Dynamic behavior of flexible rectangular fluid containers

Thin-Walled Structures 66 (2013) 23–38

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Dynamic behavior of flexible rectangular fluid containers S. Hashemi a,1, M.M. Saadatpour a, M.R. Kianoush b,n a b

Department of Civil Engineering, Isfahan University of Technology, Iran Department of Civil Engineering, Ryerson University, Toronto, Canada

a r t i c l e i n f o

abstract

Article history: Received 24 June 2012 Accepted 1 February 2013 Available online 22 March 2013

An analytical method is proposed to determine the dynamic response of 3-D rectangular fluid containers with four flexible walls, subjected to seismic ground motion. By applying Rayleigh–Ritz method using the vibration modes of flexible plates, fluid–structure interaction effects on the dynamic responses of fluid containers are considered. A mechanical model, which takes into account the deformability of the tank wall, is developed. The maximum seismic loading of the base at the tank and a section immediately above it can be predicted by this model. Accordingly, a 2-D simplified model is proposed to evaluate pressure distribution on the flexible tank wall. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Rectangular tank Fluid–structure interaction Flexible plate Rayleigh–Ritz Seismic design Pressure distribution curve

1. Introduction Large-capacity ground-supported tanks are used to store a variety of liquids, such as water for drinking and fire fighting, petroleum, chemicals, liquid natural gas and nuclear spent fuel assemblies. So, satisfactory performance of tanks during an earthquake is crucial for modern facilities. Based on observations from previous earthquakes, it is concluded that liquid storage tanks can be subjected to large hydrodynamic pressures during earthquakes. Consequently, high stresses can cause buckling failure in the steel tanks. In concrete tanks, due to the large inertial mass of concrete, the stresses could be large and result in cracking, leakage or even collapse of the structure. The poor performance of some of these structures in past earthquakes has led engineers and researchers to study this problem, and to improve the behaviors of these structures. There are some numerical and a few analytical methods available that have been used for dynamic analysis of concrete rectangular liquid storage tanks. Hoskins and Jacobsen [1] published the first report on analytical and experimental observations of rigid rectangular tanks under a simulated horizontal earthquake excitation. Housner [2,3] developed the most commonly used analytical model for estimating the dynamic response of a rigid rectangular tank. This model, with some modifications, has been adopted in most of the current codes and standards.

n

Corresponding author. Tel.: þ1 416 979 5000x6455; fax: þ1 416 979 5256. E-mail addresses: [email protected] (S. Hashemi), [email protected] (M.M. Saadatpour), [email protected] (M.R. Kianoush). 1 Currently at: Department of Civil Engineering, Ryerson University, Toronto, Canada. 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.02.001

The 1964 Alaska earthquake [4] caused the first large-scale damage to tanks of modern design of its time and initiated many investigations into the dynamic characteristics of flexible containers. Several studies were carried out to investigate the dynamic interaction between the deformable wall in the tank and the liquid, and showed that the seismic response of a flexible tank may be substantially greater than that of a similarly rigid tank. A three-degrees-of-freedom (3 DOF) model of the groundsupported cylindrical tank was developed by Haroun [4], the application of which resulted in design charts used to estimate sloshing, impulsive and rigid masses. Yang [5] and Veletsos [6] studied the effect of wall flexibility on the magnitude and distribution of hydrodynamic pressures and associated tank forces. It was found that, for tanks with realistic flexibility, the impulsive forces are considerably higher than those in rigid walls. For rectangular tanks, Haroun [7] presented a very detailed method of analysis on the typical system of loadings. The hydrodynamic pressures were calculated by a classical potential flow approach. The formula of hydrodynamic pressures only considered the rigid wall condition. This may be due to the fact that rectangular fluid containers are usually made of reinforced or pre-stressed concrete and may be considered quite rigid dynamically. Nevertheless, there are containers of this type for which flexibility must be taken into account in their dynamic response analysis, such as very large reinforced concrete structures used for the storage of nuclear spent fuel assemblies or pre-stressed concrete water tanks [8]. Some numerical methods that consider wall flexibility have been used for the dynamic analysis of rectangular liquid storage tanks. Dogangun et al. [9] and Dogangun and Livaoglu [10]

24

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

investigated the seismic response of liquid-filled rectangular storage tanks using the 3-D Lagrangian fluid finite element. Park et al. [11] and Koh et al. [12] studied the seismic response of rectangular tanks with four flexible walls by using a 3-D coupled boundary element–finite element method. Ghaemmaghami and Kianoush [13] investigated the dynamic behavior of concrete rectangular tanks using the FEM in 2D space. Moreover, there are a few analytical methods that have been used for dynamic analysis of rectangular liquid storage tanks. Kim et al. [14] studied dynamic behavior of 3-D rectangular flexible fluid containers using the Rayleigh–Ritz method. In their study, only a pair of walls, orthogonal to the direction of the applied ground motion is assumed to be flexible, while the other pair remains rigid. Chen and Kianoush [15] proposed a simplified method using the generalized SDOF system to study the dynamic response of liquid storage tanks. In the analytical methods that have so far been used for dynamic analysis of rectangular liquid storage tanks, neither the effect of sloshing nor the effect of wall flexibility have been appropriately considered. While the most common standards and codes currently used for the design of rectangular liquid tanks do not consider the variation of hydrodynamic pressure in the horizontal direction, it is confirmed that the hydrodynamic pressure, even for the case of fairly thick walls, is much amplified in the middle of the wall and shows the distributon of 2-D spatial variation. In other words, hydrodynamic pressure varies not only in the vertical direction but also in the horizontal direction over the wall surface. In this paper, an analytical method is presented to investigate the dynamic response of flexible 3-D rectangular liquid storage tanks with flexible walls on all four sides, subjected to horizontal seismic ground motion. Fluid–structure interaction effects on the dynamic responses of partially filled fluid containers, incorporating wall flexibility, are accounted for in evaluating impulsive pressure. The impulsive pressure distribution in which boundary conditions are satisfied is solved by the method of separation of variables using the superposition principle. Solutions based on 3D modeling of the rectangular containers are obtained by applying the Rayleigh–Ritz method, using the vibration modes of flexible plates with suitable boundary conditions. Trigonometrical functions that satisfy boundary conditions of the storage tank such that the flexibility of the wall is thoroughly considered are used to define the admissible vibration modes. The analysis is then performed in the time domain. The influence of tank flexibility on the convective response components is generally small and may be neglected. Therefore, an analytic procedure is developed for evaluating convective pressure and surface displacements in a similar rigid tank. The main objective of this study is to close the gap between analytical studies and practical design considerations. This is to provide the practicing engineers with a simple and sufficiently accurate tool for estimating seismic response of rectangular tanks. A mechanical model, which takes into account the deformability of the tank wall, is developed. The parameters of such a model can be obtained from developed charts and the maximum seismic loading of the base of a deformable rectangular tank and a section immediately above it can be predicted by means of a response spectrum characterizing the design earthquake. Accordingly, a 2D simplified model for pressure distribution curve on flexible tank wall is suggested to improve code formulas for seismic design of liquid storage tanks.

