# The Study of Seismic Response on Accelerated Contained Fluid

Authors: Kamila Kotrasova; Eva Kormanikova
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

Ground-supported cylindrical tanks are strategically very important structures used to store a variety of liquids. This paper presents the theoretical background of fluid effect on tank when a fluid container is subjected to horizontal acceleration. Fluid excites the hydrodynamic (impulsive and convective) pressures, impulsive and convective (sloshing) actions. Seismic response of cylindrical fluid filling tanks fixed to rigid foundations was calculated for variation of the tank slenderness parameter. The calculating procedure has been adopted in Eurocode 8.


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## Body

## 1. Introduction

Liquid-containing cylindrical tanks are strategically very important structures used in water distribution systems and in industries for storing water for firefighting, petroleum, oil and liquefied natural gas, chemical, toxic, flammable, and radioactive liquids. These tanks are mainly of two types: ground-supported cylindrical tanks (concrete, prestressed concrete, and steel) and elevated tanks [1–3].Seismic analysis of liquid-containing tanks requires special consideration and is different from analysis of buildings [4–7]. The fluid inside of tanks exerts hydrodynamic forces on the tank walls and base. The knowledge of liquid hydrodynamic pressures developed during an earthquake is important for design of tanks, because stresses of the tank wall depend on distribution of loading along the wall height. The first solution structure of a problem was by Westergaard (1933), who determined the pressures on a rectangular dam subjected to horizontal acceleration. Hopkins and Rodriquez (1934) gave the first report on analytical and experimental observations of rigid rectangular tanks under a simulated horizontal earthquake excitation. Jacobsen analyzed, in 1949 and 1951, a rigid cylindrical liquid-containing tank and cylindrical pier surrounded by liquid, subjected to horizontal acceleration. Housner (1963) derived the expressions for distribution of hydrodynamic pressure on a rigid tank wall due to lateral base excitation and considered impulsive and convective components of hydrodynamic pressure. Veletsos (1984) obtained also the distribution of hydrodynamic pressure on rigid as well as flexible wall.Historically, mechanical models were developed for tank with rigid walls. Housner [8] simplified the method of hydrodynamic analysis in system tank-liquid and introduced the concept of two dynamic components: dynamic impulsive and convective masses. Housner established spring-mass system, which simulates the impulsive and convective mode of vibration of tank-fluid system modified by Graham and Rodriquez (1952) and Epstein (1976). Wozniak and Michell (1978) generalized the Housner model for short and slender tanks. Velestos and Yang (1977) used a different approach to arrive at a similar type of mechanical model for circular rigid tanks. Haround and Housner (1981) and Velestos (1984) developed mechanical model for flexible tanks. Malhotra et al. [9] simplified the flexible model of Velestos (1984) and studied the dynamic behavior of unanchored and base-isolated liquid cylindrical storage tanks [10, 11].The seismic design of circular tanks is recommended by various codes of practices; their implementation strategy is rather varied leading to significantly different design forces in some cases [12]. Of the best structural engineering design codes that tackle fluid tank systems are [12–14] the American Concrete Institute, ACI 350.3, Eurocode 8 [15], the Standards Association of New Zealand, NZS, and Indian code IS 1893–1984 [16]. These codes address ground-supported circular and rectangular concrete tanks with fixed or flexible bases.European Committee for Standardization prepared code Eurocode 8 (2006). Part 4 [15] of this code is related to tanks, silos, and pipelines. Eurocode 8 uses approach of Veletsos (1984) to get hydrodynamic pressure distribution in circular tanks. Eurocode 8 mentions mechanical model of Veletsos and Yang (1977) as an acceptable procedure for rigid circular tanks. Models of Haroun and Housner (1981) and Veletsos (1984) are described along with the procedure of Malhotra et al. (2000) [9] for flexible circular tanks. Model of Housner (1963) is suggested for rigid rectangular tanks.In regard to the combination procedure of the impulsive forces and the convective forces, Eurocode 8 suggests the absolute sum combination procedure of the impulsive forces and the convective forces, while the ACI, NZS, and IS suggest the Square Root Sum of the Squares method (SRSS) [14, 16, 17].In this paper, analysis for the seismic design of fluid filling in unanchored rigid vertical circular tanks on-ground, fixed to the foundation, presented by Eurocode 8 is provided [5, 15].


