Experimental and Analytical Investigations of Rectangular Tuned Liquid Dampers (TLDs)

Transcription

1 Experimental and Analytical Investigations of Rectangular Tuned Liquid Dampers (TLDs) By Hadi Malekghasemi A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Department of Civil Engineering University of Toronto Copyright by Hadi Malekghasemi 2011

2 Experimental and Analytical Investigations of Rectangular Tuned Liquid Dampers (TLDs) Hadi Malekghasemi Master of Applied Science Department of Civil Engineering University of Toronto 2011 Abstract A TLD (tuned liquid damper) is a passive control devise on top of a structure that dissipates the input excitation energy through the liquid boundary layer friction, the free surface contamination, and wave breaking. In order to design an efficient TLD, using an appropriate model to illustrate the liquid behaviour as well as knowing optimum TLD parameters is of crucial importance. In this study the accuracy of the existing models which are able to capture the liquid motion behaviour are investigated and the effective range of important TLD parameters are introduced through real-time hybrid shaking table tests. ii

3 Acknowledgments It is with immense gratitude that I acknowledge the support and help of my supervisor Dr. Oya Mercan, whose encouragement, supervision and guidance from the preliminary to the concluding part of my research enabled me to develop an understanding of the subject. Besides my supervisor, I would like to thank the other member of my thesis committee: Dr. Oh-Sung Kwon for his continued interest and encouragement. I would like to extend my thanks to Dr. Constantin Christopoulos who provided the shaking table for the research experiments and my colleague Ali Ashasi Sorkhabi who helped during the experimental part of the study. Lastly, and most importantly, I wish to thank my parents without whom I would never reach this stage of my life. They bore me, raised me, supported me, taught me, and loved me. To them I dedicate this thesis. iii

4 Table of Contents Chapter 1 Introduction Seismic Protection Systems Conventional Systems Isolation Systems Supplemental Damping Systems Active Systems Semi-Active Systems Passive Mechanism Tuned Liquid Damper History Tuned Liquid Column Dampers Tuned Sloshing Damper Active TLDs TLDs in Practice Scopes of This Study and Outline of the Thesis... 9 Chapter 2 Literature Review Chapter 3 Analytical Models Solving Liquid Equations of Motion Sun s Model iv

5 3.2 Equivalent TMD Models Yu s Model Sloped Bottom Shape Xin s Model Chapter 4 Experimental Results and Analysis Testing Method Test Setup TLD Subjected to Predefined Displacement History TLD-Structure Subjected to Sinusoidal Force Mass Ratio Damping Ratio TLD-Structure Subjected to Ground Motions Chapter 5 Summary and Conclusion Chapter 6 Refrences Appendix A Solution of the Basic Equations for Sun s Model A.1 Non-dimeesionalization of Basic Equations A.2 Discretization of Basic Equations A.3 Runge-Kutta-Gill Method Appendix B Matlab Codes and Simulink B.1 Simulink for Xin s Model B.1.1 The Embedded function B.2 Matlab Code for Sun s Model v

6 B.3 Matlab Code for Yu s Model Appendix C List of Symbols and Acronyms vi

7 List of Tables Chapter Seismic Protection Systems Types of Passive Dampers... 4 Chapter Parameters for Experiments Introduced in Chapter Parameters for Experiments Introduced in Chapter Parameters for Experiments Introduced in Chapter vii

8 List of Figures Chapter Dimensions of the Rectangular TLD Schematic of SDOF System with a TLD Attached to It Schematic of the a) TLD and b) Equivalent NSD Model Displacement Time History to Calculate A DOF System a) Structure with TLD b) Structure with NSD Model Schematic for Determining the NSD Parameters Equivalent Flat-Bottom Tank Chapter Schematic of the Hybrid Testing Method Experimental Setup Hysteresis Loops for Different β Values a Destructive Interface of Sloshing and Inertia Forces at β= b Constructive Interface of Sloshing and Inertia Forces at β= Structural Displacement and Acceleration with and Without TLD Structural Displacement and Acceleration Reduction Comparison Between Experimental Results and Analytical Predictions for F=3N Comparison Between Experimental Results and Analytical Predictions for F=5N viii

9 4.9 Comparison Between Experimental Results and Analytical Predictions for F=8N The Effect of Mass Ratio on TLD-Structure Behaviour Acceleration and Displacement Reduction for Different Mass Ratios a Displacement Increase Due to Undesirable TLD Forces for 5% Mass Ratio b Displacement Time History for 3% Mass Ratio The Effect of Damping Ratio on TLD-Structure Behaviour Acceleration and Displacement Reduction for Different Damping Ratios Structural Response with and Without TLD under El Centro Earthquake Structural Response with and Without TLD under Kobe Earthquake Structural Response with and Without TLD under Northridge Earthquake Hybrid Test Results and Sun s Model Predictions under El Centro Earthquake Hybrid Test Results and Sun s Model Predictions under Kobe Earthquake Hybrid Test Results and Sun s Model Predictions under Northridge Earthquake Hybrid Test Results and Yu s Model Predictions under El Centro Earthquake TLD Hybrid Test Results and Yu s Model Predictions under Kobe Earthquake Hybrid Test Results and Yu s Model Predictions under Northridge Earthquake ix

10 4.24 Hybrid Test Results and Xin s Model Predictions under El Centro Earthquake Hybrid Test Results and Xin s Model Predictions under Kobe Earthquak Hybrid Test Results and Xin s Model Predictions under Northridge Earthquake Appendix A A.1 Schematic of Discretized Tank with Respect to x Appendix B B.1 Simulink for Xin s Model x

11 1 Chapter 1 1. Introduction Increasing demand in constructing flexible high rise buildings that have relatively low damping properties has attracted attention to find efficient and economical ways to reduce the structural motion under dynamic loads (e.g. due to wind or earthquake). Various systems are proposed to increase structural resistance against lateral loads. Following is a summary of common seismic protection systems. 1.1 Seismic Protection Systems Three categories of seismic protection systems have been implemented, as shown in Table 1.1 (Christopoulos and Filiatrault 2006). Table 1.1: Seismic Protection Systems Conventional Systems Isolation Systems Supplemental Damping Systems Passive Damper Active/Semi-Active Dampers Flexural Plastic Hinges Shear Plastic Hinges Yielding Braces Elastomeric Metallic Braces Lead Rubber Friction Tuned-Mass High-Damping Rubber Viscoelastic Tuned-Liquid Metallic Viscous Variable Damping Lead-Extrusion Tuned-Mass Variable Stiffness Friction Pendulum Tuned-Liquid Piezoelectric Self-centering Rheological

12 Conventional Systems These systems are based on traditional concepts and use stable inelastic hysteresis to dissipate energy. This mechanism can be reached by plastic hinging of columns, beams or walls, during axial behaviour of brace elements by yielding in tension or buckling in compression or through shear hinging of steel members Isolation Systems Isolation systems are usually employed between the foundation and base elements of the buildings and between the deck and the piers of bridges. These systems are designed to have less amount of lateral stiffness relative to the main structure in order to absorb more of the earthquake energy. A supplemental damping system could be attached to the isolation system to reduce the displacement of the isolated structure as a whole Supplemental Damping Systems Supplemental damping system can be categorized in three groups as passive, active and semi-active systems. These dampers are activated by the movement of structure and decrease the structural displacements by dissipating energy via different mechanisms Active Systems. Active systems monitor the structural behaviour, and after processing the information, in a short time, generate a set of forces to modify the current state of the structure. Generally, an active control system is made of three components: a monitoring system that is

13 3 able to perceive the state of the structure and record the data using an electronic data acquisition system; a control system that decides the reaction forces to be applied to the structure based on the output data from monitoring system and; an actuating system that applies the physical forces to the structure. To accomplish all these, an active control system needs continuous external power source. The loss of power that might be experienced during a catastrophic event may render these systems ineffective Semi-Active Systems. Semi-active systems are similar to active systems except that compared to active ones they need less amount of external power. Instead of exerting additional forces to the structural systems, semi-active systems control the vibrations by modifying structural properties (for example damping modification by controlling the geometry of orifices in a fluid damper). The need for external power source has also limited the application of semi-active systems Passive Mechanism. Passive systems dissipate part of the structural seismic input energy without any need for external power source. Their properties are constant during the seismic motion of the structure and cannot be modified. Passive control devices have been shown to work efficiently; they are robust and cost-effective. As such, they are widely used in civil engineering structures. The main categories of the passive energy dissipation systems can be seen in Table 1.2 (Christopoulos and Filiatrault 2006).