Fig. 1. 3-D model of rectangular container.

incompressible and non-viscous liquid of depth HL, as shown in Fig. 1. The side lengths and height of the tank are 2Lx, 2Ly and Hs, respectively. The walls of the tank are considered as thin plates made of linearly elastic, homogeneous and isotropic material and are assumed to perform transverse bending deflection but no inplane deformation. The motion of the liquid is assumed to be frictionless and irrotational so that the velocity distribution of the liquid may be represented as a gradient of the velocity potential: ux ¼

@F , @x

uy ¼

@F , @y

uz ¼

@F @z

ð1Þ

According to the theory of fluid dynamics, the liquid velocity potential should satisfy the Laplace equation: @2 F @2 F @2 F þ 2 þ 2 ¼0 @x2 @y @z

ð2Þ

And the hydrodynamic pressure at any point and time is given by pðx,y,z,tÞ ¼ rl

@F @t

ð3Þ

in which rl is the mass density of the liquid. Considering the walls of the tank to be impermeable and no cavitation on the liquid–wall interface, the liquid adjacent to the wall must move with it by the same velocity. The consistent conditions between the liquid motion and the tank-wall deflection, w(x, y, z, t), can be given as     @F @w @F @w ¼ , ¼ ð4Þ    @t x ¼ 7 Lx @t y ¼ 7 Ly @x x ¼ 7 Lx @y y ¼ 7 Ly The rigid bottom condition of the tank is  @F ¼0 @z z ¼ 0

ð5Þ

Assuming that the liquid makes a small amplitude oscillation, the linearized sloshing condition at the free liquid surface ðz ¼ HL Þ is @2 F @F ¼0 þg @z @t 2

ð6Þ

3. Sloution of impulsive pressure 2. Tank geometry and basic equations A rectangular tank with four flexible vertical walls of uniform thickness ts and a horizontal rigid bottom is partially filled with

The coupling between liquid sloshing modes and wall vibrational modes is weak; consequently, for the analysis it is sufficient to consider the two uncoupled systems separately. This includes

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

the liquid–wall system and the free surface gravity waves in a similar rigid tank [4–6,14,16,17]. The solution for F will be expressed as the sum of the impulsive component, Fi, and the convective component, Fc. The impulsive component of the solution satisfies the actual boundary conditions along the tank wall and bottom and the condition of zero hydrodynamic pressure at z¼HL, whereas the convective component corrects for the difference between the actual boundary condition at z ¼HL and the one considered in the development of the impulsive solution. Using the superposition method the impulsive velocity potential can be analytically given in an infinite series form. One can obtain impulsive pressure due to wall deformability using Eq. (3): pf ¼ pxf þpyf

ð7Þ

where pxf satisfies the flexible wall conditions in the x-direction and the rigid wall conditions in the y-direction and the no sloshing condition on the free liquid surface. pyf satisfies the flexible wall conditions in the y direction and the rigid wall conditions in the x direction and the no sloshing condition on the free liquid surface. If the tank is subjected to an earthquake in the x direction, u€ gx ,using Eqs. (2)–(5), the condition of zero hydrodynamic pressure at z ¼HL, the method of separation of variables and considering the symmetry of both the liquid and the tank about the x–z plane and antisymmetry of them about the y–z plane (Fig. 2), the impulsive pressure can be analytically given in an infinite series form: pxf ¼ 

1 X 1 X 4rl cosðzj zÞcosðbi yÞ

xij HL Ly

i¼0j¼0

Z

Ly Z

HL

 0

pyf ¼ 

0

U

sinhðxij xÞ coshðxij Lx Þ

2

@ w ðLx ,y,z,tÞcosðzj zÞcosðbi yÞdz dy @t 2

i¼0j¼0 Lx

Z

 0

0

HL

02 2 x2ij ¼ bi 2 þ z2j , x02 ij ¼ bi þ zj

ð12Þ

( Ly ¼

2Ly ,

i¼0

Ly ,

ia 0

ð13Þ

The hydrodynamic pressure on the tank wall of a similar rigid € tank, pgx , can be obtained by substituting u€ gx ðtÞ instead of win Eq. (8): pgx ¼ Igx u€ gx ðtÞ

ð14Þ

where Igx ¼

1 X 4rl ð1Þj coszj z sinhx0j x ð2j þ1Þpx0j coshx0j Lx j¼0

ð15Þ

Finally, one can determine the impulsive pressure using the following equation: pi ¼ pf þ pgx

ð16Þ

4. Admissible functions It is important to develop a set of suitable admissible functions to describe the vibration of the tank wall in the Rayleigh–Ritz method. For this purpose, the transverse motion of the flexible walls is expressed as a linear combination of admissible functions: M X N X

V f mn ðtÞCH m Cn

ð17Þ

m¼1n¼1

ð9Þ

in which

zj ¼ ð2j þ 1Þp=2HL

ð11Þ

wðx,y,z,tÞ ¼

0 0 l cosðzj zÞsinðbi xÞ coshðxij yÞ U 0 0 xij HL Lx sinhðxij Ly Þ

@2 w 0 ðx,Ly ,z,tÞcosðzj zÞsinðbi xÞdz dx @t 2

bi ¼ pi=Ly , b0i ¼ ð2i þ1Þp=2Lx

ð8Þ

1 X 1 X 4r

Z

25

ð10Þ

where coefficients f mn ðtÞ equal the generalized coordinates to be determined; CVn equals the nth vibration mode of a cantilever beam that is appropriate for wall displacement in a vertical direction; and CHm equals the mth eigenfunction of vibrating beams, which boundary conditions are similar to those of the wall in the horizontal direction. Because of the symmetry of both the liquid and the tank about the x–z plane and anti-symmetry of them about the y–z plane (Fig. 2(a)), only a quarter of the tank with the appropriate boundary conditions is considered. As the tank wall can only provide transverse deflection but no in-plane deformation, the vibration of the tank wall can be equivalent to the vibration of a line supported rectangular plate whose plane is shown in Fig. 2(c). The connecting line of the adjacent walls corresponds to the internal simply support condition which prevents the transverse motion of the plate but offers no resistance to the rotation. The general form for eigenfunctions of the vibrating beam of two parts of the beam shown in Fig. 2(c) can be expressed as ( X m ðxÞ, y ¼ Ly , 0 r x r Lx CHm ðx,yÞ ¼ ð18Þ Y m ðyÞ, x ¼ Lx , 0 ry r Ly By equating the determinants of the homogeneous boundary conditions to zero, eigenvalues km for the mth mode are determined in two sets of equations. The first set of equations which governs for most modes of the tank is 0

0

cotðkm Þcothðkm Þ þ tanðkm Þ þ tanhðkm Þ ¼ 0 Fig. 2. Relation of the tank wall with the line supported rectangular plate and the relation of the plate with the beam: (a) deflected shape of the tank wall, (b) equivalent model of the tank wall using symmetry, (c) a unit strip equivalent to (b).

