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Response of a flexible marine column to base excitation Vernon, Thomas A. 1984

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RESPONSE OF A FLEXIBLE MARINE COLUMN TO BASE EXCITATION by Thomas A. Vernon B.A.Sc, The University of British Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1984 © Thomas A. Vernon, 1984 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l Engineering The University of British Columbia 2324 Main Mall Vancouver, B.C. V6T 1W5 i i ABSTRACT The displacement response of a flexible, surface-piercing cylinder subjected to a unidirectional base motion i s considered i n this study. Laboratory experiments have been performed with a circular, fixed-base model using sinusoidal and scaled seismic input motion. Sinusoidal tests were designed to investigate the dependence of cylinder tip response on the ratio of base motion frequency to cylinder natural frequency and base displacement amplitude, for a fixed water depth, inertia ratio and damping ratio. Further tests with a base motion corresponding to past earthquake records were then used to determine the cylinder's response to seismic excitation. The sinusoidal test results are compared with predictions derived from analyses of the motion in terms of the f i r s t and second undamped mode shapes of a cantilever beam. The Morison equation is used to estimate hydrodynamic loads in this formulation, and three treatments of the drag term in the equation of motion are considered: neglect of drag, drag linearization and retention of the complete nonlinear form. A closed-form solution for the former two approximations i s developed, and a numerical approach is adopted for the complete nonlinear formulation. The numerical method i s used to predict the response of the column to seismic input. The dependence of cylinder tip displacement on frequency ratio and base motion amplitude follows predictable patterns of dynamic response. Peak amplitudes occur at resonance and increase with base motion amplitudes. However, this relationship is not linear because of the damping contribution of nonlinear drag forces near the free surface. i i i The numerical and linearized drag predictions agree well with the experimental response i f a suitable choice of drag coefficient i s made. The neglect of drag results i n very conservative resonant response predictions for large excitation amplitudes in which the free surface Keulegan-Carpenter number exceeds about 4. Drag forces can generally be neglected, however, in the estimation of response to earthquake motion because displacement amplitudes are small. In extreme cases, small l i f t forces can result from flow separation about the cylinder near the free surface. i v TABLE OF CONTENTS PAGE ABSTRACT i i LIST OF FIGURES v i LIST OF TABLES v i i i ACKNOWLEDGEMENTS ix 1. INTRODUCTION 1 1.1 Introduction 1 1.2 Background 2 1.3 Literature Review 6 1.3.1 Earthquake Response of Structures in Fluids .. 6 1.3.2. Response in Oscillating Flows 7 1.3.3. Hydrodynamic Drag and Damping 9 1.3.4. Related Topics 10 2. THEORETICAL DEVELOPMENT 12 2.1. Problem Definition 12 2.2. Dimensional Analysis 13 2.3. Hydrodynamic Force Formulation 16 2.4. Modal Analysis 17 2.4.1. Neglect of Drag Forces 18 2.4.2. Linearized Drag 24 2.4.3. Numerical Method with Nonlinear Drag 28 3. TESTING FACILITIES AND MODEL PARAMETERS 32 3.1. Testing F a c i l i t i e s 32 3.2. Model Parameters 33 3.3. Data Acquisition 35 V PAGE 4. DESCRIPTION OF EXPERIMENTS 36 4.1. Damping Tests and System Characteristics 36 4.2. Sinusoidal Tests 37 4.3. Seismic Motion Tests 39 5. DISCUSSION OF RESULTS 41 5.1. Response Functions from Sinusoidal Tests 41 5.2. Response Predictions for Sinusoidal Motion 42 5.3. Transverse Response 45 5.4. Response to Seismic Input 46 5.5. Numerical Prediction of Seismic Response 48 6. CONCLUSIONS AND RECOMMENDATIONS 49 6.1. Conclusions 49 6.2. Recommendations for Further Study 51 BIBLIOGRAPHY 72 APPENDIX A - MODAL ANALYSIS 76 APPENDIX B - DERIVATION OF RESPONSE AMPLITUDES 82 APPENDIX C - DESCRIPTION OF COMPUTER PROGRAM 87 APPENDIX D - INSTRUMENT CALIBRATION AND TYPICAL DATA 89 v i LIST OF FIGURES FIGURE PAGE 1 Coordinate System Definition 52 2 Mode Shapes for a Cantilever Beam 52 3 Schematic of Test F a c i l i t y 53 4 Shake Table 54 5 Testing Tank Over Shake Table 54 6 Instrumented Rod for Strain Measurement 55 7 Photo of Instrumented Rod 55 8 Free Vibration Damping in Air 56 9 Free Vibration Damping in Water 56 10 Photo of Sinusoidal Test of 4.5 Hz 57 11 Photo of Sinusoidal Test at 28 Hz 57 12 Fourier Spectra for El Centro 1940 58 13 Fourier Spectra for San Fernando 1971 58 14 Response Function for Yg/D = .020 59 15 Response Function for Y /D = .036 60 16 Response Function for Y /D = .054 61 g 17 Response Function for Y /D = .072 62 g 18 Drag Coefficient versus Reynolds Number for Various Values of K 65 19 Ratio of Y , ,Y at Resonance versus Drag Coefficient at pred/ exp Various Base Displacement Amplitudes 65 20 Numerical Simulation of the Formation of Asymmetric Vortices ... 66 21 In-line and Transverse Response Near Resonance for Y /D = .054 . 66 FIGURE v i i PAGE 22 Transverse Response Functions for Various Base Displacement Amplitudes 67 23 Tip Locus at Resonance for Y /D = .054 68 8 24 Tip Locus off Resonance for Y /D = .054 68 g 25 Response comparison for El Centro 69 26 Response comparison for San Fernando 69 27 Transverse Time History for San Fernando 70 28 Response Prediction Comparison for El Centro 70 29 Response Prediction Comparison for San Fernando 71 30 Calibration Curve for Instrumented Rod 90 31 Typical Time History at 3rd Subharmonic 91 32 Fourier Spectra for Record of Figure 31 91 33 In-line Resonance Time History at Y /d = .036 92 g 34 Transverse Resonance Time History at Y /d = .054 92 v i i i LIST OF TABLES TABLE PAGE 1 Resonant Response Ratios Y ,/Y 63 r pred exp 2 Experimental Data 64 > ix ACKNOWLEDGEMENTS I wish to express many thanks to Dr. M. Isaacson and Dr. S. Cherry for their guidance and support i n the preparation of this thesis, and to Chris Dumont for his help in the Earthquake Laboratory. 1 1. INTRODUCTION 1.1 Introduction The prediction of structural response to hydrodynamic loads i s of major importance in the design of many marine f a c i l i t i e s . As a contribution to this general subject, this thesis endeavours to Investigate methods of response prediction applicable to compliant marine columns undergoing base motion. Hydrodynamic forces are generated when relative motion exists between the fluid and structure such as occurs as a result of wave and/or current action or seismic excitation of the structure. In the case of a compliant marine structure, the fluid-structure system becomes interactive since the response and applied forces are then coupled. If the available damping is low, dynamic amplification at resonance may become important. An investigation of a quasi-resonance phenomena i s of merit in the case of seismic excitation because system damping is often low (<10% c r i t i c a l ) and natural frequencies of larger structures often tend to coincide with earthquake spectral density peaks (1-5 Hz). This proximity does not generally occur in the case of wave loading because of the lower frequencies associated with large ocean waves. The hydrodynamic forces generated by the motion of a body in a s t i l l fluid result from inertia forces associated with the acceleration of a volume of f l u i d , and from drag forces associated with flow separation. For small relative motions typical of earthquake ground motions, these forces w i l l be mainly i n e r t i a l . For flexible structures, dynamic amplification enhances the possibility of larger displacements leading to flow separation. Where separation occurs, form drag, vortex formation 2 and possibly l i f t forces can result. In the case of base motion, form drag is energy dissipative, and is treated as an added damping. However, the associated l i f t forces may synchronize with the natural frequency of the structure and in extreme cases, lead to large transverse oscillations. Such a mechanism has been the cause of failure of numerous free-standing piles in waves. For purposes of this thesis, a relatively simple system is considered: a surface-piercing circular cylindrical structure is excited with unidirectional, horizontal base motion. The in-line and transverse responses to a range of input motion amplitudes and frequencies have been recorded and the in-line responses compared to predictions based on a modal analysis assuming both linear and nonlinear hydrodynamic loading. Although several powerful methods, such as lumped-parameter modelling and finite element methods, exist for the treatment of this problem, the use of modal techniques for the continuous structure provides a convenient and straightforward approach without the need to resort to the extensive computing or modelling demanded by other methods. (See Kirkley and Murtha, 1975, for the former; Liaw and Chopra, 1973, for the l a t t e r ) . Transverse responses are not compared to any numerical predictions because a simple predictive mathematical model of l i f t force i s not generally available at this time. It must be noted, however, that transverse oscillations can fundamentally affect the near resonant response of a marine p i l e or tower. 1.2 Background Fluid-structure interaction in the marine environment i s generally assumed to occur in either a separated or unseparated (potential) flow regime. Viscous effects can be included in the former, but must be 3 neglected In potential flow solutions. The determining factor in the assumption of an appropriate flow regime is the magnitude of relative displacement between the structure and f l u i d , often expressed in terms of the Keulegan-Carpenter number, K. This is defined, for a two-dimensional sinusoidal o s c i l l a t i o n of a circular cylinder, as (Keulegan, Carpenter (1958)). K where A D amplitude of oscillation displacement cylinder diameter The Keulegan-Carpenter number indicates the significance of flow separation in the problem. For K values of roughly 10 or more, the oscillation amplitude to cylinder diameter ratio w i l l be relatively large, and the hydrodynamic loads are primarily form drag associated with the flow separation. The problem is then defined as lying within the small body regime. On the other hand, i f K is small, flow separation does not generally occur, inertia forces are dominant and potential flow theory can provide a good flow f i e l d aproximation. Overlaps of these classifications certainly exist (for a fuller account see Sarpkaya and Isaacson, 1981), and a particular problem may encompass several flow classes. The solution approach differs i n the case of large and small body problems. The former is generally formulated using potential flow theory for an inviscid f l u i d and an irrotational flow f i e l d . The velocity of any point is specified as the gradient of a scalar potential function which satisfies the Laplace equation and appropriate boundary conditions. 