Mechanical model for cylindrical flexible concrete tanks undergoing lateral excitation

Soil Dynamics and Earthquake Engineering 106 (2018) 148–162

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Mechanical model for cylindrical flexible concrete tanks undergoing lateral excitation R. Moradia, F. Behnamfara, S. Hashemib, a b

T

⁎

Department of Civil Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran Department of Civil Engineering, Yasouj University, Yasouj, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Concrete cylindrical tank Liquid-shell interaction Rayleigh–Ritz method Seismic design Mechanical model

In this paper, a mechanical model is developed for evaluating the seismic response of flexible concrete cylindrical tanks under horizontal ground motion. For obtaining the parameters of the liquid-shell interaction of this model, a semi-analytical approach is employed using the Rayleigh–Ritz method. In the developed analytical approach, the vibration modes of a deformable open-top clamped-bottom shell are considered. The shell is assumed to be thin and the first-approximation theory is applied. Based on the analytical approach, a simple yet sufficiently accurate mechanical model, in which effects of the liquid and the tank wall considered separately, is proposed for tanks completely or partially filled with liquid. Parameters of this model are illustrated in charts easy to use for design purposes. In this model, only the first circumferential and vertical modes are considered. Also, the time history of sloshing wave height and its maximum are obtained. Finally, the base shear and the overturning moment, calculated by the analytical method and proposed mechanical model, are compared with those suggested by ACI 350.3-06. The results demonstrate that the proposed mechanical model is very successful in predicting the base shear and overturning moment, but ACI 350.3-06 overestimates or underestimates the responses case by case. Therefore, this model can be utilized with confidence for estimation of the design seismic loads of concrete cylindrical tanks.

1. Introduction Liquid storage tanks, as special structures, behave differently from ordinary structures in terms of dynamic characteristics. These important structures are mostly constructed in two different shapes of cylindrical and rectangular in accordance with their use for storing various liquids such as water and oil in the form of ground, elevated, buried and semi-buried structures. Due to the large forces caused by the hydrodynamic pressure during ground motion, the correct evaluation of these containers is very important. One of the first investigations on a rigid tank under earthquake excitation was that of Hoskins and Jacobsen [1]. Although investigation of the dynamic responses of a liquid storage tank was started based on the study of the dynamic responses of a fuel tank in aerospace engineering [2] and there are some research in mechanical engineering about it [3,4], there is a difference between the research on the dynamic responses of fuel tanks in aerospace or mechanical engineering and those in civil engineering. The latter is more concerned with response in the lower frequencies as the size of such tanks is so large. Consequently, the dynamic response resulting from the lower

⁎

frequencies dominates the critical stresses and deformation of the tanks used in civil engineering practice [5]. Housner [6,7] developed a mechanical model based on a simplified analytical method for evaluating the hydrodynamic actions of a rigid cylindrical tank. This model consisted of two lamped masses, one for the impulsive and the other for the convective action. The mentioned model made the first basis for design of the cylindrical tanks with a rigid wall assumption. After the large-scale damage of 1964 great Alaska earthquake, Hanson [8] published a detailed report about the behavior of liquid storage tanks under this earthquake. After that, the importance of flexibility of tank wall was endorsed by many researchers. Veletsos and Yang [9] estimated the impulsive seismic load of flexible tanks by using a similar rigid tank assumption with modification. Instead of calculating the maximum acceleration for the rigid tank, they used the spectral acceleration associated with the natural frequency of the liquid-shell system for a similar deformable tank. They presented a simplified formula to obtain the fundamental natural frequency of the steel liquid-filled cylindrical tank. They paid special attention to the first symmetrical circumferential vibration mode. They

Corresponding author. E-mail addresses: [email protected] (R. Moradi), [email protected] (F. Behnamfar), [email protected] (S. Hashemi).

https://doi.org/10.1016/j.soildyn.2017.12.008 Received 2 June 2017; Received in revised form 8 October 2017; Accepted 3 December 2017 0267-7261/ © 2017 Elsevier Ltd. All rights reserved.

Soil Dynamics and Earthquake Engineering 106 (2018) 148–162

R. Moradi et al.

analytical method is developed to evaluate the liquid-shell interaction effects of partially filled concrete cylindrical tanks with flexible walls undergoing horizontal excitation. To obtain the impulsive pressure, the Rayleigh-Ritz method is applied, using the vibration modes of open topclamped bottom shells and n ≥ 1 for the Cos nθ circumferential modes. In the second part of this paper, a simple and accurate mechanical model is proposed by means of a response spectrum for estimating the seismic responses of concrete cylindrical tanks, in order to modify current insufficiencies of ACI 350.3 [19]. The parameters of this model are obtained based on the proposed analytical method from the developed charts and the case of a partially filled tank is also included. This aspect is not commonly accounted for in the related structural codes. Furthermore, the effects of both wall flexibility and inertia are considered in the presented analytical procedure and the proposed mechanical model. It is worth mentioning that, although the wall mass has a great importance in the seismic response of the concrete storage tanks, common codes only superficially considered its role. The time history of the sloshing wave height and its maximum is obtained with reasonable accuracy based on the convective pressure of a similar rigid tank [22].

concluded that the impulsive pressure distribution was similar for rigid and flexible tanks. However, the magnitude of hydrodynamic pressure depended on the flexibility of the tank wall. Balendra and Nash [10] modeled the steel cylindrical tanks using thin shell elements and analyzed them by the finite element method. They ignored the effect of sloshing and simplified the cylindrical fluid container by considering an empty shell with an added mass representing the liquid inside it. Haroun and Housner [11,12] presented a method for the dynamic analysis of steel cylindrical tanks full of liquid, using the finite element method. The impulsive and convective hydrodynamic pressures were expressed by the potential flow approach. Based on this research, they developed a three-degree-of-freedom model for the ground-supported flexible cylindrical tanks. The application of which resulted in design charts used to estimate sloshing and impulsive components and amount of the rigid mass. For evaluating the effective masses, only the fundamental natural mode of vibration of the deformable tank was considered. Their model disregarded the effect of the tank wall weight. Also, it does not have the ability to analyze partially filled tanks. Veletsos and Tang [13] presented an analytical solution for the Laplace equation governing the fluid for rigid cylindrical tanks subjected to a rocking base motion. They introduced a modified model in which the effect of higher sloshing modes was considered. They obtained the approximate impulsive hydrodynamic pressure on the deformable cylindrical tanks based on the formula for similar rigid tanks. Malhotra et al. [14] used the model of Veletsos et al. [9] and offered a simple model for cylindrical tanks under horizontal excitation. In their model, the effects of higher impulsive and convective modes of vibrations were considered. Shahverdiani et al. [15] investigated the behavior of concrete cylindrical tanks under harmonic excitation using the finite element method with ANSYS. They considered wall flexibility, liquid sloshing and viscosity in their study. In another study, Moslemi and Kianoush [16] investigated the behavior of concrete cylindrical tanks under horizontal ground motion using the finite element method by ANSYS. The obtained results were compared with those using ACI 350.3-06 for estimating the seismic response of liquid in cylindrical containers. Finally, they concluded that the seismic loads in this standard were estimated much larger than those obtained by the finite element model. Hashemi et al. [17,18] implemented the Rayleigh–Ritz method to present a semi-analytical method for the behavior of rectangular liquid storage tanks undergoing horizontal excitation. In their research, the flexibility of tank walls was considered. Then, based on their analytical method, a mechanical model for estimating the seismic loads was developed. The results of the aforementioned studies conducted by Moslemi and Kianoush [16] and Hashemi et al. [17,18] have shown that impulsive seismic responses such as base shear, overturning moment and distribution of the hydrodynamic pressure proposed by ACI 350.3-06 [19] are not in good agreement with numerical procedures and a lot of difference is observed in estimating the seismic responses. This drawback mainly originates from the mechanical model that is exploited in this standard. Generally, the well-known codes used for the design of cylindrical storage tanks [19–21] utilize Housner's model with the rigid wall assumption [6,7] for flexible tanks with some modification. Therefore, it is necessary to propose a mechanical model to modify the existing insufficiencies. In the present study, the Laplace equation governing the behavior of fluid in tanks is divided into two parts, namely impulsive and convective responses. To evaluate the impulsive responses on the fully or partially filled flexible containers, the liquid-shell interaction effects are considered. The convective response can be determined assuming a similar rigid tank, with minimal loss of accuracy [22]. In this study, therefore, the Rayleigh-Ritz method is employed for analysis of the liquid-shell system. It should be noted that the main focus is placed on the impulsive seismic response because most of the insufficiencies are in this action. Therefore, in the first part of this paper, a more accurate

