RESPONSE OF A FLEXIBLE MARINE COLUMN TO BASE EXCITATION by Thomas A. Vernon B.A.Sc, The University of B r i t i s h Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1984 © Thomas A. Vernon, 1984 In presenting this thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that Library s h a l l make i t freely available for reference and study. the I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my Department of C i v i l Engineering The University of B r i t i s h Columbia 2324 Main Mall Vancouver, B.C. V6T 1W5 written permission. ii ABSTRACT The displacement response of a f l e x i b l e , surface-piercing cylinder subjected to a u n i d i r e c t i o n a l base motion i s considered i n this study. Laboratory experiments have been performed with a c i r c u l a r , fixed-base model using sinusoidal and scaled seismic input motion. Sinusoidal tests were designed to investigate the dependence of cylinder t i p response on the r a t i o of base motion frequency to cylinder natural frequency and base displacement amplitude, for a fixed water depth, i n e r t i a r a t i o and damping r a t i o . Further tests with a base motion corresponding to past earthquake records were then used to determine the cylinder's response to seismic e x c i t a t i o n . The sinusoidal test results are compared with predictions derived from analyses of the motion i n terms of the f i r s t and second undamped mode shapes of a cantilever beam. The Morison equation i s used to estimate hydrodynamic loads i n this formulation, and three treatments of the drag term i n the equation of motion are considered: neglect of drag, drag l i n e a r i z a t i o n and retention of the complete nonlinear form. A closed-form solution for the former two approximations i s developed, and a numerical approach i s adopted for the complete nonlinear formulation. The numerical method i s used to predict the response of the column to seismic input. The dependence of cylinder t i p displacement on frequency r a t i o and base motion amplitude follows predictable patterns of dynamic response. Peak amplitudes occur at resonance and increase with base motion amplitudes. However, this relationship i s not linear because of the damping contribution of nonlinear drag forces near the free surface. iii The numerical and l i n e a r i z e d drag predictions agree well with the experimental response i f a suitable choice of drag c o e f f i c i e n t i s made. The neglect of drag r e s u l t s i n very conservative resonant response predictions for large excitation amplitudes i n which the free surface Keulegan-Carpenter number exceeds about 4. Drag forces can generally be neglected, however, i n the estimation of response to earthquake motion because displacement amplitudes are small. In extreme cases, small l i f t forces can result from flow separation about the cylinder near the free surface. iv TABLE OF CONTENTS PAGE ABSTRACT i i LIST OF FIGURES vi LIST OF TABLES viii ACKNOWLEDGEMENTS 1. ix INTRODUCTION 1 1.1 Introduction 1 1.2 Background 2 1.3 Literature Review 6 1.3.1 6 Earthquake Response of Structures i n Fluids .. 1.3.2. Response i n O s c i l l a t i n g Flows 7 1.3.3. Hydrodynamic Drag and Damping 9 1.3.4. Related Topics 2. 3. 10 THEORETICAL DEVELOPMENT 12 2.1. Problem D e f i n i t i o n 12 2.2. Dimensional Analysis 13 2.3. Hydrodynamic Force Formulation 16 2.4. Modal Analysis 17 2.4.1. Neglect of Drag Forces 18 2.4.2. Linearized Drag 24 2.4.3. Numerical Method with Nonlinear Drag 28 TESTING FACILITIES AND MODEL PARAMETERS 32 3.1. Testing F a c i l i t i e s 32 3.2. Model Parameters 33 3.3. Data Acquisition 35 V PAGE 4. 5. 6. DESCRIPTION OF EXPERIMENTS 36 4.1. Damping Tests and System Characteristics 36 4.2. Sinusoidal Tests 37 4.3. Seismic Motion Tests 39 DISCUSSION OF RESULTS 41 5.1. Response Functions from Sinusoidal Tests 41 5.2. Response Predictions for Sinusoidal Motion 42 5.3. Transverse Response 45 5.4. Response to Seismic Input 46 5.5. Numerical Prediction of Seismic Response 48 CONCLUSIONS AND RECOMMENDATIONS 49 6.1. Conclusions 49 6.2. Recommendations for Further Study 51 BIBLIOGRAPHY 72 APPENDIX A - MODAL ANALYSIS 76 APPENDIX B - DERIVATION OF RESPONSE AMPLITUDES 82 APPENDIX C - DESCRIPTION OF COMPUTER PROGRAM 87 APPENDIX D - INSTRUMENT CALIBRATION AND TYPICAL DATA 89 vi LIST OF FIGURES FIGURE PAGE 1 Coordinate System D e f i n i t i o n 52 2 Mode Shapes for a Cantilever Beam 52 3 Schematic of Test F a c i l i t y 53 4 Shake Table 54 5 Testing Tank Over Shake Table 54 6 Instrumented Rod for Strain Measurement 55 7 Photo of Instrumented Rod 55 8 Free Vibration Damping i n A i r 56 9 Free Vibration Damping i n Water 56 10 Photo of Sinusoidal Test of 4.5 Hz 57 11 Photo of Sinusoidal Test at 28 Hz 57 12 Fourier Spectra for E l Centro 1940 58 13 Fourier Spectra for San Fernando 1971 58 14 Response Function for Y /D = .020 59 15 Response Function for Y /D = .036 60 16 61 18 Response Function f o r Y /D = .054 g Response Function f o r Y /D = .072 g Drag Coefficient versus Reynolds Number f o r Various Values of K 19 Ratio of Y 17 g , ,Y at Resonance versus Drag Coefficient at pred/ exp Various Base Displacement Amplitudes 62 65 65 20 Numerical Simulation of the Formation of Asymmetric Vortices ... 66 21 In-line and Transverse Response Near Resonance f o r Y /D = .054 . 66 vii PAGE FIGURE 22 Transverse Response Functions for Various Base Displacement Amplitudes 23 67 Tip Locus at Resonance for Y /D = .054 68 8 24 Tip Locus o f f Resonance for Y /D = .054 g 68 25 Response comparison for E l Centro 69 26 Response comparison for San Fernando 69 27 Transverse Time History f o r San Fernando 70 28 Response Prediction Comparison for E l Centro 70 29 Response Prediction Comparison f o r San Fernando 71 30 Calibration Curve for Instrumented Rod 90 31 Typical Time History at 3rd Subharmonic 91 32 Fourier Spectra for Record of Figure 31 91 33 In-line Resonance Time History at Y /d = .036 g Transverse Resonance Time History at Y /d = .054 92 34 92 viii LIST OF TABLES TABLE 1 PAGE Resonant Response Ratios Y ,/Y pred exp Experimental Data 63 r 2 > 64 ix ACKNOWLEDGEMENTS I wish to express many thanks to Dr. M. Isaacson and Dr. S. Cherry for their guidance and support i n the preparation of t h i s t h e s i s , and to Chris Dumont for his help i n the Earthquake Laboratory. 1 1. 1.1 INTRODUCTION Introduction The p r e d i c t i o n of s t r u c t u r a l response to hydrodynamic loads i s of major importance i n the design of many marine f a c i l i t i e s . As a contribution to t h i s general subject, t h i s thesis endeavours to Investigate methods of response prediction applicable to compliant marine columns undergoing base motion. Hydrodynamic forces are generated when r e l a t i v e motion e x i s t s between the f l u i d and structure such as occurs as a r e s u l t of wave and/or current a c t i o n or seismic e x c i t a t i o n of the structure. In the case of a compliant marine structure, the f l u i d - s t r u c t u r e system becomes i n t e r a c t i v e since the response and applied forces are then coupled. the available damping i s low, dynamic amplification at resonance become important. If may An investigation of a quasi-resonance phenomena i s of merit i n the case of seismic excitation because system damping i s often low (<10% c r i t i c a l ) and natural frequencies of larger structures often tend to coincide with earthquake spectral density peaks (1-5 Hz). This proximity does not generally occur i n the case of wave loading because of the lower frequencies associated with large ocean waves. The hydrodynamic forces generated by the motion of a body i n a s t i l l f l u i d result from i n e r t i a forces associated with the acceleration of a volume of f l u i d , and from drag forces associated with flow separation. For small r e l a t i v e motions t y p i c a l of earthquake ground motions, these forces w i l l be mainly i n e r t i a l . For f l e x i b l e structures, dynamic amplification enhances the p o s s i b i l i t y of larger displacements leading to flow separation. Where separation occurs, form drag, vortex formation 2 and possibly l i f t forces can r e s u l t . In the case of base motion, form drag i s energy d i s s i p a t i v e , and i s treated as an added damping. However, the associated l i f t forces may synchronize with the natural frequency of the structure and i n extreme cases, lead to large transverse oscillations. Such a mechanism has been the cause of f a i l u r e of numerous free-standing p i l e s i n waves. For purposes of t h i s t h e s i s , a r e l a t i v e l y simple system i s considered: a surface-piercing c i r c u l a r c y l i n d r i c a l structure i s excited with u n i d i r e c t i o n a l , horizontal base motion. The i n - l i n e and transverse responses to a range of input motion amplitudes and frequencies have been recorded and the i n - l i n e responses compared to predictions based on a modal analysis assuming both linear and nonlinear hydrodynamic Although several powerful methods, such as lumped-parameter loading. modelling and f i n i t e element methods, exist for the treatment of this problem, the use of modal techniques for the continuous structure provides a convenient and straightforward approach without the need to resort to the extensive computing or modelling demanded by other methods. (See K i r k l e y and Murtha, 1975, for the former; Liaw and Chopra, 1973, for the l a t t e r ) . Transverse responses are not compared to any numerical predictions because a simple predictive mathematical model of l i f t generally a v a i l a b l e at t h i s time. force i s not It must be noted, however, that transverse o s c i l l a t i o n s can fundamentally affect the near resonant response of a marine p i l e or tower. 1.2 Background Fluid-structure i n t e r a c t i o n i n the marine environment i s generally assumed to occur i n either a separated or unseparated (potential) flow regime. Viscous e f f e c t s can be included i n the former, but must be 3 neglected In p o t e n t i a l flow solutions. assumption The determining factor i n the of an appropriate flow regime i s the magnitude of r e l a t i v e displacement between the structure and f l u i d , often expressed i n terms of the Keulegan-Carpenter number, K. This i s defined, for a two-dimensional sinusoidal o s c i l l a t i o n of a c i r c u l a r cylinder, as (Keulegan, Carpenter (1958)). K where A amplitude of o s c i l l a t i o n displacement D cylinder diameter The Keulegan-Carpenter separation i n the problem. number indicates the significance of flow For K values of roughly 10 or more, the o s c i l l a t i o n amplitude to cylinder diameter r a t i o w i l l be r e l a t i v e l y large, and the hydrodynamic loads are primarily form drag associated with the flow separation. small body regime. The problem i s then defined as lying within the On the other hand, i f K i s small, flow separation does not generally occur, i n e r t i a forces are dominant and potential flow theory can provide a good flow f i e l d aproximation. Overlaps of these c l a s s i f i c a t i o n s c e r t a i n l y exist (for a f u l l e r account see Sarpkaya and Isaacson, 1981), and a p a r t i c u l a r problem may encompass several flow classes. The s o l u t i o n approach d i f f e r s i n the case of large and small body problems. The former i s generally formulated using potential flow theory for an i n v i s c i d f l u i d and an i r r o t a t i o n a l flow f i e l d . The v e l o c i t y of any point i s specified as the gradient of a scalar potential function which s a t i s f i e s the Laplace equation and appropriate boundary conditions. 