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An experimental study of the seismic forces on submerged structures Pegg, Neil Gordon 1983

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EXPERIMENTAL STUDY OF THE SEISMIC FORCES ON SUBMERGED STRUCTURES by NEIL GORDON PEGG B.Sc, The University Of Guelph, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT 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 t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1983 © N e i l Gordon Pegg, 1983 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t fre e l y available for reference and study. I further agree that permission for extensive copying of t h i s 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 written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 16 August 1983 i i Abstract In t h i s investigation, the dynamic c h a r a c t e r i s t i c s of a submerged cylinder were determined by performing vibration tests on a model underwater. These c h a r a c t e r i s t i c s are expressed in terms of the added mass and damping values of the c y l i n d e r . Such quantities are required in the design of offshore structures in seismic zones. Sinusoidal tests were used to determine these values as a function of excitation frequency. The frequency range was varied from 0.5 to 6.0 Hertz, which i s the primary range of interest of most earthquakes. The testing was c a r r i e d out in the Seismic Simulation Laboratory of the Department of C i v i l Engineering at the University of B r i t i s h Columbia. The experimental values of added mass and damping versus frequency were compared with the values produced using po t e n t i a l flow theory. The experimental and th e o r e t i c a l r e s u l t s were found to agree very c l o s e l y . The t h e o r e t i c a l added mass and damping values were then used to develop the frequency transfer function for the base shear developed in the cylinder as a result of an input acceleration record. To check the v a l i d i t y of t h i s t h e o r e t i c a l l y derived transfer function, the base shear was measured for a given random acceleration input and compared to the r e s u l t s obtained using the th e o r e t i c a l transfer function. The transfer function derived from Fourier transforms of the random test records, as well as the transfer function developed through sinusoidal tests were also compared to the t h e o r e t i c a l transfer function; the agreement was good. This study i s r e s t r i c t e d to structures which f a l l into the large body or wave d i f f r a c t i o n regime. This means that f l u i d separation does not occur and Laplace's equation for potential flow can be used in solving the problem with the assumption of i n v i s c i d f l u i d and i r r o t a t i o n a l flow. The th e o r e t i c a l solution used in t h i s work contemplates complete free surface boundary conditions, which account for the production of surface waves in the physical problem. These boundary conditions are usually ignored in other studies of this problem, as they increase the d i f f i c u l t y of the solution. Part of the work for t h i s thesis involved the design and construction of testing apparatus and procedures to be employed in the studies of seismic e f f e c t s on offshore structures. This aspect of the research i s described in some d e t a i l . The study reported in t h i s thesis confirms that an exi s t i n g potential theory wave d i f f r a c t i o n program can be used to accurately determine the added mass and added damping values for application in the aseismic design of offshore structures. These parameters can then be applied to evaluate the transfer function for such systems. iv Table of Contents Abstract i i L i s t of Tables v L i s t of Figures v i Acknowledgements . . . v i i I. INTRODUCTION 1 1 . BACKGROUND 1 2. REVIEW OF LITERATURE ON EXPERIMENTAL STUDIES 6 3. OBJECT AND SCOPE OF INVESTIGATION 13 11 . THEORY 16 1. DEFINITION OF THE PROBLEM 16 2. ASSUMPTIONS AND CONDITIONS OF FLUID STATE 18 3. REVIEW OF THEORETICAL STUDIES 20 4. EQUATION OF MOTION 24 5. DIMENSIONAL ANALYSIS OF HYDRODYNAMIC FORCE 26 6. DESCRIPTION OF AXIDIF COMPUTER PROGRAM 28 7. DERIVATION OF ADDED MASS AND DAMPING FROM EXPERIMENT 29 8. DEVELOPMENT OF TRANSFER FUNCTION 34 9. DERIVATION OF TRANSFER FUNCTION FROM EXPERIMENTS 35 I I I . MODEL AND TESTING APPARATUS 37 1. DEVELOPMENT OF TESTING APPARATUS 37 2. DESIGN OF MODEL 43 3. DATA MEASUREMENT 46 IV. DESCRIPTION OF EXPERIMENTS 49 1. SINUSOIDAL TESTS 50 2. RANDOM TESTS 53 3. SURFACE PIERCING AND SUBMERGED TESTS 54 V. RESULTS AND DISCUSSION 56 1. ADDED MASS FOR SURFACE PIERCING CYLINDER 56 2. ADDED MASS FOR SUBMERGED CYLINDER 58 3. ADDED DAMPING 60 4. TRANSFER FUNCTIONS 61 5. RESONANT EFFECTS IN THE MODEL 63 6. VISCOUS EFFECTS 64 VI. CONCLUSIONS AND RECOMMENDATIONS 80 BIBLIOGRAPHY 82 APPENDIX A - SOLUTION OF LAPLACE'S EQUATION FOR ADDED MASS AND DAMPNG FOR A CYLINDER USING THE AXIDIF PROGRAM 85 APPENDIX B - MEASUREMENT AND ANALYSIS OF DATA 92 1. MEASUREMENT APPARATUS 92 A. BASE ACCELERATION 92 B. BASE DISPLACEMENT 92 C. BASE SHEAR 92 2. DATA COLLECTION 93 3. ANALYSIS OF DATA 95 A. SINUSOIDAL TESTS 95 B. RANDOM TESTS 96 L i s t of Tables Recorded Data - Surface Piercing Cylinder Recorded Data - Submerged Cylinder v i L i s t of Figures 1. Schematic of Work Performed in This Study 15 2. E f f e c t s of Surface Waves and Compressibility 23 3. Free Body Diagram of Forces Acting on Model Cylinder .30 4. Photograph of Shaking Table 40 5. Photograph of Tank and Model Apparatus 40 6. Schematic of Testing Tank 41 7. Base Seal for Model 42 8. Diagram of Model Cylinder 45 9. Photograph of Steel Shaft, Base and Strain Gauges ....48 10. Photograph of Sinusoidal Test 52 11. Photograph of Sinusoidal Test 52 12. Added Mass for Surface Piercing Cylinder 66 13. Added Mass for Submerged Cylinder 67 14. Added Damping for Surface Piercing Cylinder 68 15. Added Damping for Submerged Cylinder 69 16. Transfer Function for the Surface Piercing Cylinder Derived from Sinusoidal Tests 72 17. Transfer Function for the Submerged Cylinder Derived from Sinusoidal Tests 73 18. Transfer Function for Surface Piercing Cylinder: E l Centro N-S, 1940 74 19. Transfer Function for Surface Piercing Cylinder: San Fernando S74W, 1971 75 20. Transfer Function for Submerged Cylinder: E l Centro N-S, 1940 76 21. Frequency Spectrum of Output Base Shear on Surface Piercing Cylinder 77 22. Frequency Spectrum of Output Base Shear on Submerged Cylinder 78 23. Comparison of Time Series Output for the E l Centro N-S 1 940 Earthquake Record 79 24. Wheatstone Bridge - Strain Gauge Configuration 98 25. Example of Data From Sinusoidal Tests 99 26. Example of Data From Random Earthquake Tests 100 27. Fourier Amplitude Spectra for Sinusoidal Data of Base Shear and Acceleration 101 28. Spectras of Base Shear and Acceleration and Transfer Function Derived from them 102 v i i Acknowledgement Upon completion of t h i s thesis I would l i k e to express many thanks to my wife, Deborah and parents for their endless support. I would also l i k e to acknowledge the considerable guidance offered to me by my supervisors, Dr. S. Cherry and Dr. M. Isaacson as well as the technical s t a f f in the Department of C i v i l Engineering for their e f f o r t s in bringing about these experiments. 1 I . INTRODUCTION 1 . BACKGROUND The subject of f l u i d - s t r u c t u r e interaction has been studied for many years. The fact that structures react d i f f e r e n t l y to a given loading when located in water rather than in a i r has been the topic of much research. Marine engineers and hydrodynamicists have examined t h i s problem quite thoroughly, p a r t i c u l a r l y as i t pertains to ship design and coastal structures. More recently, s t r u c t u r a l engineers have become seriously involved in t h i s important problem as a result of the large increase in offshore construction. Structures which pose a po t e n t i a l danger to the environment, such as o i l r i g s and storage tanks, are being b u i l t in continuously harsher locations and an accurate analysis of a l l forces acting on such structures i s e s s e n t i a l . Offshore structures undergo loading as a result of waves, currents, wind, operating machinery and seismic a c t i v i t y . Much work has been done on evaluating wave and current loading on submerged structures, but only recently have e f f o r t s been directed to determining the forces r e s u l t i n g from seismic loading. O i l r i g s and other offshore structures are being b u i l t or proposed for construction in increasing numbers in various seismic zones, including the east and west coasts of Canada. This has led to the need for more research into their design for t h i s environment. The present study, which is intended as a contribution to t h i s general problem, is 2 concerned with evaluating the hydrodynamic forces resulting from the seismic motions of a structure in water. These hydrodynamic forces result from the moving body having to displace and accelerate a volume of f l u i d in addition to i t s own mass, and from drag forces developed at the surface of the moving body. The force r e s u l t i n g from the body having to accelerate a volume of f l u i d i s an i n e r t i a l force and i s treated as an 'added mass' that i s hypothetically attached to the body's own mass and i s l i n e a r l y proportional to the body's acceleration. The drag force consists of form drag, which i s a result of f l u i d separation from the body, and skin f r i c t i o n , which occurs between the f l u i d and the body surface. The drag force di s s i p a t e s energy from the system and i s therefore treated as a damping force. This term i s generally not l i n e a r l y proportional to the structure's v e l o c i t y and produces a nonlinear problem. Methods of l i n e a r i z i n g the damping and including i t as an 'added damping' term are used in some cases [24,26], Additional 'added damping', which can be taken as being l i n e a r l y proportional to the v e l o c i t y , comes from the structure producing waves by i t s motion and d i s s i p a t i n g energy. This term i s s i g n i f i c a n t in some problems. These two terms, added mass and added damping, and the manner in which they are derived from the hydrodynamic force w i l l be discussed further in chapter two. The determination of these two dynamic c h a r a c t e r i s t i c s for structures in water i s the object of much research done in t h i s area. 3 The type of force which predominates - either form drag or i n e r t i a , determines the type of solution which can be used to solve for the hydrodynamic forces. The shape of the body, v i s c o s i t y of the f l u i d , and the r e l a t i v e motion between the body and the f l u i d , determine the amount of drag force present. The i n e r t i a force depends on the body dimensions, f l u i d density and frequency of the body motion. In the study of wave forces on structures there are two separate regimes of behaviour depending on the predominant type of force [26]: 1) small body regime, and 2) large body regime. The small body regime i s one in which s i g n i f i c a n t flow separation occurs and the form drag forces are large. Structures which f a l l into t h i s class are those whose dimensions, shape and r e l a t i v e f l u i d - s t r u c t u r e motion result in f l u i d separation. This occurs i f the cross section of the structure i s small in r e l a t i o n to the r e l a t i v e motion between the structure and the water. This class of problem i s mainly concerned with wave loading, as the wave length may be large in r e l a t i o n to the body cross section. Structures with sharp edges or other abrupt changes in cross section also induce flow separation and may f a l l into t h i s regime. In the solution of thi s problem, the nonlinear drag term i s the predominant force and the analysis i s performed by means of the well known Morison equation [21]: 4 F = O.SpDCjjUlUl + 0.25p7rD2Cm(dU/dt) (1.1) where, F i s the f l u i d force, p i s the f l u i d density, D i s the body cross section (diameter), U is the r e l a t i v e v e l o c i t y between the structure and the f l u i d , i s the drag c o e f f i c i e n t and C m i s the i n e r t i a c o e f f i c i e n t . Due to the nonlinear nature of the problem and the d i f f i c u l t y in attaining accurate drag and i n e r t i a c o e f f i c i e n t s , which must be determined empirically, t h i s solution i s usually d i f f i c u l t to obtain accurately. The large body regime i s concerned with structures and r e l a t i v e f l u i d - s t r u c t u r e motions which do not cause flow separation. In t h i s class of problem the i n e r t i a forces predominate. Form drag i s not present as there i s no flow separation. Although some drag force may resu l t from skin f r i c t i o n , t h i s i s usually quite small. In general, the drag term i s neglected or assumed to be small and vary l i n e a r l y with the v e l o c i t y of the structure. Structures whose cross sections are large in re l a t i o n to the r e l a t i v e f l u i d - s t r u c t u r e motion, and whose changes in shape are smooth, such that flow separation i s not induced, f a l l into this regime. Another designation for t h i s class of problem i s the ' d i f f r a c t i o n regime' as the incoming wave t r a i n , having a wave length which i s not too much larger than the body cross section, i s interrupted and d i f f r a c t e d by the body [26]. In the case of a structure undergoing motions, either from earthquake or wave 5 loading, the waves radiated by the structure motion result in energy d i s s i p a t i o n which can be represented as an additional damping term. This damping i s usually much larger than any drag damping from skin f r i c t i o n and i s taken to be l i n e a r l y proportional to the structure v e l o c i t y . This energy d i s s i p a t i o n i s considered as an 'added damping' which acts in addition to the structural damping; i t i s dependent on the structure dimensions, t o t a l water depth and frequency of structure motion. When flow separation does not occur, viscous e f f e c t s can usually be ignored, and the resulting l i n e a r problem i s expressed by Laplace's equation for potential flow with the assumption of i r r o t a t i o n a l flow. If li n e a r i z e d kinematic and dynamic free surface boundary conditions are included in the analysis, the added damping due to surface wave production i s incorporated in the solution [4,14,19,26]. The equations and solution governing t h i s problem are given in appendix A. Structures which have some l o c a l flow separation may also be studied in thi s regime but the eff e c t of the degree of flow separation on the solution must be considered. The large body regime i s easier to analyze than the small body regime, as the hydrodynamic force and thus the added mass and damping can be evaluated t h e o r e t i c a l l y using Laplace's equation for potential flow. Most e x i s t i n g t h e o r e t i c a l and experimental studies for the earthquake design of offshore structures have been performed on the class of structures which s a t i s f y the large 6 body ' d i f f r a c t i o n ' regime. There are two reasons for t h i s : f i r s t , the problem can be solved a n a l y t i c a l l y by potential flow theory and second, the degree of r e l a t i v e motion between the structure and the f l u i d in seismic loading i s not usually very large, so that the assumption of no flow separation i s v a l i d . This w i l l be discussed further in chapter two. The present study i s concerned with the earthquake loading problem and therefore w i l l be r e s t r i c t e d to structures which s a t i s f y the large body regime. 2. REVIEW OF LITERATURE ON EXPERIMENTAL STUDIES In 1779, Pierre Louis Gabriel Du Buat conducted some experiments on pendulums underwater [27]. He noted that t h e i r periods of motion were d i f f e r e n t from the corresponding results obtained for the same pendulums tested in a i r . He explained t h i s in terms of an added mass ef f e c t acting on the pendulums. Since that time, t h i s added mass ef f e c t has been studied for a variety of shapes and by a variety of methods. Several experiments have been performed in which a body on a f l e x i b l e support was set into free v i b r a t i o n in a i r and in water and i t s natural frequencies in these environments were measured [5,6,8,20,27,28]. With the assumption that the support s t i f f n e s s remains constant, the two frequency values were compared and the added mass taken as the difference in the mass values calculated from the measured frequencies. Thus, i f w and m represent the frequency and mass of a system respectively and the subscripts A and w the i r respective 7 values in a i r and in water, the added mass, ma i s obtained by equating the st i f f n e s s e s in both mediums such that: w A 2 m A = ww 2 mw (1.2) from which mw = wA2n>A (1.3) w ~2 w yie l d s the t o t a l mass in water. The added mass i s then determined from: m a = mw-mA (1.4) In 1955, Stelson and Mavis [27], conducted experiments of th i s type. They suspended cylinders, spheres and rectangles from a f l e x i b l e beam, set them into free vibr a t i o n and determined added mass quantities for the f i r s t mode frequencies in the manner described above. There was good agreement between their results and those obtained from a potenti a l flow solution. In 1960, Clough [6] performed si m i l a r experiments on hori z o n t a l l y oriented cylinders, plates and rectangles. By changing the length of the f l e x i b l e supports attached to these models he was able to r e a l i z e a set of systems with varying 8 f i r s t mode natural frequencies. He measured the added mass by performing tests which excited the f i r s t mode response of his models; his results also agreed clo s e l y with those predicted from pot e n t i a l flow theory. His testing was done by mounting a stationary water tank over a shaking table excited by a pendulum s t r i k i n g the edge of the table. In addition, Clough made measurements on a f l e x i b l e , v e r t i c a l cantilever model and evaluated the added mass corresponding to second mode vibrations by adding weights to the model in a i r to reproduce the same natural period as was measured underwater. He measured damping values as well in free v i b r a t i o n tests and found increased damping when the models were submerged. Clough also came to the important conclusion that i t was unlikel y that the st r u c t u r a l vibrations r e s u l t i n g from seismic loading would be large enough to induce flow separation, thus enabling one to use pote n t i a l flow theory in solving t h i s problem. In the free vibration experiments discussed above, the dependence of the added mass and damping on the actual base exci t a t i o n was not considered. The e x c i t a t i o n may vary in amplitude and frequency. In applying a potential flow solution, the added mass and damping must be independent of amplitude, since i t i s assumed that no flow separation occurs. This fact was checked in the present experimental study and found to be v a l i d . However, the added mass and damping values do depend on exc i t a t i o n frequency [4,19,26,31]. This i s because the amount of energy required to produce the surface 9 waves caused by structural motion varies with wave frequency, which i s the same as the structure's excitation frequency. The present study i s concerned with exploring how these parameters vary with excitation frequency. A dimensional analysis [20,26] of the problem (see Chapter 2.3) c l e a r l y i l l u s t r a t e s the frequency dependence of the hydrodynamic force. The problem i s governed by a second order d i f f e r e n t i a l equation with variable c o e f f i c i e n t s , representing the frequency dependent added mass and damping terms. In 1965, McConnell and Young [20] investigated the dependence of added mass and damping on the Stokes number, wa2/v, for a sphere in a bounded f l u i d . Here, w i s the excitation frequency, a i s the radius of the sphere and v i s the kinematic v i s c o s i t y of the surrounding f l u i d . They performed harmonic tests varying both w and v to give the added mass and damping as a function of the Stokes number. Although t h i s study was concerned mainly with the e f f e c t s of v i s c o s i t y and of an enclosing f l u i d boundary, variables which do not apply in the present problem, i t did show a s i g n i f i c a n t v a r i a t i o n in the added mass and damping with excitation frequency. These investigators i l l u s t r a t e d that for a given f l u i d , at a given excitation frequency, the problem can be resolved into a second order d i f f e r e n t i a l equation with constant c o e f f i c i e n t s , but that i f either the f l u i d properties or the excitation frequency are changed, the added mass and damping c o e f f i c i e n t s change also. In solving Laplace's equation for potential flow (where viscous e f f e c t s are 10 neglected), t h i s dependence on excitation frequency i s part of the solution, provided that f u l l kinematic and dynamic free surface boundary conditions are included in the analysis. A discussion of the boundary conditions i s contained in Appendix A. Taylor and Duncan [31], developed matrices of added mass and damping for a cylinder as functions of excitation frequency. Each element of the matrix i s represented by a graph of added mass or damping versus frequency corresponding to the d i s t o r t i o n mode of the matrix element. These were derived from potential flow theory. In the dynamic analysis of underwater structures, these 'wet' matrices from the added mass and damping are added to the corresponding 'dry' matrices and regular modal analysis follows for the structure. To v e r i f y t h e i r t h e o r e t i c a l l y derived matrices, the authors conducted experiments on a hinged cylinder capable of being deformed into f i r s t and second modes by a system of levers and cams. The model was excited sinusoidally by moving i t s top while the base was hinged to the bottom of a stationary wave tank. They concluded that their measured added mass and damping matrices were indeed a function of excitation frequency and that they agreed well with their t h e o r e t i c a l values. Perhaps the most extensive experimental study of the forces r e s u l t i n g from earthquake loading on underwater structures was c a r r i e d out on the shaking table in the Earthquake Engineering Laboratory at the University of 11 C a l i f o r n i a by Byrd in 1978 [4]. As i s the case in this present study, the table used by Byrd i s capable of sinusoidal and random motion from recorded earthquakes. This allows a wide range of excitation c h a r a c t e r i s t i c s and adds a r e a l i s t i c aspect to the study in that the models can be tested using actual earthquake records. In Byrd's study a pool l i n e r was placed over the table and supported by perimeter walls constructed independent of the table. A well instrumented model of a c y l i n d r i c a l underwater storage tank was attached to the table such that the bottom of the tank and the model underwent the same motion. The hydrodynamic forces a r i s i n g from horizontal, v e r t i c a l and ro t a t i o n a l motion were measured. Byrd performed free vibration tests to determine the natural frequency of the model in a i r and in water and sinusoidal tests to evaluate the added mass and damping terms as well as the t o t a l hydrodynamic force as a function of excitation frequency. He compared these re s u l t s to potential flow theory ignoring the free surface boundary conditions. As discussed e a r l i e r , the ommission of f u l l free surface boundary conditions results in the added mass and damping terms being independent of exc i t a t i o n frequency; they become constants for a given structure. As Byrd conducted his experiments at frequencies above 3 Hz, where the frequency dependence has been shown to be r e l a t i v e l y i n s i g n i f i c a n t [4,8], his values corresponded well with the t h e o r e t i c a l analysis. He concluded that while frequency dependence of the added mass and damping can be important for some structure 12 types at low frequency excitations, i t was not s i g n i f i c a n t for design purposes for structures whose dimensional proportions were similar to those of his model when excited over t h i s higher frequency range. As w i l l be discussed in a la t e r section, the present study investigates added mass and damping for frequencies between 0.5 and 6.0 Hz. For certain structure dimensions the frequency dependence of the added mass and damping i s quite s i g n i f i c a n t in this lower range of frequencies. Byrd also conducted random vibration tests and compared the experimentally measured base shear developed in his model to that obtained using the potential flow solution. In addition to laboratory experiments, some f u l l scale f i e l d tests on submerged structures have also been reported in the l i t e r a t u r e . Ruhl and Budhall [25], attached hydraulic actuators to an o i l r i g and applied sinusoidal forced vibrations to i t . They measured the f i r s t few mode shapes and periods and determined the damping c h a r a c t e r i s t i c s of the structure. This information i s useful for detecting damage from future earthquakes or heavy sea states by comparing the results with those obtained from similar measurements following such events. 