## 2. Seismic Analysis of Liquid-Filled Tanks

Under earthquake loading, container walls and liquid are subjected to horizontal acceleration. The seismic load acting on wall and bottom of cylindrical tanks (Figure1) can be divided into the following components:(i)
The rigid impulsive component, caused by the inertia of the liquid, if the rigid tank moves together with the foundation(ii)
The convective load component, the fluid vibration in the rigid tank (sloshing)(iii)
The impulsive flexible tank shell (e.g., steel tanks) with the liquid [12].Figure 1
Vertical circular tank.The motion of contained fluid in unanchored rigid vertical circular tanks on-ground, fixed to the foundation tank, may by expressed as the sum of two separate contributions, called “rigid impulsive” and “convective,” respectively. The “rigid impulsive” component satisfies exactly the boundary conditions at the walls and the bottom of the tank but, incorrectly, due to the presence of the waves in the dynamic response, gives zero pressure at the original position of the free surface of the fluid. The second “convective” term does not alter those boundary conditions that are already satisfied, while fulfilling the correct equilibrium condition at the free surface. Use is made of a cylindrical coordinate system,r, z, θ, with origin at the center of the tank bottom, with z vertical axis; R is the radius of fluid filling; andH is the original height of the free surface of fluid; see Figure 1. The mass density of the fluid is ρ, ξ=r/R is nondimensional radius, and ζ=z/H is dimensionless coordinate [15].


### 2.1. Rigid Impulsive Component

The seismic loads acting on wall and bottom of cylindrical tanks (Figure1) can be divided into the following components.The spatial-temporal variation of the “rigid impulsive” pressure is given by the expression(1)piξ,ζ,θ,t=Ciξ,ζρHcos⁡θAgt,where(2)Ciξ,ζ=2∑n=0∞-1nI1′νn/γνn2cos⁡νnζI1νnγξ,νn=π2n+1/2 and γ=H/R, I1· is the modified Bessel function of order 1, and I1′· is derivate that can be expressed in terms of modified Bessel function of orders 0 and 1 I1′x=dI1x/dx=I0x-I1x/x. The function Ci gives the distribution along the height of the time-dependence pressure pi. θ is angle of circumference, γ=H/R is slenderness parameter, and Agt is the free-field ground horizontal acceleration of ground with peak value denoted by ag. The result of an equivalent single-degree-of-freedom system is mi with the impulsive period Ti. The distribution pi along the height of tank wall gives the function Ci (see (2)) for ξ=1 (i.e., at the wall of the tank) and cos θ=1 (i.e., in the plane of the horizontal seismic action). Figure 2 presents the vertical distribution of impulsive pressure pi on the tank wall along the wall height, normalized to ρRag, for seven values of the slenderness parameter γ=H/R.Figure 2
Variation of vertical distribution function of the impulsive pressure along the wall height for seven values of the tank slenderness parameterγ=H/R, normalized to ρRag.For a number of purposes it is useful to evaluate the horizontal resultant of the pressure at the base of the wall, the total base shear, and the moments of the pressures with respect to an axis orthogonal to the direction of the seismic motion.Mi is the total impulsive (bending) moment, with respect to an axis orthogonal to the direction of the seismic action motion, immediately above the tank bottom and includes only the contributions of the pressures on the walls (see (7)). Mi∗ is the total impulsive (overturning) moment immediately below the tank bottom and includes the contributions of the pressures on the walls and of those on the bottom (see (5)).The horizontal impulsive pressure resultant, total impulsive base shear, at the base of the wall is(3)Qit=miAgt, where mi is called impulsive mass and indicates the mass of the contained fluid moved together with the walls and mi is given by the expression(4)mi=m2γ∑n=0∞I1νn/γνn3I1′νn/γ,where m=ρπR2H is total mass of the fluid contained in tank.The quantitiesmi are plotted in Figure 3 as function of the tank slenderness parameter γ=H/R.Figure 3
Ratiosmi/m as function of the tank slenderness parameter.The total impulsive base moment immediately below the tank bottom is given as(5)Mi∗=mihi∗A′gt,where (6)hi∗=H1/2+2γ∑n=0∞νn+2-1n+1I1νn/γ/νn4I1′νn/γ2γ∑n=0∞I1νn/γ/νn3I1′νn/γ,and total impulsive base moment immediately above the tank bottom is(7)Mi=mihiAgt,where(8)hi=H∑n=0∞-1nI1νn/γ/νn4I1′νn/γνn-1n-1∑n=0∞I1νn/γ/νn3I1′νn/γ.The quantitieshi and hi∗ are plotted in Figure 4 as function of the tank slenderness parameter γ=H/R.Figure 4
Ratioshi/H and hi∗/H as functions of the parameter tank slenderness.