2. Fundamental equations The storage tank under consideration is a cylindrical tank with a deformable wall of uniform thickness tw , open top ground-supported with a horizontal rigid bottom that is partially filled with liquid of height HL . The interior radius and height of the tank are R and HS respectively. The liquid is incompressible and non-viscous (like water and petroleum) and the wall of the tank is considered as a thin shell made of linearly elastic, homogeneous and isotropic material. Fig. 1 shows the cylindrical coordinate system. In this figure, the radial, circumferential and axial coordinates are denoted by r, θ and z, respectively and the corresponding displacements of a point on the middle surface of shell are denoted by w, v and u, respectively. The motion of the liquid is assumed to be frictionless and irrotational. Such a simplifying assumption makes the liquid motion to be governed by the Laplace equation according to the theory of fluid dynamics [23], as:

∇2 Φ =

∂ 2Φ 1 ∂Φ 1 ∂ 2Φ ∂ 2Φ + + 2 2 + =0 ∂r 2 r ∂r r ∂θ ∂z 2

(1)

in which Φ is the velocity potential function. The fluid velocity at an arbitrary point and time (t) in the direction of a generalized coordinate n, is [23]:

vn = −

∂Φ ∂n

(2)

Fig. 1. Cylindrical tank and the coordinate system.

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The hydrodynamic pressure (pressure due to vibration of liquid) at any point and time is given by [23]:

∂Φc ∂z

∂Φ pd (r , θ , z , t ) = −ρl ∂t

and along the tank wall:

(3)

=0

∂Φc ∂r

in which ρl is the mass density of the liquid. The equation of motion, Eq. (1), has to satisfy the boundary conditions. The rigid bottom condition of the tank enforces a zero velocity condition on the liquid as:

−

∂Φ ∂z

∂ 2Φ ( 2c ∂t

(4)

When the tank is subjected to a ground motion, üg (t ) , in the θ = 0 direction as shown in Fig. 1, the radial velocities of the liquid and the tank wall must be the same at r = R:

−

∂Φ ∂r

= ẇ (z , θ , t ) + u̇g (t ) cos θ r=R

+g

(5)

∞

Φi =

=

∞

2 HL

r=R

=0

H ∫0 L [u̇g (t ) cos θ + wṅ (z , t ) cos (n θ)] cos (αi z ) dz

∞

∑∑

αi I ′n (αi R)

n=1 i=1

In (αi r ) (14)

where I ′n is derivative of In with respect to r. The impulsive hydrodynamic pressure with use of Eq. (3) can be obtained by:

pi = −ρl

∂Φi ∂t

2ρl HL

=−

∞

H ∫0 L [üg (t ) Cos θ + wn̈ (z , t ) Cos (n θ)] Cos (αi z ) dz

∞

∑∑

αi I ′n (αi R)

n=1 i=1

(αi r ) Cos (αi z )

In (15)

This equation is separated into two parts as:

pi = − −

∞

(−1)i + 1ug̈ (t )

2ρl HL

∑

2ρl HL

∑ ∑

αi2 I ′n (αi R)

i=1 ∞

∞

n=1

In (αi r ) Cos (αi z ) Cos θ

H ∫0 L wn̈ (z , t ) Cos (αi z ) dz

αi I ′n (αi R)

i=1

In (αi r ) Cos (αi z ) Cos (nθ) (16)

The first term on the right, pr , is the hydrodynamic pressure on a similar rigid tank. It can be expressed as:

pr = −

2ρl HL

∞

∑ i=1

(−1)i + 1üg (t ) αi2 I ′n (αi R)

In (αi r ) Cos (αi z ) Cos θ

(17)

The second term is called pf . It is the hydrodynamic pressure due to the flexibility of the tank wall as it includes the relative response of the wall. The difference of a rigid and a flexible tank is in this part. It is expressed as:

(8)

The first term at the right is for the flexibility of the tank wall. The condition of zero impulsive pressure at z = HL : z = HL

(13)

cos (αi z )

and along the tank wall:

pi

∞

Φi

(6)

(7)

= ẇ (z , θ , t ) + u̇g (t ) cos θ

(12)

z = HL

∑ ∑ [Ani In (αi r ) Cos (αi z ) Cos (n θ)]

=0

∂Φi ∂r

z = HL

∂2Φi ∂Φ + g i) ∂t 2 ∂z

where αi = (2i − 1) π /2HL . In is the modified Bessel function of the first kind of order n, with n being number of the circumferential mode of vibration. After substituting Eq. (13) in Eq. (8), the impulsive velocity potential can be analytically determined in an infinite series form as:

Housner [6,7], Veletsos [13] and Haroun [22] divided the hydrodynamic pressure of the contained liquid into two components. One is the impulsive pressure caused by the portion of the liquid accelerating with the tank and the other one is the convective pressure caused by the rest of the liquid in the tank. This division is valid in concrete liquid storage tanks because the natural frequencies of significant convective modes are much lower than the fundamental impulsive frequencies. Therefore, study of the impulsive response of the liquid, can be separately performed from that of the convective response of liquid without significant errors. Therefore, the solution for Φ , Eq. (1), will be expressed as the sum of the impulsive component, Φi , and the convective component, Φc . The impulsive component of the solution satisfies the actual boundary conditions along the tank bottom:

z=0

= −(

n=1 i=1

3. Solution for the hydrodynamic pressure

−

∂Φc ) ∂z

Based on the method of separation of variables and by satisfying Eqs. (7) and (9), for the tank under consideration solution of Eq. (1) is expressed as:

The above equation is obtained by satisfying the kinematic and dynamic conditions. The former states that a portion of liquid that is at some time at the free surface will always remain at the free surface, and the latter is a condition imposed on the free surface through the Bernoulli equation [23].

∂Φi ∂z

(11)

3.1. Impulsive pressure

The right hand of this equation shows the absolute velocity of the tank wall that includes summation of the relative velocity and projection of ground velocity in the radial direction. The radial component of velocity, w (z , θ , t ) , can be expanded in the Fourier series along the ∞ circumference as w (z , θ , t ) = ∑n = 1 wn (z , t ) cos(nθ) . Assuming that the amplitude of liquid oscillation is small compared with HL , the linearized sloshing condition at the free liquid surface (z = HL ) is [23]:

∂ 2Φ ∂Φ +g =0 ∂t 2 ∂z

= 0 r=R

Condition of the convective pressure at z = HL is:

=0 z=0

(10)

z=0

pf = −

(9)

2ρl HL

∞

∞

∑∑ n=1 i=1

H ∫0 L wn̈ (z , t ) Cos (αi z ) dz

αi I ′n (αi R)

In (αi r ) Cos (αi z ) Cos (nθ) (18)

whereas the convective component corrects for the difference between the actual boundary condition at z = HL and the one considered in the development of the impulsive solution. Boundary condition along the tank bottom is:

For being able to determine the hydrodynamic pressure due to the flexibility of the tank wall, deflection of the shell has to be determined. It requires an analysis of the coupled liquid-shell system. 150