13 3. OBJECT AND SCOPE OF INVESTIGATION The purpose of t h i s investigation i s to determine the dynamic c h a r a c t e r i s t i c s of large offshore structures (those which c l a s s i f y for a Laplace regime solution) from underwater tests of a c y l i n d r i c a l model. Such information i s required for the seismic design of prototype systems. In the process, a testing f a c i l i t y to study the e f f e c t s of earthquakes on a variety of underwater structures was developed. Testing was performed on a simple c y l i n d r i c a l structure f a l l i n g into the large body, d i f f r a c t i o n regime, which encompasses fl u i d - s t r u c t u r e interaction problems where flow separation does not occur. This allows a potential flow solution to be used. This i s the case for most earthquake excited motions of a submerged structure, since the r a t i o of displacement to cross-section i s usually small. The added mass and damping values were determined as a function of excitation frequency through a series of sinusoidal tests ranging from 0.5 to 6.0 Hz, encompassing the range of predominant frequency components found in an earthquake record. These values were then compared to added mass and damping values derived t h e o r e t i c a l l y by solving Laplace's equation for potential flow by means of a wave d i f f r a c t i o n theory computer program available in the Department of C i v i l Engineering at the University of B r i t i s h Columbia [18]. The added mass and damping values were then used to develop a transfer function between the input base 14 acceleration and the output base shear on the c y l i n d e r . This t h e o r e t i c a l l y derived transfer function i s then compared to experimentally derived transfer functions from the input and output data taken from random motion and sinusoidal t e s t s . Also, the output base shear frequency domain spectra derived from the t h e o r e t i c a l transfer function for a given random input were compared to the output spectra measured in the random experiments. The o v e r a l l goal i s to experimentally v e r i f y the use of the t h e o r e t i c a l l y derived added mass and damping values in developing a transfer function for application in the aseismic design of structures submerged in water. Figure 1 outlines the work done in t h i s study. Some discussion of these results in comparison to other similar studies i s included, p a r t i c u l a r l y to the findings of Byrd [4], whose testing program also covered some aspects of the research presently under consideration. 15 Experimental Determination of Added Mass and Damping compare < » Theoretical Determination of Added Mass and Damping Using Computer Program Experimental Random Excitation Tests used to V e r i f y the Theoretical Transfer Function compare 4 > Theoretical Development of Transfer Function for Model using Added Mass and Damping Values from Above Use Theoretical Determination of Added Mass and Damping to Develop Transfer Function in Design of Offshore Structures Figure 1 - Schematic of Work Performed in This Study 16 I I . THEORY Many a n a l y t i c a l studies which deal with f l u i d forces on submerged structures have been performed. Most of these are concerned with moving f l u i d s or waves on stationary structures. However, there are some studies which consider the structure moving in a stationary f l u i d ; t h i s i s the si t u a t i o n in the case of a submerged structure excited by an earthquake [1,9,14,16,18,19,21,22,23,24,29,30,31,33,34,35]. The purpose of t h i s chapter i s : to further define the type of problem with which t h i s study i s concerned, to offer a brief review of previous t h e o r e t i c a l work, to discuss the t h e o r e t i c a l solution used in t h i s study, and to develop the theory which describes how the added mass, added damping and the transfer functions may be obtained from the experimental work. 1. DEFINITION OF THE PROBLEM The type of f l u i d - s t r u c t u r e interaction problem of concern in t h i s study i s that in which no flow separation w i l l occur. There have been several t i t l e s given to t h i s class of problem in e a r l i e r sections of t h i s report: large body regime, wave d i f f r a c t i o n regime, Laplace potential flow regime and no flow separation regime. This class of problem w i l l be referred to from here on in as the Laplace regime. 17 It has the following c h a r a c t e r i s t i c s : - the body cross section is large in r e l a t i o n to the r e l a t i v e f l u i d - s t r u c t u r e motion, such that no flow separation occurs. This allows the assumption of an i r r o t a t i o n a l flow. - the body s i g n i f i c a n t l y interferes with and d i f f r a c t s any incoming wave t r a i n or, as in th i s earthquake case, i f the body i s near or penetrating the surface i t may produce appreciable surface waves. Laplace's equation for potential flow can be used to solve for the hydrodynamic force on a structure exhibiting the above c h a r a c t e r i s t i c s . A common parameter used in determining whether bodies f a l l into the Laplace regime i s the Keulegan-Carpenter number, K defined as [26]: K = 2jrA (2.1) D where, A = amplitude of the r e l a t i v e displacement between f l u i d and structure, D = structure diameter This number determines the significance of the flow separation and viscous drag forces in a problem. If K i s less than 10, i n e r t i a forces predominate over drag forces. If K i s greater 18 than 10, the drag forces are predominant and the problem moves into the small body, viscous flow regime. If K i s less than 2, flow separation, and thus the drag force, i s n e g l i g i b l e . Therefore, a structure which exhibits a K value of less than 2 under either earthquake or wave loading can be studied using Laplace's equation for potential flow. Most st r u c t u r a l motions res u l t i n g from earthquakes f a l l into t h i s category, since the displacements would generally be less than 32% (A/D<1/7T) of the structure's diameter. This study pertains to r e l a t i v e l y smooth structures having well rounded shapes. Very rough surfaces or abrupt changes in cross section may result in s i g n i f i c a n t flow separation even i f the body cross section i s large in r e l a t i o n to r e l a t i v e f l u i d - s t r u c t u r e motion. 2. ASSUMPTIONS AND CONDITIONS OF FLUID STATE In analyzing for the hydrodynamic forces on a submerged body in the Laplace regime, the following assumptions of the f l u i d state are usually made: i) No Flow Separation: Separation of the f l u i d from the body i s prevented by ensuring that the Keulegan-Carpenter number i s less than 2 as discussed in section 1 of t h i s chapter. Any l o c a l i z e d separation due to abrupt changes in shape or body appendages i s neglected. With th i s s i t u a t i o n , no s i g n i f i c a n t drag forces w i l l occur. This i s also a necessary condition of the physical problem in 19 order to make assumption i i ) . i i ) I r r o t a t i o n a l Flow: This results in an ideal potential flow and allows the use of Laplace's equation for potential flow to solve the problem. i i i ) Linear Wave Theory: This i s also referred to as small amplitude wave theory. The wave height i s assumed small in comparison to the o v e r a l l water depth and wave length thus allowing the free surface boundary conditions to be l i n e a r i z e d and applied at the s t i l l water l e v e l (appendix A), reference [26], iv) S t i l l F l u i d : We assume that the water disturbances are due only to the structure motion; currents, waves and other outside disturbances of the water around the structure are neglected (and kept to a minimum in the experimental t e s t s ) . v) Incompressible F l u i d : This assumption i s v a l i d for most studies of f l u i d - s t r u c t u r e i n t e r a c t i o n . Liaw and Chopra [19] studied t h i s topic in r e l a t i o n to dams and submerged towers and came to the conclusion that for most p r a c t i c a l applications, f l u i d compressibility can be ignored. However, for some structure dimensions and for high frequency e x c i t a t i o n , the energy dissipated in f l u i d compression waves becomes s i g n i f i c a n t and f l u i d 20 compressibility must be considered. vi) Surface Wave Production: As can be seen in appendix A, ignoring surface wave radiation in the analysis s i m p l i f i e s the free surface boundary conditions and thus the solution of Laplace's equation. For some structures t h i s assumption leads to good re s u l t s ; however, for many sit u a t i o n s , t h i s condition should be included, as the surface waves produced by the structure influence the value of added mass and damping. The production of surface waves by the moving body results in the existence of the added damping and in the added mass and damping both being dependent on excitation frequency [4,19,26,31]. The e f f e c t of surface waves i s included in t h i s study in both the t h e o r e t i c a l and experimental determination of added mass and damping. 3. REVIEW OF THEORETICAL.STUDIES A b r i e f review of e x i s t i n g t h e o r e t i c a l studies pertaining to structures in the Laplace regime when excited by harmonic or random base motion i s offered here. This i s desirable in r e l a t i o n to the interpretation of the experimental data obtained in t h i s study. An analysis of the equation of motion (discussed in the next section), s i m p l i f i e d by the fact that the added mass and damping are considered to be constant (independent of structure e x c i t a t i o n ) , has been c a r r i e d out by Penzien and 2 1 Kaul, who performed regular modal and spectral analysis on offshore towers [23]. In t h i s analysis, the added mass and damping are simply added to the dry mass and damping of the structure. This approach appears to give results which are suitable for design approximations of some types of structures; i t is used quite often in practice. The solution of Laplace's equation for potential flow on o s c i l l a t i n g bodies has been studied in varying degrees of complexity. The most comprehensive investigation was done by Liaw and Chopra [19], who developed a solution for a surface piercing cylinder which incorporates water compressibility and the surface waves produced by the moving structure. A simpler solution for the same problem neglecting these e f f e c t s was presented by Helou and Tung [14,33]. The ef f e c t of including surface waves and water compressibility can be seen in Figure 2, taken from the work of Liaw and Chopra. Helou and Tung also extended their work to f u l l y submerged structures. Laplace's equation, the necessary boundary conditions and the solution used in the present investigation are discussed in appendix A. Taylor and Duncan [31], developed a design method employing added mass and damping matrices which were dependent on exc i t a t i o n frequency. These are derived from Laplace's equation for potential flow and are used in design with a regular modal analysis. They v e r i f i e d their t h e o r e t i c a l results of these matrices by experiment. In addition to the c l a s s i c a l solution of Laplace's 22 equation, studies for evaluating f l u i d forces have been performed using f i n i t e element techniques [16,19,22]. Although these investigations are often c o s t l y , due to the large number of equations to be solved, they are very useful in examining the response of bodies exhibiting complex geometry. Some recent studies [16] using f i n i t e elements have also included viscous e f f e c t s . 23 FREQUENCY RESPONSES OF SQUATTY TOWER (^/Hi«0.25) WITH l/f02 FIRST MODE RESPONSES-Y,: I-NO SURROUNDING WATER,H>0 Z-FULLY SUBMERGED TOWER, H^H. SECOND MODE RESPONSES - \ : S-NO SURROUNDING WATER, H«0 4-FULLY SUBMERGED TOWER.H^H. 0 I 2 S 4 ttCITATION FREQUENCY 5 6 7 WATER COMPRESSIBILITY INCLUDED NEGLECTED gure 2 - Ef f e c t s of Surface Waves and Compressibility from Liaw and Chopra, reference 19 24 4. EQUATION OF MOTION The equation of motion for a single degree of freedom system submerged in water i s developed in t h i s section. This i s usually the star t i n g point in any study dealing with the dynamic behaviour of a system. In general, the kinetic energy of the system gives us the i n e r t i a terms of the equation of motion. In the submerged case, the t o t a l kinetic energy i s made up of the kinetic energy of the structure, T s: T s = JmX2 (2.2) 2 where m i s the structure mass and X i s the structure v e l o c i t y , and the kin e t i c energy of the f l u i d , Tf: Tf = lm aX 2 (2.3) 2 where ma i s the mass of a volume of f l u i d set into motion by the structure. The t o t a l kinetic energy of the system T^ i s then: TT • T s + Tf = _1_(m + m a)X 2 2 (2.4) 25 Using the Lagrange method [7] of formulating the equation of motion, we may write: where X i s the structure acceleration. The damping term in the equation i s determined from the energy d i s s i p a t i o n in the system. In addition to the usual s t r u c t u r a l damping, C, experiment and theory both show that the f l u i d also contributes some damping to the system. For the Laplace regime t h i s comes mainly from the energy dissipated by the structure producing surface waves, although some damping from skin f r i c t i o n w i l l also be present. Assuming that t h i s added damping i s proportional to the structure v e l o c i t y , the t o t a l damping term i s then: where C a i s the added damping from the f l u i d . Since in our experiments i t was d i f f i c u l t to measure only the damping which was due to the f l u i d , the t o t a l damping of the system was measured and a t o t a l damping value, X= C + C &, i s used. The st r u c t u r a l damping was determined by tests in a i r and subtracted from the t o t a l damping to obtain the added (2.5) which gives the i n e r t i a term: (m + ma)X (2.6) (C + C a)X (2.7) 26 damping values used as a comparison to the th e o r e t i c a l values. Now, including the s t i f f n e s s term, K, and a forcing function term, F ( t ) , the equation of motion for a single degree of freedom system in water i s : (m + ma)X + XX + KX = F(t) (2.8) It has been shown [20,26] (see section 5) that ma and C a are dependent on the f l u i d density, the size and shape of the structure and the frequency of o s c i l l a t i o n of the body. The dynamic analysis of f l u i d - s t r u c t u r e systems usually involves the determination of the added mass and damping terms and the i r dependence on the above factors. The present study i s concerned with such an evaluation. 