DYNAMICS OF TOWERS
Liquid - Structure - Foundation Interaction
TR diss 1653
H.A. DIETERMAN
DYNAMICS OF TOWERS
Liquid-Structure-Foundation Interaction
DYNAMICS OF TOWERS
Liquid-Structure-Foundation-lnterac
.+ J's^r '
< /
o, tf *> 3 1 12 Prometho'Tolein 1
t. i C DELFT
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof.drs. P.A. Schenck, in het openbaar te verdedigen voor een commissie aangewezen door het college van Dekanen, op dinsdag 6 september 1988 te 16.00 uur.
door
HARM ANJO DIETERMAN civiel ingenieur
geboren te Nieuweschans
TR diss 1653
Dit proefschrift Is goedgekeurd door de:
Promotor : Prof.dr.ir. J. Blaauwendraad en de Copromolor: Prof.dr.ir. J.A. Battjes
ACKNOWLEDGEMENTS
This thesis was prepared while the author was employed at the Faculty of Civil Engineering of the Delft University of Technology under supervision of Prof.dr.ir. J. Blaauwendraad and Prof.dr.ir. J.A. Battjes.
For his suggestions and criticism during the realisation of part II of this thesis Prof.dr.ir. A. Verruijt is gratefully acknowledged.
The investigations leading to this monograph were started under the supervision of my former teacher Prof.dr.ir. J.C. Schnfeld. Many of the results of our discussions have found their place in this thesis.
Parts of the investigations have been carried out during a temporary stay at IBBC (Institute for Building Materials and Building Structures) of the TNO-Organization in The Netherlands. I wish to acknowledge the members of the Dynamics group in this institute for the inspiring environment.
The permission of the Rijksgebouwendienst to use the dynamic measurements from a tower in Hoorn is gratefully mentioned.
CONTENTS Pag.
Chapter 1. GENERAL INTRODUCTION
1.1 Purpose and scope 1 1.2 Previous studies 2 1.3 Method of approach 2 1.4 Results 4 1.5 Outline of the thesis 5 1.6 References 7
PART I: LIQUID-STRUCTURE INTERACTION
Chapter 2. DYNAMIC BEHAVIOUR OF THE LIQUID
2.1 Introduction 11 2.2 Continuous model 11 2 . 2 . 1 Liquid flow model 2 . 2 . 2 Boundary c o n d i t i o n s 2 . 2 . 3 General s o l u t i o n 2 . 2 . 4 Free o s c i l l a t i o n s 2 . 2 . 5 Forced o s c i l l a t i o n s 2.3 Equivalent lumped model 19 2 . 3 . 1 Model d e r i v a t i o n 2 . 3 . 2 Model s i m p l i f i c a t i o n 2.4 Conclusions 24 2.5 Appendix 24 2.6 References 26
Chapter 3. THE SUPPORTING STRUCTURE
3.1 Introduction 27 3.2 Continuous model 27 3.3 Lumped model 30 3.4 References 32
Chapter 4. COUPLING OF LUMPED LIQUID AND STRUCTURE MODELS
4.1 Introduction 33 4.2 Coupled lumped model for liquid, tank and support 33 4.3 Model validation 37 4 . 3 . 1 Introduct ion 4 . 3 . 2 Comparison to s c a l e model 4 . 3 . 2 . 1 The c a s e h /a = 0 4.3.2.2 The case h/a = 1/2 4.3.2.3 The case h/a = 1 4.3.3 Comparison to resonant frequency measurement of steel water tower
STELLINGEN
1. Uit didactische overwegingen verdient het aanbeveling de massabalans in de Lagrange benadering van de mechanica meer expliciet op te stellen.
2. Het inzicht, dat kennis van golfvoortplanting verschaft in het mechanische gedrag van lichamen, rechtvaardigt meer aandacht voor deze verschijnselen in het onderwijspakket van een "droge" civiel ingenieur.
3. Het beheersen van het gedrag van civiel-technische systemen en constructies vereist dat er in de opleiding van de civiel ingenieur aandacht wordt besteed aan zowel de systeemtheorie als aan de meet- en regeltechniek.
4. Het is niet mogelijk de eigenschappen massa en elasticiteit van een homogene een- respectievelijk driedimensionale elastische ruimte vast te stellen via een locale meting van de kracht en de kinematische grootheden.
5. Bij een synthese van deelgebieden der mechanica onstaat verwarring over de symbolen en het jargon.
6. Als een toren, die gefundeerd is op palen, dynamisch tot bezwijken wordt belast blijkt de uitdrukking "buigen of barsten" letterlijk van toepassing.
6. De huidige stand van de wetenschap maakt het mogelijk tijdens het heien het draagvermogen van een heipaal te bepalen uit het gemeten dynamische gedrag.
8. Bij relatief snelle belastingen leidt toepassing op een micro-schaal van globaal gemeten spannings-rek-reksnelheidsrelaties tot onjuiste resultaten.
9. Beschouw een star lichaam Y, dat rust op een starre halfruimte S. Tussen Y en S veronderstellen we een droge wrijvingswisselwerking. Harmonische excitatie van Y door middel van een veer kan schokkende resultaten leveren.
10. De stralingsweerstand van een harmonisch op torsie belaste plaat, die rust op een half-oneindige elastische ruimte, is gegeven in Richart e.a. [1970]. Volgens dit resultaat is de stralingsweerstand afhankelijk van de massadichtheid van de plaat. Dit is onjuist. "Vibrations of Soils and Foundations", Richart et.al., Prentice-Hall, 1970, 214 & 226.
11. In de gezondheidszorg is een aanzienlijke besparing te bereiken als de wachttijd van patinten aan de specialist in rekening wordt gebracht.
4.3.4 Comparison to observations of water towers of the classic type 4.4 Discussion 41 4.5 References 42
Chapter 5. DAMPING AND STRUCTURAL CONTROL
5.1 Introduction 43 5.2 Structural damping 44 5.3 Liquid damping 46 5 . 3 . 1 Damping of l i q u i d o s c i l l a t i o n s in u n b a f f l e d c y l i n d r i c a l t a n k s 5 .3 .2 Damping of l i q u i d o s c i l l a t i o n s in b a f f l e d c y l i n d r i c a l t a n k s 5 . 3 . 3 Model 5 .3 .4 V e r i f i c a t i o n 5.3.5 The effect of baffles on the dynamic behaviour of a water tower 5.4 Structural control 55 5 . 4 . 1 I n t r o d u c t i o n 5 .4 .2 Model 5 . 4 . 3 T h e o r e t i c a l r e s u l t s of t h e s c a l e model and e x i s t i n g water towers 5.5 Compartmentation 63 5.6 Discussion 64 5.7 Appendix 65 5.8 References 65
PART II: STRUCTURE-FOUNDATION INTERACTION
Chapter 6. FOUNDATIONS
6.1 Introduction 68
6.2 Summary of dynamics of foundations 70 6 . 2 . 1 Dynamics of non-embedded s h a l l o w f o u n d a t i o n s 6 . 2 . 2 Dynamics of embedded sha l low f o u n d a t i o n s 6 . 2 . 3 Dynamic p i l e - s o i l i n t e r a c t i o n 6 .2 .4 Dynamic p i l e - s o i l - p i l e i n t e r a c t i o n 6 . 2 . 5 Concluding remarks 6.3 Statics of the foundation components 80 6 . 3 . 1 I n t r o d u c t i o n 6.3.2 Static behaviour of axially loaded piles 6.3.3 Static behaviour of laterally loaded piles 6.3.4 Static pile group - cap interaction 6.3.5 Concluding remarks 6.4 The foundation-model 91 6.4.1 Basic equations 6.4.2 The rocking and lateral dynamic stiffness of a deep foundation 6.4.3 The rocking and lateral damping of a deep foundation
6.5 Discussion 101 6.6 References 102
Chapter 7. COUPLING OF LUMPED FOUNDATION AND STRUCTURE MODELS
7.1 Introduction 109 7.2 Structure model I l l 7.3 Coupled model of structure and foundation 114 7 . 3 . 1 In troduct ion 7 . 3 . 2 Model s i m p l i f i c a t i o n s 7 . 3 . 3 Dynamic behaviour of the coupled system 7.4 Verification of the coupled model by measurements 118 7.4.1 Introduction 7.4.2 Tower parameters 7.4.3 The foundation and soil parameters 7.4.4 Model validation 7.5 Sensitivity analyses 129 7.5.1 Introduction 7.5.2 Sensitivity of the location of and amplification at the structural
resonant frequency 7.5.3 Sensitivity of the location of and amplification at the additional
resonant frequency 7 . 5 . 4 Concluding remarks 7.6 The effect or yielding of the axial pile shaft - soil interaction on the dynamic
behaviour of towers 143 7 . 6 . 1 I n t r o d u c t i o n 7.6.2 Determination of the admittance at the pile.heads 7.6.3 Structural behaviour due to yield of axial pile shaft - soil
interaction 7 . 6 . 4 Concluding remarks 7.7 Discussion 149 7.8 Appendix 150 7.9 References 154
PART III: LIQUID-STRUCTURE-FOUNDATION INTERACTION
Chapter 8. COUPLING OF LUMPED LIQUID, STRUCTURE AND FOUNDATION MODELS
8.1 Model 157 8.2 Some aspects of llqud-structurefoundation interaction 160 8.3 Discussion 167 8.4 Appendix 167 8.5 References 170
Chapter 9. DISCUSSION AND CONCLUSIONS
9.1 Discussion 171 9.2 Conclusions 172 9.3 References 174
SUMMARY 175
SAMENVATTING 177
LIST OF SYMBOLS 180
CURRICULUM VITAE 188
- 1 -
CHAPTER 1: GENERAL INTRODUCTION
1.1 Purpose and scope
The purpose of this dissertation is to accomplish a synthesis of some methods of analysis from the fields of fluid mechanics, structural mechanics and soil mechanics concerning the dynamic behaviour of structures .
Many civil engineering structures are placed in environments where they interact not only with the soil by their foundation but also with the fluid inside or outside the structure. Examples are offshore structures, water treatment plants, water towers, etc.
In this thesis the dynamic behaviour of systems incorporating liquid-structure and structure-foundation interaction will be studied. To that end a water tower based on a rigid foundation and a tower and water tower based on a deep foundation will be examined. The model derived will incorporate the liquid-structure-foundation interaction.
Lateral and rocking vibration in a vertical plane are considered, as depicted in fig. 1.1. The study has been confined to the low frequency band of interest for the dynamic behaviour of civil engineering structures.