14 4 Table 1.2: Types of Passive Dampers Displacement-Activated Velocity Activated Motion-Activated Metallic Dampers Friction Dampers Self-Centering Dampers Viscoelastic Dampers Viscous Dampers Viscous Dampers Tuned Mass Damper Tuned Liquid Damper Displacement-activated devices absorb energy through the relative displacement between the points they connect to the structure. Their behaviour is usually independent of the frequency of the motion and is in phase with the maximum internal forces generated at the end of each vibration cycle corresponding to the peak deformations of the structure. Metallic dampers, friction dampers, self-centering dampers, and viscoelastic dampers are the main devices in this group. Velocity-activated devices absorb energy through the relative velocity between their connection points. The behaviour of these dampers is usually dependent on the frequency of the motion and out-of-phase with the maximum internal forces generated at the end of each vibration cycle corresponding to the peak deformations of the structure. This causes a lower level of design forces for structural members and foundation. Viscous and viscoelastic dampers are the typical examples in this category.

15 5 Motion-activated dampers are secondary devices that absorb structural energy through their motion. They are tuned to resonate with the main structure, but, out-of-phase from it. These dampers absorb the input energy of the structure and dissipate it by introducing extra forces to the structure; therefore, they let less amount of energy to be experienced by the structure. Tuned mass dampers (TMDs) and tuned liquid dampers (TLDs) are the examples in this category. 1.2 Tuned Liquid Damper History Since 1950s liquid dampers have been used to stabilize marine vessels or to control wobbling motion of satellites. In the late 1970s TLD has started to be used in civil engineering to reduce structural motion; Vandiver and Mitome (1979) used TLD to reduce the wind vibration of a platform. Also, Mei (1978) and Yamamoto et al. (1982) looked into structure-wave interactions using numerical methods. In the early 1980s important parameters such as liquid height, mass, frequency, and damping for a TLD attached to offshore platforms were studied by Lee and Reddy (1982). Bauer (1984) introduced a rectangular tank full of two immiscible liquids to a building structure. Kareem and Sun (1987), Sato (1987), Toshiyuki and Tanaka, and Modi and Welt (1987) were among the first researchers who suggested using TLD in civil structures.

16 6 Tuned liquid dampers (TLDs) can be implemented as an active or passive device and are divided into two main categories: tuned sloshing dampers (TSD) and tuned liquid column dampers (TLCDs) Tuned Liquid Column Dampers Tuned liquid column dampers (TLCD) combine the effect of liquid motion in a tube, which results in a restoring force using the gravity effect of the liquid, and the damping effect caused by loss of hydraulic pressure (Sakai et al. 1989). Some advantages of TLCD are: (i) it can have any arbitrary shape which helps it to be fitted in an existing structure; (ii) its behaviour is quite well understood; (iii) the TLCD damping can be controlled by adjusting the orifice opening; (iv) the TLCD frequency can be modified by adjusting the liquid column in the tube. A Double Tuned Liquid Column Damper (DTLCD) is made of two TLCDs in two directions of motion (Kim et al. 2006).Thereby DTLCD acts in more than one direction eliminating the limitation of regular unidirectional TLDC Tuned Sloshing Damper A tuned sloshing damper (TSD) dissipates energy through the liquid boundary layer friction, the free surface contamination, and wave breaking (due to the horizontal component of the liquid velocity related to the wave motion, wave crests descend as the amplitude increases; at this point simple linear models are not able to describe the liquid behaviour). A TSD

17 7 can act as a shallow or deep water damper. It is considered that waves in the range of ½>h/L>1/20 to 1/25 are shallow water waves, where h is water depth and L is wave length (Sun et al. 1992). Recent studies (Banerji et al. 2000; Seto 1996) show that a ratio equal to or less than 0.15 introduces more amount of damping corresponding to more energy dissipation. Under high amplitude excitations, shallow water TSDs dissipate a large amount of energy due to its nonlinear behaviour corresponding to wave breaking (Sun et al. 1992). On another hand, a linear behaviour can be observed for the deep water case even under high excitations (Kim et al. 2006). The liquid frequency plays an important role in the TLD behaviour. Earlier experimental studies (Sun et al. 1992) have shown that the optimum value of the liquid frequency is a value near the excitation frequency where the liquid is in resonance with the tank motion. Therefore, tuning the TLD frequency to the natural frequency of the structure will provide significant amount of energy dissipation. Mass ratio (the ratio of the mass of water to that of the whole structural levels) is another significant parameter that affects the behavior of TLD-structure system. It is shown that with a relatively small mass ratio (e.g. 4%), without contributing significantly to the overall inertia of the system, effective structural response reduction can be obtained (Banerji et al. 2000). In comparison with other passive dampers TLD has some advantages including: (i) Easy and cost-effective installation; (ii) Ease of tuning by

18 8 changing the liquid level or the tank dimensions; (iii) Ability to act as a bidirectional damper; (iv) Effective even under small-amplitude vibrations (Sun et al. 1992); (v) Can be used as the building water storage for fire emergencies etc. On the other hand, TLD has some issues such as: (i) Complex behavior due to the highly non-linear sloshing motion of the liquid (ii) Damping introduced by the liquid itself may not be enough for some applications. To remedy this, screens (Tait 2008; Tait et al. 2007; Kaneko and Ishikawa 1999), effective tank shapes ( Deng and Tait 2009; Xin et al. 2009; Ueda et al. 1992), and triangular sticks at the bottom of the tank (You et al. 2007) have been introduced to increase the damping. (iii) Inefficiency during pulse-type ground motions (Xin et. Al 2009; Banerji et al. 2000), when the water motion does not get a chance to dissipate enough energy; (iv) The phenomenon of beating (Ikeda and Ibrahim 2005) where a fraction of the energy absorbed by TLD returns back to the structure after the excitation stops. A sloped bottom shape using density variable liquid has been proposed to help solve the last two problems (Xin et al. 2009) Active TLDs TLDs have also been investigated as active/semiactive devices by employing magnetic fluid (Abe et al. 1998; Wakahara et al. 1992), or through use of propellers (Chen and Ko 2003).

19 TLDs in Practice TLD has been employed in several civil engineering structures. The Nagasaki Airport Tower (NAT) was the first TLD installation on an actual ground structure in 1987 (Tamura et l 1995). In the other case, which is quite similar to that of the NAT, a TLD was installed in June 1987 on Yokohama Marine Tower (YMT) where the TLD is made of 39 cylindrical multilayered vessels of acryl, with a height of 0.50 m and a diameter of 0.49 m (Tamura et l 1995). Another application of the TLD to a high-rise hotel was the Shin Yokohama Prince (SYP) Hotel in Yokohama, where the design parameters which affect the TLD behaviour were investigated (Tamura et l 1995; Wakahara et al. 1992). Tuned Liquid Dampers have also been implemented on bridges such as: Ikuchi Bridge and Sakitama Bridges in Japan (Kaneko and Ishikawa 1999). 1.3 Scope of This Study and Outline of the Thesis This study focuses on sloshing type of tuned liquid dampers. As commonly done in the literature the abbreviation TLD is employed here for this type of dampers and water is considered as the liquid inside the TLD. To enable efficient use of TLDs in suppressing the structural vibrations several models with different levels of complexity have been proposed in the literature. On the other hand, in order to design an effective TLD, its influential parameters such as frequency and mass must be tuned in a way to significantly reduce the structural response. The main aim of this

20 10 study is to check the accuracy of selected models under different conditions (i.e., different levels of excitation frequency, amplitude etc.) and investigate the effect of selected TLD parameters that affect their response using real-time hybrid pseudo-dynamic testing method. Chapters 1 and 2 provide background information and a through literature review, respectively. Various existing analytical models are considered in Chapter 3 and among them three models are selected and explained in detail. The procedure of implementation of the recommended models is also presented in this chapter. In Chapter 4 a series of real-time hybrid pseudo-dynamic tests are carried out; and based on the test results, the accuracy of each selected model is investigated. Also the effect of important TLD parameters on the response reduction efficiency are explored. Chapter 5 provides the summary and conclusions of this study. Appendix A illustrates the procedure of solving the basic equations of Sun s Model. Appendix B shows the MATLAB codes and Simulink model provided to solve the suggested models in Chapter 3.