ð19Þ

The second set of equations which may govern for some modes of the tank which has particular lengths is 0

sinðkm Þ ¼ cosðkm Þ ¼ 0

ð20Þ

26

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38 0

where km and km are the shape parameters which are in the relationship: 0

km ¼ km

Ly Lx

ð21Þ

Eigenfunctions corresponding to the first set of equations (Eq. (19)) are     km x km x X m ðxÞ ¼ sin ð22Þ =sinðkm Þsinh =sinhðkm Þ Lx Lx  0   0  k y k y 0 0 Y m ðyÞ ¼ cos m =cosðkm Þ þ cosh m =coshðkm Þ Ly Ly

ð34Þ

where Ms and Kare the mass matrix and stiffness matrix of the flexible wall that is modeled as a plate, respectively. rs and B are the mass density of the liquid and the area of a quarter of the tank wall shown in Fig. 2 (b), respectively. The external forces acting on the wall due to ground motion u€ gx ðtÞ include (i) the distributed inertia force of the wall which is given by

ð23Þ F gx ¼ rs t s u€ gx ðtÞ

and eigenfunctions corresponding to the second set of equations (Eq. (20)) are   km x X m ðxÞ ¼ sin ð24Þ =cosðkm Þ Lx  0  k y 0 Y m ðyÞ ¼ cos m =sinðkm Þ Ly

ð35Þ

and (ii) the hydrodynamic pressure on the tank wall. The work done by these external loads during arbitrary virtual displacement dw can be expressed as Z Z dW ¼ F gx dw dAþ pi ðx,y,z,tÞ dw dA ð36Þ B

ð25Þ

For the cantilever beam condition that is appropriate for wall displacement in the vertical direction, the vibration functions for the nth mode is          g z g z g z g z CVn ¼ cosh n cos n sn sinh n sin n ð26Þ HL HL HL HL

B

where A is the area of a quarter of the tank wall shown in Fig. 2 (b) which is in contact with liquid. A system of ordinary differential equations can be obtained by application of Hamilton’s principle: Z t Z t dðEK EP Þdt þ dW dt ¼ 0 ð37Þ 0

0

Therefore, the matrix equations which govern the earthquake response of the undamped liquid-wall system are determined:

in which

sn ¼

The kinetic energy of the wall is given by Z  2 r ts @w 1 T EK ¼ s dB ¼ f_ Ms f_ 2 2 B @t

cosðgn Þ þ coshðgn Þ sinðgn Þ þ sinhðgn Þ

ð27Þ

and the eigenvalues (gn ) for the nth mode are determined using Eq. (28): cosðgn ÞUcoshðgn Þ þ1 ¼ 0

ð28Þ

ðMs þ Mliquid Þf€ þ K f ¼ Ix u€ gx in which Z Ix ¼ rs t s

Hs

Z

0

Ly

CT dy dzþ

0

ð38Þ

Z

Hl

0

Z

Lx

Igx CT dx dz þ

0

Z

Hl 0

Z

Ly

Igx CT dy dz

0

ð39Þ and 5. Analysis method Mliquid ¼

Z

þ

ð29Þ

HL



C ¼ ½CH1 CV1

CH1 CV2    CH1 CVN

CH2 CV1    CH2 CVN    CHM CV1

CHM CV2    CHM CVN 

ð30Þ T

f ¼ ½f 11 ðtÞ

f 12 ðtÞ    f 1N ðtÞ

f 21 ðtÞ    f 2N ðtÞ    f M1 ðtÞ

f M2 ðtÞ    f MN ðtÞ

ð31Þ The strain energy of the wall is expressed by 8 ! !9 < @2 w @2 w 2 @2 w @2 w @2 w = þ 2 2ð1nÞ  dy dz 2 @z @y2 @z2 @y @z ; 0 0 : @y 8 !2 !9 Z Z D Hs Lx < @2 w @2 w @2 w @2 w @2 w = þ 2ð1 n Þ  þ dx dz 2 2 0 @z2 @x2 @z2 @x @z ; 0 : @x Hs

Z

Z

0

U

sinhðxij xÞ dA coshðxij Lx Þ

A

0

CT

0 4rl cosðzj zÞsinðbi xÞ coshðxij yÞ dA U 0 0 xij Hl Lx sinhðxij Ly Þ

Cðx,Ly ,z,tÞcosðzj zÞsinðb0i xÞdz dx

ð40Þ

Mliquid is the added mass matrix due to the effect of the liquid. Ms and K are the mass matrix and stiffness matrix of the flexible wall that is modeled as a plate, respectively. This matrix equation of motion can be solved by the mode-superposition method. By equating the determinant of the left hand side of Eq. (38) to zero, frequencies of impulsive modes of tank and eigenvectors pertinent to them are obtained:

Ly

f¼

K X

/k qk

ð41Þ

k¼1

ð32Þ where D and n are the flexural rigidity of the tank wall and the Poisson’s ratio, respectively. Using Eq. (29), EP can be rewritten as 1 T EP ¼ f Kf 2

0

xij Hl Ly

CðLx ,y,z,tÞcosðzj zÞcosðbi yÞdz dy Z

i¼0j¼0 Lx

4rl cosðzj zÞcosðbi yÞ

HL

0 0 1 X 1 X

Z

in which

Z

Z



Using Eq. (17), the transverse motion of the flexible wall can be rewritten as w ¼ Cf

Ly

CT

A

i¼0j¼0

5.1. Impulsive response

D EP ¼ 2

1 X 1 Z X

ð33Þ

/k is the modal vector pertinent to the kth mode of vibration of the tank. K ¼ M  N is the number of modes. Introducing damping into Eq. (38) and using Eq. (41), results in q€ k þ 2zf ok q_ k þ o2k qk ¼ bk u€ gx

ð42Þ

wherezf , ok and bk are damping ratio, frequency and modal participation factor of the kth impulsive mode, respectively. The complete time history of qk ðtÞ for each mode (k¼1,2, y, K) and its time derivatives can be computed by a step-by-step

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

27

application (Interpolation of Excitation [18]). Once they are obtained, the displacement, acceleration and the impulsive pressure can be calculated using Eqs. (41), (17) and (7).

of the inertia of the oscillating liquid is neglected in this expression. The following relationship can be obtained using Eqs. (46) and (54):

5.2. Convective response

Zðx,y,tÞ ¼ Lx

1 X 2ð1Þj Aj ðtÞ

l2j g

j¼0

The convective pressure can be evaluated with reasonable accuracy by considering the tank wall to be rigid. Rather than evaluating the convective component of the pressure, pc, from the associated component of the velocity potential Fc function, it is convenient to evaluate it directly by requiring that pc satisfy Laplace’s equation: @2 pc @2 pc @2 pc þ þ ¼0 @x2 @y2 @z2 along with the boundary conditions:   @pc  @pc  ¼ 0, ¼0 @x x ¼ 7 Lx @y y ¼ 7 Ly

ð43Þ

ZðLx ,y,tÞ ¼ Lx

pc ðx,y,z,tÞ ¼ rl Lx

1 X

Dj Aj ðtÞsinðaj xÞcoshðaj zÞ

ð46Þ

j¼0

l2j g

j ¼ 1,2,. . .