Fig. 1.1 Types of towers studied; lateral and rocking vibration
The study of the interaction of a structure with its environment is of special interest, since the internal damping of most structures is small. When mechanical energy is transferred to the liquid or soil and converted to non-mechanical forms, the damping of the structural vibrations may increase considerably.
- 2 -
1.2 Previous studies
In the literature the dynamic loading of structures surrounded by liquid is often determined with the neglect of the dynamic liquid-structure interaction, see Faltinsen [1985]. The dynamic behaviour of liquid-containing structures is mostly modelled by considering a defective form of interaction, see Dieterman [1986].
The dynamic structure-foundation interaction is extensively studied for vibrating machine-foundations. However the structure (machine) is modelled mostly as a rigid body, see Richart, Hall and Woods [1970] or Moore [1985] . Structure-foundation interaction is also studied in earthquake engineering, see Tajimi [1977]. Then the system is often modelled by a modal analysis or finite element method (F.E.M.) approach. A more extensive review of the literature concerned, will be given chapter 6.
An earlier study of coupled dynamic behaviour of a structure filled with liquid and based on a foundation has been elaborated by Haroun and Housner [1981] and Haroun [1983]. The dynamic behaviour of a ground-based liquid storage tank with flexible walls has been investigated. The results are not applicable in our study.
A theoretical study concerning soil-structure interaction of tower structures based on a rigid footing attached to the surface of an elastic halfspace has been elaborated by Warburton [1978] .
Studies with respect to liquid-structure-foundation interaction, in which the structure is surrounded by the liquid has been elaborated by Warburton and Hutton [1978]. They considered the dynamic interaction of idealized off-shore structures on a visco-elastic half-space. Other studies have been found in the area of earthquake analysis of gravity dams including dam-water-foundation-rock interaction by Chopra [1981] and together with Fenves [1984].
The dynamic behaviour of water towers is often studied by simple models, such as two degrees of freedom (fourth order) spring-mass systems or double pendulums, see Van Uchelen [1970] . Recent literature on water tower dynamics has not been found.
The dynamics of liquid-filled tanks has been thoroughly investigated in space vehicle dynamics, because of the serious stability problems, that originate from the liquid oscillations during flight, see Abramson [1966] . This reference is still an excellent survey of the literature about the subject.
One of the early attempts to describe liquid - tank wall interaction has been elaborated by Miles [1957], who considered a prescribed wall motion. Bauer [1962] and [1963] considered the effect of tank wall bending on the liquid dynamics to model liquid - tank wall interaction. Wen-Wha Chu [1963] considered the breathing vibrations with an assumed mode shape of a liquid in an elastic tank.
- 3 -
The dynamic fluid effects in liquid-filled flexible cylindrical tanks have been investigated by Fischer [1979]. His analysis showed that the loading of the wall by the fluid will be somewhat higher than in the rigid tank due the elasticity of the tank wall.
Some of the existing literature on damping of oscillating liquid in tanks will be surveyed in chapter 5.
1.3 Method of approach
In this thesis we use lumped (discrete) mechanical models for the liquid, the structure and the foundation and from that we compose a global lumped mechanical model for the total system.
Analytic solutions are used for the dynamic behaviour of the liquid, the structure and the foundation, considered as continuous systems. The analytical results for each of the component parts will be equated to the corresponding results for lumped (discrete) mechanical models for the liquid, structure and foundation. Analytic expressions will follow for the parameters of the lumped models. Using frequency-, convergence-and symmetry-considerations the lumped models will be simplified. The resulting models for the liquid, structure and foundation will be coupled by elimination of the common dynamic and kinematic quantities. The damping of liquid, structure and soil has been incorporated.
Mostly the dynamics of structures is modelled by approaches such as modal analysis or F.E.M.. The reasons for employing the particular method stated are as follows.
Using a F.E.M. approach the equations follow from the degrees of freedom considered and the solution can be determined by use of varia-tional methods, e.g. Galerkin functionals. However studying a system consisting of liquid, structure and soil, additional problems originate. The dynamic equations of fluids often are derived by the Eulerian approach, where the dynamic equations of structures and soil are derived mostly by the Lagrangian approach. This results in difficulties in coupling the fluid- and structure-dynamics. The different time scales in liquid- and structural vibrations may lead to additional numerical difficulties. A F.E.M. approach is expensive from the point of view of evaluation of the results. A sensitivity analysis of the dynamic behaviour will demand high costs.
In a modal approach the model parameters have to be estimated, requiring the skills of a trained engineer. When all the problems just mentioned have been tackled, a relatively complex model results, in which the damping is not easily included, since this is not distributed proportionally over the system.
Another approach of modelling the dynamic behaviour of coupled systems follows from the analogy of the equations of liquid motion and those of
- 4 -
mechanical systems, see Schonfeld [1953] . In this approach the system is described by a set of first order differential equations, which may be coupled by elimination of the common kinematic and dynamic quantities. This approach is followed in this thesis and results in a relatively simple model, by which the dynamic behaviour of the structure including the liquid-structure and structure-foundation interaction can be studied in the frequency domain. Since the parameters of the model have been determined analytically, the coupled model can be used to determine the sensitivity of the dynamics for all the parameters involved. Such analysis can be managed by use of a personal computer.
Non-linear elastic behaviour in the structure and foundation can be studied by including non-linear constitutive relations. Non-linear behaviour of the liquid can be incorporated in the lumped liquid model. Then a numerical approach is required and calculations have to be done in time domain.
1.4 Results
The principal results derived show that the interaction of the tower, liquid and deep foundation leads to the originating of additional resonant frequencies in addition to the structural resonant frequencies. The additional resonant frequencies due to the structure-liquid interaction are in the neighbourhood of the liquid natural frequencies. For existing water towers the lowest additional resonant frequencies are mostly smaller than the lowest structural resonant frequency. The lowest additional resonant frequency due to the structure-foundation interaction is mostly larger than the lowest structural resonant frequency.
The dynamic behaviour of the water tower model is verified by measurements at a scale model and by measurements at existing water towers, whereas the structure-foundation model is verified by measurements at a tower and its deep foundation. The agreement between theory and measurement is good.
The damping of the sloshing liquid in the tank and the structural damping of the tower are very small, resulting in large amplifications at the resonant frequencies. Therefore additional damping devices such as baffling of the tank wall have been considered. This type of damping device is used in liquid tanks of space vehicles to suppress the propel-lant oscillations, avoiding dynamic instabilities during flight. It will be shown that tuning of the liquid height in the tank and the additional damping due to baffles result in passive control of the dynamics of the water tower.
A sensitivity analysis of the tower-foundation model has been elaborated, showing the effects of variations in foundation stiffness and damping on the location of and the amplification at the resonant frequencies
- 5 -
of the coupled system. The analysis shows that the amplification at the lowest structural and additional resonant frequencies is decreasing to relatively small values due to sufficient soil damping. The practical meaning of this result is verified theoretically.
1.5 Outline of (he thesis
The thesis has been divided into three main parts, dealing with dynamic liquid-structure, structure-foundation and liquid-structure-foundation interaction respectively.
In Part 1 the liquid-structure interaction is studied by considering the dynamic behaviour of a water tower subjected to harmonic lateral and rocking excitations. The water tower under consideration is axisym-metric and composed of a liquid-filled tank supported by a slender column.
In chapter 2 the dynamic behaviour of the liquid in a ring-corn-partmented cylindrical (annular) tank will be determined, using techniques as presented by Abratnson [1966) . However the boundary conditions at the tank circumference will be satisfied in a different way. The resulting forces and moments acting on the tank by the liquid-oscillations will be derived. Next we determine the same quantities exerted by a discrete mechanical representation of the liquid, which has been developed by Bauer, see Abramson [1966]. Equating the resulting forces and moments, analytic expressions follow for the parameters of the elements of the lumped model. In practical situations only a few elements suffice for an adequate representation of the resulting forces and moments, so the model can be further simplified.
In chapter 3 the dynamic behaviour of the slender supporting structure is determined starting from an Euler-Bernoulli beam model. Using symmetry properties and frequency considerations a sixth order model is derived in the frequency band of interest. Next a sixth order lumped model is composed, incorporating the relevant mechanisms. By equating the transmission properties of both models the parameters of the lumped model are determined analytically.
In chapter 4 the liquid and support model are coupled taking into account the relevant tank parameters. A lumped ninth order model results. Comparison of the dynamic behaviour of the theoretical model with measurements from a scale model and measurements from existing water towers show good agreement.
In chapter 5 the structural and liquid damping are incorporated in the lumped model. The damping due to the liquid and the structure is very small, resulting in large amplifications at the resonant frequencies. Therefore damping devices such as baffles on the tank wall are studied. The effect of the damping due to ring baffles on the dynamic behaviour of the structure is determined. The additional damping due to
- 6 -
the baffles opens the possibility of passive structural control by tuning the liquid height in the tank. The effect of compartmentation of the tank on the dynamic behaviour of the structure is discussed.
In Part II the structure-foundation interaction is studied by considering the dynamic behaviour of a tower based on a foundation subjected to harmonic lateral and rocking excitations.
In chapter 6 a survey of the literature concerning the dynamic and static behaviour of foundations is given. Further a lumped foundation model is introduced, representing the rocking and lateral dynamics. The dynamic stiffness and damping parameters of the lumped model will be determined, using results from continuous models.
In chapter 7 an analytical model is derived, by which the dynamic behaviour of towers can be studied in the low frequency band, based on the model derived in chapter 4. The structure model is coupled to the foundation model by elimination of the common dynamic and kinematic quantities and the dynamic behaviour of the coupled system is determined. The dynamic behaviour of the analytic model is confirmed by measurements from a tower-foundation system. An extensive sensitivity analysis concerning the effects of the foundation stiffness and damping parameters on the locations of and the amplification at the resonant frequencies of the coupled system is carried out. A large decrease of the amplification at the resonant frequencies of the structure-foundation system due to yield of the pile-shaft-soil interaction is shown theoretically.
In Part III the dynamic behaviour of a water tower based on a deep foundation, incorporating liquid-structure-foundation interaction, is considered .
In chapter 8 the liquid-structure-foundation model is composed. It will be shown that structure-foundation interaction not only effects the structural resonant frequency, but also has a significant effect on the location of and amplification at the additional resonant frequency due to the liquid-structure interaction. This frequency is affected by structure-foundation interaction despite the absence of direct mechanical interaction between the soil and the liquid.