21 11 Chapter 2 2. Literature Review Since the early 1980s TLD has been investigated by many researchers. Lee et al. (1982) studied effective TLD parameters including liquid height, mass, frequency, and damping for a TLD attached to offshore platforms. Bauer (1984) was among the first researchers who applied TLDs to ground civil engineering structures by introducing a rectangular tank full of two immiscible liquids to decrease structural vibration. Wakahara et al. (1992) and Tamamura et al. (1995) showed the effectiveness of TLDs installed in real structures such as Nagasaki airport tower, Yokohama Marine tower, and Shin Yokohama Prince (SYP) hotel to reduce the structural vibration. Shimizu and Hayama (1986) presented a numerical model to solve for Navier-Stokes and continuity equation based on shallow water wave theory. They descritized the main equations and solved them numerically. Sun et al. (1992) suggested a nonlinear model that utilizes the shallow water wave theory and solves Navier-Stokes and continuity equations together. Furthermore, they introduced two empirical coefficients to account for the effect of wave breaking which is a significant deficiency in many other models. Modi and Seto (1997) also proposed a numerical

22 12 study considering non-linear behaviour of the TLD. It includes the effects of wave dispersion as well as boundary-layers at the walls, floating particle interactions at the free surface, and wave-breaking. However, the analysis does not account for the impact dynamics of the wave striking the tank wall. Furthermore, at lower liquid heights, corresponding to wave breaking occurrence, the numerical analysis is not very accurate and a large discrepancy exists between numerical and experimental results. Sun et al. (1995) calibrated equivalent mass, stiffness, and damping of TLD using a tuned mass damper (TMD) analogy from experimental data of rectangular, circular, and annular tanks subjected to harmonic base excitation. Yu (1999) introduced a model based on an equivalent tuned mass damper with non-linear stiffness and damping calculated from an energy matching procedure. It is shown that the model is able to capture the TLD behaviour under large amplitude excitations and during wave breaking. Gardarsson et al. (2001) investigated the performance of a sloped-bottom TLD with an angle of 30 to the tank base. It is shown that despite the hardening spring behaviour of a rectangular TLD, the sloped-bottom one behaves as a softening spring. Also, it is observed that more liquid mass participates in sloshing force in the slopped-bottom case leading to more energy dissipation.

23 13 Reed et al. (1998) investigated the TLD behaviour under large amplitude excitations through experiments and compared the results with a numerical model based on non-linear shallow water wave equations. It was observed that the TLD frequency response increases as the amplitude of excitation increases and TLD behaves as a hardening spring. Also, it was captured that to achieve the most robust system, TLD frequency should be tuned to a value less than structural response frequency; so, the actual non-linear TLD frequency matches the structural response. Olson and Reed (2001) investigated the sloped-bottom TLDs using nonlinear stiffness and damping model developed by Yu (1999). The softening spring behaviour of the sloped-bottom system was confirmed. Also, it is concluded that the sloped-bottom tank should be tuned slightly higher than the fundamental frequency of the structure to introduce the most effective damping. Xin et al. (2009) proposed a density variable TLD with sloping bottom and experimentally investigated it on a ¼ -scale, three-story structure. The density-variable control system had been observed to be more effective and more robust than a corresponding flat bottom, plane water TLD in decreasing story drift and floor acceleration of the structure. Yamamoto and Kawahara (1999) used arbitrary Lagrangian-Eulerian (ALE) form of Navier-Stokes equations to predict the liquid motion.

24 14 Improved-balancing-tensor-diffusivity and fractional steps methods were employed to discritize and solve the Navier-Stokes equations in space and Newmark s β method was used in time domain to predict the TLDstructural interaction response. However the model did not verified with experiments. Siddique and Hamed (2005) presented a new numerical model to solve Navier-Stokes and continuity equations. They mapped irregular, timedependent, unknown physical domain onto a rectangular computational domain where the mapping function is unknown and is determined during the solution. It is indicated that the algorithm can accurately predict the sloshing motion of the liquid undergoing large interfacial deformations. However, it is unable to predict the deformations in the case of surface discontinuity such as existing of screens or when wave breaking occurs. Kareem et al. (2009) presented a model for TLD using sloshing-slamming (S 2 ) analogy, which consist of a combination of the dynamic features of liquid sloshing and slamming impact and is able to capture the behaviour for both low and high amplitudes of excitations. However, experimental results do not have a good agreement with the proposed model. Li et al. (2002) solved continuity and momentum fluid equations for shallow liquid using finite element method. They simplified the threedimensional problem into a one-dimensional problem that simplifies the

25 15 computation procedure. However the model was not verified with experiments. Frandsen (2005) developed a fully nonlinear 2-D σ-transformed finite difference model based on inviscid flow equations in rectangular tanks. Results were presented for small to steep non-breaking waves at a range of tank depth to length ratios representing deep to near shallow water cases. However, the model is not able to capture damping effects of liquid and the shallow water wave behaviour. Warnitchai and Pinkaew (1998) proposed a mathematical model of TLDs that includes the non-linear effects of flow-dampening devices. Experimental investigations with a wire mesh screen device were carried out. With the introduction of the flow dampening device an increase in sloshing damping and the non-linear characteristic of the damping was observed; and the slight reduction in sloshing frequency agreed well with the model predictions. Kaneko and Ishikawa (1999) proposed an analytical model that is able to account for the effect of submerged nets on the TLD behaviour based on shallow water wave theory. It is shown that the optimal damping factor can be achieved by nets and the structural vibration will reduce more in presence of nets. Tait (2008) developed an equivalent linear mechanical model that accounts for the energy dissipated by the damping screens for both

26 16 sinusoidal and random excitation. The model is validated using experimental tests and a preliminary design procedure is suggested for a TLD equipped with damping screens. Cassolato et al. (2010) proposed inclined slat screens to increase the TLD damping ratio. They calculated the pressure-loss coefficient for inclined screens and estimated the energy dissipated by the screens. A model to predict the fluid steady-state response was also developed. It was observed that a TLD equipped with adjustable inclined screens could introduce a constant damping ratio over a range of excitation amplitudes. It was also captured that increasing the screen angle decreases the TLD damping ratio. Li and Wang (2004) suggested multiple TLDs to reduce multi-modal responses of tall buildings and high-rise structures to earthquake ground motion excitations. TLDs were tuned to the first several natural periods of structure. It was shown both theoretically and experimentally that having the same mass as a single TLD, multi-tlds are more effective to reduce structural motion up to 40%. Koh et al. (1995) also investigated multiple liquid dampers tuned to several modal frequencies of the structure. It was shown that multiple TLDs provide a better vibration control than TLDs tuned to a particular modal frequency. Furthermore, they captured the TLD dependency on the nature of excitation and the significant effect of the TLD s position on the vibration response.