4 The pressure may be obtained from the potential using the unsteady Bernoulli equation and thus once the potential function has been determined, the hydrodynamic loads may be calculated from an integration of the pressure around the body. This type of approach is applicable to the determination of wave or seismic loading of large structures such as o i l storage tanks or gravity platform bases where A/D in (1.1) is small. Local flow separation due to sharp geometries can occur on large structures and leads to inaccuracies in solutions obtained from potential flow assumptions. The significance of these errors should be given consideration in such cases. In the small body regime, viscous effects are included in the hydrodynamic force derivation. Experimental evidence suggests that viscous forces resulting from flow separation can become important at Keulegan-Carpenter numbers greater than about 2 (A/D - 0.3 in (1.1)) for oscillating flows about bluff bodies. Vortex shedding and l i f t forces can occur for K > 5 (A/D > .7). Although seismic base motion amplitudes are usually small such that K < 5 over most portions of a structure, the viscous forces may become significant near the free surface. To include viscous effects, the hydrodynamic loads are usually formulated using the well known Morison equation, deriving from the work of Morison, et a l (1950). This equation i s based on the simultaneous addition of an inertia force associated with inviscid fluid acceleration and a drag force analogous to that i n a steady flow, and is given in simplest form for a stationary circular cylinder as: F' = C m D O (1.2) 5 Here, F' is the f l u i d force per unit length, p i s the f l u i d density, D is the body diameter, U and U are the f l u i d velocity and acceleration respectively (using the dot notation to imply time differentiation), and Cffl and Cj are inertia and drag coefficients respectively. The nonlinearity of the forcing function i s apparent in the drag term. The force coefficients C,, C depend on the dimensionless flow and structure d m parameters such as local Reynolds number, Keulegan-Carpenter number and surface roughness. These coefficients are not fundamental constants and must be determined experimentally. The problem considered i n this study encompasses both flow regimes. Near the structure base, the displacement is essentially the base motion input. In the case of seismic input these displacements are generally less than 20 cm and for an average size pile or platform leg, the maximum Keulegan-Carpenter number would then be roughly 0.5 or less. However, many marine structures are highly dynamic, with fundamental frequencies of oscillation typically i n the range 0.2 to 1.0 Hz, a range of frequency also typical of earthquake spectral intensity maxima (See Figures 12 and 13). The proximity of these frequencies leads to the possible quasi-resonance condition commonly associated with seismic excitation of large structures. Because dynamic amplification at resonance can be of order 10 or more, a Keulegan-Carpenter number of 5 can be attained near the free surface. This value indicates that flow separation and possibly l i f t forces may become important in the problem. A bottom fixed marine structure can consequently span both flow regimes: a potential flow near the base (and nodes If we consider the response to be modal) and a separated flow in regions of large structural displacements. The present study uses the Morison equation to estimate 6 hydro-dynamic loads, since that formulation w i l l essentially be valid in the case of small structures, for the entire span, aside from effects of wave generation near the free surface. 1.3 Literature Review A number of more recent investigations of structure response to hydrodynamic loads are relevant to the present study. These have been divided here into several broad categories: response to seismic loading, response in oscillating flows, damping in fluids and additional miscellaneous topics. 1.3.1. Earthquake Response of Structures in Fluids Clough (1960) performed vibration tests of fle x i b l y mounted cylindrical shapes in air and water. The study determined added mass and damping values for f i r s t and second mode natural frequencies using the different frequencies of oscillation in the two mediums. Clough found agreement between the experimental values of added mass and those derived from potential flow theory. The response of a pseudo-continuous structure subjected to base loading was also investigated. However, the applied displacement in these tests was generated by a pendulum striking the edge of a table and did not simulate r e a l i s t i c seismic input. Clough concluded that structural oscillations resulting from seismic excitation would not be large enough to cause flow separation. If this conclusion is accepted, potential flow theory can be used to predict marine structure response during earthquakes. A comprehensive analytical study of the response of elastic structures surrounded by a f l u i d using potential flow theory was 7 presented by Liaw and Chopra (1973). The authors u t i l i z e d a modal analysis technique in combination with a f i n i t e element method to predict structural response. No experimental verifications were given in this study. Much of the work on the seismic response of marine structures has been concerned with large, non-elastic structures such as submerged o i l tanks. In these cases, most authors use a potential flow formulation similar to that used for load prediction in the wave diffraction regime. Such studies include those by Byrd (1978), Tung (1979), Mei (1979), Westermo (1980), and Isaacson (1983). A review of many of the methods used by the above authors, as well as others, i s presented by Eatock Taylor (1981), for the case of dams and offshore structures. Kirkley and Murtha (1975) have studied the response of offshore structures i n earthquakes, comparing several techniques of response prediction using lumped-parameter models. A Morison forcing function was used and a direct integration procedure compared with an uncoupled approximation. Their results indicate that the linearization technique i s accurate in higher frequency ranges, but unconservative for low frequencies. The same authors have presented response spectra for offshore structure design based on the numerical solution of the linearized equations (Kirkley and Murtha, 1975). 1.3.2. Response of Flexible Structures in Oscillating Flows There exists a very significant volume of literature concerning the hydrodynamic loads imposed on structures in oscillating flows. However, very l i t t l e of this work actually concerns elastic structure response to these loads. 8 An early work by Selna and Cho (1972) presented a numerical integration and time step technique for discretized models of marine structures. This approach can incorporate the nonlinear coupled drag characteristics, although such direct integration methods are very costly for more than a few degrees of freedom. Blevins (1977) has considered the in-line response of continuous structures in oscillatory flows with zero and nonzero means. He provides a solution based on a Fourier expansion of the linearized equation of motion. An assumption of small structure response i n comparison to the fl u i d motion i s inherent i n this linearization and solution approach. Such an assumption cannot be made in an analysis of seismic response and other linearization techniques must be used. A number of authors have presented numerical methods for the response solution using integration of the motion equations derived for discretized models. These include Sawaragi et a l (1977), and Anagnostopoulos (1982), who uses an uncoupled Morison forcing function. These numerical techniques are expensive, but very powerful and the limitations on their accuracy are imposed mainly by uncertainty in the input data. The transverse response of el a s t i c a l l y restrained cylinders in oscillatory flows has been investigated by several authors. Sarpkaya (1979, 1981), has performed extensive experimentation relevant to this subject and presented correlations of transverse response with system parameters such as reduced velocity and Keulegan-Carpenter number. Similar correlations were presented by Isaacson and Maull (198l) for linear mode models in progressive waves. Zedan and Yeung (1980) investigated pile dynamics under lock-on conditions in waves at moderate 9 Keulegan-Carpenter numbers. Lock-on was found to occur over only a narrow bandwidth and to be highly interactive with the in-line response. McConnel and Park (1982) oscillated e l a s t i c a l l y restrained cylinders in a s t i l l fluid and concluded that the frequency ratio parameter is more of a controlling factor in transverse response than K or reduced velocity as suggested by other authors. Skop and Griffin (1975) presented an analytical model of the fluctuating l i f t force coefficient and used this model in conjunction with modal superposition to obtain transverse response predictions. The model predictions were compared to experiments and the results follow at least similar trends. 1.3.3. Hydrodynamic Drag and Damping The drag forces acting on marine structures are nonlinear and, in a dynamic system, coupled to the structure response. Because of this coupling the drag force can be expressed in terms of an exciting force, associated with the incident flow and a damping force associated with the structural motion. The estimation of such damping forces can be very important when resonant response is a possibility. Several authors have considered the treatment of these exciting forces and damping terms. Dao and Penzien (1980) have presented a comprehensive comparison of methods of handling nonlinear drag forces using discretized models. These methods again involve assumptions on the relative magnitudes of the displacements (uncoupling) or in some cases assume harmonic response. The linearization techniques are modified to give an approximation of the response to non-harmonic motion. Moe and Verley (1980) have considered the hydrodynamic damping of offshore structures in various flow situations. They conclude that the 10 use of the Morison equation may give unconservative response predictions because the normally assumed values of drag coefficients are too high. This i s a result of an assumption of separated flow inherent i n the Morison approach. Separated flow does not occur where motion amplitude to diameter ratios are small and in such cases drag damping i s overestimated by a choice of drag coefficient based on that assumption. Sugiyama and Ito (1981) have considered the treatment of nonlinear drag damping as a function of vibration amplitude for a single degree of freedom system. A polynomial function of Reynolds number was found appropriate for the drag coefficient. The function was determined empirically from experiments. 1.2.4. Miscellaneous Related Topics Two related areas of investigation are the prediction of marine riser response, and the non-deterministic approach to offshore structure response. Kirk, et a l (1979) have developed a normal mode solution in the frequency domain for a marine riser subjected to wave loading and top excitation from horizontal motions of the support platform. The nonlinear drag term was found to be very important at resonance conditions. A dynamic analysis of a multi-tube riser has been given by Grecco and Utt (1982). Chakrabarti and Frampton (1982) have presented a comprehensive review of the many techniques used in the analysis of marine risers . These solutions are invariably based on numerical methods because of the complexity of the governing equations of motion. Non-deterministic methods of response analysis have been presented by Malhotra and Penzien (1970), Penzien and Dao (1980). Other references and a review of the principles of non-deterministic analyses can be found in Sarpkaya and Isaacson (1981). 