International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 6308(Print) ISSN 0976 6316(Online) Volume 1 Number 1, May - June (2010), pp. 27-45 IAEME, http://www.iaeme.com/ijciet.html International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 6308(Print), IJCIET I A E M E DYNAMIC RESPONSE OF ELEVATED LIQUID STORAGE ABSTRACT ELASTIC TANKS WITH BAFFLE Damodar Maity Associate Professor Department of Civil Engineering IIT Kharagpur, India E-Mail: dmaity@civil.iitkgp.ernet.in C. Naveen Raj Department of Civil Engineering IIT Guwahati, India Indrani Gogoi Associate Professor Department of Civil Engineering NITK Surathkal, India The dynamic response of elevated baffled liquid storage tanks, including the hydrodynamic interaction, is presented in this paper. Both the tank and fluid are discretized by finite elements. The tank wall and baffle are made of elastic, isotropic material. The liquid in the tank is considered as water. The equation of the liquid motion is expressed in terms of pressure by neglecting the viscosity and nonlinear term of the Navier-Strokes equation. The fluid is modeled with twenty seven node solid (3-D) elements whereas the elastic tank wall and its baffle are modeled with nine node (2-D) shell elements using ADINA FSI, a multi-physics simulation program. The response of the coupled system is obtained considering fluid structure interaction effects. Hydrodynamic pressure generated in the fluid-structure interface and the displacement of the tank wall is evaluated. A parametric study has been carried out to study the effectiveness of a baffle for damping liquid oscillations in order to evaluate efficient baffle configuration in an elevated tank. 27
Key words: Elevated tank; baffle; fluid-structure interaction; hydrodynamic pressure; seismic response 1. INTRODUCTION In the recent years there has been an upsurge of unrest in the safety of all structures, components and facilities that can produce both primary and secondary health hazards in the event of an earthquake. The structural integrity of liquid operational and liquid retaining structures is one of the prime concerns. Liquid storage tanks are very important components of industrial and agricultural facilities used for storing toxic or inflammable liquids and water respectively. Observations from available field reports on the structural performance of tanks during recent earthquakes indicate that the steel tanks, rather than concrete tanks, are more susceptible to damage and eventual collapse (Hamdan 2000). Failure of tanks and their accessories is not only limited to the immediate danger to nearby human lives, but also to a large extent leads to serious consequences and long-term environmental damages. Unlike buildings, predicting seismic behavior of liquid containing tanks are (i) affected by hydrodynamic force on tank walls and base exerted by liquid during seismic excitation and (ii) liquid containing tanks are less ductile and have less redundancy. In big cities, sometimes swimming pools are constructed on the terrace of the buildings, which may disturb the structural integrity as a whole during the earthquake, if hydrodynamic pressure is not taken into account properly while designing the structural system. Heavy damages were caused to liquid storage tanks especially elevated tanks by strong earthquakes such as Niigata in 1964, Alaska in 1964 and Parkfield in 1966. A review of seismic codes on liquid containing tanks carried out by Jaiswal et al. (2007) concludes that all the codes design for higher seismic force for liquid containing tank as they possess low ductility and redundancy. A lower value of response modification factor is prescribed as compared to a building system. However, there is a large variation in assigning of response modification factor for different types of tanks. When a tank like container, partly filled with liquid is excited by external forces, sloshing takes place. Sloshing is the term used to describe the motion of a liquid in a partially filled tank. The hydrodynamic pressure is developed due to the liquid movement 28
and the pressure is exerted on the tank wall. The tank wall should therefore be strong enough to withstand these hydrodynamic forces in addition to hydrostatic pressure. An additional structural member called baffle can be provided to control sloshing. For ease of installation, a disk-type baffle with inner hole is being widely used for liquid storage tanks. Both experimental and theoretical studies related to the vibration of liquid in a stored container have been reported in the literature. A numerical investigation was carried out by Haroun and Tayel (1985) on partly filled axi-symmetric tanks in which the shell was modeled by finite elements and liquid region treated analytically. Natural frequencies and mode shapes were evaluated. Hwang (1989) studied the dynamic response of liquid storage tank during to earthquake by combining boundary element method and finite element method. The tank wall 3 and the fluid domain were treated as two substructures of the total system. The boundary element method was used to find the hydrodynamic forces associated with small amplitude excitations. A baffle is a supplementary structural member, which supplies a sort of