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Hydrodynamic interactions between ice masses and large offshore structures Cheung, Kwok Fai 1987

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HYDRODYNAMIC INTERACTIONS BETWEEN ICE MASSES AND LARGE OFFSHORE STRUCTURES by KWOK FAI CHEUNG B.A.Sc. University of Ottawa, 1985. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1987 © Kwok Fai Cheung, 1987 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the The University of B r i t i s h Columbia, I agree that the Library s h a l l make i t fre e l y a v a i l a b l e 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 i s understood that copying or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: March, 1987 ABSTRACT The objective of the work described in t h i s thesis i s to evaluate the s i g n i f i c a n c e of the ambient f l u i d on the motion of an ice mass in the v i c i n i t y of an offshore structure and during the subsequent impact mechanism. Models for iceberg d r i f t are f i r s t reviewed. The changes in flow f i e l d around an ice mass d r i f t i n g in a current near an offshore structure are investigated by pot e n t i a l flow theory. The proximity effects and current interactions are generalized by introducing the added mass and convective force c o e f f i c i e n t s for the ice mass. A two-dimensional numerical model based on the boundary element method i s developed to calculate these c o e f f i c i e n t s over a range of separation distances up to the point of contact. A numerical model based on i c e properties and geometry is developed to simulate the impact force acting on the structure. Both the 'contact-point' added masses estimated in t h i s thesis and the t r a d i t i o n a l l y assumed f a r - f i e l d added masses are used in the impact model separately. The r e s u l t s from the two cases are compared and the c r u c i a l roles played by the ambient f l u i d during impact are discussed. F i n a l l y , a number of related topics i s proposed for further studies. i i Table of Contents ABSTRACT i LIST OF TABLES iv LIST OF FIGURES v ACKNOWLEGEMENTS v i i i 1. LITERATURE REVIEW 1 1.1 Introduction 1 1.2 F a r - F i e l d Phase 2 1.3 Near-Field Phase 3 1.4 Contact Phase 4 1.5 Improvements 6 2. HYDRODYNAMIC FORCES ON TWO CYLINDERS 7 2.1 Introduction 7 2.2 Theoretical Formulation 9 2.2.1 Governing Equations 9 2.2.2 Hydrodynamic Force Coe f f i c i e n t s 12 2.3 Numerical Solution 16 2.3.1 Distributed Source Method 16 2.3.2 Numerical Formulation 18 2.3.3 S i n g u l a r i t i e s 20 2.4 Results and Discussions 22 2.4.1 Comparison with A n a l y t i c a l Results 23 2.4.2 Kinematics and Dynamics of F l u i d Flow 25 2.4.3 Added Mass during Impact 27 2.4.4 Discussion of Results 29 3. ICEBERG IMPACT LOAD 33 3.1 Introduction 33 3.2 Ice Crushing Model 34 i i i 3.2.1 Uniaxial Compressive Strength 35 3.2.2 Cha r a c t e r i s t i c Strain Rate 37 3.2.3 Indentation Factor 38 3.2.4 Contact Factor 39 3.2.5 Shape Factor 40 3.3 Impact Model 40 3.3.1 Foundation Model 41 3.3.2 Added Masses with Both Cylinders Moving ...43 3.3.3 Added Masses: from 2-D to 3-D 44 3.3.4 Mathematical Formulation 45 3.3.5 Numerical Procedure 47 3.4 Results and Discussions 48 3.4.1 Added Mass E f f e c t s 49 3.4.2 Parameter Study 51 4. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STUDY ...54 4.1 Conclusions 54 4.2 Recommendations for Further Study 55 APPENDIX A: LIST OF SYMBOLS 57 APPENDIX B: REFERENCES 61 iv LIST OF TABLES 3.1 Basic case used in impact analysis 65 3.2 S o i l properties (Adapted from Croteau, 1983) 65 3.3 Comparison of ice penetrations 66 3.4 Added mass e f f e c t s : energy and impulse 66 v LIST OF FIGURES 2.1 Two-cylinder problem: d e f i n i t i o n sketch 67 2.2 Two-cylinder problem: i l l u s t r a t i o n of numerical 67 procedure 2.3 Added mass c o e f f i c i e n t for V x and 68 2.4 Added mass c o e f f i c i e n t for U 68 A. 2.5 Added mass c o e f f i c i e n t for U 69 2.6 Convective force c o e f f i c i e n t for V 2 and V 2 69 x y 2.7 Convective force c o e f f i c i e n t for U 2 70 2.8 Convective force c o e f f i c i e n t for U 2 70 y 2.9 Convective force c o e f f i c i e n t for V U 71 A A 2.10 Convective force c o e f f i c i e n t for V U 71 y y 2.11 Convective force c o e f f i c i e n t for U U , V U and 72 x y y x V U x y 2.12 Comparison with analytic r e s u l t s : C g 1 73 2.13 Comparison with analytic r e s u l t s : C 2 a n c * C ^ ?3 2.14 Comparison with analytic r e s u l t s : C v 1 74 2.15 Comparison with analytic r e s u l t s : C 2 and C ^ 7 ^ 2.16 D i s t r i b u t i o n of normalized unit potential around 75 cylinder 1 due to i t s own motion in the x di rect ion 2.17 D i s t r i b u t i o n of normalized unit potential around 76 cylinder 2 due to the motion of cylinder 1 in the x d i r e c t i o n 3.1 Mathematical model of ice element (Powell et a l , 77 1 985) v i 3.2 Normalized uniaxial compressive strength vs 77 st r a i n rate (Adapted from Bohon and Weingarten, 1985) 3.3 Indentation test (Schematic sketch) 78 3.4 Indentation factor for b r i t t l e f a i l u r e : 78 comparison between experimental and t h e o r e t i c a l results 3.5 Mathematical model of foundation ( E l a s t i c 79 half-space) 3.6 Added mass c o e f f i c i e n t for horizontal motions of 80 a f l o a t i n g c i r c u l a r cylinder in f i n i t e depth of water (Yeung,l98l) 3.7 Impact configuration 81 3.8 Mathematical model of impact (Hydrodynamic force 81 not shown) 3.9 Added mass e f f e c t s : displacement of structure 82 3.10 Added mass e f f e c t s : impact force on structure 82 3.11 Variation of ice strength: displacement of 83 structure 3.12 Variation of ice strength: impact force on 83 structure 3.13 Variation of impact v e l o c i t y : displacement of 84 structure 3.14 Variation of impact v e l o c i t y : impact force on 84 structure 3.15 Variation of iceberg d r a f t : displacement of 85 structure v i i 3.16 V a r i a t i o n of iceberg d r a f t : impact force on 85 structure v i i i ACKNOWLEGEMENTS I wish to express my sincere gratitude to my supervisor, Dr. M. Isaacson, for his valuable advice and guidance during the preparation of t h i s t h e s i s . I greatly appreciate his e f f o r t and time in reviewing the preliminary draft of thi s thesis and the valuable suggestions to improve the content. Thanks are also due to Dr. N.D. Nathan for reviewing t h i s thesis. The f i n a n c i a l support in the form of a research assistantship from the Natural Sciences and Engineering Research Council of Canada i s g r a t e f u l l y acknowledged. •ix 1 . LITERATURE REVIEW 1.1 INTRODUCTION The c o l l i s i o n forces which ari s e during impact of an ice mass on an offshore structure are an important consideration rin offshore design in the A r c t i c region. In order to determine the impact load on the structure, a knowledge of the location, v e l o c i t y and acceleration of the ice mass before c o l l i s i o n i s necessary. These are influenced by the ice mass and structure geometries, current, wave, wind, c o r i o l i s force as well as the location of the ice mass r e l a t i v e to the structure. The presence of a large offshore structure has s i g n i f i c a n t effects on the d r i f t motions of nearby ice masses. Since the current and wave f i e l d s are modified for d i f f e r e n t separation distances between the two objects, the wave d r i f t force, wave-frequency and zero-frequency added masses, convective forces and l o c a l current v e l o c i t y have to be recalculated at d i f f e r e n t locations of the ice mass. With a time stepping procedure, such calculations eventually lead to the impact angle and v e l o c i t y with which the impact load is estimated. A number of research studies have been directed towards these two-body and component problems. However, the technical l i t e r a t u r e on these topics i s widely scattered, and therefore, a summary of some representative papers and research reports is presented in th i s chapter. The studies 1 2 in t h i s area can be c l a s s i f i e d into three phases according to the locations of the ice mass r e l a t i v e to the structure: namely, F a r - F i e l d , Near-Field and Contact Phases. 1.2 FAR-FIELD PHASE The F a r - F i e l d Phase concerns iceberg d r i f t in open water, where the separation between the iceberg and the structure is large and the interactions between them are n e g l i g i b l e . The trajectory depends mainly on iceberg properties and geometry, and environmental conditions such as current, wave, wind and the c o r i o l i s e f f e c t . A number of numerical models, based on d i f f e r e n t assumptions, have been proposed. These models can be broadly c l a s s i f i e d as dynamic, kinematic, and s t a t i s t i c a l in nature depending on the approach adopted (Hsiung and Aboul-Azm, 1982; Mountain, 1980; Sodhi and El-Tahum, 1980). In addition, a hybrid model which combines dynamic and s t a t i s t i c a l methods to predict iceberg d r i f t was developed by G a s k i l l and Rochester (1984). A study concerned with the accuracy of iceberg d r i f t models was conducted by A r c t i c Sciences (1984), in which they found that the major source of uncertainty i s associated with current information and ice geometry, which are seldom available in an accurate manner. Therefore they concluded that an accurate model in predicting long term iceberg d r i f t i s not l i k e l y to be available in the near future. 3 1.3 NEAR-FIELD PHASE The r e a l i z a t i o n that ice mass motions in the v i c i n i t y of a large offshore structure do not depend s o l e l y on environmental conditions, has focussed the studies on iceberg d r i f t onto the Near-Field Phase, in which the motions of icebergs are influenced by the structure. The disturbance to the flow f i e l d around an ice mass caused by an offshore structure leads to hydrodynamic interactions and eventually influences the d r i f t motion of the ice mass. In a more fundamental study, Yamamoto (1976) obtained an analytic solution for the forces on an a r b i t a r y number of c i r c u l a r cylinders subjected to a uniformly accelerating current in a two-dimensional flow. For the s p e c i a l case of two cylinders, one moving and one fixed, i t i s found that an a t t r a c t i n g force always acts on the f l o a t i n g one toward the fixed one and the zero frequency added mass varies s i g n i f i c a n t l y as the two are close together. The two-body problem in pure wave action was investigated by Ohkusu (1974), Van Oortmerssen (1979) and Loken (1981), where the wave frequency added mass and damping c o e f f i c i e n t s were found to vary with the separation distance between them. The hydrodynamic interaction between several v e r t i c a l bodies of revolution in waves was studied by Kokkinowrachos, Thanos and Z i b e l l (1986) and the model was extended to predict the motions of an ice mass near an offshore structure. 