12 2. THEORETICAL DEVELOPMENT The purpose of this chapter is to formalize the problem under consideration and to develop the methods of response prediction used in this study. A dimensional analysis is presented which defines the relevant parameters in the problem. The modal analysis method of response prediction is developed for three forms of the hydrodynamic drag force in the equation of motion; zero, linearized and nonlinear drag forces. A closed-form solution is obtained for the zero and linearized drag cases; a numerical approach i s given for the nonlinear form. 2.1. Problem Definition This study i s concerned with the general subject of response prediction for submerged flexible structures undergoing base motion. As a fundamental case characterizing this problem, a single, flexible, surface-piercing circular cylinder is considered. For a constant mass, length and diameter of cylinder, and constant f l u i d properties, the displacement response w i l l be a function of base motion amplitude and frequency. This dependence is investigated using sinusoidal input. The response of flexible columns to earthquakes is then investigated using scaled seismic inputs. The solution approaches used in this study necessitate a number of assumptions about the flow and structure characteristics in the problem. These assumptions include the following: 1) The problem exists in the small body flow regime where the Morison equation can be used to predict hydrodynamic loads. 2) The structure can be considered as a uniform cantilever beam and appropriate beam theory used to develop the equation of motion. The response is assumed to be primarily in the f i r s t mode and the undamped mode shapes are assumed to adequately describe the response. 3) The column i s free standing, with no tip mass and i s v i r t u a l l y submerged. The c r i t i c a l response is taken as that at the free surface. 4) Seismic base motion amplitude i s typical of that recorded on firm ground at moderate distances from the focus of a strong motion earthquake but i s limited to a unidirectional and horizontal component. 5) Base motion i s prescribed, circumventing the problem of s o i l structure Interaction. With these assumptions, response prediction methods are developd using the Morison equation to estimate hydrodynamic loads and a modal analysis to solve the equation of motion. Response predictions are compared to experimental results from model tests. 2.2. Dimensional Analysis Dimensional analysis can be used to define the dimensionless parameters important in a particular problem. The present study intends to model the displacement response of a flexible marine column undergoing base motion. The relevant independent variables are then characteristics of the structure, f l u i d and motion. The variables are: D diameter of cylinder f fundamental frequency of structure in water 14 m structure mass per unit length damping ratio in air v kinematic viscosity of f l u i d P fluid density H depth of fl u i d U velocity amplitude at base (for sinusoidal motion) t time f forcing frequency (for sinusoidal motion) Dimensional analysis w i l l provide seven independent dimensionless groups from these ten variables. Using conventional groups, time-invariant representative values of cylinder tip response can be written as: Here, Y and Z are in-line and transverse displacement amplitudes respectively, of the column t i p . The dynamic properties of the structure are represented by the non-dimensional groups f/fn» the frequency ratio, £, the damping ratio, and m/pD2, the inertia ratio. The natural frequency and damping ratio may be taken as values either in a i r or water. The Keulegan-Carpenter number K = U/fD, and Reynolds number, Re = UD/v, characterize the motion at the structure base. In general, the applicability of model test results depends upon the similtude observed between prototype and model values of these dimensionless groups. As i s common in many fluid-structure modelling Y/D.Z/D = f l ( f / f n , C, D/H, m/pD2, K, Re) 15 problems, Reynolds number similtude Is d i f f i c u l t to maintain and i t s significance is not investigated in this study. A constant damping ratio is also d i f f i c u l t to maintain because structural damping i s not easily controlled. However, damping ratios of both model and prototype are expected to be low (< 10% c r i t i c a l ) . For a specific choice of cylinder and water depth, the damping ratio, diameter to depth ratio D/H, and inertia ratio are constants. The non-dimensional in-line response investigated here can then be described as: Y/D = f 2 ( f / f n , U/fD) The dependence of the response on the frequency ratio results in a dimensionless transfer type function. The Keulegan-Carpenter number characterizes the response dependence on the amplitude of base motion. Although K is defined at the structure base, the response dependence on K is important only at higher values of this parameter such as occurs at the free surface at resonance. The choice of dimensionless groups is not unique, and in-line and transverse response can be expressed in terms of alternate parameters such as the reduced velocity U = U/f D = K/(f / f ) , for which ' r n n correlations exist in the case of rigid cylinder response (Sarpkaya, 1979). However, the reduced velocity i s more suitable at larger values of K. In his investigations of the hydroelastic oscillations of a rig i d cylinder, Sarpkaya has also included a roughness factor, Kr/D and combined the Reynolds and Keulegan-Carpenter numbers as Re/K = D2/vT. 16 Isaacson & Maull (1981) and Sarpkaya (1979) have expressed transverse displacements of cylinders in harmonically oscillating flow as a function of the parameter A = (U/f nD) 2/(m£/pD 2). Similar correlations may exist for the case of flexible column motion; however, these correlations are much more d i f f i c u l t to obtain experimentally and are not addressed in this thesis. 2.3 Hydrodynamic Force Formulation The Morison equation i s used i n this study to estimate the hydrodynamic loads acting on the structure as i t oscillates in water. With reference to Figure 1, the most general form of the equation, for a body moving in a uniform fluid flow f i e l d is given as (Sarpkaya & Isaacson, 1981). F» = PV(1 + Ca)U - PVC aY t + j p^ApCU-Y^IU-Yj (2.1) Here, F' is the force per unit length, p the fluid density, U(x,t) • ** represents the flow f i e l d , Y , Y and Y are the total body motions as defined in Figure 1, the dot notation implying time derivative, V is the displaced volume of the body per unit length, A p is the projected width normal to the plane of motion and C n and C are empirical drag and added mass coefficients. The added-mass and drag components are seen to depend on the relative motion between structure and f l u i d . In the present case, the undisturbed fluid is stationary. Equation (2.1) therefore reduces to 17 (2.2) where the projected width Ap in (2.1) has been replaced by the diameter D of the cylinder. 2.4 Modal Analysis The cylindrical marine tower considered in this study can be considered as a uniform cantilever beam and the governing differential equation of motion developed from appropriate beam theory (see Appendix A). For the support motion problem, defined in Figure 1, the differential equation in terms of total displacement can be separated into two components; 6(t), the known input at the structure base, and Y(x,t) the displacement relative to the base. The dif f e r e n t i a l equation including the structural damping term, can be written as (Clough and Penzien, 1975). where the superscript denotes differentiation with respect to x, ^ ( x ) i s the mass per unit length, C is the coefficient of internal damping and p(x,t) i s the forcing function here defined as in Equation (2.2). E i s the elastic modulus and I the moment of inertia of the section. The same separation technique applied to total displacement can be applied to the forcing function p(x,t) since both drag and added mass depend on total relative motion between f l u i d and structure. Equation (2.3) can then be m (x)Y(x,t) + EIY (x,t) + CIY V(x,t)Y(x,t) = - mQ(x) 6(t) + p(x,t) (2.3) 18 rewritten as mY + CIY l vY + EIY iv = - m5 - ^ . P C D D | Y 4 | ( Y - ^ ) (2.4) in which m now includes the added mass, m = m + p—-.— C o a (2.5) and the dependence of Y on x and t is now implied. Equation (2.4) is a coupled nonlinear differential equation and must be solved numerically or simplified by making certain assumptions about the relative importance of the various terms. Two simplification methods and a numerical approach are considered. 2.4.1 Neglect of Drag Forces For small amplitudes of response and forcing, the nonlinear term in Equation (2.4) is small in comparison to the i n e r t i a l terms. If this term i s neglected, Equation (2.4) becomes linear and modal analysis can be used to obtain a solution for the response. In the case of harmonic base motion forcing 6(t), a closed-form solution i s obtainable. In the present study, a solution including only the two lowest modes (See Figure 2) i s developed, because the large frequency increments between modes precludes any significant contribution from higher modes. In principle, a solution involving any number of modes can be developed. The familiar techniques of modal analysis are applied in the linear aproximation (see Clough and Penzien, 1975). A separable series solution i s assumed: Y(x,t) = I <j>r(x) 5 r ( t ) (2.6) 19 where £ r(t) * s a time dependent amplitude function and ^ ( x ) i s the r-th mode shape of a cantilever beam given by Ax A x Ax A x <j)r(x) = cosh — c o s —£ o r(sinh - j - sin -y- ) (2.7) Here, £ is the length of the cantilever, A is a solution of cosAcoshA + 1 = 0 (2.8) and sinhA -sinA o = (2.9) r coshA +cosA r r Substituting Equation (2.6) into the linear (C Q = 0) form of Equation (2.4) and using orthogonality of mode shapes to decompose the partial d i f f e r e n t i a l equation (see Appendix A) we obtain the system of single degree of freedom equations i r ( t ) + 25 w I (t) + w 25 r(t) = P (t) (2.10) r Here P^ (t) is the generalized i n e r t i a l load. An assumption of r equivalent viscous damping has been assumed for the structural damping, with the damping ratio given as (2.11) 20 and the generalized i n e r t i a l load given as where P x (t) - o r6(t) (2.12) <l t \* t \a (2.13) J m(x)(}>r(x)dx o « = —n /Vx)^ (x)dx o For harmonic excitation given by 6(t) = Y sin ut (2.14) the generalized i n e r t i a l forcing function i s given as PT (t) = ct u>2Y sin uit (2.15) I r g x ' r In this case, a steady state solution can be found directly by substitution of the harmonic form in (2.10). This yields a co2Y sin(oot - 6 ) 5 r(t) = - — S - = - a rA rsin(oot-9 r) (2.16) u r L d - ( " / w r ) 2 ) 2 + ( 2 ? r ^ r ) 2 ] where the phase angle 6^  is given by 2£ u)/u 6 r = t a n _ 1 ^ l4^T)2) ( 2 * 1 7 ) r and w2Y A = r 1/2 w 2i(l-(u)/u) ) 2 ) 2 + (2C u/u ) 2 J (2.18) r r r r 21 The response of the column relative to the base is then Y(x,t) = - I • (x) ocrArsin(wt - 0 ) (2.19) r=l and the total response is Y t = Y(x,t) + 6(t) (2.20) For the uniform cantilever beam used in the present study I *r<X>d* (2.21) i <t>2(x)dx r o where Xf and are defined as in Equation (2.8) and (2.9) (See Blevins, 1977). The tip response can be obtained by evaluating <J>r(JO and substituting this result, with those of (2.16) and (2.19), into (2.20). Using a f i r s t mode approximation, this yields = 2 [ I 2 + 0.25 Y 2 + Y S i c o s e J 1 / 2 (2.22) tip L i g g 1 J where the bar indicates amplitude, and Q\ is as in (2.17). The addition of higher mode contributions i s straightforward but algebraically inconvenient because of the increasing number of phase angles which must 22 be combined (See Appendix B). The two mode approximation is given as Y., = 2 [ I 2 + I 2 + 0.25 Y 2 + Y (I cos Bj- l 2 c o s 6 2) tip 1 2 g g 1 + 25^2 cos (0! - e 2 ) ] 1 / 2 (2.23) and 6f is as in (2.17). As discussed earlier, the large modal frequency separation makes the inclusion of more than the two lowest modes unnecessary in this analysis. The functional dependence of tip displacement on the frequency ratio parameter can be generated from Equation (2.23) by simple calculation at discrete frequencies and specified forcing amplitudes and compared to experimental