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3.1.1. Analysis of the liquid-shell system using the Rayleigh–Ritz method In this section, for obtaining the impulsive hydrodynamic pressure caused by the relative wall deflection, pf , the Rayleigh–Ritz method is used. Actually, the liquid-shell interaction is the main part of the analysis of a cylindrical tank fully or partially filled with liquid. By analyzing the liquid-shell system, the so-called pf portion of the impulsive hydrodynamic pressure can be calculated. Because of symmetry, displacements of the shell in the directions θ and z are expanded in Fourier series as follows:

εzn =

∑ un (z, t ) Cos (nθ)

One can write the displacement vector, rn = {u flexible shell in the matrix form as:

n

(21)

where rTn (z , t ) = un (z , t ) as:

vn (z , t )

wn (z , t ) and Sn can be expressed

0 0 ⎤ ⎡ cos nθ Sn = ⎢ 0 sin nθ 0 ⎥ ⎢ 0 cos nθ ⎥ ⎦ ⎣ 0

(22)

Epn =

) + sin (

ψv j = ψw j = Cosh (

λj z HS

HS

) − σj (cosh (

) − Cos (

λj z HS

λj z HS

) − cos (

) − σj (Sinh (

λj z HS

λj z HS

K Sn =

λj z HS

))

HS

∫0 ∫0

2π

εnT D ε nR dθ dz =

1 T q K Snfn 2 n

HS

∫0 ∫0

2π

BnT D Bn R dθ dz

(35)

0 0

υ 1 0

1−υ 2

0 0 0

0 0 0

0

0

2 tw 12

2 υ tw 12

0

0

2 υ tw 12

2 tw 12

0

0

0

0

0 0 0

⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ 2⎥ (1 − υ) tw ⎥ 24 ⎦

δEpn = δfTn K Snfn

cosh (λj ) + cos (λj ) sinh (λj ) + sin (λj )

(26)

E KSn =

1 2

0 ⎤ ⎡ ψun 0 Tn (z ) = ⎢ 0 ψvn 0 ⎥ ⎢ ⎥ 0 ψwn ⎦ ⎣ 0

HS

∫0 ∫0

2π

m (z ) rṅ T rṅ R dθ dz =

1 ̇T f n MSn fṅ 2

(38)

where: (27)

fvn

(37)

The kinetic energy of the cylindrical shell wall, corresponding to the nth circumferential mode, is expressed by:

Also, rn in Eq. (21) can be expanded to:

rn (z , t ) = Tn (z ) fn (t )

(36)

(25)

and the eigenvalue (λj ) for the jth mode is determined from:

Cosh (λj ) . Cos (λj ) + 1 = 0

(34)

in which E and υ are the modulus of elasticity and Poisson's ratio of the concrete. The variation of the potential energy has the form:

where:

where (fn (t ))T = {fun

1 2

⎡1 ⎢υ ⎢0 ⎢ E tw ⎢ 0 D= 1 − υ2 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎣

(24)

σj =

(33)

and:

))

) − Sin (

(32)

where:

(23)

where ψwn , ψvn and ψun represent the 1 × m vectors consisting of admissible functions in r, θ and z directions, f wn , f vn and f un are m × 1 generalized coordinate vectors, and m is the number of admissible functions that are used for the expansion. The admissible functions that satisfy the boundary condition of an open top and clamped bottom shell in the vertical direction are associated with the vibration modes of a cantilever beam as [24,25]:

λj z

⎤ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ∂2 ⎥ ∂z 2 ⎥ 1 ∂2 − 2 2 ⎥ R ∂θ ⎥ ∂2 2 − R ∂z ∂θ ⎥ ⎦ 1 R

The potential energy equivalent to the elastic strain energy of the cylindrical shell wall, corresponding to the nth circumferential mode, is expressed by:

m

m = ∑ j = 1 ψvn j (z ) f vn j (t ) = ψvn (z ) f vn (t ) m = ∑ j = 1 ψwn j (z ) fwn j (t ) = ψwn (z ) fwn (t )

0

εn = L Sn Tn fn = Bn fn

un (z , t ) = ∑ j = 1 ψun j (z ) f un j (t ) = ψun (z ) f un (t )

HS

2 ∂vn 2 ∂2wn − R ∂z R ∂z ∂θ

Substituting Eqs. (21) and (27) into Eq. (31) results in:

The Rayleigh–Ritz method needs a set of admissible functions to describe the vibration of the tank wall. These functions are required to satisfy only the geometric boundary conditions. Then the responses are expressed as:

ψu j = sinh (

(29)

(31)

∂ 0 ⎡ ∂z ⎢ 1 ∂ ⎢0 R ∂θ ⎢ ⎢1 ∂ ∂ ⎢ R ∂θ ∂z L=⎢ 0 ⎢0 ⎢ 1 ∂ ⎢0 R2 ∂θ ⎢ 2 ∂ ⎢0 R ∂z ⎣

wn} , of a

vn

rn = Sn rn

λj z

κ zθn =

εn = L rn

(20) T

wn (z , t )

1 ∂vn 1 ∂2wn − 2 R2 ∂θ R ∂z 2

1 ∂un ∂v + n R ∂θ ∂z

where L is a differential operator matrix that is defined by:

∑ vn (z, t ) Sin (nθ) n=1

vn (z , t )

κ θn =

εzθn =

where εθn , εzn and εzθn are the normal and shear strains, respectively, at the middle surface of the shell and κ θn , κ zn and κ zθn are curvatures in the nth circumferential mode. In matrix form:

∞

v (z , θ , t ) =

1 ∂vn ( + wn ) R ∂z

(30)

(19)

n=1

εθn =

∂2wn ∂z 2

κ zn = −

∞

u (z , θ , t ) =

∂un ∂z

MSn = πR

fwn} and:

∫0

HS

m (z ) TTn Tn dz

(39)

in which MS and K S are the mass matrix and stiffness matrix of the shell, respectively. m (z ) is the mass per unit area of the tank wall, i.e., (m (z ) = ρs tw ). The variation of the kinetic energy has the form:

(28)

T

δE KSn = δf ̇ n MSn q̇ n

Due to Novozhilov [26], based on the first-approximation theory, the strain component for the nth circumferential mode of a homogeneous and linearly elastic thin shell can be expressed as:

(40)

The work done by the liquid-shell interface pressures ( pf ) during the arbitrary virtual displacement δwn can be expressed as: 151

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R. Moradi et al. HL

∫0 ∫0

δwliquid =

2π

Pf (R, θ , z , t ). δwn. cos nθ R dθ dz

each of the circumferential modes n are uncoupled [22,24], it is appropriate to eliminate the subscript n for abbreviation. By use of variation of energy and application of the Hamilton's principle [22]:

(41)

Substituting pf from Eq. (18) into Eq. (41) results in:

δwliquid

δ ∞

=−

∞

2πRρl In (αi r ) ( HL αi I ′n (αi R)

∑∑ n=1 i=1

(

∫0

HL

∫0

HL

ẅn (z , t ) cos (αi z ) dz )

HL

δwn (z , t ) cos (αi z ) dz )

ẅn (z , t ) cos (αi z ) dz = (

∫0

HL

HL

Integration by parts to these terms results in:

̈ (t ) ψwn (z ) cos (αi z ) dz ) fwn

as

(45)

ϕk is the modal vector pertinent to the k mode of vibration of the tank. Introducing damping into Eq. (54) and using Eq. (55) with the assumption of uncoupling of modes results in:

(46)

∞

∑ ∑ Cn i (Tˆni)T Tˆni

qk̈ + 2ζ f ωk qk̇ + ωk2 qk = −βk üg (t )

(47)

n=1 i=1

(FTg . δ rn) R dθ dz +

HL

∫0 ∫0

2π

(pr δw cosθ) R dθ dz (48)

where FTg = −ρS tw { 0 − sinθ cosθ } üg (t ) and pr was introduced in Eq. (17) and: g δw = −δ f T (Peff + Preff ) ug̈ (t ) = δ f T Peff