27 17 Tait et al. (2005, 2007) conducted a study on 2D TLDs behaviour. They subjected the TLD to both 1D and 2D horizontal excitations. The sloshing response of the water in the tank was characterized by the free surface motion, the resulting base shear force, and evaluation of the energy dissipated by the sloshing water. Results showed a decoupled behaviour for the 2D TLD which allows rectangular tanks to be used as 2D TLDs and simultaneously reduce the dynamic response of a structure in two perpendicular modes of vibration. Tait and Deng (2009) introduced models of triangular-bottom, slopedbottom, parabolic-bottom, and flat-bottom tanks using the linear long wave theory. The energy dissipated by damping screens and the equivalent mechanical properties including effective mass, natural frequency, and damping ratio of the TLDs were compared for different tank geometries. It was shown that the normalized effective mass ratio (the liquid mass that participates in sloshing) for a parabolic-bottom tank and a sloped-bottom tank with a sloping angle of 20 deg are larger than the normalized effective mass ratio of triangular-bottom and flat-bottom tanks. Idir et al. (2009) derived the natural frequency of the water sloshing wave for various tank bottom shapes from the equivalent flat bottom tank using the linear wave theory. The frequency formula was shown to be accurate at weak excitations, particularly for V and arc bottom-shaped tanks. Banerji et al. (2000) studied the effectiveness of the important TLD parameters based on the model introduced by Sun et al. (1992). The

28 18 optimum value of depth, mass and frequency ratios standing for the depth of water to the length of tank, the mass of water to the mass of the structure and the frequency of tank to the structural frequency were found via experiments. Subsequently, a practical TLD design procedure is suggested to control the seismic response of structures. Chang and Gu (1999) conducted a theoretical and experimental study to achieve optimal TLD properties installed on the top of a tall building and subjected to vortex excitations (that is a special case of wind excitation). A series of wind tunnel experiments corresponds to different TLD geometries were performed. They proposed a TLD frequency ranges between 0.9 and 1.0 of that of the building model and a mass ratio of 2.3%. Samanta and Banerjy (2010) theoritically modified TLD configuration where the TLD rests on an elevated platform that is connected to the top of the building through a rigid rod with a flexible rotational spring at its bottom. Since for particular values of rotational spring flexibility the rotational acceleration of the rod is in phase with the top structural acceleration, the TLD was subjected to larger amplitude acceleration than the traditional fixed bottom one and its efficiency was increased. Modi and Akinturk (2002) focused on the installation of two-dimensional wedge-shaped obstacles to amplify TLD energy dissipation efficiency. The optimum obstacles geometry was determined through a parametric free vibration study. It was shown that the damping factor can be increased by approximately 19.8% in the optimum condition.

29 19 Abe et al. (1998) presented an active TLD which consists of magnetic fluid activated by electromagnets. A rule-based control law of active dynamic vibration absorbers was employed due to nonlinear behaviour of sloshing. It was observed that active TLD is more effective to reduce structural vibration and at the same time less sensitive to the error in tuning. Ikeda (2010) investigated the influence of the two rectangular tanks configuration on a two-storey structure response. In one case, one tank was installed at the top and another at the second story, the second case was putting one tank at the top and the last one was installing both tanks at the top. He concluded that multiple tanks are less effective in reducing structural response. Ikeda (2003) investigated the TLD nonlinear behaviour attached to a linear structure subjected to a vertical harmonic excitation. He observed that by selecting an optimal liquid level, a liquid tank can be used as a damper to suppress vertical sinusoidal excitations. Ikeda and Ibrahim (2005) studied the nonlinear random interaction of an elastic structure with liquid sloshing dynamics in a cylindrical tank subjected to a vertical narrow-band random excitation. Four regimes of liquid surface motion were observed and uni-modal sloshing modeling found to reasonably investigate the structure-tld interaction. Biswall et al. (2003, 2004) investigated the effects of annular baffles on cylindrical TLD

30 20 behaviour. It was captured that the sloshing frequencies of liquid in the flexible-tank baffle system are lower than those of the rigid system. Tait et al. (2005) performed a range of experimental studies to investigate the TLD behaviour in terms of the free surface motion, the resulting base shear forces, and the energy dissipated by a TLD with slat screens based on a linear (Fediw et al. 1995) and non-linear (kaneko and Ishikawa 1999) model. It is observed that the non-linear model is able to accurately describe the response while the linear model is an appropriate tool for an initial estimate of the energy dissipating characteristics of a TLD. Furthermore, larger liquid depth to tank length values and using multiple screens had been investigated based on experimental results. Also, a method is presented to determine the loss coefficient of screens. Lieping et al. (2008) suggested using Distributed tuned liquid dampers (DTLDs) to fill the empty space inside the pipes or boxes of cast-in-situ hollow reinforced concrete (RC) floor slabs to increase structural damping ratio. Lee et al. (2007) performed a real-time hybrid pseudodynamic (PSD) test to evaluate the TLD performance. In this method the structure was modeled in a computer and the tank was tested physically. They compared results with the conventional shaking table test and indicated that the performance of the TLD can be accurately evaluated using the

31 21 real-time hybrid shaking table testing method RHSTTM without the physical structural model.

32 22 Chapter 3 3. Analytical Models In this chapter selected models that are considered in this study are introduced. Since the liquid behaviour is highly nonlinear, considering nonlinearity is of crucial importance. Also, using shallow water in tanks leads to the wave breaking occurrence under various excitation amplitude and frequency combinations where the liquid surface is no longer continuous. Therefore, the models presented in this chapter were selected with the nonlinearity and wave breaking in mind for rectangular tanks filled with water. Additionally, there are some researchers who considered slopped bottom shape tanks (Olson and Reed 2001; Xin et al. 2009; Gardarsson et al. 2001) and introduced models that are able to account for slopped bottom shapes, one of these models is also included in this study. There are two common approaches that have been used to model the liquid-tank behaviour. In the first one the dynamic equations of motion are solved, whereas in the second approach the properties of the liquid damper are presented by equivalent mass, stiffness and damping ratio essentially modeling the TLD as an equivalent TMD (Tuned Mass Damper).

33 Solving Liquid Equations of Motion Several researchers have investigated the liquid behaviour based on solving the liquid equations of motion. The assumptions they made along with numerical methods they used to solve the liquid equations of motion have a significant effect on their prediction. Ohyama and Fujji (1989) were among the first who introduced a numerical model for the TLD. Using potential flow theory their model was able to take care of nonlinearity; however, computational time was the main problem with this model (Sun et al. 1992). Kaneko and Ishikawa (1999) used an integrating scheme to solve continuity and Navier-Stokes equations without any consideration for wave breaking. Zang et al. (2000) used a linearized form of Navier-Stokes equations. Fediew et al. (1995) assumed that the derivative and higher orders of the velocity and wave height can be neglected due to small values of velocity and wave height; however, this assumption works for weak excitations or when the frequency of excitation is away from that of the TLD (Lepelletier and Raichlen 1988). Ramaswamy et al. (1986) solved nonlinear Navier-Stokes equations using Lagrangian description of fluid motion and finite element method. The model has some physical problems involving sloshing dynamics of inviscid and viscous fluids. Although the model is based on nonlinear equations, but, considering only small amplitude excitations, they assumed a linear behaviour of the liquid sloshing. Yamamoto and Kawahara (1999) used arbitrary Lagrangian- Eulerian (ALE) form of Navier-Stokes equations to predict the liquid

34 24 motion. The model tends to be unstable in the case of large amplitude sloshing. To solve the instability problem a smoothing factor is considered and the accuracy is highly dependent to the value of this factor that varies from zero to one with no clear outline for the selection. Siddique and Hamed (2005) presented a new numerical model to solve Navier-Stokes and continuity equations. Although it is indicated that the algorithm can accurately predict the sloshing motion of the liquid under large excitations, the model is unable to predict the deformations in the case of surface discontinuity where screen exist or wave breaking occurs. Frandsen (2005) developed a fully nonlinear 2-D σ-transformed finite difference model based on inviscid flow equations in rectangular tanks. The model was not able to capture damping effects of liquid and shallow water wave behaviour Sun s Model Sun et al. (1992) introduced a model to solve nonlinear Navier-Stokes and continuity equations. A combination of boundary layer theory and shallow water wave theory is employed and resulting equations were solved using numerical methods. An important aspect of this model is that it considers wave breaking under large excitations by means of two emprical coefficients. In what follows, a summary of this model will be provided. The rigid rectangular tank shown in figure 3.1 with the length 2aa, width bb and the undisturbed water level h is subjected to a lateral displacement xx ss. The liquid motion is assumed to develop only in the xx zz plane. It is also

