frequency are given in Figure 8. CONCLUSIONS
The following conclusions can be observed from the results of the analysis: 1. Soil-structure interaction causes a decrease in the natural frequency of the system. It is seen that the maximum response of the system occurs in lower frequency values. It is apparently due to the soil-structure interaction effect. 2. Embedment affects the tank dynamic behaviour by reduction of the maximum response. It increases the natural frequency of the tank as well. The effect shows a maximum in a different frequency form the convective component that is due to impulsive modes. 3. In slender tanks, the resonant response moves to a frequency other than the impulsive mode frequency. It is due to the effect of convective modes of vibration that is more significant in slender tanks. REFERENCES
1. ASCE, ‘Fluid-Structure Interaction During Seismic Excitation’, a report by the ASCE, Committee on Seismic Analysis of the Nuclear and Materials of the Struc. Div.1984 2. Epstein, H.L. ‘Seismic Design of the liquid Storage Tanks’, J of Eng. Mech. Div. ASCE 1976 Vol. 102 (1659-1673) 3. Fang, W. ‘ Handbook of Foundation Engineering’. VNR 1991 4. Haroun, M. ‘Seismic Large Amplitude Sloshing’ Proc. of Conference on Seismic Eng. (418-427) ASCE, 1989 5. Haroun, M. A. and Ellaithy H. M. ‘Model for flexible tanks undergoing rocking’ J. of Eng. Mech. Div. ASCE, 1985 Vol. 111, 143-157 6. Housner, G.W. ‘The Dynamic Behavior of Water Tanks’, Bulletin of the Seismological Society of America 1963, Vol. 53 (2), 381-387 7. Novak, M. ‘ Vertical Vibrations of Embedded Footings’ J. of Soil Mech. and Foundation Eng. Div. ASCE 1972, Vol. 98 (1291-1311) 8. Novak, M. ‘ Torsional and coupled Vibrations of Embedded Foundations’ Earthquake Eng. and Structural Dynamics 1973 Vol. 22 (1-12) 9. Novak, M. ‘ Response of Embedded Foundations to vertical Vibration’ J of Soil Mech. And Foundation Eng. Div. ASCE 1973 Vol. 99 (863-883) 10. Novak, M. ‘ Dynamic Soil Reactions in the Plane strain Case’ J of Soil Mech. And Foundation Eng. Div. ASCE, 1978 Vol. 104 (953-959) 11. Veletsos, A.S. Tang, Y. ‘Soil-Structure Interaction Effects for Laterally Excited Liquid Storage Tanks’, Earthquake Engineering and Structural Dynamics, 1990. Vol. 19, 473-496 12. Veletsos, A.S. ‘Seismic response and design of liquid storage tanks’, Guidelines for the Seismic Design of Oil and Gas Pipeline Systems, Technical Council on lifeline Earthquake Engineering, ASCE, New York, 1984, pp 225-370 and 443-461 13. Veletsos, A.S. Tang, Y. ‘Rocking response of liquid storage tanks’ J. of Eng. Mech. Div. ASCE, 1987 Vol. 113, 1774-1792
Figure 7a.
Figure 7c.
Figure 7b.
Figure 7d.
Figure 7- Amplitude Ratios vs. Frequency ratio for different H/R values of the ground Tank
Figure 8 - Amplitude Ratios vs. Frequency ratio for different embedment values of the ground tank With H/R = 1