4 The pressure may be obtained from the p o t e n t i a l using the unsteady Bernoulli equation and thus once the potential function has been determined, the hydrodynamic loads may be calculated from an integration of the pressure around the body. the determination This type of approach i s applicable to of wave or seismic loading of large structures such as o i l storage tanks or gravity platform bases where A/D i n (1.1) i s small. Local flow separation due to sharp geometries can occur on large structures and leads to inaccuracies i n solutions obtained flow assumptions. from potential The s i g n i f i c a n c e of these errors should be given consideration i n such cases. In the small body regime, viscous e f f e c t s are included i n the hydrodynamic force derivation. Experimental evidence suggests that viscous forces r e s u l t i n g from flow separation can become important at Keulegan-Carpenter numbers greater than about 2 (A/D o s c i l l a t i n g flows about b l u f f bodies. can occur for K > 5 (A/D > .7). - 0.3 i n (1.1)) for Vortex shedding and l i f t forces Although seismic base motion amplitudes are u s u a l l y small such that K < 5 over most portions of a structure, the viscous forces may become s i g n i f i c a n t near the free surface. viscous e f f e c t s , the hydrodynamic loads are u s u a l l y formulated To include using the well known Morison equation, deriving from the work of Morison, et a l (1950). This equation i s based on the simultaneous addition of an i n e r t i a force associated with i n v i s c i d f l u i d acceleration and a drag force analogous to that i n a steady flow, and i s given i n simplest form for a stationary c i r c u l a r cylinder as: F' = C m DO (1.2) 5 Here, F' i s the f l u i d force per unit length, p i s the f l u i d density, D i s the body diameter, U and U are the f l u i d v e l o c i t y and acceleration respectively (using the dot notation to imply time d i f f e r e n t i a t i o n ) , and C ffl and Cj are i n e r t i a and drag c o e f f i c i e n t s r e s p e c t i v e l y . The n o n l i n e a r i t y of the forcing function i s apparent i n the drag term. The force c o e f f i c i e n t s C,, C depend on the dimensionless flow and structure d m parameters such as l o c a l Reynolds number, Keulegan-Carpenter number and surface roughness. These c o e f f i c i e n t s are not fundamental constants and must be determined experimentally. The problem considered i n t h i s study encompasses both flow regimes. Near the structure base, the displacement i s e s s e n t i a l l y the base motion input. In the case of seismic input these displacements are generally less than 20 cm and for an average size p i l e or platform l e g , the maximum Keulegan-Carpenter number would then be roughly 0.5 or l e s s . However, many marine structures are highly dynamic, with fundamental frequencies of o s c i l l a t i o n t y p i c a l l y i n the range 0.2 to 1.0 Hz, a range of frequency also t y p i c a l of earthquake spectral i n t e n s i t y maxima (See Figures 12 and 13). The proximity of these frequencies leads to the possible quasi- resonance condition commonly associated with seismic excitation of large structures. Because dynamic amplification at resonance can be of order 10 or more, a Keulegan-Carpenter number of 5 can be attained near the free surface. lift This value indicates that flow separation and possibly forces may become important i n the problem. A bottom fixed marine structure can consequently span both flow regimes: a potential flow near the base (and nodes If we consider the response to be modal) and a separated flow i n regions of large s t r u c t u r a l displacements. The present study uses the Morison equation to estimate 6 hydro-dynamic loads, since that formulation w i l l e s s e n t i a l l y be v a l i d i n the case of small structures, for the entire span, aside from effects of wave generation near the free surface. 1.3 L i t e r a t u r e Review A number of more recent investigations of structure response to hydrodynamic loads are relevant to the present study. divided here into several broad categories: These have been response to seismic loading, response i n o s c i l l a t i n g flows, damping i n f l u i d s and a d d i t i o n a l miscellaneous t o p i c s . 1.3.1. Earthquake Response of Structures i n Fluids Clough (1960) performed v i b r a t i o n tests of f l e x i b l y mounted c y l i n d r i c a l shapes i n a i r and water. The study determined added mass and damping values f o r f i r s t and second mode natural frequencies using the different frequencies of o s c i l l a t i o n i n the two mediums. Clough found agreement between the experimental values of added mass and those derived from potential flow theory. continuous The response of a pseudo- structure subjected to base loading was a l s o investigated. However, the applied displacement i n these tests was generated by a pendulum s t r i k i n g the edge of a table and d i d not simulate r e a l i s t i c seismic input. Clough concluded that structural o s c i l l a t i o n s r e s u l t i n g from seismic e x c i t a t i o n would not be large enough to cause flow separation. If this conclusion i s accepted, potential flow theory can be used to predict marine structure response during earthquakes. A comprehensive a n a l y t i c a l study of the response of e l a s t i c structures surrounded by a f l u i d using p o t e n t i a l flow theory was 7 presented by Liaw and Chopra (1973). The authors u t i l i z e d a modal analysis technique i n combination with a f i n i t e element method to predict s t r u c t u r a l response. No experimental v e r i f i c a t i o n s were given i n t h i s study. Much of the work on the seismic response of marine structures has been concerned with large, non-elastic structures such as submerged o i l tanks. In these cases, most authors use a p o t e n t i a l flow formulation similar to that used for load prediction i n the wave d i f f r a c t i o n regime. Such studies include those by Byrd (1978), Tung (1979), Mei (1979), Westermo (1980), and Isaacson (1983). A review of many of the methods used by the above authors, as well as others, i s presented by Eatock Taylor (1981), for the case of dams and offshore structures. Kirkley and Murtha (1975) have studied the response of offshore structures i n earthquakes, comparing several techniques of response prediction using lumped-parameter models. A Morison forcing function was used and a direct integration procedure compared with an uncoupled approximation. Their r e s u l t s indicate that the l i n e a r i z a t i o n technique i s accurate i n higher frequency ranges, but unconservative for low frequencies. The same authors have presented response spectra for offshore structure design based on the numerical solution of the linearized equations ( K i r k l e y and Murtha, 1.3.2. 1975). Response of F l e x i b l e Structures i n O s c i l l a t i n g Flows There exists a very s i g n i f i c a n t volume of l i t e r a t u r e concerning the hydrodynamic loads imposed on structures i n o s c i l l a t i n g flows. However, very l i t t l e of t h i s work a c t u a l l y concerns e l a s t i c structure response to these loads. 8 An e a r l y work by Selna and Cho (1972) presented a numerical integration and time step technique for d i s c r e t i z e d models of marine structures. This approach can incorporate the nonlinear coupled drag c h a r a c t e r i s t i c s , although such direct integration methods are very c o s t l y for more than a few degrees of freedom. Blevins (1977) has considered the i n - l i n e response of continuous structures i n o s c i l l a t o r y flows with zero and nonzero means. He provides a solution based on a Fourier expansion of the linearized equation of motion. An assumption of small structure response i n comparison to the f l u i d motion i s inherent i n this l i n e a r i z a t i o n and solution approach. Such an assumption cannot be made i n an analysis of seismic response and other l i n e a r i z a t i o n techniques must be used. A number of authors have presented numerical methods for the response solution using integration of the motion equations derived for d i s c r e t i z e d models. Anagnostopoulos These include Sawaragi et a l (1977), and (1982), who uses an uncoupled Morison forcing function. These numerical techniques are expensive, but very powerful and the limitations on their accuracy are imposed mainly by uncertainty i n the input data. The transverse response of e l a s t i c a l l y restrained cylinders i n o s c i l l a t o r y flows has been investigated by several authors. Sarpkaya (1979, 1981), has performed extensive experimentation relevant to this subject and presented correlations of transverse response with system parameters such as reduced v e l o c i t y and Keulegan-Carpenter number. Similar correlations were presented by Isaacson and Maull (198l) for linear mode models i n progressive waves. Zedan and Yeung (1980) investigated p i l e dynamics under lock-on conditions i n waves at moderate 9 Keulegan-Carpenter numbers. Lock-on was found to occur over only a narrow bandwidth and to be highly interactive with the i n - l i n e response. McConnel and Park (1982) o s c i l l a t e d e l a s t i c a l l y restrained cylinders i n a s t i l l f l u i d and concluded that the frequency r a t i o parameter i s more of a c o n t r o l l i n g factor i n transverse response than K or reduced v e l o c i t y as suggested by other authors. Skop and G r i f f i n (1975) presented an a n a l y t i c a l model of the fluctuating l i f t force c o e f f i c i e n t and used this model i n conjunction with modal superposition to obtain transverse response p r e d i c t i o n s . The model predictions were compared to experiments and the results follow at least similar trends. 1.3.3. Hydrodynamic Drag and Damping The drag forces acting on marine structures are nonlinear and, i n a dynamic system, coupled to the structure response. Because of this coupling the drag force can be expressed i n terms of an exciting force, associated with the incident flow and a damping force associated with the s t r u c t u r a l motion. The estimation of such damping forces can be very important when resonant response i s a p o s s i b i l i t y . Several authors have considered the treatment of these exciting forces and damping terms. Dao and Penzien (1980) have presented a comprehensive comparison of methods of handling nonlinear drag forces using d i s c r e t i z e d models. methods again involve assumptions These on the r e l a t i v e magnitudes of the displacements (uncoupling) or i n some cases assume harmonic response. The l i n e a r i z a t i o n techniques are modified to give an approximation of the response to non-harmonic motion. Moe and Verley (1980) have considered the hydrodynamic damping of offshore structures i n various flow situations. They conclude that the 10 use of the Morison equation may give unconservative response predictions because the normally assumed values of drag c o e f f i c i e n t s are too high. This i s a r e s u l t of an assumption of separated flow inherent i n the Morison approach. Separated flow does not occur where motion amplitude to diameter r a t i o s are small and i n such cases drag damping i s overestimated by a choice of drag c o e f f i c i e n t based on that assumption. Sugiyama and Ito (1981) have considered the treatment of nonlinear drag damping as a function of vibration amplitude for a single degree of freedom system. A polynomial function of Reynolds number was found appropriate for the drag c o e f f i c i e n t . The function was determined empirically from experiments. 1.2.4. Miscellaneous Related Topics Two related areas of investigation are the prediction of marine r i s e r response, and the non-deterministic approach to offshore structure response. Kirk, et a l (1979) have developed a normal mode solution i n the frequency domain for a marine r i s e r subjected to wave loading and top e x c i t a t i o n from horizontal motions of the support platform. nonlinear drag term was conditions. The found to be very important at resonance A dynamic analysis of a multi-tube r i s e r has been given by Grecco and Utt (1982). Chakrabarti and Frampton (1982) have presented a comprehensive review of the many techniques used i n the analysis of marine r i s e r s . These solutions are i n v a r i a b l y based on numerical methods because of the complexity of the governing equations of motion. Non-deterministic methods of response analysis have been presented by Malhotra and Penzien (1970), Penzien and Dao (1980). Other references and a review of the p r i n c i p l e s of non-deterministic analyses can be found in Sarpkaya and Isaacson (1981). 12 2. THEORETICAL DEVELOPMENT The purpose of this chapter i s to formalize the problem under consideration and to develop the methods of response p r e d i c t i o n used i n this study. A dimensional analysis i s presented which defines the relevant parameters i n the problem. The modal analysis method of response prediction i s developed for three forms of the hydrodynamic drag force i n the equation of motion; forces. zero, l i n e a r i z e d and nonlinear drag A closed-form solution i s obtained for the zero and l i n e a r i z e d drag cases; a numerical approach i s given for the nonlinear form. 2.1. Problem D e f i n i t i o n This study i s concerned with the general subject of response prediction for submerged f l e x i b l e structures undergoing base motion. As a fundamental case characterizing t h i s problem, a s i n g l e , f l e x i b l e , surface-piercing c i r c u l a r cylinder i s considered. For a constant mass, length and diameter of cylinder, and constant f l u i d properties, the displacement response w i l l be a function of base motion amplitude and frequency. This dependence i s investigated using sinusoidal input. The response of f l e x i b l e columns to earthquakes i s then investigated using scaled seismic inputs. The s o l u t i o n approaches used i n t h i s study necessitate a number of assumptions about the flow and structure c h a r a c t e r i s t i c s i n the problem. These assumptions include the following: 1) The problem exists i n the small body flow regime where the Morison equation can be used to predict hydrodynamic loads. 