35 25 assumed that the liquid is incompressible, irrotational fluid, and the pressure is constant on the liquid free surface. z η x h a a Figure 3.1: Dimensions of the Rectangular TLD The continuity and two-dimensional Navier-Stokes equations that are employed to describe liquid sloshing are defined as + = 0 (1) + uu + ww = 1 ρρ pp + vv uu 2 xx uu zz 2 xx ss (2) + uu + ww = 1 pp ρρ + vv ww 2 xx ww gg (3) zz2 where uu(xx, zz, tt) and ww(xx, zz, tt) are the liquid velocities relative to the tank in the xx and zz direction, respectively, gg is the gravity acceleration, pp is the pressure, ρρ denotes the density and vv represents the kinematic viscosity of the liquid. Because of the relatively small viscosity of the liquid, the friction is only appreciable in the boundary layers near the solid boundaries of the tank. The liquid outside the boundary layers is

36 26 considered as potential flow and the velocity potential can be expressed as (Sun 1991) ΦΦ(xx, zz, tt) = gggg 2ωω cosh kk(h + zz) cos(kkkk ωωωω) (4 1) cosh(kkh) where kk is wave number and H is defined as (Sun 1991) HH = 2ηη sin(kkkk ωωωω) (4 2) Based on the shallow water wave theory, potential ΦΦ is assumed as (Shimisu and Hayama 1986) ΦΦ(xx, zz, tt) = ΦΦ (xx, tt). cosh kk(h + zz) (4 3) The boundary conditions are described as uu = 0 on the end walls (xx = ±aa); ww = 0 on the bottom (zz = h); ww = + uu on the free surface (zz = h); and pp = pp 0 = constant on the free surface (zz = h) where ηη(xx, tt) is the free surface elevation. ΦΦ (xx, tt) in equation (4-3) can be determined by the boundary conditions. Then, using equation (4-3), ww and its differentials are expressed in terms of uu. Since the liquid depth is shallow, the governing equations are integrated with respect to z from bottom to free surface to obtain:

37 27 (φφφφ) + hσ = 0 (5) + 1 TT HH 2 uu + CC ffff 2 gg + gghσσσσ 2 ηη xx 2 = CC dddd λλλλ xx ss (6) Where σσ = tanh(kkh) /kkh,φφ = tanh kk(h + ηη) /tanh(kkh),tt HH = tanh(kk(h + ηη)), uu and ηη are the independent variables of these equations. λλ in equation (6) is a damping coefficient accounting for the effects of bottom, side wall and free surface, and is determined as (Sun et al. 1989): λλ = 1 8 (ηη + h) 3ππ ωω llvv 1 + 2h + SS (7) bb In which S stands for a surface contaminating factor and a value of one corresponding to fully contaminated surface is used in this model (Sun et. Al 1992). ωω ll is the fundamental linear sloshing frequency of the liquid and can be found as (Sun 1991) ωω ll = ππππ 2aa tanh ππh (8) 2aa CC ffff and CC dddd in equation (6) are employed to account for wave breaking when (ηη > h). These coefficients are initially equal to a unit value, and when wave breaking occurs, CC ffff takes a constant value equal to 1.05 as suggested by Sun et al. (1992). CC dddd depends on xx ss mmmmmm that is the maximum displacement experienced by the structure at the location of the TLD, when there is no TLD attached; and it can be found as CC dddd = 0.57 h2 ww ll 2aa xx ss mmmmmm (9)

38 28 Equations (5) and (6) are discretized in space by finite difference method and solved simultaneously using Runge-Kutta-Gill method to find u and η. Knowing η the force introduced at the walls of the TLD can be described as [29]: FF = ρρρρρρ 2 [(ηη nn + h) 2 (ηη 0 + h) 2 ] (10) where ηη nn and ηη 0 are the free surface elevations at the right and left tank walls, respectively. To consider TLD-Structure interaction, a single-degree-of-freedom (SDOF) structure with TLD is considered as shown in figure 3.2. The equation of motion of the TLD-structure system subjected to a ground acceleration (aa gg ) is or mm ss xx ss + cc ss xx ss + kk ss xx ss = aa gg mm ss + FF (11) xx ss + 2ww ss ξξ ss xx ss + ww ss 2 xx ss = aa gg + FF mm ss (12) where mm ss, cc ss, kk ss, ξξ ss and ww ss are structural mass, damping coefficient, stiffness, damping ratio and natural frequency respectively, xx ss represents structural relative displacement to the ground which is meanwhile the displacement experienced by the TLD, aa gg is ground acceleration and F is TLD base shear due to sloshing force on the TLD wall that is given by equation (10).

39 29 Equations (5), (6) and (12) must be solved simultaneously in order to find the response of the SDOF structure equipped with TLD. A step-by-step procedure is employed where knowing the structural acceleration at each F TLD x s m s C s F TLD x s K s /2 K s /2 a g K s m s a g C s Figure 3.2: Schematic of SDOF System with a TLD Attached to It step, equations (5) and (6) are solved using Runge-Kutta-Gill method and F is calculated based on η. Then, using F, the SDOF response is calculated using Runge-Kutta-Gill method from equation (12) and next step acceleration is found to be used in the next step calculations. Appendix A provides information on the discretization technique and application of Runge-Kutta-Gill method.

40 Equivalent TMD Models Another approach to investigate TLD behaviour is replacing the TLD by its equivalent TMD and finding the effective TMD properties such as stiffness, damping ratio, and mass that can properly describe TLD characteristics. These equivalent properties are found through experimental procedures. Sun et al. (1995) found equivalent TMD properties base on nonlinear Navier-Stokes equations and shallow water wave theory. However, theexperimental cases presented in this study are limited. Casciati et al. (2003) proposed a linear model which can interpret frustum-conical TLDs behaviour for small excitations. The model is not able to capture high amplitude excitations and instability problems occur near resonance. Tait (2008) developed an equivalent linear mechanical model that accounts for the energy dissipated by the damping screens for both sinusoidal and random excitation Yu s Model. Yu (1997) and Yu et al. (1999) modeled the TLD as a solid mass damper that can capture nonlinear stiffness and damping of the liquid motion. This mechanical model can capture the behaviour of the TLD in a broad range of excitation amplitudes and can be a good TLD design tool. An equivalent Nonlinear-Stiffness-Damping (NSD) model is proposed through an energy matching procedure when the dissipated energy by the equivalent NSD model is matched by that of the TLD. Figure 3.3 shows the schematic of the characterized SDOF model of the TLD; kk dd, cc dd, and mm dd refer to the

41 31 stiffness, damping coefficient, and mass of the NSD model, respectively. A challenge in this model is the determination of the NSD parameters to describe TLD behavior. As it is shown in figure 3.3, the NSD model used in simulation is based on introducing the interaction force made by liquid sloshing inside the tank. Considering the TLD as an equivalent linear system, this force can be characterized by its amplitude and phase. Energy dissipation per cycle is found by equation (13) and nondimensionalized version is provided as equation (14). EE ww = FF ww dddd TT ss (13) EE ww = EE ww (1/2)mm ww (wwww) 2 (14) Where, dddd shows integration over the shaking table displacement per cycle, FF ww is the force generated by the liquid sloshing motion in the tank, mm ww refers to the mass of the liquid, w is the excitation angular frequency of TLD Excitation a) FF ww kk dd cc dd mm dd Excitation b) FF dd Figure 3.3: Schematic of the a) TLD and b) Equivalent NSD Model