4 The problem of the motions of an ice mass due to waves and a current near an offshore structure was investigated by Isaacson (1986) (See also Hay & Company, 1986; Isaacson and Dello S t r i t t o , 1986). In t h i s model, the nonlinear interactions between waves and the current were assumed to be i n s i g n i f i c a n t , and the wave and current f i e l d s were treated separately. The wave-induced o s c i l l a t i o n s were obtained as a solution to the two-body d i f f r a c t i o n problem and the d r i f t t r a j e c t o r y , accounting for current drag, wave d r i f t forces and added mass e f f e c t s , was obtained through a time-stepping procedure. A similar problem was studied experientally by Sakvalaggio and Rojansky (1986), where tests were made to simulate the impact of d i f f e r e n t wave driven icebergs with a fixed caisson type structure. Both studies found no impact occured for small ice mass. 1.4 CONTACT PHASE The Contact Phase i s characterized by the impact of an ice mass on an offshore structure. This includes studies of ice properties, crushing mechanism as well as numerical models. Depending on the complexity of the problem, an energy method or dynamic analysis may be used to estimate the impact load. The energy method i s the most fundamental procedure in treating ice mass impact problems on r i g i d structure-foundation systems. In the most common and simplest approach, the i n i t i a l kinetic energy of the ice mass i s equated to the energy dissipated by crushing of the 5 ice. With the assumption of constant ice strength, the maximum impact force and ice indentation can be estimated d i r e c t l y from the energy balance equation (Johnson and Nevel, 1985; Gershunov, 1986). In a more rigorous approach, the energy equation is solved incrementally, and the ice impact force is obtained as a function of ice indentation distance (Cox, 1985). A comprehensive study using dynamic models was done by Croteau (1983) where the ice properties, s t r u c t u r a l f l e x i b i l i t y and structure-foundation interactions were modelled. With his numerical models, three d i f f e r e n t types of ice-structure interactions were investigated: earthquake responses of an offshore structure surrounded by a thin ice sheet, migrating of a large ice floe across a slender f l e x i b l e member and the impact of an iceberg on a massive structure. Various results were presented as functions of time. An experimental scheme and a combined kinematic-dynamic model concerning ice mass impact on an offshore structure were developed by Salvalaggio and Rojansky (1986). The key parameters in the model were i n d e n t i f i e d through a s e n s i t i v i t y analysis and the results were correlated with a p r o b a b i l i t i s t i c model. The structure design loads for ice impact were estimated by Johnson and Nevel (1985). In their study, an energy based mathematical model for predicting ice impact forces was developed, and the design loads were obtained by 6 using the Monte Carlo d i s t r i b u t i o n technique and s t a t i s t i c a l input d i s t r i b u t i o n . However, this model t e l l s nothing about the force v a r i a t i o n during the period of contact which i s of p r a c t i c a l importance in design. 1 . 5 IMPROVEMENTS Nowadays, numerical models for iceberg motions near an offshore structure and the subsequent impact mechanism are under extensive research and development. However, the studies of hydrodynamic interactions between the two objects are mainly directed to wave actions. The changes in flow f i e l d around a d r i f t i n g ice mass in the v i c i n i t y of a large offshore structure and during the impact process are ignored in most models. Some empirical relations or arbitary assumed constants are usually used to account for these e f f e c t s . Results obtained by such models are not f u l l y r e l i a b l e and sometimes give r i s e to misleading design c r i t e r i a . It i s attempted here to r e c t i f y this oversight. The instantaneous hydrodynamic c o e f f i c i e n t s which characterize the interaction between a d r i f t i n g ice mass and a fixed large offshore structure in a uniform current are formulated using p o t e n t i a l flow theory and solved by the boundary element method. The r e s u l t s are then applied to an impact model to estimate the iceberg impact load on structure. 2. HYDRODYNAMIC FORCES ON TWO CYLINDERS 2.1 INTRODUCTION Hydrodynamic forces a r i s e whenever there are motions between an object and i t s ambient f l u i d . In poten t i a l flow, these forces can be broadly c l a s s i f i e d into two categories, namely, temporal and convective forces. The former are induced by temporal acceleration of an object in the ambient f l u i d and are l i n e a r l y proportional to the magnitude of acceleration. The l a t t e r , associated with the v e l o c i t y squared term in the Bernoulli equation, are caused by the rel a t i v e motion of the ambient f l u i d at the surface of the body. Introducing the added mass and convective force c o e f f i c i e n t s , the hydrodynamic forces can thereby be expressed in terms of the ve l o c i t y and acceleration of the body. The added mass can be interpreted as the mass of a certain volume of f l u i d that i s accelerated with an immersed body. The added mass of a general body i s d i r e c t i o n a l l y dependent, since the amount of f l u i d accelerated with the body depends on the precise flow geometry. In p r i n c i p l e , every f l u i d p a r t i c l e accelerates to some extent as the body accelerates, and the added mass can be considered as a suitably weighted integration of the entire f l u i d mass surrounding the body. As a body moves r e l a t i v e to the ambient f l u i d , the re l a t i v e f l u i d v e l o c i t y varies along the surface of the body 7 8 and so does the pressure associated with the v e l o c i t y squared term in the Bernoulli equation. The integration of such a pressure d i s t r i b u t i o n constitutes the convective force. Unless the flow geometry is symmetric, the convective force i s usually non-zero. Since t h i s force i s related to the r e l a t i v e v e l o c i t y of the object, i t i s sometimes referred as a damping force. However, from the hydrodynamic point of view, t h i s force i s c a l l e d a convective force in t h i s t h e s i s . Because of the changing flow geometry when a body moves r e l a t i v e to the ambient f l u i d in the v i c i n i t y of another stationary body, the hydrodynamic forces acting on i t vary markedly. Such proximity effects can be generalized by expressing the hydrodynamic c o e f f i c i e n t s as functions of the i r r e l a t i v e geometry. The hydrodynamic forces at th i s stage or at the contact phase are of p r a c t i c a l interest, since the d r i f t motions of a fl o a t i n g body and the subsequent impact load depend considerably on these forces. In t h i s chapter, the added mass and convective force c o e f f i c i e n t s in two-dimensions are formulated by potential flow theory and solved by the boundary element method. A solution i s presented for the special case of two c i r c u l a r c y l i n d e r s , one fixed and one moving, in an unsteady uniform current. The boundary element solution is then compared with that obtained by Yamamoto (1976) on the basis of the c i r c l e theorem. 9 2 . 2 THEORETICAL FORMULATION The two-dimensional problem considered corresponds to two cylinders of arbitary sections, one fixed and one moving in an unsteady uniform current of an i n f i n i t e f l u i d . In three-dimensions, this may be considered to correspond to two v e r t i c a l cylinders extending from the free surface to the seabed in water of constant depth when free surface effects are n e g l i g i b l e . 2 . 2 . 1 GOVERNING EQUATIONS The f l u i d i s assumed incompressible and i n v i s c i d , and the flow is assumed to be i r r o t a t i o n a l . As far as real f l u i d is concerned, such assumptions are unsatisfactory and flow separation occurs in most s i t u a t i o n s . However, flow separation on the downstream side of the stationary cylinder has no s i g n i f i c a n t e f f e c t on the hydrodynamic aspects of the approaching cyl i n d e r . Furthermore, when the moving cylinder moves close to current speed, the ef f e c t s of flow separation are n e g l i g i b l e . The hydrodynamic force on the moving cylinder can reasonably be estimated by potential flow theory, and therefore the flow can be described by a vel o c i t y p o t e n t i a l , s a t i s f y i n g the Laplace equation in the f l u i d domain, 0 : V2<i> = 0 within ( 2 . 1 ) With reference to F i g . 2 . 1 , the centroid of the stationary cylinder (cylinder 2 ) i s fixed at the o r i g i n of a 1 0 right-handed Cartesian coordinate system, while the other cylinder (cylinder 1) i s moving freely with an unsteady v e l o c i t y V = ^ v x ' V y ^ * Both cylinders are subjected to an unsteady uniform current U = ( u x,Uy). The v e l o c i t y potential, i s made up of a component associated with the incident current $ , a scattered p o t e n t i a l $ u due to the incident current acting on the two c y l i n d e r s , and a component $ v due to the motion of the moving cylin d e r near the stationary cylinder. * = * c + * u ' + $ v (2.2) The p o t e n t i a l corresponding to the incident current <i>c i s known and i s given in terms of the current components U"x and U as: y $ = U x + U y (2.3) c x yJ The p o t e n t i a l is also subjected to boundary conditions on the c y l i n d e r surfaces and in the f a r - f i e l d . Since the f l u i d i s taken as i n v i s c i d , the tangential v e l o c i t i e s on the cylinder surfaces may be non-zero. However, the normal f l u i d v e l o c i t y r e l a t i v e to each cylinder surface i s zero. Thus, the boundary conditions on cylinder surfaces are given by: 3$ V on T, — = (2.4) d n 0 on r2 where n i s the distance normal to the surface and V i s the n cylinder v e l o c i t y in the d i r e c t i o n n. F i n a l l y , in the far 11 f i e l d , t h e i n f l u e n c e o f t h e c y l i n d e r s i s r e q u i r e d t o d i m i n i s h s o t h a t $ i s e q u a l t o # c . The p o t e n t i a l s <i>u a n d <i>v a r e e a c h p r o p o r t i o n a l t o U a n d V r e s p e c t i v e l y , a n d c a n f u r t h e r be d e c o m p o s e d i n t o c o m p o n e n t s p r o p o r t i o n a l U , U , V a n d V , s o t h a t t h e f l o w x ^ x y p o t e n t i a l # c a n be e x p r e s s e d a s : $ = U (0 +x) + U (0 +y) - V 0 - V 0 (2.5) x Yux y v u y 1 x * v x y v y \*---J> 0 u x a n d 0 may be i n t e r p r e t e d a s t h e s c a t t e r e d p o t e n t i a l s g e n e r a t e d by t h e two c y l i n d e r s when b o t h a r e f i x e d a n d s u b j e c t e d t o a n i n c i d e n t c u r r e n t o f u n i t m a g n i t u d e . T o g e t h e r w i t h t h e u n i t i n c i d e n t c u r r e n t p o t e n t i a l s , x a n d y , t h e f i r s t two t e r m s i n E q . 2.5 r e p r e s e n t a u n i f o r m c u r r e n t s c a t t e r i n g a r o u n d t w o s t a t i o n a r y c y l i n d e r s . 0 y x a n d 0 v ^ , a r e t h e u n i t s c a t t e r e d p o t e n t i a l s a s s o c i a t e d w i t h t h e x a n d y c o m p o n e n t s o f t h e m o t i o n o f c y l i n d e r 1 i n a n o t h e r w i s e s t i l l f l u i d a n d i n t h e p r e s e n c e o f c y l i n d e r 2 w h i c h i s s t a t i o n a r y . The b o u n d a r y c o n d i t i o n E q . 2 . 4 , c a n a l s o be e x p r e s s e d i n t e r m s o f t h e s e f o u r u n i t p o t e n t i a l s . S u b s t i t u t i n g E q . 2.5 a n d e x p r e s s i n g V n i n t e r m s o f v e l o c i t y c o m p o n e n t s , V a n d V , E q . 2.4 b e c o m e s : 30 30 30 30 TJ ( JIJUX + n ) + u ( _Zuy. + n ) _ v _Zvx _ V J!Vy x 3n x y 3n Y x 3n Y 3n V n + V n on T , = x x y y (2.6) 0 w h e r e n x a n d n ^ a r e t h e d i r e c t i o n c o s i n e s w i t h r e s p e c t t o 1 2 the x and y direc t i o n s . Since t h i s holds for a r b i t r a r y values of V"x, V^, U x and U^, the boundary condition defined in Eq. 2.6 can be resolved into four uncoupled expressions: 9*vx _ " nx ° n F l 3n 0 on T (2.7) 2 30 -n on T, — ^ = y (2.8) 9n o on T 2 3</» — ^ - -n on T, and T 2 (2.9) 3n 30 ^ = -n on and T 2 (2.10) 3n v In the calcu l a t i o n of added masses, the temporal de r i v a t i v e of the velocity potential $ i s required. Since the unit potentials are functions of space (x,y) only, d i f f e r e n t i a t i o n of Eq. 2.5 with respect to time gives: 3$ — = U (0 +x) + U (<t> +y) - V 0 - V 0 (2.11) g t x yux y uy 1 x^vx y vy Once the unit potentials 0 , 0 , 0 and d> have been r ux uy vx vy evaluated, the velocity potential and i t s temporal de r i v a t i v e are defined. The hydrodynamic force acting on the moving cylinder can readily be obtained. 