2) The structure can be considered as a uniform cantilever beam and appropriate beam theory used to develop the equation of motion. The response i s assumed to be primarily i n the f i r s t mode and the undamped mode shapes are assumed to adequately describe the response. 3) The column i s free standing, with no t i p mass and i s v i r t u a l l y submerged. The c r i t i c a l response i s taken as that at the free surface. 4) Seismic base motion amplitude i s t y p i c a l of that recorded on firm ground at moderate distances from the focus of a strong motion earthquake but i s limited to a u n i d i r e c t i o n a l and horizontal component. 5) Base motion i s prescribed, circumventing the problem of s o i l structure Interaction. With these assumptions, response prediction methods are developd using the Morison equation to estimate hydrodynamic loads and a modal analysis to solve the equation of motion. Response predictions are compared to experimental results from model t e s t s . 2.2. Dimensional Analysis Dimensional analysis can be used to define the dimensionless parameters important i n a particular problem. The present study intends to model the displacement response of a f l e x i b l e marine column undergoing base motion. The relevant independent variables are then c h a r a c t e r i s t i c s of the structure, f l u i d and motion. The variables are: D diameter of cylinder f fundamental frequency of structure i n water 14 m structure mass per unit length damping r a t i o i n a i r v kinematic v i s c o s i t y of f l u i d P f l u i d density H depth of f l u i d U v e l o c i t y amplitude at base ( f o r sinusoidal motion) t time f forcing frequency ( f o r sinusoidal motion) Dimensional analysis w i l l provide seven independent dimensionless groups from these ten v a r i a b l e s . Using conventional groups, time- invariant representative values of cylinder t i p response can be written as: Y/D.Z/D = f l ( f / f , C, D/H, m/pD , K, Re) 2 n Here, Y and Z are i n - l i n e and transverse displacement amplitudes respectively, of the column t i p . are The dynamic properties of the structure represented by the non-dimensional groups f/f » the frequency r a t i o , n £, the damping r a t i o , and m/pD , the i n e r t i a r a t i o . 2 The natural frequency and damping r a t i o may be taken as values either i n a i r or water. UD/v, The Keulegan-Carpenter number K = U/fD, and Reynolds number, Re = characterize the motion at the structure base. In general, the a p p l i c a b i l i t y of model test r e s u l t s depends upon the similtude observed between prototype and model values of these dimensionless groups. As i s common i n many f l u i d - s t r u c t u r e modelling 15 problems, Reynolds number similtude Is d i f f i c u l t to maintain and i t s significance i s not investigated i n this study. A constant damping r a t i o i s also d i f f i c u l t to maintain because s t r u c t u r a l damping i s not e a s i l y controlled. However, damping ratios of both model and prototype are expected to be low (< 10% c r i t i c a l ) . For a s p e c i f i c choice of cylinder and water depth, the damping r a t i o , diameter to depth r a t i o D/H, and i n e r t i a r a t i o are constants. The non-dimensional i n - l i n e response investigated here can then be described as: Y/D = f ( f / f , 2 n U/fD) The dependence of the response on the frequency r a t i o r e s u l t s i n a dimensionless transfer type function. The Keulegan-Carpenter number characterizes the response dependence on the amplitude of base motion. Although K i s defined at the structure base, the response dependence on K i s important only at higher values of this parameter such as occurs at the free surface at resonance. The choice of dimensionless groups i s not unique, and i n - l i n e and transverse response can be expressed i n terms of alternate parameters such as the reduced v e l o c i t y U = U/f D = K/(f / f ) , for which ' r n n correlations exist i n the case of r i g i d cylinder response (Sarpkaya, 1979). of K. However, the reduced v e l o c i t y i s more suitable at larger values In his investigations of the hydroelastic o s c i l l a t i o n s of a r i g i d cylinder, Sarpkaya has also included a roughness factor, K /D r and combined the Reynolds and Keulegan-Carpenter numbers as Re/K = D /vT. 2 16 Isaacson & Maull (1981) and Sarpkaya (1979) have expressed transverse displacements of cylinders i n harmonically o s c i l l a t i n g flow as a function of the parameter A = (U/f D) /(m£/pD ). 2 2 n for the case of f l e x i b l e column motion; Similar correlations may exist however, these correlations are much more d i f f i c u l t to obtain experimentally and are not addressed i n this thesis. 2.3 Hydrodynamic Force Formulation The Morison equation i s used i n t h i s study to estimate the hydrodynamic loads acting on the structure as i t o s c i l l a t e s i n water. With reference to Figure 1, the most general form of the equation, f o r a body moving i n a uniform f l u i d flow f i e l d i s given as (Sarpkaya & Isaacson, 1981). F» = PV(1 + C )U - V C Y a P a + j t p^ApCU-Y^IU-Yj (2.1) Here, F' i s the force per unit length, p the f l u i d density, U(x,t) ** • represents the flow f i e l d , Y , Y and Y are the t o t a l body motions as defined i n Figure 1, the dot notation implying time derivative, V i s the displaced volume of the body per unit length, A normal to the plane of motion and C mass c o e f f i c i e n t s . n and C p i s the projected width are empirical drag and added The added-mass and drag components are seen to depend on the r e l a t i v e motion between structure and f l u i d . the undisturbed f l u i d i s stationary. In the present case, Equation (2.1) therefore reduces to 17 (2.2) where the projected width Ap i n (2.1) has been replaced by the diameter D of the cylinder. 2.4 Modal Analysis The c y l i n d r i c a l marine tower considered i n t h i s study can be considered as a uniform cantilever beam and the governing d i f f e r e n t i a l equation of motion developed from appropriate beam theory (see Appendix A). For the support motion problem, defined i n Figure 1, the d i f f e r e n t i a l equation i n terms of t o t a l displacement can be separated into two components; 6 ( t ) , the known input at the structure base, and Y(x,t) the displacement r e l a t i v e to the base. The d i f f e r e n t i a l equation including the s t r u c t u r a l damping term, can be written as (Clough and Penzien, 1975). m (x)Y(x,t) + EIY (x,t) + CIY ( x , t ) Y ( x , t ) = V - m (x) 6(t) + p(x,t) (2.3) Q where the superscript denotes d i f f e r e n t i a t i o n with respect to x, ^ ( x ) i s the mass per unit length, C i s the c o e f f i c i e n t of i n t e r n a l damping and p(x,t) i s the forcing function here defined as i n Equation (2.2). the e l a s t i c modulus and I the moment of i n e r t i a of the section. E is The same separation technique applied to t o t a l displacement can be applied to the forcing function p(x,t) since both drag and added mass depend on t o t a l r e l a t i v e motion between f l u i d and structure. Equation (2.3) can then be 18 rewritten as mY + CIY Y + EIY lv iv = - m5 - ^. C D|Y4 P D (2.4) |(Y-^) i n which m now includes the added mass, m = m o + p—-.— C (2.5) a and the dependence of Y on x and t i s now implied. Equation (2.4) i s a coupled nonlinear d i f f e r e n t i a l equation and must be solved numerically or simplified by making certain assumptions about the r e l a t i v e importance of the various terms. Two s i m p l i f i c a t i o n methods and a numerical approach are considered. 2.4.1 Neglect of Drag Forces For small amplitudes of response and forcing, the nonlinear term i n Equation (2.4) i s small i n comparison to the i n e r t i a l terms. If this term i s neglected, Equation (2.4) becomes l i n e a r and modal analysis can be used to obtain a solution for the response. In the case of harmonic base motion forcing 6 ( t ) , a closed-form solution i s obtainable. In the present study, a solution including only the two lowest modes (See Figure 2) i s developed, because the large frequency increments between modes precludes any s i g n i f i c a n t contribution from higher modes. In p r i n c i p l e , a solution involving any number of modes can be developed. The f a m i l i a r techniques of modal analysis are applied i n the linear aproximation Clough and Penzien, 1975). Y(x,t) = I <j>(x) 5 ( t ) r r (see A separable series solution i s assumed: (2.6) 19 where £ (t) r * sa time dependent amplitude function and ^ ( x ) i s the r - t h mode shape of a cantilever beam given by Ax <j) (x) = cosh — c o s r Ax —£ Ax o (sinh -j- Here, £ i s the length of the c a n t i l e v e r , A cosAcoshA Ax sin -y- ) r (2.7) i s a solution of +1=0 (2.8) and sinhA -sinA o = r (2.9) coshA +cosA r r Substituting Equation Equation (2.6) into the linear ( C = 0) form of Q (2.4) and using orthogonality of mode shapes to decompose the p a r t i a l d i f f e r e n t i a l equation (see Appendix A) we obtain the system of single degree of freedom equations i ( t ) + 25 w I (t) + w 5 ( t ) = P (t) 2 r r (2.10) r Here P^ (t) i s the generalized i n e r t i a l load. r An assumption of equivalent viscous damping has been assumed for the s t r u c t u r a l damping, with the damping r a t i o given as (2.11) 20 and the generalized i n e r t i a l load given as P (t) - o 6 ( t ) x (2.12) r where t \* t(x)dx \a J< m(x)(}> o (2.13) l r = —n « /Vx)^(x)dx o For harmonic excitation given by 6(t) = Y s i n ut (2.14) the generalized i n e r t i a l forcing function i s given as P (t) = ct u>Y s i n uit I r g r 2 T x (2.15) ' In this case, a steady state solution can be found d i r e c t l y by substitution of the harmonic form i n (2.10). a coY 2 5 (t) r = - — u S sin(oot - 6 ) - Ld-("/ r w = - a A sin(oot-9 ) r r r (2.16) ) ) +(2? ^ ) ] 2 r This yields 2 2 r r where the phase angle 6^ i s given by 2£ u)/u 6 r = t a n _ 1 ^ l4^T)r 2) ( 2 * 1 7 ) and wY 2 A r = w i(l-(u)/u) ) ) + (2C u/u ) J r r r r 2 2 2 2 1/2 (2.18) 21 The response of the column r e l a t i v e to the base i s then Y(x,t) = - I • (x) oc A sin(wt - 0 ) r=l r (2.19) r and the t o t a l response i s Y t = Y(x,t) + 6(t) (2.20) For the uniform cantilever beam used i n the present study I *r< > * X (2.21) d i <t>(x)dx o r 2 where X and f 1977). are defined as i n Equation (2.8) and (2.9) (See Blevins, The t i p response can be obtained by evaluating <J>(JO and r substituting this r e s u l t , with those of (2.16) and (2.19), into (2.20). Using a f i r s t mode approximation, t h i s yields = 2 [ I + 0.25 Y + Y S i c o s e J / i g g 2 tip L 2 1 1 (2.22) 2 J where the bar indicates amplitude, and Q\ i s as i n (2.17). The addition of higher mode contributions i s straightforward but a l g e b r a i c a l l y inconvenient because of the increasing number of phase angles which must 22 be combined (See Appendix B). The two mode approximation i s given as Y., tip = 2[I 2 1 + I 2 2 + 0.25 Y 2 g + Y (I cos Bj- g 1 + 25^2 cos (0! - e ) ] l 2 c o s 6 ) 2 (2.23) 1 / 2 2 and 6 i s as i n (2.17). f As discussed e a r l i e r , the large modal frequency separation makes the i n c l u s i o n of more than the two lowest modes unnecessary i n t h i s analysis. The functional dependence of t i p displacement on the frequency r a t i o parameter can be generated from Equation (2.23) by simple calculation at discrete frequencies and specified forcing amplitudes and compared to experimental results. For a random input function such as seismic base motion the formulation cannot be reduced to give a closed-form solution i n the time domain and must be integrated numerically to obtain the time s e r i e s . this case, i t i s usually more convenient In to work i n the frequency domain, using Fast Fourier Transform methods commonly a v a i l a b l e . Because time series were desired, this study uses numerical integration to obtain the response. Here again, the addition of higher mode contributions i s straightforward. However, i n the present case, seismic input frequencies 23 would not r e s u l t i n a s i g n i f i c a n t second mode v i b r a t i o n contribution because of the high second fundamental frequency of the model, and a f i r s t mode approximation i s used. For design purposes, a base shear prediction F^(t) can r e a d i l y be obtained once the structure response i s known, from an integration over the structure span given by F (t) b = /£m(x)Y (x,t)dx t and s i m i l a r l y for base moment. (2.24) Response spectra can be obtained i n the usual manner f o r these and other functions for s p e c i f i e d input motions. There are several assumptions inherent i n the above approach which must be noted. F i r s t i s the assumption, used i n a l l parts of t h i s study, of a constant added mass. Other authors have shown that added mass i s i n fact a function of mode shape and frequency (Liaw and Chopra 1973; Byrd 1978; Pegg 1983). However, i t i s generally accepted that the error introduced by neglecting these dependencies i s minimal for slender structures vibrating i n the lower mode shapes, for which the present analysis i s concerned. The second assumption involves the neglect of two forms of energy dissipation; viscous drag and surface wave generation. Near resonance conditions where t i p o s c i l l a t i o n s can be large, these terms w i l l not be i n s i g n i f i c a n t . For this reason, the above approach w i l l overestimate the response of the structure, the largest discrepancies occurring at resonance. i s thus overconservative. This method of response prediction 24 The q u a n t i f i c a t i o n of damping due to surface wave generation i s difficult. That term i s r e a d i l y accounted for as a free surface boundary condition i n the p o t e n t i a l flow solution, but has no d i r e c t interpretation i n the small body regime. An approximation as viscous damping ignores the frequency dependence of t h i s term although resonance is perhaps the only c r i t i c a l may have merit. frequency and hence such an approximation Because, i n general, t h i s damping cannot be separated from the viscous drag damping i n the experiments, the t o t a l external damping i n the analysis i s controlled by the choice of drag c o e f f i c i e n t and wave generation damping values are not s p e c i f i e d . A l o g i c a l approach to the problem of including drag forces i s to uncouple or l i n e a r i z e the nonlinear drag term i n Equation (2.4). However, the commonly used practise of uncoupling the drag terms (See Blevins, 1977) i s not an alternative i n an analysis of base motion problems because the r e l a t i v e motion which i s neglected i n such an approach i s exactly the structure response. Linearization techniques are tractable i n c e r t a i n problems and one such method i s considered i n the next section. 