42 32 the shaking table (equation (8)), A is the amplitude of the sinusoidal excitation and the denominator of (14) is the maximum kinetic energy of the water mass treated as a solid mass. Non-dimensional energy dissipation of the NSD model EE dd is determined based on NSD model behaviour when it is subjected to harmonic base excitation with frequency ratio β. The non-dimensionalized amplitude FF dd, and phase φφ that describe the interaction force of the NSD model and are calculated as 1 + 4ξξ 2 dd 1 ββ ξξ 2 dd ββ 6 FF dd = 1 + 4ξξ 2 (15) dd 2 ββ 2 + ββ 4 φφ = tttttt 1 2ξξ dd ββ ξξ 2 dd ββ2 (16) where ββ = ff ee ff dd is the excitation frequency ratio, ff ee is the excitation frequency, ff dd = (1/2π) kk dd mm dd is the natural frequency of the NSD, ξξ dd = cc dd cc cccc is the damping ratio of the NSD model, cc cccc = 2mm dd ww dd is the critical damping coefficient, ww dd = 2ππff dd is the linear fundamental natural angular frequency, and mm dd, kk dd and cc dd are the mass, stiffness and damping coefficients of the NSD model, respectively. The non-dimensional energy dissipation of the NSD model at each excitation frequency is defined as: EE dd = 2ππ FF dd ssssssss (17) EE dd is fitted to EE ww over high-frequency dissipation range of the frequency using least-squares method. In this procedure mm dd = mm ww, and assuming

43 33 initial values for ξξ dd and ff dd, the stiffness and damping coefficients are determined. The results are analyzed through two ratios; the first is frequency shift ratio as defined by: ξξ = ff dd ff ww (18) where ff ww stands for the linear fundamental frequency of the liquid and is defined as: ff ww = 1 2ππ ππππ 2aa tanh ππh (19) 2aa where h is the undisturbed height of the water and a is the half length of the tank. The second ratio is the stiffness hardening ratio κκ that is defined as κκ = kk dd kk ww (20) where kk ww = mm ww (2ππff ww ) 2 The above matching scheme is applied to a set of experimental tests in order to evaluate the equivalent stiffness and damping ratio for the NSD model. The equivalent stiffness and damping ratio are investigated as a function of the wave height, water depth, amplitude of excitation and the tank size. Non-dimensional value of the amplitude was found to be the most suitable parameter to describe the stiffness and damping ratio. This value is described as: ᴧ = AA 2aa (21)

44 34 where AA is the amplitude of excitation and a is the half length of the tank in the direction of motion. To calculate AA, as it is shown in figure 3.4, each time the displacement curve crosses the time axis, the maximum displacement during the previous half cycle x max, i-1 is calculated and the absolute value of that is considered as AA for the i th half cycle in order to find ᴧ. x i-1 i x max, i-1 time x max, i Tank Displacement Figure 3.4: Displacement Time History to Calculate AA After finding the corresponding values of ξξ and κ from equations (18) and (20), they are plotted versus ᴧ and the best-fitted curve is found in order to find the equations for damping ratio and stiffness hardening ratio. Yu (1997) and Yu et al. (1999) obtained the damping ratio as ξξ dd = 0.5 ᴧ 0.35 (22) As stiffness hardening ratio changes considerably before ᴧ = 0.03 (corresponding to weak wave breaking) and then starts to grow up sharply after ᴧ = 0.03 (corresponding to strong wave breaking), Yu (1997) and Yu et al. (1999) obtained the equation for κκ is obtained as

45 35 κκ = ᴧ ᴧ 0.03 weak wave breaking (23) κκ = 2.52 ᴧ 0.25 ᴧ 0.03 strong wave breaking (24) Finally, as it is shown in figure 3.5, a two-degree-of-freedom model is considered to investigate the interaction of TLD-structure system when a TLD is attached to a SDOF structure. In this model AA in equation (21) is found from the structural displacement where the TLD is attached (usually top floor). So, each time the displacement curve crosses zero axis stiffness and damping ratio of the NSD model are updated based on equations (22), (23) and (24) corresponding to the top structural displacement. Figure 3.6 illustrates the schematic for stiffness and damping parameter updating of the NSD model. The equations of motion are presented in matrix form as mm dd 0 xx dd + cc dd 0 mm ss xx ss cc dd cc dd cc ss + cc xx dd + kk dd dd xx ss kk dd kk dd xx dd kk ss + kk dd xx = 0 (25) ss FF ee where mm ss, cc ss, kk ss, xx ss, xx ss and xx ss are the mass, damping, stiffness, relative displacement, velocity and acceleration of the structure, respectively. The same parameters with the subscripts d refer to the NSD model. The parameters mm ss, mm dd, cc ss, and kk ss are assumed to be given in this procedure.

46 36 a) b) Figure 3.5: 2-DOF System: a) Structure with TLD b) Structure with NSD Model Given Constants: mm ss, kk ss, cc ss, mm dd Given Function: FF ee Stop Initial Conditions: ii = 1, xx ss, xx ss Yes Get FF ee,ii ii = ii + 1 No End Equation (25) xx ss Zero-Crossing No Yes Find AA Calculate cc dd and kk dd Equations (22), (23) and (24) Figure 3.6: Schematic for Determining the NSD Parameters

47 Sloped Bottom Models There are studies on the effect of changing the tank bottom shape from rectangular to sloped bottom pattern. Gardarsson et al. (2001) investigated the performance of a sloped-bottom TLD with an angle of 30 to the tank base. It is observed that more liquid mass participates in sloshing force in the slopped-bottom case leading to more energy dissipation. Olson and Reed (2001) investigated the sloped-bottom TLDs using non-linear stiffness and damping model developed by Yu (1999). It is shown that the sloped-bottom tank should be tuned slightly higher than the fundamental frequency of the structure to introduce the most effective damping. Tait and Deng (2009) showed that the normalized effective mass ratio for a sloped-bottom tank with a sloping angle of 20 is larger than the normalized effective mass ratio of flat-bottom tanks Xin s Model Xin (2006) and Xin et al. (2009) proposed a model that is capable to investigate sloping-bottom TLD based on the linearized shallow water wave equation (Gardarsson 1997), using the velocity potential function and wave height equation suggested by Wang (1996) and liquid damping introduced by Sun et al. (1995) and Sun (1991). As it is shown in figure 3.7, an equivalent flat-bottom TLD model is proposed to simulate the sloping-bottom tank. The total contact area between water and the tank for the sloped-bottom case and the equivalent flat-bottom tank is kept equal (the length LL of the equivalent flat-bottom tank is equal to the total length

48 38 of the sloping bottom). The maximum water depth HH remains the same when the equivalent width of the flat-bottom tank BB is decreased in order to keep the total volume VV ww of the sloshing water the same as that of the sloped-bottom tank. BB is defined as (Xin 2006): BB = VV ww HH LL (26) The horizontal control force FF jj (tt) applied to the jj tth floor of the building structure by a sloping-bottom TLD is equal to the resultant of the fluid dynamic pressures on the left and right walls of the flat-bottom TLD tank; and is expressed as (Xin et al. 2009) FF jj (tt) = ρρvv ww XX jj (tt) + xx gg(tt) + yy (tt) 8LL ππhh ππ 3 tttttth HH LL (27) zz zz xx xx HH mmmmmm HH = HH mmmmmm LL LL Figure 3.7: Equivalent Flat-Bottom Tank where ρρ is the water density, XX jj (tt) represents the relative acceleration on the jj tth building floor with respect to the base of the building, xx gg(tt) refers to the ground acceleration at the base of the building, and yy (tt) is the first modal acceleration of water sloshing. The modal response of water sloshing can be determined as yy (tt) + 2ξξ ww yy (tt) + ww 2 yy(tt) = XX jj (tt) xx gg(tt) (28)

49 39 where and ξξ = 2ww vv 1 + 2HH BB + SS LL 4ππHH gghh (29) ww 2 = ππππ LL tanh ππhh LL (30) in which vv refers to the kinematic viscosity of the liquid and S (=1.0 )) is the surface contamination factor ((Sun et. Al 1992)). In order to consider TLD-structure interaction, the structural response to the ground motion and force FF jj (tt) is written in the matrix form as MMXX + CCXX + KKKK = MMMMxx gg(tt) + II ff FF jj (tt) (31) Where M, C, and K are the mass, damping coefficient, and stiffness matrices of the structure; XX, XX, and XX represent the relative acceleration, velocity, and displacement vectors with respect to the base of the building; II is the earthquake influence vector with unity for all elements; and II ff is the TLD influence vector with zero elements except for the element corresponding to the jj tth floor of the building where the TLD is attached that is unity. Knowing the initial condition, equation (28) is solved and the tank acceleration is calculated and used in equation (27) to find the interface force. Having the interface force and using equation (31) structural displacement and acceleration are found for next step calculations.