5. DIMENSIONAL ANALYSIS OF HYDRODYNAMIC FORCE As was stated in section 4, for the case of structures in the Laplace regime, the added mass and damping are dependent on the f l u i d , body size and shape, and the frequency of ex c i t a t i o n . The added mass and damping are derived from the hydrodynamic force and i t can be shown by a dimensional analysis that these quantities are dependent on the above factors. In general, for any type of r i g i d body f l u i d - s t r u c t u r e problem, the f l u i d force can be given as [26]: 27 F = f ( d,, H, D, Re ) (2.9) pgdD2 L L L where, F = f l u i d force p = f l u i d density g = gra v i t a t i o n a l acceleration H = wave height D = structure cross-section (diameter) d = water depth L = wave length; for the earthquake problem t h i s represents excitation frequency Re = Reynolds number = VD/u where v is kinematic v i s c o s i t y and V i s v e l o c i t y Using the f l u i d assumptions discussed in section 2.2 for th i s regime of problem, the dimensionless parameters Re and ' H/L disappear, as viscous e f f e c t s are assumed ne g l i g i b l e and the wave height i s assumed s u f f i c i e n t l y small for lin e a r wave theory to apply. The dimensional equation for the hydrodynamic force then becomes: F = f ( _d, _D ) (2.10) pgcS 2 L L It i s seen here that the f l u i d force i s dependent on the f l u i d density, structure size and frequency of ex c i t a t i o n . A similar analysis was carried out by McConnell and Young [29], who came to the same conclusion. 28 6. DESCRIPTION OF AXIDIF COMPUTER PROGRAM AXIDIF i s the name of a computer program developed at the University of B r i t i s h Columbia for studying f l u i d forces on offshore structures. The theory and method used in the program to solve such problems are taken from references 26 and 18. It calculates t h e o r e t i c a l values of added mass and damping for r i g i d body, axisymmetric structures in the Laplace regime as a function of excitation frequency. AXIDIF was developed for the purpose of determining wave loading on structures, but the added mass and damping values derived from i t are v a l i d also in the case of the base motion problem, as discussed in Appendix A. AXIDIF solves Laplace's equation using wave d i f f r a c t i o n theory. The ve l o c i t y potentials for the incoming and r e f l e c t e d wave trains and for the radiated waves due to the structure motions are determined separately and then combined for the t o t a l v e l o c i t y p o t e n t i a l . The approach used i s based on a boundary element method involving an axisymmetric Green's function [26,18], The f u l l kinematic and dynamic free surface boundary conditions are incorporated in t h i s way and thus the dependence of the added mass and damping on excitation frequency i s included. The goal of t h i s study i s to v e r i f y experimentally the added mass and damping values determined from AXIDIF. Once t h i s has been done the program can then be used with confidence to provide these values for the design of f u l l scale structures f a l l i n g in the Laplace regime. 29 7. DERIVATION OF ADDED MASS AND DAMPING FROM EXPERIMENT As w i l l be discussed in la t e r sections, sinusoidal and random tests were used in the experimental program. The sinusoidal tests were conducted over a range of frequencies lying between 0.5 and 6.0 Hz in order to determine the added mass and damping c o e f f i c i e n t s as a function of excitation frequency. In each frequency t e s t , the forces acting on the test cylinder were determined by measuring i t s base shear r e s u l t i n g from a known sinusoidal input acceleration. The base shear V ( t ) , for an acceleration excitation a ( t ) , given by: a(t) = acos(wt) (2.11) where a i s acceleration amplitude and w i s the excitation frequency i s : V(t) = Vcos(wt + 0 ) (2.12) where V i s the base shear amplitude, and <t> i s the phase s h i f t between the acceleration and base shear records. A free body diagram of the forces acting on the cylinder i s shown in Figure 3. Here, F f ( t ) i s the f l u i d force on the cyli n d e r , ma(t) i s the i n e r t i a force of the cylinder with m being the dry mass of the cyl i n d e r , V(t) i s the base shear and M(t) the base moment acting on the cylinder at any time t. 30 F f ( t ) m a ( t ) <-V ( t ) M ( t ) F i g u r e 3 - F r e e Body D i a g r a m o f F o r c e s A c t i n g on M o d e l C y l i n d e r F o r a g i v e n s i n u s o i d a l d i s p l a c e m e n t X ( t ) = X c o s ( w t ) ( 2 . 1 3 a ) w here X i s t h e a m p l i t u d e o f d i s p l a c e m e n t , t h e v e l o c i t y i s : X ( t ) = - X w s i n ( w t ) ( 2 . 1 3 b ) a n d t h e a c c e l e r a t i o n i s : a ( t ) = X ( t ) = - w 2 X c o s ( w t ) ( 2 . 1 3 c ) 31 where w2X represents the acceleration amplitude a, previously defined. Taking equilibrium of forces on the free body diagram (Figure 3), the resulting equation i s : Ff(t) + V(t) = matt) (2.14) and applying equations 2.12 and 2.13c to 2.14 gives: F f + Vcos(wt + 4>) = -mw2Xcos(wt) (2.15) The moment on the cylinder was not considered in th i s a n a l y s i s . The base shear can be resolved into i t s components: V(t) = Vcos(wt + <t>) = V,cos(wt) + V 2sin(wt) (2.16) where V, = Vcostf and V 2 = Vsin0. Introducing t h i s into equation 2.15 gives: -mw2Xcos(wt) = V,cos(wt) + V 2sin(wt) + F f (2.17) For sinusoidal input, the f l u i d force Ff i s sinusoidal and can be resolved to represent added mass and added damping as follows: 32 Sinusoidal F l u i d Force / \ Kinetic Energy Energy Dissipation I I Inertia Force Damping Force I I In Phase With Motion In Quadrature With Motion Added Mass = m. Added Damping = C a The f l u i d force can then be represented as: F f = F,cos(wt) + F 2sin(wt) = m X + XX (2.18) Applying equation 2.18 along with 2.13, equation 2.17 becomes: -mw2Xcos(wt) = V,cos(wt)+V2sin(wt)+m_w2Xcos(wt)+XwXsin(wt) (2.19) Re-arranging 2.19 yi e l d s the equation of motion for t h i s problem, similar to equation 2.8, only lacking the s t i f f n e s s term KX, since the cylinder i s r i g i d : 33 -(m + ma) w2Xcos (wt) - XwXsin(wt) = V^ostwt) + V 2sin(wt) (2.20) On solving for the added mass ma, and the t o t a l damping X, which includes both structural and f l u i d damping, one obtains: - (in + mn)w2X = V, or ma = V^., - m = JL, - m (2.21 ) w2X a and -XwX = V 2 or X = zS 2 X = _2.2w (2.22) wX a The same tests were f i r s t performed in a i r to determine the 'dry' mass and damping values. It should be noted here that ma and X are functions of both e x c i t a t i o n frequency w, and displacement X. By invoking the f l u i d assumptions of section 2, the problem s a t i s f i e s the Laplace regime so that the displacement i s eliminated as an influencing variable. This fact was checked in t h i s study experimentally by varying X at a constant frequency w, and determining m a and X for our model. The experiments were performed to v e r i f y the added mass and damping values derived from the Laplace equation. It i s important, therefore, that any forces which a r i s e as a result of viscous action do not have a large influence on the added mass and damping values derived from our model te s t s . Since we are dealing with a rea l f l u i d which does have v i s c o s i t y , i t was expected that the measured added mass and damping values would have some dependence on displacement. This condition 34 was, in fact, observed. The ef f e c t was small, however, and the experiments therefore represented the c h a r a c t e r i s t i c s of potential flow reasonably well. 8. DEVELOPMENT OF TRANSFER FUNCTION The transfer function r e l a t i n g cylinder base shear V in water, with input acceleration a, i s a useful design parameter. The transfer function for the r i g i d cylinder model of t h i s experiment or for any r i g i d submerged structure can be derived as follows. The equation of motion for t h i s case, from equations 2.14 and 2.18, can be written: Letting X = Xexp(iwt) and V = Vexp(iwt), where X and V are both amplitudes, we get: (m + m a)(iw) 2Xexp(iwt) + X(iw)Xexp(iwt) = Vexp(iwt) (2.24) which gives: The transfer function r e l a t i n g base shear to displacement i s then: (m + ma)X + XX = V (2.23) [ -(m + ma)w2 + Xiw ]X = V (2.25) V = /(m + ma)w4 + X2w2 X (2.26) 35 U s i n g a = w 2X, t h e t r a n s f e r f u n c t i o n r e l a t i n g b a s e s h e a r t o a c c e l e r a t i o n i s : |H(w) | = V a = 1 /(m + m j 2 w 2 + X 2 (2.27.) — cL W where m a and X a r e f u n c t i o n s o f f r e q u e n c y w, a s s t a t e d e a r l i e r i n s e c t i o n 5. T h e s e v a l u e s c a n be d e r i v e d f r o m e x p e r i m e n t o r t h e o r y . 9. DERIVATION OF TRANSFER FUNCTION FROM EXPERIMENTS As s t a t e d i n s e c t i o n 7, t h i s s t u d y i n v o l v e d s i n u s o i d a l a n d random t e s t i n g . The s i n u s o i d a l t e s t s were u s e d t o d e t e r m i n e a d d e d mass an d d a m p i n g v a l u e s a s a f u n c t i o n o f f r e q u e n c y , w h i c h were c o m p a r e d t o t h e t h e o r e t i c a l l y d e r i v e d v a l u e s . They were a l s o u s e d t o d e v e l o p t h e t r a n s f e r f u n c t i o n by t a k i n g t h e r a t i o o f t h e i n p u t t o t h e o u t p u t a t e a c h f r e q u e n c y v a l u e ( e q u a t i o n 2 . 2 8 ) . H(w) = V(w) ( 2 . 2 8 ) a (w) The random t e s t s were u s e d a l s o t o d e v e l o p , e x p e r i m e n t a l l y , t h e t r a n s f e r f u n c t i o n r e l a t i n g b a s e s h e a r t o i n p u t b a s e a c c e l e r a t i o n f o r t h e c y l i n d e r m o d e l . T h i s l a t t e r t r a n s f e r f u n c t i o n was t h e n u s e d t o c h e c k t h e v a l i d i t y o f t h e t r a n s f e r f u n c t i o n d e r i v e d t h e o r e t i c a l l y . The a d d e d mass an d d a m p i n g v a l u e s o f t h e t h e o r e t i c a l l y d e t e r m i n e d t r a n s f e r f u n c t i o n w e r e d e r i v e d u s i n g t h e A X I D I F c o m p u t e r p r o g r a m . 36 In the random tests, earthquake excitations a ( t ) , were used to excite the cylinder in water. The random base shear V ( t ) , was recorded. The power spectral densities, Sa(w) and Sv(w) corresponding to a(t) and V(t) respectively, were calculated from: Sv(w) = [ ; v ( t ) e " i w t d t ] 2 (2.29) -OO and Sa(w) = [ Ja(t ) e ~ i w t d t ] 2 (2.30) -oo These power spectral densities were evaluated using a Fast Fourier Transform (FFT) computer program. From random analysis theory [2,7], the transfer function i s then calculated as: |H(w)|2 = Sv(w) (2.31) Sa(w) The measured output base shear spectrum, Sv(w), was also compared to the spectrum derived t h e o r e t i c a l l y . This was accomplished by multiplying the input base acceleration spectrum of the earthquake record by the t h e o r e t i c a l l y derived transfer function. The experimental base shear spectrum was determined by performing a Fourier analysis of the recorded output data. 37 I I I . MODEL AND TESTING APPARATUS It was the intention in this study to create experiments which represented a structure undergoing base motion in the Laplace regime of flu i d - s t r u c t u r e interaction. The conditions necessary to qu a l i f y for t h i s regime were outlined in chapter two. As described in the following sections, a model and testing apparatus were developed which enabled these conditions to be s a t i s f i e d . 