2.4.2. Linearized Drag Forces The nonlinear drag term can be l i n e a r i z e d to obtain a l i n e a r added damping i n the equation of motion. This method, based on a Fourier expansion or error minimization procedure, r e s u l t s i n amplitude dependent damping and forcing terms, which necessitates an i t e r a t i v e solution technique. For continuous structures, the amplitude and hence the damping i s p o s i t i o n a l l y dependent, and this dependence prevents a complete modal decomposition as i n the purely i n e r t i a l formulation. However, a f i r s t mode approximation can be made as follows. 25 The drag term i n Equation (2.4) can be written as: P (x,t) = - V j Y j (2.25) Kp = j PC D (2.26) D where D To l i n e a r i z e (2.25), l e t |Y |Y t t = A Y (x,t) t (2.27) where A includes the p o s i t i o n a l dependence, A = A(x). For a zero mean Gaussian process i t can be shown that minimizing the mean square error of Equation (2.27) for x constant yields |Y |Y t t = 1.2 a- (2.28) where 0£ i s the root-mean-square of the t o t a l r e l a t i v e v e l o c i t y and i s a function of p o s i t i o n because of the dynamic responses. For a harmonic process, (2.28) becomes 1.2 0£ = a Y and a = 8/3n t (2.29) 26 where the bar represents amplitude. Using a f i r s t mode approximation, we write the p o s i t i o n a l dependence as Y (x) - 5 + 4 . ! (x) ?! (2.30) t Combining the above, we have an expression for the l i n e a r i z e d drag force, -aides' P (x,t) = D + <h (x)5i ) ( 6 ( t ) + 4,! (x)li ( t ) ) (2.31) The generalized drag load becomes, after manipulation P (t) = - ^ D {6 6 (t) Bi + 3 2 (6 ki (t) + 6 (t)4i ) +5i£i(t)6 } 3 where 3i = i = J* 4,3 (x) dx 3 32 Combining (2.10), (2.32) 4)1 (x) dx (2.33) = /* <t> (x) dx (2.32) 2 with the generalized i n e r t i a force P j ( t ) i n Equation r and reorganizing, the equation of motion becomes 27 + C— C (2.34) •Pl(t) Taking due account of phases, (See Appendix B), P i ( t ) Is defined as (aK a Y (e a)Y +g Q) ] sin(a)t-6 ) [(ma^S!)^ ? 1 2 D 1 ^ g i g 1 2 LD 2 mp where 9 LD = t a ? n o>Y m I, g <' > 2 36 i s the phase angle inherent i n the forcing function due to drag damping and Cg = s t r u c t u r a l damping c o e f f i c i e n t C Q = added damping from drag forces Assuming viscous damping, the linearized drag formulation gives s 5 + a V [6B2+SiB j 3 2w m3? n * (2.37) From equations (2.35) and (2.37) i t can be seen that both force and damping are amplitude dependent. The s o l u t i o n amplitude of (2.34) i s given i n the usual form as 28 2 5i - 2 11/2 [(m3iO) Y ) +(aK^toY (a)Y ^ + 5 i t S 2 ) ) ] 2 2 m3 wi[(l-(a)/io ) ) 2 2 2 1 + (25i0)/a) ) ] 2 (2.38) 1 / 2 1 A solution of (2.38) can be obtained by choosing an i n i t i a l value o f 5l, t y p i c a l l y zero, and i t e r a t i n g to a prescribed s o l u t i o n accuracy. The t i p response i s then given, as before, as Y(t) = 4 > i ( £ K i ( t ) + 6(t) tip (2.39) and the amplitude of cylinder t i p motion i s Y„. = 2 [ I + 0.25 Y tip l g 2 2 + Y 5 cos(Q + %n)] 1/2 g l i LD / J (2.40) As before, Q\ i s the phase angle a r i s i n g from s t r u c t u r a l damping: 0! = 2.4.3. tan" 1 250/n>i) { } l-Cw/G)!) (2.41) 2 Numerical Method The nonlinear form drag term can be included i n a numerical solution of the equation of motion. Beginning again with the separated form of the equation, we have, f o r the motion r e l a t i v e to the base 29 m I 5<|>(x) + CI I l < t . r=l r=l r r r - - m6(t) - I r=l iV r l + ( x ) + 6(t) r I 5 ^ (x) r=l ( x ) + EI r where again m includes the added mass. V r | I 54> (x) + 6 r=l r (2.42) Making a f i r s t mode approximation (2.43) Y(x,t) = <h(x) € i ( t ) and assuming that S(t) « Y(x,t) i n regions where drag forces are s i g n i f i c a n t , we have il<n<(>i(x) + £iCI<h (x) + 5iEI()) (x) lv (2.44) lv 1 = - {m6(t) + K |li<(>i(x)|c: <t. (x)} D 1 1 Multiplying by <i>i(x), integrating over the length and rearranging, we obtain, l\ /* CI <j) (x)<t) (x)dx + K | 5 | 5 i / ^ i ( x ) d x /* m^(x)dx + lv 1 + ?! /* EI <h(x)<h (x) dx - - 6(t) ±V D 1 1 m<p (x)dx 1 (2.45) Orthogonality concepts can be used to reduce the fourth order terms i n Equation (2.44); however, the nonlinear term cannot be s i m p l i f i e d 30 further. Performing the integrations, and assuming viscous damping, a single degree of freedom equation i s obtained: 33 ?l(t) + 2Ci<o ? (t) 4 — K j l i C t O l ^ C t ) + 0 ) ^(t) = (2.46) 2 1 1 -a ^ t ) where a i s given by Equation (2.21) and 3 and 3 are defined as i n 2 3 Equation (2.33) Equation (2.46) can be solved numerically using one of a number of solution algorithms for nonlinear d i f f e r e n t i a l equations. In the present study a fourth order Runge-Kutta integration method i s used. This method reduces Equation (2.46) to two f i r s t order equations by appropriate substitution (See Appendix C). The time h i s t o r y response i s generated by time stepping the integration process. This method i s p e r f e c t l y general and can be used for random or sinusoidal forcing functions. For numerical s t a b i l i t y and accuracy, the time step i s chosen as approximately a tenth of the smallest forcing function period. The numerical analysis can be extended to include higher mode contributions and the base v e l o c i t y i n the drag term which was o r i g i n a l l y assumed small. For a two mode formulation, Equation (2.43) i s replaced by Y(x,t) = 4>l(*Kl(t) + 4> (xH (t) 2 2 (2.47) 31 The nonlinear term i n t h i s case cannot be separated into time and p o s i t i o n dependent functions and integrals involving E,^ must be evaluated at each time step. The same problem r e s u l t s from the i n c l u s i o n of the base v e l o c i t y i n the drag term i n a f i r s t mode approximation. In that case, a solution i s sought for the equation ll + j^ { m S ( t ) K + m g D 2*510)! + KJJ J* <h (x)|4>i(x)li + 6(t)|dx (2.48) rX, , • • , J* +i(x)|4>i(x)e + 6(t)|dx + 0)i5i = - ai6(t) 2 1 An exact solution to Equation (2.48) can evidently be found only by iteration. The combination of a Runge-Rutta method, i t e r a t i o n and time stepping can be time consuming and hence expensive. applicable However, i n the frequency range, a two mode solution approximation, which included the base v e l o c i t y , was found to d i f f e r very l i t t l e from a f i r s t mode approximation, which neglected base v e l o c i t y . approach i s to use the previously the i n t e g r a l evaluation. An a l t e r n a t i v e calculated value of the 5 variable i n If the time step i s small, this method provides very close agreement with an iterated solution and decreases the cost s i g n i f i c a n t l y . by Anagnostopoulos A similar approach for a d i s c r e t i z e d system was used (1981). second mode contribution Because the i n c l u s i o n of base v e l o c i t y and a does not s i g n i f i c a n t l y a l t e r the predicted solution, the response predictions for the nonlinear drag case are generated from the solution of Equation (2.46). 32 3. TESTING FACILITIES AND MODEL PARAMETERS The present study attempts to model and p r e d i c t the response of a s l e n d e r c a n t i l e v e r marine s t r u c t u r e undergoing h o r i z o n t a l base motion. The e x i s t i n g facilities at the U n i v e r s i t y of B r i t i s h Columbia Earthquake L a b o r a t o r y used f o r t h i s purpose i n c l u d e a shaking t a b l e , water tank and data a c q u i s i t i o n system. A s u i t a b l e model, i n s t r u m e n t a t i o n system, and t e s t procedure must be chosen t o be c o m p a t i b l e with these f a c i l i t i e s . 3.1 Testing All Facilities t e s t s were performed i n the Earthquake L a b o r a t o r y o f the Department of C i v i l E n g i n e e r i n g at U.B.C. The l a b o r a t o r y facilities i n c l u d e a 3.3m x 3.3m s i n g l e degree o f freedom shaking t a b l e ( F i g u r e 4) supported by an MTS d r i v e system and a PDP-11 mini-computer. The shaking t a b l e o p e r a t e s i n a s i n g l e h o r i z o n t a l d i r e c t i o n i n the frequency range 030 hz, with very l i m i t e d motion c a p a b i l i t y at the h i g h e r f r e q u e n c i e s . The MTS system can e x c i t e the t a b l e with a v a r i e t y of waveforms, o f variable or frequency and amplitude, as w e l l as random motions from other d i s c r e t i z e d records. were used i n these t e s t s . S i n u s o i d a l and s e i s m i c base motion seismic input The PDP-11 p r o v i d e s the data a c q u i s i t i o n system with a p r o c e s s i n g c a p a b i l i t y o f 17 channels i n any one t e s t . channels were used i n these t e s t s . Four The output s i g n a l s are processed through an analogue t o d i g i t a l c o n v e r t e r and s t o r e d d i r e c t l y on f l o p p y discs. The water tank s t r a d d l e s the shake t a b l e so as not to be i n f l u e n c e d by i t s v i b r a t i o n is lined ( F i g u r e 5 ) . The tank can be f i l l e d to a depth of lm and with h o r s e h a i r - t y p e mats to damp waves r a d i a t e d by the c y l i n d e r 33 and thereby minimize the e f f e c t s of r e f l e c t i o n on cylinder response. The model to be tested was attached to the table through a hole i n the tank bottom sealed with a natural rubber diaphragm. This rubber seal e f f e c t i v e l y allows the f u l l table displacements of 15 cm peak to peak at low frequencies. A schematic of the test f a c i l i t y i s presented i n Figure 3. 3.2 Model Parameters Model c h a r a t e r i s t i c s must be chosen c a r e f u l l y i n order to obtain experimental results which are applicable to prototype s i t u a t i o n s . As discussed i n Chapter 2, dimensional analysis provides the relevant nondimensional parameters i n the problem. In the present study, Reynolds number s i m i l a r i t y was not maintained and no attempt was made to maintain consistent damping r a t i o s . An attempt remaining dimensionless groups: r a t i o D / H , i n e r t i a r a t i o m/pD 2 was made to e f f e c t i v e l y model the the frequency r a t i o f / f > geometric n and Keulegan-Carpenter number K. The choice of model i s controlled by the s t r u c t u r a l parameters, f / f , D / H , n and to a lesser extent by the damping r a t i o C« Seismic e x c i t a t i o n i s severest i n the frequency range 0-20 hz, whereas t y p i c a l structure fundamental frequencies are 0.5 - 2.0 hz. a r e a l i s t i c r a t i o of f / f n i s 0-10. the shaking table, a model f Thus In order to maintain t h i s r a t i o using of 1-3 hz i s required. Prototype values of D/H vary but are usually 1/20 -1/30. This r a t i o is limited i n the model tests by the lm depth to which the water tank can be f i l l e d . 1/20. For a model diameter of 5cm, maximum submergence gives D/H = With the diameter known from the geometric scaling, the sectional mass, m, can be found for a s p e c i f i c i n e r t i a r a t i o m / p D . 