results. For a random input function such as seismic base motion the formulation cannot be reduced to give a closed-form solution in the time domain and must be integrated numerically to obtain the time series. In this case, i t is usually more convenient to work in the frequency domain, using Fast Fourier Transform methods commonly available. Because time series were desired, this study uses numerical integration to obtain the response. Here again, the addition of higher mode contributions i s straightforward. However, in the present case, seismic input frequencies 23 would not result i n a significant second mode vibration contribution because of the high second fundamental frequency of the model, and a f i r s t mode approximation i s used. For design purposes, a base shear prediction F^(t) can readily be obtained once the structure response is known, from an integration over the structure span given by F b(t) = /£m(x)Yt(x,t)dx (2.24) and similarly for base moment. Response spectra can be obtained in the usual manner for these and other functions for specified input motions. There are several assumptions inherent in the above approach which must be noted. First i s the assumption, used in a l l parts of this study, of a constant added mass. Other authors have shown that added mass is in fact a function of mode shape and frequency (Liaw and Chopra 1973; Byrd 1978; Pegg 1983). However, i t is generally accepted that the error introduced by neglecting these dependencies is minimal for slender structures vibrating in the lower mode shapes, for which the present analysis i s concerned. The second assumption involves the neglect of two forms of energy dissipation; viscous drag and surface wave generation. Near resonance conditions where tip oscillations can be large, these terms w i l l not be insignificant. For this reason, the above approach w i l l overestimate the response of the structure, the largest discrepancies occurring at resonance. This method of response prediction is thus overconservative. 24 The quantification of damping due to surface wave generation i s d i f f i c u l t . That term is readily accounted for as a free surface boundary condition in the potential flow solution, but has no direct interpretation in the small body regime. An approximation as viscous damping ignores the frequency dependence of this term although resonance is perhaps the only c r i t i c a l frequency and hence such an approximation may have merit. Because, in general, this damping cannot be separated from the viscous drag damping in the experiments, the total external damping in the analysis i s controlled by the choice of drag coefficient and wave generation damping values are not specified. A logical approach to the problem of including drag forces i s to uncouple or linearize the nonlinear drag term in Equation (2.4). However, the commonly used practise of uncoupling the drag terms (See Blevins, 1977) is not an alternative in an analysis of base motion problems because the relative motion which i s neglected in such an approach is exactly the structure response. Linearization techniques are tractable in certain problems and one such method i s considered in the next section. 2.4.2. Linearized Drag Forces The nonlinear drag term can be linearized to obtain a linear added damping in the equation of motion. This method, based on a Fourier expansion or error minimization procedure, results in amplitude dependent damping and forcing terms, which necessitates an iterative solution technique. For continuous structures, the amplitude and hence the damping is positionally dependent, and this dependence prevents a complete modal decomposition as in the purely i n e r t i a l formulation. However, a f i r s t mode approximation can be made as follows. 25 The drag term in Equation (2.4) can be written as: P D(x,t) = - V j Y j (2.25) where Kp = j PCD  (2.26) To linearize (2.25), let |Yt|Yt = A Y t(x,t) (2.27) where A includes the positional dependence, A = A(x). For a zero mean Gaussian process i t can be shown that minimizing the mean square error of Equation (2.27) for x constant yields |Yt|Yt = 1.2 a- (2.28) where 0£ is the root-mean-square of the total relative velocity and i s a function of position because of the dynamic responses. For a harmonic process, (2.28) becomes 1.2 0£ = a Yt (2.29) and a = 8/3n 26 where the bar represents amplitude. Using a f i r s t mode approximation, we write the positional dependence as Y t(x) -5+4.! (x) ?! (2.30) Combining the above, we have an expression for the linearized drag force, P D(x,t) = -aides' + <h (x)5i ) ( 6(t) + 4,! (x)l i (t)) (2.31) The generalized drag load becomes, after manipulation P D(t) = - ^ {6 6 (t) Bi + 3 2 (6 ki (t) + 6 (t)4i ) +5i£i(t)6 3} (2.32) where 3 i = 4)1 (x) dx i 3 = J* 4,3 (x) dx (2.33) 32 = /* <t>2 (x) dx Combining (2.32) with the generalized inertia force Pj(t) in Equation r (2.10), and reorganizing, the equation of motion becomes 27 C + C— •Pl(t) (2.34) Taking due account of phases, (See Appendix B), Pi(t) Is defined as [ ( m a ^ S ! ) ^ (aK Da 1Y g ( e ia)Y g + g 2Q) 2] 1sin(a)t - 6 L D ) ? 1 ^ 2 mp where 9LD = t a n ? mo>Y I, <2'36> g is the phase angle inherent in the forcing function due to drag damping and Cg = structural damping coefficient C Q = added damping from drag forces Assuming viscous damping, the linearized drag formulation gives 5s + a V [6B2+SiB3j 2w m3? n * (2.37) From equations (2.35) and (2.37) i t can be seen that both force and damping are amplitude dependent. The solution amplitude of (2.34) i s given i n the usual form as 28 5i -[(m3iO)2Y )2+(aK^toY (a)Y ^ + 5 i t S 2 ) ) 2 ] 2 11/2 m3 2 wi[(l - (a)/io 1) 2) 2 + (25i0 ) / a ) 1 ) 2 ] 1 / 2 (2.38) A solution of (2.38) can be obtained by choosing an i n i t i a l value of 5l, typically zero, and iterating to a prescribed solution accuracy. The tip response i s then given, as before, as Y(t) = 4 > i ( £ K i(t) + 6(t) (2.39) tip and the amplitude of cylinder tip motion i s Y„. =2 [ I 2 + 0.25 Y 2 + Y 5 cos(Q + %n)]1/2 (2.40) tip l g g l i LD / J As before, Q\ is the phase angle arising from structural damping: 250/n>i) 0! = tan" 1 { } l - C w / G ) ! ) 2 (2.41) 2.4.3. Numerical Method The nonlinear form drag term can be included in a numerical solution of the equation of motion. Beginning again with the separated form of the equation, we have, for the motion relative to the base 29 m I 5r<|>r(x) + CI I l r<t. r i V(x) + EI I 5 r^ V(x) r=l r=l r=l - - m6(t) - I l r + r ( x ) + 6(t) | I 5r4> (x) + 6 (2.42) r=l r=l where again m includes the added mass. Making a f i r s t mode approximation Y(x,t) = <h(x) €i(t) (2.43) and assuming that S(t) « Y(x,t) in regions where drag forces are significant, we have il<n<(>i(x) + £iCI<h l v(x) + 5iEI()) 1 l v(x) (2.44) = - {m6(t) + KD|li<(>i(x)|c:1<t.1(x)} Multiplying by <i>i(x), integrating over the length and rearranging, we obtain, l\ /* m^(x)dx + /* CI <j)1(x)<t)1lv(x)dx + K D | 5 1 | 5 i/^i(x)dx + ?! /* EI <h(x)<h±V(x) dx - - 6(t) m<p1(x)dx (2.45) Orthogonality concepts can be used to reduce the fourth order terms i n Equation (2.44); however, the nonlinear term cannot be simplified 3 0 further. Performing the integrations, and assuming viscous damping, a single degree of freedom equation is obtained: 33 ?l(t) + 2Ci<o 1? 1(t) 4 — K j l i C t O l ^ C t ) + 0 ) 2 ^(t) = (2.46) - a ^ t ) where a is given by Equation (2.21) and 3 2 and 3 3 are defined as i n Equation (2.33) Equation (2.46) can be solved numerically using one of a number of solution algorithms for nonlinear differential equations. In the present study a fourth order Runge-Kutta integration method is used. This method reduces Equation (2.46) to two f i r s t order equations by appropriate substitution (See Appendix C). The time history response i s generated by time stepping the integration process. This method i s perfectly general and can be used for random or sinusoidal forcing functions. For numerical s t a b i l i t y and accuracy, the time step is chosen as approximately a tenth of the smallest forcing function period. The numerical analysis can be extended to include higher mode contributions and the base velocity in the drag term which was originally assumed small. For a two mode formulation, Equation (2.43) is replaced by Y(x,t) = 4>l(*Kl(t) + 4>2(xH2(t) (2.47) 31 The nonlinear term in this case cannot be separated into time and position dependent functions and integrals involving E,^ must be evaluated at each time step. The same problem results from the inclusion of the base velocity i n the drag term in a f i r s t mode approximation. In that case, a solution is sought for the equation ll +mj^ { 2*510)! + KJJ J* <h (x)|4>i(x)li + 6(t)|dx (2.48) S ( t ) KD rX, , • • , 2 + m g J* +i(x)|4>i(x)e1 + 6(t)|dx + 0)i5i = - ai6(t) An exact solution to Equation (2.48) can evidently be found only by iteration. The combination of a Runge-Rutta method, iteration and time stepping can be time consuming and hence expensive. However, in the applicable frequency range, a two mode solution approximation, which included the base velocity, was found to differ very l i t t l e from a f i r s t mode approximation, which neglected base velocity. An alternative approach i s to use the previously calculated value of the 5 variable i n the integral evaluation. If the time step is small, this method provides very close agreement with an iterated solution and decreases the cost significantly. A similar approach for a discretized system was used by Anagnostopoulos (1981). Because the inclusion of base velocity and a second mode contribution does not significantly alter the predicted solution, the response predictions for the nonlinear drag case are generated from the solution of Equation (2.46). 32 3. TESTING FACILITIES AND MODEL PARAMETERS The present study attempts to model and predict the response of a slender c a n t i l e v e r marine structure undergoing horizontal base motion. The e x i s t i n g f a c i l i t i e s at the University of B r i t i s h Columbia Earthquake Laboratory used for t h i s purpose include a shaking table, water tank and data a c q u i s i t i o n system. A suitable model, instrumentation system, and test procedure must be chosen to be compatible with these f a c i l i t i e s . 3.1 Testing F a c i l i t i e s A l l tests were performed i n the Earthquake Laboratory of the Department of C i v i l Engineering at U.B.C. The laboratory f a c i l i t i e s include a 3.3m x 3.3m single degree of freedom shaking table (Figure 4) supported by an MTS drive system and a PDP-11 mini-computer. The shaking table operates i n a sing l e h o r i z o n t a l d i r e c t i o n i n the frequency range 0-30 hz, with very limited motion c a p a b i l i t y at the higher frequencies. The MTS system can excite the table with a va r i e t y of waveforms, of variable frequency and amplitude, as well as random motions from seismic or other d i s c r e t i z e d records. Sinusoidal and seismic base motion input were used i n these t e s t s . The PDP-11 provides the data a c q u i s i t i o n system with a processing c a p a b i l i t y of 17 channels i n any one t e s t . Four channels were used i n these t e s t s . The output signals are processed through an analogue to d i g i t a l converter and stored d i r e c t l y on floppy d i s c s . The water tank straddles the shake table so as not to be influenced by i t s v i b r a t i o n (Figure 5). The tank can be f i l l e d to a depth of lm and i s l i n e d with horsehair-type mats to damp waves radiated by the cylinder 33 and thereby minimize the effects of reflection on cylinder response. The model to be tested was attached to the table through a hole in the tank bottom sealed with a natural rubber diaphragm. This rubber seal effectively allows the f u l l table displacements of 15 cm peak to peak at low frequencies. A schematic of the test f a c i l i t y i s presented in Figure 3. 