Chapter 9 contains the discussion and the general conclusions.
- 7 -
1.6 References
ABRAMSON H.N.,"The Dynamic Behaviour of Liquids in Moving Containers, Washington DC, NASA Rpt. SP-106, 1966, ch. 2 and 6.
BAUER H.F.,"Theory of Liquid Sloshing in Compartmented Cylindrical Tanks Due to bending Excitation", MTP-AERO-62-61, NASA-MSFC, 1962.
BAUER H.F.,"Theory of Liquid Sloshing in Compartmented Cylindrical Tanks Due to bending Excitation", AIAA J., vol.1, 7, 1963, 1590-1596.
CHOPRA A.K.,"Earthquake Analysis of Concrete Gravity Dams Including Dam-Water-Foundation Rock Interaction", J. Earthq. Engng. and Struc. Dyn., 9, 1981, 363-383.
DIETERMAN H.A.,"An Analytically Derived Lumped-Impedance Model for the Dynamic Behaviour of a Watertower", Ing. Archiv, 56, 1986, 265-280.
FALTINSEN O.M.,"Hydrodynamic Loads on Marine Structures",(eds. NIORDSON F.I., OLHOFF N.) Proc. of XVIth IUTAM Conf., Denmark, Nrth. Hoi. Pbl. Comp., 1985, 117-134.
FENVES G. and CHOPRA A.K.,"Earthquake Analysis of Concrete Gravity Dams Including Reservoir Bottom Absorption and Dam Water Foundation Rock Interaction", J. Earthq. Engng. and Struc. Dyn., 12, 1984, 663-680.
FISCHER D.,"Dynamic Fluid Effects in Liquid-Filled Flexible Cylindrical Tanks", J. Earth. Struc. Engng., 7, 1979.
HAROUN M.A. and HOUSNER G.W.,"Dynamic Interaction of Liquid Storage Tanks and Foundation Soil", Proc. 2nd.ASCE/EMD Spec. conf. dyn. resp. struc, Atlanta, 1981.
HAROUN M.A.,"Vibration Studies and Tests of Liquid Storage Tanks", J. Earthq. Engng. and Struct. Dyn., 11, 1983, 179-206.
MILES J.W.,"On the Sloshing of Liquid in a Liquid in a Flexible Tank", J. of Appl. Mech, 57-A-1Z, 1957.
MOORE P.J.,"Analysis and design of foundations for vivrations", Balkema, Rotterdam, 1985.
RICHART F.E., HALL J.R. and WOODS R.D.,"Vibration of Soils and Foundations", Prentice-Hall, Englewood Cliffs, NJ, 1970.
SCHONFELD J.C.,"Analogy of Hydraulic, Mechanical, Acoustic and Electric Systems", Appl. Sci. Res., B3, 1953.
TAJIMI H.,"Seismic Effects on Piles", Proc. 9th Int. Conf. Soil Mech., State-of-the-Art-Report spec. Session, Tokyo, 1977, 1-12.
VAN UCHELEN H.,"Oscillation Problems with Water Towers", De Ingenieur, 9, 1970, B13-17 (in Dutch).
WARBURTON G.B.,"Soil-Structure Interaction for Tower Structures", J. Earthq. Engng. and Struc. Dyn., 6, 1978.
WARBURTON G.B. and HUTTON S.G.,"Dynamic Interaction for Idealized Offshore Structures", J. Earthq. Engng. and Struc. Dyn.,6, 1978, 557-567.
WEN-HWA CHU,"Breathing Vibrations of a Partially Filled Cylindrical Tank-Linear Theory", J. Appl. Mech., 1963, 532-536.
- 9 -
PART I: LIQUID-STRUCTURE INTERACTION
In this part the dynamic behaviour of a water tower subjected to harmonic lateral and rocking excitations is studied. The water tower under consideration is axisymmetric and composed of a liquid-filled ring-com-partmented cylindrical (annular) tank supported by a slender column.
The dynamic behaviour of such towers is often studied by simple models, such as two degrees of freedom (fourth order) spring-mass systems or double pendulums, see Van Uchelen [1970] . In such models, the solid mass of the tank and the "solid" mass of the liquid, supported by the column are represented by one spring-mass system (pendulum) ; the liquid mass oscillating with respect to the tank is represented by the other spring-mass system (pendulum).
The models just mentioned have some serious deficiencies. In the first place the coupling of liquid and structure is not properly modelled. Furthermore rotational and gravitational effects cannot be studied in these simple models. Scaling up the tank dimensions all these deficiencies are of increasing importance. For a water tower discussed by Steenbergen and Den Hond [1968], these effects add up to deviations of 40 percent in the location of the resonant frequencies, not to mention the dynamic response.
Recent literature on the dynamic behaviour of water towers has not been found.
In the following a lumped model of the water tower representing its dynamic behaviour will be derived. The model incorporates the liquid-structure interaction and does not suffer from the deficiencies mentioned above. The parameters of the elements of the lumped liquid and structure models will be determined analytically by use of continuous models for liquid and support. The resulting lumped models will be simplified by convergence-, symmetry- and frequency considerations. In the simplified model the liquid damping and structural damping will be incorporated. The damping of liquid sloshing in the first mode in a cylindrical tank has been determined on a mixed theoretical and experimental basis by several researchers, see Silverman and Abramson [1966]. The structural damping is of hysteretic nature mostly and is introduced in the model as a complex stiffness. Hysteretic damping may be modelled by an equivalent viscous damper, of which the parameter is frequency dependent.
The damping of the supporting structure is usually small. This also applies for the damping of the sloshing liquid in tanks of dimensions as usual in water towers. So devices increasing the damping of the structural vibrations are worthwile to be considered in order to diminish the high amplifications at the resonant frequencies of the water tower. The effect of ring baffles on the dynamic behaviour of the tower dynamics
- 10 -
will be examined. Then the structural energy transferred to the liquid by the dynamic interaction will be converted to non-mechanical forms by the baffles resulting in damping of the structural vibrations. The transfer of energy from tower to liquid can be affected by the liquid height in the tank.
- 11 -
CHAPTER 2: DYNAMIC BEHAVIOUR OF THE LIQUID
2.1 Introduction
The dynamic behaviour of the liquid in a tank due to a harmonic lateral and rocking excitation will be determined by Stokes' theory of short waves of small amplitude and using techniques as presented by Abramson [1966]. The resulting Laplace equation is solved by the method of separation of variables, which results in sets of eigenvalues to be determined from the boundary conditions. The solution of Laplace's equation for the eigenvalues zero is used to fulfil the non-homogeneous boundary conditions. The values of the other eigenvalues follow from the homogeneous boundary conditions. Yet there are several possibilities in the order of fulfilling the boundary conditions. Different orders of fulfilment result in solutions represented by different series, of which the convergence characteristics vary. For liquid oscillations in filled tanks the series-solution of the velocity potential converges faster, when the non-homogeneous conditions are satisfied before the homogeneous ones.
From the resulting potential the pressure distribution in the liquid can be derived. Integration of the pressure-distribution over the wet tank circumference gives the resulting forces and moments acting on the tank by the oscillating liquid.
The natural frequencies of the liquid motion follow from fulfilling the surface condition of the liquid. Since each of the natural frequencies of the liquid can be represented by a second order spring-mass system, a lumped mechanical model of the liquid can be determined. We use a lumped mechanical representation as developed by Bauer, see Abramson [1966]. The resulting forces and moments on the tank due to a harmonic lateral and rocking excitation of the lumped model will be derived and equated to the quantities derived by the continuous model. Then analytic expressions follow for the parameters of elements of the lumped model. In the practical range of liquid heights in water tanks and the frequency band considered it is shown that a few elements are enough to represent the resulting forces and moments. Therefore the lumped model can be further simplified.
2.2 Continuous model
The tank is supposed to be ring-compartmented (annular), as shown in fig. 2,1. The annulus of inner radius b and outer radius a is partly filled with liquid to a height h (at rest). The tank is supposed to be rigid.
- 12 -
An Eulerian coordinate system Oxyz is used with the center of gravity of the undisturbed liquid being the origin. We will apply cylindrical coordinates r, 9 and z denoting the radial, azimuthal and axial coordinate respectively.
Fig. 2.1 Tank-geometry; relevant dimensions; coordinate system
2.2.1 Liquid flow model
The liquid in the tank is assumed to be homogeneous and incompressible and the flow to be irrotational. The tank is excited by harmonic lateral and rocking movements of small amplitude. Linearization results in the use of Stokes' theory of short waves of small amplitude to calculate the movements of the liquid, see Stoker [1957] or Abramson [1966] . The potential is connected to the liquid velocity by:
grad 4> = U* (2.1)
Incompressibility of the liquid gives Laplace's equation:
V2 = 0 (2.2)
The Navier-Stokes equations simplify to:
3jt> + gz (2-3)
p represents the pressure at (r,6,z) at time t, g the acceleration of gravity and p the mass density of the liquid. The time-dependent Bernoulli constant has been included in .
- 13 -
2.2.2 Boundary conditions
At the boundary-surface the common ve loc i ty v of l iquid and boundary
surface in the d i rec t ion normal to the surface i s equal to the der ivative of normal to the surface:
4>=vn (2.4) 3n
The free surface z = Ti(r,0,t) is connected to the potential by:
2 = | 4 a t z = h/2 0 t OZ
For the free surface it follows using also (2.3) with p = 0:
^ * + g H= 0 at z = h/2 (2.6) 3t2 3z
Using the linear model (2.1) - (2.6) the forces and moments exerted on the tank by the liquid due to harmonic lateral and rocking movements of the tank can be determined. Therefore a lateral excitation of the axi-symmetric tank given by:
^ = V ^ e i m (2-7)
and a rotational excitation about the y-axis given by:
- 14 -
2.2.3 General solution
The general solution of Laplace's equation can be determined using the method of separation of variables, as shown in e.g. Abramson [1966]:
(() (r,9,z,t;M.;m) = (cos m8 + A sin m 8) [B Jm(|x ) + C Ym(u, )]
* z _ . , z, iOt x {D cosh p- + E sinh p-} e ,2 jn
For every value of the parameters p and m, which belong to the set of complex numbers p,m e p,(2.11) satisfies (2.2). In this solution A,B,C,D and E will be considered as functions of p and m to be determined from the boundary conditions. The solution (2.11) has to fulfil the non-homogeneous boundary conditions (2.9) and (2.10) and the homogeneous condition (2.6). To fulfil the non-homogeneous conditions the asymptotic solution of (2.2) at the eigenvalue zero is used. Since we have failed in deriving this solution directly from (2.11), the asymptotic solution of (2.2) in Cartesian coordinates is used, see Dieterman [1982]. Transforming that solution to the cylindrical coordinates results in:
^ ( r . e . z . t j O i O ) = {a, + b , r cos 6 + c , r s i n 8 + d , z + e , r 2 s i n 9 cos 8
+ f, rz cos 8 + g, rz sin 8} e f 2 12)
The coefficients a, - g, are constants. The particular solution (2.12) satisfies Laplace's equation as can be seen directly from substitution. Fulfilling the boundary conditions we first consider the constants A(p.,m), B(p,m) etc. in (2.11) for all values p,m 4= 0 and afterwards as far as necessary a, - g, from (2.12) for p = 0, m = 0, resulting in the solution:
(r,8,z,t) = Z 4>(r,8,z,t;p.;m) + , (r,8,z, t ;0;0) (2.13) P,m40
We are merely interested in the resulting forces and moments exerted on the entire tank wall. Only the modes for m = 1 contribute to these quantities, as follows from the orthogonality relations of the trigoniome-tric functions on the interval (0,2ir).