1. DEVELOPMENT OF TESTING APPARATUS The te s t i n g was performed in the Seismic Simulation Laboratory of the C i v i l Engineering Department at U.B.C. This laboratory contains a single degree-of-freedom shaking table capable of a maximum peak-to-peak displacement of six inches. A PDP-11 mini-computer i s used to operate the table. It i s capable of ex c i t i n g the table with sinusoidal frequencies from 0.0 to 30.0 Hz and with simulated earthquake records which are stored on tape. The table was used to produce the sinusoidal and random base excitations on the model. The PDP-11 was also used to tabulate and process a l l of the data recorded in the experiments. Figure 4 shows a photograph of the shaking table. A water tank was constructed to straddle the table so that the model could be set into motion underwater. The goal in designing the tank was to create the required f l u i d conditions for testing and to enable the model to be properly 38 excited through the base by the table. The required f l u i d conditions were: i) no wave r e f l e c t i o n from the walls. i i ) no water disturbances caused by the table motion other than through the test model. i i i ) no viscous interaction between the tank walls and the model. In addition to these requirements, i t was also desired to construct a f a c i l i t y which could incorporate future testing of a variety of models and experiments. Two similar studies involving experimental tests [4,6], used a water tank with the shaking table acting as the floor of the tank. It was f e l t that t h i s approach would not be suitable in the present test case since the U.B.C. table moves by rocking on four legs and thus has a small v e r t i c a l component which would cause water disturbances. It was decided to construct the tank independent of the table with only the model base in contact with the moving table. As a re s u l t , the table was designed to completely straddle the table and to be supported on the laboratory floor surrounding i t . A hole was cut in the center of the tank f l o o r , through which the model could be fastened to the table. Figure 5 i s a photograph of the tank and model apparatus. Figure 6 shows a schematic of the tank. The tank dimensions are 1 2 X 1 3 X 4 feet and i t consists of a steel frame with plywood sheathing. A p l a s t i c pool l i n e r i s used as 39 a seal. Horse hair f i l t e r s are placed around the inside perimeter of the tank to dissipate the surface water waves and prevent them from being r e f l e c t e d back from the walls. The tank i s supported by ten legs around i t s perimeter, which rest on the floor surrounding the table. Since the model was attached to the shaking table through a hole in the tank f l o o r , i t was necessary to design a watertight seal at the assembly base to allow for the motion of the model. Figure 7 shows the sealing arrangement used. An aluminum plate was fastened to a r i g i d wood block which was mounted to the table through a plywood sheet. The model base was attached to th i s plate. Another aluminum plate was sealed into the bottom of the tank at i t s center through which an 18 inch hole was placed. A round rubber sheet was then fastened between the base plate on the table and the plate on the bottom of the tank by means of a stain l e s s steel sealing ring. This formed a seal between the tank and the table allowing f u l l transfer of table motion to the model. To prevent f l u i d disturbances from t h i s mounting apparatus, an aluminum sealing ring, which reduced the hole diameter to 9 inches, was used to seal the rubber to the tank f l o o r . Over t h i s aluminum ring, a smaller 14 inch diameter plexiglass disc was attached to the model support. This e f f e c t i v e l y kept the water set into motion by the mounting apparatus from disturbing the water surrounding the model. Figure 5 - Photograph of Tank and Model Apparatus F i g u r e 6 - Schematic of T e s t i n g Tank Stainless Steel Shoft Support Stoinless Steel Sealing Ring Rubber Sealing Membrone Aluminum Sealing Ring Aluminum Plate - Tank Bottom •Aluminum Plate - Table Motion •Wooden Block - Table Motion -Shaking Table Scale= 1"= 8" 43 2. DESIGN OF MODEL The test model was selected to s a t i s f y the requirements for a potential flow si t u a t i o n . The available AXIDIF computer program yie l d s a theoretical solution to Laplace's equation for axisymmetric bodies. Accordingly, a c y l i n d r i c a l model was chosen for the tests, since t h i s i s the simplest shape for representing the Laplace regime and, as well, i t i s quite common in offshore construction. Furthermore, the few exis t i n g experimental studies which are available for comparison purposes also used c y l i n d r i c a l shapes. The dimensions of the cylinder were chosen so as to meet the requirements of no flow separation (Keulegan-Carpenter Number < 2). This corresponds to a maximum allowable displacement to diameter r a t i o of A/D < 1/TT or D/A > ir (see chapter 2). The maximum base amplitude used in the tests was 1.5 inches. An 11 inch diameter cylinder was chosen, such that D/A = 11/1.5 = 7.3 > jr. The cylinder model i s shown in Figure 8. It i s 22 inches high and made of aluminum. The c y l i n d r i c a l s h e l l i t s e l f i s meant to be r i g i d . It consists of a 3/32 inch outer s h e l l with a 3/4 inch plate at the top and a 1/8 inch plate on the bottom. The cylinder i s attached r i g i d l y to the top of a 1 and 1/4 inch diameter steel shaft which i s fastened at the bottom to the shaking table through a stainless s t e e l base. The cylinder i s made water tight by sealing the 4 inch diameter hole through which the shaft passes in i t s bottom with a rubber membrane. The res u l t of t h i s arrangement i s 44 that a l l force on the outside of the cylinder i s transferred to the top of the shaft. The shaft then acts as an end loaded cantilever through which we can measure the t o t a l force on the c y l i n d e r . The c r i t e r i a for designing the model were: i) to have i t act as a r i g i d cylinder i i ) to have the s t e e l shaft f l e x i b l e enough to measure strains at a l l load l e v e l s i i i ) to have the natural frequency of the system high enough so as not to cause any resonant interference with the test frequencies used (preferably above 20 HZ). Meeting these c r i t e r i a proved to be quite d i f f i c u l t . It was not possible to select a steel shaft which was f l e x i b l e enough to measure small strains yet s t i f f enough to have a natural frequency above 20 Hz. The design f i n a l l y s e t t l e d on had a natural frequency in water of about 16 Hz (determined from free v i b r a t i o n t e s t s ) ; the small strains developed were measured with the aid of considerable electronic a m p l i f i c a t i o n . 45 Lead Wires f Aluminum Top Strain Gauges |"Aluminum Flange §2 Aluminum Can li"Steel Shaft ^ Aluminum Bottom Rubber Membrane Seoling Flange Seoling Ring Scale • \" = 4 " F i g u r e 8 - Diagram of Model C y l i n d e r 46 3. DATA MEASUREMENT As discussed in chapter 2, the data necessary for determining the added mass and damping values in these tests was the input base acceleration record, a, and the resulting cylinder base shear record, V. In addition to these values, the displacement of the base, X, also was measured to keep track of i t s value during the tests and as a check on the acceleration record through the simple harmonic r e l a t i o n a=-w2X, where w i s the harmonic input frequency. The displacement was measured by means of an LVDT situated on the shaking table. The acceleration was recorded by an accelerometer attached to the table. This l a t t e r record was used in analysis instead of the known exci t a t i o n record in order to account for any discrepencies between the input command motion and the actual recorded table motion. The base shear was measured by a Wheatstone bridge arrangement using four s t r a i n gauges mounted on the steel shaft. The steel shaft, base and stra i n gauges are shown in the photograph of Figure 9. This system measured the strains at the top and bottom of the shaft, from which the moments at these points could be calculated. Thus M = eEI (3.1) y where M = moment e = s t r a i n measured 47 E = Young's modulus of the shaft material I = moment of i n e r t i a of the shaft cross section y = distance between the neutral axis and the surface of the shaft The Wheatstone bridge set up also yielded the difference between the top and bottom moments, after which the base shear could be determined from: base height of shaft ( 3 . 2 ) The model was ca l i b r a t e d i n i t i a l l y through s t a t i c load tests which correlated a given base shear value with a voltage output from the bridge. The c a l i b r a t i o n curve was l i n e a r . The required data from the tests were the amplitudes of the base shear, V, the base acceleration, a, and the phase s h i f t between these variables, <j>. More information on how the data were recorded and processed to give the above values i s provided in Appendix B. 48 Figure 9 - Photograph of Steel Shaft, Base and Strain Gauges 49 IV. DESCRIPTION OF EXPERIMENTS The purpose of th i s study was to experimentally determine the dynamic c h a r a c t e r i s t i c s of submerged structures due to seismic loading. The dynamic c h a r a c t e r i s t i c s of interest are the added mass and damping due to the f l u i d . The th e o r e t i c a l derivation of the frequency transfer function, H(w), re l a t i n g base shear to base acceleration was also tested experimentally. Of pa r t i c u l a r interest in t h i s study i s the frequency dependence of the added mass and damping values, which i s s i g n i f i c a n t at the lower end of the frequency scale and in surface piercing structures which produce surface waves. The testing consisted of two phases: 1) Sinusoidal tests between 0.5 and 6.0 Hz which i s the frequency range of importance in most recorded earthquakes. 2) Random motion tests, using records of actual earthquakes, to confirm the t h e o r e t i c a l l y derived transfer function between input base acceleration and output base shear on the cylinder The tests were carr i e d out with the cylinder in two sit u a t i o n s : 1) Surface Piercing 50 2) Submerged to a depth of one times the cylinder radius 1. SINUSOIDAL TESTS The f i r s t set of sinusoidal tests were done at frequency leve l s of 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, and 6.0 Hz. After these were analyzed i t was r e a l i z e d that the greatest fluctuation in added mass and damping values occurred below 2.5 Hz, so additional testing at increments of 0.25 Hz was conducted in t h i s range. Photographs of the sinusoidal testing being performed are shown in Figures 10 and 11. The analysis for added mass and damping was based on linear wave theory (chapter 2). This was checked v i s u a l l y in the testing and controlled by reducing the input amplitude i f any peaking or nonlinear wave c h a r a c t e r i s t i c s appeared. At the lower frequencies t h i s was not a problem and quite large amplitudes could be used. However, at the higher frequencies the amplitudes had to be kept small in order to prevent the waves from breaking and becoming nonlinear. This situation would probably be re f l e c t e d in real earthquake loading, as displacements at higher frequencies are usually not large. In applying Laplace's equation, i t i s necessary that the added mass and damping values be independent of displacement, which means that viscous effects are considered to be n e g l i g i b l e . In r e a l i t y , of course, water i s viscous and some eff e c t s on the results are to be expected. This was checked 5 1 by running tests at a given frequency for several displacement amplitudes and determining added mass and damping values. The results of t h i s study are shown in Figures 12 and 14, where more than one value of added mass and damping i s noted at a given frequency. When the results were f i r s t analyzed, i t was suspected that at higher frequencies there was some amplification in the acceleration of the top of the cylinder r e l a t i v e to the base value, presumably as a res u l t of approaching a resonant condition for the model. This would result in the cylinder, undergoing a rocking mode rather than a pure t r a n s l a t i o n a l mode. This was checked by repeating the tests with an accelerometer attached to the top of the cylinder as well as to the bottom. A small increase in the top acceleration was noted at higher frequencies and could be accounted for through the resonant amplification factor: *top= a b a s e [ i 1 ( ^ 2 ] (4.1) where w i s the excitation frequency and wn i s the natural frequency of the model in water (16 Hz). A correction was applied to account for t h i s small ef f e c t when analyzing the data, as discussed in the next chapter. Figure 11 - Photograph of Sinusoidal Test 53 2. RANDOM TESTS As was shown in chapter 2, the transfer function for the model can be determined experimentally from random tes t i n g . The transfer function can also be evaluated experimentally from a series of sinusoidal tests. The former approach serves as a v e r i f i c a t i o n of the theoretical transfer function in pseudo earthquake loading. The base acceleration, a ( t ) , and base shear, V ( t ) , time h i s t o r i e s were recorded for input excitations of the E l Centro N-S, 1940 and San Fernando S74W, 1971 earthquakes. The earthquake data were taken from taped records of the actual earthquakes. In order to keep the displacements within the l i m i t s of the shaking table and of the model during t e s t i n g , the earthquake records were scaled in amplitude to an acceptable l e v e l . Again the wave condition was monitored v i s u a l l y ; no nonlinear c h a r a c t e r i s t i c s were observed. The time series data were transferred into the frequency domain in the form of power spectral density functions by way of a Fast Fourier Transform program. These power spectral density functions were then employed to calculate an experimental transfer function, which was used to check the theo r e t i c a l values. Also, the spectral density of the output derived using the the o r e t i c a l transfer function was compared to the spectral density of the recorded output data. The time series derived from this theoretical spectral density of the base shear output was also compared to the time series output recorded during the random test. 54 3. SURFACE PIERCING AND SUBMERGED TESTS The frequency dependence of the added mass and damping values results from energy dissipation in the system due to the production of surface waves. Byrd [ 4 ] and Liaw and Chopra [ 1 9 ] , discussed this e f f e c t . This frequency dependence i s accounted for in the a n a l y t i c a l determination by incorporating f u l l dynamic and kinematic free surface boundary conditions in the solution of Laplace's equation. Frequency dependence i s most s i g n i f i c a n t for surface piercing structures at low freqencies. As the structure i s submerged, and as the frequency increases, i t has less tendency to produce surface waves, and the frequency dependence becomes n e g l i g i b l e . Liaw and Chopra show t h i s t h e o r e t i c a l l y in solving Laplace's equation for p o t e n t i a l flow and Byrd shows thi s experimentally in his tests on models which are submerged and are excited at higher frequencies. Byrd defined a factor, 27rg/w2, which i s the wavelength for a wave of frequency w, such that i f the depth of submergence of the structure i s greater than t h i s value, the effect of surface waves diminishes. This factor also gives an indication of the effect of frequency on the hydrodynamic force. For lower values of frequency, the factor i s large, indicating higher frequency dependence and t h i s reduces quickly for increasing frequency values. In order to explore t h i s condition, tests were performed on both a surface piercing cylinder and a cylinder submerged to a depth of one times i t s radius under the surface. It was 55 expected, and indeed observed, that the frequency dependence of the added mass and damping for the surface piercing cylinder was much more s i g n i f i c a n t than for the submerged case. The frequency values tested (0.5 - 6.0 Hz) were also in the frequency range necessary to investigate this dependence. Byrd's tests were c a r r i e d out above 3 Hz. Earthquakes can be expected to contain considerable power below th i s frequency so that an investigation below th i s 3 Hz l i m i t was considered to be desirable. 