### 2.2. Convective Component

The spatial-temporal variation of the “convective” (sloshing) pressure component is given by the expression(9)pcξ,ζ,θ,t=ρ∑n=1∞ψncos⁡νnγζJ1νnξcos⁡θAcnt,where(10)ψn=2Rλn2-1J1λncosh⁡λnγ,J1 is Bessel function of the first order, and λn are the roots of the first-order Bessel function of the first kind (λ1=1.8412; λ2=5.3314; λ3=8.5363, λ4=11.71, λ5=14.66, and λ5+i=λ5+5i (i=1,2,…)). Acnt is acceleration time-history of the response of a single-degree-of-freedom oscillator having a circular frequency ωcn given by (11)ωcn=gλntanh⁡λnγR,so (12)Tcn=2πgλntanh⁡λnγ/R,and damping ratio appropriate for the sloshing of the fluid.Only the first oscillating, or sloshing, mode and frequency of the oscillating liquid (n=1) need to be considered in expression for design purposes (13)ωc1=4.2R,in which the oscillation yields periods of the order for the usual values of R are few seconds.Figure5 shows the functions of sloshing pressure distributions along the height of tank for the tank slenderness parameters γ=H/R: 0.5, 1.0, and 3.0, the first three modes.Figure 5
Variation of convective pressures along the height for values of the tank slenderness parameterγ=H/R: 0.5, 1.0, and 3.0.The horizontal resultants of the convective pressure are the total base shear and the total convective moments. The total convective base shear is given as(14)Qct=∑n=1∞mcnAcnt,where(15)mcn=m2tanh⁡λnγλnγλn2-1.mcn is thenth modal convective mass and hcn is its height; see Figure 10.(16)hcn=H1+1-cosh⁡λnγλnγsin⁡λnγ.Figures6 and 7 show the quantities mci as function of the tank slenderness parameter γ=H/R:(i)
fori = 1, 2, 3, and 4 in Figure 6(ii)
only fori = 2, 3, and 4 in Figure 7.Figure 6
Ratiosmci/m as function of the tank slenderness parameter fori = 1, 2, 3, and 4.Figure 7
Ratiosmci/m as function of the tank slenderness parameter fori = 2, 3, and 4.The total convective (overturning) moment immediately below the bottom plate of the tank is(17)Mc∗t=∑n=1∞mcnAcnthcn∗=∑n=1∞Qcnthcn∗,where(18)hcn∗=H1+2-cosh⁡λnγλnγsin⁡λnγ.The total convective (bending) moment in tank wall immediately above the bottom plate is(19)Mct=∑n=1∞mcnAcnthcn=∑n=1∞Qcnthcn,where (20)hcn=H1+1-cosh⁡λnγλnγsin⁡λnγ.The quantitieshci and hci∗ as function of the tank slenderness parameter γ=H/R are seen in Figures 8 and 9, the quantities hci in Figure 8 and the quantities hci∗ in Figure 9.Figure 8
Ratioshci/H as functions of the tank slenderness parameter.Figure 9
Ratioshci∗/H as functions of the tank slenderness parameter.Figure 10
Liquid-filled tank modeled by generalized single-degree-of-freedom systems.The convective component of the response may be obtained from that of oscillators having massesmcn, attached to the rigid tank through springs having stiffness kn=ωn2mcn. The tank is subjected to the ground acceleration time-history Ag(t) and the masses responds with accelerations Acnt. hcn∗ or hcn is the level where the oscillator needs to be applied in order to give the correct value of Mcn∗ or Mcn, respectively.