The subset p follows from the chosen sequence in meeting the boundary conditions. In the literature little attention has been paid to the sequence in which the boundary conditions are satisfied. In translated Russian work from Kantorowitsch and Krylow [1956] we found some discussion. Yet there are several possibilities in the order of fulfilling the
- 15 -
boundary conditions. The standard procedure is to treat every boundary condition separa
tely, considering the others homogeneous and in the end superposing the partial results. Another way is to fulfil the boundary conditions in the solution successively, see Boyce and Diprima [1969]. In this procedure there is another choice of order, viz. meeting the homogeneous condition first and the non-homogeneous ones subsequently, or this order can be reversed. Eventually mixed order procedures are possible.
Preliminary investigation in which both orders have been elaborated for a rectangular tank, showed that the rate of convergence of the solution depends on the height-radius ratio (h/a) of the liquid in the tank. For the h/a - ratio of interest in watertowers an increase of the rate of convergence is obtained when we first fulfil the non-homogeneous boundary conditions. For a rectangular tank the results of the investigation have been mentioned in the appendix 2.5.
2.2.4 Free oscillations
Calculating the free oscillations of the liquid for V = 0 and W = 0 in (2.9) and (2.10), the free surface condition (2.6) gives the natural frequencies of interest (m=l), with X a = u :
- 16 -
resulting in
- 17 -
in which:
-* n A [-5- - bC,(bXn)] ^ n ,,'tl, L- _ b A = "An with k = n i a (2.22)
(Oq - i) a a - k2) and
mt = p-na^fl - k2) (the liquid mass) (2.23)
The force in the y-direction is zero due to the axi-symmetric tank-geometry. The first terra in (2.21) represents the moment due to the liquid, treated as a rigid body.
Rotational excitation
The liquid movement due to the rotational excitation of the tank W e around the y-axis follows when the solution (2.11) fulfils the boundary conditions (2.6), (2.9) and (2.10) with V = 0.
Now we have non-homogeneous boundary conditions at the walls and bottom. The last two terms in the asymptotic solution (2.12) can be used to satisfy either the wall or the bottom conditions. Treating the wall conditions first, the series (2.16) can be used to fulfil the bottom condition. Meeting the surface condition the potential, which satisfies (2.2), (2.6), (2.9) and (2.10) f or V = 0, follows as:
4 f c - W . ' c o . e . i m [ H + XnC '(Xnr) r a z n=0 ~"77 ( n - 1) cosh \nh
*{( .L+ h)cosh\n(z + h,_ i sinh^zsinh^ ( 2 2 4 )
a 2 2 ci 2
- _L sinhKn(- z)}] ^n **
Using (2.3) and accounting for the free surface displacement with respect to the tank, which is given by:
s(r,e,t) = T)(r,8,t) - rWcos 0 (ifl)-1eint (2.25)
the pressure distribution follows. Appropriate integration of the pressure over the walls gives the force on the tank in the x-direction and the moment around the y-axis, with the results:
- 18 -
r x = - l m t g - e - i n m t a W e n Z n - - { ( , + i ) t a n h \ n h + x n n=0 xna hn 2 2 n
+ _2_ ( l - 1 ) } ] Xnh coshX n h ( 2 2 g )
and:
Qv = - i n n , t g W e i m a * + k 2 ) - inm We i 0 t [jV + l_(i + k2) y l n 4h t 12 a2 4
oo 7* + 2 _H_ { ( h - 3J_ - _4_)tanhXnh + _ _ _ J ( 2 - J f _ )
n= Xna2 4 n 2 X2nh Xn cosh Xnh hfl2
+ ( 5 g - _ ! )} (2.27) n2xnh xn
In the derivation of (2.27) the result:
Z Jj- S - M = i ^ 2 (2.28) n= Xna2 V 4
has been used; a proof of this relation has been given in the appendix 2.5.
The resulting force and moment for the case of lateral excitation ((2.20 and 2.21) respectively) are equal to results found in Abramson [1966] . The results for the case of rocking motion (2.26 and 2.27) differ from the results stated in Abramson [1966], due to some misprints in the results given in that reference and due to further refinements in our result (2.27) by use of (2.28).
For values of il w and b/a | 1 it can be shown by rewriting the cylinder function C, (X b) as:
c A b) - 2 fJo(V>> - ^ n ) , ,l n ' TrXnb V J0(Xnb) -J 2(X nb) /
that the terms in the infinite series of the resulting forces and moments converge faster than (X a ) - 3 . The roots of (2.15) for large values of n are proportional to n. So the series converge and besides we can conclude that only a few terms of the series are sufficient to represent the resulting forces and moments acting on the tank. This will be confirmed by the mechanical representation for the liquid-movement developed in the following.
- 19 -
2.3 Equivalent lumped model
2.3.1 Model derivation
We now use a lumped model representing the dynamic behaviour of the liquid. This model will later be coupled to a lumped tank and support model. As follows from modal analysis of (un)daroped systems the equations of motion can be uncoupled, so the continuous system is represented by an infinite series uncoupled spring-mass-systems.
The lumped liquid model will be subjected to harmonic lateral and rocking excitations and the resulting forces and moments on the tank will be calculated. These results are equated to the results from the continuous model, from which the model parameters will follow analytically. The lumped model will be simplified substantially.
h/2
V 2 Cn/2 _A A A A A A I 1_A A A A A A V V W V n rVVVVV
c o / 2 :mo V 2
AAAAA/H H V W W - e .g .
V J V
t h n
IN -"v
Fig. 2.2 Discrete mechanical representation of liquid; center of gravity; n spring-mass-systems; non-oscillating mass; rotational inertia; points of application of spring-mass systems.
In Abramson [1966] several lumped representations of liquid movement have been derived for tanks of other geometry than we are studying. Next to the pendulum-model, the model shown in fig. 2.2 has been considered.
The lumped spring-mass-systems m , c act on the tank walls at a vertical coordinate h with respect to the center of gravity of the undisturbed liquid (e.g.), n = 0,1,. The rigid body with mass m and rotatory inertia J represents the "solid" part of the liquid. Its vertical coordinate is h .
Further the constraint is introduced, that the natuarl frequency n ) _ 1 of the spring mass system n is e<
quency of the liquid (2.14) for n = 0,1,2, (mncn) 1 of the spring mass system n is equal to the n natural fre-
un " (rancn> (2.29)
The total mass in the analogy should be equal to the total liquid mass m. :
- 20 -
oo m + m_ = mt (2.30)
v n=o " L We keep the level of the center of gravity unchanged, which r e su l t s in:
m v h v = n 0 m n h n < 2 3 1 >
The tank is subjected to harmonic l a t e r a l and rocking exci ta t ions of small amplitude as in the continuous case.
The resu l t ing forces and moments, exerted on the tank due to the lumped system, are determined using a non- iner t ia l frame connected to the tank. Neglecting second and higher order terms, the force on the tank walls due to the springs and r ig id body i s :
F*= "n?0mn (*n + * + h"' i t ) " mv(x ' - h v V + mtS9t ( 2 3 2 )
in which x'n i s the accelerat ion of m with respect to the tank;
x (= iflV e ) i s the accelerat ion of the tank with respect to Oxyz and
8^ (= iflWe ) i s the angular accelerat ion of the tank with respect
to Oxyz. The l a s t term in (2.32) i s due to the grav i ta t ion
The moment exerted on the tank r e s u l t s from the balance of angular momentum as:
Q y= -< J v + n i 0 m n h n + m v h v > t - n S o ' n n h n x n + Z^&Xn
Using the momentum balance of the n mass and the kinematic equation
x n = x + x n + h n 6 t
x n can be expressed in V and W as :
i n t ' n _ {V+ (h + _ ) W } (2-34)
ton- n ) n
Substitution in (2.32) and (2.33) using (2.29) - (2.33), neglecting quadratic terms and terms of higher degree, gives for the force:
2 1
Fx=-inn,teimt{l+Z> 1 ^ w ' - n V ^ i j x t n=0 m i 2 n=Om . ^ 7 n^1 vyi
n2 " ( 2 . 3 5 )
and for t h e moment:
Q y = - i n e i m t { J v + | o m n h n m X + m JVY g / f l 2 ) 2}W
" m t ( - V 1 ) ft2
' n = m t ( i ^ - l ) ( 2 . 3 6 , ft2
Equating the right-hand side of (2.35) to those of (2.20) and (2.26) leads to analytic expressions for the following model parameters: - The model masses and compliances:
mn = 1'tA- 1) tanh nh n . 0 1 n l n ! ; -r n - o,i (2.37) mt n \ nh
A and mt are defined in (2.22) and (2.23). n L The "solid" mass follows from (2.23) and (2.30) and the compliances
c n follow from (2.29) and (2.14), n = 0,1 - The vertical coordinates
h = !? - 2_tanh ^o!? n = 0 , 1 R , n 2 Xn 2 < 2 - 3 8 )
and t h e c o o r d i n a t e h v f o l l ows from (2.31) and ( 2 . 3 7 ) .