56 V. RESULTS AND DISCUSSION 1. ADDED MASS FOR SURFACE PIERCING CYLINDER The added mass values were determined as a function of frequency through a series of sinusoidal tests as discussed in chapter 4. The base acceleration and base shear time history records were processed as discussed in appendix B to obtain the peak values and phase s h i f t s . These values were then used as shown in chapter 2 to calculate the added mass. The theo r e t i c a l values of added mass versus frequency were determined using the computer program AXIDIF, which solves Laplace's equation for potential flow (chapter 2 and appendix A). Figure 12 shows the added mass versus frequency for the surface piercing cylinder. The added mass i s plotted as a dimensionless value, n^/pr 3, where p i s water density and r i s the radius of the cylin d e r . The agreement between experiment and theory i s very good. It i s important f i r s t to note the large fluctuation in the curve below 2.5 Hz. The frequency dependence of the added mass i s quite evident at frequencies less than t h i s value. As the frequency increases above 2.5 Hz, the added mass tends to be a constant, independent of frequency. This agrees well with the experimental work of Byrd [4], who ca r r i e d out tests above 3.0 Hz and obtained a constant value of added mass, independent of frequency. This also agrees well with the 57 t h e o r e t i c a l work of Liaw and Chopra [19], whose results for hydrodynamic force were previously shown in Figure 2. The radiation of surface waves by a moving structure i s c l a s s i f i e d as a dispersive type of energy propagation [2]. This means that the ve l o c i t y of the energy propagation wave i s dependent on the frequency of o s c i l l a t i o n of the structure. The propagation velocity increases with the excitation frequency for the case of surface wave production by a moving structure. The phase s h i f t between the v e l o c i t y of the structure and the ve l o c i t y of the propagating waves also varies. It i s t h i s phase s h i f t that causes the large fluctuation in added mass and damping at the low end of the frequency scale and not at the higher end. This was noted in the experiments, where the phase s h i f t between the structure acceleration (velocity) and the f l u i d force on the structure exhibited the same peaking tendencies as the added mass and damping curves - starting at 0° for 0. Hz, r i s i n g to a peak at about 1.0 Hz and then dropping back to 0° as the frequency increased. The degree of fluctuation in the added mass values also depends on the depth of the structure. The dependence on frequency i s more s i g n i f i c a n t for shallow structures, where a greater percentage of the body surface i s affected by wave action. For the surface p i e r c i n g case i t might be expected that the frequency dependence would be most s i g n i f i c a n t for shallow, large diameter structures which have a high percentage of surface area in contact with surface waves. The 58 e f f e c t would be least s i g n i f i c a n t for t a l l , deep, small diameter structures (but s t i l l f a l l i n g within the Laplace regime). This e f f e c t i s apparent i f one uses the AXIDIF program to solve the problem t h e o r e t i c a l l y for various sizes of c y l i n d e r s . It should also be noted in Figure 12 that there is more than one value of added mass plotted for most of the frequency values tested. This arises from the dependence of added mass on displacement, which i s related to the fact that water i s viscous, and not a true ideal i n v i s c i d f l u i d . As discussed in chapters 2 and 4, th i s was checked by running several sinusoidal tests at d i f f e r e n t displacement amplitudes for a single frequency value. Of course, t h i s v a r i a t i o n does not show up in theory, as the solution of Laplace's equation for potent i a l flow imposes the assumption of i n v i s c i d flow. As can be seen, the ef f e c t i s quite small and i t appears that our tests s a t i s f i e d the requirements of potential flow quite well, and that the viscous e f f e c t s were n e g l i g i b l e . From Figure 12, i t may be concluded that the added mass values can be accurately derived from theory using the AXIDIF program. 2. ADDED MASS FOR SUBMERGED CYLINDER Tests were also performed on a submerged cylinder and the data were processed in the same manner as for the surface piercing cases. Fewer tests were ca r r i e d out, since these were primarily intended for comparison with the surface 59 piercing cylinder. Also, since the added mass did not fluctuate s i g n i f i c a n t l y with frequency, i t was not necessary to have a fine v a r i a t i o n in the frequency. Figure 13 shows the results of these t e s t s . There i s good agreement between the experimentally and t h e o r e t i c a l l y derived added mass values. By comparing Figure 13 with Figure 12, i t may be concluded that the frequency dependence of the added mass for the submerged case i s n e g l i g i b l e . This agrees well with the findings of Byrd [4], and Liaw and Chopra [19], Physi c a l l y , t h i s can be explained by the fact that the structure i s unable to produce any surface waves when i t i s submerged. This was v e r i f i e d in the tests, during which no apparent surface disturbances of the water were observed. The values derived t h e o r e t i c a l l y can be used quite s a t i s f a c t o r i l y in c a l c u l a t i n g the transfer function for the submerged cyli n d e r . Also, since added mass i s e s s e n t i a l l y frequency independent in t h i s case, a t h e o r e t i c a l solution ignoring the free surface kinematic and dynamic boundary conditions should give good values for the added mass of a submerged cylinder. This solution i s much easier to obtain, and since i t i s independent of frequency, only has to be solved once for a given structure. 60 3. ADDED DAMPING The added damping, which results from the energy dissipated in producing surface waves, was determined from the sinusoidal data according to the method described in chapter 2. Figures 14 and 15 show the results for the surface piercing cylinder and the submerged cylinder respectively. The experimental and the o r e t i c a l values agree quite well. As for the added mass values, the peaks occur at the low end of the frequency range. Figure 14 shows the influence of displacement amplitude at any one frequency on the added damping. As noted in the added mass re s u l t s , t h i s i s related to the influence of f l u i d v i s c o s i t y . In comparing Figures 14 and 15, i t i s noted that the damping values for the surface piercing case are much greater than for the submerged case. This i s to be expected, since the surface waves diminish as the structure i s submerged. The graphs are plotted in dimensionless values, C a/wpr 3, where Ca i s the added damping in kg/s, kg i s kilograms, s i s seconds, w i s frequency, p i s ~water density and r i s the radius of the cylin d e r . The peak value for the surface pierci n g cylinder, Figure 14, corresponds to about 3.5% of c r i t i c a l damping for the model. The peak for the submerged case i s less than 1.0% of c r i t i c a l damping. 61 4. TRANSFER FUNCTIONS Tests using both sinusoidal and earthquake records were used to develop experimental transfer functions as discussed in chapter 2. These experimentally derived transfer functions were then used to check the v a l i d i t y of the theoreti c a l transfer functions determined from the solution of Laplace's equation (see chapter 2.8). Figures 16 and 17 show the results for the surface piercing and submerged cylinders respectively from the sinusoidal t e s t s . The comparison i s quite good in both cases. For the surface piercing cylinder (Figure 16), the peaks from the sinusoidal tests are larger than the corresponding t h e o r e t i c a l values. Figures 18 and 19 show the comparison of the experimental and t h e o r e t i c a l transfer functions for the surface piercing cylinder when subjected to the E l Centro N-S 1940 and the San Fernando S74W, 1971 earthquakes respectively. Figure 20 shows the comparison between these functions for the submerged case for the E l Centro earthquake record. The surface piercing cylinder transfer function i s quite frequency dependent. For input accelerations in the lower frequency range ( < 3 Hz), t h i s becomes important. If the solution was obtained ignoring the wave radiation boundary condition, and thus ignoring frequency dependence, the res u l t i n g forces from input accelerations below 3 Hz would be unconservative. The analysis would not accurately represent the real s i t u a t i o n . 62 As would be expected, the transfer function for the submerged cylinder does not show as much frequency dependence as the surface piercing case. Figures 21 and 22 show the spectra of the output base shear for the surface piercing and submerged cylinders respectively. The s o l i d l i n e s indicate the results obtained by multiplying the spectrum of the input acceleration record by the t h e o r e t i c a l l y derived transfer function (equation 2.27). The broken l i n e s are the spectra of the base shear recorded in the random t e s t s . The t h e o r e t i c a l transfer functions predict good r e s u l t s . Comparison between experimental and t h e o r e t i c a l results i s generally considered to be best performed in the frequency domain for random tests [2], However, the time series output derived t h e o r e t i c a l l y was also compared to the time series output data of the E l Centro test (Figure 23). This was obtained by multiplying the complex frequency spectrum of the input acceleration by the complex transfer function derived from theory and then performing an inverse Fourier transform to obtain the output time s e r i e s . The agreement between experiment and theory was quite good in a l l cases. This establishes the v a l i d i t y of using the t h e o r e t i c a l AXIDIF computer program for developing transfer functions for offshore structures. 63 5. RESONANT EFFECTS IN THE MODEL The theoretical solution applies to a cylinder undergoing r i g i d body acceleration in water. As discussed in chapter 3, the model had to be f l e x i b l e enough to measure the hydrodynamic forces developed, yet s t i f f enough to act as a r i g i d body. The natural frequency of the model in water, measured from free v i b r a t i o n tests, was 16 Hz. At the higher frequency range, towards 6 Hz, the cylinder moved in a rocking mode, while the cylinder shape remained r i g i d , and, as a res u l t , some amplification of the acceleration at the top of the cylinder with respect to the base acceleration was noted. This amplification was small and could be determined by equation 4.1; t h i s fact was checked by measuring the accelerations at the top and base of the cylinder. To correct for t h i s condition, the acceleration at the center of gravity of the cylinder was determined from: and used in the calc u l a t i o n s for the added mass and damping. Here, aegis the acceleration of the center of gravity of the cylinder and 0.64 i s the r e l a t i o n of the position of the center of gravity to the height of the cylin d e r . ( 5 . 