(49)

Where: g T (Peff ) = πRtw ρS { 0 − ∫0

HS

ψvn ∫0

HS

ψwn }

(50)

∞

(Preff )T =

∑ Ci Tˆi

(51)

i=1

(56)

where ζ f , ωk and βk are damping ratio, frequency, and modal participation factor of the k-th impulsive mode, respectively. The complete time history of qk (t ) for each mode (k = 1, …, 3 m) and its time derivatives can be computed by a step-by-step numerical scheme. Once they are obtained, the displacement and the impulsive pressure can be calculated using Eqs. (21) and (16). A distinguishing feature of this study is how it calculates pf . As mentioned earlier, this part of the hydrodynamic pressure corresponds to deformation of the tank wall. Thus, it is important to predict such a deformation with good accuracy. Haroun [22] used the finite element method to perform the mentioned estimation. But in this study, use is made of the Rayleigh–Ritz method for the same purpose. Opposite to the approximate finite element approach, the semi-analytical Rayleigh–Ritz procedure has the advantage of the shape functions and their higher derivatives being continuous all over the shell area. This is important especially for calculating wn̈ (z , t ) and the impulsive pressure. The admissible functions are used for exploring the dynamic behavior of a shell with the pressure of liquid being applied on it. It is shown that the results are in good agreement with similar research works. 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 tank-wall flexibility on the sloshing frequencies can be ignored. In such cases, the studies on the impulsive response of the tank-wall vibration and the convective response of the liquid oscillation can be separately considered, without significant errors.

2πRρ I (α r )

and Cn i = H α lI ′n (αi R) . Mliquid is the added mass matrix due to the effect L i n i of liquid. The effective force vector can be considered as the sum of two components: the distributed inertia force of the shell associated with the ground acceleration vector as shown in Fig. 2, and the hydrodynamic pressure on a rigid tank wall ( pr ) associated with the relative displacement vector. Effect of pf is already included in Mliquid . The virtual work done by these external loads can be expressed as: 2π

(55) th

where:

HS

∑ ϕk qk k=1

−δfnT (Mliquid)n fn̈

∫0 ∫0

(54)

3m

f=

Therefore:

δw =

(53)

This equation can be solved directly by numerical integration. However, in analyzing the earthquake response of linear structures, it is generally more efficient to use the modal superposition to evaluate the seismic response because it uses only a few lowest modes. Therefore, Eq. (54) is solved by the modal analysis method. Then the response can be decomposed into its modal components as:

(44)

ˆ = {0 0 T } T ni ni

∞

+ Mliquid) f ̈ + K S f − peff ] = 0

(MS + Mliquid) f ̈ + K S f = peff

(43)

in which:

(Mliquid)n =

(52)

So, the governing matrix equation of the earthquake response of the undamped liquid-shell system is determined as:

ˆ f̈ ẅn (z , t ) cos (αi z ) dz = T ni n

δwliquid =

(EK − EP + w ) dt = 0

(42)

Tni is introduced To simplify expressions, H Tni = ∫0 L ψwn (z ) cos (αi z ) dz ; then Eq. (43) can be rewritten as:

∫0

t

δf T [(MS

Using Eq. (23), one can write:

∫0

∫0

ˆ are C and T ˆ when n = 1. Since the displacement of where Ci and T i ni ni

3.2. Convective pressure To calculate the convective pressure, pc , it is convenient to evaluate it directly by requiring that pc satisfies the Laplace equation instead of ∇2 Φc = 0 with the boundary conditions of Eqs.(10), (11) and (12), as:

∇2 pc =

∂2pc ∂r 2

+

∂2pc 1 ∂pc 1 ∂ 2p + 2 2c + =0 r ∂r r ∂θ ∂z 2

(57)

Therefore, the boundary conditions of Eqs. (10) and (11) can be converted with the use of Eq. (3) as below:

Fig. 2. Components of the ground acceleration.

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∂pc ∂z

∂pc

=0

horizontal excitation (θ = 0 ) can be obtained. The maximum sloshing wave height, ηmax , is computed using a response spectrum as:

=0

∂r

z=0

(58)

r=R

The solution of Eq. (57) is expressed as [23]: ∞

pc =

ηmax =

γj

γj

∑ ⎡⎢Bj (t ) J1 ⎛ R r ⎞ cosh ⎛ R z ⎞ cos ( θ) ⎤⎥ j=1

⎝

⎣

⎠

⎝

⎠

⎦

∞

∑ j=1

⎟

∑ [Bj̈ (t )cosh μj +

R

j=1

ηmax = R

r Bj (t )sinh μj ] J1 ⎛γj ⎞ = ⎝ R⎠

−2g ug̈ (t ) R HL

∞

I (αi r ) 1 (αi R )

i=1

i

∑

( ) × cosh ( ) × A (t ) γ −1 cosh (γ )

r J1 γj R

z γj R

J1 (γj )

H jR

j=1

j

2 j

∫0

t

ug̈ (t ) sin [ωjc (t − τ )] dτ

(62)

(63)

and:

ωjc =

g γj R

tanh (γj

HL ) R

(64)

Damping of convective responses (ζ c ) can easily be introduced into Eq. (63). The first sloshing natural frequency determined using Eq. (64) is identical to what is proposed by ACI 350.3-06. The hydrodynamic pressure at any point and time is given by:

p (r , θ , z , t ) = pi + pc = pr + pf + pc

(65)

where pr , pf and pc are, respectively, pressure on a similar rigid tank, pressure due to flexibility of the tank wall and the convective pressure that can be evaluated using Eqs. (17), (18) and (62), respectively. 3.3. Sloshing wave height Vertical displacement, η (r , θ , t ) , of the liquid at the free surface is calculated by using the hydrostatic pressure [23] as:

ρl g η (r , θ , t ) = pc

Sa c g

(66)

z = HL

Solution of the above equation, η (r , θ , t ) , at z = HL , is given by the following expression:

2R η (r , θ , t ) = g

∞

∑ J =1

( )× r

J1 γj R

Aj (t )

J1 (γj )

γ j2 − 1

(67)

The sloshing wave height at the wall r = R and θ = 0 can be obtained by:

η (R, θ , t ) =

2R g

∞

∑ j=1

Aj (t ) γ j2 − 1

(71)

In this section, the accuracy and validity of the present method is investigated. A computer program has been written to determine the validity of the theoretical formulation developed for the cylindrical flexible tank under horizontal excitation. The tank under study has a height, HS = 12 m ; wall thickness, tw = 0.5 m ; water depth, HL = 11 m ; diameter, D = 34 m . The material properties of the concrete are its density, ρS = 2400 kg / m3 ; Young's modulus, E = 24.86 GPa ; and the Poisson's ratio, υ = 0.16 . The horizontal component of 1940 El-Centro earthquake scaled to a peak acceleration of 0.4 g is used as [16] an input motion in the horizontal direction. The scaled earthquake component is shown in Fig. 3. The tank is assumed to be fixed to the ground, the damping ratios for the impulsive and convective components are assumed to be 5% and 0.5% respectively [16]. The maximum impulsive hydrodynamic pressure along the height of the tank for rigid and flexible walls is presented in Fig. 4 and is compared with the results of Moslemi and Kianoush [16] and ACI 350.3-06 [19]. It is worth noting that the ACI hydrodynamic pressure values have been calculated assuming I = 1 and R = 1, where I is the “Importance factor” and R is the “Response modification coefficient”. For code calculations, a design spectrum is used having a peak ground acceleration of 0.4 g corresponding to the horizontal seismic excitation utilized in the present example. The mapped spectral accelerations SS and S1 of 1.5 g and 0.6 g, respectively, are selected for the Imperial Valley location. Moreover, a site comprising a hard rock that is categorized as the site class A as per ASCE7-10 is assumed in the code calculations [16]. As can be observed in Fig. 4, the comparison shows that the results of the present work are in good agreement with those obtained by Moslemi and Kianoush [16] for rigid and flexible tanks. Calculation for

where Aj (t ) represent the acceleration of an undamped equivalent single-degree-of-freedom system with the natural frequency ωjc equal to that of the jth sloshing mode of vibration of the liquid excited by a base acceleration, ug̈ (t ) . It can be calculated using the Duhamel's integral as:

Aj (t ) = ωjc

(70)

4. Verification of the proposed analytical method

With solving this equation and calculating Bj (t ) , the convective hydrodynamic pressure can be expressed as:

pc = −2R ρL cos θ

(69)

That is the same as ACI 350.3-06 formula used to determine the vertical surface displacement.