passive control during earthquake motion. Baffle has been introduced recently with the objective of improving the seismic safety and reducing the risk of damage or failure of thin walled cylindrical liquid storage tanks. The effects of a rigid baffle on the seismic response of liquid in a rigid cylindrical tank was studied by Gedikli and Erguven (1999). The parametric eigen characteristics of baffled cylindrical liquid storage tank was investigated by Cho et. al. (2002) with the coupled structural acoustic finite element method. Biswal et al. (2004) demonstrated coupled formulation for the free vibration analysis of liquidfilled cylindrical tank - baffle system that can be used to compute the low frequencies associated with liquid sloshing modes and high frequencies associated with the coupled vibration modes. Maity et al. (2009) showed the effective location of baffle and its dimension to obtain minimum response of the tank-water coupled system under seismic excitations. Eswaran et al. (2009) investigated the effect of ring baffles on liquid sloshing for partially filled cubic tank. The effect of various factors like depth of liquid, tank geometry, the amplitude and nature of the tank motions on sloshing severity were studied. The results obtained were further used to verify the effectiveness of numerical simulation technique. 29
Following the development of FEA tools used for structural analysis, significant advances in the fluid analysis have empowered the fluid analysts with a number of commercial CFD codes that are robust and efficient for general fluid flow analysis (Freitas 1995, Bathe et al. 1995, Zhang et al 2003, Bathe and Zhang 2004). The Lagrangian-Eulerian formulation in the fluid system can efficiently be solved by using ADINA FSI, a multi-physics simulation program (Andersson and Andersson 1997, Bathe et al 1999, Wang 1999, Panigrahy et al 2009). Moreover, recent efforts have resulted in a very smart design of the user-interface such as providing effective ways to generate meshes, set modeling assumptions, and state boundary conditions. The present study focuses on the study on the response of elevated liquid storage tank with the presence of baffle. The steel tank and the water domain are discretized by finite elements. The tank material is considered as isotropic and elastic. The tank wall and baffle are discretized by nine node shell elements. The liquid in the present analysis is water and is assumed to be inviscid, linearly compressible and is under small amplitude of excitation. Pressure degree of freedom is taken for fluid element. The fluid domain is modeled using twenty seven noded elements. The fluid and the tank domain are treated as two sub-structures coupled through their interface by an iterative scheme. The dynamic response of elevated baffled liquid storage tank has been studied extensively to study the influence of baffle on the response control of the coupled system. 2. MATHEMATICAL MODELING The results of any dynamic analysis depend on the approximation involved in the development of the mathematical models for the systems. It is not possible always to obtain a closed form analytical solution for many engineering problems. With the advent of faster generation computers, one of the most powerful numerical techniques that have been developed in the realm of engineering analysis is the finite element method. The method, being general, can be used for the analysis of liquids and solids of complex shapes and complicated boundary conditions. In the present study, nine noded shell elements for the tank wall and twenty seven noded 3D solid elements for the liquid are chosen (Fig. 1 and Fig. 2) for the analysis. Three noded 3-d frame elements (Fig. 1) have been considered to model the steel column and bracing of the elevated tank. 30
2.1 Mathematical Modeling of Liquid: Using the principles of classical mechanics, the motion of continuous fluid medium in a fixed Cartesian coordinate frame of reference can be expressed using the Eulerian approach in terms of mass, momentum and energy as The characteristic properties of the medium are considered as functions of time and space in the frame of reference. In the above equations, t is the time, ρ is the density, v is the velocity vector, fb is the body force vector of the fluid medium, τ is the stress tensor and given by τ = ( p +λ.v)i + 2µe (4) Here, p is hydrodynamic pressure distribution in excess of the hydrostatic pressure; µ and λ are the two coefficients of fluid viscosity. The heat flux q and the specific rate of heat generation qb are neglected in the present problem. E is the specific total energy and is defined as Here b is the specific kinetic energy and e is the velocity strain tensor, which may be expressed as: written as The body forces included in fb (Eq. 2-3) is the gravitational force which may be where g is the gravitational acceleration vector. Since the flows are basically incompressible, a constant density (ρ) throughout the governing equations has been 31