2.2.2 HYDRODYNAMIC FORCE COEFFICIENTS As potential flow theory i s used, the hydrodynamic pressure acting on the moving surface of cylinder 1 i s given by the unsteady Bernoulli equation (Milne-Thomson, 1938): 13 3$ p p = " P u — - q r 2 + C (2.12) w 9t 2 r where C i s a constant when the cylinder is not rotating. As far as the resultant force acting on the body is concerned, C w i l l not contribute to the f i n a l results and can be taken as zero for convenience, q^ i s the tangential velocity of f l u i d at the body surface r e l a t i v e to the moving surface i t s e l f and so is given by: Or = ^ " ("Vy + V y n x ) ( 2 ' 1 3 ) where V n and V n correspond to the clockwise and counter x y y x clockwise tangential v e l o c i t y components respectively while d i f f e r e n t i a t i o n i s performed in the counter clockwise d i r e c t i o n . Substituting the expression for the flow potential <t> given in Eq. 2.5 gives 30 30 r^ = " V — " nv> " V + nx> 3<6 30 + U ( — ^ - n ) + U ( — + n ) (2.14) x ar y y 3r x The resultant hydrodynamic force components acting on the moving body in the i - t h d i r e c t i o n are given by: „(i> = - I pni< dr i i /• 3$ P f p i — n.dr + -H I q 2n.dT i = 1,2 (2.15) w / r i 3 t 1 2 /r, r 1 where i=1,2 denotes the force components in the x and y directions respectively. The f i r s t and second terms on the 1 4 right-hand side of Eq. 2.15 represent the temporal and convective forces respectively. Expanding the temporal and convective terms with Eqs. 2.11 and 2.14, Eq. 2.15 becomes: F ( 0 = AlPw[-Ci? V, - C « Vy + (eg + C®)V, + (CU + C$)UV} + RiP.[{C$ V? + 2cg V,V, + Cj{ V») + ( C « + 2<7<? tf.tf, + c" tfy2) - 2(eg V,U z + eg V,ff, + C« 7yff, + C<J, V,U,)] { 2 ] e ) in which (2.17) cg = — f yn,dT (2.18) r(0 _ (2.19) J " f 4>vyflidT (2.20) (2.21 ) (2.22) 1 I ,dd>m 2Ri JTl ( dT n y ) 2 n , ( fr (2.23) cl;' = _L / _ 2i?j J r i v dT n ' ) ( ar + (2.24) J L / (£*2 + 2i?! / F l v ar n z ) 2 n , - < f T (2.25) i ' r , a ^ 2i?! JTi [ dT (2.26) 1 /" , 3 ^ / F l 1 ar n ' ) ( a r + n i ) n ) (fr (2.27) 2^1 Jpj v ar n , )2 n , - < f T (2.28) 15 ^^Mf^-^*^ (2-30) where the C a's and c v ' s are the added mass and convective force c o e f f i c i e n t s corresponding to the appropriate acceleration and ve l o c i t y terms. In Eq. 2.16, the C Q c o e f f i c i e n t s represent forces due to the incident current i f i t were not modified by the presence of the two cyl i n d e r s . On the other hand, the C c o e f f i c i e n t s corresponding to U"x and represent forces associated with the scattered potential. The resultant forces due to the incident current are due to both terms so that proportional to (C +C ) which is sometimes referred to o a as the i n e r t i a c o e f f i c i e n t . For i representing the x and y directions, a t o t a l of 8 added mass and 20 convective force c o e f f i c i e n t s are required •to describe the hydrodynamic force acting on the moving cylinder. However, because of the orthogonality properties of potential functions, the t o t a l number of coe f f i c i e n t s may be greatly reduced i f the flow geometry is symmetric about any of the axes. Under any circumstance, the C Q c o e f f i c i e n t s can readily be obtained by closed-form or numerical integrat ions. 16 2.3 NUMERICAL SOLUTION As long as the two cylinders are c i r c u l a r and are not touching each other, the problem can be solved by a number of c l a s s i c a l methods, such as the method of images (Kelvin, 1847) and the c i r c l e theorem (Miln-Thomson, 1938). The method of images can be applied to solve the problem of two c i r c u l a r cylinders of the same size moving c o l i n e a r l y . Yamamoto (1976) applied the c i r c l e theorem to formulate and solve the problem of multiple arbitary moving c i r c u l a r cylinders in an uniformly accelerating flow. However, most solutions by c l a s s i c a l methods lack generality and so are not suitable for most p r a c t i c a l situations. For more general geometries, several numerical methods may be used. Among these methods, the d i s t r i b u t e d source method is frequently used in f l u i d mechanics and i s used in this thesis. The method was f i r s t described in t h i s context by von Karman (1927) and Lamb (1932). Since t h i s method involves the d i s c r e t i z a t i o n of the boundary of a domain, i t is sometimes referred as the boundary element method. 2.3.1 DISTRIBUTED SOURCE METHOD Consider a potential flow problem in which the solution has to s a t i s f y the Laplace equation and the f a r - f i e l d condition. Since any individual source or combinations of sources w i l l s a t i s f y the Laplace equation and the f a r - f i e l d condition, i t is natural to represent the boundary by a 17 continuous d i s t r i b u t i o n of sources. The source strength d i s t r i b u t i o n i s defined in such a way that the one remaining boundary condition on the body surface is also s a t i s f i e d . Representing the boundary by a continuous d i s t r i b u t i o n of sources, the unit p o t e n t i a l at any point x = (x,y) in the domain can be obtained by summing the contributions from each source and i s given by the following integral over the entire boundary, T 0(x) = — I f(*)G(x,£) dr (2.33) 2TT Jr in which G(x,£) = ln(s) = In /(x-£) 2 + (y-r?) 2 (2.34) where j[ = ( £ , 1 7 ) i s the coordinate of a point in the boundary T, G(x,j[) is the Green's function corresponding to a source of unit strength located at £ and f(ji) is the source strength at £. Eq. 2.33 s a t i s f i e s the Laplace equation and the f a r - f i e l d conditions. The remaining boundary condition on the body surface can be obtained by d i f f e r e n t i a t i n g Eq. 2.33 with respect to n, distance normal to the boundary, 30 1 1 r 3G - — (x) = - - f(x) + — I f(£) — (x,£) dr (2.35) 3n 2 2ir Jr 3n where x i s now defined as the point at which the boundary condition is enforced. The second term on the right-hand side of Eq. 2.35 represents the sum of contributions from each source to the unit normal velocity at x» whereas the f i r s t term i s due to the source at x i t s e l f and arises from 18 a careful integration of thi s s i n g u l a r i t y (See, for example, Brebbia, 1980). The boundary condition equates 3<£/3n to a known expression as in Eqs. 2.7 to 2.10. The only unknown in Eq. 2.35 is the source strength d i s t r i b u t i o n function, f(£), which can be obtained numerically through a d i s c r e t i z a t i o n procedure. 2.3.2 NUMERICAL.FORMULATION The integral equation, Eq. 2.35, i s used to develop an approximate numerical procedure to solve for the source strength d i s t r i b u t i o n function. The boundary T i s divided into N segments each of length Ar.., and the unknown source strength f(J.j) i s assumed to be constant within each segment (Fig. 2.2). Thus, Eq. 2.35 i s s a t i s f i e d at the centre of each segment and the integral equation i s reduced to N linear equations: 1 1 N 3G d<p - - f(x.) + — L f(£.) — (x.,£.) A r . = - — (x.) (2.36) 2 - 1 2TT j=i 3 3n _ 1 3 J 3n _ 1 or in matrix form [B]{f} = {b} (2.37) in which 5.. A r . 3G B., = - + - J - — (x.,£.) (2.38) x l 2 2TT 3n 1 1 and bj are the right-hand sides given in Eqs. 2.7 to 2.10. 5. . is the Kronecker delta and s. . i s the distance between 19 points and J_j . 9G/9n is given simply as - ( 9G/9s)n^ • 1 ^ , in which n^ i s the normal vector at x ^  , and l ^ j i s the unit vector for the l i n e , s ^ , joining points x^ and as shown in Fig 2.2. By obtaining the derivative of G, Eq. 2.38 can be written in a more convenient form for programming: 6.. A r . 1 B. . = -. - - J - n . - l . . (2.39) x l 2 2TT 1 ^ When i=j, the term 5 ^ accounts for the ve l o c i t y at a segment center due to the source d i s t r i b u t i o n on the same segment and thus the second term of Eq. 2.39, which would otherwise be singular, i s instead taken as zero. Once the matrix [B] has been evaluated, the source strength d i s t r i b u t i o n {f} can be solved from Eq. 2.37 by any standard matrix solution routine. With {f} now known, the pote n t i a l d i s t r i b u t i o n i s then obtained from the numerical form of Eq. 2.33 [<t>] = [A]{f} (2.40) in which A r . - A. . = —=- G(x.,£.) for i ? j . (2.41) x l 2n ~l 1 When i=j, a s i n g u l a r i t y occurs in Eq. 2.41, but the value of A ^ can s t i l l be evaluated. Consider each element being made up of a continuous d i s t r i b u t i o n of sources with constant strength. The contribution from these sources to the pote n t i a l at the center of the element can be obtained by closed form integration in this case and i s given by: 20 Ar. /Ar. A. . = —i- (In —=• - 1 ) (2.42) 1 1 2TT 2 With the appropriate expressions for {b} given by the boundary conditions in Eqs. 2.7 to 2.10, the four unit potentials, 0 , 0 , 0 and 0 can be determined. The ^ ^ux' uy vx rvy added mass and convective force c o e f f i c i e n t s may then be readily obtained: C ( i ) = — {0,} T[n. HAD (2.43) a A, 1 C V U ; = { F tn .HAr } (2.44) v 2R, 9T 9T 1 where 0, and 0 2 are the appropriate unit potentials corresponding to those defined in Eqs. 2.19 to 2.32. With a l l c o e f f i c i e n t s determined, the t o t a l hydrodynamic force acting on a cylinder moving r e l a t i v e to i t s ambient f l u i d near another stationary cylinder can be calculated from Eq. 2.16. 2.3.3 SINGULARITIES Theoretically speaking, the formulation given so far i s v a l i d for a l l separation distances except when the two cylinders are just touching. However, when the gap, e, between the two cylinders i s very small, say for e/R, less than 0.02, where R, i s the radius of cylinder 1, the matrix [B] in Eq. 2.37 tends to be singular. The unit potentials obtained through the inversion of [B] may not be unique, and the hydrodynamic c o e f f i c i e n t s are unstable. 21 Examining the formulation, a l l entries in [B], except the diagonal terms, contain terms l i k e 1 / s i j (EcI« 2.39), where s^j i s the distance between a source and the point at which boundary condition i s enforced (Fig 2.2). When the separation distance i s very small, there exist a few very small values of s ^ across the gap between the two cyl i n d e r s . Consequently, the corresponding entries in [B] have r e l a t i v e l y large values compared to a l l other entries. This gives r i s e to d i f f i c u l t i e s in the matrix inversion routine and constitutes the main reason for the occurrence of s i n g u l a r i t i e s in [B], In the above formulation, s i n g u l a r i t i e s (source points) are placed on the boundaries and therefore [B] becomes singular when the gap i s very small. If the sources are placed on an a u x i l i a r y boundary away from the physical boundaries T, the minimum value of s^j can be controll e d . The shape of the a u x i l i a r y boundary can be varied u n t i l an optimum configuration i s obtained. As the results are very s e n s i t i v e to the location of the a u x i l i a r y boundary, t h i s method involves a minimization procedure and an i t e r a t i v e scheme must be adopted (see, for example, Han, 86). In any event, the use of an au x i l i a r y boundary has not been adopted in t h i s t h e s i s . 