3.2 Model Parameters Model charateristics must be chosen carefully i n order to obtain experimental results which are applicable to prototype situations. As discussed in Chapter 2, dimensional analysis provides the relevant non-dimensional parameters in the problem. In the present study, Reynolds number similarity was not maintained and no attempt was made to maintain consistent damping ratios. An attempt was made to effectively model the remaining dimensionless groups: the frequency ratio f/f n> geometric ratio D / H , inertia ratio m/pD2 and Keulegan-Carpenter number K. The choice of model is controlled by the structural parameters, f / f n , D / H , and to a lesser extent by the damping ratio C« Seismic excitation i s severest in the frequency range 0-20 hz, whereas typical structure fundamental frequencies are 0.5 - 2.0 hz. Thus a r e a l i s t i c ratio of f/fn is 0-10. In order to maintain this ratio using the shaking table, a model f of 1-3 hz is required. Prototype values of D/H vary but are usually 1/20 -1/30. This ratio is limited in the model tests by the lm depth to which the water tank can be f i l l e d . For a model diameter of 5cm, maximum submergence gives D/H = 1/20. With the diameter known from the geometric scaling, the sectional mass, m, can be found for a specific inertia ratio m/pD 2 . Typical proto-type values are 0.8-2.0, giving m of 20-50 g/cm for a diameter of 5 cm. 34 Full scale damping ratios are usually 3 - 10%. The parameter cannot easily be controlled in the model and measured model values of 3.8% for f i r s t mode and 6.6% for the second mode were considered acceptable. The modulus-mass characteristics of dynamic prototypes are d i f f i c u l t to model. Convenient geometric scaling often requires elastic moduli which are not available, particularly when continuous models are desired. Sectional distortion can be used in some cases where the dynamic behaviour is primarily flexural vibration. Alternatively, sectional or linear mode models could be considered. Neither of these methods w i l l give accurate modal response which is desired in this investigation. Therefore, a continuous model of high mass and low stiffness was sought. After several t r i a l s were made with hollow plastic pipe f i l l e d with heavy materials, a solid Teflon cylinder with the following characteristics was fin a l l y chosen as the test specimen: Material : Teflon (polytetrafluoroethylene) Modulus : 9.9 x 106 kg/cm.sec Diameter : 5 cm Length free standing : 99 cm Sectional Mass : 43 g/cm 4.6 hz f (air) 28.7 hz Water depth H 97 cm Free height 2 cm 35 m/pD2 1.72 Modal damping r a t i o s 1 3.8% 2 6.6 % D/H 1/20 Although Teflon i s susceptible to creep at low stresses, and exhibits h y s t e r i t i c response behaviour, i t was f e l t that i t s use i n dynamic tests would be f e a s i b l e . 3.3 Data A c q u i s i t i o n The test instrumentation consisted of a v e r t i c a l strain-gauged aluminum bar connected to the cylinder t i p with a l i g h t spring as shown i n Figures 6 and 7. Horizontal motion of the cylinder t i p induces flexure i n the bar which can be measured with the s t r a i n gauges. This system measures displacement response both i n - l i n e and transverse to the d i r e c t i o n of base motion. Two s t r a i n gauges and a bridge are used for each d i r e c t i o n . Although a four gauge system would have increased the signal to noise r a t i o , the system performed adequately at a l l but the smallest outputs. The instrumented bar and spring were p e r i o d i c a l l y c a l i b r a t e d with s t a t i c displacements and a l i n e a r c a l i b r a t i o n was main-tained. Table displacements and accelerations were measured with the l i n e a r voltage d i f f e r e n t i a l transducer (LVDT) and accelerometer attached to the shaking table. The four channels monitored i n each test were processed through an analogue-to-digital converter and stored d i r e c t l y on d i s c s . The output sampling rate was set v i a the a c q u i s i t i o n software to give minimum sampling frequencies of approximately 20 times the testing frequency. 36 4. EXPERIMENTS The objective of the tests performed i n this study was to investigate the behaviour of a compliant marine structure excited by uni-d i r e c t i o n a l horizontal base motion. The experimentally determined response can then be used as the basis f or comparison and evaluation of the response predictions discussed i n Chapter 2. If the predicted response using one of the three approaches i s accurate, then that method would be adequate for prediction of earthquake loads and displacements. (As well, the test cylinder i s instrumented to obtain transverse displacements and although no comparisons w i l l be made to any p a r t i c u l a r theory, i t i s informative to note the c h a r a c t e r i s t i c s of t h i s motion). The tests and comparisons were of two types: sinusoidal test s , using frequencies i n the range 2 to 28 hz, and random motion using actual scaled seismic records. A l l tests were performed with a water depth of 97 cm and freeboard of 2 cm. 4.1 Damping Tests and System C h a r a c t e r i s t i c s The damping c h a r a c t e r i s t i c s of the model and model-instrumentation system are extremely important i n terms of r e a l i s t i c a l l y modelling the dynamic behaviour of a prototype structure. With t h i s f a c t i n mind, the i n i t i a l tests conducted were free v i b r a t i o n tests i n a i r and water to determine the system damping. It was hoped that the natural damping of the Teflon dowel plus the damping introduced by the v e r t i c a l spring would be low enough so as to reproduce the range of t y p i c a l prototype damping r a t i o s , usually about 5% of c r i t i c a l . Figure 8 shows the record obtained of a free v i b r a t i o n test i n a i r , Figure 9 the r e s u l t of a s i m i l a r test i n 37 water. Second mode free vibration frequency tests were also performed by oscillating the model at 28 hz and suddenly stopping the excitation, to determine a second modal damping ratio. An approximate value could be determined from this method. The log decrement was used to find the damping ratio, C, in each mode and an averaged result gave c,\= 3.8% c r i t i c a l in the f i r s t mode and £2 = 6.6% in the second mode. The tests in water were to determine the effect of added damping on the natural period of the model. A small decrease in natural frequency was noted and this reduced frequency was used for the response predictions. The instrumentation system used in these tests, as described in Chapter 3, has certain characteristics which should be noted. Whereas i t would be beneficial in terms of response accuracy to have very light spring forces acting on the model t i p , the strain-gauge system used required a moderate spring tension to give recoverable readings at low oscillation amplitudes. This fact in turn necessitated the use of a spring system with at least a nominal mass which resulted in a system resonance at approximately 22 hz, close to the second mode frequency of the model. Although this problem may have been correctable to some extent, observations Indicated that second mode effects did not contribute significantly to the magnitude of cylinder tip response, even though their presence was noted in the output signals (See Appendix D) at certain subharmonic frequencies. 4.2 Sinusoidal Tests The sinusoidal tests were performed at four amplitudes of base displacements: 1.0, 1.9, 2.8 and 3.6 mm. The choice of amplitude was 38 l i m i t e d by the possible destruction of the model from large amplitude resonances. At each amplitude of base motion, approximately 13 frequen-c i e s were used i n the range 2 to 28 hz. The higher frequencies could not be combined with the larger amplitudes because of l i m i t s on the shaking table system. The higher frequency tests were an attempt to induce second mode response of the model. Unfortunately the second mode frequency was very close to a resonant frequency i n the instrumentation system, as previously discussed, and the output was then obviously i n er r o r . This problem was compounded by the intense water spray generated by the cylinder motion which caused excessive noise i n the signal output at high frequencies. The accurate measurement of second mode response using ex i s t i n g f a c i l i t i e s appears f e a s i b l e only i f the model fundamental frequency i s lower than the value of 4.5 hz obtained i n the present t e s t s . Second mode response could be important for structures with low fundamental frequencies (<1.5 hz). However, the addition of higher mode contributions i n general decreases the structure t i p displacements, but increases those at more deeply submerged points. The combined response remains everywhere less than the f i r s t mode t i p response and hence i s not c r i t i c a l . I n - l i n e and transverse displacements were recorded i n most t e s t s . The transverse displacements showed a very peaked frequency response centering on the fundamental resonant frequency (discussed further i n Chapter 5). V i r t u a l l y no output was attainable outside a 1.5 hz band-width centered at resonance and the transverse s i g n a l was not recorded at frequencies outside this range. The output sampling rate was set at 39 frequencies of 20- 50 times the driving frequency and 5 seconds of steady state motion were recorded in each test. At low amplitudes of response, f i l t e r s were required to reduce noise in the signal. A 10 hz low-pass f i l t e r was used at test frequencies below 4.5 hz, a 40 hz low-pass f i l t e r on a l l other tests. Figure 10 shows a test at the fundamental resonance (4.5 hz); Figure 11 a test at the second mode resonance (28 hz). Note the intense water spray which affected response measurements near the second mode frequency. 