2 type values are 0.8-2.0, giving m of 20-50 g/cm Typical proto- for a diameter of 5 cm. 34 F u l l scale damping r a t i o s are usually 3 - 10%. The parameter cannot e a s i l y be controlled i n the model and measured model values of 3.8% for f i r s t mode and 6.6% for the second mode were considered acceptable. The modulus-mass c h a r a c t e r i s t i c s of dynamic prototypes are d i f f i c u l t to model. Convenient geometric scaling often requires e l a s t i c moduli which are not available, p a r t i c u l a r l y when continuous models are desired. Sectional d i s t o r t i o n can be used i n some cases where the dynamic behaviour i s primarily f l e x u r a l v i b r a t i o n . l i n e a r mode models could be considered. A l t e r n a t i v e l y , sectional or Neither of these methods w i l l give accurate modal response which i s desired i n this investigation. Therefore, a continuous model of high mass and low s t i f f n e s s was sought. After several t r i a l s were made with hollow p l a s t i c pipe f i l l e d with heavy materials, a s o l i d Teflon cylinder with the following c h a r a c t e r i s t i c s was f i n a l l y chosen as the test specimen: Material : Teflon (polytetrafluoroethylene) Modulus : 9.9 x 10 kg/cm.sec Diameter : 5 cm Length free standing : 99 cm Sectional Mass : 6 43 g/cm 4.6 hz f (air) 28.7 hz Water depth H 97 cm Free height 2 cm 35 m/pD 1.72 Modal damping r a t i o s 1 3.8% 2 2 6.6 % 1/20 D/H Although T e f l o n i s s u s c e p t i b l e to creep a t low s t r e s s e s , exhibits hysteritic response dynamic t e s t s would be 3.3 Data The behaviour, i t was felt and that i t s use i n feasible. Acquisition t e s t instrumentation c o n s i s t e d of a v e r t i c a l aluminum bar connected i n F i g u r e s 6 and 7. to the c y l i n d e r H o r i z o n t a l motion strain-gauged t i p with a l i g h t of the c y l i n d e r s p r i n g as shown t i p induces f l e x u r e i n the bar which can be measured with the s t r a i n gauges. system measures displacement d i r e c t i o n of base motion. each d i r e c t i o n . signal calibrated tained. The with s t a t i c t r a n s v e r s e to the s t r a i n gauges and a b r i d g e are used f o r the system performed instrumented bar and would have i n c r e a s e d the adequately at a l l but the s p r i n g were p e r i o d i c a l l y d i s p l a c e m e n t s and a l i n e a r calibration was main- Table d i s p l a c e m e n t s and a c c e l e r a t i o n s were measured with the l i n e a r voltage d i f f e r e n t i a l to the shaking t a b l e . processed discs. Two A l t h o u g h a f o u r gauge system to n o i s e r a t i o , smallest outputs. response both i n - l i n e and This through The t r a n s d u c e r (LVDT) and f o u r channels monitored i n each an a n a l o g u e - t o - d i g i t a l c o n v e r t e r and The output sampling r a t e was g i v e minimum sampling frequency. accelerometer attached test were stored d i r e c t l y on s e t v i a the a c q u i s i t i o n software to f r e q u e n c i e s of approximately 20 times the testing 36 4. EXPERIMENTS The o b j e c t i v e of the t e s t s performed i n t h i s study was i n v e s t i g a t e the behaviour of a compliant marine d i r e c t i o n a l h o r i z o n t a l base motion. s t r u c t u r e e x c i t e d by response p r e d i c t i o n s d i s c u s s e d i n Chapter 2. response u s i n g one of the t h r e e approaches would be adequate uni- The e x p e r i m e n t a l l y determined response can then be used as the b a s i s f o r comparison the to and e v a l u a t i o n of I f the p r e d i c t e d i s a c c u r a t e , then t h a t method f o r p r e d i c t i o n of earthquake loads and d i s p l a c e m e n t s . (As w e l l , the t e s t c y l i n d e r i s instrumented to o b t a i n t r a n s v e r s e d i s p l a c e m e n t s and a l t h o u g h no comparisons w i l l be made to any particular t h e o r y , i t i s i n f o r m a t i v e to note the c h a r a c t e r i s t i c s of t h i s m o t i o n ) . The t e s t s and comparisons were of two types: sinusoidal tests, u s i n g f r e q u e n c i e s i n the range 2 to 28 hz, and random motion u s i n g a c t u a l scaled seismic records. 97 cm and f r e e b o a r d of 2 4.1 A l l t e s t s were performed of with a water depth cm. Damping T e s t s and System C h a r a c t e r i s t i c s The damping c h a r a c t e r i s t i c s of the model and m o d e l - i n s t r u m e n t a t i o n system are extremely important i n terms of r e a l i s t i c a l l y dynamic behaviour of a p r o t o t y p e s t r u c t u r e . initial t e s t s conducted determine the were f r e e v i b r a t i o n the system damping. I t was m o d e l l i n g the With t h i s f a c t i n mind, the t e s t s i n a i r and water to hoped that the n a t u r a l damping T e f l o n dowel p l u s the damping i n t r o d u c e d by the v e r t i c a l spring of would be low enough so as to reproduce the range of t y p i c a l p r o t o t y p e damping r a t i o s , u s u a l l y about of 5% of c r i t i c a l . F i g u r e 8 shows the r e c o r d o b t a i n e d a f r e e v i b r a t i o n t e s t i n a i r , F i g u r e 9 the r e s u l t of a s i m i l a r t e s t i n 37 water. Second mode free v i b r a t i o n frequency tests were also performed by o s c i l l a t i n g the model at 28 hz and suddenly stopping the excitation, to determine a second modal damping r a t i o . determined from this method. An approximate value could be The log decrement was used to find the damping r a t i o , C, i n each mode and an averaged r e s u l t gave c,\= 3.8% c r i t i c a l i n the f i r s t mode and £2 = 6.6% i n the second mode. The tests i n water were to determine the e f f e c t of added damping on the natural period of the model. A small decrease i n natural frequency was noted and t h i s reduced frequency was used for the response predictions. The instrumentation system used i n these tests, as described i n Chapter 3, has c e r t a i n c h a r a c t e r i s t i c s which should be noted. Whereas i t would be b e n e f i c i a l i n terms of response accuracy to have very l i g h t spring forces acting on the model t i p , the strain-gauge system used required a moderate spring tension to give recoverable readings at low o s c i l l a t i o n amplitudes. This fact i n turn necessitated the use of a spring system with at least a nominal mass which resulted i n a system resonance at approximately 22 hz, close to the second mode frequency of the model. Although this problem may have been correctable to some extent, observations Indicated that second mode e f f e c t s did not contribute s i g n i f i c a n t l y to the magnitude of cylinder t i p response, even though their presence was noted i n the output signals (See Appendix D) at certain subharmonic 4.2 frequencies. Sinusoidal Tests The sinusoidal tests were performed at four amplitudes of base displacements: 1.0, 1.9, 2.8 and 3.6 mm. The choice of amplitude was 38 l i m i t e d by the p o s s i b l e d e s t r u c t i o n o f the model from l a r g e resonances. At each amplitude of base motion, amplitude a p p r o x i m a t e l y 13 f r e q u e n - c i e s were used i n the range 2 t o 28 h z . The h i g h e r f r e q u e n c i e s c o u l d not be combined with the l a r g e r amplitudes because table of l i m i t s on the shaking system. The h i g h e r frequency t e s t s were an attempt response of the model. t o induce second mode U n f o r t u n a t e l y the second mode frequency was v e r y c l o s e to a resonant frequency i n the i n s t r u m e n t a t i o n system, as p r e v i o u s l y d i s c u s s e d , and the output was then o b v i o u s l y i n e r r o r . problem This was compounded by the i n t e n s e water spray generated by the c y l i n d e r motion frequencies. existing which caused e x c e s s i v e n o i s e i n the s i g n a l output a t h i g h The a c c u r a t e measurement o f second mode response u s i n g facilities appears f e a s i b l e only i f the model fundamental frequency i s lower than the v a l u e o f 4.5 hz o b t a i n e d i n the p r e s e n t tests. Second mode response c o u l d be important f o r s t r u c t u r e s with low fundamental f r e q u e n c i e s (<1.5 h z ) . However, the a d d i t i o n of h i g h e r mode c o n t r i b u t i o n s i n g e n e r a l d e c r e a s e s the s t r u c t u r e t i p d i s p l a c e m e n t s , but increases remains those at more deeply submerged p o i n t s . The combined response everywhere l e s s than the f i r s t mode t i p response and hence i s not critical. I n - l i n e and t r a n s v e r s e d i s p l a c e m e n t s were r e c o r d e d i n most The t r a n s v e r s e d i s p l a c e m e n t s showed a very peaked frequency c e n t e r i n g on the fundamental Chapter 5 ) . resonant frequency ( d i s c u s s e d tests. response further i n V i r t u a l l y no output was a t t a i n a b l e o u t s i d e a 1.5 hz band- width c e n t e r e d a t resonance and the t r a n s v e r s e s i g n a l was not recorded at f r e q u e n c i e s o u t s i d e t h i s range. The output sampling r a t e was s e t at 39 frequencies of 20- 50 times the d r i v i n g frequency and 5 seconds of steady state motion were recorded i n each test. filters At low amplitudes of response, were required to reduce noise i n the s i g n a l . A 10 hz low-pass f i l t e r was used at test frequencies below 4.5 hz, a 40 hz low-pass filter on a l l other t e s t s . Figure 10 shows a test at the fundamental resonance (4.5 hz); Figure 11 a test at the second mode resonance (28 hz). Note the intense water spray which affected response measurements near the second mode frequency. 4.3 Seismic Motion Tests The acceleration records of two actual earthquakes, E l Centro N-S (1940) and San Fernando N21E (1971) were used as random motion input. These records were chosen because they have reasonably representative frequency spectra of earthquakes i n the P a c i f i c region (See Figures 12 and 13) and they are r e l a t i v e l y severe. The San Fernando record i n particular contains s i g n i f i c a n t energy i n the region of the fundamental frequency of the test cylinder (4.5 hz), as might occur from s o i l filtering. The d i g i t i z e d acceleration input data can be scaled to an appropriate l e v e l on the MTS system. In this investigation i t was desired to keep peak-to-peak base displacements of the model geometrically similar to those which can occur i n actual strong motion seismic a c t i v i t y (40 - 60 cm). This resulted i n 0.15 f u l l scale displacements i n the E l Centro test and 1.5 f u l l scale displacements i n the San Fernando t e s t . A peak and time-averaged Keulagan-Carpenter number i s then maintained between prototype and model since that non- AO dimensional group i s proportional to the r a t i o of displacement to cylinder diameter. Because the actual earthquake time scale i s maintained the test accelerations are scaled by the same factors. tests were performed at peak-to-peak displacements of 3.5 2.0) and 2.8 cm ( ase ~ K i n t h e E 1 C e n t r o a n d S a n F e r n cm a n d o D The ^'Sase tests respect i v e l y . The random motion test data were sampled at a frequency which would allow direct use of table displacement and acceleration records in the numerical program described i n Chapter 2. approximation this frequency was 50 hz. For the f i r s t mode This gives a time step of seconds, or approximately 1/11 of the fundamental period, a value s u f f i c i e n t l y small to assure numerical accuracy and stability. 0.02 41 5. 5.1 DISCUSSION OF RESULTS Response Functions from Sinusoidal Tests The sinusoidal test data as recorded are presented In Table 2. The frequency dependence of cylinder t i p response can be presented i n non-dimensional form f o r the sinusoidal tests and predictions. These comparison p l o t s , sometimes termed receptance functions, are presented i n the Figures 14, 15, 16 and 17 for the tested r a t i o s of base amplitude to diameter r a t i o Y /D. g The frequency r a t i o f / f n i s defined i n terms of the fundamental frequency of the cylinder i n water. r e s u l t s are indicated on these f i g u r e s . Several important As anticipated, the response near resonance i s very peaked and dynamic amplifications on the order of 15 - 20 occurred at the fundamental frequency. Unfortunately, as discussed i n Chapter 4, a true second mode response was not obtainable i n part because of l i m i t a t i o n s on the shake table and p a r t l y because large system resonances led to erroneous data above a frequency r a t i o of about 4.0. The general trend i n the data towards a second response peak i s evident i n the figures. For a l i n e a r system, the dynamic a m p l i f i c a t i o n i s a system constant dependent on the s t i f f n e s s and damping parameters and independent of the amplitude of applied motion or force. The nonlinear behaviour of the system considered here i s apparent i n the amplitude dependence of the dynamic a m p l i f i c a t i o n factor Y/Y . This factor ranges from a high of 19 at Y /D = 0.02, decreasing s t e a d i l y with amplitude increases to a low of 13 at Y /D = 0.072 (See Figures 14 to 17). This decrease i s a r e s u l t of the increased damping associated with the occurrence of flow separation near the cylinder t i p , and to increased surface wave generation at the larger forcing amplitudes. An increase i n the viscous damping r a t i o of approximately 45% i s required to achieve the t o t a l a m p l i f i c a t i o n reduction. Although this i s a large increase, the absolute damping values are quite small and the change i s of the order of 1.5% critical. However, as we note from the peaked response c h a r a c t e r i s t i c , the response i s very s e n s i t i v e to small changes i n damping. 