22 2.4 RESULTS AND DISCUSSIONS A FORTRAN program, HYDYFC, based on the above formulation i s developed to calculate the 8 added masses and the 20 convective force c o e f f i c i e n t s for one cylinder moving in the v i c i n i t y of a second fixed c y l i n d e r . In the program, a mesh with variable facet sizes can be generated automatically, since smaller facets are usually required at some locations when the separation distance between the two cylinders i s small. In the results presented, both the ice mass and the structure are assumed to be c i r c u l a r cylinders with r a d i i equal to R, and R2 respectively. Although the program may be applied in general to two cylinders of a r b i t r a r y section, attention here i s given to a more fundamental case. This is intended to contribute to the understanding of the hydrodynamics developed for an ice mass d r i f t i n g near an offshore structure. In order to make use of the orthogonality properties of p o t e n t i a l functions so as to reduce the number of c o e f f i c i e n t s involved, the centroid of the ice mass has been considered to l i e on the x-axis. For t h i s case, the number of nonzero force c o e f f i c i e n t s is reduced to 13 and the rel a t i o n s among them are summarized as: C < 1 ' = C < 2 > a1 a2 c <: i ) = c ( 1 ) v 1 v3 C < 2 ) = 2C < 2 ) = 2C <2> v5 v8 v9 23 C a 3 , } ' C a 4 2 ' ' C v 4 1 > ' Cv6 '' C v 7 1 > a n d C v i r j 1 ) r e m a i n n ° n z ^ o while a l l other c o e f f i c i e n t s are zero. Because of the properties of the mesh layout scheme, this arrangement also y i e l d s more accurate results than others. Rearranging Eq. 2.16, the resultant hydrodynamic force acting on cylinder 1 i s given by the following equations. p U ) - [- Ca1^x + ( Ca2 + 1 )<V + C v 2 U x 2 + C v 3 U y 2 " 2 C v 4 V x U x " 2 C v 5 V y U y ] ( 2 > 4 5 ) f ( Y ) = V i [- Ca1^y + ( Ca3 + 1 ) 0 y ] + " w R i C v 6 ( 2 U x U y " V y U x " V V ( 2 ' 4 6 ) The 3 added mass and 6 convective force c o e f f i c i e n t s are plotted in Figs. 2.3 to 2.11 as functions of the r e l a t i v e separation distances between the two cylinders, e/R 1 f and radius r a t i o R 2/Ri. 2.4.1 COMPARISON WITH ANALYTICAL RESULTS The boundary element solutions obtained in th i s thesis for the case of two c i r c u l a r cylinders are compared with those obtained by Yamamoto (1976), who formulated and solved the problem using the c i r c l e theorem. In his formulation, the two c i r c u l a r cylinders are represented by two doublets in an unsteady uniform flow, while an i n f i n i t e number of 24 tiny (auxiliary) doublets are used to enforce the boundary conditions on the surfaces of the two cylinders. An analytic solution in terms of an i n f i n i t e series was obtained, and the f i r s t 40 terms of t h i s were used to compute the required c o e f f i c i e n t s . The convergence of t h i s series i s very fast except when the two cylinders are close to touching. The added mass c o e f f i c i e n t s for cylinder 1 due to i t s acceleration near cylinder 2 and due to an accelerating current passing them are compared in Figs. 2.12 and 2.13 respectively. The convective force c o e f f i c i e n t s for cylinder 1 induced by i t s motion near cylinder 2 and by a current passing them are compared in Figs. 2.14 and 2.15 respectively. Although the approaches to the problem are t o t a l l y d i f f e r e n t in the two methods, both results agree i d e n t i c a l l y . However, when the separation distance is small, for e/R, less than 0.02, the boundary element solutions become unstable and dependent on the mesh layout. On the other hand, Yamamoto's results also diverge in this range and become undefined when the two cylinders are touching. The other convective force c o e f f i c i e n t s in the two approaches are expressed in d i f f e r e n t formats and so no comparison has been attempted for them. However, i t i s expected that the accuracy of the whole set of convective force c o e f f i c i e n t s could be r e f l e c t e d through the comparison of the above two cases. 25 2.4.2 KINEMATICS AND DYNAMICS OF FLUID FLOW With the numerical integrations defined by Eqs. 2.17 to 2.32, the complicated hydrodynamic interactions between the two cylinders are generalized by a set of dimensionless c o e f f i c i e n t s . The resultant hydrodynamic forces acting on the moving cylinder are simply given by Eq. 2.16. In order to interpret these c o e f f i c i e n t s adequately, i t i s desirable to look into the actual kinematics and dynamics of the flow. For purposes of i l l u s t r a t i o n , the discussion i s directed to the case when cylinder 1 i s moving along the x axis near cylinder 2 and both cylinders are of the same radius. The corresponding unit potentials are normalized by the cylinder radius and are shown in Figs. 2.16 and 2.17. As mentioned in Section 2.1, the hydrodynamic pressures in such interactions can be c l a s s i f i e d into two categories, namely, temporal and convective pressures. The former can be interpreted as the f l u i d pressure r e s i s t i n g the acceleration of a body and is proportional to the potential on the body surface. The l a t t e r i s induced by the r e l a t i v e motions of f l u i d p a r t i c l e s on the body surface and is proportional to the square of the r e l a t i v e tangential v e l o c i t y on the body surface. When the two cylinders are far apart, the poten t i a l d i s t r i b u t i o n on the surface of cylinder 1 i s almost sinusoidal from which the added mass and convective force c o e f f i c i e n t s are approximately equal to one and zero respectively. This agrees with the values predicted by 26 c l a s s i c a l methods for an i s o l a t e d c i r c u l a r cylinder. When the two cyl i n d e r s are close together, cylinder 2 acts as a blockage to the flow generated by cylinder 1. Instead of moving in the usual directions, the f l u i d p a r t i c l e s are now forced to flow along the periphery of cylinder 2. Therefore, a higher energy is required for cylinder 1 to produce the same motion as i t does in the f a r - f i e l d . In other words, when the acceleration i s in the posi t i v e d i r e c t i o n , the upstream surface of cylinder 1 i s experiencing a higher pressure in accelerating the same amount of f l u i d . The integration of the increased pressure gives an additional force proportional to the magnitude of the acceleration but acts in the opposite d i r e c t i o n . Such an increase in temporal force i s manipulated as an increase in added masses. As the forward motion of the f l u i d i s blocked by cylinder 2, the f l u i d flow i s forced into the directions normal to the motions of cylinder 1. When the gap between the two cylinders decreases, these effects become more pronounced and the tangential v e l o c i t i e s on the two surfaces increase rapidly. This phenomenon i s represented by sharp increases of pot e n t i a l gradients at the corresponding locations of the two cy l i n d e r s , which eventually leads to the increase of convective force attracting the two cylinders together. The interactions between the two cylinders are mainly l o c a l e f f e c t s . The sharp increases in the potential 27 d i s t r i b u t i o n s are mainly confined in the region where the two boundaries are close. The potential d i s t r i b u t i o n s on the other sides of the two cylinders remain f a i r l y constant for various separation distances. Although the model becomes singular when the two boundaries are touching, the temporal f l u i d pressure and the tangential v e l o c i t y at the surface of the two cylinders are f i n i t e and steady despite the discontinuity at the point of contact. Therefore, most of the hydrodynamic c o e f f i c i e n t s at the point of contact can reasonably be estimated by extrapolation of known values over a range of separation distances. 2.4.3 ADDED MASS DURING IMPACT As the decceleration term i s much higher during impacts, the magnitude of convective forces compared to that of temporal forces is small. Therefore, i t i s reasonable to ignore the convective forces during impacts. Since the problem i s set up in numerical form, i t i s not d i f f i c u l t to evaluate the mathematical solution for the added masses after contact. Appropriate boundary conditions can be defined at the boundary of the two merging cylinders and a solution may be obtained by the outlined procedure. Such an approach i s sa t i s f a c t o r y for 0 U X r # u v and 0 y v > but not for 0 V X « Under th i s mathematical model, the t o t a l volume of the two bodies decreases as they merge. The rate of decrease of the s o l i d volume i s proportional to the rate of indentation. In actual impacts, despite some i n i t i a l 28 e l a s t i c compression, the t o t a l volume of ice remains f a i r l y constant in the system. The crushing and s p a l l i n g of ice only create l o c a l disturbances in the f l u i d . However, the major problem of the model is not with any decrease in volume of the ice mass, but rather with the f l u i d flow induced by decreasing ice volume in the system. Consider an external boundary containing the two cylinders. As cylinder 1 indents into cylinder 2, the s o l i d volume diminishes simultaneously, and f l u i d must flow through the external boundary in order to replenish the lost volume. Therefore, the merging cylinders behave in part as a sink. Consequently, the unit p o t e n t i a l , <PVX, obtained by t h i s model relates to two flow components: one induced by the motions of cylinder 1, and the other corresponding to a flow towards the system replenishing the lost volume. The integration of 0 v x over the boundary of cylinder 1 leads to the corresponding added mass c o e f f i c i e n t . Since such a flow regime i s u n r e a l i s t i c , the added mass calculated from i t i s not expected to represent a r e a l i s t i c value for application to an impact model. Calculations have indicated that for the conditions of interest here, the ice penetration is about two orders of magnitude smaller than the overa l l geometry of the ice feature being considered. Therefore, the o v e r a l l impact configuration remains f a i r l y close to that of the i n i t i a l contact. Due to the decrease in contact surface with the ambient f l u i d , the added mass is expected to drop s l i g h t l y 29 during indentation. As a conservative and reasonable approach, the added mass at the point of i n i t i a l contact i s used in impact models. 2.4.4 DISCUSSION OF RESULTS In order to i l l u s t r a t e the proximity e f f e c t s for th i s two-cylinder problem, the solutions for two c i r c u l a r cylinders placed along the x-axis are studied. For th i s c particular case, the number of added mass and convective force c o e f f i c i e n t s reduces to 3 and 6 respectively as already indicated. The resultant hydrodynamic force acting on cylinder 1 i s given by Eqs. 2.45 and 2.46. In order to interpret the physical significance of the two hydrodynamic force equations, two fundamental cases are discussed in d e t a i l . Case I: One cylinder moving near another stationary cylinder in an otherwise still fluid. The hydrodynamic force equations for th i s case can be expressed -in a more convenient form by vector notation. £ = " Pw A l Ca1 ^ + Pw R l C V1<|V| 2,0) (2.47) The f i r s t term i s the temporal force acting on cylinder 1 in res i s t i n g i t s acceleration near cylinder 2. C a 1 i s the added mass c o e f f i c i e n t which i s independent of the d i r e c t i o n of acceleration and i s a function of the separation distance only. The second term i s the convective force acting on cylinder 1 due to the asymmetric d i s t r i b u t i o n of tangential 30 v e l o c i t y on i t s surface and always acts to at t r a c t the cylinders together regardless of the motion d i r e c t i o n . The solution for a cylinder moving near a stationary plane is mathematically equivalent to two equal cylinders moving symmetrically about the l i n e of symmetry and can be simply obtained by the method of superposition, or approximated by a large radius r a t i o , R 2 / P M . Considering Figs. 2.3 and 2.6, C , and C , increase as 3 3 a 1 v1 the distance between the two cylinders decreases and as the radius r a t i o increases. Thus, the effects of a stationary cylinder or plane boundary on the hydrodynamic behaviour of a nearby d r i f t i n g cylinder are to increase i t s added mass and to induce an att r a c t i n g force on i t toward the stationary object. Case II: Two stationary cylinders in a uniform current. This i s equivalent to the case when two cylinders move in the negative d i r e c t i o n plus an unscattered incident current in the p o s i t i v e d i r e c t i o n . The resultant hydrodynamic forces acting on cylinder 1 are given by: p U ) " " « A ' ( C a 2 + 1 ) K + V ' [ C v 2 U x 2 + C v 3 U y 2 ] ( 2 ' 4 8 )  p ( Y > = ^ ' ( C a 3 + 1 ) °y + 2 p w R i C v 6 U x U y ( 2 ' 4 9 ) The terms ( c a 2 + 1 ^ a n d ^ C a 3 + 1 ^ a r e k n o w