AN INNOVATIVE TYPE OF TUNED LIQUID DAMPER

Transcription

1 10NCEE Tenth U.S. National Conference on Earthquake Enineerin Frontiers of Earthquake Enineerin July 21-25, 2014 Anchorae, Alaska AN INNOVATIVE TYPE OF TUNED LIQUID DAMPER R. O. Ruiz 1, D. Lopez-Garcia 2 and A. A. Taflanidis 3 ABSTRACT This paper introduces a new type of Tuned Sloshin Damper (TSD) havin a relatively simple, easy-to-model behavior and capable of effectively controllin earthquake vibrations. It consists of a traditional TSD with addition of a floatin roof. Since the roof is much stiffer than water, it prevents wave breakin, hence makin the response linear even at lare amplitudes. The roof also facilitates the incorporation of supplemental devices (either passive, semi-active or active) with which the level of vibration suppression can be substantially aumented. This new proposed TSD, from now on denoted as Tuned Liquid Damper with Floatin Roof (TLD-FR), maintains the traditional advantaes of TSD (low cost, easy installation and tunin) but its numerical characterization is much simpler because the floatin roof only allows one deree of freedom performance. An efficient numerical scheme, based on potential flow in which its dynamic behavior is expressed as a second order lineal system of equations, is discussed for the modelin of the TLD-FR. Preliminary results reveal that structures with TLD-FR can achieve sinificant reductions in their seismic response. This reduction is sensitive to the desin characteristics of the TLD-FR, the fundamental frequency and the introduced dampin. The overall study shows that the TLD-FR is an efficient alternative to reduce seismic responses on structures offerin low cost, and easy-to-model and tune behavior. 1 Ph.D. candidate, Dept. of Civil & Environmental En. & Earth Sciences, University of Notre Dame, USA & Dept. of Structural & Geotechnical En., Pontificia Universidad Catolica de Chile, Chile 2 Associate Professor, Dept. of Structural & Geotechnical En., Pontificia Universidad Catolica de Chile, Chile 3 Associate Professor, Dept. of Civil & Environmental En. & Earth Sciences, University of Notre Dame, USA Ruiz RO, Lopez-Garcia D, Taflanidis AA. An innovative type of tuned liquid damper. Proceedins of the 10 th National Conference on Earthquake Enineerin, Earthquake Enineerin Research Institute, Anchorae, AK, 2014.

2 Tenth U.S. National Conference on Earthquake Enineerin Frontiers of Earthquake Enineerin July 21-25, 2014 Anchorae, Alaska An Innovative Type of Tuned Liquid Damper R. O. Ruiz 1, D. Lopez-Garcia 2 and A. A. Taflanidis 3 ABSTRACT This paper introduces a new type of Tuned Sloshin Damper (TSD) havin a relatively simple, easy-to-model behavior and capable of effectively controllin earthquake vibrations. It consists of a traditional TSD with addition of a floatin roof. Since the roof is much stiffer than water, it prevents wave breakin, hence makin the response linear even at lare amplitudes. The roof also facilitates the incorporation of supplemental devices (either passive, semi-active or active) with which the level of vibration suppression can be substantially aumented. This new proposed TSD, from now on denoted as Tuned Liquid Damper with Floatin Roof (TLD-FR), maintains the traditional advantaes of TSD (low cost, easy installation and tunin) but its numerical characterization is much simpler because the floatin roof only allows one deree of freedom performance. An efficient numerical scheme, based on potential flow in which its dynamic behavior is expressed as a second order lineal system of equations, is discussed for the modelin of the TLD-FR. Preliminary results reveal that structures with TLD-FR can achieve sinificant reductions in their seismic response. This reduction is sensitive to the desin characteristics of the TLD-FR, the fundamental frequency and the introduced dampin. The overall study shows that the TLD-FR is an efficient alternative to reduce seismic responses on structures offerin low cost, and easy-to-model and tune behavior. Introduction In modern urban areas there is an increasin trend to build tall, liht and slender structures. Since this kind of structures have low levels of structural dampin, addition of supplemental devices is often necessary to control undesirable vibrations. In this reard, Tuned Sloshin Dampers (TSD) have been shown to be able to successfully control vibrations induced by winds [1-3], and have the potential to effectively control earthquakes-induced vibrations [4-7]. When compared to traditional Tuned Mass Dampers (TMDs), they are attractive because of lower installation costs, bidirectional control capabilities, easier tunin processes and possible alternative uses (e.., the mass of water could be used to suppress fire emerencies). Their dynamic behavior, thouh, is typically hihly nonlinear due to wave breakin phenomena, and their inherent level of dampin is typically less than the optimum level [8]. While hiher levels of dampin can be attained by addin submered obstacles, the resultin behavior is very difficult to model and, consequently, to reliably predict. Thouh TSDs have been primarily implemented to control wind-induced vibrations, many researches are currently investiatin how to develop practical TSDs than could be used to effectively control seismically induced vibrations. Current research on this topic aims to: (a) develop relatively simple analytical tools to model wave breakin and dampin [9-11], (b) 1 Ph.D. candidate, Dept. of Civil & Environmental En. & Earth Sciences, University of Notre Dame, USA & Dept. of Structural & Geotechnical En., Pontificia Universidad Catolica de Chile, Chile 2 Associate Professor, Dept. of Structural & Geotechnical En., Pontificia Universidad Catolica de Chile, Chile 3 Associate Professor, Dept. of Civil & Environmental En. & Earth Sciences, University of Notre Dame, USA Ruiz RO, Lopez-Garcia D, Taflanidis AA. An innovative type of tuned liquid damper. Proceedins of the 10 th National Conference on Earthquake Enineerin, Earthquake Enineerin Research Institute, Anchorae, AK, 2014.