in whichSac is the maximum of A0 ðtÞ, or the spectral acceleration corresponding to the natural frequencies oc ¼ ocj 9j ¼ 0 and damping zc . Therefore, 1 Sac X 2 Lx 2 g j¼0l

ð58Þ

j

where Zmax is the maximum surface displacement near the wall. On making use of identities (Standard Math Tables, [19]): 1 X 2 2 j ¼ 0 lj

¼ lim x-0

tanhðxÞ ¼1 x

Zmax rðSac =gÞLx

lj ¼ ð2j þ 1Þp=2

ð47Þ

aj ¼ lj =Lx

ð48Þ 2

ð57Þ

ð59Þ

Eq. (58) results in

in which

Dj ¼ 2ð1Þj =½lj coshðlj mÞ

ð56Þ

When the surface displacement near the wall is maximum, using Eq. (52) and by means of a response spectrum, one can conclude

ð45Þ

The solution of Eq. (43) such that it satisfies the boundary conditions defined by Eqs. (44) and (45) may be expressed as

1 X 2Aj ðtÞ j¼0

Zmax r

 @pc  ¼0 @z z ¼ 0

ð55Þ

Vertical surface displacement of the liquid near the wall x¼  Lx can be determined from:

Aj ðtÞ rSac ,

ð44Þ

sinðaj xÞ

ð49Þ

ð60Þ

This is the same as ACI 350.3 formula used to determine the vertical surface displacement.

6. Convergence and validity verification

where

m ¼ HL =Lx

ð50Þ

and Aj ðtÞ ¼ ocj

Z

t

0

h i u€ gx ðtÞ sin ocj ðttÞ dt

ð51Þ

whereAj ðtÞrepresents the instantaneous pseudo-acceleration of an undamped SDOF system which has a circular natural frequency ocj equal to that of the jth sloshing mode of vibration of the liquid and is excited by a base acceleration u€ gx ðtÞ.

oc2 j ¼ g aj tanhðaj H L Þ

ð52Þ

Damping of convective responses (zc ) can be introduced into Eq. (51) easily. The first sloshing frequency determined using Eq. (52) is equal to its quantity as measured using Eurocode-8 2006. The hydrodynamic pressure at any point and time is given by pðx,y,z,tÞ ¼ pf þ pc þpgx

ð53Þ

where pc , pf and pgx are, respectively, convective pressure, pressure due to flexibility, and pressure on a similar rigid tank. 5.3. Surface displacements The vertical displacement,Zðx,y,tÞ, of an arbitrary point at the liquid surface is determined from:  ð54Þ pðx,y,tÞz ¼ H ¼ rl g Zðx,y,tÞ L

which effectively states that the hydrodynamic pressure at z ¼HL is equal to the weight of the liquid column above. The contribution

In this section, some numerical examples are presented to investigate the convergence and validity of the present method. A computer program was first written to determine the validity of the theoretical formulation in computing seismic responses of the flexible storage tank. The dimensionless natural frequencies (O) of wall vibration are expressed as

O ¼ ðrs o2 ts H4s =DÞ1=2

ð61Þ

whereD is the flexural rigidity of the tank wall: D ¼ Et3s =½12ð1n2 Þ

ð62Þ

where E is the Young’s modulus of the tank wall. The first six dimensionless natural frequencies of wall vibration (impulsive modes) of the empty cubic tank (2Lx ¼ 2Ly ¼ Hs ) and a cubic tank fully filled with liquid which has a mass ratio of the liquid to the tank-wall equal to 5 are presented in Table 1 and compared with those presented by Kim and Dickinson [20] and Zhou and Liu [21]. Five terms of the static beam functions are used in each direction. Good agreements are observed for all cases. The convergence study on the fundamental impulsive frequencies with respect to the term number of the static beam functions is shown in Table 2. It is seen that the impulsive frequencies monotonically decrease with the increase of the term number of the static beam functions. The results show rapid convergence to the present method. The present method is now applied to investigate seismic response characteristics of 3-D rectangular fluid storage tanks. Since rectangular tanks are used most often for the wet-type storage of nuclear spent fuel assemblies, a typical dimension for those tanks is selected for the next example [12]: height of the

28

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

Table 1 Comparison on the first six impulsive frequencies of the empty and filled cubic tank. Dimensionless natural frequency

Empty cubic tank

O1 O2 O3 O4 O5 O6

Cubic tank filled with liquid

Present study

Zhou and Liu (2007)

Kim and Dickinson (1987)

Present study

Zhou and Liu(2007)

17.619 36.049 52.076 71.206 74.377 106.297

17.632 36.045 52.133 71.208 74.576 107.531

17.577 36.104 52.069 71.463 74.579 107.13

8.212 18.011 26.742 38.484 42.441 60.47

8.48 18.175 26.212 37.494 42.931 58.016

Table 2 The convergence study on the fundamental impulsive frequencies with respect to the term number of the static beam functions. Fundamental dimensionless natural frequency

Empty cubic tank Cubic tank filled with liquid

Term number of the static beam functions in each direction 11

22

33

44

55

17.8625 8.8104

17.6582 8.2506

17.6324 8.2221

17.6317 8.2198

17.6194 8.2118

15000

Resultant force, kN

10000 5000 0 50001000015000-

0

2

4

6

8

10

Time, sec Fig. 3. Time history of the resultant force acting on the long side wall.

0

Depth (m)

wall, Hs ¼ 10 m; wall thickness, ts ¼1 m; water depth, HL ¼ 9 m; length of the short side wall, 2Lx ¼20 m; and length of the long side wall, 2Ly ¼50 m, and typical material properties for the concrete tanks: density, rs ¼2400 kg/m3; Young’s modulus, E¼2.1  1010 N/m2; and the Poisson’s ratio, n ¼0.17. The N–S component of the 1940 El Centro earthquake records is used as an input motion in the x direction. The tank is assumed to be fixed to the ground, and to have 3% structural damping. The time history of the resultant force acting on the long side wall of the tank model is presented in Fig. 3 and can be compared with those presented by Koh et al. The comparison shows that they are in good agreement and the maximum of the resultant force predicted by the present method (9460 kN) is very close to that obtained by Koh et al. (9244 kN). The accuracy of the present method can be further verified by pressure distribution over the long side wall when the resultant forces reach their peak values. The predictions by the proposed method are compared with the results from the coupled 3-D BEM-FEM [12] and a Lagrangian fluid finite element [10] as shown in Fig. 4. It is worthy here to note that hydrodynamic pressure at the time when the equivalent resultant force is maximum can be much amplified in the middle length of the wall in rectangular tanks and shows the distribution of 2-D spatial variation over the surface of the wall as shown in Fig. 5.

3

Rigid Tank 6

present work Koh et al. (1998) Dogangun and Livaglu (2004)

9 0

10 20 30 Hydrodynamic Pressure (kPa)

40

Fig. 4. Comparison of pressure distributions along the height of the middle crosssection of the long side wall.