assumed except in the continuity equation. The non-conservative form of the continuity equation in slightly compressible flows becomes where, ρm is the fluid density with the compressibility and ρ is the density at p = 0 and thus can be relating as: Here, k is the bulk modulus of elasticity of the fluid. Thus for the small amplitude of motion and with the absence of body force, the continuity and momentum equations of the fluid can be simplified to: 2.2 Mathematical Modeling of Solid: Material for solid parts is assumed to be isotropic elastic with small displacements and strains. The tank wall and its baffle (Figure 1) are modeled using the standard Lagrangian formulation for displacement and strain, which is as follows: where, is the ijth components of the Cauchy stress tensor for (i, j = 1, 2, 3), is the displacement component in the co-ordinate i direction and is the mass density. 2.3 Modeling of FSI Problems: In case of fluid-structure interaction (FSI) problems, fluid forces are applied on the structure and the structural deformation changes the fluid domain. Difficulties arise in the FSI analysis not only because of the non-linear governing equations for fluid; but also because of description of the governing equations for fluid and structure in different coordinates. The computational domain is divided into fluid and structural domain, where 32
a fluid model and a structural modelare defined respectively, through their material data, boundary conditions etc. The interaction occurs along the interface of the two domains. One can perform simulations and predict many physical phenomena when these two models are coupled together. The coupling conditions at the fluid-structure interfaces are: Here, f is the traction vector acting on the structure surface, and n is the unit vector normal to the interface. Using an Arbitrary-Lagrangian-Eulerian coordinate system (ALE) i.e., eqs. (10)- (11) for fluid; eq. (12) for solid and application of the boundary conditions eq. (13) to the discrete finite element equations of the fluid and the structure, the finite element equations of the coupled fluid-structure system can be expressed as where, are the solution vectors of the coupled system. Xf and Xs are the fluid and solid vectors defined at the fluid and solid nodes respectively. Ff and Fs are the finite element equations corresponding to the fluid and structure model. The decoupled fluid and solid equations can be represented by respectively. 3. RESULTS AND DISCUSSIONS The types of problems addressed through this formulation are that of a rectangular thin walled elevated water filled tank and is subjected to external excitations. The tank wall is considered in this case to be elastic in nature. 3.1 Validation for empty tank without baffle To validate the correctness of the developed models a benchmark problem has been considered and results are compared with the results available in the literature. A cylindrical ground supported circular steel tank of modulus of elasticity, 33
Poisson s ratio, υ = 0.3 and mass density, has been considered for analysis and compared with the results obtained by Haroun and Tayel (1985). The thickness of tank wall and its radius is considered as 25.4 mm and 18.29 m respectively. Figure 1 Finite element discretization of Elevated tank Figure 2 Finite element discretization of fluid domain Free vibration analysis of the above tank has been carried out to validate the developed finite element model. The variations of natural frequencies with various heights to radius ratio of the tank have been studied and the results are tabulated in Table 1 along with the results obtained from literature by Haroun and Tayel (1985). In Table 1, 34
H/R represents the ratio of height to radius of the tank. A convergence study has been done to find the results with a desired level of accuracy. For example, in case of H/R = 0.67, the discretization has been taken as 5 40 in which the results are converging sufficiently. The comparison of results at R = 18.29 m, show that the frequencies obtained by the proposed model agree quite well with those obtained in literature. Table 1 Variation of fundamental frequency of empty tank 3.2 Response of elevated tank The dynamic analysis of elevated tank has been carried out to study the stability of the developed modeling. The geometry and material properties of the tank are considered as follows: Tank: length = 6 m, width = 6 m, height = 4 m, Thickness = 20 mm, column spacing = 4 m, column height = 15 m. The tank wall is made of steel having the material properties of modulus of elasticity, Poisson s ratio, υ =0.3 and density. The mass density and velocity of sound in water are assumed as and 1438.7 m/s respectively. The above dimensions and material properties has been considered throughout the study unless it is mentioned. The horizontal displacements of the elevated tank under different sinusoidal excitation frequencies with unit amplitude of acceleration are plotted in Figures 3 to 5. The frequencies under consideration are the fundamental natural frequency along with some different frequencies. It is observed that the displacement is increasing continuously (Figure 4) when the tank is excited at its natural frequency as expected. The first three mode shapes of the elevated empty tank have been plotted in Figure 6. 35
Figure 3 Displacement at top of tank due to sinusoidal ground acceleration for f = 0.8 cps Figure 4 Displacement at top of tank due to sinusoidal ground acceleration for f=2.134cps Figure 5 Displacement at top of tank due to sinusoidal ground acceleration for f = 4 cps 36