1 ) 64 6. VISCOUS EFFECTS The assumption of i n v i s c i d f l u i d is made when applying Laplace's equation for potential flow. The model and experiments were designed to s a t i s f y the requirements of this s i t u a t i o n as c l o s e l y as possible. However, water is viscous and i t was anticipated that t h i s condition might influence the experimental r e s u l t s . The v i s c o s i t y of the water could be expected to cause: i) the added mass and damping values to exhibit a small dependence on displacement amplitude i i ) some additional damping due to skin f r i c t i o n drag forces The dependence on displacement amplitude has already been discussed in sections 1 and 3 of t h i s chapter. This effect did show up, but i t was quite small and could be neglected in the analysis. Any viscous damping forces which were present would be included in the added damping measured in the experiments. To v e r i f y that t h i s viscous term was quite small in r e l a t i o n to the t o t a l damping, the t o t a l drag force on the cylinder (which for t h i s case i s the drag force from skin f r i c t i o n only) was estimated by employing the approximate expression [26]: F d= l/2C dA plp|u|u (5.2) 65 where; F(j= viscous drag force on the cylinder C<j = drag c o e f f i c i e n t , taken equal to 1.0 Ap = projected area of cylinder p = density of water 1 = length of the cylinder |u| = absolute value of u ii = peak r e l a t i v e v e l o c i t y between the cylinder and the water The nonlinear term, |u|u, was l i n e a r i z e d using |u| = {/8/i:)a^1 for small amplitudes [26], where a^= root mean square of the ve l o c i t y which equals u//2~ for sinusoidal motion. This viscous drag term was calculated for each of the tests and found to be small in r e l a t i o n to the t o t a l damping term which consisted of both the wave radiation and viscous damping terms: i t had a maximum value of 9% of the t o t a l damping value and was less than 5% for most of the tests. The assumption of i n v i s c i d flow therefore seems to be quite reasonable for these t e s t s . 66 Figure 12 - Added Mass for Surface Piercing Cylinder 67 -e ©- THEORY -4 +-EXPERIHENT See Table 2 For Plot Values r^: 1 — —"cr — i ( 1 1 1 1 1 1 1 r ^ - i i i i i ~ i 0 0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 FREQUENCY (HZ) F i g u r e 13 - Added Mass f o r Submerged C y l i n d e r 68 -a? o THEORY -+ +-EXPERIHENT See T a b l e 1 F or P l o t V a l u e s FREQUENCY (HZ) F i g u r e 14 - Added Damping f o r S u r f a c e P i e r c i n g C y l i n d e r 69 -e © — T H E O R Y -+ +-EXPERIDENT See T a b l e 2 F o r P l o t V a l u e s F i g u r e 15 - Added Damping f o r Submerged C y l i n d e r 70 Experiment Theory Base Added Added Added Added Frequency Shear / Acceleration I )isplacement Mass Damping Mass Damping (Hertz) 00 (m/s*) (cm) 0 .6 6 .61 0.165 1 .75 11.2 1.1+3 11.9 1 .3 13.6 0.311+ 2.31 12.3 0 .63 5.5 0.131+ 1 .38 11.21+ 1.1+3 0.75 15.6 0.361+ 1 .75 11.5 2.02 12.0 2 . 3 . 1.0 39.8 0 .907 2.37 11.1+ 6 . 0 10.9 1+.1+ 2 2 . 7 0 .513 1.30 11.6 5.5 10.8 0.236 0.6ll+ 12.1 5-8 19-6 0.1+13 1 .09 11.. 7 6.7 26 .9 0.587 1.51+ 11.5 6 .6 1.25 2 0 . 7 0 .633 1.05 7.7 1+.12 8.2 1+.9 2 0 . 9 0.61+3 1 .08 7-5 1+.36 H+.3 0.1+31 0.723 7-8 1+.31 1.5 17-2 0.577 0.69 6 .6 1+.11+ 6.5 3.1+ 15 -1 0.527 0.606 6.1+ 3.5 2 0 . 0 0.673 0.782 6 .6 3 .9 1.75 39-6 1.1+8 1.21+ 6.1 2 . 0 6 .3 2 . 0 2 . 0 17-5 0.61+1 0.1+16 6.5 1.2 6.1+ 1.2 35-1+ 1.32 0.859 6 .3 1.1+ 10.6 0.391 0.26 6 . 3 1.1 50.5 1 .81 1 .21 6.7 1.5 28.2 1 .02 0.665 6.5 1 .3 2.5 50.7 1 .83 0.71+ 6.7 0.35 6 .8 0.5 3 .0 1+0.6 1 .39 0.1+02 7 .3 0.03 7 .3 0.25 81.6 2 .8 0 .80 7 .2 0 . 0 1 2 2 . 8 0.7!+ 0.22 7.7 0.07 1+.0 35.8 1.17 0.181 7.61+ 0 . 0 7-7 0.08 53.8 1.75 0.27 7.65 0 . 0 8 59.7 1 .96 0.30 7.7 0.03 5 .0 77 .1 2.1+3 0.23 8.2 0.12 7.9 0.01+ 8 0 . 9 2 . 6 l 0.236 7-9 0 . 0 7 6 . 0 2.1+ 0.209 8 . 0 0 . 0 1+0.3 1 .27 0.120 8.2 0 . 0 6 . 0 51+.8 1.65 0.105 8.6 0 . 0 8 .1 0.03 51+.5 1 .7 0.106 8 .1 0 . 1 N=Newtons m=meters s=seconds cm=centimeters kg=kilograms p=density r=radius v=frequency T a b l e I - Recorded Data - S u r f a c e P i e r c i n g C y l i n d e r 71 Experiment Theory Frequency Base Shear Acceleration Displacement Added Mass Added Damping Added Mass Added Damping (Hertz) (N) (m/s2) (cm) (^) 'swf r3) 0 . 6 3.66 0.098 1.03 10.2 0.23 10.8 0.1+2 1 . 0 9 .85 0.265 0.679 1 0 . 0 1.3k 10.6 1.1+2 2 . 0 32.1+ 0.902 0.587 9.6U 0.2 10.1+ 0.02 3 . 0 1+1+.7 1.30 0.35 9-8 0 . 0 10.2 0 . 0 1+.0 Ik.k 2.00 0.297 10.2 0 . 0 10.1+ 0 . 0 5 .0 75.5 2 .00 O.llh 10.5 0 . 0 10.3 0 . 0 6 . 0 9h.6 2.1+0 0.133 10.6 0 . 0 10.7 0 . 0 N=Newtons m=meters s=seconds cm=centimeters kg=kilograms Q=density r=radius v=frequency Table II - Recorded Data - Submerged Cylinder 72 Figure 16 - Transfer Function for the Surface Piercing Cylinder Derived from Sinusoidal Tests 73 & — I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r— 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 FREQUENCE (HZ) Figure 17 - Transfer Function for the Derived from Sinusoidal Submerged Tests Cylinder 74 Figure 18 - Transfer Function for Surface Piercing Cylinder: E l Centro N-S, 1940 75 Figure 19 - Transfer Function for Surface Piercing Cylinder: San Fernando S74W, 1971 76 "i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.0 0.8 16 2.4 3.2 4.0 4.8 5.6 6.4 FREQUENCY (HZ) Figure 20 - Transfer Function for Submerged Cylinder: E l Centro N-S, 1940 77 Figure 2 1 - Frequency Spectrum of Output Base Shear on Surface Piercing Cylinder 78 Figure 22 - Frequency Spectrum of Output Base Shear on Submerged Cylinder THEORY EXPERIMENT F i g u r e 23 - C o m p a r i s o n o f Time S e r i e s O u t p u t f o r t h e E l C e n t r o N-S 1940 E a r t h q u a k e R e c o r d 80 VI. CONCLUSIONS AND RECOMMENDATIONS The experimental and th e o r e t i c a l values for the added mass, added damping and transfer functions agreed very well. This means that, for structures which meet the c h a r a c t e r i s t i c s of the Laplace regime (chapter 2), the added mass and damping values calculated from the AXIDIF computer program can be used in design to evaluate the response of a structure as a result of seismic e x c i t a t i o n . For surface piercing structures, the frequency dependence of the added mass and damping due to the production of surface waves i s quite s i g n i f i c a n t . In solving for these values t h e o r e t i c a l l y , the f u l l kinematic and dynamic free surface boundary conditions should be included in the solution to account for t h i s frequency dependence. The submerged structure tests showed that the frequency dependence of the added mass and damping becomes less s i g n i f i c a n t as the structure i s submerged below the surface. Solutions which do not include the surface wave e f f e c t s would probably be quite s a t i s f a c t o r y when solving for the f l u i d forces for most f u l l y submerged structures. Several areas of investigation are recommended for future study: determine added mass and damping v&lues as a function of mode shape for f l e x i b l e c ylinders. 81 eva luate f l u i d f o r c e s on s t r u c t u r e s whose dimensions and motions approach the s m a l l body regime, where v i s c o u s e f f e c t s may be impor tant . determine e x p e r i m e n t a l l y the added mass and damping fo r s t r u c t u r a l shapes other than c y l i n d e r s , p o s s i b l y to support numer ica l methods for c a l c u l a t i n g these parameters . study more i n t e n s i v e l y the e f f e c t of submergence on the frequency dependence of the dynamic c h a r a c t e r i s t i c s . 82 BIBLIOGRAPHY 1. Bea,R.G., Audibert,J.M., Akkay,M.R., 'Earthquake Response of Offshore Platforms', ASCE Structural D i v i s i o n , Feb. 1979. 2. Bendat,J., Piersol,A., 'Random Data Analysis and Measurement Prodedures', John A. Wiley and Sons, Toronto, 1971. 3. Bury,M.R.C, Domone,P.L., 'The Role of Research in the Design of Concrete Offshore Structures', OTC 1949, 1974. 4. Byrd,R.f 'A Study of the F l u i d Structure Interaction of Submerged Tanks and Caissons in Earthquakes', EERC -78/08, May 1978. 5. Chandrasedaran,A., Saini,S.S., Malhatra,M.M., 'V i r t u a l Mass of Submerged Structures',Journal of the Hydraulics D i v i s i o n , ASCE, May 1972. 6. Clough,R.W., 'Effects of Earthquakes on Underwater Structures', Proceedings, Second World Conference on Earthquake Engineering, Roorkee, India, pp 161-171, 1970. 7. Clough,R.W., Penzian,J., 'Dynamics of Structures', McGraw-Hill Inc., New York, 1975. 8. Dong, R.G., 'Effective Mass and Damping of Submerged Structures', U.S. Department of Commerce, Lawrence Livermore Laboratory, Livermore, C a l i f o r n i a , UCRL-52342, A p r i l 1978. 9. Dungar,R., Eldred.,P.J., 'The Dynamic Response of Gravity Platforms', Earthquake Engineering and Structural Dynamics, March-April 1978,pp 123-138. 10. Enochson,E., Otnes,R., 'Programming and Analysis for D i g i t a l Time Series Data', U.S. Department of Defence, 1968. 11. Finn,L.D., ' A New Deepwater Offshore Platform - The Guyed Tower', OTC 2688, 1976. 12. Gerwick,B., 'Application of Concrete D r i l l i n g and Production Caissons to Seismic Areas', OTC 2408, 1975. 13. Gibson,R., Wang,H., 'Added Mass of P i l e Groups', ASCE-Waterways, 1977. 83 14. Helou, Amin Habib, 'Seismic Analysis of Submerged Underwater O i l Storage Tanks', PhD Thesis, North Carolina State University, Raleigh, N.C., 1981. 15. Idress,I.M., Cluff,L.S., Patwardhan,A.S., 'Microzonation of Offshore Areas - An Overview', Microzonation 2nd International Conference, Volume III, 1980. 16. Ir a n i , M.B., 'F i n i t e Element Analysis of Viscous Flow and Rigid Body Interaction', Master's Thesis, University of B r i t i s h Columbia, Vancouver, B.C., 1982. 17. Isaacson, M. de St.Q., 'Interference E f f e c t s Between Large Cylinders in Waves', OTC 3067, 1978. 18. Isaacson, M. de St.Q., 'Earthquake Loading of Axisymmetric Offshore Structures', Proceedings, 4th Canadian Conference on Earthquake Engineering, pp 282-287, 1983. see also, Isaacson, M. de St. Q., 'Wave Ef f e c t s on Fixed and Floating Axisymmetric Structures', Coastal/Ocean Engineering Report, Department of C i v i l Engineering, University of B r i t i s h Columbia, 1981. 19. Liaw,C.Y., Chopra,A.K., 'Dynamics of Towers Surrounded by Water', EERC - 73/25, 1973. 20. McConnell,K.G., Young,D.F., 'Added Mass of a Sphere in a Bounded Viscous F l u i d ' , ASCE Mechanics D i v i s i o n , August 1965. 21. Morison,J.R., O'Brien,M.P., Johnson,J.W., Schaaf,S.A., 'The Force Exerted by Surface Waves on P i l e s ' , Petrol Transactions, AIME, Volume 189, 1950. 22. N i l r a t , F . , 'Hydrodynamic Pressure and Added Mass for Axisymmetric Bodies', EERC - 80/12, 1980. 23. Penzian,J., Kaul,M., 'Response of Offshore Towers to Strong Motion Earthquakes', Earthquake Engineering and Structural Dynamics 1, July-September 1972. 24. Penzian,J., Van Dao,B., 'Treatment of Non-Linear Drag Forces Acting on Offshore Platforms', EERC - 80/13, 1980. 25. Ruhl,J.A., Berdahl,R.M., 'Forced Vibration Tests of a Deepwater Platform', OTC 3514, 1979. 26. Sarpkaya,T., Isaacson,M. de St.Q., 'Mechanics of Wave Forces on Offshore Structures', Van Nostrand Reinhold Co., Toronto, 1981. 27. Stelson,T.E., Mavis,F.T., ' V i r t u a l Mass and Acceleration in F l u i d s ' , ASCE 2870, 1955. 84 28. S u g i y a m a , T . , I t o , M . , 'Dynamic C h a r a c t e r i s t i c s o f S t r u c t u r e s i n W a t e r ' , U n i v e r s i t y o f T o k y o P r e s s , 1981. 29. T a n a k a , Y . , Hamamoto,T., Oshima,M., O g a t a , T . , 'Dynamics an d B e h a v i o r s o f O f f s h o r e C y l i n d r i c a l T a n k s ' , J a p a n N a t i o n a l Commitee f o r T h e o r e t i c a l a n d A p p l i e d M e c h a n i c s , pp 357, 1980. 30. T a y l o r , R.E., 'A R e v i e w o f H y d r o d y n a m i c L o a d A n a l y s i s f o r Submerged S t r u c t u r e s E x c i t e d by E a r t h q u a k e s ' , E n g i n e e r i n g S t r u c t u r e s , V olume 3, J u l y 1981. 3 1 . T a y l o r , R.E., D u n c a n , P . E . , ' F l u i d - I n d u c e d I n e r t i a a n d Damping i n V i b r a t i n g O f f s h o r e S t r u c t u r e s ' , A p p l i e d Ocean R e s e a r c h , Volume 2, 1980. 32. To,N.M., 'The D e t e r m i n a t i o n o f S t r u c t u r a l D ynamic P r o p e r t i e s o f T h r e e B u i l d i n g s . i n V a n c o u v e r f r o m A m b i e n t V i b r a t i o n S u r v e y s ' , M a s t e r ' s T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , B.C. 1981. 33. T u n g , C . C , ' H y d r o d y n a m i c F o r c e s on Submerged V e r t i c a l T a n k s U nder G r o u n d E x c i t a t i o n ' , A p p l i e d O cean R e s e a r c h , V o l 1, No. 2, pp 7 5 - 7 8 , 1979. 34. Van M a r c h e , E . e t a l , ' E s t i m a t i o n o f D y n a m i c C h a r a c t e r i s t i c s o f Deep Ocean Tower S t r u c t u r e s ' , M a s s . I n s t i t u t e o f T e c h n o l o g y , C a m b r i d g e , M a s s . , J u n e 1972. 35 . W a r b u r t o n , G . B . , H u l t o n , S . G . , 'Dynamic I n t e r a c t i o n f o r I d e a l i z e d O f f s h o r e S t r u c t u r e s ' , E a r t h q u a k e E n g i n e e r i n g a n d S t r u c t u r a l D y n a m i c s - 6, pp 5 5 7 - 5 6 7 , November-December 1978. 