Equat ing t h e r i g h t - h a n d s i d e of (2 .36) t o t h o s e of (2.21) and (2 .27) u s i n g (2.28) l e a d s t o a n a l y t i c e x p r e s s i o n s f o r the r o t a t o r y - i n e r t i a : - t he r o t a t o r y - i n e r t i a J v fo l lows a s :
J t o t = J v + n ? n m n h n + mv K = at**{Z* + 7 ( 1 + ^ n=0 " " v v l i 2 a2 4
4 E _JQ_ f n _ i ) ( i - 1 _ tanh i n ! \ (2 .39) n = V a 2 V Xn" 2
n
The f i r s t two terms on t h e r i g h t s i d e of (2 .39)
*
- 22 -
2.3.2 Model simplification
As indicated by the convergence of the series in the continuous results, the lumped model can be simplified. However, the modelparameters are still functions of h/a, k = b/a and the roots p. = X a of (2.15) .
The roots of (2.15) are tabulated in tabel 2.1 for various values of k = b/a. In order to calculate the modelparameters as functions of h/a, we choose k = 0.1, a datum from the water tower discussed by Steenbergen and Den Hond [1968] . Since liquid heights in water towers are restricted, we confine the h/a interval of interest to 0 < h/a < 1.2.
i i i i Table 2.1. The roots or J, (k(l)Y, (p.) - J, (p.)Y, (kp.) for various values of k - b/a.
\ k _ 0.01 0.1 0.4 0.7 0.9
1 . 8 4
5 . 3 2
8 . 5 3
1 1 . 6 9
1 . 8 0
5 . 1 3
8 . 2 0
1 1 . 3 6
1 . 4 6
5 . 6 5
1 0 . 6 8
1 5 . 8 5
1 . 1 8
1 0 . 5 9
2 1 . 0 0
3 1 . 4 6
1 . 0 5
3 1 . 4 4
6 2 . 8 4
9 4 . 2 6
The relative model masses (n = 0,1,2, .. .) and the relative mass of the rigid body have been determined as function of h/a and plotted in fig. 2.3.
1 2 -3 -4 -5 6 -7 -8 -9 10 1-1 1-2 h/a
Fig. 2.3 Relative masses for k = 0.1
From the results (2.37), (2.22) and (2.17) it can be shown that the masses m n are proportional to (Xna)~3 Their rapid decrease with increasing n is illustrated in fig. 2.3. The liquid movement is primarily determined by one spring-mass (n = 0) and by the rigid body with mass m v . only for small values of h/a, when nearly all the liquid oscil-
- 23 -
lates, the system nij.Cj might be of some interest. The rapid decrease of the importance of higher modes can be understood physically by considering the standing wave forms of those modes. The equivalent mass of the "solid" liquid increases approximately in proportion to h/a. These interpretations are valid for the force on the tank due to translational excitation only since otherwise the vertical coordinates h n (n = 0,1 ) and h v are of importance too.
Fig. 2.4 Relative heights for k =0.1
For k = 0.1 several relative vertical coordinates have been plotted in fig. 2.4. In spite of the fact that hy/h -oo as h/a 0, m v h y and m yh v are finite. In the h/a interval 0.5 < h/a < 1.2 the most important system m n,c 0 and the rigid body are located on either side of the level of the center of gravity. For large values of h/a all systems m,c are located below the liquid surface, however their influence diminishes rapidly due to mn/m
- 24 -
The last model parameter Jtot = Jv + ^ mnhn + mv nv n a s been Plotted for k = 0.1 in fig. 2.5, normalized with the rotatory inertia Js of an equal volume of solidified liquid, see (2.40). The rotatory inertia J^ot consists of the spin inertia Jv of the rigid body and the rotatory inertias of the masses with respect to the center of gravity of the undisturbed liquid.
2.4 Conclusions
The liquid movement due to harmonic lateral and rocking excitation of an annular tank has been derived and the resulting forces and moments acting at the tank due to the liquid oscillation have been determined. The parameters of an equivalent lumped mechanical model representing the dynamic behaviour of the liquid have been analytically determined.
Damping can be easily incorporated in the lumped model. The effects of damping and compartmentation will be studied in chapter 5.
Tanks of other geometry can be studied by determining the relevant parameters.
2.5 Appendix
Results of convergence comparison.
A lateral excitation as given by (2.7) of a rectangular tank of length 1 is considered. Eqs. (2.1) to (2.6) apply. The solution for the velocity potential for different sequences of fulfilling of the boundary conditions will be given. Difference in convergence will be shown.
When the nonhomogeneous boundary conditions are fulfilled before the homogeneous ones the asymptotic solution at the eigenvalue X = 0 can be used for adaption of the nonhomogeneous conditions, see Dieterman [1982] . The homogeneous boundary conditions can be fullfilled by use of the parameterset X. = kir/1 (k = 1,2,...). For the potential results:
iOt, . 2? 4n zi coshXk(z + -) , ] ,
- 25 -
number of roots X^ (k = 0,1,...) are imaginary. The solution is, see Dieterman [1983] .
v iflt ?? 2 sinh 2Xkh {cos Xk(x - 1) - cos Xkx} cosh Xk(z + n) ^ k = 1X k(2X kh + sinh 2Xkh) sin Xkl cosh X kh
Comparison of the coefficients of n terms of both the series mentioned above shows that for h/a > 1 the convergence is faster, in case the non-homogeneous boundary conditions are first fulfilled. In case h/a < 1 the reverse applies.
Proof of result (2.28)
The series (2.16) is uniformly convergent on the interval (0,a]
o _ _i = I A C , (X r) a n=0 n ' n
I t fo l lows t h a t :
b
Using:
r 3 d r = Z a A r 2 c , ( X n r ) dr / n=0 n h J ' n
( I )
a x n r 2 c , ( r ) d r = X n aC, (Xna) - X n b c , (Xnb)
b X n
i t fo l lows from (I) and c, (^ n b) = 2/Tr\ na:
Z _ ^ a { J _ - bCl(Xnb)} = [ i r ' f n = 0 X n irXn 4 b
With (2.22) t h i s r e s u l t s i n :
0 0 A * * , . 1 + k 2 E _D_ (J2L- i ) n=0 \ 2 2 n z
By use of (2.37) t h i s can be r e w r i t t e n t o :
_%__ = at1 + k ) n= m t h " 4h
- 26 -
2.6 References
ABRAMOWITZ M. and STEGUN J.A.,"Handbook of Mathematical Functions", New York, Dover Publ., 1965, ch. 9 and 11.
ABRAMSON H.N.,"The Dynamic Behaviour of Liquids in Moving Containers, Washington DC, NASA Rpt. SP-106, 1966, ch. 2 and 6.
BOYCE W.E. and DIPRIMA CD. , "Elememtary Differential Equations and Boundary Value Problems", Nwe York, J. Wiley & Sons, 1969.
DIETERMAN H.A.,"Liquid Movements in Rectangular Tanks", TUD Rpt., 1982, 11-13 (in Dutch).
DIETERMAN H.A.,"Physically Oriented Solution of Oscillating Walls in Liquid", TUD Rpt., 1983, (not published), 16-17. (in Dutch).
KANTOROWITSCH, L.W.and KRYLOW,"Naherungsmethoden der Hoherenen Analyse" Dt. Verlag Wissenschaften, 1956, Ch. 1.
SILVERMAN S and ABRAMSON H.N.,"Damping of Liquid Motions and Lateral sloshing", In: "The Dynamic Behavior of Liquids in Moving Containers", (ed. ABRAMSON H.N.), NASA SP-106, Washington DC, 1966.
STEENBERGEN A.J. and DEN HOND C.H.,"Steel Water Tower in Amsterdam", Bouw, 50, 1968, 2022-2025 (in Dutch).
STOKER J.J.."Waterwaves", New York, Int. Publ. Inc., 1957, Chap. 2. VAN UCHELEN H.,"Oscillation Problems with Water Towers", De Ingenieur, 9, 1970, B13-17 (in Dutch).
WATSON G.N.,"A Treatise on the Theory of Bessel Functions", Cambridge Univ. Press, 2nd ed., 1966, 82.
- 27 -
CHAPTER 3: THE SUPPORTING STRUCTURE
3.1 Introduction
The support of the rigid tank of the water tower is supposed to be a slender shaft, i.e. its diameter-length ratio is small. Therefore the dynamic behaviour of a slender beam loaded by shear and bending will be studied. Starting from an Euler-Bernoulli beam model, a continuous transmission model for the support will be developed, using symmetry properties. In the frequency band considered the continuous model can be simplified to a sixth order system. Then a lumped model of the sixth order will be derived, of which the parameters follow analytically by comparison of its transmission properties to the continuous results.
3.2 Continuous model
Because of the slenderness of the support the effects of shear deflection and rotatory inertia can be neglected, and we will use the Euler-Bernoulli beam model. We could use the Timoshenko beam model as well.
The support has a length /, its cross-section has an area A and an elastic moment of inertia I. The mass density of the material be p s and E be Young's modulus. We refer to an Eulerian coordinate system with the center of gravity of the support at rest being the origin, see fig. 3.1. In this way we fully utilize the symmetric properties of the support.
w(|,t)
ii q(e
v ( . t ) - f ( ? , t ) f ( ? , t )
a-Fig. 3.1 Support geometry; coordinate system; sign convention
We further use a Lagrangian coordinate system O; connected to the material cross-sections of the support and at rest coinciding with Oz.
Let f(?,t) be the force in cross-section ?, impressed by the matter at
- 28 -
these quantities are complex, bold characters are used: F,, Fa, etc.. The amplitude of these quantities are denoted by F,, Qt etc.. Similarly the displacement amplitudes are x, , x2. Forming the balances of linear momentum, angular momentum, using constitutive equations and kinematics results in non-linear equations. For "small" disturbances with respect to a stationary state the equations can be linearised to:
_ + m T _ = 0 m = p A, mass per unit length 3? X 3 t I S l - w = o 3? (3.1) 8_3 + f = o H ow + c dq _ 0 c (El)"'1 , bending compliance/length H X 3 t Z
For harmonic excitations ( e s?e i" t) of the support the characteristic equation is:
s" - IVnijCj = 0
Defining:
P= (fl^jCj)1'4 and T = B2/ncI{=(mI/CI)"2}
the general solution of (3.1) is:
v(|,t) = ( A cos BE + B sin PC + C cosh p? + D sinh p?) eiftt
w({j,t) = p(- A sin p? + B cos P? + C sinh p? + D cosh p|) e i m
q(,t) = ir(- A cos PS - B sin P + C cosh P + D sinh Pi) eiftt
f (,t) = -ipT( A sin p| - B cos PI + C sinh P? + D cosh pg) e i m
PT can be interpreted as the characteristic admittance of the support and ft/p as the phase-velocity of harmonic waves in the support. The wave-propagation velocities in the Euler-Bernoulli-beam are infinite, see Flu'gge and Zajac [1959] .