Time histories of the sloshing motion at the middle crosssection of the long side wall are presented and compared with those using the indirect BEM-FEM [12] and FEM [22] in Fig. 6. The predictions by the proposed method are virtually identical to those of mentioned references. In the following figures and tables, the term exact is used to show the results of proposed analytical method.

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

29

Fig. 5. Pressure distribution on the long side wall.

1

7.1. Hydrodynamic base shear The instantaneous hydrodynamic base shear Q(t) is given by

Height, m

0.5 Q ðtÞ ¼ 2

Z

Z

Ly

Hl

0

Ly

 p

dz dy

ð63Þ

x ¼ Lx

0

-0.5

-1

The hydrodynamic base shear due to wall deformation relative to the ground Qf (t) can be expressed as Z Ly Z Hl K X  Q f ðtÞ ¼ 2 pf  dz dy ¼ Q fk u€ k ðtÞ ð64Þ

Present Work Ghaemmaghami (2010) Koh et al. (1998) 0

2

4

0

Ly

6

8

10

Time, sec Fig. 6. Time history of liquid surface elevation at the middle cross-section of the long side wall.

where 2 Q fk ¼ 4

1 X

8rl

1 X 1 X i¼0j¼0

Z

7. Seismic design forces

Lx

Z

 0

0

k¼1

sinðzj Hl Þtanhðzj Lx Þ 2

zj Hl

j¼0

þ

The most common standards and codes currently used for the design of tanks are ACI 350.3 [23], Eurocode-8 [24] and New Zealand standard [25]. Seismic loads in these standards are based on the mechanical model derived by Housner [2,3] for rigid tanks with some modifications. The design procedure considers two response modes of the contained liquid: (1) the impulsive response in which the portion of the liquid accelerates with the tank walls and (2) the convective response caused by the portion of the liquid sloshing in the tank. It should be noted that in these codes, the importance of the effects of wall flexibility has been recognized and the corresponding increase in the acceleration coefficients has been adopted. However, the effect of wall flexibility has not been thoroughly accounted for by a reasonable and accurate method. The main objective of this part of the study is to devise a practical approach which would allow, from an engineering point of view, a simple, fast and sufficiently accurate estimate of the seismic response of rectangular storage tanks.

x ¼ Lx

HL

Z

Ly

Z

0

HL 0

CðLx ,y,zÞcosðzj zÞdz dy

0

16rl

sinðzj HL Þsinðbi Lx Þ

x0ij 2 HL Lx zj

Cðx,Ly ,zÞcosðzj zÞsinðb0i xÞdz

# dx /k bk

ð65Þ

and uk ðtÞis the solution of the differential equation: u€ k ðtÞ þ 2zf ok u_ k ðtÞ þ o2k uk ðtÞ ¼ u€ gx ðtÞ

ð66Þ

For the applicable tank, all frequencies of impulsive modes are large. Acceleration spectral spectrums accepted in codes show that the maximum accelerations which correspond to significant modes are close to each other. Therefore the mode that has the maximum Q fk is the predominant mode. The dimensionless fundamental natural frequency (Of ) that corresponds to the predominant mode of vibration is displayed for tanks completely filled with water in Fig. 7 for two different values of ts/HL, assuming normal density concrete is used. Using Eq. (61), the fundamental natural frequency (of ) that is pertinent to Of can be determined for different values of the aspect ratio. Fig. 7 shows that the first mode is predominant when Lx r 1:2Ly , otherwise another mode could become predominant. Therefore, in general, for considering the seismic response,

30

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

Fig. 7. Dimensionless fundamental natural frequency (Of ) for tanks completely filled with water (a) t s =HL ¼ 0:05 and (b) t s =HL ¼ 0:1.

it is better that the effect of all modes is considered. The value of u€ f ðtÞcorresponding to of can be obtained by the solution of Eq. (66) for the predominant mode. It is assumed that the maximum accelerations corresponding to significant modes are

equal to the maximum of u€ f ðtÞ. Therefore Eq. (64) can be rewritten approximately: Q f ðtÞ ¼ mf u€ f ðtÞ

ð67Þ

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

31

0.5 ts/HL=0.05

0.4

ts/HL=0.1 Approximte(Eq.68)

mf/mL

0.3

0.2

0.1

0 0.5

2

3.5

5

6.5

8

9.5

2Lx/HL 1.4 Exact 1.2 Cf

Approximate(Eq. 69)

1

0.8

0

1

2

3

4

5

Ly/HL Fig. 8. Values of mf/mL and correction factor (a) factors mf/mL versus ratio 2Lx/HL, (b) its correction factor versus ratio Ly/HL.

where ml is the total mass of liquid. Investigations show that when Ly =HL o 0:65, a correction factor for mf ,cf , should be considered: cf ¼ 1:250:71ðLy =HL 0:3Þ

1 0.8

mr/mL

where mf is obtained by combining Q fk for all modes (k¼1,2,y,K) by the Complete Quadratic Combination (CQC) method [26]. The value of mf can be approximated using Eq. (68) obtained by curve fitting the numerical results shown in Fig. 8. The figure shows that mf is independent of wall thickness:       mf 2Lx 2Lx ¼ tanh 0:866 = 1:732 ð68Þ ml HL HL

Analytical

0.6

ACI

0.4 0.2

ð69Þ

0 Using Eqs.(53), (63) and (67), the base shear may be expressed as Q ðtÞ ¼

1 X

mcj Aj ðtÞ þ mf u€ f ðtÞ þ mr u€ gx ðtÞ

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

2Lx/HL ð70Þ Fig. 9. Comparison of analytical and ACI solutions for mr/ml versus 2Lx/HL.

j¼0

in which 1 X ð1Þj tanhðlj =mÞsinðlj Þ mr ¼2 ml l3 =m j¼0

ð71Þ

the next modes are negligible:

j

Q ðtÞ ¼ mc u€ c ðtÞ þ mf u€ f ðtÞ þmr u€ gx ðtÞ mcj ml

¼

2 tanhðlj mÞ

ð72Þ

l3j m mcj

are equivalent masses corresponding to forces where mr , mf and associated with ground motion, wall deformation relative to the ground, and liquid sloshing, respectively. Fig. 9 shows mr obtained using Eq. (71) is in good agreement with that determined by ACI 350.3. To determine the base shear, one can consider only the first mode of convective response while

ð73Þ

where mc ¼ mcj 9j ¼ 0 and u€ c ðtÞ ¼ A0 ðtÞ is the absolute acceleration of an SDOF system which has a circular natural frequency oc ¼ ocj 9j ¼ 0 . Since the base shear and moment due to wall deformability are proportional to the relative acceleration of the wall, one can rearrange Eq. (73) in order to estimate the maximum seismic loads by means of a response spectrum: Q ðtÞ ¼ mc u€ c ðtÞ þ mf ½u€ f ðtÞ þ u€ gx ðtÞ þ ½mr mf u€ gx ðtÞ