3 develop strateies to introduce more dampin without introducin excessively complicated modelin issues, i.e. submered screen, nets, baffles, etc. [12-15]; and (c) implement active and semi-active control strateies [7,16]. Motivated by this, the current paper introduces a new type of TSD havin a relatively simple, easy-to-model behavior and capable of effectively controllin earthquakes vibrations. It consists of a traditional TSD with addition of a floatin roof. Since the roof is much stiffer than water, it prevents wave breakin, hence makin the response linear even at lare amplitudes. The roof also makes possible the addition of supplemental devices (either passive, semi-active or active) with which the level of dampin can be substantially aumented. The proposed TSD, from now on denoted as Tuned Liquid Damper with Floatin Roof (TLD- FR), is schematically shown in Fi. 1. The modelin of the TLD-FR dynamics is discussed first and then the experimental behavior of a scaled TLD-FR under harmonic and seismic excitations is presented, to evaluate the accuracy of the developed numerical model. An illustrative example of a TLD-FR actin on a 9-floor buildin is finally presented. Mathematical Modelin The liquid is modeled usin principles of mass and momentum conservation, while tank walls and bottom are considered riid. The liquid is assumed to be inviscid, incompressible, and irrotational, allowin its motion to be completely defined by a velocity potential function ϕ, whereas the roof is initially not taken into account. Body forces are assumed conservative and nonlinear terms are nelected (small displacement assumption). Thus, mass and momentum conservation equations take the form of Laplace (=0) and Bernoulli ( / t + p/ρ+π=0) equations, where p is pressure, ρ is the fluid density and Π is the potential of the conservative forces. The inertial system of reference is located at the middle of the non-perturbed free surface and an auxiliary coordinate η is defined to measure the relative displacement between the free surface and the coordinate system. The fluid model is based on an arbitrary tank eometry (Fi. 1) where, Ω represents the liquids volume, Γ o represents the non-perturbed free surface (z = 0), Γ s represents the free surface at specific time t, and Γ p represents the walls and bottom surfaces. Floatin Roof Joint Damper Γ s z z Γ o x x η η Γ p Liquid Ω Fi. 1. Scheme of a Tuned Liquid Damper with Floatin Roof Scheme to Solve the Fluid Mechanic Equations In order to resolve the Laplace Equation the boundary conditions in Eq. 1 are considered, where defines the liquid velocity for any particle, n represents the velocity projection over vector n, and n p and n o are vectors identifyin the normal directions to wet and free tank surfaces