∑ α I1′

(61)

∞

2 γ j2 − 1

2 =1 γ j2 − 1

(60)

Substituting Eq. (59) and pr form Eq. (17) in Eq. (60) gives:

gγj

j=1

Substituting Eq. (70) in Eq. (69), the maximum sloshing wave height can be written as:

2 ∂p ⎛⎜ ∂ pc + g ∂pc ⎟⎞ = ⎛−g r ⎞ 2 ∂ t ∂ t ∂t ⎠ z = H ⎝ ⎠z=H ⎝

∞

∞

∑

Sa c is the maximum of A1 (t ) or the spectral acceleration corresponding to the natural frequency ω1c and damping ζ c . Using the limiting property of the following infinite series:

(59)

in which J1 is the Bessel function of the first kind and γj is the jth root of J1′ (γ ) , the first derivative of the Bessel function of the first kind. The convective pressure can be calculated with reasonable accuracy by considering the shell wall to be rigid [11–13,22]. Therefore, Eq. (12) with the use of Eq. (3), results in: ⎜

Sa c R g

(68) Fig. 3. Horizontal record of the scaled 1940 El-Centro earthquake.

Using this equation, the time history of sloshing wave height under 153

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Fig. 4. Comparison of maximum impulsive pressure distributions along the height of the tank wall.

the rigid tank is based on Eq. (17) and for the flexible tank is based on sum of Eqs. (17) and (18). The purpose of showing the hydrodynamic pressure separately for rigid and flexible tanks is to highlight the importance of accounting for the wall flexibility. As can be seen, the value and shape of the impulsive hydrodynamic pressure in rigid and flexible tanks completely differ from each other, whereas ACI 350.3 does not consider any differences between the two. Also, the impulsive pressure in the rigid tank in the present work exactly matches the results determined by Haroun [22]. The time history of the sloshing motion at the wall for θ = 0 direction is presented and compared with the result of Moslemi and Kianoush [16] in Fig. 5. An excellent consistency between the results is observed. Fig. 6(a) and (b) show the time history of impulsive base shear under 1940 El-Centro earthquake for the rigid and flexible tanks, respectively. The maxima of the impulsive base shear of the rigid and flexible tanks are 15.1 × 106 N and 28.4 × 106 N , respectively. This means that the impulsive base shear of the flexible wall tank is 88% more than that for the similar rigid tank. As shown in Fig. 6(b) the maximum of the impulsive and impulsive plus convective base shear of the flexible tanks are 28.4 × 106 N and 29.46 × 106 N , respectively. In other words, the impulsive plus convective base shear of the flexible wall tank is only 3.7% larger than the impulsive base shear. It means that the convective pressure has a little effect on seismic responses and therefore, in the rest of this paper, the major focus will be on the impulsive responses.

Fig. 6. Time history of the impulsive base shear due to horizontal excitation.

approach, which would allow, from an engineering point of view, a simple, fast, and sufficiently accurate estimate of the seismic response of concrete cylindrical storage tanks in spite of the too approximate models of the mentioned codes. In the proposed mechanical model two effects are considered: 1) Effect of liquid consisting of pr , pf and pc that can be evaluated using Eqs. (17), (18) and (62), respectively. 2) Effect of the tank wall including its inertia and flexibility that are explained in the next sections. The proposed mechanical model is shows in the Fig. 7 and its parameters are calculated in the next sections. The only available threemass model is that of Haroun for steel tanks. However, there are important differences between the two models as follows: 1) Parameters of the Haroun's model [11] are obtained based on an approximate finite element analysis while in the proposed mechanical model the parameters are calculated based on a more accurate Rayleigh–Ritz method. 2) Parameters in the Haroun's model are for steel tanks [11]. There, the maximum value of tw / R is only 0.004, where in concrete tanks

5. The proposed mechanical model The main objective of this part of the study is to devise a practical

Fig. 7. The proposed mechanical model of a flexible cylindrical tank.

Fig. 5. Comparison of the sloshing height time histories.

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minimum of the same value is 0.01! The similar parameter in the proposed mechanical model is associated with thickness of concrete tanks. 3) In the Haroun's model [11] the case of partial fulfillment is not considered while height of liquid can be varied in the proposed model. 4) The Haroun's model [11] does not account for the effect of wall mass while this is very important in the concrete tanks and is considered in the current model. 5.1. Effect of liquid 5.1.1. The hydrodynamic base shear The instantaneous hydrodynamic base shear QL (t ) is given by: HL

∫0 ∫0

QL (t ) =

2π

p r = R Cosθ R dθ dz

(72)

By substituting pr from Eq. (17) in Eq. (72), the associated value in a similar rigid tank is:

Qr L (t ) =

HL

∫0 ∫0

2π

prL

r=R

cosθ R dθ dz = mr L ug̈ (t )

(73)

where:

mr L 2 = mL R HL2

∞

I1 (αi R) αi3 I ′1 (αi R)

∑ i=1

(74)

in which mL and mr L are the total mass of liquid in the storage tank and the equivalent mass corresponding to the hydrodynamic force associated with the ground motion, respectively. Fig. 8 shows that mr L obtained using Eq. (74) is in good agreement with that determined by ACI 350.3-06. Substituting pf in Eq. (72), the hydrodynamic base shear due to wall deformation relative to the ground can be determined as:

QfL (t ) =

HL

∫0 ∫0

2π

3m

pfL

r=R

cosθ R dθ dz =

∑ QfLk qk̈ (t ) k=1

(75)

in which: k QfL =[

2 ρl R π HL

∞

∑ i=1

(−1)iI1 (αi R) ˆ Ti] ϕk αi2 I ′1 (αi R)

(76)

It is necessary to mention that in the proposed mechanical model only the first mode of vibration is used in the modal analysis. The hydrodynamic base shear can be expressed by:

QfL (t ) = mfL uf̈ (t )

(77)

where mfL is the equivalent mass corresponding to the hydrodynamic forces associated with wall deformation relative to the ground and uf (t ) Fig. 9. Impulsive frequency parameter of the liquid-shell system (ωf HL ρ/ E ) for completely or partially filled tanks.

is the solution of the differential equation below:

uf̈ (t ) + 2ζ f ωf u̇f (t ) + ωf2 uf (t ) = −üg (t )

(78)

Comparing Eq. (78) and Eq. (56) results in:

qk̈ (t ) = βk uf̈ (t )

(79)

By substituting Eq. (76) in Eq. (75) and comparing the result with Eq. (77), mfL can be obtained:

mf

L

mL

=

2 β1 HL2 R

∞

[∑ i=1

(−1)iI1 (αi R) ˆ Ti]. ϕ1 αi2 I ′1 (αi R)

(80)

In Fig. 9, the frequency parameter (ωf HL ρS / E ) of tanks completely or partially filled with water versus the diameter to liquid height ratio (D / HL ) are shown and compared with those determined by ACI 350.3-

Fig. 8. Comparison of mr L using the present method and ACI 350.3.