ð74Þ

32

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

And subsequently, the maximum base shear can be estimated by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð75Þ 9Q max 9 ¼ ðmc sac Þ2 þ ðmf saf Þ2 þ ½ðmr mf Þðu€ gx Þmax 2 in which Sac and Saf are the spectral accelerations corresponding to the natural frequenciesoc andof , respectively. 7.2. Hydrodynamic base moment The overturning moment M(t) induced on a section of the tank immediately above its base is given by Z Ly Z Hl  p z dz dy ð76Þ MðtÞ ¼ 2 0

Ly

x ¼ Lx

The overturning moment due to wall deformation relative to the ground,Mf ðtÞ, applied to the bottom of the wall using Eq. (7) can be expressed as Z Ly Z Hl Q X  pf  z dz dy ¼ M fk u€ k ðtÞ ð77Þ M f ðtÞ ¼ 2 0

Ly

where 2 M fk ¼ 4

1 X

8rl

x ¼ Lx

Rðzj Hl Þtanhðzj Lx Þ

1 X 1 X

0

Lx

Z

0

0

HL

Z

0

HL 0

CðLx ,y,zÞcosðzj zÞdz dy



1 1 X ð1Þj tanhðlj =mÞRðlj Þ X ð1Þj tanhðlj =mÞsinðlj Þ hr ¼ = 4 HL l l3 j¼0 j¼0 j

2

x0ij HL Lx z2j



Cðx,Ly ,zÞcosðzj zÞsinðb0i xÞdz dxÞ /k bk

ð78Þ

ð82Þ

j

c

hj csc hðlj mÞcothðlj mÞ ¼ þ1 HL lj m

ð83Þ

The numerical values of hr are determined from Fig. 11 that are compared with those obtained using ACI 350.3. One may consider only the first mode of convective response to determine the overturning moment: MðtÞ ¼ mc hc u€ c ðtÞ þ mf hf u€ f ðtÞ þmr hr u€ gx ðtÞ

ð84Þ

c wherehc ¼ hj 9j ¼ 0 .

The maximum overturning moment applied to the bottom of the wall is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9M max 9 ¼ ðmc hc sac Þ2 þ ðmf hf saf Þ2 þ½ðmr hr mf hf Þðu€ gx Þmax 2 ð85Þ Using Eqs.(74) and (84) a mechanical model that is equivalent to the rectangular liquid storage tank has been developed and shown in Fig. 12. In this model: m0 ¼ mr mf

ð86Þ

and h0 ¼ ðmr hr mf hf Þ=m0

Rðzj HL Þsinðbi Lx Þ

l

Z

Ly

0

@16r

i¼0j¼0



Z

zj 3 Hl

j¼0

þ

k¼1

concentrated:

ð87Þ

A clear distinction must be made between the foundation moment M0 (t) and the moment M(t) induced on a section of the tank immediately above its base. The latter moment is due exclusively to the hydrodynamic wall pressure, whereas the

in which R is defined as a function: ð79Þ

0.5

for all modes (k ¼1, mf hf can be determined by combining 2, y, K) using CQC method. The value of hf is obtained by dividing it by mf for different values of the aspect ratio, and is shown in Fig. 10. A proper function is fitted with this curve approximately:

0.4

Rð Þ ¼ cosð Þ þ ð Þsinð Þ1

hf =HL ¼ 0:580:12 tanh½2:5ðLx =HL 0:25Þ

MðtÞ ¼

0.3 0.2

ð80Þ

The results show that the effect of variation of thicknesses or Ly =HL on hf =HL is negligible. Using Eqs. (53) and (76), MðtÞ can be expressed as 1 X

hr/HL

M fk

c mcj hj Aj ðtÞ þ mf hf u€ f ðtÞ þ mr hr u€ gx ðtÞ

ð81Þ

j¼0

0.1

Analytical ACI

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 2Lx/HL Fig. 11. Comparison of analytical and ACI solutions for hr/HL versus 2Lx/HL.

c

where hj , hf and hr are, respectively, the heights at which the jth convective component of the liquid mass, mcj , equivalent mass corresponding to force associated with ground motion, mr , and equivalent mass corresponding to force associated with wall deformation relative to the ground,mf , are considered to be

0.7 0.6

hf/HL

0.5 0.4 0.3

Exact Approximate

0.2 0.1 0 0.25

1

1.75

2.5

3.25

4

4.75

Lx/HL Fig. 10. Comparison of exact and approximate solutions for hf/HL versus Lx/HL.

Fig. 12. Mechanical model of flexible tank.

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

foundation moment M0 (t) also incorporates the effect of the hydrodynamic pressure exerted on the tank base. The foundation moment M0 (t) is then given by

3.5

M 0 ðtÞ ¼ MðtÞ þ DMðtÞ

2.5

where the moment increment DMðtÞ induced by the pressure on the tank base is given by Z Ly Z Lx  DMðtÞ ¼ p x dx dy ð89Þ Ly

Lx

4 3 h'r/HL

ð88Þ

Q X

Analytical ACI

2 1.5 1 0.5

z¼0

0

The hydrodynamic base moment due to wall deformation relative to the ground,M 0f , using Eq. (53) can be expressed as M 0f ¼

33

ðM fk þ DM fk Þu€ k

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 2Lx/HL

ð90Þ

Fig. 14. Comparison of factors h0 r/HL versus ratio 2LX/HL.

!

Table 3 Comparison of factors hC/HL, mC/mL and h0 C/HL versus ratio 2Lx/HL for rectangular tanks.

k¼1

where 2

1 X

DMfk ¼ 4

8rl

Lx zj tanhðLx zj Þ

zj Hl

j¼0 1 X 1 X

þ

0

16rl

Lx

Z

HL

 0

0

Ly

Z

0

HL 0

0

CðLx ,y,z,tÞcosðzj zÞdz dy

2Lx/HL

hC/HL

0

sinðbi Lx Þbi Lx cosðbi Lx Þ 02 x02 ij H L Lx bi

i¼0j¼0

Z

Z

3

# Cðx,Ly ,z,tÞcosðzj zÞsinðb0i xÞdz dx /k bk

ð91Þ

1 2 4 6 8

h0 c/HL

mC/mL

ACI

Analytical

ACI

Analytical

ACI

Analytical

0.709 0.583 0.524 0.511 0.506

0.708 0.582 0.524 0.511 0.506

0.263 0.485 0.695 0.765 0.793

0.257 0.473 0.677 0.743 0.771

0.736 0.858 1.986 3.989 6.814

0.736 0.859 1.992 4.001 6.834

mf hf can be determined by combining ðM fk þ DM fk Þ for all modes 0 (k¼1,2, y,K) using the CQC method. hf has been obtained by dividing it by mf for different values of the aspect ratio, and is shown in Fig. 13. A proper function is fitted with this curve approximately: ( 0 0:58, Lx =HL r 1:2 hf ð92Þ ¼ 0:77ðLx =HL Þ=tanhð1:732Lx =HL Þ, Lx =HL 4 1:2 HL 0

Using Eqs.(53), (88) and (89), M 0 ðtÞ can be expressed as M 0 ðtÞ ¼

1 X

0c

0

0

mcj hj Aj ðtÞ þ mf hf u€ f ðtÞ þ mr hr u€ gx ðtÞ

ð93Þ

j¼0

in which 0c

hj

Hl 0

¼

hr ¼ HL

2csc hðlj mÞcothðlj mÞ

lj m P1

j¼0

þ1

ð94Þ

3

ð1Þj =lj ½ðRðlj Þtanhðlj =mÞtanhðlj =mÞ=ðlj ÞÞ þ ð1=mÞ P1 3 j j ¼ 0 ðð1Þ tanhðlj =mÞsinðlj Þ=ðlj ÞÞ

ð95Þ

0

The numerical values of hr can be determined from Fig. 14 which are compared with those obtained using ACI 350. One may consider only the first mode of convective response to determine

4 3

ð96Þ

0c hj 9j ¼ 0 .