4 n o / z at o and n p 0 at p (1) The Galerkin s method is applied to the Laplace equation; for this purpose it is converted first in a weak formulation and then the Third Green s identity is applied to include the boundary condition in an explicit way. A typical FEM discretization is applied to the auxiliary coordinate η and velocity potential ϕ. In this context, denote by ϕ the vector with the values of the nodal velocity potential in the volume Ω, and by η s the vector that contains the values of the nodal displacements of the free surface. ϕ is partitioned into components associated and not to the free surface (ϕ s and ϕ r respectively), leadin ultimately to a linear system of ordinary differential equations iven by Eq. 3. A static condensation is then performed in order to express Eq. 3 only in terms of the surface variables. G 0 ηs D ss D sr s D ; d ss D T sr ; T 0 G N d ηnη o N N o Drs Drr r Drs Drr (3) For solvin the Bernoulli Equation, it is necessary to define the potential Π. For this purpose, the round acceleration u is introduced as part of Π to avoid time-dependent boundary conditions due to the use of an inertial reference system. The Bernoulli Equation is then particularized at the free surface nodes and combined with the Eq. 3 to obtain Eq. 4. Here, I is the identity matrix, X s is a vector containin the x coordinate of each node located at the free surface and p s is the vector with the pressure at the free surface. -1 I η I X u p / 0 D G η + I η I X u 1/ p (4) s s s s s s s s This numerical procedure ultimately expresses the dynamic behavior of liquid tanks as a second order lineal system of equations, where the independent variables are the vertical displacements of liquid at the free surface and the excitation is directly related to round acceleration. More details may be found in the paper by Ruiz and Lopez-Garcia [17]. Inclusion of the Floatin Roof After this first step, the effect of the floatin roof is taken into account by the incorporation of a flexible beam or shell located at the free surface of the liquid. The modelin is performed by a traditional FEM approach; the equations are manipulated by a condensation process in order to express the dynamics only in terms of the vertical displacements. An important assumption is that the free surface of the liquid and the beam have the same vertical displacement. The dampin of potential external devices is incorporated by addin the damper force at the proper nodes. In that sense, and for a linear dampin, the equation of motion of the floatin roof is expressed by Eq. 5. Mf η s + Cf η s + Kf ηs G p s (5) Here, M f and K f are the condensed mass and stiffness matrices of the floatin roof, while C f is a matrix that contains the linear coefficients of the external dampers. Note that the only physical connection between liquid and the floatin roof is throuh the pressure in the solid-liquid interface. Combination of Eq. 4 and Eq. 5 leads then to the coupled system

5 D G G Mf ηs G Cf ηs I G Kf ηs IXs u M η C η K η m u a s a s a s x (6) Throuh this approach the dynamic behavior of TLD-FR is expressed as a second order lineal system of equations, where the independent variables are the vertical displacements of the floatin roof and the excitation is directly related to round acceleration. It is important to note that traditional tuned liquid dampers can be described by an Equation such as Eq. 6 only if they operate in a linear way. Pressure on the Walls The Bernoulli Equation is evaluated at walls and combined with Eq. 4 to obtain the dynamic pressure over surface Γ p. Then, Q is defined as a transformation matrix, in order to express ϕ p in function of ϕ s, such that ϕ p =Q ϕ s, Eq.7. Here, p p, ϕ p and X p are vectors with the pressures, velocity potentials and x coordinates of the nodes located at the surface Γ p, respectively. p X u p Q D G η X u (7) -1 p p p p s p Nodal forces are computed via FEM scheme applied to the solid in contact with Γ p. For the element i the horizontal nodal forces F (i) p are iven by Eq. 8. Where α (i) is the anle between the surface Γ p and the x-y plane. After computin all nodal forces, the system is expressed by F R p. Let m be the number of nodes on Γ p. Then, the dimension of R p is m x m and the p p p dimension of p p is m x 1. Note that an easy way to obtain the total horizontal force transmitted F 1 1 F S F. to the round is by multiplyin F p by a 1 x m vector of ones, such that () i F sin N N d p (8) ( i) ( i) ( i) T ( i) ( i) ( i) p η η o p o -1 F S R p QD Gηs S Rp Xp u A η s Bu (9) Experimental Setup The tests were conducted in a 6-DOF shakin table with a maximum weiht capacity of 1440k under both harmonic and seismic excitations. A rectanular-base lass tank is used as TLD-FR. The tank rests on the shakin table by three rollers and the horizontal motion is constrained by two load cells. An ad hoc chassis was built around the lass tank to increase the resistant and to attach the sensors and dampers. Two accelerometers measure the absolute acceleration of the shakin table and the tank. The floatin roof is made by 1.5in thick of expanded polystyrene foam board and it is attached to the chassis by two spherical join located at the midspan. The location of joins enforces only one-directional operation and prevents undesired contacts between the floatin roof and walls. Two LVDT sensors measure at different points the vertical displacement of the floatin roof. The rotation of the floatin roof is obtained as an interpolation of these values by assumin a riid behavior. In order to control the dampin of the system, two air dampers are installed between the chassis and the floatin roof; the dampin coefficient is adjustable in a rane of 0 to 0.35N/mm/s. This confiuration allows a direct comparison between the mathematical model and the experimental results; moreover, it allows provin the validity of the p p