n a s t^e i n e r t i a coef f ic i e n t s . When the flow i s in the x dire c t i o n and the two cylinders are close enough, the f l u i d motion in the gap 31 between the two cylinders i s stagnated. An asymmetric d i s t r i b u t i o n of f l u i d v e l o c i t y i s found on the surface of cylinder 1 and so i s the convective pressure associated with th i s v e l o c i t y d i s t r i b u t i o n . The integration of such pressure d i s t r i b u t i o n gives r i s e to a force p u l l i n g the two cylinders apart and i s represented by a negative convective force c o e f f i c i e n t , On the other hand, the force induced by an accelerating current in the x d i r e c t i o n decreases as the two cylinders get closer and the corresponding i n e r t i a c o e f f i c i e n t , ( C a 2 + 1 ) , decreases from i t s f a r - f i e l d value of two. When the flow i s in the y d i r e c t i o n , due to the curvature of the two cylinders, the f l u i d v e l o c i t y in the gap increases markedly and gives r i s e to a convective force a t t r a c t i n g the two cylinders together. The corresponding convective force c o e f f i c i e n t C ^ increases rapidly as the gap between the two cylinders decreases. However, when the two cylinders are touching, the passage of f l u i d between them is stopped and the position of the maximum ve l o c i t y and minimum pressure switch to a point farthest from the contact location. Consequently, the corresponding convective force acts in the opposite d i r e c t i o n and C v^ changes sign when the cylinders are in contact (Yamamoto, Nath and S l o t t a , 1974). With the two fundamental cases mentioned, the general idea about the hydrodynamic forces acting on an ice mass d r i f t i n g in the v i c i n i t y of an offshore structure is i l l u s t r a t e d . Obviously, the interactions become more 32 complicated, and a l l of the coupling terms w i l l be involved i f the ice mass i s moving with a current. Such hydrodynamic interactions are important to a d r i f t model in predicting the impact location and v e l o c i t y . For example, as the added mass increases with decreasing separation distance, a r e l a t i v e l y small ice mass may slow down and change d i r e c t i o n before any impact could occur. The added mass of an iceberg just before impact on an offshore structue i s also an important consideration. For example, the added mass c o e f f i c i e n t at contact increases from the f a r - f i e l d value of 1.0 to 1.6 for R 2/Ri equal to two. In contrast to the current design procedure where the f a r - f i e l d value of added mass is used in an impact model, a 60% increase in added mass gives r i s e to a substantial increase in impact energy and may lead to modifications of the design load. 3. ICEBERG IMPACT LOAD 3.1 INTRODUCTION C o l l i s i o n s between large ice features and massive gravity-type offshore structures in the A r c t i c region present an important engineering problem. In many cases, the iceberg impact load a f f e c t s the design and governs the ov e r a l l s t a b i l i t y of the structure. Although this problem i s being extensively investigated, a completely s a t i s f a c t o r y design method has not yet been obtained. In most impact models, important parameters such as the crushing force and added mass of icebergs have not been adequately simulated, and instead various empirical formulae derived from related cases are usually used. Results obtained by these models are not f u l l y r e l i a b l e and sometimes give r i s e to misleading design c r i t e r i a . An attempt i s made here to predict the ice impact loads on offshore structures using a more r e a l i s t i c approach. The r e d e f i n i t i o n of the ice crushing formula by Bohon and Weingarten (1985) i s f i r s t addressed. With the redefined ice crushing formula, a simple "head-on type" impact model based on the dynamic equilibrium of the ice mass and structure during impact i s developed. The 'contact-point' added mass c o e f f i c i e n t s estimated in Chapter Two are correlated with 3-dimensional cases and input to the impact model. The r e s u l t s are then compared 33 34 with those obtained by using f a r - f i e l d added mass during the indentation process. The damage to the structure in the two cases i s quantified by calcul a t i n g the impulse and energy absorbed by the structure during the impact. 3.2 ICE CRUSHING MODEL The crushing force of ice i s one of the most important parameters in impact models. In order to estimate the crushing load adequately, a r e a l i s t i c and consistent crushing model i s necessary. T r a d i t i o n a l l y , the force in such an interaction i s given by the experimentally based Korzhavin formula (Korzhavin, 1962) F = M I K D t a (3.1) where ac i s the uniaxial compressive strength of ice at the f a i l u r e s t r a i n rate, D is the c h a r a c t e r i s t i c width of the ice-structure interface and t represents the thickness of the ice feature. The variables M, I and K are the shape factor, indentation factor and contact factor respectively. These are dimensionless parameters used to correlate the uni a x i a l strength of ice with the crushing force due to mu l t i - a x i a l impact. As implied by the structure of the formula, the pressure d i s t r i b u t i o n over the contact area i s assumed to be uniform. It i s obvious that the Korzhavin formula i s only applicable to head-on impacts, but most authors s t i l l apply i t to eccentric impacts, which correspond to a skewed rather 35 than a uniform pressure d i s t r i b u t i o n . In order to account for the l o c a l ice pressure d i s t r i b u t i o n , the ice should be reasonably modelled by the so-called "ice element" (Fig 3.1) or the non-linear f i n i t e element (Powell et a l , 1985; Bercha et a l , 1985). Since the Korzhavin formula and the ice element give similar results for head-on type c o l l i s i o n s , the former i s used in thi s t h e s i s . Although Eq. 3.1 i s simple-looking, the d e f i n i t i o n s of i t s parameters are not c l e a r . Many discussions in the l i t e r a t u r e are contradictory, and i s d i f f i c u l t to obtain a set of universally accepted d e f i n i t i o n s of these parameters. In t h i s section, an ice crushing model based on the Korzhavin formula and the most up-to-date l i t e r a t u r e i s summarized. 3.2.1 UNIAXIAL COMPRESSIVE STRENGTH The uniaxial compressive strength of ice is a function of grain type, grain or i e n t a t i o n , temperature, s a l i n i t y as well as the st r a i n rate. Based on laboratory-grown ice (granular texture), Bohon (1984) showed that the plots of uniaxial strength versus s t r a i n rate for ice specimens of d i f f e r e n t temperatures and s a l i n i t i e s are s i m i l a r . The peak strengths of a l l specimens occur approximately at a s t r a i n rate of 2 x 10"3 s~ 1. The same observations for sea ice were also reported by Schwarz (1971). This s t r a i n rate i s known as the reference or tr a n s i t i o n s t r a i n rate which defines the t r a n s i t i o n between 36 d u c t i l e and b r i t t l e f a i l u r e s . With increases of s t r a i n rate beyond the reference s t r a i n rate, the specimen f a i l s in a b r i t t l e mode at a lower pressure. On the other hand, i f the ice specimen i s subjected to a low rate of loading, the specimen creeps and f a i l s in a d u c t i l e mode. Based on their experimental results and the reference s t r a i n rate of 2 x 10"3 s _ 1 , Bohon and Weingarten (1985) proposed a curve to define the uniaxial strength for a l l kinds of ice as function of s t r a i n rate e (Fig 3.2): a o[0.3 - 0.451og(14.29e)] for e > 2X10" 3 s" 1 ac = (3.2) a o[0.3 + 0.351og(5xl0 4e)] for e < 2X10" 3 s" 1 where a 0 i s the unaxial compressive strength of the corresponding ice specimen measured at the reference s t r a i n rate of 2 x 10"3 s" 1. Although Eq. 3.2 was developed on the basis of experiments using laboratory-grown ice, i t s a p p l i c a b i l i t y to ice of a l l d i f f e r e n t types has been validated by p l o t t i n g the curve together with published results for granular and columnar sea ice. The data points were found to be adequately bounded by the curve, so that Eq. 3.2 can be considered to form an upper l i m i t to the unaxial strength of di f f e r e n t kinds of ice. 37 3.2.2 CHARACTERISTIC STRAIN RATE In a uniaxial compressive test, the s t r a i n rate i s obviously defined as the rate of change of a specimen's length divided by that length. In an indentation test (Fig. 3.3), an indentor i s pushed into an ice sheet of f i n i t e thickness, and multi-axial compression a r i s e s . In this case, the s t r a i n rate is t r a d i t i o n a l l y given by: V e = — (3.3) 4D where V and D are the v e l o c i t y and diameter of the indentor. Unlike the unaxial test, the interactions L in the indentation test are multiaxial and the c h a r a c t e r i s t i c str a i n rate should depend on the thickness of the ice sheet as well as the width of the indentor. With the s t r a i n rate defined by Eq. 3.3, the t r a n s i t i o n s t r a i n rate for the indentation test was found to be a function of aspect r a t i o . This causes a serious drawback, since the a p p l i c a b i l i t y of the Korzhavin equation i s based on the constant reference str a i n rate of the unaxial t e s t . To overcome t h i s problem, a new formula for the c h a r a c t e r i s t i c s t r a i n rate obtained in an indentation test was proposed by Bohon and Weingarten (1985). This i s ' V/(4D) for D/t < 0.5 e =< VD/t 2 for 0.5 < D/t < 2.0 (3.4) , 2V/t for D/t > 2.0 where t is the thickness of the ice sheet. Using t h i s 38 formula, the peak indentation pressures for specimens of di f f e r e n t aspect r a t i o s have been found to occur at approximately 2 x 10~3 s" 1, which coincides with the reference st r a i n rate for unaxial strength. Consequently, the correlation between the multiaxial and un i a x i a l strength in the Korzhavin equation becomes more consistent. An examination of Eq. 3.4 reveals d i f f e r e n t physical effects for d i f f e r e n t aspect r a t i o s . For small aspect r a t i o s f a i l u r e is r e s t r i c t e d to the plane of the ice sheet, so that the c h a r a c t e r i s t i c s t r a i n rate i s proportional to V/D. On the other hand, for large aspect r a t i o s , f a i l u r e tends to occur in the out-of-plane d i r e c t i o n so that the ch a r a c t e r i s t i c s t r a i n rate i s a function of t rather than D. For intermediate aspect r a t i o s , both D and t are appropriate c h a r a c t e r i s t i c lengths so that the s t r a i n rate depends on both of them. 3.2.3 INDENTATION FACTOR The indentation factor accounts for the fact that the interaction i s multiaxial rather than uniaxial and i s usually expressed as a function of the aspect r a t i o of the ice-indentor interface. Physically, i t can be interpreted as the r a t i o of the peak indentation pressure at a given aspect r a t i o to the peak uniaxial strength of i c e . With the new d e f i n i t i o n of s t r a i n rate, both peaks occur approximately at the str a i n rate of 2 x 10~3 s" 1. 39 On the basis of a p l a s t i c l i m i t analysis and assumed yi e l d i n g c r i t e r i a , Croteau (1983) expressed the the o r e t i c a l indentation factor as a function of the aspect r a t i o . For d u c t i l e f a i l u r e , an in-plane Prandtl-type f a i l u r e i s assumed, and the indentation factor can be approximated by: 0.80 I = 3.0 + < 4.5 (3.5) D/t For b r i t t l e and t r a n s i t i o n s t r a i n rates, an out-of-plane or fla k i n g type f a i l u r e i s assumed, and the indentation factor i s given by the following approximate expression: 0.32 1 = 1.2+ < 3.0 (3.6) D/t A similar r e s u l t for b r i t t l e f a i l u r e was obtained experimentally by Bohon and Weingarten (1985). The comparison between the two results i s shown at Fig 3.4. 