85 APPENDIX A SOLUTION OF LAPLACE'S EQUATION FOR ADDED MASS AND  DAMPNG FOR A CYLINDER USING THE AXIDIF PROGRAM In t h i s study, the experimental results for added mass and damping were compared with the solution obtained from a wave d i f f r a c t i o n theory computer program c a l l e d AXIDIF [18]. The solution for the forces on submerged structures due to earthquake loading i s d i r e c t l y related to the wave loading case as the same added mass and damping values determined by the computer programs are needed to account for the f l u i d -structure i n t e r a c t i o n . AXIDIF i s for axisymmetric structures only, and i s considerably more economical in terms of computer costs than a program for any a r b i t r a r i l y shaped body. The t h e o r e t i c a l development of the AXIDIF program given here i s e s s e n t i a l l y that of reference [18]. The solution i s based on a boundary element method involving an axisymmetric Green's function. A sinusoidal, u n i d i r e c t i o n a l base motion, Xexp(-iwt) i s applied to a r i g i d axisymmetric structure of c y l i n d r i c a l coordinates, (r,0,z), where X i s a complex amplitude, w i s the excitation frequency, r i s the r a d i a l coordinate , z i s the v e r t i c a l coordinate and 8 i s the angle measured from the di r e c t i o n of motion. The f l u i d i s considered to be incompressible and i n v i s c i d and the flow i r r o t a t i o n a l . The f l u i d motion can then be described by the v e l o c i t y potential s a t i s f y i n g Laplace's equation: 86 32<i> 3r 2 1_3$ r3r 32<*> 3 2* = dz1 0 (A.1) The assumption of incompressible f l u i d to the case of a body vibrating in water i s discussed at some length by Liaw and Chopra [19], For most structure dimensions and frequencies of vibration t h i s assumption is quite v a l i d but for some cases, water compressibility should be considered. With the assumption of small amplitude motion and the f l u i d assumptions discussed in chapter 2, the usual l i n e a r i z e d boundary conditions are applied to the d i f f e r e n t i a l equation. The relevent boundary conditions are: 1. 3j?(r,O,0,t) = O (A.1a) 3z defines the v e l o c i t y condition normal to the ocean fl o o r at z=0 2. 3$(R,z, 6, t)= -iw cosa cost? (A.1b) 3n s p e c i f i e s that the f l u i d p a r t i c l e motion and the motion of the structure i s the same at the structure boundary; n i s the d i r e c t i o n normal to the structure surface and a i s the d i r e c t i o n of n in re l a t i o n to the horizontal axis. 3. 3f$(r,H,0,t)=-g_3f (r,H,0,t) (A.1c) 3t 2 3z describes a l i n e a r i z e d free surface condition 87 including dynamic and kinematic boundary conditions; H=total depth of water. 4. j)$(r,z,0,t) = 3$(r ,z , 7 r,t) (A.1d) stip u l a t e s symmetry about 0=0 plane The v e l o c i t y potential i s harmonic and proportional to the amplitude of motion, $Xexp(-iwt). In the boundary int e g r a l method, the unknown pote n t i a l , $(x), at the general point, x=(r , 0,z), i s represented as due to a source d i s t r i b u t i o n over the structure's surface S 0, and i s thus expressed as: $(x) = J _ f f(x)G(x,y)dS (A.2) Here, f(x) i s a source strength d i s t r i b u t i o n function, G(x,y) i s a Green's function for the general point x due to a source of unit strength at y, and the integration i s c a r r i e d out for a l l points y over S 0. G i s i t s e l f chosen to s a t i s f y the Laplace equation, the seabed and l i n e a r i z e d free surface boundary conditions, and the radiation condition. This ensures that $ also s a t i s f i e s these equations, and i t remains for f to be chosen so as to ensure that the boundary condition on the structure surface i s s a t i s f i e d . Boundary condition Al.b equating the f l u i d v e l o c i t y normal to the structure surface to the v e l o c i t y of the 88 structure surface, together with equation A.2 gives r i s e to a surface integral equation for f: -J_f(x) +_J_ J f (y) 3G(x,y)dS = -iw cosa cos0 (A.3) 2 4TT ' 9n Here, n i s measured from the point x, and the integration i s ca r r i e d out over the point y. In equation A.3, x l i e s on the structure surface and may be defined by the coordinates (s,0), where s i s the surface coordinate and y may be defined by corresponding coordinates (s',0'). Because of the structure's axisymmetry, the functions f and G for points on the structure surface may be expanded as Fourier s e r i e s : CO *(s,t5) = I * m(s)cosm0 (A.4) m=l f(s,0) = ^  f m(s)cosm0 (A.5) m=l G(s,0,s',0') =°E G m(s,s')cosm(0-0') (A.6) m=l and only the terms corresponding to m=1 w i l l be required here. Substituting equations A.5 and A.6 into A.3, algebraic manipulation y i e l d s a set of l i n e integral equations, of which the equation corresponding to m=1 i s : - f ^ s ) + _Lf f ! (s' )R(s' ) 3Gj (s,s' )ds' = 2iwcosa (A.7) 2 s« 9n 89 Here, s 0 i s the structure's entire contour described by s, and R(s') i s the structure's radius at s'. In a numerical solution to equation A.7, the contour s 0 i s d i s c r e t i z e d into N short segments with the function f, taken to be uniform over each segment, and equation A.7 i s applied at the centre of each segment. Thus equation A.7 may be approximated by a matrix equation: N L Ajfcf}^ 1) = -2iwcosa for j = 1,2,...N (A.8) k—X where ffc( 1) denotes f ^ s f c ) . Expressions for the matrix c o e f f i c i e n t s Ajk are given by Isaacson [18]. Once the source strengths ffc( 1)are determined, the potential i t s e l f can be obtained by a d i s c r e t i z e d form of equation A.2. The necessary Fourier c o e f f i c i e n t at the j-th segment centre can be approximated as: ^ ( s - j ) = 1/2 I f k ( 1 ) C j k for j = 1,2,...N (A.9) k—1 Once more, Isaacson [18], provides expressions for the co e f f i e n t s C j ^ . Now that the potential function $, i s known, the hydrodynamic loads on the structure may be evaluated. 90 The hydrodynamic pressure p acting on the structure surface i s given by the l i n e a r i z e d Bernoulli equation, p = iwp $ exp(-iwt), where p i s the f l u i d density. Thus the horizontal force F/ ^ exp(-iwt) and overturning moment F 2 ^ ^ exp(-iwt) due to the f l u i d may be expressed as: F j ( f ) = _ i w p j s $rijds , for j = 1,2 (A.10) where n t = cosacosfl n 2 = zcosacosf? - rsinacosa Substituting the Fourier expansion of equation A.5, and integrating with respect to 6, we obtain -wiwp ^.Ljcr^njk*! (% ) for j = 1,2 (A. 1 1 ) where i s the length of the k-th segment, and "ik = cosqc n 2 k = zkcos (% ) - rksin(cqc) The f l u i d forces Fj(^) are conveniently expressed in terms of added masses n ^ j , and damping c o e f f i c i e n t s Xj, by taking: F j ( f ) = w2maj + iwXj (A.12) in which m aj and Xj may be retrieved by separating the real and imaginary parts of F j ( f ) . It i s emphasized that % j and 91 Xj are frequency dependent variables. Many authors [4,19,4 and 33], set the free surface boundary condition, equation A.1c, equal to 0: 9 2 * ( r , H , e , t ) =0 (A.13) a t " 5 This greatly s i m p l i f i e s the solution but neglects any surface wave e f f e c t s and results in the solution being independent of the exc i t a t i o n frequency. These e f f e c t s can be important for some structures as discussed in chapter 5. 92 APPENDIX B MEASUREMENT AND ANALYSIS OF DATA 1 . MEASUREMENT APPARATUS A. Base Acceleration The measurement of the base input to the cylinder was made with an accelerometer fastened d i r e c t l y to the shaking table. A K i s t l e r MD 305A 50g accelerometer, in conjunction with a servoamplifier, was used for this purpose. B. Base Displacement The displacement of the table (and hence the base of the cylinder) was recorded as a check on the acceleration measurements. These measurements were taken with the LVDT, which i s attached permanently to the arm of the hydraulic jack exciting the table. C. Base Shear The shear developed at the base of the cylinder was measured using s t r a i n gauges. Four s t r a i n gauges were used on the shaft of the model arranged in a f u l l Wheatstone Bridge (Figure 24). The bridge was set up to measure the difference between the average s t r a i n at the top of the shaft and at the bottom of the shaft. The base shear i s d i r e c t l y proportional to t h i s difference in s t r a i n : 93 V = eEI (B.1) yH where: V = base shear E = modulus of e l a s t i c i t y of the shaft (steel in t h i s case) I = moment of i n e r t i a of shaft e = difference between the average s t r a i n at the top and at the bottom of shaft y = distance from neutral axis to outer f i b e r of shaft H = height of shaft between s t r a i n gauges The constant El/yH was evaluated by a load c a l i b r a t i o n test of the model prior to conducting the experiments; in t h i s t e s t , e was measured for known values of V and EL/yH was calculated from B.1. The base shears and thus the s t r a i n gauge output voltages varied over a wide range of values, being very small at low loads to quite large at high load l e v e l s . As a r e s u l t , i t was necessary to use a variable amplifier to boost and condition the data signals to a suitable l e v e l for recording on the PDP-11 mini computer. 2. DATA COLLECTION The experiments were ca r r i e d out in the Earthquake Simulation Laboratory of the Department of C i v i l Engineering at the University of B r i t i s h Columbia. This f a c i l i t y i s equipped with a PDP-11 mini computer with disc drive, backed by an RT-11 operating system. It i s capable of handling 17 channels of input; the tests required only three. Each 94 channel i s equipped with a variable amplifier and a variable f i l t e r to bring the generated signals up to recordable l e v e l . The data for the base shear V, base acceleration a, and base displacement X, were recorded onto a floppy d i s c . Each sinusoidal test was recorded over a ten second period at a sampling rate of 100 samples per second. To aid in smoothing the data, the f i l t e r s were set at cutoff values of at least twice the test frequency. A t y p i c a l set of results from the sinusoidal tests i s shown in Figure 25. As can be seen the plots are not pure sinusoids. This was caused by imperfections in the shaking table system, which produce small, high frequency vibrations other than those desired in the test. This problem cannot be corrected and must be compensated for in the analysis by using Fourier spectra as described in the next section. The random tests were recorded in the same manner. Real earthquake records were fed into the table through the PDP-11 system to provide the random ex c i t a t i o n . Figure 26 i s an example of the base shear, acceleration, and displacement recorded during a test of the surface pierc i n g cylinder using the 1940 N-S E l Centro record. The data for the random tests were f i l t e r e d at 50 Hz to eliminate high frequency noise from the system. 95 3. ANALYSIS OF DATA A. Sinusoidal Tests The sinusoidal tests provided information on the amplitudes of the base shear V, and the base acceleration a, and the phase s h i f t <f>, between these variables. The table displacement X, was used as a check on the acceleration through the simple harmonic r e l a t i o n a=-w2X. The added mass and damping were then derived from t h i s information as discussed in Chapter 2. If the data were purely sinuoidal i t would be quite easy to determine the above values; however, as can be seen in Figure 25, t h i s was not the case. To iso l a t e the peak value at the test frequency from the data, a Fourier analysis was used to produce Fourier spectra. Fourier amplitude and phase spectra were produced for the base shear and the base acceleration of each test (see example, Figure 27). The required amplitudes, V and a, and the phase s h i f t , <t>, were then taken d i r e c t l y from the spectra, the phase s h i f t being the difference between the phase values of V(t) and a ( t ) . A given record in the time domain: x(t)=Xcos(wct + t9<>) = X0exp(i(w„t + e,)) (B.2) can be transformed into the frequency domain by taking i t s Fourier transform: F(w)= Jx(t)exp(-iwt)dt = X£Xp(i$) fexp(i(w 0-w)t)dt (B.3) 96 This may be written as: F(w) = X,exp(i6>j6(w-wJ where /exp(iwt)dt = 6(w) (B.4) which on expansion becomes: F(w) = X0cosc906(w-w<,)+X0 isin^>6(w-w0) (B.5) Then the Fourier Amplitude = i/Re2 + Im2 = |F(w) | « X o V/cos 20 o + sin2~£ 6(w-wJ = X06(w-wJ (B.6) and the phase angle = 0 = tan' 1 _Im = tan' 1X n sing6(w-wj Re Xocos06(w-w,) = tan" 1 (tanf?,) = 6B (B.7) This analysis i s true for each frequency component, An cos(wnt + dn) in the data. B. Random Tests Random tests were performed in which the cylinder base shears and accelerations were measured when using the E l Centro 1940 N-S and the San Fernando S74W, 1971 earthquakes as the input e x c i t a t i o n . The parameter of interest here was the frequency transfer function r e l a t i n g the input acceleration and the output base shear. 97 Assuming a stationary process, which may be taken as reasonable for at least part of the earthquake records, the frequency transfer function can be derived from the spectral densities of the input and output records [7]: Sv = |H(w)|2 (B.8) Sa where: Sv i s the spectral density of the output base shear Sa i s the spectral density of the input acceleration record |H(w)| i s the amplitude of the frequency transfer function The base shear V ( t ) , and the acceleration a ( t ) , data were run through a Fast Fourier Transform program (FFT) from which the power spectral densities were calculated (see Figure 28). The graph of the r a t i o of the power spectral density values, |H(w)|2, at each frequency produces the frequency transfer function. 98 77 B R . E R G r A R B °4 (R+R)L R R B = R C = R D € F = AR € -2E 0 (R + R6) E R G F D | ^ s stroin E0=meosured woltoge model R s stroin gouge resistance R 6 = golvanometer resistance E =excitotion voltage F =gouge foctor Figure 24 - Wheatstone Bridge - Strain Gauge Configuration 99 Figure 25 - Example of Data From Sinusoidal Tests 100 Figure 26 - Example of Data From Random Earthquake Tests 101 e p c i ^ a a ) p s q a s p B B | 9 a y e i S | " a 2.0 3.0 4.0 5.0 FREQUENCY (HZ) 6.0 7.0 8.0 Figure 27 - Fourier Amplitude Spectra for Sinusoidal Data of Base Shear and Acceleration 102 FREQUENCY (HZ) CP FREQUENCY (HZ) Figure 28 - Spectras of Base Shear and Acceleration and Transfer Function Derived from them 

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