For small harmonic excitations of the support the differences in amplitudes AV = V2 - V, are expressed in terms of the mean values V = (V2 + Vt)/2, etc. using the quantities Vm, etc. in the middle section ( = 0) . These obey the equations:
- 29 -
Vm = A + C; Wm = ((B + D) ; Qm = iT (-A + C) and Fm = - ip T ( -B + D)
Solving for the constants A,B,C and D and introducing the shor t notat ion:
T , = cos(pf/2) + cosh(P//2) and T, 2 = sin(P//2) sinh(p//2)
r e su l t s in :
V = T0Vm/2 + iT,Qm/2T
AV= T W ^ P - iT3Fm/pT
W = T0Wm/2 - iT,Fm/2T
AW= -pT3Vm - ipT2Qm/T (3.2)
Q = -iTT,Vm/2 + T0Qm/2
AQ = -iTTaWn/p - T2Fm/p
F = iTT,Wm/2 + T0Fm/2
AF=-ipTT2Vm + pT3Qm
Result (3.2) shows that AV,W and F are f(Wm, F m ) , AW, V and Q are f(Vm ,Qm) , e t c . . Elimination of the quan t i t i e s Vm, e t c . of the middle section r e su l t s in:
AV= l ( t a n M + tanh M) w - i ^ t a n ^ - tanh f) F
AW = -p( tan P~ - tanh y ) V - i^ ( tan ^ + tanh ^ ) Q
AQ = - i l ( t a n y - tanh ^ ) W + (tan ^ + tanh ^ ) F
AF = -ipT(tan ~ + tanh y ) V + p(tan ^L - tanh ^ ) Q
For ex is t ing water towers of the type studied here, the i n t e r e s t i ng frequency-domain i s such tha t :
^ = ! ( ! C 1 ) " 4 < 1
and usually:
1 2 ^
Expanding tan - and tanh in power s e r i e s of p/ and neglect ing 0(P/)4
- 30 -
and higher order terms results in:
Av' AW AQ AF
0
n2msc/ 12
0
i O m s
/
0
iftm s< 12 0
0
- i O C
0
ft msC/
iCld 12 0
- /
0
V
w
Q
F 12
(3.3)
We have introduced: C = /Cj and m g = /m-j. The result (3.3) for the support model can be rewritten into:
W-
1 I n2
ft mcl 6
/ 1
i f tC/ /3 - i f tC/ /2
- i f tC//2 - iOC
iftm
i f tm s C/ 6
i f tm J /2
iftms//2 iftms//3
ln !
r2
% V 1 w.
(3.4)
If ft 0 familiar results follow. If mg t 0 known relations follow, see Fertis [1973]. Since compliance and mass are equally distributed over the support, the occurrence of comparable mass and compliance elements in the matrix is plausible.
We did not succeed in expressing the quantities at one end of the support in terms of those at the other end. A possible physical explanation is the fact that e.g. not both F, and V, etc. can be prescribed in wave propagation.
The determinant of the matrix in (3.4) is a polynomial of the sixth degree in ft so to this approximation the support can be realised mechanically by a sixth order system.
3.3 Lumped model
We now develop a lumped model representing the dynamic behaviour of the slender support in the frequency-domain under consideration. Equating the result to the continuous result (3.3) analytic expressions for the model parameters follow.
Analyzing the transmission mechanisms in the support following from the momentum balances, constitutive equations, kinematics and geometry, we realise a lumped model, as shown in fig. 3.2. The model consists of two massless rigid bars connected mutually and to their neighbourhood by a rotational hinge. Each bar has a length 1/2. In the rotational hinge the mass and bending compliance have been concentrated. The system contains three springs and three masses and since each element is described by a first order differential equation the system is of the sixth order.
- 31 -
y//////////////^////////'
Fig. 3.2 Lumped mechanical representation of support
The mass and compliance distribution over the model follows from symmetry considerations. Writing ocms/2 for the lumped masses at the extreme ends, with a to be determined, the mass in the middle joint is (1 - oc)ms. Likewise the compliance distribution is -yC/2 at the ends and (1 - -y)C in the middle. The constants a and -y will be determined comparing the harmonic response of the lumped model to the result (3.3) . The harmonic response of the model has been expressed in AV, V, A W , W, etc. as before. Forming the balances of linear momentum and angular momentum, and using constitutive equations, kinematics and geometry results in:
AV
AW
AQ
AF
o ;
- ( l - 9 (l-"r)n2msC//4 0
0 - i a f l m / /4
-iOtn o
0 - i 7 f i C / / 4
- iftc o
0 - /
(1-a) (1-7) a mCl/4
(3.5)
Compared with (3.3), agreement to the present order of approximations 0(P/)4, is obtained if:
a = 1/3 and -y = 1/3.
The mass- and compliance distribution over the support have been given in fig 3.3.
- 32 -
Fig. 3.3 Mass- and compliance distribution over the lumped support model
For the Timoshenko-beam-model a principally equivalent derivation can be given. The lumped model has to be extended by shear springs near the joints and putting the rotatory inertia in the rigid beams. It leads to the same order of deviations. The derivation and the results are not presented here.
3.4 References
FERTIS D.G.,"Dynamics and Vibration of Structures", New York, J. Wiley & Sons, 1973, ch. 6.6.
FLUEGGE W. and ZAJAC E.E.,"Bending Impact Waves in Beams", Ing. Arch., 28, 1959, 59-70.
- 33 -
CHAPTER 4: COUPLING OF LUMPED LIQUID AND STRUCTURE MODELS
4.1 Introduction
In the preceding chapters we have developed lumped representations for the liquid and the support. Taking into account the relevant parameters of the tank, a lumped representation of the water tower can be derived. The complete model is shown in fig. 4.1. Only the spring-mass systems representing the two lowest liquid natural frequencies have been sketched. If higher frequencies play a role, the model can easily be extended. Damping has not been considered yet. This will be incorporated in chapter 5.
The resulting model will be verified by analysis of measurements from a scale model, of resonant frequency measurements from a slender steel water tower and of observed vibrations of 37 water towers of the classic type.
r 1"
C j / 2
Cqj/2
|
mi C j / 2 H HWvW-
c .g . H ^ \-\ARAAA-
m T 1 1 "V ' "v 1
MMUmJHU IJIMMMMMMMmiZWI
V J R
'/////////////////////////////////////////A
Fig. 4.1 Lumped mechanical representation of water tower frequency band of interest
4.2 Coupled discrete model for liquid, tank and support
We denote the mass of the tank as m^, i t s sp in - ine r t i a as J p and the ver t i ca l location of i t s centre of gravi ty with respect to the tank bottom as h R .
The force acting on the tank due to the l iquid i s given by (2.35) as :
file:///-/aRaaa
- 34 -
x t l n=0 mi 2 ' n=Omi^? T n ^ t , w, ft2
"n - 1 . n' (2.35)
and the moment by (2.36) as:
int 2? ,, ,_, S? ro (h + g/n 2)* ine [{Jv+ ? "-nhn * m yh 2 + m Z n n } W n~ n = 0 m t (J^.- 1)
n x V mn ( hn + Stil2), 1 n=0 m. a
n'
n (2.36)
The force and moment acting on the tank due to the support are F2 and Q2. The balances of linear and angular momentum of the tank have been formed. The results are expressed in the quantities of the upper cross-section of the support, accounting for the difference in the locations of the centers of gravity of the tank and of the liquid at rest. A matrix relation between F2, Q2, V2 and W 2 results:
Yn Yi2
Y12 Y22 W, (4.1)
The matrix in (4.1) will be denoted as V . The elements Y ^ of V are:
' ->{ , ( - , *> * .,(-5 * ? -'-;;'.">>}, n2 2 n2 n-o (u^/n2- i)
Y = iCl\ (m + nu)h n + S n R n \ and 21 l R t R "= (con2/n2- o '
( 4 . 2 )
Y2 2 = i n ( J R + J t o t + r a R h R( h R + -5- 2 ) + "> thR(h + . )
+ ~ mn(jj + hn+g/n')(h,l+hn+g/n') | n= (co /n2- 1)
The parameters of the liquid presented in these expressions have been derived in chapter 2, see (2.14), (2.37), (2.38) and (2.39). As shown in chapter 2 only the lowest liquid modes are of significance, so the expressions for Y J J can be simplified in our case.
- 35 -
The model equation (3.4) for the dynamic behaviour of the support in the low frequency band has been derived in chapter 3.
W . 2.
1
/ - i O C / / 3
-C/ /2
6 S 1
-iilCl/2
- i i l c
iftm s
ifims/ /2
1
I f tVc/
iftm s / /2
i f lm s / /3
/
1
2
W , 1,
(3.4)
Eq. (3.4) together with (4.1) and (4.2) for the tank and liquid represent the dynamics of the water tower for small harmonic lateral and rocking excitations. Introducing the submatrices:
s = 1 ln!, O mC/ M
iflm
the vectors :
- H C =
and
- inc/ /3 -inc//2
U
'v.' w i
-mci/z - i O C
for i =
the equation (3.4) can be rewritten as:
El-u MR]
iOin//2
iOmJ /2 iClmJ/3
which may simplify calculations considerably. Eliminating F2 and Q 2 from (3.4) using (4.1) four equations in six
unknowns F,, Q,, V,, W,, V2 and W 2 result. If only the two lowest liquid modes are incorporated and the masses
at the end of the support are lumped to the tank and foundation respectively (see fig. 4.1), the model is of ninth order in case of a stationary clamped support base (V, = 0; W, = 0 ) . The natural frequencies of the model follow from (3.4), (4.1) and (4.2) eliminating F2 and Q2. It follows that:
-J_n2c/4(YuY22 -Yi2Y2i) + i ClCjlHVii + Y12+Y21 + Y22) + 1 = 0 12 v 3 2~7 " p
(4.3)
Since the first term is of the order (mR + mt) m R _ 1 (p/)4 with respect to
- 36 -
the other terms it can often be neglected. Equation (4.3) can be verified by the calculation of the natural
frequencies of a rigid body (mass mR, rotatory inertia JR) supported by an elastic beam (bending compliance Cjl) . We found:
04mRJRC^/'' - 4n2CI/(mR/2+ 3JR) + 13 = 0 (4.4)
If we consider the case h = h R = 0 in the model (3.4), (4.1) and (4.2), equation (4.3) simplifies to (4.4).