3.2.4 CONTACT FACTOR The contact factor accounts for incomplete contact at the ice-structure i n t e r f a c e . More precisely, i t i s a quantified evaluation of the bri t t l e n e s s of the crushing mechanism. Although t h i s factor depends on the type of ice and other environmental conditions, i t is usually expressed as a function of s t r a i n rate only. Obviously, the contact factor is unity for creeping d u c t i l e f a i l u r e , in which the ice sheet and the indentor are in perfect contact. However, different values have been proposed for b r i t t l e f a i l u r e . Bohon and Weingarten defined 40 the contact factor as the r a t i o of the normalized indentation pressure to the normalized u n i a x i a l strength, with which the contact factor for b r i t t l e f a i l u r e was found to vary from 0.2 to 0.3. Michel and Toussaint (1977) suggested that the contact factor ranges from 0.3 to 0.35 for b r i t t l e f a i l u r e s and is equal to 0.6 for d u c t i l e and tra n s i t i o n f a i l u r e s . As far as impact problems are concerned, ice masses f a i l in the b r i t t l e mode and a contact factor of 0.3 i s commonly used in most impact models. 3.2.5 SHAPE FACTOR The shape factor accounts for the shape of the indentor and applies primarily to narrow structures such as bridge piers. For ice-structure interfaces with aspect r a t i o s greater than 5 or for indentors with f l a t surfaces, the shape factor i s 1.0. For a c i r c u l a r surface, the shape factor can be taken approximately equal to 0.9, and for a 90° wedge-shaped indentor, i t i s about 0.7. 3.3 IMPACT MODEL The impact force exerted by ice on an offshore structure is usually a function of time and is dynamic in nature. In most cases, a s t a t i c analysis i s not s u f f i c i e n t to describe such interactions adequately and therefore dynamic models are usually used. Generally speaking, a complete dynamic model for ice mass impact on an offshore structure involves a 41 reasonable representation of the interaction between the ice mass, the structure and i t s foundation, as well as the ambient f l u i d . The r e s u l t i n g non-linear problem requires a step-by-step solution of the equations of motion. A comprehensive analysis involving the f i r s t three parameters l i s t e d above was carried out by Croteau (1983). In his approach, the ice-structure-foundation interaction was represented by an elaborate numerical model, whereas the e f f e c t s of the ambient f l u i d were represented by a constant added mass, corresponding to that of an isolated f l o a t i n g c y l i n d e r . The same assumption has been made in most other impact models described in the l i t e r a t u r e . It i s not intended here to extend Croteau's model, but rather to adopt a r e l a t i v e l y simple dynamic model and use t h i s to i l l u s t r a t e the importance of the hydrodynamic e f f e c t s during ice-structure impacts. 3.3.1 FOUNDATION MODEL General speaking, the motions of a structure s i t t i n g on a non-rigid foundation are composed of three modes, namely, t r a n s l a t i o n a l , rocking and t o r s i o n a l . In the present model, the head-on c o l l i s i o n of an ice mass on a massive offshore structure i s considered, so that the t i l t i n g and bending of the structure are taken as negligible and the response of the structure i s assumed to occur in the t r a n s l a t i o n a l mode only. This i s s u f f i c i e n t to i l l u s t r a t e the general procedure and the extension to take account of t i l t i n g , which may be 42 just as important, i s straight forword in p r i n c i p l e . From continuum mechanics, a dynamic foundation system is modelled as a massless r i g i d disk resting on a uniform half-space as indicated in F i g . 3.5. The material of the half-space can be e l a s t i c or v i s c o e l a s t i c from which the added s o i l mass, equivalent dashpot and spring system are evaluated. With the assumption of an e l a s t i c material, a set of equivalent s o i l parameters independent of the excitation frequency have been described by Newmark and Rosenblueth (1971) and Clough and Penzien (1975). They express the equivalent spring constant, viscous damper and added s o i l mass as: \ - v 2 k = 18.2G R (3.7) s s ( 2-„) 2 c = 1.08/k p R3 (3.8) me = 0.28p R3 (3.9) where R i s the radius of the plate, G g i s the shear modulus of the s o i l , v i s the Poisson's r a t i o of the s o i l and p is s the s o i l density. More refined foundation models with v i s c o e l a s t i c foundation material have been given by Veletsos and Verbic (1973), and Luco (1974, 1976), where the equivalent s o i l parameters were given as functions of excitation frequency. The added s o i l mass i s derived through the assumption of zero pressure acting on the foundation s o i l , with the structure represented by a r i g i d weightless disk. The t o t a l 4 3 mass of the structure-foundation system i s then usually taken as the sum of the mass of structure and the added s o i l mass. In other words, the influence of the overburden pressure on the added s o i l mass is assumed to be n e g l i g i b l e (Veletsos and Verbic, 1973; Clough and Penzien, 1975). 3.3.2 ADDED MASSES WITH BOTH CYLINDERS MOVING The added masses of an iceberg d r i f t i n g near an offshore structure are derived in Chapter Two with the assumptions of a two-dimensional flow and of a stationary structure. For a structure with a non-rigid foundation, the added masses of the structure as well as coupling added masses are required in the impact model. There are two sets of coupling added masses. These are the added mass on the iceberg due to the structure's motions and the added mass on the structure due to the ice mass' motions. P h y s i c a l l y , they can be interpreted as the temporal forces acting on one object due to the acceleration of another nearby object. Although some new parameters are required, i t i s not necessary to modify the existing computer program to generate them. Indeed, they can be obtained from the ex i s t i n g r e s u l t s by reciprocal relations similar to those used in s t r u c t u r a l mechanics. For example, given a certa i n structure to ice mass radius r a t i o R 2/Ri, and the centre to centre distance L, the added mass c o e f f i c i e n t of the structure i s given by that of the ice mass for the r a t i o of 44 R,/R2 and the same L. S i m i l a r l y , the added mass of the ice mass due to the motion of the structure in the x direction can be obtained by subtracting C , from C 0 for the ratio of a i az R2/R,. The other coupling term, the added mass on structure due to the motion of the ice mass in the x di r e c t i o n can be obtained in the same manner but with the r a t i o R,/R2. 3.3.3 ADDED MASSES: FROM 2-D TO 3~D In Chapter 2, the treatment of added mass was based on a two-dimensional flow. In order to use the impact model, added masses for a three-dimensional flow, taking account of f i n i t e water depths and ice mass d r a f t s , are required. Although a three-dimensional model can be developed to implement the task, as an approximate but reasonable approach, the added masses for the three-dimensional case can be obtained by multiplying t h e i r two-dimensional counterparts by appropriate c o r r e l a t i o n factors. The added mass c o e f f i c i e n t s for a f l o a t i n g c i r c u l a r cylinder in a f i n i t e depth of water have been computed by Yeung (1981) and are used for t h i s purpose. These results are summarized in F i g . 3.6. The three-dimensional added mass c o e f f i c i e n t for a cylinder extending to seabed i s unity and is equivalent to the two-dimensional case. As the clearance between the cylinder and the seabed increases, the added mass c o e f f i c i e n t decreases accordingly. In both the two-dimensional and three-dimensional cases, the added mass c o e f f i c i e n t for an isola t e d structure, 45 represented as a c i r c u l a r cylinder extending from the seabed to the free surface, i s unity. As an ice mass i s d r i f t i n g nearby, the added mass c o e f f i c i e n t of the structure increases from unity. Therefore, when applying the co r r e l a t i o n c a l c u l a t i o n to the two-dimensional added mass for a structure, only the quantity in excess of one i s affected. 3.3.4 MATHEMATICAL FORMULATION In most ice mass impact problems, the ice mass and the offshore structure are represented by two c i r c u l a r c y l i n d e r s , one fl o a t i n g and the other fixed and extending from the seabed to the free surface (see F i g . 3.7). The external forces acting on the ice mass during impact include convective forces, wave forces, current drag, wind drag, etc. Although a l l of these are important in d r i f t models, these forces are generally i n s i g n i f i c a n t when compared to ice impact forces. The ice-structure interactions are represented by a mathematical model as indicated in F i g . 3.8. The equations of dynamic equilibrium of the ice mass and the structure are given respectively by: (M,+M1,)x, + M I 2 x 2 + F = 0 (3. 10) M21X1 + (M2+m +/i 2j)x, + c x 2 + k x 2 - F = 0 (3.11) where M21 are the coupling added masses, and M 1 1 f M22 46 are the added masses on the ice and the structure due to their own motions. The ice crushing force, F, is a function of the r e l a t i v e v e l o c i t y and displacement of the two objects which can be considered as a non-linear internal damping device. As a l l external forces are ignored, the driving force for the impact and indentation i s from the i n i t i a l k inetic energy of the ice mass. Such energy is dissipated throughout the process and i s characterized by the v e l o c i t y of the ice mass just before impact. In order to obtain a solution, Eqs. 3.10 and 3.11 are expressed in matrix form: [M]{x} + [C]{x] + [K]{x} = {F} (3.12) in which [M] = M i 2 M21 M2+m +M22 (3.13) [C] = (3.14) [K] = (3.15) t \ -F (F} = F (3.16) From the technical point of view, the ice crushing force, F, 47 is now considered as an externally applied force to the system rather than as an internal damping term. Eq. 3.12 i s non-linear and a time-stepping procedure i s adopted for i t s solut ion. 3.3.5 NUMERICAL PROCEDURE As discussed in Sections 3.1 and 3.2, the ice crushing force i s highly nonlinear and so i s Eq. 3.12. Many numerical schemes involving time-stepping procedures have been developed for the analysis of such nonlinear problems. However, most of these methods require a knowledge of the applied force as an input to the analysis. Since the impact force i s unknown beforehand, an i t e r a t i v e scheme i s combined with the time-stepping procedure in order to estimate the change of impact force during each time step. As a simple but accurate procedure, the lin e a r acceleration method is used to give the solution at an advanced time (t+At) in terms of the solution up to time t. Since the change in the applied force, AF, i s unknown at the f i r s t i t e r a t i o n of each time step, an arbitary assumed value, usually zero, is used. At the end of each i t e r a t i o n , the displacements, v e l o c i t i e s and accelerations of the system are calculated, from which the impact force at (t+At) can be evaluated, leading to AF during t h i s time step. If the difference between the i n i t i a l l y assumed value and the calculated value is greater than a prescribed tolerance, the new AF i s used as an input to the next i t e r a t i o n u n t i l 48 sat i s f a c t o r y convergence i s obtained and one then proceeds to the next time step. Since a l l minor forces such as viscous, convective and drag forces are neglected in the formulation, the model i s good when the impact force i s r e l a t i v e large. While the ice mass i s not in contact with the structure, the minor forces become governing terms of motions and any subsequent calculations by t h i s model are u n r e a l i s t i c . Therefore, the time-stepping procedure stops when the ice mass is f u l l y stopped in front of the structure after a l l the kinetic energy i s dissipated. 