5.2 Response Predictions for Sinusoidal Motion The methods of response prediction used i n t h i s study have been developed i n Chapter 2. The simplest approach neglects the nonlinear drag force term i n the Morison equation and a closed-form solution i s obtained for the resulting linear equation of motion. The neglect of drag forces i n the formulation e f f e c t i v e l y lowers the damping by eliminating the drag damping present i n the actual system. This method consequently overestimates the response of the cylinder t i p . The discrepancy w i l l increase with forcing amplitude due to the increasing importance of drag forces near the free surface. The response function for this solution method i s compared with other methods and the experimental data i n Figures 14 to 17. of predicted response to measured response at resonance Y The r a t i o s ,/Y for pred exp the four amplitudes used i n the tests' are presented i n Table 1. The r e s u l t s of t h i s method exhibit the c h a r a c t e r i s t i c s anticipated. Only for the lowest amplitude case, Figure 14, does this method provide an accurate estimation of structure response. forces in the experiment were n e g l i g i b l e . In that case, nonlinear drag Because the experimental data for second mode response i s not considered v a l i d , for reasons discussed, 43 i t i s i m p o s s i b l e t o make c o n c l u s i o n s r e g a r d i n g t h e a c c u r a c y o f the second mode response p r e d i c t i o n based on t h i s method. The second p r e d i c t i o n method i n c l u d e s a l i n e a r i z e d form o f t h e M o r i s o n drag term i n the e q u a t i o n of m o t i o n . A c l o s e d - f o r m s o l u t i o n can be o b t a i n e d f o r a f i r s t mode response assumption. The response function generated by t h i s method i s i n c l u d e d i n F i g u r e s 14 t o 17 and the resonant response r a t i o s a r e t a b u l a t e d i n T a b l e 1. I t i s e v i d e n t from Table 1 t h a t the a c c u r a c y of a l i n e a r i z e d drag p r e d i c t i o n i s dependent on the c h o i c e o f d r a g c o e f f i c i e n t , (Lj, as c o u l d be a n t i c i p a t e d . The i n i t i a l c h o i c e o f d r a g c o e f f i c i e n t was based on an e x t r a p o l a t i o n o f e x i s t i n g c o r r e l a t i o n s of t h i s c o e f f i c i e n t w i t h Reynolds number and K e u l e g a n C a r p e n t e r number as i n d i c a t e d i n F i g u r e 18 ( a f t e r Sarpkaya and I s a a c s o n ) . A c y l i n d e r t i p Reynolds number of o r d e r l& was assumed. Accordingly a drag c o e f f i c i e n t o f 0.7 was s e l e c t e d as a r e p r e s e n t a t i v e v a l u e f o r t h e range of K and Re expected near the c y l i n d e r t i p and was used i n both t h e l i n e a r i z e d d r a g and n u m e r i c a l s o l u t i o n s . For Cp = 0.7, the l i n e a r i z e d drag p r e d i c t i o n s a r e v e r y u n c o n s e r v a t i v e a t resonance and d i v e r g e from t h e e x p e r i m e n t a l r e s u l t s w i t h i n c r e a s i n g base motion a m p l i t u d e ( T a b l e 1 ) . I n t h i s c a s e , drag f o r c e l i n e a r i z a t i o n r e s u l t s i n an o v e r e s t i m a t e o f drag damping. A more a c c u r a t e and n o n - d i v e r g e n t response e s t i m a t i o n i s o b t a i n e d w i t h C^ = 0.25 and an o p t i m a l s o l u t i o n i s reached w i t h - 0.20. The l i n e a r i z e d method appears t o g i v e a c c u r a t e response p r e d i c t i o n s f o r t h i s when an a p p r o p r i a t e drag c o e f f i c i e n t i s used. Of c o u r s e , t h i s i n f o r m a t i o n would not g e n e r a l l y be known a p r i o r i . drag problem 44 As a t h i r d approach, the nonlinear drag term was included i n a numerical solution of the equation of motion derived from a f i r s t mode response assumption. A drag c o e f f i c i e n t of = 0.7 was again used i n i t i a l l y and the results are presented i n Figures 14 to 17 and Table 1. The response predictions for = 0.7 are once more unconservative, but consistent over the range of base displacement amplitudes. This indicates that the numerical method can provide good resonant response estimates i f the drag c o e f f i c i e n t i s chosen c o r r e c t l y . Figure 19 shows the relationship between resonant response amplitude ratios and drag c o e f f i c i e n t used i n the numerical approach. Evidently a very low drag c o e f f i c i e n t i s required to obtain an accurate prediction. This i s a result of the low Reynolds number associated with the motion away from the free surface. The amplitude dependence of the drag terra i s apparent from the v a r i a t i o n of response ratios at fixed C^. would be amplitude invariant. In a linear system, this r a t i o In the present case a general requirement for agreement of experimental and numerical response i s an increasing drag c o e f f i c i e n t for increasing base motion amplitude. However, the interaction of transverse and i n - l i n e o s c i l l a t i o n s at the largest test amplitude caused a reversal i n this trend. Based on Figure 19 a drag c o e f f i c i e n t of about 0.3 would give reasonable r e s u l t s except i n the case of the lowest input amplitude. 45 5.3 Transverse Response The cylinder response near resonance at the larger base displacement r a t i o s , Y /D, was influenced by l i f t S forces. The occurrence and magnitude of l i f t forces i s a function of the degree of flow separation, hence the Keulegan-Carpenter number. For the two largest base motion amplitudes, (2.8 and 3.6 mm), the free surface Keulegan-Carpenter numbers based on peak response values, were 5.1 and 6.2 respectively. The l i t e r a t u r e suggests that vortex shedding (Figure 20) and l i f t forcing can occur for K > 5.0 (Sarpkaya, 1979). Thus, i t could be expected that forces would be present near the free surface. experiments these l i f t lift In the present forces trigger ovalling of the cylinder t i p . The interaction of i n - l i n e and transverse o s c i l l a t i o n s decreases the i n - l i n e response over a narrow band-width centered at resonance, from the response expected with no l i f t forces present. The i n t e r a c t i o n i s evident i n the results presented i n Table 2 for Y /D = .054 and .07 2. 8 As resonance (4.5 hz) i s approached, transverse and i n - l i n e amplitudes increase. of Between 4.3 and 4.7 hz, the i n - l i n e response drops as a result the energy d i s s i p a t i o n i n transverse o s c i l l a t i o n s . Maximum i n - l i n e response occurs, i n t h i s case, at a frequency below resonance. Figure 21 indicates such response characteristics near resonance for Y^/D = 0.054. The very narrow bandwidth of this phenomenon i s evident i n the transverse response function presented as Figure 22. The response at the lowest Y /D r a t i o (0.20) was negligible and i s not shown i n the figure. The displacement responses of the cylinder t i p are combined as t i p l o c i i n Figures 23 and 24. Figure 23, a resonant response at Y /D = 0.54, shows a c l e a r l y defined orientation of the o s c i l l a t i o n axes. c h a r a c t e r i s t i c was evident i n a l l the responses which included large This 46 transverse components, and the axes appeared stable only i n such orientations. This s t a b i l i t y may indicate either a hydrodynamic interaction favouring that orientation, such as a s p e c i f i c l i f t force phase, or simply some inherent structural c h a r a c t e r i s t i c or both. However, i f the cylinder exhibited a tendency to o s c i l l a t e diagonally because of a material defect, this behaviour would presumably be evident in the response i n a i r . Also, the response would be altered i f the cylinder were rotated i n i t s base clamp. Because only i n - l i n e response occurred i n a i r tests and cylinder rotation had no e f f e c t on the preferred axes orientations i t can be concluded that the directions of the e l l i p t i c axes are controlled by a hydrodynamic mechanism. The exact details of this interactive mechanism are no doubt complex and beyond the scope of the present study. 5.4 Response to Seismic Input The displacement response of the cylinder during earthquakes was tested using r e a l earthquake accelerograms for base motion input. These seismic records were scaled to give maximum displacements consistent with geometric scaling while the time frame was maintained f u l l scale. As a r e s u l t , accelerations i n the test are lower than recorded values. Two earthquake records were used i n the tests; the E l Centro 1940, N-S component and the San Fernando, 1971, N21E component. The test input was scaled to give a maximum base displacement r a t i o , Y /D, of about 0.4, consistent with strong motion earthquake displacements. The E l Centro earthquake energy i s distributed f a i r l y evenly over a range of frequencies (see Figure 12) and this particular record i s often treated as white noise input. Such input would not be expected to produce large dynamic responses. Figure 25 shows the t i p response for the E l Centro input, and indicates that dynamic response of the cylinder i s minimal during t h i s earthquake. negligible during this test. Transverse o s c i l l a t i o n s were also The hydrodynamic loads i n this case could be estimated quite adequately by a pseudo-static a n a l y s i s . whereas most of the energy i n the El Centro earthquake i s equipartitioned at frequencies below 3 hz, the San Fernando 1971 earthquake has a large spectral peak near 4.5 Hz, the fundamental frequency of the cylinder (see Figure 13). Such spectral peaks r e s u l t from s o i l f i l t e r i n g or r e f l e c t i o n interaction of white noise earthquakes such as E l Centro. The magnitude of t h i s peak i s a l s o s i g n i f i c a n t l y greater than any peaks in the El Centro record (Figure 12). This earthquake provides a better i l l u s t r a t i o n of the e f f e c t s of coincidence between structure natural frequency and earthquake spectral density peaks. The base input and i n - l i n e t i p response for the San Fernando earthquake are compared i n Figure 26. It i s apparent that dynamic amplification i s s i g n i f i c a n t i n this case because of the effects. The resulting t i p displacements resonant and v e l o c i t i e s are s u f f i c i e n t to cause l i f t forces which r e s u l t i n transverse o s c i l l a t i o n s during the peak excitation period. A transverse displacement presented Although the transverse o s c i l l a t i o n s are small, i n Figure 27. time h i s t o r y i s they can contribute to the stresses on the structure and should be included i n a seismic design load estimation. develop from a longer exposure to resonant Larger responses could frequencies. This could be a p o s s i b i l i t y under c e r t a i n conditions, although the scaling of the San Fernando record used here, i n conjunction with the proximity of cylinder 48 natural frequency and spectral density peak probably represents a reasonably extreme event. 5.5 Numerical Prediction of Seismic Response With several small modifications, the numerical time step program discussed i n Chapter 2 can be used to predict the response of the cylinder to seismic input. The actual base motion i s used as input and the integration performed at each time step. A linear approximation with no drag loading can be obtained by setting the drag c o e f f i c i e n t to zero. The response time h i s t o r i e s presented i n Figures 28 and 29 were generated using a drag c o e f f i c i e n t of 0.2, i . e . assumming that non-linear effects would be minimal. A l t e r n a t i v e l y , the drag c o e f f i c i e n t could be set to zero and the s t r u c t u r a l damping increased s l i g h t l y to account for the hydrodynamic damping i f the l a t t e r i s low. Figures 28 and 29 indicate that the response prediction using the numerical solution approach i s quite accurate. The peak responses are i n close agreement, and there are only minor differences i n phase. The accuracy obtained at this low value of drag c o e f f i c i e n t indicates that viscous e f f e c t s are almost n e g l i g i b l e i n seismic response, a conclusion previously reached by other researchers (Clough 1960; 1973; Byrd 1978). Liaw and Chopra 49 6. 