0 hc

¼ The maximum base moment applied to the where bottom of the wall is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 0 0 ð97Þ 9M 0max 9 ¼ ðmc hc sac Þ2 þ ðmf hf saf Þ2 þ½ðmr hr mf hf Þðu€ gx Þmax 2

2.5 h'f/HL

the base moment: 0 0 0 M0 ðtÞ ¼ mc hc u€ c ðtÞ þ mf hf u€ f ðtÞ þ mr hr u€ gx ðtÞ

Exact Approximate

3.5

Fig. 15. The model suggested for pressure distribution.

2 1.5

To develop an equivalent mechanical model for determining 0 0 0 base moment, Fig. 12 can be used by replacing hc , hf and h0 with hc , hf and h0 , respectively:

1 0.5 0 0.25

0

1

1.75

2.5

3.25

4

Lx/HL Fig. 13. Comparison of factors h0 f/HL versus ratio Lx/HL.

4.75

0

0

h0 ¼ ðmr hr mf hf Þ=m0

ð98Þ

Table 3 shows that hc/HL, h0 c/HL and mc/ml obtained from the present work are in good agreement with those determined by ACI 350, however, the suggested formulas are easier.

34

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

0.8 0.75

hmax/HL

0.7 0.65 0.6 Exact approximate response

0.55 0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Lx/HL Fig. 16. Factors hmax/HL versus ratios Lx/HL.

7.3. Pressure distribution curve While the most common standards and codes currently used for the design of 3-D rectangular liquid tanks do not consider the variation of impulsive pressure in the horizontal direction, it has been confirmed that the impulsive hydrodynamic pressure, even for the case of fairly thick walls, is much amplified in the middle of the wall and shows the distributon of 2-D spatial variation. In other words, impulsive hydrodynamic pressure varies not only in the vertical direction but also in the horizontal direction over the wall surface. After some investigation for different values of thickness, aspect ratio and different earthquakes, the model shown in Fig. 15 is suggested for the impulsive pressure distribution curve on the walls perpendicular to the earthquake direction when the base shear is at the peak value. In this model, when y Z0, one can write: ( pi ðy,zÞ ¼

½pmax yðpmax pE Þ=Ly z0 =h1 , pE y=Ly þ ½pmax þðp0 pmax Þðz0 h1 Þ=hmax ð1y=Ly Þ,

z0 r h1 z0 4 h1

ð99Þ

(I) hmax is the height at which maximum pressure on the centreline of the walls perpendicular to the earthquake direction occurs. To evaluate hmax, only the predominant impulsive mode has been considered and the Golden Section Search Technique has been used for optimization. It can be determined from Fig. 16 or the approximate formula: ( 0:7930:13Lx =HL , Lx =HL o1:1 hmax ¼ ð100Þ 0:65, Lx =HL Z1:1 HL

Fig. 17. The bf is the perpendicular distance between the point of action of the equivalent load and the center-line of those walls.

0.6 Exact

0.5 0.4 bf/Ly

where h1 ¼ HL hmax . To draw the pressure distribution curve, it may be necessary to introduce two new parameters:

0.3 0.2 0.1 0 0

0.5

1

1.5

0 that,hf ,hf

and hmax have been It is worthy here to note determined for different thicknesses, ts ¼(0.05–0.1)Hs, which are applied to concrete rectangular tanks and the results show that the effect of wall thickness can be ignored. Moreover, the results have shown that the effect of variation of Ly =HL on the above mentioned parameters is negligible; Therefore, an average of the results for different Ly =HL is shown in Figs. 10, 13 and 16. (II) T(t) is the torsion on the middle of the foundation, if the impulsive hydrodynamic pressure on only half of one wall that

2 LY/HL

2.5

3

3.5

4

Fig. 18. Factors bf/Ly versus ratios Ly/HL.

is perpendicular to the earthquake direction is considered:

TðtÞ ¼

Z

Ly 0

Z 0

Hl

 pi 

x ¼ Lx

y dz dy

ð101Þ

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

35

Table 4 Comparison on the impulsive earthquake response obtained by different methods. Variable

ACI 350.3-06

Exact

Mechanical model

Fundamental natural frequency,of (Hz) Maximum surface displacement near the wall, Zmax (m) Maximum impulsive base shear,Q i,max (N) Maximum impulsive overturning moment, M i,max (N m)

16 0.49 42.6  106 159.6  106

12.96 0.456 24.7  106 114.5  106

13 0.49 29  106 131.9  106

Fig. 19. Comparison between pressure distributions on the long side wall: (a) approximate and (b) EXACT.

36

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

Using Eqs.(99),(63), (76) and (101) one may write 8 h1 Ly ðpmax þpE Þ þhmax Ly ðp0 þ2pE þ pmax Þ ¼ Q i,max > >  > < h2 Ly h1 Ly ðpmax þpE Þ hmax þ h31 þ max ðp0 þ 3pE þ2pmax Þ ¼ M i,max 3 > > 2

 > : Ly ðp max þ 2pE Þh1 þðp0 þ pmax þ4pE Þhmax ¼ T max 12

Part of this torsion that is due to wall deformation relative to the ground Tf (t) can be expressed as T f ðtÞ ¼

Z

Ly

Z

0

Hl 0

 pf 

Q X

y dz dy ¼ x ¼ Lx

T fk u€ k ðtÞ

ð102Þ

k¼1

ð106Þ

The parameter ðmf =4Þbf in which bf is the perpendicular distance between the point of action of the equivalent load due to wall deformation imposed on the walls perpendicular to the earthquake direction, and the centre-line of those walls (Fig. 17), can be determined by combining T fk for all modes (k¼1,2, y, K) using the CQC method. bf has been obtained by dividing it by ðmf =4Þ for different values of the aspect ratios, and is shown in Fig. 18. An approximate equation is suggested for determining bf: bf ¼ Ly

(

0:460:07ðLy =HL 0:3Þ,

Ly =HL r1

0:41,

Ly =HL 41

where maximum impulsive base shear, Q i,max , maximum impulsive overturning moment, M i,max , and T max can be determined using Eqs. (75), (85) and (105), respectively; however, the effect of convective pressure is not considered in Eqs. (75) and (85). Once the linear system of equations (Eq. (106)) is solved, numerical values for pmax ,p0 and pE are derived and the pressure distribution curve can be drawn easily. 7.4. Numerical example