6 1-DOF and linearity hypothesis. The liquid used is water, and the base tank dimension is 0.8m of lenth in the excitation direction (L) by 0.4m of width (W). Different heihts of water (H) and dampin ratios (ζ) are handlin in this study. Each different confiuration was identified as Tank A (H=0.2L and ζ=3.6%), Tank B (H=0.2L and ζ=7.7%), Tank C (H=0.3L and ζ=3.7%), Tank D (H=0.3L and ζ=8.4%), Tank E (H=0.4L and ζ=3.5%) and Tank F (H=0.4L and ζ=8.2%). The external dampers were activated only for the Tanks B, D and F. The dampin actin on Tanks A, C and E comes from an intrinsic source, i.e. friction, dra o similar. Load Cell Disp. Sensor Damper Accel. Sensor Shakin Table Fi. 2. Scheme of the experimental setup Performance of the TLD-FR The mathematical model is implemented in 2D since all experimental tests were conducted settin only one-directional excitations. The fluid inside the tank is meshed by a Delaunay Trianulation Alorithm where the size of the trianles is controlled by imposin a maximum admissible ede size. A pulse input is devised to calculate the experimental natural frequency and dampin factor. Additionally, three different methods are used to compare the natural frequency of the TLD-FR. The first method is an analytical expression, which is valid only for rectanular tanks. The second method is the Housner Model, where analytical solutions are used to obtain an equivalent linear system where the fluid is replaced by a mass riidly attached to the walls and a mass attached to the wall throuh a linear sprin. Reference [17] offers details on both these analytical procedures. It is important to mention that these two methods predict the fundamental sloshin frequency of liquid tank; the floatin roof is not taken into account. Finally, the third method corresponds to the model explained in the previous section. The estimation of the fundamental sloshin period is practically the same for the three methods and the experimental results for all studied cases. The maxima difference in the period estimation is around 2%, i.e., for Tank A, the fundamental period obtained by analytic method, Housner s model, proposed model and experimental characterization is sec, sec, sec and 1.36 sec respectively. Moreover, the results indicate that the fundamental frequency is not sinificantly affected by the presence of the floatin roof. Also, it is observed that the system has a sinificant inherent dampin due possible to the presence of dra or friction forces between floatin the roof and walls. The harmonic response is studied by imposin different harmonic displacements and accelerations at the base of the TLD-FR. Here, the Normalized Transmitted Force and the Amplitude Ratio are respectively introduced as: n / ; max 100/ F F t M u t max max AR t x H (10)

7 Fi. 3. Harmonic Response of the TLD-FR for confiuration A and B The normalized transmitted force is an indicator of the force amplification due the sloshin effects. The amplitude ratio relates the maximum amplitude of the floatin roof with the water heiht. It is observed that if the excitation period is much reater than the natural period then F n is close to 1, indicatin that the total mass of liquid (M) moves in phase with the excitation. Results iven by the experiments and mathematical model also show a local minimum of F n, this local minimum is related to in-and out-of-phase movements of the impulsive and convective liquid mass respect to the excitation. For periods close to resonance it was only possible to set a 10mm harmonic displacement excitations; reater displacements enerate nonsafe floatin roof oscillation. The vertical red dashed lines indicate the location of the first three sloshin frequencies of a tank without floatin roof. The absence of maxima at second and third natural frequencies indicates that TLD-FR could be describes as a sinle-dof system. Additionally, different excitation amplitudes do not sinificantly affect the normalized transmitted force and the roof amplitude ratio; thus the dynamic behavior of the TLD-FR can be considered as linear (no amplitude dependence). Althouh only results for confiuration A and B are present in Fi 3, it is important to mention that similar behavior is observed for all confiurations. Different round motions were then used to study the TLD-FR behavior under seismic excitations. In particular, Fi 4 shows the response of the TLD-FR under the Melipilla earthquake recorded in Chile in In eneral, the numerical results are in ood areement with the experimental data since difference are close to 6% in terms of the maximum values. Some differences in the amplitudes are evident when the response starts to decrease, i.e., for t > 25sec. For the confiuration A (TLD-FR without external dampers), the differences could be related to the inherent nonlinear dampin mechanism (dra or friction) and the linear dampin assumption in the mathematical model, Fi 4. These differences are reater when the TLD-FR has the