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Table 1 Comparison of the factors mc /mL and hc/HL versus the ratio D /HL for cylindrical tanks. mc /mL

D/HL

4 5 6 7 8

hc/HL

Analytical

ACI

Analytical

ACI

0.6622 0.7148 0.7479 0.7698 0.7849

0.6678 0.7207 0.7541 0.7762 0.7913

0.53255 0.52144 0.51512 0.51122 0.50864

0.53251 0.52141 0.51510 0.51120 0.50864

5.33 in spite of its value of 1.84 in the first mode. This results in a much smaller value of the mass ratio in Eq. (82). Therefore, the base shear due to convective hydrodynamic pressure can be rewritten as:

Qc (t ) = mc1 A1 (t ) = mc uc̈ (t )

(83)

in which:

mc D 3.68 = 0.228 ( ) tanh [ ] mL HL D / HL

(84)

By comparing Eq. (84) with ACI 350.3-06, a good agreement can be observed (see Table 1). The small difference itself is due to estimation of the roots of the equations at γj = 1.84. Using Eqs. (73), (77), and (83), one may express the total hydrodynamic base shear as:

QL (t ) = mc uc̈ (t ) + mrL ug̈ (t ) + mfL uf̈ = Qc (t ) + QiL (t )

(85)

where QiL (t ) is the impulsive base shear. Since the base shear due to wall deformability is proportional to the relative movement of the wall, one can rearrange Eq. (85), to estimate the maximum seismic loads by means of a response spectrum:

QiL = [mrL − mfL] ug̈ (t ) + mfL [uf̈ (t ) + üg (t )]

(86)

Subsequently, the maximum base shear can be estimated by:

(QL)max = mc Sac + (mrL − mfL)(ug̈ (t ))max + mfL Saf = (Qc )max + (QiL)max

(87)

in which Sa c and Sa f are the spectral accelerations corresponding to the natural frequencies ωc and ωf , respectively. In comparison, in Haroun's model [11] the SRSS rule has been used to calculate the base shear while in this model, use of the absolute sum of the responses results in better accuracy. 5.1.2. The hydrodynamic overturning moment The instantaneous hydrodynamic overturning moment ML (t ) due to the seismic motion can be expressed as:

Fig. 10. Values of mfL / mL for tanks: (a) 100%, (b) 80%, (c) 60% filled with liquid.

ML (t ) = 06. One can use Fig. 10 to obtain values of mf L / mL for tanks with different thicknesses. It is necessary to mention that in this mechanical model the case of partially filled tanks can also be dealt with. Although, the other mechanical models [11, 13 and ACI 350.3] assume that the tank is full. After substituting the expression of pc from Eq. (62) in Eq. (72), the convective hydrodynamic base shear can be given as:

Mr L (t ) =

hr L = HL

(81)

where:

mcj mL

=

(88)

HL

∫0 ∫0

2π

prL

r=R

cosθ R z dθ dz = mr L hrL ug̈ (t )

(89)

∞

∞

∑ f (αi) R (μi ) /∑ f (αi) μi Sin (μi ) i=1

i=1

(90)

where μi = αi HL and R (μi ) = ( μi ) Sin (μi ) + Cos (μi ) − 1 and f (αi ) is the following function:

2γj

1 D ] ( ) tanh [ D / HL γj (γ j2 − 1) HL

p r = R Cosθ R z dθ dz

hrL is the effective height of the equivalent mass corresponding to the resultant of the hydrodynamic force associated with the ground motion, mrL . It is calculated as:

∑ mcj Aj (t ) j=1

2π

Substituting the hydrodynamic pressure pr yields:

∞

Qc (t ) =

HL

∫0 ∫0

(82)

f (αi ) =

One can consider only the first mode of convective response and neglect the higher modes [22]. For instance, in the second mode γj =

(−1)i + 1I1 (αi R) αi4 I ′1 (αi R)

(91)

The numerical values of hrL / HL can be determined from Fig. 11 and 156

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R. Moradi et al.

Fig. 11. Values of the analytical hrL/ HL and comparison with ACI 350.3.

can be compared to those obtained using ACI 350.3. The hydrodynamic overturning moment due to wall deformation relative to the ground, MfL (t ) , applied to the bottom of wall can be express as: HL

∫0 ∫0

MfL (t ) =

K

2π

pf

cosθ R z dθ dz =

r=R

∑ M fLk qk̈ (t ) = mfL hfL uf̈ (t ) k=1

(92) where: k M fL =[

∞

2 ρl R π HL

∑ i=1

I1 (αi R) R (μi ) ˆ Tni] ϕk αi3 I ′1 (αi R)

(93)

The value of hfL in Eq. (92) can be obtained as:

hfL HL

=

3m

∞

3m

∞

k=1

i=1

k=1

i=1

∑ ( ∑ αi f (αi) R (μi ) Tˆni) ϕk / ∑ ( ∑ αi f (αi) μi Sin (μi ) Tˆni) ϕk (94)

In the proposed mechanical model, one can use only the first mode of vibration without considerable loss of accuracy as will be shown in the numerical examples, as:

hfL HL

∞

∞

i=1

i=1

ˆ ) ϕ /( ∑ α f (α ) μ sin (μ ) T ˆ = ( ∑ αi f (αi ) R (μi ) T i 1 i i i i 1) ϕ1

(95)

Fig. 12 can be used to evaluate the values of hfL for different heights of fill. After substituting the expression for pc in Eq. (88), the convective overturning moment can be derived as:

Mc (t ) =

HL

∫0 ∫0

2π

∞

pc

r=R

cosθ R z dθ dz =

Fig. 12. hfL/ HL for different tanks: (a) 100%, (b) 80%, (c) 60% filled with liquid.

∑ mcj hcj Aj (t ) j=1

(96) overturning moment can be estimated by:

where hcj is the height at which the jth convective component of the liquid mass is considered to be concentrated. It is determined as:

hcj HL

ML (t ) = mc hc uc̈ (t ) + mrL hrL ug̈ (t ) + mfL hfL uf̈ = Mc (t ) + MiL (t )

(100)

2γj

=1−

cosh [ D / H ] − 1

Subsequently, the maximum overturning moment of a tank subjected to earthquake can be estimated by:

L

2γj

2γj

D / HL

sinh [ D / H ] L

(97)

By using only the first mode of convective response, the overturning moment due to hydrodynamic convective pressure can be determined as:

Mc (t ) = mc1 hc1 A1 (t ) = mc hc uc̈ (t )

(ML)max = mc hc Sac + (mrL hrL − mfL hfL)(ug̈ (t ))max + mfL hfL Saf = (Mc )max + (MiL)max

Whereas in the Haroun's model [11] the SRSS rule has been used for the same purpose, use of the absolute sum of the maximum responses gives better results in this study. Using Eqs. (87) and (101), a mechanical model equivalent to the liquid inside a concrete cylindrical tank has been developed and shown in Fig. 7. In this model:

(98)

where: 3.68

cosh [ D / H ] − 1 hc L = 1 − 3.68 3.68 HL sinh [ ] D/H D/H L

L

(101)

(99)

m 0L = mrL − mfL

This equation has been adapted by ACI 350.3-06 (see Table 1). The 157

(102)

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R. Moradi et al.

Fig. 13. mfS / mL for different tanks: (a) 100%, (b) 80%, (c) 60% filled with liquid. Fig. 14. hfS / HL for different tanks: (a) 100%, (b) 80%, (c) 60% filled with liquid.

h 0L = (mrL hrL − mfL hfL)/ m 0L

(103)

QrS (t ) =

HS

∫0 ∫0

5.2. The effect of tank wall

in which:

In this part, determining the inertial and flexibility effects of the tank wall on the dynamic responses of the tank are intended. While this effect is negligible for a steel storage tank and has not been considered in the available mechanical models, it is believed in this study to be considerable for dynamic analysis of concrete liquid storage tanks.