### 2.3. Mechanical Model

The dynamic analysis of a liquid-filled tank may be carried out using the concept of generalized single-degree-of-freedom (SDOF) systems representing the impulsive and convective modes of vibration of the tank-liquid system as shown in Figure10. For practical applications, only the first convective mode of vibration needs to be considered in the analysis of mechanical model.


## 2.1. Rigid Impulsive Component

The seismic loads acting on wall and bottom of cylindrical tanks (Figure1) can be divided into the following components.The spatial-temporal variation of the “rigid impulsive” pressure is given by the expression(1)piξ,ζ,θ,t=Ciξ,ζρHcos⁡θAgt,where(2)Ciξ,ζ=2∑n=0∞-1nI1′νn/γνn2cos⁡νnζI1νnγξ,νn=π2n+1/2 and γ=H/R, I1· is the modified Bessel function of order 1, and I1′· is derivate that can be expressed in terms of modified Bessel function of orders 0 and 1 I1′x=dI1x/dx=I0x-I1x/x. The function Ci gives the distribution along the height of the time-dependence pressure pi. θ is angle of circumference, γ=H/R is slenderness parameter, and Agt is the free-field ground horizontal acceleration of ground with peak value denoted by ag. The result of an equivalent single-degree-of-freedom system is mi with the impulsive period Ti. The distribution pi along the height of tank wall gives the function Ci (see (2)) for ξ=1 (i.e., at the wall of the tank) and cos θ=1 (i.e., in the plane of the horizontal seismic action). Figure 2 presents the vertical distribution of impulsive pressure pi on the tank wall along the wall height, normalized to ρRag, for seven values of the slenderness parameter γ=H/R.Figure 2
Variation of vertical distribution function of the impulsive pressure along the wall height for seven values of the tank slenderness parameterγ=H/R, normalized to ρRag.For a number of purposes it is useful to evaluate the horizontal resultant of the pressure at the base of the wall, the total base shear, and the moments of the pressures with respect to an axis orthogonal to the direction of the seismic motion.Mi is the total impulsive (bending) moment, with respect to an axis orthogonal to the direction of the seismic action motion, immediately above the tank bottom and includes only the contributions of the pressures on the walls (see (7)). Mi∗ is the total impulsive (overturning) moment immediately below the tank bottom and includes the contributions of the pressures on the walls and of those on the bottom (see (5)).The horizontal impulsive pressure resultant, total impulsive base shear, at the base of the wall is(3)Qit=miAgt, where mi is called impulsive mass and indicates the mass of the contained fluid moved together with the walls and mi is given by the expression(4)mi=m2γ∑n=0∞I1νn/γνn3I1′νn/γ,where m=ρπR2H is total mass of the fluid contained in tank.The quantitiesmi are plotted in Figure 3 as function of the tank slenderness parameter γ=H/R.Figure 3
Ratiosmi/m as function of the tank slenderness parameter.The total impulsive base moment immediately below the tank bottom is given as(5)Mi∗=mihi∗A′gt,where (6)hi∗=H1/2+2γ∑n=0∞νn+2-1n+1I1νn/γ/νn4I1′νn/γ2γ∑n=0∞I1νn/γ/νn3I1′νn/γ,and total impulsive base moment immediately above the tank bottom is(7)Mi=mihiAgt,where(8)hi=H∑n=0∞-1nI1νn/γ/νn4I1′νn/γνn-1n-1∑n=0∞I1νn/γ/νn3I1′νn/γ.The quantitieshi and hi∗ are plotted in Figure 4 as function of the tank slenderness parameter γ=H/R.Figure 4
Ratioshi/H and hi∗/H as functions of the parameter tank slenderness.