6.1 CONCLUSIONS AND RECOMMENDATIONS Conclusions The displacement response of a f l e x i b l e , surface-piercing c i r c u l a r cylinder to sinusoidal and random base motion has been investigated. The variables considered i n the study have been the frequency r a t i o and nondimensional base motion amplitude. Experimental response has been compared to predictions based on modal analyses using a Morison equation representation of the hydrodynamic forces. Three treatments of the nonlinear drag term i n the r e s u l t i n g equation of motion have been considered: neglect of the drag terra, l i n e a r i z a t i o n , and inclusion i n i t s nonlinear form. Based on results discussed i n Chapter 5, conclusions are made as follows: Cylinder response to small base displacements can be accurately predicted neglecting f l u i d drag forces. However, the nonlinear drag term becomes increasingly important at higher displacement amplitudes. In the present case, this term i s equivalent to a nonlinear added damping which reduces the dynamic amplification as input motion amplitude increases. Neglect of this term leads to overly conservative response predictions. The method of drag l i n e a r i z a t i o n developed herein can give accurate response estimates i f a suitable drag c o e f f i c i e n t i s used i n the analysis. A suitable c o e f f i c i e n t w i l l be lower than that chosen on the basis of anticipated cylinder t i p motion, a value which gives consistently unconservative r e s u l t s . Numerical solution of the equation of motion including the nonlinear drag term can give accurate response predictions for a suitable choice of drag c o e f f i c i e n t . This method i s numerically much more time consuming 50 than the linearized drag approach, and suffers from the same limitations on accuracy. The drag l i n e a r i z a t i o n technique appears, then, to be a more r e a l i s t i c approach for response prediction i n the case of sinusoidal motion. The ground displacements of strong motion earthquakes with spectral density peaks near structure resonance frequencies can be amplified to l e v e l s which cause flow separation near the free surface. Small transverse o s c i l l a t i o n s are evident i n such cases and this additional response would contribute to the stresses i n the structure. However, i t is concluded that the i n - l i n e response can be adequately predicted neglecting drag forces. The more common l i n e a r analysis methods i n the frequency domain can thus be used i n the solution of the seismic response problem, as an alternative to the time step integration method used here. Inherent i n the above conclusions i s the f e a s i b i l i t y of using modal analysis with a Morison type forcing function i n the problem formulation. In fact, this approach appears to be quite tractable and can give good results with the inclusion of only the f i r s t mode. It should be noted that the present experiments have generated data, such as modal damping ratios which are then used as input for numerical predictions. In the design process this data i s not known a p r i o r i and estimations of such quantities as added mass and structural damping must be made. These estimations w i l l undoubtedly affect the accuracy of response predictions based on any of the techniques presented i n this study. 51 6.2 Recommendations for Further Study The work of this study could be extended into several areas i n terras of the base motion problem. «» These areas might include: Investigation of possible c o r r e l a t i o n of nonlinear drag and added viscous damping over s p e c i f i c frequency and amplitude ranges. correlations would allow the accurate Such prediction of response using only linear terras i n the equation of motion. «> Similar studies of response for multi-leg structure with various deck-leg o linkage arrangements. A similar dynamic response analysis which includes a base system for modelling s o i l structure i n t e r a c t i o n . Y . (t) tip Y(x,t) Y (x,t) t P(x,t) 6(t) Figure 1. Coordinate System D e f i n i t i o n Figure 2. Mode Shapes For a Cantilever Beam WATER TANK SPRING SHAKING TABLE Figure 3. Schematic of Test F a c i l i t y Figure 5. Testing Tank Over Shaking Table 55 TO M I K E MWLiritu S T U m SAUGES ! ' . - SIDE » l U M i m m HOC (25>m SIU*«E: TO C T l l U D f K T l » Figure 6. Instrumented Rod For S t r a i n Measurement Figure 7. Photo of Instrumented Rod 56 57 Figure 10. Photo of Sinusoidal Test at 4 . 5 H z . Figure 1 1 . Photo of Sinusoidal Test at 28Hz. 53 Figure 13. Fourier S p e c t r a f o r San Fernando 1971 59 a o EXPERIMENTAL NON-LINERR DRAG LINEARIZED DRAG NO DRAG Dynamic Amplification fl.O T 1 0.8 1 1 16 1 1 2.4 Figure 14. 1 1 1 1 1 (Y /Y ) = 19 t g max 1 9.2 4RATIO .D 4.8 FREQUENCY F/F 1 In 1 5.6 1 1 6.4 Response Function For Y /D =.020 1 1 1.2 r— 9.3 60 o s EXPERIMENTAL NON-LINERR DRAG LINEARIZED DRAG NO ORflC . " - a o EXPERIMENTAL NON-LINERR DRAG LINEARIZED DRAG NO DRAG Figure 16. Response Function For Y /D =.054 62 a o EXPERIMENTAL NON-LINERR DRRG LINEARIZED DRAG NO DRAG Figure 17. Response Function For Y /D =.072 CD = .7 Y /D g CD = .25 CD = .20 NO DRAG LINEARIZED LINEARIZED LINEARIZED CD = .7 NONLINEAR .02 1.07 .65 .85 .91 .74 .036 1.30 .60 .90 .97 .76 .054 1.53 .5 7 .92 1.00 .79 .072 1.55 '.49 .84 .93 .75 TABLE 1 : Resonant Response Ratio Y v ,/Y pred exp 64 IN-LINE RESPONSE Y /D = .02 g Y /D = .036 g RESPONSE Y /D = .054 g Y /D = .072 g FREQ. RESPONSE FREQ. 2.0 3.0 1.5 2.3 2.0 3.0 2.8 4.5 2.0 3.9 2.0 5.2 3.0 6.5 3.0 8.5 3.5 4.0 3.4 6.0 3.5 4.0 6.7 3.5 9.3 13.8 4.0 4.5 4.7 19.3 10.4 4.3 17.9 4.3 19.5 38.0 3.5 4.0 4.3 30.5 48.0 4.5 30.1 4.5 26.0 4.5 25.0 5.0 6.0 8.5 3.9 4.7 5.0 4.7 5.0 4.7 5.0 22.0 9.0 6.0 20.0 20.4 11.0 6.0 9.0 5.6 9.0 2.9 6.0 18.1 15.8 6.8 14.0 2.2 9.0 4.8 18.0 22.0 5.0 6.5 14.0 18.0 28.0 4.4 7.6 23.0 FREQ. RESPONSE 14.0 18.0 28.0 7.1 14.0 23.8 FREQ. 14.0 18.0 RESPONSE 12.3 24.0 12.0 7.0 10.0 12.0 TRANSVERSE RESPONSE Y /D = .02 g 4.5 4.7 2.3 1.4 Y /D - .036 g Y /D = .054 g : g 4.0 2.1 4.0 1.7 4.0 2.7 4.3 4.5 4.7 5.0 3.5 15.6 4.3 4.5 4.7 5.0 6.0 28.0 25.0 2.4 4.3 4.5 4.7 5.0 5.1 8.0 33.0 1.6 1.0 FREQUENCIES ARE IN Hz, RESPONSE VALUES IN mm. TABLE 2 Y /D = .072 Sinusoidal Test Data f = 4.5 Hz 29.0 20.0 2.5 •••10 _i 0.1 Figure 18. I I I I 0.3 I I I I I 1 J. J I 3 , I 4 I 5 ' ' ' ' ' 10 15 Drag Coefficient versus Reynolds Number For Various Values of K (after Sarpkaya and Isaacson) 66 (2P Figure 20. Numerical Simulation of the Formation of Asymmetric Vortices (after Sarpkaya and Isaacson) J » e TRANSVERSE i + IN-LINE a. m l ST U 1 1 IM 1 1 IBS 1 1 (L92 1 p 1 1 1 1 IX ID 104 FREQUENCY R f l T I O F/F1N 1 1 1M 1 1 1J2 1 1 1JI r12 Figure 21. In-line and Transverse Response Near Resonance For Y /D =.054 g 67 Figure 22. Transverse Response Functions For Various Base Displacement Amplitudes 68 V I Figure 24. Tip Locus off Resonance For Y /D =.054 69 Figure 25. Figure Response 26. Comparison Response For Comparison El For Centro San Fernando 70 f i g u r e 28. Response P r e d i c t i o n Comparison For El Centro 71 Figure 29. Response P r e d i c t i o n Comparison For San Fernando 72 BIBLIOGRAPHY Anagnostopoulos, S., 1982, "Dynamic Response of Offshore Platforms to Extreme Waves Including F l u i d Structure Interaction", Engineering Structures, V o l . 4, pp 179-185. Blevins, R.D., 1977, "Flow Induced Vibrations", Van Nostrand Reinhold Co., New York. Blevins, R.D., 1979, "Formulas f o r Natural Mode Shape and Frequency", Van Nostrand Reinhold, New York. Byrd, R., 1978, "A study of the F l u i d Sructure Interaction of Submerged Tanks and Caissons i n Earthquakes", Earthquake Engineering Research Report 78/08, May 1978. Chakrabarti, S.K. and Frampton, R., 1982, "Review of Riser Analysis Techniques", Applied Ocean Research, V o l . 4, No. 2, pp. 73-90. Clough, R.W., 1960, "Effects of Earthquakes on Underwater Structures", Proceedings, Second World Conference on Earthquake Engineering, Tokyo 1960, V o l . I l l , pp. 815-831. Clough, R. and Penzien, J . , 1975, "Dynamics of Structures", McGraw-Hill. Eatock-Taylor, R., 1981, "A Review of Hydrodynamic Load Analysis f o r Submerged Structures Excited by Earthquakes", Engineering Structures, V o l . 3, pp. 131-139. Fish, P.R., et a l . , 1980, "Fluid-structure Interaction i n Morison's Equation f o r the Design of Offshore Structures", Engineering Structures, V o l . 2, pp. 15-26. Grecco, M. and Utt, M., 1982, "Dynamic Analysis of a Multi-Tube Production Riser", Proceedings, Ocean Structural Dynamics Symposium, Oregon State University, pp. 289-305. 73 Isaacson, M., 1983, "Earthquake Loading on Axisymmetric Offshore Structures", Proceedings, Fourth Canadian Conference on Earthquake Engineering, Vancouver, Canada. Isaacson, M. and Maull, D., 1981, "Dynamic Response of V e r t i c a l P i l e s i n Waves", Proceedings, Hydrodynamics i n Ocean Engineering, Trondheim, Norway. Keulegan, G.H. and Carpenter, L.H., 1958, "Forces on Cylinders and Plates in an O s c i l l a t i n g F l u i d " , Journal of Research of the National Bureau of Standards, V o l . 30, No. 5, pp. 423-440. Kirk, C.L., et a l . , 1979, "Dynamic and Static Analysis of a Marine Riser", Applied Ocean Research, V o l . 1, No. 3, pp. 125-135. Kirkley, O.M., Ph.D. 1973, "Earthquake Response of Fixed Offshore Structures", Thesis, U n i v e r s i t y of I l l i n o i s , Urbana, U.S.A. Kirkley, O.M. and Murtha, J . , 1975, "Earthquake Response of Offshore Structures", Proceedings, C i v i l Engineering i n the Oceans I I I , pp. 865-879. Liau, C.Y. and Chopra, A., 1973, "Dynamics of Towers Surrounded by Water", Earthquake Engineering Research Centre Report, 73/25. Malhotra, A.K. and Penzien, J . , 1969, "Stochastic Analysis of Offshore Tower Structures", Earthquake Engineering Research Center Report, 69/6. Mei, C.C., et a l . , 1979, "Exact and Hybrid-Element Solutions for the Vibration of a Thin E l a s t i c Structure Seated on the Seafloor", Applied Ocean Research, V o l . 1, No. 2, pp. 79-88. Moe, G. and Verley, R.L.P., 1980, "Hydrodynamic Damping of Offshore Structures i n Waves and Currents", Proceedings, 12th Offshore Technology Conference, Houston, pp. 37-44. 74 Morison, J.R., et a l . , 1950, "The Force Exerted by Surface Waves on P i l e s " , Transactions, American Institute of Mining and Metalurgical Engineers, V o l . 189, pp. 149-154. Murtha, J.P. and Kirkley, O.M., 1975, "Response Spectra for Ocean Structures", Proceedings, 7th Offshore Technology Conference, pp. 985-990. Newnark, N.M. and Rosenbleuth, E., 1971, "Fundamentals of Earthquake Engineering, Prentice-Hall Inc., Englewood C l i f f s , Pegg, N.G., 1983, N.J. "An Experimental Study of the Seismic Forces on Submerged Structures", MASc Thesis, University of B r i t i s h Columbia, Vancouver, Canada. Penzien, J . and Kaul, M.K., Motion Earthquakes", 1972, "Response of Offshore Towers to Strong Earthquake Engineering and Structural Dynamics, Vol. 1, pp. 55-68. Sarpkaya,T., 1979, "Lateral O s c i l l a t i o n s of Smooth and Sand-Roughened Cylinders i n Harmonic Flow", Mechanics of Wave-Induced Forces on Cylinders (ed. T.L. Shaw), Pitman, London, pp. 421-435. Sarpkaya, T. and Isaacson, M., 1981, "Mechanics of Wave Forces on Offshore Structures", Van Nostrand Reinhold, New York. Sawaragi, T. et a l . , 1977, "Dynamic Behaviour of a Circular P i l e due to Eddy Shedding i n Waves", Coastal Engineering i n Japan, V o l . 20, pp. 109-120. Selna, L. and Cho, D., 1972, "Resonant Response of Offshore Structures", Journal of Waterways, Harbours and Coastal Engineering Division ASCE, V o l . 98, WW1, pp. 15-24. Skop, R.A. and G r i f f i n , O.M., 1975, "Vortex-Excited O s c i l l a t i o n s of E l a s t i c Cylinders", Proceedings, C i v i l Engineering i n the Oceans I I I , Delaware, pp. 535-545. Stark, P., 1970, "Introduction to Numerical Methods", Macmillan Company, New York. Stelson, T.E. and Mavis, F.T., 1955, " V i r t u a l Mass and Acceleration i n Fluids", Transactions ASCE 2870, pp. 518-530. Sugiyama, T. and Ito, M., 1981, "Dynamic Characteristics of Structures i n Water", Theoretical and Applied Mechanics, V o l . 30, University of Tokyo Press, pp. 373-379. Tung, C.C., 1979, "Hydrodynamic Forces on Submerged V e r t i c a l C i r c u l a r C y l i n d r i c a l Tanks Under Ground Excitation", Applied Ocean Research, Vol. 1, No. 2, pp. 75-78. Van Dao, B. and Penzien, J . , "Treatment of Nonlinear Drag Forces Acting on Offshore Platforms", Earthquake Engineering Research Center Report, 80/13, University of C a l i f o r n i a , Berkeley, C a l i f o r n i a , Warburton, G.B. and Hutton, S.G., 1978, USA. "Dynamic Interaction for Idealized Offshore Structures", Earthquake Engineering and Structural Dynamics, V o l . 6, pp. 557-567. Westermo, B.D., 1980, "Hydrodynamic Interaction of E l a s t i c Structures", Proceedings, 7th World Conference on Earthquake Engineering, Istanbul, Turkey, V o l . 6, pp. 129-132. Zedan, M.F. and Yeung, J.Y., 1980, "Dynamic Response of a Cantilever P i l e to Vortex Shedding i n Regular Waves", Proceedings, 12th Offshore Technology Conference, pp. 45-59. 