(a) 1 st Mode (b) 2 nd Mode (c) 3 rd Mode Figure 6 Mode shapes of elevated tank 3.3 Effect of baffle on elevated tank under sinusoidal acceleration To study the effect of baffle on elevated tank, the baffle is located at mid height of water level. The baffle location is arrived on the basis of previous work carried by Maity et al (2009) at which substantial amount of response reduction is expected. The material properties of the baffle are same as in case of tank wall. A sinusoidal ground acceleration of frequency 6cps, is applied at the ground. The dynamic responses due to both horizontal and vertical sinusoidal loadings are shown in Figures 7 to 10. It is observed that the developed hydrodynamic pressures as well as the tank displacements are reducing due to the presence of an elastic baffle. The deformation pattern of the water filled baffled tank under horizontal sinusoidal ground excitation at different time instant has been plotted in Figure 11. The time period (T) of the excitation is taken as 0.1676 cps in this analysis. The deformation patterns are plotted at time, t = 0, T/16, 2T/16, 3T/16 and 4T/16 secs respectively shows the behaviour of the elevated tank. Figure 12 shows deformation of the same tank at different time instant due to sinusoidal excitation in vertical direction. 37
Figure 7 Top displacement of the tank wall due to horizontal sinusoidal acceleration Figure 8 Hydrodynamic pressure at the bottom of tank due to horizontal sinusoidal acceleration Figure 9 Top displacement of the tank wall due to vertical sinusoidal acceleration 38
Figure 10 Hydrodynamic pressures at the bottom of tank due to vertical sinusoidal acceleration (a) t = 0 sec. (b) t = T/16 sec. (c) t = 2T/16 sec. (d) t = 3T/16 sec. (e) t = 4T/16 sec Figure 11 Deformation pattern at different time instant due to horizontal sinusoidal excitation (a) t = 0 sec. (b) t = T/16 sec. (c) t = 2T/16 sec. (d) t = 3T/16 sec. (e) t = 4T/16 sec. Figure 12 Deformation pattern at different time instant due to vertical sinusoidal excitation 39
3.3 Effect of baffle on elevated tank under seismic excitation The seismic response of the same tank has been plotted in Figures 13 to Figure 16 to study the effectiveness of the baffle. Results presented are used to depict the distinction between the response obtained with and without the presence of baffle under horizontal and vertical El Centro (N-E) horizontal earthquake excitation. It is observed that both the displacement and hydrodynamic pressure are well controlled if the baffle is present in the water filled tank. Figures 17and 18 show the hydrodynamic pressure distribution of fluid domain at a particular time instant (in this case 10 sec.) due to El Centro horizontal earthquake excitation without and with the presence of baffle respectively. It is observed by comparing above two figures that the magnitude of the hydrodynamic pressure becomes considerably less if the baffle is present inside the water tank. Same observations have been noticed in Figures 19 and 20 while the tank is subjected to El Centro earthquake excitation in the vertical direction. Figure 13 Top displacement of the tank wall due to horizontal earthquake acceleration Figure 14 Hydrodynamic pressures at the bottom of tank due to horizontal earthquake acceleration 40
Figure 15 Top displacement of the tank wall due to vertical earthquake acceleration Figure 16 Hydrodynamic pressures at the bottom of tank due to vertical earthquake acceleration Figure 17 Hydrodynamic pressure distribution of water domain due to El Centro horizontal earthquake (without baffle) 41
Figure 18 Hydrodynamic pressure distribution of water domain due to El Centro horizontal earthquake (with baffle) Figure 19 Hydrodynamic pressure distribution of water domain due to El Centro vertical earthquake (without baffle) 42
Figure 20 Hydrodynamic pressure distribution of water domain due to El Centro vertical earthquake (with baffle) 4. CONCLUSIONS The objective of the present investigation is to gain an understanding of the behavior of the coupled elevated baffled tank-water system. The FE model using ADINA program is developed, in which a general time domain analysis of the coupled system is carried out considering fluid-structure interaction effects. The displacements in the tank and the pressure developed in the water domain are computed using the developed model under external excitations. Study has been carried out under horizontal as well as vertical seismic excitation to have a comparative response of the baffled tank-water system. The fluid-structure interaction effects are achieved by indirect coupling of the two systems viz., the elevated baffled tank and water domain. The elevated water tanks are, in general, very susceptible to earthquakes. The parametric study reveals that the presence of baffle in the elevated water tank reduces the dynamic response of the coupled system to a large extent. The response results depict that baffles may be used effectively as controlling devices for the control of tank displacements and hydrodynamic forces during earthquake excitations. 43
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