8 external dampin device, but here these differences are related to the nonlinearity of the air dampers. In spite of that, the approximations are still close to the measurements. A sequence of photos was taken when the round motion was applied to the TLD-FR and Fi. 5 shows the instants of the maximum displacement in a window of 20 to 24 seconds (the system without floatin roof is also presented, for the same excitation and time-period). The first row of photoraphs corresponds to the TLD-FR (confiuration A) while the second row corresponds to a traditional TSD. The only difference between them is that one has the floatin roof and the other not. It is easy to appreciate that the TLD-FR has a more predictable (linear) behavior. Fi. 4. Seismic Response of the TLD-FR for confiuration A and B Fi. 5. Photoraph sequence of a TLD-FR and TLD under seismic excitation Dynamic Response Model for TLD-FR-Structure System Consider a n-derees of freedom structure with one TLD-FR (also shown in Fi.7 later). The damper could be modeled directly by Eq. 6 takin into account m-derees of freedom or

9 could be modeled by 1-deree of freedom via a modal reduction. The overnin equations for the coupled structure- TLD-FR system are: M η C η K η m u Lu a s a s a s x T M u Cu K u L F M R u ; F A η B u Lu where u is a n x 1 vector with the displacement of each floor; M, C and K are the n x n mass, dampin and stiffness matrices. R is the n x 1 vector with the influence coefficients for the excitation and L is a 1 x n vector definin the damper s floor position. It is necessary defined some stratey to determine the optimum parameters of the TLD-FR. These parameters are essentially the external dampin C ext and the fundamental period a of the TLD-FR. For their desin the stochastic response is considered here, with the performance described by the mean square statistics x 2 of the chosen structural quantities, such as displacements, drifts, accelerations etc. Since Eq. 11 is linear, the mean square displacement of the independent variables could be calculated by Eq. 12, where Sü corresponds to the power spectral density function of the round motion and H x () corresponds to the transfer function between the round acceleration and the system displacements. 2 2 x (1/ 2 ) H x u S d (12) Illustrative Implementation For the illustrative implementation an elastic structure with a rectanular TLD-FR is considered. The structure corresponds to a 9-story benchmark buildin establish by Ohtori et al. [18]. The structure has 2.1 sec of natural period and 2% of dampin, the mass of the TLD-FR was setup in 2%. The natural frequency of the TLD-FR depends primarily on the tank eometry; althouh is not part of this work, it is possible to show that different eometries miht lead to the same natural frequency. In particular, for rectanular tanks, the natural frequency depends of the tank lenth L and the water heiht H. Then, the admissible domain Q is defined by all possible values of C ext, L and H and the optimization problem is formulated in order to minimize the mean square displacement value of the structure displacement x 2, defined as: s (11) Find H L C ext Q to minimize 2 x ; Such that 1mL 6m Q 0.4 H / L 1 0 N / m / s Cext N / m / s In order to solve the optimization problem, the structure is modeled by the first only mode, whereas the stochastic excitation is chosen as a while noise excitation. The optimum values are found as C ext =3000N/s/m, L=3.1m and H=0.4. This optimization process can be also used to present curves of iso-reduction if some eometric parameter is fixed: in Fi 6. the iso-lines representin the reduction percentae of the structure displacement for values of H equal to 0.4 and 0.6 are presented. The optimum damper is then used to evaluate the seismic response of the 9-story structure

10 under 38 different round motions chosen from the NGA database. Results are presented in Fi 7. The horizontal axis represents the maximum displacement of the 9 th floor without the TLD-FR while the vertical axis contains the maximum displacement for the structure with the damper. The continuous line indicates the border in which the structure with/without the TLD-FR has the same response. Displacement reduction is achieved only for the cases below this line. In eneral, the proposed implementation/optimization scheme leads to a damper confiuration that enerates sinificant reduction for most round motions. It is interestin to note that the cases that lie above the continuous line (no advantaes from addin the damper) correspond to round motions with impulsive characteristics, for which it is enerally acknowleded that mass dampers face challenes in efficiently reducin structural responses. Fi. 6. Curves of iso-response for two different rectanular tank confiurations. Fi. 7. Structure equipped with a TLD-FR and its displacements under different round motions. Conclusions An alternate TLD confiuration, denoted as Tuned Liquid Damper with Floatin Roof (TLD-FR) was studied. The basis of this confiuration lies on the introduction of a floatin roof that covers the free surface. A mathematical model based on FEM was developed and validated with experimental results. The results indicate that under harmonic excitations the TLD-FR does not present a sinificant amplitude dependency as other tuned liquid dampers. Furthermore, the system can be adequately modeled as havin only one deree of freedom, since the floatin roof prevents the wave breakin and enforce a linear behavior even in lare amplitudes conditions. As