4.3 Seismic Motion Tests The acceleration records of two actual earthquakes, El Centro N-S (1940) and San Fernando N21E (1971) were used as random motion input. These records were chosen because they have reasonably representative frequency spectra of earthquakes in the Pacific region (See Figures 12 and 13) and they are relatively severe. The San Fernando record in particular contains significant energy in the region of the fundamental frequency of the test cylinder (4.5 hz), as might occur from s o i l f i l t e r i n g . The digitized acceleration input data can be scaled to an appropriate level on the MTS system. In this investigation i t was desired to keep peak-to-peak base displacements of the model geometrically similar to those which can occur in actual strong motion seismic a c t i v i t y (40 - 60 cm). This resulted in 0.15 f u l l scale displacements in the El Centro test and 1.5 f u l l scale displacements in the San Fernando test. A peak and time-averaged Keulagan-Carpenter number is then maintained between prototype and model since that non-AO dimensional group is proportional to the ratio of displacement to cylinder diameter. Because the actual earthquake time scale is maintained the test accelerations are scaled by the same factors. The tests were performed at peak-to-peak displacements of 3.5 cm 2.0) and 2.8 cm ( K Dase ~ i n t h e E 1 C e n t r o a n d S a n F e r n a n d o tests respect ively. The random motion test data were sampled at a frequency which would allow direct use of table displacement and acceleration records in the numerical program described in Chapter 2. For the f i r s t mode approximation this frequency was 50 hz. This gives a time step of 0.02 seconds, or approximately 1/11 of the fundamental period, a value sufficiently small to assure numerical accuracy and st a b i l i t y . '^Sase 41 5. DISCUSSION OF RESULTS 5.1 Response Functions from Sinusoidal Tests The sinusoidal test data as recorded are presented In Table 2. The frequency dependence of cylinder tip response can be presented in non-dimensional form for the sinusoidal tests and predictions. These comparison plots, sometimes termed receptance functions, are presented in the Figures 14, 15, 16 and 17 for the tested ratios of base amplitude to diameter ratio Y /D. The frequency ratio f/f is defined in terms of g n the fundamental frequency of the cylinder in water. Several important results are indicated on these figures. As anticipated, the response near resonance is very peaked and dynamic amplifications on the order of 15 - 20 occurred at the fundamental frequency. Unfortunately, as discussed in Chapter 4, a true second mode response was not obtainable in part because of limitations on the shake table and partly because large system resonances led to erroneous data above a frequency ratio of about 4.0. The general trend in the data towards a second response peak i s evident in the figures. For a linear system, the dynamic amplification i s a system constant dependent on the stiffness and damping parameters and independent of the amplitude of applied motion or force. The nonlinear behaviour of the system considered here is apparent in the amplitude dependence of the dynamic amplification factor Y/Y . This factor ranges from a high of 19 at Y /D = 0.02, decreasing steadily with amplitude increases to a low of 13 at Y /D = 0.072 (See Figures 14 to 17). This decrease is a result of the increased damping associated with the occurrence of flow separation near the cylinder t i p , and to increased surface wave generation at the larger forcing amplitudes. An increase in the viscous damping ratio of approximately 45% is required to achieve the total amplification reduction. Although this is a large increase, the absolute damping values are quite small and the change is of the order of 1.5% c r i t i c a l . However, as we note from the peaked response characteristic, the response is very sensitive to small changes in damping. 5.2 Response Predictions for Sinusoidal Motion The methods of response prediction used in this study have been developed in Chapter 2. The simplest approach neglects the nonlinear drag force term in the Morison equation and a closed-form solution is obtained for the resulting linear equation of motion. The neglect of drag forces in the formulation effectively lowers the damping by eliminating the drag damping present in the actual system. This method consequently overestimates the response of the cylinder t i p . The discrepancy w i l l increase with forcing amplitude due to the increasing importance of drag forces near the free surface. The response function for this solution method is compared with other methods and the experimental data in Figures 14 to 17. The ratios of predicted response to measured response at resonance Y ,/Y for pred exp the four amplitudes used in the tests' are presented in Table 1. The results of this method exhibit the characteristics anticipated. Only for the lowest amplitude case, Figure 14, does this method provide an accurate estimation of structure response. In that case, nonlinear drag forces in the experiment were negligible. Because the experimental data for second mode response is not considered valid, for reasons discussed, 43 i t i s impossible to make conclusions regarding the accuracy of the second mode response p r e d i c t i o n based on t h i s method. The second p r e d i c t i o n method i n c l u d e s a l i n e a r i z e d form of the Morison drag term i n the equation of motion. A closed-form s o l u t i o n can be obtained f o r a f i r s t mode response assumption. The response f u n c t i o n generated by t h i s method i s included i n Figures 14 to 17 and the resonant response r a t i o s are tabulated i n Table 1. I t i s evident from Table 1 that the accuracy of a l i n e a r i z e d drag p r e d i c t i o n i s dependent on the choice of drag c o e f f i c i e n t , (Lj, as could be a n t i c i p a t e d . The i n i t i a l choice of drag c o e f f i c i e n t was based on an e x t r a p o l a t i o n of e x i s t i n g c o r r e l a t i o n s of t h i s c o e f f i c i e n t with Reynolds number and Keulegan-Carpenter number as i n d i c a t e d i n Figure 18 ( a f t e r Sarpkaya and Isaacson). A c y l i n d e r t i p Reynolds number of order l& was assumed. Accordingly a drag c o e f f i c i e n t of 0.7 was s e l e c t e d as a re p r e s e n t a t i v e value f o r the range of K and Re expected near the c y l i n d e r t i p and was used i n both the l i n e a r i z e d drag and numerical s o l u t i o n s . For Cp = 0.7, the l i n e a r i z e d drag p r e d i c t i o n s are very unconservative at resonance and diverge from the experimental r e s u l t s with i n c r e a s i n g base motion amplitude (Table 1). In t h i s case, drag f o r c e l i n e a r i z a t i o n r e s u l t s i n an overestimate of drag damping. A more accurate and non-divergent response e s t i m a t i o n i s obtained with C^ = 0.25 and an optimal s o l u t i o n i s reached with - 0.20. The l i n e a r i z e d drag method appears to give accurate response p r e d i c t i o n s f or t h i s problem when an appropriate drag c o e f f i c i e n t i s used. Of course, t h i s i n f o r m a t i o n would not g e n e r a l l y be known a p r i o r i . 44 As a third approach, the nonlinear drag term was included in a numerical solution of the equation of motion derived from a f i r s t mode response assumption. A drag coefficient of = 0.7 was again used i n i t i a l l y and the results are presented in Figures 14 to 17 and Table 1. The response predictions for = 0.7 are once more unconservative, but consistent over the range of base displacement amplitudes. This indicates that the numerical method can provide good resonant response estimates i f the drag coefficient i s chosen correctly. Figure 19 shows the relationship between resonant response amplitude ratios and drag coefficient used in the numerical approach. Evidently a very low drag coefficient is required to obtain an accurate prediction. This is a result of the low Reynolds number associated with the motion away from the free surface. The amplitude dependence of the drag terra is apparent from the variation of response ratios at fixed C^. In a linear system, this ratio would be amplitude invariant. In the present case a general requirement for agreement of experimental and numerical response is an increasing drag coefficient for increasing base motion amplitude. However, the interaction of transverse and in-line oscillations at the largest test amplitude caused a reversal in this trend. Based on Figure 19 a drag coefficient of about 0.3 would give reasonable results except in the case of the lowest input amplitude. 45 5.3 Transverse Response The cylinder response near resonance at the larger base displacement ratios, Y /D, was influenced by l i f t forces. The occurrence and S magnitude of l i f t forces i s a function of the degree of flow separation, hence the Keulegan-Carpenter number. For the two largest base motion amplitudes, (2.8 and 3.6 mm), the free surface Keulegan-Carpenter numbers based on peak response values, were 5.1 and 6.2 respectively. The literature suggests that vortex shedding (Figure 20) and l i f t forcing can occur for K > 5.0 (Sarpkaya, 1979). Thus, i t could be expected that l i f t forces would be present near the free surface. In the present experiments these l i f t forces trigger ovalling of the cylinder t i p . The interaction of in-line and transverse oscillations decreases the in-line response over a narrow band-width centered at resonance, from the response expected with no l i f t forces present. The interaction i s evident in the results presented in Table 2 for Y /D = .054 and .07 2. As 8 resonance (4.5 hz) is approached, transverse and in-line amplitudes increase. Between 4.3 and 4.7 hz, the in-line response drops as a result of the energy dissipation in transverse oscillations. Maximum in-line response occurs, in this case, at a frequency below resonance. Figure 21 indicates such response characteristics near resonance for Y^ /D = 0.054. The very narrow bandwidth of this phenomenon is evident in the transverse response function presented as Figure 22. The response at the lowest Y /D ratio (0.20) was negligible and is not shown in the figure. The displacement responses of the cylinder tip are combined as tip loci in Figures 23 and 24. Figure 23, a resonant response at Y /D = 0.54, shows a clearly defined orientation of the oscillation axes. This characteristic was evident in a l l the responses which included large 46 transverse components, and the axes appeared stable only in such orientations. This s t a b i l i t y may indicate either a hydrodynamic interaction favouring that orientation, such as a specific l i f t force phase, or simply some inherent structural characteristic or both. However, i f the cylinder exhibited a tendency to oscillate diagonally because of a material defect, this behaviour would presumably be evident in the response in a i r . Also, the response would be altered i f the cylinder were rotated in its base clamp. Because only in-line response occurred in air tests and cylinder rotation had no effect on the preferred axes orientations i t can be concluded that the directions of the e l l i p t i c axes are controlled by a hydrodynamic mechanism. The exact details of this interactive mechanism are no doubt complex and beyond the scope of the present study. 5.4 Response to Seismic Input The displacement response of the cylinder during earthquakes was tested using real earthquake accelerograms for base motion input. These seismic records were scaled to give maximum displacements consistent with geometric scaling while the time frame was maintained f u l l scale. As a result, accelerations in the test are lower than recorded values. Two earthquake records were used in the tests; the El Centro 1940, N-S component and the San Fernando, 1971, N21E component. The test input was scaled to give a maximum base displacement ratio, Y /D, of about 0.4, consistent with strong motion earthquake displacements. The El Centro earthquake energy is distributed f a i r l y evenly over a range of frequencies (see Figure 12) and this particular record is often treated as white noise input. Such input would not be expected to produce large dynamic responses. Figure 25 shows the tip response for the El Centro input, and indicates that dynamic response of the cylinder i s minimal during this earthquake. Transverse oscillations were also negligible during this test. The hydrodynamic loads in this case could be estimated quite adequately by a pseudo-static analysis. whereas most of the energy in the El Centro earthquake is equipartitioned at frequencies below 3 hz, the San Fernando 1971 earthquake has a large spectral peak near 4.5 Hz, the fundamental frequency of the cylinder (see Figure 13). Such spectral peaks result from s o i l f i l t e r i n g or reflection interaction of white noise earthquakes such as El Centro. The magnitude of this peak i s also significantly greater than any peaks in the El Centro record (Figure 12). This earthquake provides a better il l u s t r a t i o n of the effects of coincidence between structure natural frequency and earthquake spectral density peaks. The base input and in-line tip response for the San Fernando earthquake are compared in Figure 26. It is apparent that dynamic amplification is significant in this case because of the resonant effects. The resulting tip displacements and velocities are sufficient to cause l i f t forces which result in transverse oscillations during the peak excitation period. A transverse displacement time history is presented in Figure 27. Although the transverse oscillations are small, they can contribute to the stresses on the structure and should be included in a seismic design load estimation. Larger responses could develop from a longer exposure to resonant frequencies. This could be a po s s i b i l i t y under certain conditions, although the scaling of the San Fernando record used here, in conjunction with the proximity of cylinder 48 natural frequency and spectral density peak probably represents a reasonably extreme event. 