76 APPENDIX A MODAL ANALYSIS The methods of response prediction used i n this study are based on the technique of modal a n a l y s i s . obtained This technique uses the mode shapes from the solution of the governing equations of motion for an Euler beam to describe the dynamic response of a continuous structure undergoing external loading. Modal analysis e f f e c t i v e l y reduces the governing p a r t i a l d i f f e r e n t i a l equation to a system of ordinary d i f f e r e n t i a l equations for which solutions e x i s t . The response functions are l i n e a r l y superimposed to describe the o v e r a l l response. This method i s very useful f o r linear systems and can be applied successfully to nonlinear problems provided that the degree of nonlinearity i s small. The p a r t i a l d i f f e r e n t i a l equation describing the f l e x u r a l motion of a slender beam, neglecting the e f f e c t s of damping, shear deformation and rotatory i n e r t i a i s m (x)Y(x,t) + (EIY"(x,t))" = p(x,t) o where m Q (Al) i s the mass of the beam per unit length, y i s the normal displacement from the l o n g i t u d i n a l a x i s , x i s the coordinate along the beam axis and p the externally applied load per unit length (See Clough and Penzien, 1975). In general, E and I can be functions of position, however, i n the case of a prismatic beam, the equation becomes, for free vibrations m Y(x,t) + E I Y ( x , t ) = 0 iV (A2) 77 subject to the appropriate geometric and kinematic boundary conditions, Variable separation i s used to solve Equation (A2); l e t Y(x,t) - 4>(x)5(t) (A3) Substituting (A3) into (A2) and re-arranging, we have iv, „ -m (x) _ o S(t) _ u a • (x) EI ? ( t ) 4> (A4) 4 The solution to the fourth order d i f f e r e n t i a l equation defines the p o s i t i o n a l dependence given i n general as <j> (x) = Asinax + Bcosax + Csinhax + Dcoshax (A5) where the c o e f f i c i e n t s are determined from the boundary conditions. The v i b r a t i o n frequency can be related to the constant 'a' by l e t t i n g 2 0) m * - TT* in the solution for the time dependent amplitude ( A 6 ) £(t) For the cantilever beam, the appropriate boundary conditions are expressed as 78 . , Geometric r o n m o ( H / 4> (0) = 0 ^'(o) = 0 (A7) Natural • .,* ^ ' ° Q These conditions are used to obtain a system of equations from (A5). The c o e f f i c i e n t matrix of this system i s set equal to zero for n o n t r i v i a l solutions resulting i n the transcendental equation 1 + cos a l cosh a l = 0 For a particular solution (A8) = a^i of (A8), the ratio of c o - e f f i c i e n t s i n the system of equations derived from (A3) and (A7) becomes cosX X, + coshX SL sinX % + sinhX i. n n n K * J and the nth mode shape function i s then given by Xx <P (x) = cosh -£ n Xx cos -£ O r Xx (sinh — sin Xx —) (A10) It can readily be shown that the natural modes of beams with c l a s s i c a l boundary conditions are orthogonal over the span of the beam, where orthogonality, as defined herein, implies 0 m * n <j> (x>j> (x)dx = { o £/2m=n I / n m (All) The free v i b r a t i o n frequency associated determined from (A6), with a p a r t i c u l a r mode shape i s as (A12) - " The *2 "o time dependent amplitude function becomes K (t) = A n n 1 sinw t + B n n 1 cosu t n where the c o e f f i c i e n t s A^ , B^ conditions. The (A13) v i n (A13) are determined by the complete solution to (A2) i s the sum / initial of the modal solutions, Y(x,t) = ) (A n=l The sinw t + B 1 1 COSUJ t) q> (x) (A14) formulation of the more general problem of beam vibration w i l l include damping and external f o r c i n g . formed from the substitution of (A14) including damping. ao I m<|> n=l In that case, the solution i s into the general equation of motion For the present problem, this y i e l d s oo oo . (x) 5 (t) + 1 CI<t> (x)£ (t) + I EI^ (x)S (t) n=l n=l v n n n = (x,t) P (A15) 80 Multiplying by <J> (x), integrating over the length of the beam, and using m ( A l l ) , we obtain M J n ( t ) + C n^n ( t ) + K n n ? ( t ) ^ = * ( x ) p n ( x » t ) d (A16) x o where M C n i s the equivalent mass, given here simply as the t o t a l mass, m£, n i s the generalized modal damping c o e f f i c i e n t and K stiffness. K = n r i s the generalized For orthogonal mode shapes, i t can be shown that 2 0) M n n v (A17) If the system damping i s approximated by viscous damping, as i s usually done, equation (A16) can be written as 2 5 (t) + 25 w £ (t) + u> 5 (t) =P (t) n ' n n n n n n v v v v (A18) ' where the equivalent viscous damping ratio for each mode has been defined as , g m n fn (A19) 2m n and t h i s term w i l l c o n t a i n any damping c o n t r i b u t i o n present generalized i n the f o r c i n g f u n c t i o n P ( t ) , g i v e n as n / %(x,t)<j> (x)dx P (t) n -2 (A20) Solutions to (A18) can be found using either convolution integrals, Laplace transforms or numerical methods. Although i n p r i n c i p a l , a solution based on modal analysis can include contributions from many modes, generally only the lowest modes are Important i n the analysis slender beams at low frequencies. 82 APPENDIX B : DERIVATION OF RESPONSE AMPLITUDES For the nth damped single degree of freedom system defined as E 2 + 2c u) £ + u> E - P ( t ) n n n n n n (Bl) n a steady state amplitude i s given as f . I* n ? or (> 1 [ (l-(w/w ) ) +(£ w/w ^ l n n n 2 n L B2 2 1 J ' 2 In the case of base e x c i t a t i o n given by (5 (t) = Y sinwt (B3) the generalized i n e r t i a l forcing function becomes P I = ct ID Y n g 2 T n (B4) and a solution to (Bl) including only i n e r t i a forces i s given then as £ where n = a A sin(u)t-© ) n n n (B5) 83 u> Y = * u [(l-(w/w ) ) n n' ' 1 2 A n 2 2 1v (B6) 2 v + ( 2 ; u/d n n v / J )] 1 / 2 and 0 2? w/u) = tan" ! — M l-(u>/ f n (B7) 1 n U For two modes, we have the modal and base motions as Si (t) = ? i sin(o)t-<pi ) (B8) £ (t) = S sin(wt^> ) 2 6 (t) 2 2 = Y sinw t 8 The cylinder t i p response i s given by combining and expanding (B8) to yield Y(x,t) = <Pi (£ )Ci (sinwtSi - cosw t s i n ^ ) 2 + <t>2 (*• (siao tcosS2 ~ cosw tsin32 ) + Y sinwt g For the present formulation • l (A ) - 2 (B9) 84 4>2(*> = "2 (BIO) Using (BIO) i n (B9) and c o l l e c t i n g terms, Y = [(25icosOi - 25 cos© 2 2 + Y ) + (2S sinG - 2 ^ i s i n 0 i ) ] 2 2 2 (Bll) 1 / 2 2 Simplifying Y . = 2k + \ tip 1 2 2 2 + 0.25Y g + Y (£icos0i - £ c o s © ) g 2 - 2Ci"i2Cos(0i-O )] 2 2 (B12) 1 / 2 2 In the case of l i n e a r i z e d drag, the generalized forcing function complex. i s more The sum of the generalized i n e r t i a l and drag components i s given as aK P(t) = 2 g + 3 (li^Y 2 with D sinwt aiU) Y l g 1 m3 2 oa Y cosu)t 2 2 8 + coliY coswt) + (B13) 85 01 " J * <t>i(x)dx 3 - /* <t>(x)dx (B14) 2 2 33 = /* <t> (x)dx 3 i The components o f (B13) involving I i are included as viscous damping, leaving the forcing function as P(t) = aiu Y 2 8 sincot + —-—& m& (8].u>Y + e li)cosu>t 2 (B15) 8 2 which can be rewritten i n amplitude form. The response i s then given, as before, as il OJ The constituent 2 [(l-(u)/u>i) ) 2 2 (B16) + (2tiu/ui) \ 2 l 1 2 responses of the cylinder t i p are Y(A,t) = 4> (A)5 sin(u>t-e 1 1 LD - ©x) (B17) 6(t) = Y sinut g 86 where aK^uY (BiUY • = tan'l + B E) 2 ^ — § (B18) moj^Y B i g ©1 as i n (B7) Combining and simplifying, we obtain Y tip = 2 + 0 , 2 5 Y g 2 + Y g ^l c o s ( l 0 + e LD )J 1/2 (B19) 87 APPENDIX C DESCRIPTION OF COMPUTER PROGRAM The numerical method used to solve the linearized and nonlinear d i f f e r e n t i a l equation derived i n Chapter 2 involves a small Fortran IV program. The program requires the following input to be contained in a data f i l e attached to the main program: 1) I n i t i a l values of displacement and v e l o c i t y . Zero i n i t i a l values are u s u a l l y assumed since t h i s allows a stable s o l u t i o n to be reached more quickly. 2) Length of structure, diameter, e l a s t i c modulus, moment of i n e r t i a , sectional mass and damping r a t i o . 3) Frequency of o s c i l l a t i o n , i n hertz, and amplitude of base motion, for sinusoidal motion. For random forcing, these variables are replaced by the appropriately d i s c r e t i z e d input. 4) Drag c o e f f i c i e n t 5) Number of time steps, and the time increment. The program calculates the natural frequencies and c o e f f i c i e n t s of the generalized equation of motion using cantilever beam theory. second order d i f f e r e n t i a l equation i s replaced by two f i r s t The order equations which are solved using a fourth order Runge-Kutta integration 88 scheme. A time h i s t o r y response i s generated f o r the t i p displacement by time stepping. Time increments of approximately a tenth of the minimum forcing function period are recommended f o r numerical accuracy. At high frequencies, modulated responses are evident for many cycles N > 600 for low damping values. Near the resonant frequency, the output s t a b i l i z e s reasonably quickly (N < 300). 89 APPENDIX D INSTRUMENT CALIBRATION AND TYPICAL DATA The strain-gauged rod and spring system was calibrated p e r i o d i c a l l y by s t a t i c displacement t e s t s . Figure 30 shows a t y p i c a l c a l i b r a t i o n curve indicating a linear relationship between bridge voltage and i n - l i n e and transverse cylinder t i p displacement. A time history and Fourier spectra for a test at 9 Hz and g / ° Y .036 are presented i n Figure 31 and 32 respectively. = These records i l l u s t r a t e the presence of the second mode response at c e r t a i n forcing frequencies. Such response was only noticeable i n the 9 and 14 Hz records, frequencies near subharmonics of the second modal frequency, 28 Hz. It should be noted that the single mode approximations developed i n t h i s study w i l l not reproduce t h i s type of response. The average half-peak-to-peak response of the actual record was used as representative response i n the receptance functions rather than the Fourier amplitudes of the output at the forcing frequency. Typical i n - l i n e and transverse records a t resonance (4.5 Hz) are presented i n Figures 33 and 34 respectively. 90 F i g u r e 30. C a l i b r a t i o n Curve f o r the Instrumented Rod 91 O 03 -1 Figure 31. Typical Time History at 3rd Subharmonic ID 0.0 4.0 B.O 12.0 16.0 FREQUENCY Figure 32. 20..0 24.0 28.0 32.0 (HZ) Fourier Spectra For Record of Figure 31 36.0 Figure 34. Transverse Resonance Time History at Y /D =.054
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Response of a flexible marine column to base excitation Vernon, Thomas A. 1984
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Title | Response of a flexible marine column to base excitation |
Creator |
Vernon, Thomas A. |
Publisher | University of British Columbia |
Date Issued | 1984 |
Description | The displacement response of a flexible, surface-piercing cylinder subjected to a unidirectional base motion is considered in this study. Laboratory experiments have been performed with a circular, fixed-base model using sinusoidal and scaled seismic input motion. Sinusoidal tests were designed to investigate the dependence of cylinder tip response on the ratio of base motion frequency to cylinder natural frequency and base displacement amplitude, for a fixed water depth, inertia ratio and damping ratio. Further tests with a base motion corresponding to past earthquake records were then used to determine the cylinder's response to seismic excitation. The sinusoidal test results are compared with predictions derived from analyses of the motion in terms of the first and second undamped mode shapes of a cantilever beam. The Morison equation is used to estimate hydrodynamic loads in this formulation, and three treatments of the drag term in the equation of motion are considered: neglect of drag, drag linearization and retention of the complete nonlinear form. A closed-form solution for the former two approximations is developed, and a numerical approach is adopted for the complete nonlinear formulation. The numerical method is used to predict the response of the column to seismic input. The dependence of cylinder tip displacement on frequency ratio and base motion amplitude follows predictable patterns of dynamic response. Peak amplitudes occur at resonance and increase with base motion amplitudes. However, this relationship is not linear because of the damping contribution of nonlinear drag forces near the free surface. The numerical and linearized drag predictions agree well with the experimental response if a suitable choice of drag coefficient is made. The neglect of drag results in very conservative resonant response predictions for large excitation amplitudes in which the free surface Keulegan-Carpenter number exceeds about 4. Drag forces can generally be neglected, however, in the estimation of response to earthquake motion because displacement amplitudes are small. In extreme cases, small lift forces can result from flow separation about the cylinder near the free surface. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-24 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062464 |
URI | http://hdl.handle.net/2429/24958 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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