## 2.2. Convective Component

The spatial-temporal variation of the “convective” (sloshing) pressure component is given by the expression(9)pcξ,ζ,θ,t=ρ∑n=1∞ψncos⁡νnγζJ1νnξcos⁡θAcnt,where(10)ψn=2Rλn2-1J1λncosh⁡λnγ,J1 is Bessel function of the first order, and λn are the roots of the first-order Bessel function of the first kind (λ1=1.8412; λ2=5.3314; λ3=8.5363, λ4=11.71, λ5=14.66, and λ5+i=λ5+5i (i=1,2,…)). Acnt is acceleration time-history of the response of a single-degree-of-freedom oscillator having a circular frequency ωcn given by (11)ωcn=gλntanh⁡λnγR,so (12)Tcn=2πgλntanh⁡λnγ/R,and damping ratio appropriate for the sloshing of the fluid.Only the first oscillating, or sloshing, mode and frequency of the oscillating liquid (n=1) need to be considered in expression for design purposes (13)ωc1=4.2R,in which the oscillation yields periods of the order for the usual values of R are few seconds.Figure5 shows the functions of sloshing pressure distributions along the height of tank for the tank slenderness parameters γ=H/R: 0.5, 1.0, and 3.0, the first three modes.Figure 5
Variation of convective pressures along the height for values of the tank slenderness parameterγ=H/R: 0.5, 1.0, and 3.0.The horizontal resultants of the convective pressure are the total base shear and the total convective moments. The total convective base shear is given as(14)Qct=∑n=1∞mcnAcnt,where(15)mcn=m2tanh⁡λnγλnγλn2-1.mcn is thenth modal convective mass and hcn is its height; see Figure 10.(16)hcn=H1+1-cosh⁡λnγλnγsin⁡λnγ.Figures6 and 7 show the quantities mci as function of the tank slenderness parameter γ=H/R:(i)
fori = 1, 2, 3, and 4 in Figure 6(ii)
only fori = 2, 3, and 4 in Figure 7.Figure 6
Ratiosmci/m as function of the tank slenderness parameter fori = 1, 2, 3, and 4.Figure 7
Ratiosmci/m as function of the tank slenderness parameter fori = 2, 3, and 4.The total convective (overturning) moment immediately below the bottom plate of the tank is(17)Mc∗t=∑n=1∞mcnAcnthcn∗=∑n=1∞Qcnthcn∗,where(18)hcn∗=H1+2-cosh⁡λnγλnγsin⁡λnγ.The total convective (bending) moment in tank wall immediately above the bottom plate is(19)Mct=∑n=1∞mcnAcnthcn=∑n=1∞Qcnthcn,where (20)hcn=H1+1-cosh⁡λnγλnγsin⁡λnγ.The quantitieshci and hci∗ as function of the tank slenderness parameter γ=H/R are seen in Figures 8 and 9, the quantities hci in Figure 8 and the quantities hci∗ in Figure 9.Figure 8
Ratioshci/H as functions of the tank slenderness parameter.Figure 9
Ratioshci∗/H as functions of the tank slenderness parameter.Figure 10
Liquid-filled tank modeled by generalized single-degree-of-freedom systems.The convective component of the response may be obtained from that of oscillators having massesmcn, attached to the rigid tank through springs having stiffness kn=ωn2mcn. The tank is subjected to the ground acceleration time-history Ag(t) and the masses responds with accelerations Acnt. hcn∗ or hcn is the level where the oscillator needs to be applied in order to give the correct value of Mcn∗ or Mcn, respectively.