5.5 Numerical Prediction of Seismic Response With several small modifications, the numerical time step program discussed in Chapter 2 can be used to predict the response of the cylinder to seismic input. The actual base motion is used as input and the integration performed at each time step. A linear approximation with no drag loading can be obtained by setting the drag coefficient to zero. The response time histories presented in Figures 28 and 29 were generated using a drag coefficient of 0.2, i.e. assumming that non-linear effects would be minimal. Alternatively, the drag coefficient could be set to zero and the structural damping increased slightly to account for the hydrodynamic damping i f the latter is low. Figures 28 and 29 indicate that the response prediction using the numerical solution approach is quite accurate. The peak responses are in close agreement, and there are only minor differences in phase. The accuracy obtained at this low value of drag coefficient indicates that viscous effects are almost negligible in seismic response, a conclusion previously reached by other researchers (Clough 1960; Liaw and Chopra 1973; Byrd 1978). 6. CONCLUSIONS AND RECOMMENDATIONS 49 6.1 Conclusions The displacement response of a flexible, surface-piercing circular cylinder to sinusoidal and random base motion has been investigated. The variables considered in the study have been the frequency ratio and non-dimensional base motion amplitude. Experimental response has been compared to predictions based on modal analyses using a Morison equation representation of the hydrodynamic forces. Three treatments of the nonlinear drag term in the resulting equation of motion have been considered: neglect of the drag terra, linearization, and inclusion in i t s nonlinear form. Based on results discussed in Chapter 5, conclusions are made as follows: Cylinder response to small base displacements can be accurately predicted neglecting fluid drag forces. However, the nonlinear drag term becomes increasingly important at higher displacement amplitudes. In the present case, this term is equivalent to a nonlinear added damping which reduces the dynamic amplification as input motion amplitude increases. Neglect of this term leads to overly conservative response predictions. The method of drag linearization developed herein can give accurate response estimates i f a suitable drag coefficient is used in the analysis. A suitable coefficient will be lower than that chosen on the basis of anticipated cylinder tip motion, a value which gives consistently unconservative results. Numerical solution of the equation of motion including the nonlinear drag term can give accurate response predictions for a suitable choice of drag coefficient. This method is numerically much more time consuming 50 than the linearized drag approach, and suffers from the same limitations on accuracy. The drag linearization technique appears, then, to be a more r e a l i s t i c approach for response prediction in the case of sinusoidal motion. The ground displacements of strong motion earthquakes with spectral density peaks near structure resonance frequencies can be amplified to levels which cause flow separation near the free surface. Small transverse oscillations are evident in such cases and this additional response would contribute to the stresses in the structure. However, i t is concluded that the in-line response can be adequately predicted neglecting drag forces. The more common linear analysis methods in the frequency domain can thus be used in the solution of the seismic response problem, as an alternative to the time step integration method used here. Inherent in the above conclusions is the f e a s i b i l i t y of using modal analysis with a Morison type forcing function in the problem formulation. In fact, this approach appears to be quite tractable and can give good results with the inclusion of only the f i r s t mode. It should be noted that the present experiments have generated data, such as modal damping ratios which are then used as input for numerical predictions. In the design process this data is not known a priori and estimations of such quantities as added mass and structural damping must be made. These estimations will undoubtedly affect the accuracy of response predictions based on any of the techniques presented in this study. 51 6.2 Recommendations for Further Study The work of this study could be extended into several areas in terras of the base motion problem. These areas might include: «» Investigation of possible correlation of nonlinear drag and added viscous damping over specific frequency and amplitude ranges. Such correlations would allow the accurate prediction of response using only linear terras in the equation of motion. «> Similar studies of response for multi-leg structure with various deck-leg linkage arrangements. o A similar dynamic response analysis which includes a base system for modelling s o i l structure interaction. Y t ( x , t ) 6 ( t ) Y . (t) t i p Y (x,t) P(x,t) Figure 1. Coordinate System Definition Figure 2. Mode Shapes For a Cantilever Beam WATER TANK S P R I N G SHAKING TABLE Figure 3. Schematic of Test F a c i l i t y Figure 5. Testing Tank Over Shaking Table 55 TO M I K E MWLiritu STUm SAUGES ! ' . - SIDE » lUMimm HOC (25>m SIU*«E: TO C T l l U D f K T l » Figure 6. Instrumented Rod For Strain Measurement Figure 7. Photo of Instrumented Rod 5 6 57 Figure 10. Photo of Sinusoidal Test at 4 .5Hz . Figure 1 1 . Photo of Sinusoidal Test at 28Hz. 53 F i g u r e 13. F o u r i e r S p e c t r a f o r San Fernando 1971 59 a o EXPERIMENTAL NON-LINERR DRAG LINEARIZED DRAG NO DRAG Dynamic Amplification (Y /Y ) = 19 t g max T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r— 0.8 16 2.4 9.2 4.D 4.8 5.6 6.4 1.2 9.3 fl.O FREQUENCY RATIO F/F In Figure 14. Response Function For Y /D =.020 60 o s EXPERIMENTAL NON-LINERR DRAG LINEARIZED DRAG NO ORflC . a o EXPERIMENTAL NON-LINERR DRAG " LINEARIZED DRAG - NO DRAG Figure 16. Response Function For Y /D =.054 62 a o EXPERIMENTAL NON-LINERR DRRG LINEARIZED DRAG NO DRAG Figure 17. Response Function For Y /D =.072 CD = .7 CD = .25 CD = .20 CD = .7 Y /D g NO DRAG LINEARIZED LINEARIZED LINEARIZED NONLINEAR .02 1.07 .65 .85 .91 .74 .036 1.30 .60 .90 .97 .76 .054 1.53 .5 7 .92 1.00 .79 .072 1.55 '.49 .84 .93 .75 TABLE 1 : Resonant Response Ratio Y ,/Y v pred exp 64 IN-LINE RESPONSE Y /D = .02 g Yg/D = .036 Yg/D = .054 Yg/D = .072 FREQ. RESPONSE FREQ. RESPONSE FREQ. RESPONSE FREQ. RESPONSE 2.0 1.5 2.0 2.8 2.0 3.9 2.0 5.2 3.0 2.3 3.0 4.5 3.0 6.5 3.0 8.5 3.5 3.4 3.5 6.7 3.5 9.3 3.5 12.3 4.0 6.0 4.0 13.8 4.0 19.5 4.0 30.5 4.5 19.3 4.3 17.9 4.3 38.0 4.3 48.0 4.7 10.4 4.5 30.1 4.5 26.0 4.5 25.0 5.0 8.5 4.7 18.1 4.7 20.0 4.7 22.0 6.0 3.9 5.0 15.8 5.0 20.4 5.0 24.0 9.0 2.9 6.0 6.8 6.0 11.0 6.0 12.0 14.0 2.2 9.0 4.8 9.0 5.6 9.0 7.0 18.0 5.0 14.0 4.4 14.0 7.1 14.0 10.0 22.0 6.5 18.0 7.6 18.0 14.0 18.0 12.0 28.0 23.0 28.0 23.8 TRANSVERSE RESPONSE Y /D = .02 g Y /D g - .036 Y /D g = .054 Yg/D = .072 4.5 2.3 4.0 2.1 4.0 1.7 4.0 2.7 4.7 1.4 4.3 3.5 4.3 6.0 4.3 8.0 4.5 15.6 4.5 28.0 4.5 33.0 4.7 1.6 4.7 25.0 4.7 29.0 5.0 1.0 5.0 2.4 5.0 20.0 5.1 2.5 FREQUENCIES ARE IN Hz, RESPONSE VALUES IN mm. f = 4.5 Hz TABLE 2 : Sinusoidal Test Data _ i I I I I I I I I I 0.1 0.3 1 J. J •••10 I , I I ' ' ' ' ' 3 4 5 10 15 Figure 18. Drag Coefficient versus Reynolds Number For Various Values of K (after Sarpkaya and Isaacson) 66 ( 2 P Figure 20. Numerical Simulation of the Formation of Asymmetric Vortices (after Sarpkaya and Isaacson) J » e TRANSVERSE i + IN-LINE a. m l ST 1 1 1 1 1 1 1 1 p 1 1 1 1 1 1 1 1 1 r-U IM IBS (L92 I X ID 104 1M 1J2 1JI 12 FREQUENCY R f l T I O F/F1N Figure 21. In-line and Transverse Response Near Resonance For Y /D =.054 g 67 Figure 22. Transverse Response Functions For Various Base Displacement Amplitudes 68 V I Figure 24. Tip Locus off Resonance For Y /D =.054 69 F i g u r e 2 5 . R e s p o n s e C o m p a r i s o n F o r E l C e n t r o F i g u r e 2 6 . R e s p o n s e C o m p a r i s o n F o r S a n F e r n a n d o 70 f igure 28. Response Predict ion Comparison For El Centro 71 Figure 29. Response Predict ion Comparison For San Fernando 72 BIBLIOGRAPHY Anagnostopoulos, S., 1982, "Dynamic Response of Offshore Platforms to Extreme Waves Including Fluid Structure Interaction", Engineering Structures, Vol. 4, pp 179-185. Blevins, R.D., 1977, "Flow Induced Vibrations", Van Nostrand Reinhold Co., New York. Blevins, R.D., 1979, "Formulas for Natural Mode Shape and Frequency", Van Nostrand Reinhold, New York. Byrd, R., 1978, "A study of the Fluid Sructure Interaction of Submerged Tanks and Caissons in Earthquakes", Earthquake Engineering Research Report 78/08, May 1978. Chakrabarti, S.K. and Frampton, R., 1982, "Review of Riser Analysis Techniques", Applied Ocean Research, Vol. 4, No. 2, pp. 73-90. Clough, R.W., 1960, "Effects of Earthquakes on Underwater Structures", Proceedings, Second World Conference on Earthquake Engineering, Tokyo 1960, Vol. I l l , pp. 815-831. Clough, R. and Penzien, J., 1975, "Dynamics of Structures", McGraw-Hill. Eatock-Taylor, R., 1981, "A Review of Hydrodynamic Load Analysis for Submerged Structures Excited by Earthquakes", Engineering Structures, Vol. 3, pp. 131-139. Fish, P.R., et a l . , 1980, "Fluid-structure Interaction in Morison's Equation for the Design of Offshore Structures", Engineering Structures, Vol. 2, pp. 15-26. Grecco, M. and Utt, M., 1982, "Dynamic Analysis of a Multi-Tube Production Riser", Proceedings, Ocean Structural Dynamics Symposium, Oregon State University, pp. 289-305. 73 Isaacson, M., 1983, "Earthquake Loading on Axisymmetric Offshore Structures", Proceedings, Fourth Canadian Conference on Earthquake Engineering, Vancouver, Canada. Isaacson, M. and Maull, D., 1981, "Dynamic Response of Vertical Piles in Waves", Proceedings, Hydrodynamics in Ocean Engineering, Trondheim, Norway. Keulegan, G.H. and Carpenter, L.H., 1958, "Forces on Cylinders and Plates in an Oscillating Fluid", Journal of Research of the National Bureau of Standards, Vol. 30, No. 5, pp. 423-440. Kirk, C.L., et a l . , 1979, "Dynamic and Static Analysis of a Marine Riser", Applied Ocean Research, Vol. 1, No. 3, pp. 125-135. Kirkley, O.M., 1973, "Earthquake Response of Fixed Offshore Structures", Ph.D. Thesis, University of I l l i n o i s , Urbana, U.S.A. Kirkley, O.M. and Murtha, J., 1975, "Earthquake Response of Offshore Structures", Proceedings, C i v i l Engineering i n the Oceans III, pp. 865-879. Liau, C.Y. and Chopra, A., 1973, "Dynamics of Towers Surrounded by Water", Earthquake Engineering Research Centre Report, 73/25. Malhotra, A.K. and Penzien, J., 1969, "Stochastic Analysis of Offshore Tower Structures", Earthquake Engineering Research Center Report, 69/6. Mei, C.C., et a l . , 1979, "Exact and Hybrid-Element Solutions for the Vibration of a Thin Elastic Structure Seated on the Seafloor", Applied Ocean Research, Vol. 1, No. 2, pp. 79-88. Moe, G. and Verley, R.L.P., 1980, "Hydrodynamic Damping of Offshore Structures in Waves and Currents", Proceedings, 12th Offshore Technology Conference, Houston, pp. 37-44. 