50 40 In this study, since the experiments were done for a rectangular tank, these equations are solved for the equivalent rectangular tank in order to investigate this model s accuracy. The Simulink used to solve these equations is presented in Appendix B.

51 41 Chapter 4 4. Experimental Results and Analysis 4.1 Testing Method In this study the real-time hybrid Pseudodynamic (PSD) testing method has been employed to investigate the TLD behaviour under a range of structural parameters and load cases. Hybrid PSD testing method combines computer simulation with physical by testing part of the structure physically (experimental substructure) coupled with a numerical model of the remainder of the structure (analytical substructure). When the experimental substructure has load rate dependent vibration characteristics as in the case of TLD, the hybrid PSD test needs to be performed dynamically in real-time. By employing real-time hybrid PSD test in this study, the structure-tld interaction has been investigated by only physically testing the TLD as the experimental substructure and a wide range of TLD-structure system properties were easily investigated by modifying the parameters of the structure as the analytical substructure. As it is shown in figure 4.1, the whole system is divided into the experimental (TLD) and analytical (structure) substructures. TLD is tested physically and the interaction force is measured using a load cell. The response of the structure considering also the measured interaction force from the TLD under the specified external loading is calculated numerically using Simulink and Real-time Workshop. In this study the

52 42 analytical substructure is modeled as a single degree of freedom oscillator. The displacement command generated by the Simulink model is imposed on the shaker. The software/hardware communication and synchronization issues are taken care of by using the WinCon/Simulink interface. Interaction Force TLD Interaction Force TLD Experimental Substructure Shaker TLD Analytical Substructure Displacement Commend Figure 4.1: Schematic of the Hybrid Testing Method 4.2 Test Setup Figure 4.2 shows a picture of the test setup. The shaker table consists of a 1 Hp brushless servo motor driving a 12.7 mm lead screw. The lead screw drives a circulating ball nut which is coupled to the 457x457 mm table. The table itself slides on low friction linear ball bearings on 2 ground hardened shafts and has 76.2 mm stroke. The shaker comes with WinCon software, the real-time control software that runs Simulink models in real-time. The built-in control laws are able to impose harmonic or preset earthquake historical data under displacement control. In this study a velocity feed forward component was added to improve the tracking of the command displacements by the shaker.

53 43 The load cell is a 22.2 N (5 lb) load cell that can carry compression and tension loads. The tank is made of plexi-glass that has dimensions of 464 mm (length), 305 mm (width). A water height of 40 mm which corresponds to Hz of sloshing frequency of the tank (based on equation (8)) was selected there the weight of the TLD was 5.64 kg. Figure 4.2: Experimental Setup As it is shown in figures 4.1 and 4.2, the tank is placed on greased ball bearings to eliminate friction. Special attention was given to keep the tank in the perfectly horizontal position. Only a few degrees out of horizontal position was observed to introduce large amount of error in the measured

54 44 restoring force. Two rollers are also placed at the two sides of the tank in order to keep its movement in one direction. 4.3 TLD Subjected to Predefined Displacement History This section summarizes the results where the TLD was subjected to displacement histories with amplitude of 20 mm and various frequencies to cover a range of β from 0.5 to 1.5. The frequency ratio β, as previously defined, is the ratio of the frequency of loading to the sloshing frequency of the tank. By considering the energy dissipated in each case (see figure 4.3), the effective value for β was obtained. As can be seen in figure 4.3 energy dissipated by the TLD increases until β<1.2, and starts to decrease for values of β>1.2, rendering β=1.2 as the effective frequency in terms of energy dissipation. To shed some light into the TLD energy dissipation behavior, another set of experiments were performed. In these tests, the water inside the TLD was replaced with an equivalent solid mass while the TLD was imposed to the same predefined displacement histories. The measured restoring forces in these tests correspond to the inertia component of the interface force. By subtracting the inertia component from the interface force, the sloshing force was calculated for each frequency ratio. Figure 4.4a shows the inertia and sloshing force components of the interface force for β=1.5. It can be seen that these components have a destructive interface where they almost cancel each other resulting in very little if not nonexistent

55 Figure 4.3: Hysteresis Loops for Different β Values 45

56 46 energy dissipation for this frequency (see figure 4.3). As it is shown in figure 4.4b for β=1.2, on the other hand, the inertia and sloshing force components have constructive interface leading to an efficient energy dissipation as described earlier. Figure 4.4a: Destructive Interface of Sloshing and Inertia Forces at ββ = 1.5 Figure 4.4b: Constructive Interface of Sloshing and Inertia Forces at ββ = 1.2

57 TLD-Structure Subjected to Sinusoidal Force In this section, using real-time hybrid PSD method, the TLD-structure system was investigated under a series of sinusoidal force. To be able to observe weak and strong wave breaking behavior, three different force amplitudes (i.e., 3 N, 5 N, and 8 N) were used while the forcing frequency adjusted to be the same as the structural frequency (see table 4.1). In addition to the forcing function amplitude, a range of structure to TLD sloshing frequency ratio (α) from 0.5 to 1.5 was considered in the hybrid simulations. The TLD properties were kept unchanged; to obtain the aforementioned range of α, the structural stiffness in the analytical substructure was adjusted. The mass of the structure and the structural damping ratio remained unchanged (see table 4.1). Table 4.1: Parameters for experiments introduced in Chapter 4.4 Structural Mass (m) Structural Stiffness (k) Structural Damping Coefficient (c) Structural Damping Ratio (ξ) Structural Frequency (f) Sinusoidal Force Amplitude Sinusoidal Force Frequency 564 kg (1% Mass Ratio) Adjusted to Change f Adjusted to Keep ξ Constant 0.20% Adjusted to Change α 3N, 5N, and 8 N Equal to f Structural displacements in the form of deformations relate to the damage of the structural members during seismic events. On the other hand, nonstructural components (ceiling- wall attachments, furniture etc) may experience considerable inertial forces due to floor accelerations. Figure

58 present the displacement/ acceleration versus frequency response graphs for 3 different force levels considering the structure with and without the TLD. Figure 4.5: Structural Displacement and Acceleration with and Without TLD Figure 4.6 represents the data from figure 4.5 in terms of percent reductions of displacements and accelerations. From these figures it can be concluded that, for the forcing levels considered corresponding to weak and strong wave breaking, TLD is remarkably efficient in reducing the displacement and acceleration response around the frequency ratio α near

59 49 Figure 4.6: Structural Displacement and Acceleration Reduction one, where the tank is in resonance with the structure. The results from the experiments with the sinusoidal forcing function were also used to investigate the accuracy of the models that were selected and presented in chapter 3. In each case, the error between the experiment and model prediction was quantified by: EEEEEE = (FF mm FF pp ) 2 αα ii (32) where FF mm stands for measured values and FF pp represents predicted values from the models.

60 50 Figure 4.7: Comparison Between Experimental Results and Analytical Predictions for FF = 3NN Figure 4.7 presents the comparison between experiment and numerical model predictions for the force level of FF = 3NN Since no wave breaking was observed, all the models were able to capture the TLD behavior reasonable well. Considering the entire range of α, the error quantified by equation (32) for Sun s and Xin s models is around 2cm, and for Yu s model it is 2.7 cm. For the range of αα between 0.9 and 1.1, which is the range where TLDs are tuned in the design practice, Yu s and Xin s model introduce more accurate predictions with 0.7 and 0.6 cm error while Sun s model has 1.8 cm error in this range.

61 51 Figure 4.8: Comparison Between Experimental Results and Analytical Predictions for FF = 5NN In the case of FF = 5NN, where some wave braking near αα = 1 occurs, Yu s and Xin s models have a good prediction while Sun s model overestimates the displacement. For α smaller than 0.8 the models do not agree well with experimental results while for α larger than 1.3 Sun s model agrees well with experimental results and Xin s model overestimates the displacement. Although Sun s model has accounted for wave breaking in its formulation, it is unable to accurately estimate liquid behaviour for α values near one where some wave breaking is observed. Overall Xin s model generates more accurate results with 3 cm error in comparison with Yu s and Sun s model with about 4.5 cm error. For αα between 0.9 and 1.1, Yu s and Xin s model accumulate an error of 0.8 and 0.6 cm, respectively; whereas Sun s model has less accurate predictions with 3.7 cm error.