11 the TLD-FR has an important inherent dampin, related to the friction or dra in the ap between the walls and the roof, more studies are necessary to determine the exact nature of this dampin. Experimental results also showed ood areement between mathematical model prediction and the seismic response of the TLD-FR. The numerical study considered indicated that the TLD-FR can provide sinificant reduction of seismic responses. Ultimately the TLD-FR should be considered an attractive seismic protection device since: a) has a reliable linear behavior, b) it is equivalently a sinle-dof, c) supplemental dampin or semi-active devices can be easily used to control the roof response, d) has the same advantaes of traditional TLD and e) its mathematical model is relatively easy to implement and understand. Acknowledments This research is supported by the Vicerrectoria de Investiacion of the Pontificia Universidad Catolica de Chile throuh the Instructor Becario Scholarship and by the Department of Civil & Environmental Enineerin & Earth Sc. of the University of Notre Dame. Referencies [1] Fujii K, Tamura Y, Sato T, et al. Wind-induced vibration of tower and practical applications of tuned sloshin damper. Journal of Wind Enineerin and Industrial Aerodynamics 1990; 33(1-2): [2] Tamura Y, Fujii K, Ohtsuki T, et al. Effectiveness of tuned liquid dampers under wind excitation. Enineerin Structures 1995; 17(9): [3] Kareem A. Reduction of wind induced motion utilizin a tuned sloshin damper. Proceedins of the 6th US National Conference on Wind Enineerin Part 2 (of 2) 1990; 36(2): [4] KOH C, MAHATMA S, WANG C. Theoretical And Experimental Studies On Rectanular Liquid Dampers Under Arbitrary Excitations. Earthquake Enineerin & Structural Dynamics 1994; 23(1): [5] Banerji P, Murudi M, Shah A H, et al. Tuned liquid dampers for controllin earthquake response of structures. Earthquake Enineerin & Structural Dynamics 2000; 29(5): [6] Banerji P, Samanta A. Earthquake vibration control of structures usin hybrid mass liquid damper. Enineerin Structures 2011; 33(4): [7] Zahrai S M, Abbasi S, Samali B, et al. Experimental investiation of utilizin TLD with baffles in a scaled down 5-story benchmark buildin. Journal of Fluids and Structures 2012; 28: [8] Soon T T, Darush G F. Passive Enery Dissipation Systems in Structural Enineerin. [9] Tait M J. Modellin and preliminary desin of a structure-tld system. Enineerin Structures 2008; 30(10): [10] Love J S, Tait M J. Nonlinear simulation of a tuned liquid damper with dampin screens usin a modal expansion technique. Journal Of Fluids And Structures 2010; 26(7-8): [11] Maravani M, Hamed M S. Numerical modelin of sloshin motion in a tuned liquid damper outfitted with a submered slat screen. International Journal For Numerical Methods In Fluids 2011; 65(7): [12] Modi V J, Munshi S R. An efficient liquid sloshin damper for vibration control. Journal of Fluids and Structures 1998; 12(8): [13] Modi V J, Akinturk A. An efficient liquid sloshin damper for control of wind-induced instabilities. Journal of Wind Enineerin and Industrial Aerodynamics 2002; 90(12-15): [14] Biswal K C, Bhattacharyya S K, Sinha P K. Dynamic characteristics of liquid filled rectanular tank with baffles. Journal of the Institution of Enineers (India): Civil Enineerin Division 2003; 84(2): [15] Kaneko S, Ishikawa M. Modelin of tuned liquid damper with submered nets. Journal Of Pressure Vessel Technoloy-Transactions Of The ASME 1999; 121(3): [16] Shan C-Y, Zhao J-C. Periods and enery dissipations of a novel TLD rectanular tank with anle-adjustable baffles. Journal of Shanhai Jiaoton University (Science) 2008; 13(2): [17] Ruiz R O, Lopez-Garcia D. A computationally efficient numerical model for the seismic analysis of liquid storae tanks. Vienna Conress on Recent Advances in Earthquake Enineerin and Structural Dynamics 2013; id 381. [18] Ohtori Y, Christenson R E, Spencer B F, et al. Benchmark control problems for seismically excited nonlinear buildins. Journal of Enineerin Mechanics-Asce 2004; 130(4):