2 tw HS ρS mr S = mL R HL ρL

2π

tw ρS üg (t ) Rdθdz = mr S üg (t )

(104)

(105)

The base shear corresponding to the wall mass due to the relative wall deformation can be obtained as:

QfS (t ) =

5.2.1. The base shear corresponding to the wall mass In a similar rigid tank, the base shear corresponding to the wall mass can be expressed as:

where: 158

HS

∫0 ∫0

2π

K

tw ρS ẅ cosθRdθdz =

∑ QfSk qk̈ (t ) k=1

(106)

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Fig. 15. Sloshing height time history.

Table 2 Comparison between the maximum impulsive responses. Difference percentage relative to the analytical results are also shown in parentheses.

Base shear (kN)

Proposed analytical method

Proposed mechanical model

ACI 350.3-06

7230

7440 (+ 2.9%)

9220 (+

18,790

19,150 (+ 1.9%)

20,750 (+

247

225

27.5%)

Overturning moment (kN m) Sloshing height at the wall (mm)

10.5%) 225

Fig. 16. Time history of impulsive base shear and overturning moment due to horizontal excitation.

Table 3 Comparison between maximum responses corresponding to the wall mass. Difference percentage relative to the analytical results are also shown in parentheses.

Base shear (kN)

Proposed analytical method

Proposed mechanical model

ACI 350.3-06

8230

8460 (+ 2.8%)

5000 (−

33,140

33,490 (+ 1%)

17,520 (−

Table 4 Comparison on the convective responses obtained by different methods. Proposed analytical method

Proposed mechanical model

ACI 350.3-06

821

818

827

2498

2490

2520

39%)

Overturning moment (kN m)

k QfS = [4Rtw ρS

∫0

Hs

Base shear (kN) Overturning moment (kN m)

47%)

{ 0 0 ψ w} dz ] ϕk

(107)

MS (t ) = MrS (t ) + MfS (t )

Therefore:

QfS (t ) = mfS uf̈ (t )

= ∫0

HS

∫0

2π

tw ρS ug̈ (t ) R z dθ dz + ∫0

HS

∫0

2π

tw ρS wCosθRdθdz ̈

= mrS hrS ug̈ (t ) + mfS hfS uf̈ (t )

(108)

(111) In the above equations, mrS and mfS are the total weight of the wall and the wall equivalent mass corresponding to the resultant of the forces associated with relative wall deformations, respectively. Using Eqs. (104) and (108), the base shear due to the tank wall can be expressed as:

hrS and hfS are, respectively, the heights at which the equivalent masses corresponding to the resultant of the wall forces associated with ground motion, mrS , and wall deformation relative to the ground, mfS , are considered to be concentrated. hrS = HS /2 , the numerical values of hfS / HL are shown for tanks that are 100%, 80% and 60% full of liquid in Fig. 14. Eq. (111) can be rewritten as:

QS (t ) = mrS ug̈ (t ) + mfS uf̈ = [mrS − mfS ] ug̈ (t ) + mfS [uf̈ (t ) + ug̈ (t ))] (109)

MS (t ) = (mrS hrS − mfS hfS ) ug̈ (t ) + mfS hfS ( uf̈ (t ) + ug̈ (t ))

The maximum value of this equation can be expressed as:

(QS )max = (mrS − mfS ) ug̈ (t )max + mfS Saf

(112)

The maximum amount of the overturning moment at the base corresponding to the wall mass can therefore be estimated by means of a response spectrum as:

(110)

Fig. 13 shows mfS / mL for tanks 100%, 80% and 60% filled with liquid corresponding to different values of tw / R .

(MS )max = (mrS hrS − mfS hfS )(ug̈ )max + mfS hfS Saf

5.2.2. The overturning moment corresponding to the wall mass The overturning moment corresponding to the wall inertia and flexibility can be expressed as:

It is to be noted that in other available mechanical models [11,13,14], effect of the tank wall has been ignored since they have been mainly developed for steel tanks. In this study it is considered as above. In the next section it will shown that how such an effect is 159

(113)

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Table 5 Comparison of seismic responses obtained by different methods, HL = 4.5. Difference percentage relative to the analytical results are also shown in parentheses.

Effect of Liquid Maximum impulsive base shear (kN) Maximum impulsive overturning moment (kN m) Effect of wall mass Maximum base shear (kN) Maximum overturning moment (kN m)

Proposed analytical method

Proposed mechanical model

ACI 350.3-06

3140

3330 (+ 6%)

4530 (+ 44%)

5930

6240 (+ 5.2%)

7650 (+ 29%)

7250

7500 (+ 3.5%)

6980 (− 3.7%)

28,510

28,990 (+ 1.7%)

24,430 (− 14%)

Table 6 Comparison of the seismic responses obtained by different methods, HL = 10. Difference percentage relative to the analytical results are also shown in parentheses.

Effect of Liquid Maximum impulsive base shear (kN) Maximum impulsive overturning moment (kN m) Maximum surface displacement near the wall (mm) Effect of wall mass Maximum base shear (kN) Maximum overturning moment (kN m) Sum of two effects Maximum total base shear (kN) Maximum total overturning moment (kN m)

Proposed analytical method

Proposed mechanical model

ACI 350.3-06

17,360

17,470 (+ 0.6%)

17,790 (+ 2.5%)

74,140

74,150 (0%)

66,720 (− 10%)

393

403

403

10,540

10,780 (+ 2.3%)

7280 (− 31%)

62,990

63,830 (+ 1.3%)

40,060 (− 36.4%)

27,890

28,250 (+ 1.3%)

25,080 (− 10%)

137,940

137,980 (0%)

106,770 (− 22.6%)

hydrodynamic pressure) in the proposed mechanical model (Fig. 7) can be estimated by means of a response spectrum as:

Table 7 Comparison on the convective responses obtained by different methods.

Convective Base shear (kN) Convective Overturning moment (kN m)

Proposed analytical method

Proposed mechanical model

ACI 350.3-06

1245

1240

1250

6779

6770

6800

Qmax = mc Sac + (m 0L + m 0S ) üg (t )max + (mfL + mfS ) Saf Mmax = mc hc Sac + (m 0L + m 0S ) h 0 üg (t )max + (mfL + mfS ) hf Saf

(119)

6. Numerical examples 6.1. Example 1

important for concrete tanks. Using Eqs. (110) and (113), properties of the proposed mechanical model of Fig. 7 associated with the wall of the cylindrical concrete tank can be calculated. In this model:

m 0S = mrS − mfS

(114)

h 0S = (mrS hrS − mfS hfS )/ m 0S

(115)

An open top shallow tank is considered whose dimensions are: D = 50 m, HL = 6 m, HS = 7 m, tw = 0.7 m and its impulsive and convective damping ratios are 5% and 0.5%, respectively. The tank is assumed to be 86% full of water and to be subjected to the North- South component of the 1940 El-Centro earthquake having a PGA = 0.313 g. The fundamental natural frequency of sloshing is obtained from Eq. (64) to be ωc = ωjc j = 1 = 0.547 rad/ s resulting in the spectral acceleration Sac = 0.009g from the response spectrum of the same earthquake. The time history of sloshing height at the wall is calculated using Eq. (68) and shown in Fig. 15. The maximum sloshing heights are 247 mm and 225 mm, based on the present analytical method with 10 modes and ACI 350.3-06 respectively. The fundamental impulsive frequency for this tank is determined from Fig. 9 to be ωf = 107.65 rad/ s . The spectral acceleration for this frequency is Saf = 0.576g . The fundamental impulsive frequency

Also, in this model h 0 and hf can be obtained by:

h0 =

hf =

(118)

m 0L h 0L + m 0S h 0S m 0L + m 0S

(116)

mfL hfL + mfS hfS mfL + mfS

(117)