62 52 Figure 4.9: Comparison Between Experimental Results and Analytical Predictions for FF = 8NN For FF = 8NN, where wave breaking was captured during almost all frequency ratios, the accuracy of all the models suffer. Sun s model overestimates the structural displacement for the entire range of the frequency ratio. For α near one, Yu s and Xin s models seem to match well with experiment results and Yu s model continues to have a good agreement with experiment for α larger than one. Overall, Yu s and Xin s model show more accurate results with 7 cm error and Sun s model has a less accurate prediction with an accumulated error of 8.5 cm. For αα between 0.9 and 1.1, Xin presents a more accurate model with 1.5 cm error in comparison with Yu s and Sun s model with 3 and 6 cm error, respectively.

63 53 Considering all three load cases and the ranges of the frequency ratios, Yu s model provides reasonable predictions in both weak and strong wave breaking and in a broad range of frequency ratios. Xin s model presents good results near αα = 1 and overestimates the displacement for α larger than 1.2. Sun s model can predict the TLD behaviour in the absence of wave breaking, i.e. FF = 3NN, however overestimates the displacements in the case of wave breaking. 4.5 Mass Ratio The TLD efficiency under a range of mass ratios (the ratio of the mass of water to that of the structure) has been investigated in terms of structural displacement and acceleration reduction. Table 4.2: Parameters for Experiments Introduced in Chapter 4.5 Structural Mass (m) Structural Stiffness (k) Structural Damping Coefficient (c) Structural Damping Ratio (ξ) Structural Frequency (f) Sinusoidal Force Amplitude Sinusoidal Force Frequency Adjusted to Change Mass Ratio Adjusted to Keep f constant Adjusted to Keep ξ Constant 0.20% Hz Adjusted to Keep Structural Disp. (w/o TLD) Constant Hz. As it is shown in table 4.2, the structural stiffness, mass and damping coefficient were changed in order to capture different mass ratios varying from 0.5% to 5% while the damping ratio remains constant as well as the structural frequency which is equal to the tank and forcing frequency. The

64 54 amplitude of the applied sinusoidal force has been also changed in a way to reach to the same steady state amplitude in the absence of TLD. Figure 4.10: The Effect of Mass Ratio on TLD-Structure Behaviour Figure 4.11: Acceleration and Displacement Reduction for Different Mass Ratios

65 55 It can be seen from figure 4.10 and 4.11 that the efficiency in reducing the displacements and accelerations increase as the mass ratio increases up to 3%. For larger mass ratios (i.e., up to 5 %), although the response of the structure with TLD is reduced in comparison to the structure without TLD (see figure 4.10), there is a reduction in the efficiency in comparison to the TLD s with mass ratio less than 3% (see figure 4.11). Noting that the increasing the mass ratio from 1.5% to 3% increases the efficiency in displacement acceleration reduction only by 10% while considerably increasing the mass of the water that needs to be employed, from a practical point of view 1.5% mass ratio can be recommended as the optimum value. Figure 4.12a: Displacement Increase Due to Undesirable TLD Forces for 5% Mass Ratio

66 56 Figure 4.12b: Displacement Time History for 3% Mass ratio Another interesting phenomenon that was observed for the mass ratio of 5% was beating (see figure 4.12). Kareem and Yalla (2000) concluded that the off-diagonal mass terms in the coupled mass matrix of the damper-structure system was responsible for this phenomenon. Figure 4.12b shows the TLD displacement in the absence of beating. 4.6 Structural Damping Ratio The effect of structural damping ratio on the TLD behaviour is investigated here. Damping ratio was varied from 0.2% to 5% as the typical range of damping for building structures. Table 4.3: Parameters for Experiments Introduced in Chapter 4.6 Structural Mass (m) 564 kg (1% Mass Ratio) Structural Stiffness (k) N/m Structural Damping Coefficient (c) Adjusted to Change ξ Structural Damping Ratio (ξ) Changed from 0.2% to 5% Structural Frequency (f) Sinusoidal Force Amplitude Sinusoidal Force Frequency Hz 9 N Hz

67 57 As it is shown in Table 4.3, structural mass and stiffness were kept constant in order to have the structural frequency constant and equal to the tank and forcing frequency. The force amplitude had constant amplitude equal to 9 N during all tests. In figure 4.13, as the structural damping ration increases the effectiveness of TLD in reducing the structural displacements decreases. For the case considered, when the structure has 5% inherent damping, its displacement response with and without TLD is almost the same. It may be because, when the structural damping is already high, the TLDstructure system does not go through large displacements, where TLD does not get the chance to dissipate energy. Figure 4.13: The Effect of Damping Ratio on TLD-Structure Behaviour In the case of acceleration, for the case considered, an increase in the accelerations for the system with TLD was observed for structural damping ratios more than

68 58 Therefore, it can be concluded that TLD is more effective for structures with low damping ratios. As the structural damping ratio increases TLD not only ceases to become effective in reducing the displacements, it can also amplify structural accelerations (see figure 4.14). It needs to be pointed out that to establish boundaries for the effective damping ratio ranges; an extensive study with different force levels is required. Figure 4.14: Acceleration and Displacement Reduction for Different Damping Ratios 4.7 TLD-Structure Subjected to Ground Motions Here the TLD-structure system is subjected to three well known ground motions. The effectiveness of TLD and the accuracy of the selected

69 59 models in predicting the response under seismic loading are investigated. El Centro, Kobe and Northridge earthquakes have been used and due to the shaking table displacement limitations, each record was scaled down by 0.3, 0.1 and 0.05 factors, respectively. As can be seen in figures 4.15 to 4.17, the TLD is quite effective in reducing structural response in terms of both displacement and acceleration. However, it is noted that it takes a while for TLD to take effect (for the liquid to be set in motion and dissipate energy) and the first peak displacement of the time history remain unaffected by the existence of TLD for all three earthquakes. This is expected as TLDs have been considered ineffective under impulse type sudden loading (Xin et al. 2009). 4.15: Structural Response with and Without TLD under El Centro Earthquake

70 : Structural Response with and Without TLD under Kobe Earthquake 4.17: Structural Response with and Without TLD under Northridge Earthquake

71 61 Additionally the accuracy of the selected models to predict TLD-structure response under seismic loading was investigated and shown in figures 4.18 to As it is shown in figure 4.18 to 4.20, Sun s model prediction for Northridge ground motion matches well with experimental results. In the case of El Centro and Kobe ground motions Sun s model overestimates the displacements, but the model has a reasonable prediction of the accelerations. 4.18: Hybrid Test Results and Sun s Model Predictions under El Centro Earthquake

72 : Hybrid Test Results and Sun s Model Predictions under Kobe Earthquake 4.20: Hybrid Test Results and Sun s Model Predictions under Northridge Earthquake

73 63 Figure 4.21 to 4.23 indicate that, although on the conservative side, Yu s model have a better agreement (less error) compared to the other two models under seismic loading. Also, the acceleration of Yu s model agrees very well with the real-time hybrid test results. 4.21: Hybrid Test Results and Yu s Model Predictions under El Centro Earthquake

74 : Hybrid Test Results and Sun s Model Predictions under Kobe Earthquake 4.23: Hybrid Test Results and Yu s Model Predictions under Northridge Earthquake

75 65 As can be seen from figures 4.24 to 4.26, Xin s model is the least accurate among the three selected models in predicting the response under seismic loading. The displacement comparison between Xin s model and the realtime hybrid test result reveals that this model underestimates the displacements in the earlier times of the time history followed by overestimation. Phase and amplitude inaccuracy in both displacement and acceleration comparisons are apparent. 4.24: Hybrid Test Results and Xin s Model Predictions under El Centro Earthquake