74 Morison, J.R., et a l . , 1950, "The Force Exerted by Surface Waves on Piles", Transactions, American Institute of Mining and Metalurgical Engineers, Vol. 189, pp. 149-154. Murtha, J.P. and Kirkley, O.M., 1975, "Response Spectra for Ocean Structures", Proceedings, 7th Offshore Technology Conference, pp. 985-990. Newnark, N.M. and Rosenbleuth, E., 1971, "Fundamentals of Earthquake Engineering, Prentice-Hall Inc., Englewood C l i f f s , N.J. Pegg, N.G., 1983, "An Experimental Study of the Seismic Forces on Submerged Structures", MASc Thesis, University of British Columbia, Vancouver, Canada. Penzien, J. and Kaul, M.K., 1972, "Response of Offshore Towers to Strong Motion Earthquakes", Earthquake Engineering and Structural Dynamics, Vol. 1, pp. 55-68. Sarpkaya,T., 1979, "Lateral Oscillations of Smooth and Sand-Roughened Cylinders in Harmonic Flow", Mechanics of Wave-Induced Forces on Cylinders (ed. T.L. Shaw), Pitman, London, pp. 421-435. Sarpkaya, T. and Isaacson, M., 1981, "Mechanics of Wave Forces on Offshore Structures", Van Nostrand Reinhold, New York. Sawaragi, T. et a l . , 1977, "Dynamic Behaviour of a Circular Pile due to Eddy Shedding in Waves", Coastal Engineering in Japan, Vol. 20, pp. 109-120. Selna, L. and Cho, D., 1972, "Resonant Response of Offshore Structures", Journal of Waterways, Harbours and Coastal Engineering Division ASCE, Vol. 98, WW1, pp. 15-24. Skop, R.A. and G r i f f i n , O.M., 1975, "Vortex-Excited Oscillations of Elastic Cylinders", Proceedings, C i v i l Engineering in the Oceans III, Delaware, pp. 535-545. Stark, P., 1970, "Introduction to Numerical Methods", Macmillan Company, New York. 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Zedan, M.F. and Yeung, J.Y., 1980, "Dynamic Response of a Cantilever Pile to Vortex Shedding in Regular Waves", Proceedings, 12th Offshore Technology Conference, pp. 45-59. 76 APPENDIX A MODAL ANALYSIS The methods of response prediction used in this study are based on the technique of modal analysis. This technique uses the mode shapes obtained from the solution of the governing equations of motion for an Euler beam to describe the dynamic response of a continuous structure undergoing external loading. Modal analysis effectively reduces the governing partial differential equation to a system of ordinary differential equations for which solutions exist. The response functions are linearly superimposed to describe the overall response. This method is very useful for linear systems and can be applied successfully to nonlinear problems provided that the degree of nonlinearity i s small. The partial differential equation describing the flexural motion of a slender beam, neglecting the effects of damping, shear deformation and rotatory inertia is where mQ is the mass of the beam per unit length, y is the normal displacement from the longitudinal axis, x i s the coordinate along the beam axis and p the externally applied load per unit length (See Clough and Penzien, 1975). In general, E and I can be functions of position, however, in the case of a prismatic beam, the equation becomes, for free vibrations mo(x)Y(x,t) + (EIY"(x,t))" = p(x,t) (Al) m Y(x,t) + EIY i V(x,t) = 0 (A2) 77 subject to the appropriate geometric and kinematic boundary conditions, Variable separation is used to solve Equation (A2); let Y(x,t) - 4>(x)5(t) (A3) Substituting (A3) into (A2) and re-arranging, we have iv, „ -m a 4 (A4) 4> (x) _ o S(t) _ u • (x) EI ?(t) The solution to the fourth order differential equation defines the positional dependence given i n general as <j> (x) = Asinax + Bcosax + Csinhax + Dcoshax (A5) where the coefficients are determined from the boundary conditions. The vibration frequency can be related to the constant 'a' by letting 2 0) m * - TT* ( A 6 ) in the solution for the time dependent amplitude £(t) For the cantilever beam, the appropriate boundary conditions are expressed as 78 r o n m o ( . H / , 4> (0) = 0 Geometric ^'(o) = 0 Natural • .,* ^  ' ° Q (A7) These conditions are used to obtain a system of equations from (A5). The coefficient matrix of this system i s set equal to zero for nontrivial solutions resulting in the transcendental equation 1 + cos al cosh al = 0 (A8) For a particular solution = a^i of (A8), the ratio of co-efficients in the system of equations derived from (A3) and (A7) becomes cosX X, + coshX SL n sinX % + sinhX i. K* J n n and the nth mode shape function is then given by X x X x X x X x <Pn(x) = cosh -£ cos -£ O r (sinh — sin —) (A10) It can readily be shown that the natural modes of beams with classical boundary conditions are orthogonal over the span of the beam, where orthogonality, as defined herein, implies I 0 m * n / <j> (x>j> (x)dx = { (All) o n m £ / 2 m = n The free vibration frequency associated with a particular mode shape is determined from (A6), as - (A12) " *2 " o The time dependent amplitude function becomes K (t) = A 1 sinw t + B 1 cosu t (A13) n n n n n v / where the coefficients A^ , B^ in (A13) are determined by the i n i t i a l conditions. The complete solution to (A2) i s the sum of the modal solutions, Y(x,t) = ) (A 1 sinw t + B 1 COSUJ t) q> (x) (A14) n=l The formulation of the more general problem of beam vibration w i l l include damping and external forcing. In that case, the solution i s formed from the substitution of (A14) into the general equation of motion including damping. For the present problem, this yields ao oo . oo I m<|> (x) 5 (t) + 1 CI<t>n(x)£n(t) + I E I ^ v ( x ) S n ( t ) = P(x,t) (A15) n=l n=l n=l 80 Multiplying by <J>m(x), integrating over the length of the beam, and using ( A l l ) , we obtain M J n ( t ) + C n ^ n ( t ) + K n ? n ( t ) = ^ * n ( x ) p ( x » t ) d x (A16) o where Mn is the equivalent mass, given here simply as the total mass, m£, C n is the generalized modal damping coefficient and K r is the generalized stiffness. For orthogonal mode shapes, i t can be shown that 2 K = 0) M (A17) n n n v If the system damping is approximated by viscous damping, as is usually done, equation (A16) can be written as 2 5 (t) + 25 w £ (t) + u> 5 (t) =P (t) (A18) n v ' n n n n n v n v v ' where the equivalent viscous damping ratio for each mode has been defined as , m fn (A19) g n 2m n and this term w i l l contain any damping contribution present i n the generalized f o r c i n g function P n ( t ) , given as / %(x,t)<j> (x)dx P n(t) - -2 (A20) Solutions to (A18) can be found using either convolution integrals, Laplace transforms or numerical methods. Although i n principal, a solution based on modal analysis can include contributions from many modes, generally only the lowest modes are Important i n the analysis slender beams at low frequencies. 82 APPENDIX B : DERIVATION OF RESPONSE AMPLITUDES For the nth damped single degree of freedom system defined as 2 E + 2c u) £ + u> E - P n(t) (Bl) n n n n n n a steady state amplitude is given as f . I* 1 (B2> n ? or [ (l-(w/w ) 2 )2+(£ w/w ^ l 1 ' 2 n L n n n J In the case of base excitation given by (5 (t) = Y sinwt (B3) the generalized i n e r t i a l forcing function becomes PT = ct ID 2 Y (B4) I n n g and a solution to (Bl) including only inertia forces is given then as £ = a A sin(u)t-© ) (B5) n n n n where 83 u>2 Y 1 A = * (B6) n u 2 [(l-(w/w ) 2 ) 2 + ( 2 ; u/d ) ] 1 / 2 n 1 v v n' ' v n n / J and 2? w/u) 0 = tan" 1! — M (B7) n l-(u>/U f n For two modes, we have the modal and base motions as Si (t) = ?i sin(o)t-<pi ) £ 2 (t) = S 2sin(wt^> 2) ( B 8 ) 6 (t) = Y sinw t 8 The cylinder tip response is given by combining and expanding (B8) to yield Y(x,t) = <Pi (£ )Ci (sinwtSi - cosw tsin^ 2 ) + <t>2 (*• (siao tcosS2 ~ cosw tsin32 ) + Y sinwt g (B9) For the present formulation • l (A ) - 2 84 4>2(*> = "2 (BIO) Using (BIO) in (B9) and collecting terms, Y = [(25icosOi - 25 2cos© 2 + Y ) 2 + (2S 2sinG 2 - 2 ^ i s i n 0 i ) 2 ] 1 / 2 (Bll) Simplifying Y . = 2 k 2 + \ 2 + 0.25Y2 + Y (£icos0i - £ 2cos© 2) tip 1 2 g g - 2Ci " i 2 C o s(0i-O 2)] 1 / 2 (B12) In the case of linearized drag, the generalized forcing function i s more complex. The sum of the generalized i n e r t i a l and drag components i s given as a KD P(t) = a i U ) 2 Y sinwt l g 1 oa2Y2cosu)t g m32 8 + 3 2(li^Y + coliY coswt) + (B13) with 85 01 " J* <t>i(x)dx 3 2 - /* <t>2(x)dx (B14) 33 = /* <t>3i(x)dx The components of (B13) involving Ii are included as viscous damping, leaving the forcing function as P(t) = aiu2Y sincot + —-—& (8].u>Y + e2li)cosu>t (B15) 8 m&2 8 which can be rewritten in amplitude form. The response is then given, as before, as i l (B16) OJ 2 [(l-(u)/u>i)2)2 + (2tiu/ui)2\l 1 2 The constituent responses of the cylinder tip are Y(A,t) = 4>1(A)51sin(u>t-eL D - ©x) 6(t) = Y sinut g (B17) 86 where aK^uY (BiUY + B2E) • = tan ' l ^ — § (B18) moj^ Y B i g ©1 as in (B7) Combining and simplifying, we obtain Y t i p = 2 + 0 , 2 5 Y g 2 + Y g ^ l c o s ( 0 l + e L D ) J 1 / 2 (B19) 87 APPENDIX C DESCRIPTION OF COMPUTER PROGRAM The numerical method used to solve the linearized and nonlinear di f f e r e n t i a l equation derived in Chapter 2 involves a small Fortran IV program. The program requires the following input to be contained in a data f i l e attached to the main program: 1) I n i t i a l values of displacement and velocity. Zero i n i t i a l values are usually assumed since this allows a stable solution to be reached more quickly. 2) Length of structure, diameter, elastic modulus, moment of inertia, sectional mass and damping ratio. 3) Frequency of oscillation, in hertz, and amplitude of base motion, for sinusoidal motion. For random forcing, these variables are replaced by the appropriately discretized input. 4) Drag coefficient 5) Number of time steps, and the time increment. The program calculates the natural frequencies and coefficients of the generalized equation of motion using cantilever beam theory. The second order differential equation is replaced by two f i r s t order equations which are solved using a fourth order Runge-Kutta integration 88 scheme. A time history response i s generated for the tip displacement by time stepping. Time increments of approximately a tenth of the minimum forcing function period are recommended for numerical accuracy. At high frequencies, modulated responses are evident for many cycles N > 600 for low damping values. Near the resonant frequency, the output stabilizes reasonably quickly (N < 300). 89 APPENDIX D INSTRUMENT CALIBRATION AND TYPICAL DATA The strain-gauged rod and spring system was calibrated periodically by static displacement tests. Figure 30 shows a typical calibration curve indicating a linear relationship between bridge voltage and in-line and transverse cylinder tip displacement. A time history and Fourier spectra for a test at 9 Hz and Yg/° = .036 are presented in Figure 31 and 32 respectively. These records il l u s t r a t e the presence of the second mode response at certain forcing frequencies. Such response was only noticeable in the 9 and 14 Hz records, frequencies near subharmonics of the second modal frequency, 28 Hz. It should be noted that the single mode approximations developed in this study w i l l not reproduce this type of response. The average half-peak-to-peak response of the actual record was used as representative response in the receptance functions rather than the Fourier amplitudes of the output at the forcing frequency. Typical in-line and transverse records at resonance (4.5 Hz) are presented in Figures 33 and 34 respectively. 90 Figure 30. C a l i b r a t i o n Curve for the Instrumented Rod 91 O 03 -1 Figure 31. Typical Time History at 3rd Subharmonic I D 0.0 4.0 B.O 12.0 16.0 20..0 24 .0 28.0 32.0 36.0 FREQUENCY (HZ) Figure 32. Fourier Spectra For Record of Figure 31 F i g u r e 34. T r a n s v e r s e R e s o n a n c e T i m e H i s t o r y a t Y /D =.054 


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