External excitations can easily be incorporated. We first consider a lateral harmonic displacement at the base of the model, since such a loading has been exerted on a scale model of a water tower, of which the dynamic behaviour is measured. The theoretical model and measurement will be compared in the next section. With W, = 0 and neglecting 0(p/)4 and higher order terms, the transmission function v2/v, follows as:
. _ ,2/Yiz . Y22\
vi 1 + i n c ^ Y Y u + Yia+ Yai + Y22) _ J_f t 2 c^ 4 (YnY22 - Yi2Y2i) v 3 27 ; 2 / 12 l (4 .5)
This function also expresses the transmission function of the displacement-amplitudes at the support top and base: x2/x, . For il -0 the static solution results.
For designers the transmission function expressing the moment Q, due to an exciting force F0 at the level of the upper cross-section of the support may be of interest and this function can be determined by forming the momentum balance of the tank. Assuming a clamped base again and denoting the matrix in (4.1) as Y. , it can be shown that:
(CYL-I)[V| =[5] (4.6, in which I represents the unit matrix. Solving for F2 and Q 2 and using the equation Q, = / F2 + Q 2 from (3.4) it follows that:
/ { l + l O C ! ^ +*?)}
F0 1 + i n c V 3 ( Y i i + Yia+ Yai + Y22) - i_U2C?/4 (YnY2 2 - Yi2Y2i) 1 v 3 2? / 2 / 12 i (4 .7)
Q,
- 37 -
4.3 Model validation
4.3.1 Introduction
The theoretical model developed in the foregoing will be verified on the basis of: (a) Measurements at a scale model, to examine the dynamic response,
following from (3.4), (4.1) and (4.2). (b) Resonant frequency measurements at a steel water tower. (c) Observed vibrations of 37 steel water towers of the classic type.
4.3.2 Comparison to scale model
A steel water tower in Amsterdam discussed by Steenbergen and Den Hond [2.1968J is of the type analysed in this paper. The data of the tower are: a = 7 m ; k = 0.1; h R = 6 m; A = 0.25 i2; / = 2 0 m ; m R ~ 105 kg;
5 10" kgm2; and Cj 10" (Nm2)"1 , see fig. 4.2.a.
Fig. 4.2 Fotograph of water tower in Amsterdam (a) and of scale model (b)
A (scale) model of this tower has been made, see fig. 4.2.b. However the outer tank radius, support-length and empty tank mass have been chosen such that the ratio of the lowest liquid natural frequency to the empty tower natural frequency is about 0.5. This particular choice has been made for two reasons. First the chosen frequency ratio is familiar in
- 38 -
existing towers and second we could measure potential coupling phenomena. The empty tank mass has been overscaled in the model for technical reasons. The data of the model are:a = 0.23 m; k = 0.10; h R = 0.19 m; m R = 19 kg; A = 3.14 10-" ms; JR ~ o . 5 kgm*; CI=i;3.3 10"4 (Nm*)-i and 7 = 1.20 m The support was made of steel, the basin of perspex.
The scale model has been excited at its base by small lateral excitations, without rotation (W, = 0). Dynamic response to harmonic as well as stochastic inputs have been measured using X-Y recording and correlation equipment. Responses have been measured for h/a = 0 , 1/2, 1/3 and 1. The measurements have been reported by Dieterman [1974].
The amplification factor of the amplitude of the tank-movement with respect to the base-movement has been plotted in fig. 4.3 for h/a = 0, 1/2 and 1. Both theoretical and measured values have been represented.
We will discuss the cases h/a = 0 , 1/2 and 1 respectively. 10'
103
10'
10 '
.01
M i = 0.5
*.1 -11 = 0.5
r = 0
h/a 0
0.5 1
measurements O A
a 1 H 1 1 1 1 1 h"1 1 1 1 1 r
0 0.4 0.8 1.2 1.6 2 2.4
Fig. 4.3 Theoretical and measured amplification ratio of tank deflection and base deflection versus frequency of scale model for h/a = 0, 0.5 and 1
1 2.8
flHz)
4.3.2.1 The case h/a = 0 (no liquid)
In the frequency band of interest the theoretical model gives one resonant frequency at f2 = 2.0 Hz (f, will be defined in the following). The theoretical and measured response are in good agreement except for the amplification at the resonant frequency. This is rational since
- 39 -
damping has not been considered yet. In the following we will call f2 the structural resonant frequency, also when it shifts due to the liquid in the tank for h/a > 0. The frequencies f, and f3 are additional resonant frequencies due to liquid-structure interaction.
4.3.2.2 The case h/a = 1/2
In the frequency band under consideration the theoretical model gives in addition to the (shifted) structural resonant frequency f2= 1.95 Hz, additional resonant frequencies at f, = 1.02 Hz, f3= 2.5 Hz, f4 = 3.03 Hz and at f5 = 3.55 Hz. The last two ones have not been plotted, because no amplification was measured at those frequencies. The effect of the higher liquid modes is negligible, as shown in chapter 2. In the following the theoretical response will therefore be calculated by including the two lowest liquid modes only. The absence of the peaks in the measurements at these frequencies can be explained by liquid damping.
In the frequency band up to and including f2 good agreement is shown between theory and measurements, except for the amplification at f, and f2. In the frequency band 2.1 - 3.0 Hz (and also around f4 and f5) differences occur between the theoretical and measured amplifications. The explanation can be found in damping. Since no amplification has been measured at f4 and f9 and measured amplification is relatively small at f3 we mainly confine ourselves to the frequency band containing f, and
Special attention is asked for the dip-frequency fS = 1.18 Hz between f, and f2 as well as f,* = 2.38 Hz between f2 and f3. Calculating the two lowest liquid natural frequencies using (2.14) and table 2.1 we find (O/2TT = 1.20 Hz and o),/2Tr = 2.39 Hz. So the two lowest liquid natural frequencies are located near the dips. At the lowest dip-frequency fj, we observed that tower and liquid-movement were in antiphase, resulting in a stationary tank and lateral moving support base.
4.3.2.3 The case h/a = 1
The structural resonant frequency has been shifted to f2 = 1.83 Hz and f, to 0.99 Hz. The dip-frequency f is located at 1.32 Hz, i.e. nearby the present location of the lowest liquid natural frequency, which is shifted to 1.38 Hz. The value of ft is still located near 2.38 Hz. Remarks made in the case h/a = 1/2 are also valid now.
The two lowest resonant frequencies derived from a double spring mass, as described in the introduction of part I, using the parameter values following from (2.29) and (2.37): (cs = /'/3EI = 1.9 10"4 m/N -compliance support; mR + m v = 37.2 kg; c 0 = 9.15 10"2m/N; m Q = 14.7 kg)
- 40 -
leads to f, = 0.5 Hz and f2 = 2.4 Hz, so differences up to 50% of the correct value.
We notice the decreasing measured response at fa and f3 and the increasing response at f,. The maximum amplification of the filled tower is smaller than it is of the empty tower. This phenomenon will be studied more extensively in the next chapter.
4.3.3 Comparison to resonant frequency measurement of steel water tower
A steel water tower discussed by van Uchelen [2.1970] is of the type studied. This water tower is located at the campus of the Technical University of Eindhoven.
Fig. 4.4 Fotograph of water tower in Eindhoven
The data of the tower are: a = 3 m; k = 0.23; h R = 2.5 m; m R ~ 2 104 kg A = 4.4 10~2m2; JR ~ 2 105 kg; Cj ~ 4.4 10"10 (Nm2)-1 and / ~ 15 m. The structural resonant frequency in the empty case (h/a = 0), following from our model, is f2 = 1.19 Hz while 1.17 Hz was measured.
The tank of this tower has partly been sector-compartmented though the sectors are mutually in open connection. So it is a damping device as well. Due to the sector-compartmentation the first liquid natural frequency increases. A rough estimate of the shift based on experimental data found in Abramson [2.1966] is 20%. With this datum we find for f2 = 0.7 Hz while 0.67 Hz was measured, see van Uchelen [2.1970]. A double spring mass model would predict 1 Hz. The amplification at the
- 41 -
resonant frequency f, is nearly suppressed due to the damping action of the open sectors.
4.3.4 Comparison to observations of water towers of the classic type
At the end we introduce a qualitative confirmation of our results. Carder [1936] has compiled observations of vibrations of thirty-seven steel water towers. The towers are classic i.e. they are supported by spatial truss structures. These structures are weak by shear deflection in the same order of magnitude as modern towers are weak by bending. So the location of the additional and structural resonant frequencies are more or less the same. The observed vibrations were, see Carder [1936] : {1} The socalled fundamental translatory motion with a period of about
1.6 s (0.67 Hz). This resonant frequency decreases as h/a increases. Tower and water oscillate as an elastic unit. Wind is the motivating force.
{2} A long wave, having a period of about 3 s (0.33 Hz) . Origin is in question, appears when a strong wind is blowing.
{3} Short riders having a period of about 0.38 s. It is a torsional vibration.
The vibration {1} is the equivalent of the structural resonant frequency f2 in our model. The vibration {2} is the additional resonant frequnency fi in our model. The tankdiameter of the observed towers was about 6 m, hence the lowest natural frequency of the liquid follows from (2.14) and Xoa = Ho ~ 1.84 and is 0.38 Hz. This explains the observed value of 0.33 Hz. The vibration {3} has not been observed at modern towers. This is explained by the relatively small polar moment of inertia of the support of these towers. The torsional natural frequency of the tower, discussed by Steenbergen and Den Hond [2.1968], calculated with a result from Den Hartog [1956], is f ~0.01 Hz (Relevant tower data: ksnaft ~ 3 . 8 10" N/m Jp, tank - 4 106 kg2)
4.4 Discussion
A lumped model has been derived, of which the parameters are analytically determined. The mathematical description of the model represents the dynamic behaviour of water towers of the type under consideration in the frequency band of interest. The model has been verified by measurements at a scale model, and at existing water towers.
Double spring-mass and pendulum models as discussed by Van Uchelen [1.1970] roughly explain the general features of the dynamic behaviour of a water tower, but numerical values derived from these models are unreliable.
- 42 -
In the water tower model derived, damping can easily be incorporated. The model developed could be used in analyzing the dynamic be
haviour of other liquid containing structures such as chemical plants. The dynamics of slender structures can also be studied by the model derived. This will be shown in chapter 7.
The model can further be used to study structure-foundation interaction and liquid-structure-foundation interaction as will be shown in the chapters 7 and 8, respectively.