## 2.3. Mechanical Model

The dynamic analysis of a liquid-filled tank may be carried out using the concept of generalized single-degree-of-freedom (SDOF) systems representing the impulsive and convective modes of vibration of the tank-liquid system as shown in Figure10. For practical applications, only the first convective mode of vibration needs to be considered in the analysis of mechanical model.


## 3. Numerical Example and Results

In this study fluid filling of ground-supported cylindrical rigid tank is analyzed. The characteristics of fluid filling areH=2 m andR, inner radiuses of cylindrical tanks, whereR depended on tank slenderness parameter γ=H/R.For the tank slenderness parameterγ=0.3 is the inner radius of tankR 6.667 m,(i)
forγ=0.5⇨R is 4 m,(ii)
forγ=0.7⇨R=2.857 m,(iii)
forγ=1⇨R=2 m,(iv)
forγ=2⇨R=1 m,(v)
forγ=3,R is given 0.667 m.The material characteristic of fluid filling (H2O) is density ρw=1,000 kg/m3. As the excitation input we consider horizontal earthquake load given by the accelerogram of the earthquake in Loma Prieta, California (18.10.1989), Figure 11.Figure 11
Accelerogram of Loma Prieta, California.The elastic response spectra for the Loma Prieta accelerogram in California were used for calculation of seismic response in the given examples, cylindrical fluid filling. The elastic response spectra drawn for damping 0.5%, 2%, 4%, 5%, 7% can be seen in Figure12. The impulsive spectral accelerations are obtained from a 5% damped elastic response spectrum and the convective spectral accelerations are obtained from a 0.5% damped elastic response spectrum.Figure 12
The response spectra for the Loma Prieta accelerogram from Figure11.The peak values of accelerated fluid pressure onto wall are located at the bottom of the container wall and presented in Figure13 as functions of the tank slenderness parameter γ. The peak values of pressures of accelerated fluid at the bottom of the tanks wall are bigger in short and large-scale tanks compared with pressures in narrow and high (slender) tanks.Figure 13
The peak value of fluid pressure in [kPa] on the bottom of tank wall as functions of the tank slendernessγ.The fluid pressure is zero pressure at the original position of the free surface of the fluid in the static situation. The peak values of accelerated fluid pressure onto wall at the original position of the free surface are shown in Figure14 as function of the tank slenderness parameter.Figure 14
The peak value of fluid pressure in [kPa] on the tank wall at the original position of the fluid free surface as function of the tank slenderness parameterγ.Figures15 and 16 show(i)
comparison of the total massm in [kg] of the fluid contained in tank as function of the tank slenderness parameter γ,(ii)
comparison of component ratio of impulsive and convective fluid masses contained in tank as functions of the tank slenderness parameterγ. If the tank slenderness parameter γ is smaller, then the component of convective fluid mass is bigger and the component of impulsive fluid mass is smaller.Figure 15
Comparison of the total mass of the fluid contained in tank as function of the tank slenderness parameterγ.Figure 16
Comparison of the total mass components (impulsive and convective) of the fluid contained in tank as functions of the tank slenderness parameterγ.Dependence ofhi, hc, and hi∗ on the tank slenderness parameter is seen in Figure 17; they correlate well with hi, hc, and hi∗ in Figures 4, 5, and 7.Figure 17
Comparison ofhi, hc, and hi∗ as functions of the tank slenderness parameter γ.Figure18 documents comparison of the total base shearV in [kN] of the fluid contained in tank as function of the tank slenderness parameter γ. Figure 19 shows comparison of component ratio of impulsive and convective component of base shearV on tank as functions of the tank slenderness parameter. If the tank slenderness is smaller, then the component of convective base shear is bigger and the component of impulsive base shear is smaller.Figure 18
Comparison of the total base shear as function of the tank slenderness parameterγ.Figure 19
Comparison of the base shear components (impulsive and convective) as functions of the tank slenderness parameterγ.Figure20 shows comparison of the total bending momentsM and overturning moments M∗ in [kNm] of the fluid contained in tank as functions of the tank slenderness parameter γ.Figure 20
Comparison of the total bending moments as functions of the tank slenderness parameterγ.Comparison of component ratio of impulsive and convective fluid moments in tank as functions of the tank slenderness parameterγ can be seen in Figures 21 and 22. Figure 21 shows comparison of total the bending momentsM and Figure 22 describes comparison of total overturning moments M∗. If the tank slenderness is smaller, then the components of convective fluid moments are bigger and the components of impulsive fluid moments are smaller.Figure 21
Comparison of bending moment components (impulsive and convective) as functions of the tank slenderness parameterγ.Figure 22
Comparison of overturning moment components (impulsive and convective) total base shear as functions of the tank slenderness parameterγ.