ð103Þ

Consider the tank discussed in the preceding example but with wall thickness of ts ¼ 0.6 m. The tank is assumed to be completely full of water and its impulsive and convective responses are damped at 5% and 0.5%, respectively. The maximum ground acceleration of the N–S component of the 1940 El Centro earthquake records is ðu€ gx Þmax ¼ 0:313g. The fundamental natural frequency of sloshing, oc ¼ ocj 9j ¼ 0 ¼ 1:19 rad=s, is obtained from Eq. (52); and consequently, the spectral acceleration corresponding to it for a damping ratio of zc ¼ 0:5 % can be found from spectral response (Sac ¼ 0:049g). The fundamental impulsive frequency of liquid–wall vibration for values ofLx =HL ¼ 1, Ly =HL ¼ 2:5 and t s =HL ¼ 0:06 is determined

The effect of variation of Lx =HL or thicknesses on bf is negligible for the applicable concrete rectangular tank. Using Eqs.(53) and (101),TðtÞ can be expressed as TðtÞ ¼

  Ly 1 mf bf u€ f ðtÞ þ mr u€ gx ðtÞ 4 2

ð104Þ

The maximum of T(t) is given by

1 4

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmf bf saf Þ2 þ½ðmr Ly =2mf bf Þðu€ gx Þmax 2

ð105Þ

0

0

2 Depth (m)

2 Depth (m)

4 6

4 6 8

8 10

10 0

10 20 30 40 50 60 70 80 Hydrodynamic Pressure (kPa)

0

10 20 30 40 50 60 70 Hydrodynamic Pressure (kPa)

80

0

2

Depth (m)

9T max 9 ¼

4

6

8

10 0

10 20 30 40 50 60 70 80 Hydrodynamic Pressure (kPa)

Fig. 20. Comparison of pressure distributions along the height of the different cross-sections of the long side: (a) y¼ 25; (b) y¼12.5; and (c) y¼ 0.

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

from Fig. 7 by interpolation; Of ¼ 2:5andof ¼ 13 rad=s. The spectral acceleration for a damping ratio of zf ¼ 5 % is Saf ¼ 1:045g. The remaining parameters can be obtained from Eqs. (68) and (80): mf =ml ¼ 0:271, hf ¼ 0:464HL ¼ 4:64 m, and from Figs. 9 and 11: mr =ml ¼ 0:542, hr ¼ 0:4, HL ¼ 4 m. The exact results of maximum surface displacement near the wall, maximum impulsive base shear and maximum impulsive overturning moment induced on a section of the tank immediately above its base, respectively, are determined using Eqs. (56), (63) and (76). However, the effect of convective pressure is not considered in Eqs. (63) and (76), and are compared with the approximate results obtained by the proposed mechanical model and approximate results determined by ACI 350.3 in Table 4. The approximate results obtained by the proposed mechanical model are in good agreement with the exact results.

37

It is worthy here to note that, mr and mf of a steel cylindrical tank are close to each other [4]. Therefore, for those tanks, Eq. (75) is the same as ACI 350.3 formula. For concrete rectangular tank, the walls are not as flexible as the steel cylindrical tanks and mf is about one half of mr (assuming Ly =HL Z0:65). As a result, the proposed mechanical model obtains smaller base shear and more accurate base shear and base moment than ACI 350.3. Using Eq. (105), T max ¼ 77:7  106 N m and using Eqs. (103) and (100), bf ¼ 0:41 Ly ¼ 10:25and hmax ¼ 6:63 m. Solution of Eq. (106) results in pE ¼ 19937 Pa, pmax ¼ 61435 Pa and p0 ¼ 32292 Pa. The pressure distribution curve produced by the proposed model and that obtained by using the ACI 350.3 are compared with the exact pressure distribution curve in Figs. 19–21. The comparison shows that the results obtained by the proposed model are in good agreement with the exact results.

Hydrodynamic Pressure (kPa)

80 70 50 40 30 20 10 0

Hydrodynamic Pressure (kPa)

ApproximateResponse ExactResponse ACI

60

-52

25

70 60 50 40 30 ApproximateResponse ExactResponse ACI

20 10 0 -25

Hydrodynamic Pressure (kPa)

0 Y

0 Y

25

60 50 40 30 20

ApproximateResponse ExactResponse ACI

10 0 -25

0 Y

25

Hydrodynamic Pressure (kPa)

50 40 30 20

ApproximateResponse ExactResponse ACI

10 0 -52

0 Y

25

Fig. 21. Comparison of pressure distributions along the length of the different cross-sections of the long side: (a) z¼ 0; (b) z¼ 2.5; (c) z¼ 5; and (d) z ¼ 7.5.

38

S. Hashemi et al. / Thin-Walled Structures 66 (2013) 23–38

8. Conclusions

References

Analytical solution methods are developed that can be used for the analysis of the dynamic behavior of a partially filled rectangular fluid container under horizontal ground excitations. In concrete liquid storage tanks, the natural frequencies of significant sloshing modes are much lower than the fundamental impulsive frequencies. Consequently, the effect of the surface waves on the impulsive frequencies and the effect of the tankwall flexibility on the sloshing frequencies can be ignored. In such cases, the studies on the impulsive response of the tank-wall vibration and the sloshing response of the liquid oscillation can be separately considered without significant errors. Using the combination of the superposition method and the method of separation of variables, the exact analytical solution of the impulsive pressure is derived. Solutions based on 3-D modeling of the rectangular containers are obtained by applying the Rayleigh–Ritz method, using the vibration modes of flexible plates with suitable boundary conditions. Trigonometrical functions that satisfy boundary conditions of the storage tank, such that the flexibility of the wall can be thoroughly considered, are used to define the admissible vibration modes. The analysis is then performed in the time domain. Moreover, an analytical procedure is developed for deriving a simple formula that estimates convective pressure and surface displacements in a similar rigid tank. The validity and convergence of the proposed methods are confirmed through numerical examples. The predictions by the proposed method are in good agreement with the results obtained from a Lagrangian fluid finite element, FEM, and the results obtained by using the coupled BEM-FEM. The results of this study show that hydrodynamic pressure distributions for assuming rigid and flexible walls differ from each other in magnitude and in shape. The hydrodynamic pressure in the middle of the wall for flexible storage tanks are generally larger than for rigid storage tanks and shows the distributon of 2-D spatial variation. In other words, hydrodynamic pressure varies not only in the vertical direction but also in the horizontal direction over the wall surface. A mechanical model, which takes into account the deformability of the tank wall, is developed and its parameters can be obtained from charts. The maximum seismic response of the base of a deformable rectangular tank and a section immediately above it can therefore be estimated by means of a response spectrum. Accordingly, a 2-D simplified model for pressure distribution curve on flexible tank wall is suggested to improve code formulas for seismic design of liquid storage tanks. Comparison with the exact solution of the problem confirms the validity of the method. The simple and sufficiently accurate method presented in this paper which estimates the seismic response and pressure distribution of flexible rectangular storage tanks is recommended to be considered as design rules in design codes and standards.

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