4.5 References
CARDER D.S.,"Observed Vibrations of Steel Water Towers", Bull. Seism. SOC. Am., 26, 1936, 69-88.
DEN HARTOG J.P.,"Mechanical Vibrations", New York.McGraw Hill,1956, 430. DIETERMAN H.A.,"The Dynamic Behaviour of a Water Tower Excited Harmonically and Stochastically"; Mast.Thes., Techn. Univ. Delft, 1974, (in Dutch).
- 43 -
CHAPTER 5: DAMPING AND STRUCTURAL CONTROL
5.1 Introduction
In the preceding chapters we found that sloshing of the liquid in the tank of a tower results in additional resonant frequencies next to the structural ones. Excitation of the structure at one of the resonant frequencies may be critical to the stability and structural integrity of the tower, when the damping in the system is small.
In the modelling of the dynamic behaviour of the tower the effect of damping has been neglected sofar, as shown by the infinite amplification at the natural frequencies of the theoretical model in fig. 4.3. The measurements at the scale model as presented in fig. 4.3 show finite amplitudes at the resonant frequencies. More specific, the maximum amplification at the lowest structural resonant frequency decreases at increasing liquid height tank-radius ratio (h/a), where the amplification at the additional resonant frequencies increases. In order to study these phenomena the damping will be incorporated in the model.
The structural damping is small, since towers are mostly made of steel or concrete. Damping ratios, defined as percentage of the critical damping of an equivalent spring mass-system, are smaller than 1%, resulting in large amplification at the resonant frequencies of the tower. So every additional dissipation of mechanical energy or tranfer of energy to other modes is welcome. In this respect the liquid in the tank and the foundation are of special interest. The damping contribution of the liquid will be studied now, while the damping contribution of the foundation will be studied in the chapters 6 and 7.
The sloshing of liquid as well as compartmentation of the tank can be used to reduce the structural amplification. By the sloshing liquid mechanical energy will be dissipated. By the compartmentation the liquid motion can be phased and energy can be transferred to other (liquid) modes.
The liquid damping in the tank originates from the free surface boundary layer (small tanks), fluid viscosity, turbulence, boundary layer friction and interchange of energy between different sloshing modes. Some of these mechanisms are essentially non-linear.
Only a few theories are available for the determination of damping in tanks analytically, see Miles [1956], [1958], [1961], Case and Parkinson [1957] and Nelson [1960]. The principal knowledge of damping in tanks results from extensive experimental studies. Since the resonance of liquid in fuel tanks of space vehicles causes serious stability and integrity problems, an extensive theoretical and experimental survey in the field of liquid damping in tanks has been executed in space vehicle research, see Silverman and Abramson [1966]. However the experimental studies have been concerned with tanks of rela-
- 44 -
tive small diameter, so the conclusions drawn from these experiments do not apply directly to tanks of water towers.
The liquid damping in an unbaffled tank of dimensions as usual for water towers is small. It is therefore necessary to consider means to provide increasing damping of the liquid motion, by devices such as movable or floating cans or mats, diaphragms, fixed solid/perforated stiff or elastic baffles. Some baffle types are shown in fig. 5.1. For water towers
Annular ring baffle in cylindrical tank
Vertical baffles in cylindrical tank
Fig. 5.1
the ring baffle is of special interest, because of the substantial damping increase at relatively low weight penalties.
Damping is usually introduced linearly by adding viscous dampers to a lumped representation of the liquid behaviour as derived in chapter 2. This way of introduction assumes that the dynamic behaviour of each of the liquid modes can be described by a linear, viscously damped spring-mass-system. This is a good approximation in the lowest liquid mode. For higher modes this may not apply, however whatever the nature of the damping is, it can be represented by an equivalent viscous damper.
In the following the effects of ring baffles and ring- and sector compartmentation will be discussed.
Using a linear model it will be shown that adjustment of the depth of the liquid and placing of ring baffles on the tank walls near the free liquid surface result in passive structural control of the tower. In fact an optimal energy transfer in the structure-liquid system will result in a relatively large dissipation of the mechanical energy around the baffles in the liquid.
5.2 Structural damping
The structural damping in the theoretical model will be located in the support, since the tank has been assumed as rigid. The nature of damping may be dry friction, viscous friction, radiation, hysteresis. The constitutive relations may be very complex and nonlinear. The material damping of steel and concrete is predominantly of hysteretic nature and
- 45 -
can be easily included in the lumped model (3.4) of the support by the introduction of a complex stiffness or complex compliance:
{Ei(i + i-yr1 (5.1)
In fact one assumes that the coefficient of damping of an equivalent viscous damper is frequency dependent and that its value is proportional to the stiffness per frequency. Substitution of (5.1) in (3.4) results in a damped support model valid in the low frequency band. Substitution of (5.1) in (4.5) results in an extended transmission function for the water tower excited laterally at a non-rotating base:
' 2 =
l + i iClCjlH^n + Y12 + Y2i + Y 2 2 ) + r y h ^ V _ T 27 "7*' n
(5.2)
The function (5.2) can be checked by the "common sense" cases y. - and y. 0, resulting in proper transmissions.
The modulus of the complex transmission function (5.2) can be easily determined.
Fig. 5.2 Dynamic behaviour of empty water tower model, including structural (hysteretic) damping (theory vs. experiment)
The transmission function (x-j/x,) measured at the scale-model for h/a (empty) and the modulus of (5.2) coincide very well for "V ~ 0.015, representing the hysteretic damping. See fig. 5.2.
- 46 -
Determination of an equivalent viscous damping ratio is rather arbitrary, because the system has to be simplified to a second order system, of which the stiffness and mass can be estimated as a ratio only. With the damping ratio y = b/b , in which b represents the damping
v cr coefficient and b is the critical damping, a value of Y. ~ 0.004 can cr v be derived.
5.3 Liquid damping
5.3.1 Damping of liquid oscillations in unbaffled cylindrical tanks
The viscous damping of liquid in an unbaffled non-annular circular cylindrical tank, denoted as cylindrical in the following, has been evaluated on a mixed theoretical and experimental basis by a number of researchers independently, see Miles [1956], Case and Parkinson [1957] and Mikishev and Dorozhkin [1963] . The effects of the liquid depth, oscillation amplitude, kinematic viscosity and surface tension on the damping were specifically investigated by the last contributors. Their result for the logaritmic decrement 8 of the amplitude of the free surface oscillation in the lowest liquid mode can be written as:
_. 1/2 -3/4 -1/4 r 0.318 I (* ~ a) .11 ,, , 8 = 4.98 v a g [ l + --ox { + l)J (5.3) sinh(1.84g) cosh (1.84 5)
where v is the kinematic viscosity. For h/a ;> 1 the term between the square brackets approximates to one. Results equivalent to (5.3) were derived by the other contributors mentioned.
For linear second order systems the damping ratio y can be determined from the logaritmic decrement 8, since they are related by:
-y = 8/(2ir) (5.4)
Substitution of practical values considering water in a tank with a radius of a few meters results in very small damping. The contribution (5.4) will be incorporated in the model for an unbaffled tank.
5.3.2 Damping of liquid oscillations in baffled cylindrical tanks
Extensive studies of damping devices in liquid containing tanks have been executed in space vehicle research. The damping contribution of devices as moveable and floating cans and mats, diaphragms, fixed solid/perforated stiff or elastic baffles of cruciform and/or ring type have been investigated. The fixed ring baffles are of special interest for water towers, because of the simple technical construction and the
- 47 -
relatively large damping they cause at relatively low weight penalties. The ring baffles concern both symmetrical and asymmetrical ring segments. We will pay attention to the first type only. Cruciform baffles result in a relatively smaller energy loss than ring baffles, see fig. 5.3. They will not be considered here.
O I I I
I View A-A View B-B Fig. 5.3 Cruciform (left) and ring (right) baffle configurations in a circular
cylinder
The damping caused by ring baffles in liquid-filled cylindrical tanks has been studied theoretically by Miles [1958] and his results have been extended by O'Neill [1960] and modified by Bauer [1962] . The liquid, oscillating in its lowest mode, produces a wave having its maximum amplitude at the wall. The direction of the flow in the vicinity of the wall is essentially vertical, and thus normal to an annular ring located at a small distance below the free surface, see fig. 5.4. Energy dissipation results from wave motion being opposed by the ring pressure C_.pv2/2, where v is the local wave velocity producing the pressure and C_ is the local drag coefficient.
Assuming that (a) the liquid is oscillating in its fundamental mode and linearized potential theory accurately describes the flow except in a small region near the ring; (b) The local flow in the region of the ring is unaffected by the free surface or the tank bottom; (c) The natural frequency is accurately approximated by the potential flow solution for the undamped case; (d) the drag coefficient is independent of the angular coordinate, Miles showed that the damping ratio in the lowest mode for the ring damping in a cylindrical tank can be expressed by:
0.473 e C!| CD (5.5)
where d , a and C, are related to the tank geometry as shown in fig. 5.4
and z represents the oscillation amplitude. The damping is still a
- 48 -
function of the unknown z/a, but variations of this parameter will be studied.
JL z T
~* w r
1
l .-j
T
J Tank area: Tra2 Annular ring area: Cv&' Fig. 5.4 Annular ring in cylindrical tank; parameters
Miles also presented an expression for the drag coefficient, based upon certain experimental data from Keulegan and Carpenter [1956] as:
,UmT\-1/2 {2 < ffiT < 20} (5.6)
where U is the time-wise maximum velocity, T the period and D the plate width. For large liquid depths, h/a > 2, and for the dominant sloshing mode in a cylindrical tank the expression for Cn can be rewritten (employing the present notation) as, see Miles [1956]:
Cn = 15 [ ?1 e' C, (5.7)
Substitution of (5.7) into (5.5) yields the damping ratio:
-4.6^ 3/2 ,z l'2 y= 2.83 e a Cj (f) (5.8)
Using this expression the damping ratio of a baffled tank can be approximated. It should further be noted that the value of 7 in (5.8) is very sensitive to the location of the ring baffle with respect to the free surface represented by the parameter d /a.
Bauer [1962] extended the results of Miles by considering a tank having a number of ring baffles. The effects of various baffle spacing have been studied, showing the dependance of the liquid damping on the spacing. He further accounted for the condition during sloshing where part of the baffle emerges from the liquid. It results in a lower damping ratio for small values of ds/a. Some results are shown in fig. 5.5.
- 49 -
Decreasing liquid level *" d, /a
Damping factor for a series of ring
Recommended