## 4. Conclusions

The seismic analysis of liquid storage containers is highly complex problem. Therefore, the knowledge of earthquake fluid effect on solid domain of tanks and appropriate modeling of the liquid storage tanks is essential for dynamic analysis and seismic response evaluation of earthquake resistance structure/facility, tanks. The dynamic behavior for liquid-containing structures has to use different approach as for solid structure. The weight of storage liquid varies in time because of variable liquid storage level; accordingly there exists sloshing of fluid which influences the response of the tank. The thoroughgoing dynamic analysis of contained liquid is necessary. The knowledge of forces acting onto containers, pressures in solid of tanks, surface sloshing process, and maximal height of liquids wave during an earthquake plays essential role in reliable and durable design of earthquake resistance containers. The analysis of a coupled multiphysics system is frequently required nowadays, to understand the behavior of the system. The motion of contained fluid in tanks may by expressed as the sum of two separate contributions, impulsive and convective actions. The mechanical model represents the dynamic analysis of fluid-filled tank, using the concept of generalized SDOF systems, representing the impulsive and convective (sloshing) modes of vibration. This paper summarizes the results of a comprehensive analytical investigation concerning the seismic analysis of ground-supported cylindrical tank-fluid filling effects on the tanks, subject to a horizontal component of earthquake ground motion. The tanks were excited by acceleration of Loma Prieta earthquake in California. Basic responses of the interest were the peak values of fluid pressures, the total base shear, the total bending, and the overturning moments of fluid as functions of the tank slenderness parameter and their ratios of impulsive and convective effects. In this study fluid filling of ground-supported cylindrical rigid tank is analyzed. The geometrical characteristics of fluid filling (H2O, density ρw=1,000 kg/m3) forH = 2 m, whereR represents the inner radiuses of cylindrical tanks depended on tank slenderness parameter γ=H/R. Inner container radiuses are given as 6.667 m, 4 m, 2.857 m, 2 m, 1 m, and 0.667 m of fluid filling.It is seen from results in Section3 that(i)
the hydrodynamic pressures are bigger in short and large-scale tanks (tanks with smaller tank slenderness parameter) than in narrow high (slender) tanks;(ii)
the impulsive components of hydrodynamic pressures give maximum values on the bottom of tank wall and they obtain bigger values in narrow high (slender) tanks;(iii)
the convective components of hydrodynamic pressures receive the maximum values on the tank wall at the original position of the fluid free surface and they are bigger in short and large-scale tanks than in narrow high (slender) tanks;(iv)
the total mass of fluid, the total base shear, and the total moments are bigger in narrow slender tanks; these results are due to bigger volume of contained liquid;(v)
the impulsive component of fluid masses obtains bigger values in narrow and slender tanks;(vi)
the bigger values of the convective fluid masses are in short and large-scale tanks;(vii)
the impulsive component of base shear receives bigger values in narrow slender tanks and on the other sides the convective fluid base shear is bigger in short and large-scale tanks;(viii)
the bigger values of fluid impulsive bending and overturning moments are given in narrow and slender tanks;(ix)
the convective components of moments (bending and overturning) give the bigger values in short and large-scale tanks.The knowledge of fluid effect on solid domain of tanks and their interdependence is, due to earthquake, significant for safe and economic design of earthquake-resistant structure/facility, containers.



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