Therefore, the maximum amount of the base shear and overturning moment corresponding to the total effect (i.e., tank wall and

Table 8 Comparison of the seismic responses obtained by different methods, HL = 8. Difference percentage relative to the analytical results are also shown in parentheses. Proposed analytical method Effect of Liquid Maximum base shear (kN) Maximum overturning moment (kN m) Effect of tank wall Maximum base shear (kN) Maximum overturning moment (kN m) Sum of two effects Maximum base shear (kN) Maximum overturning moment (kN m)

Proposed mechanical model

ACI 350.3-06

8850

8950 (1.1%)

9150 (3.4%)

29,700

30,040 (0.2%)

27,450 (− 9%)

8950

9180 (2.5%)

5140 (− 42.5%)

51,990

52,840 (1.6%)

28,290 (− 46%)

17,800

18,130 (1.8%)

14,290 (− 20%)

81,940

82,880 (1.1%)

53,750 (− 32%)

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and compared with those obtained by the proposed mechanical model and ACI 350.3 in Table 7. In the next round of analysis, the height of liquid in the tank is decreased to HL = 8 m and the seismic responses are collected in Table 8. Comparing values in Tables 6, 8 show that in some cases, ACI 350.3 underestimates the hydrodynamic and total responses. In the above examples, it can be realized that ACI 350.3 does not estimate the seismic responses accurately. This is because in ACI 350.3, the effect of wall flexibility has not been thoroughly accounted for by a reasonable and accurate method. In the proposed mechanical model, a more rational method is used to consider the deformability of the tank wall. Therefore, it is in good agreement with the proposed analytical procedure.

according to ACI 350.3 is ωf = 93.3 rad/ s and the spectral acceleration for this frequency is Saf = 0.605g . The remaining parameters can be obtained from Fig. 10 to 14 to be mf L / mL = 0.079 , mr L / mL = 0.139 , hf L / HL = 0.489, hr L / HL = 0.399 . To consider the effect of wall mass, the following parameters should be determined as: mf S / mL = 0.0918, mr S / mL = 0.157 , hf S / HL = 0.815. The base shear and overturning moment corresponding to liquid and wall masses are determined using the proposed analytical method and are compared with those obtained by the proposed mechanical model and ACI 350.3 in Tables 2, 3. It is worth noting that the ACI 350.3 responses are calculated assuming Importance factor (I) and Response modification coefficient (R) equal to 1. It should be noticed that the Haroun's model [11] cannot be utilized in solving this example because here tw / R = 0.028 while in the Haroun's model for steel tanks [11] the maximum usable value of tw / R is only 0.004. Moreover, in his model effect of tank wall inertia and stiffness are not considered. In ACI 350.3 the tank wall effects are considered but not in an appropriate way (see Table 3). The time histories of the hydrodynamic impulsive base shear and overturning moment obtained by the analytical method are shown in Fig. 16. The convective base shear and overturning moment are determined using the analytical method (for the first mode) and compared with those obtained by the proposed mechanical model and ACI 350.3 in Table 4. It is worth noting that if all the modes of convective component are employed, the total mass of liquid and tank wall will be equal to ∞ ∑ j = 1 mcj + m 0L + m 0S + mfL + mfS in which mcj , m 0L and m 0S can be derived using Eqs. (82), (102) and (114). To more investigate the proposed mechanical model, the height of liquid in the previous tank is decreased to HL = 4.5. The fundamental impulsive frequency of liquid-shell vibration for this case is calculated by the present method to be ωf = 126.22 rad/ s , and by ACI 350.3 to be ωf = 133.06 rad/ s . Results of response analysis are shown in Table 5. The purpose of this part of the example is to show the effect of partial fulfillment of tank. Results show that how with decreasing the height of liquid, difference of the result of ACI350.3 and those of the presented analytical procedure increases whereas estimations of the proposed mechanical model are very good. Comparing values in Tables 2–5, one can conclude that ACI 350.3 overestimates the hydrodynamic seismic responses and underestimates the effect of wall mass. Tables 3, 4 show that effect of the wall mass is considerable in concrete tanks. In this example, Table 3 shows that the base shear and overturning moment corresponding to the wall mass are 113.8% and 176.4% of the hydrodynamic impulsive base shear and overturning moment, respectively.

7. Conclusions The first part of this paper focused on determination of the hydrodynamic pressure in a cylindrical tank induced by horizontal seismic excitation. This pressure was evaluated by solving the Laplace equation governing behavior of fluid in a cylindrical tank satisfying its corresponding boundary conditions. For simplicity, the Laplace equation has been solved by discretizing it into two components called the impulsive and convective parts. With the aid of separation of variables for solving each component, the total hydrodynamic pressure can be determined using three parts. One part is because of the effect of ground motion associated with a similar rigid tank ( pr ) and the other is due to the relative wall deformation ( pf ). Finally, the last part is the convective pressure ( pc ). Portions of the hydrodynamic pressure ( pr and pc ) have a closed form solution while the other portion, pf , depends on wall deflection and needs to be analyzed using a liquid-shell system. The Rayleigh-Ritz method has been employed to solve the liquid-shell system by applying the vibration modes of open top-clamped bottom flexible shells. It is almost clear that the latter one is the most complicated part of the hydrodynamic pressure and the study focused mainly on it. The proposed analytical approach was compared with other available studies for verification. The maximum sloshing wave height due to the convective pressure was obtained and confirmed by comparing it with the ACI 350.3-06 standard. In the second part of this paper, a three-mass mechanical model for concrete cylindrical storage tanks was developed to modify the available mechanical models. Parameters of the model were calculated based on the proposed analytical approach but only for the Cos θ -type vibration mode and the first mode of vertical vibration, because the effects of other circumferential and vertical modes were negligible. Whereas other available mechanical models assume the tank to be full of water, both cases of completely or partially filled tanks and the mass of tank are considered in the proposed model. The results proved the importance of the wall mass in the concrete tanks. Finally, with solving some numerical examples, the proposed mechanical model was compared with the analytical method and ACI 350.3-06. The results show that the developed mechanical model is in good agreement with the analytical approach. On the other hand, ACI 350.3 overestimates the seismic responses in some cases and underestimates them in others. That means this standard cannot correctly account for the effect of deformability of the tank wall as well as the effect of the tank mass. In addition, this study shows that the hydrodynamic pressure distributions for rigid and flexible walls differ from each other in value and in shape. The simple mechanical model presented in this study can be a potential alternative for the mechanical model of ACI 350.3-06 for the effect of wall flexibility on the hydrodynamic pressure and the effect of tank mass on seismic responses of flexible cylindrical concrete tanks.

6.2. Example 2 An open top tall tank is considered whose dimensions are: D = 33 m, HL = 10 m, HS = 11 m, tw = 0.8 m. The tank is assumed to be 91% filled with water and to be subjected to the North- South component of the 1940 El-Centro earthquake similar to Example 1. Also impulsive and convective responses are damped by 5% and 0.5%, respectively. The fundamental impulsive frequency of liquid-shell vibration for this tank is determined from Fig. 9 to be ωf = 100.81 rad/ s and the spectral acceleration for this frequency is determined as Saf = 0.610g . The fundamental impulsive frequency can also be found using ACI 350.3 as ωf = 101.73rad/ s . The remaining parameters can be obtained from Fig. 10 to 14 to be mf L / mL = 0.319, mr L / mL = 0.362 , hf L / HL = 0.454 , hr L / HL = 0.399 , mf S / mL = 0.163, mr S / mL = 0.256, hf S / HL = 0.661. The base shear and overturning moment corresponding to liquid and wall masses and the total base shear and overturning moment are determined using the proposed analytical method and compared with those obtained by the proposed mechanical model and ACI 350.3 in Table 6. The convective base shear and overturning moment are determined using the analytical method (for the first mode)

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