3.4 RESULTS AND DISCUSSIONS A FORTRAN program, ICEIMP, based on the above assumptions and formulation, has been developed to calculate the displacement of the structure and the impact load in the time domain. Results obtained by using f a r - f i e l d and contact-point added masses are compared, and the implications on design c r i t e r i a are discussed in d e t a i l . A s e n s i t i v i t y analysis i s then c a r r i e d out to investigate the r e l a t i v e effectiveness of a number of important parameters on the response of the system. Since the number of variables involved is large, a complete dimensionless analysis on a l l parameters i s i n e f f i c i e n t . In order to cut down the number of variables, unless stated Otherwise, a l l analyses w i l l be done on a basic case as given in Table 3.1. Also, two different types 49 of foundation corresponding to weak and firm s o i l s are used to investigate the eff e c t s of the underlying s o i l on the response of the system (Table 3.2). 3.4.1 ADDED MASS EFFECTS Ice penetrations for the two assumed values of added mass and the two sets of assumed s o i l properties are compared in Table 3.3. The penetration in each case i s about two orders of magnitude smaller than the size of the ice mass and the structure. Therefore, the o v e r a l l geometry during impact remains f a i r l y constant and the use of the contact-point added mass throughout the impact process i s j u s t i f ied. An inspection of the results for the same type of s o i l indicates that the ice penetration i s larger but the structure displacement (Fig. 3.9) is smaller when the contact-point added mass i s used. Since the added mass at contact i s larger than the f a r - f i e l d value, i t i s obvious that the former gives r i s e to a higher incoming energy and therefore the penetration has to be greater in order to dissipate additional energy by ice crushing. At the same time, as the ice mass deccelerates due to the resistance of the structure, the surrounding f l u i d i s accelerated in such a way that a resultant temporal force always acts on the structure against i t s displacement (Figs. 2.16 and 2.17).. This force is proportional to the deceleration of the ice mass during impact and i s incorporated into the impact model 50 through the coupling added mass as described previously. These hydrodynamic e f f e c t s increase the ef f e c t i v e s t i f f n e s s of the foundation and lead to a smaller displacement of the structure. Comparisons in F i g . 3.10 indicate that the maximum impact forces are e s s e n t i a l l y the same under the d i f f e r e n t assumptions of added, mass and foundation. However, the duration of contact and the extent of ice penetration vary in each case. Therefore, the energy dissipated in ice crushing and the impulse transmitted to the structure are d i f f e r e n t . The energy and impulse (Table 3.4) calculated by using the contact-point added mass are about 20% higher than those obtained by using the f a r - f i e l d added mass. As the structure i s assumed to be a r i g i d body in the impact model, the energy, impulse and maximum impact force can be considered to serve as measures of the severity of l o c a l damage. In other words, using the f a r - f i e l d added mass instead of the added mass at contact leads to an underestimation of l o c a l damage by as much as 20% in thi s p a r t i c u l a r case. Very often such an underestimation may be c r u c i a l to the design of the overall structure and foundation system. The contact-point added mass w i l l be r e l a t i v e l y high and i t s e f f e c t s r e l a t i v e l y more s i g n i f i c a n t when the ice mass i s smaller than the structure (Fig. 2.3) and when the draft of the ice mass becomes close to the water depth (Fig. 3.6). If the design case f a l l s into one or both of 51 these categories, the added mass at contact should be used in an impact model in order to obtain a more r e a l i s t i c r e s u lt. 3.4.2 PARAMETER STUDY In order to study the r e l a t i v e e f f e c t s of various parameters on the response of the system, some of the si g n i f i c a n t parameters are treated as variables, taken one at a time, to the basic case considered so f a r . Results obtained by varying each parameter in turn are b r i e f l y discussed. Foundation Strength. For a given choice of added mass, the ice penetration and energy dissipated in crushing are less for a weak foundation, since more energy i s devoted to r i g i d body motions of the structure rather than crushing of the ice. While the type of foundations has no s i g n i f i c a n t effect on the maximum impact force, the displacement of a structure depends considerably on the foundation strength (Figs 3.9 and 3.10). The displacement of the structure in this p a r t i c u l a r case i s four times higher for the weak foundation compared to the firm foundation. Uniaxial Strength of Ice. For the case corresponding to the use of the contact-point added mass and the firm foundation, Figs. 3.11 and 3.12 show displacement and impact force variations with time for d i f f e r e n t values of the uniaxial strength of the ice. Since the size and the i n i t i a l v e l o city of the ice mass are constant, the ki n e t i c energy of 52 the incident ice mass i s the same in a l l cases, and therefore the energy dissipated in crushing and the impulse transmitted to the structure also remain the same. However, the structure displacement and impact force increase with ice strength so that the stronger ice dissipates energy at a higher rate and over a shorter duration, which leads to a more intense impact. Initial Velocity. For the same basic case based on the use of the contact-point added mass and the firm foundation, Figs. 3.13 and 3.14 show the corresponding results for d i f f e r e n t values of impact velocity of the ice mass. As the kinetic energy of the ice mass at impact is proportional to the square of the impact ve l o c i t y , i t i s obvious that the response of the system i s very sensitive to variations of the impact v e l o c i t y . Response parameters such as the contact time, impact force, ice penetration and structure displacement a l l increase appreciably with the impact v e l o c i t y , emphasizing the need to estimate a reasonable impact v e l o c i t y . Iceberg Draft. Results obtained by applying d i f f e r e n t iceberg drafts to the basic case are shown in Figs. 3.15 and 3.16. The increase in iceberg draft not only increases the impact energy but also increases the strength of the ice mass. As the iceberg f i r s t touches the structure, the r e l a t i v e l y small aspect r a t i o of the contact interface causes ice f a i l u r e confined to the horizontal plane, and a r e l a t i v e l y high f a i l u r e pressure is then required. As the 53 penetration progresses, the width of the contact area increases and the ice tends to f a i l in the out-of-plane d i r e c t i o n at a lower pressure. The aspect r a t i o dependence of ice strength i s accounted for in the ice crushing formula through the indentation factor. As an iceberg with a large draft impacts on an offshore structure, the aspect r a t i o of the contact interface remain f a i r l y small during the whole process. Therefore, a higher ice crushing pressure i s expected in the interaction and the response i s more intense. 4. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STUDY 4.1 CONCLUSIONS The hydrodynamic interactions between a d r i f t i n g and a stationary cylinder in close proximity are formulated by potential flow theory and solved by the boundary element method. These proximity effects on the d r i f t i n g cylinder are generalized by 8 added mass and 20 convective force c o e f f i c i e n t s , which in turn may be expressed as functions of the separation distance between the two cylinders. For the case of two c i r c u l a r cylinders located on the x-axis, only 3 added mass and 6 convective force c o e f f i c i e n t s are required to describe the interactions. These c o e f f i c i e n t s are found to vary s i g n i f i c a n t l y when the separation distance i s about twice the diameter of the d r i f t i n g cylinder and the interaction culminates when the two cylinders are in contact. For instance, for two cylinders of approximately the same size, both the added mass and the y component i n e r t i a c o e f f i c i e n t s increase by approximately 50% when the two cylinders are close to touching in r e l a t i o n to the values when the cylinders are far apart. At the same time, the x component i n e r t i a c o e f f i c i e n t becomes less than one in r e l a t i o n to the f a r - f i e l d value of two. When the separation distance i s less than about 2% of the radius of the ice mass, the numerical procedure becomes unstable. The added masses at this stage are estimated by an 54 55 extrapolation of known values over a range of separation distances. Three-dimensional added masses are obtained by multipling two-dimensional added masses by a correction factor. Although the added mass after i n i t i a l contact is expected to decrease s l i g h t l y , as a conservative and reasonable approach, the added mass at i n i t i a l contact should be used in impact models. A numerical model i s developed to simulate the response of an offshore structure subjected to the head-on impact of an ice mass. In contrast to re s u l t s based on the t r a d i t i o n a l l y assumed f a r - f i e l d added mass, a more severe impact and smaller structure displacement are obtained when the contact-point added mass i s used. These effects become more pronounced when the ice mass size i s of the same order or smaller than that of the structure, or when the iceberg draft i s close to the water depth. F i n a l l y , a parameter study i d e n t i f i e s the impact v e l o c i t y to be the most i n f l u e n t i a l factor in ice impact problems. 4.2 RECOMMENDATIONS FOR FURTHER STUDY Further studies in ice mass interactions with offshore structures could be made in several areas. In t h i s thesis, the t h e o r e t i c a l formulation of the hydrodynamic interaction between the two cylinders is based on potential flow theory. As ideal f l u i d i s assumed, i t is desirable to compare the results obtained here with experimental work. Also, the use of the c o r r e l a t i o n factor 56 to convert an added mass c o e f f i c i e n t from two-dimensions to three-dimensions needs to be v e r i f i e d by a more rigorous approach. In most design procedures in current use, the impact v e l o c i t y of an ice mass is obtained from s t a t i s t i c a l data and i s used d i r e c t l y in impact models. As the impact v e l o c i t y i s the most i n f l u e n t i a l parameter in an impact model, i t should be estimated in a more r a t i o n a l manner. Work has been done in predicting the motion of an ice mass near a large offshore structure (Isaacson, 1986). Many aspects of these studies could be extended, such as the modification of the model with the results obtained in t h i s thesis, the inclusion of m u l t i - d i r e c t i o n a l and i r r e g u l a r wave input and the comparison with experimental work and f i e l d measurements. In most circumstances, the impact of an ice mass on an offshore structure is eccentric. The pressure d i s t r i b u t i o n on the contact area is no longer uniform and the Korzhavin equation can not be applied. In order to account for the l o c a l ice pressure d i s t r i b u t i o n , the ice should be modelled by the so-called "ice element" (Powell et a l , 1985; Bercha et a l , 1985). The foundation model should then be extended to include the torsional mode of motion to account for the tangential component of the impact force. For r e l a t i v e l y slender structures, bending and t i l t i n g modes may be important and should be included in the analysis. APPENDIX A: LIST OF SYMBOLS : cross-sectional area of ice mass : cross-sectional area of structure : c o e f f i c i e n t as defined by Eqs. 2.17 and 2.18 : added mass c o e f f i c i e n t in i - t h d i r e c t i o n : s o i l damping c o e f f i c i e n t (horizontal mode) : Convective force c o e f f i c i e n t in i - t h direction : damping matrix in impact model : water depth : width of ice-structure interface : gap between structure and ice mass i f positi v e ; indentation of structure into ice mass i f negative : source strength d i s t r i b u t i o n function : ice impact force : force vector in impact model : resultant hydrodynamic force in i - t h direction : Green's function : shear modulus of s o i l : draft of ice mass : indentation factor (Korzhavin formula) : contact factor (Korzhavin formula) : equivalent spring constant for foundation (horizontal mode) : s t i f f n e s s matrix in impact model : unit vector for l i n e s.. 1 D 57 58 M : shape factor (Korzhavin Formula) M, : mass of iceberg M2 : mass of structure M s : added s o i l mass (horizontal mode) [M] : mass matrix in impact model N : number of elements n^ : unit normal vector at point i nx / n v : direction cosines of normal with respect to x and y axes p : hydrodynamic pressure q r : re l a t i v e tangential v e l o c i t y of f l u i d at ice or structure surface R : radius of foundation R, : radius of ice mass R2 : radius of structure s^j : distance between sources i and j t : thickness of ice sheet in indentation test Ux'lTy : x and y vel o c i t y components of incident current U : velocity vector of incident current V x,Vy : x and y velocity components of ice mass motion V : velocity vector of ice mass motion {x} : displacement vector in ice-structure impact model x,,x 2 : displacements of ice mass and structure in colinear impact model x : position vector : position vector of source i : Kronecker delta : length of element i : s t r a i n rate of ice : added mass of ice due to i t s own motion : added mass of ice due to structure's motion : added mass of structure due to ice's motion : added mass of structure due to i t s own motion : Poisson's r a t i o of foundation s o i l : position vector : position vector of source j : density of foundation s o i l : density of water : uniaxial compressive strength of ice measured at reference s t r a i n rate of 2X10" 3 s~' : uniaxial compressive strength of ice measured at a given s t r a i n rate : v e l o c i t y p o t e n t i a l : v e l o c i t y p o t e n t i a l associated with incident current : scattered p o t e n t i a l associated with current : potential associated with motion of ice : scattered unit potentials associated with x and y components of current : unit potentials associated with x and y components of ice's motion : f l u i d domain boundaries of domain boundary of ice mass boundary of structure f i r s t and second derivative with respect to time APPENDIX B: REFERENCES Ar c t i c Sciences Ltd. (1984) "Iceberg Modelling off Canada's East Coast: A Review and Evaluation". Bercha, F.G., Brown, T.G. and Cheung, M.S. (1985), "Local Pressure in Ice-Structure Interactions", Proc. Conf. A r c t i c '85, San Francisco, pp.1243-1251 . Bohon W.M. (1984), "The Calculation of Ice Force on A r c t i c Structure", Proc. 3rd I n t l . Offshore Mechanics and A r c t i c Engineering Symposium, New Orleans, Vol.3, pp.187-195. Bohon, W.M. and Weingarten, J.S. (1985), "Forces Exerted by Ice F a i l u r e in Crushing", Proc. Conf. A r c t i c '85, San Francisco, pp.456-464. Brebbia, C.A. (1980), "The Boundary Method for Engineers", Pentech Press, London, 189 pp. Cheema, P.S. and Ahuja H.N. (1977), "Computer Simulation of Iceberg D r i f t " , Proc. Offshore Tech. Conf., Houston, Paper No. OTC 2951, pp.565-572. Clough, R.W. and Penzien, J. (1975),"Dynamics of Structures", McGraw H i l l , New York, 634pp. Croteau, P. (1983), "Dynamic Interactions Between Floating ice and Offshore Structures", Earthquake Engineering Research Center, University of C a l i f o r n i a , Berkeley, Cal., Report No. UCB/EERC-83/06. Faltinsen, O.M. and Michelsen, F.C. (1974), "Motion of Large Bodies at Zero Froude Number", Proc. I n t l . Sym. on Dynamics of Marine Vehicles and Offshore Structures, Univ. College, London, pp. 91-106. G a s k i l l , H.S. and Rochester, J. (1984), "A New Technique for Iceberg D r i f t Prediction", Cold Regions Science and Technology, Vol. 8, pp.223-234. Gershunov, E.M. (1986), " C o l l i s i o n of Large Floating Ice Feature with Massive Offshore Structure", J . Waterway, Port, Coastal and Ocean Divis i o n , ASCE, Vol.112, No.3, August, pp.390-401. Han, P.S., Olson, M.D. and Johnston, R.L. (1984), "A Galerkin Boundary Element Formulation with Moving S i n g u l a r i t i e s " , Engineering Computations, Vol.1, No.3, September, pp.232-236. Hay & Company Consultants Inc. (1986), "Motion and Impact of 61 62 Icebergs", Canadian Environmental Studies Revolving Funds. Hsiung, C.C. and Aboul-Azm A.F. (1982), "Iceberg D r i f t Affected by Wave Action", Ocean Engineering, V o l . 9, pp.433-439. Lamb, H. (1932), "Hydrodynamics", 6th Edition (1945), Dover Publications, New York, 738 pp. Isaacson, M. de St. Q. (1978), " V e r t i c a l Cylinders of Arbitary Section in Waves", J. Waterway, Port, Coastal and Ocean Di v i s i o n , ASCE, Vol.104, No.WW4, Paper 13973, August, pp.309-324. Isaacson, M. de St. Q. (1985), "Iceberg Interactions with Offshore Structures", Proc. Conf. A r c t i c '85, San Francisco, CA, pp.276-284. Isaacson, M. de St. Q. (1986), "Ice Mass Motions Near an Offshore Structure", Proc. 5th I n t l . Offshore Mechanics and A r c t i c Engineering Symposium, Toyko, Vol.1, pp.441-447. Isaacson, M. de St. Q. and Dello S t r i t t o , F.J. (1986), "Motion of an Ice Mass Near a Large Offshore Structure", Proc. Offshore Tech. Conf., Houston, Paper No. OTC 5085, pp.21-28. Johnson R.C. and Nevel D.E. (1985), "Ice Impact Structural Design Loads", Proc. 8th I n t l . Conf. Port and Ocean Engineering Under A r c t i c Condition, Narssarssuaq, Greenland. Kokkinowrachos, K., Thanos, I. and Z i b e l l , H.G. (1986), "Hydrodynamic Interaction between Several V e r t i c a l Bodies of Revolution in Waves", Proc. of the 5th I n t l . Offshore Mechanics and A r c t i c Engineering Symposium, Tokyo, pp.194-205. Korzhavin, K.N. (1962), "Action of Ice on Engineering Structures", U.S. Army CRREL, CRREL Translation TL260, 1971 . Loken, E.A. (1981), "Hydrodynamic Interaction Between Several Floating Bodies of Arbitary Form in Waves", Proc. I n t l . Symposium on Hydrodynamics in Ocean Engineering, Trondheim, Vol. 2, pp.745-779. Luco, J.E. (1974), "Impedence Function for a Rigid Foundation on a Layered Medium", Nuclear Engineering Design, Vol.31, No.2, January, pp.204-217. Luco, J.E. (1976), "Vibration of a Rigid Disk on a Layered 63 V i s c o e l a s t i c Medium", Nuclear Engineering and Design, Vol.36, No.3, March, pp.325-340. Michel, B and N. Toussaint (1977), "Mechanism and Theory Indentation of Ice Plates", J. Glaciology, Vol.19, No.81, pp.285-300. Milne-Thomson, L.M. (1938), "Theoretical Hydrodynamics", 5th E d i t i o n (1968), MacMillan, 600 pp. Mountain, D.G. (1980), "On Predicting Iceberg D r i f t " , Cold Region Science and Technology, Vol. 1, pp.273-282. Newman, N.J. (1977), "Marine Cambridge, Mass., 402 pp. Newmark, N.M. and Rosenblueth, Earthquake Engineering", C l i f f s , N.J., 640 pp. Hydrodynamics", MIT Press, E. (1971), "Fundamentals of Prentice H a l l , Englewood Newton, R.E. (1975), " F i n i t e Element Analysis of 2-D Added Mass and Damping". In F i n i t e Element in Fl u i d s , Vol.1, Viscous Flow and Hydrodynamics, edited by R.H. Gallagher et a l , John Wiley, N.Y. pp.219-232. Ohkusu, M. (1974), "Hydrodynamic Forces on Multiple Cylinders in Waves", Proc. I n t l . Symposium on Dynamics of Marine Vehicles and Structures in Waves, Univ. College, London, pp.107-112. Powell, G et a l . ( l 9 8 5 ) , "Ice-Structure Interaction of an Offshore Platform", Proc. Conf. A r c t i c '85, San Francisco, CA., pp.230-238. Salvalaggio, M.A. and Rojansky, M. (1986), "Importance of Wave-Driven Icebergs Impacting an Offshore Structure", Proc. Offshore Tech. Conf., Houston, Paper No. OTC 5086, pp.29-38. Sarpkaya, T and Isaacson, M. (1981), "Mechanics of Wave Forces on Offshore Structures", Van Nostrand Reinhold, New York, 651 pp. Schwarz, J . (1971), "The Pressure of Floating I c e - F i e l d on P i l e s " , I.A.H.R. Symposium 1971, Paper 6.3, Reykijavik, Iceland. Schwarz, J . and Weeks W.F. (1977), "Engineering Properties of Sea Ice", J . Glaciology, Vol.19, No.81, pp.499-531. Sodhi, D.S. and El-Tahum, M. (1980), "Prediction of an Iceberg D r i f t Trajectory During a Storm", Annals of Glaciology, Vol. 1, pp.77-82. Van Oortmerssen, G. (1979), "Hydrodynaimc Interaction 64 Between Two Structure Floating in Waves", Proc. 2nd Int. Conf. on the Behavior of Offshore Structures, BOSS '79, London, Vol. 1, pp.339-356. Veletsos, A.S. and Verbic B. (1973), "Vibration of Vi s c o e l a s t i c Foundations", International Journal Earthquake Engineering and Structure Dynamics, Vol.2, No.1, pp.87-102. Yamamoto, T. (1976), "Hydrodynamic Forces on Multiple Circular Cylinders", J. Hydraulics D i v i s i o n , ASCE, Vol.102, No. HY9, pp. 1193-1210. Yamamoto, T., Nath, J.H. and Sl o t t a , L.S.(1974), "Wave Forces on Cylinders Near Plane Boundary", J. Waterways, Harbors and Coastal Engineering D i v i s i o n , ASCE,Vol.100, NO.WW4, November, pp.345-358. Yeung, R.W. (1981), "Added Mass and Damping of a V e r t i c a l Cylinder in Finite-Depth Water", Applied Ocean Research, Vol.3, No.3, pp.1 19-133. 65 Radius of ice mass 50 m Draft of ice mass 60 m Radius of structure 50 m Height of structure 125 m Depth of water 100 m Density of water 1025 kg/m 3 Density of structure 1800 kg/m3 Impact velocity 2.0 m/s Uniaxial strength of ice 4.0 MPa Table 3.1 Basic case used in impact analysis Weak s o i l Firm s o i l Mass density (kg/m3) 1900 2200 Shear modulus (MPa) 75 350 Poisson's ratio 0.50 0.33 Table 3.2 Soil properties (adapted from Croteau, 1983) 66 F a r - f i e l d Contact-point added mass (m) added mass (m) Firm s o i l 1.43 1.59 Weak soi1 1.33 1.53 Table 3.3 Comparison of ice penetrations F a r - f i e l d Contact-point added mass added mass I n i t i a l energy (MJ) 1623 1886 Fi rm s oiI Energy dissipated (MJ) 1590 1865 Impulse transmitted (MNs) 1600 1860 Weak Soil Energy Dissipated (MJ) 1430 1765 Impulse transmitted (MNs) 1480 1810 Table 3.4 Added mass effects: energy and impulse 67 Y incident current, U Fig 2.1 Two-Cylinder Problem: Definition Sketch cylinder 1 (drifting) Fig 2.2 Two-Cylinder Problem: Illustration of Numerical Procedure 68 Fig 2.4 Added Mass Coefficient for U x T 1 I 1 I 1 1 I 1 1—' I ' I " I 1 1—> I i I 1 1 I 1 1—1 I 1 I 1 1 I 10-' 3 5 7 10- 3 5 7 10' 3 5 7 10' 3 5 7 10 J R e l a t i v e s e p a r a t i o n d i s t a n c e , e/R, Fig 2.5 Added Mass Coefficient for U v Fig 2.6 Convective Force Coefficient for V£ and V y 70 Fig 2.8 Convective Force Coefficient for UJ. 73 R l / R | » 10.0 *a1 • + B o u n d a r y e l e m e n t method Yamamoto (1976) I ' I ' M — 5 7 10-' -|—' I 1 I ' 1 I 3 5 7 10* I ' I ' M 5 7 10' H " — I ' I ' M 3 5 7 10' IO-« R e l a t i v e s e p a r a t i o n d i s t a n c e , e/R| Fig 2.12 Comparison with Analytic Results: C a j 'a2 or -a 3 + + B o u n d a r y e l e m e n t method C a 2 f o r U x (Yamamoto, 1976) C a 3 f o r Uy (Yamamoto, 1976) R 2 / R , » 10.0 10.0 ~~l ' I ' I ' M 3 5 7 10-I ' I ' M 5 7 10* ~ l ' — I ' I ' M 3 5 7 10' -1 ' — I ' I ' 1 I 3 5 7 10' R e l a t i v e s e p a r a t i o n d i s t a n c e , e/R, Fig 2.13 Comparison with Analytic Results: C ^ and C ^ 74 Fig 2.14 Comparison with Analytic Results: Cyj o R e l a t i v e s e p a r a t i o n d i s t a n c e , e/R, Fig 2.15 Comparison with Analytic Results: Cy2 and C v 3 75 Fig 2.16 Distribution of Normalized Unit Potential around Cylinder 1 due to its Own Motion in the x Direction 76 > 4-1 c o a c o T3 N (0 e u O 2 i r 160.0 180.0 Angle (degrees) Fig 2.17 Distribution of Normalized Unit Potential around Cylinder 2 due to the Motion of Cylinder 1 in the x Direction Node on Structure I n i t i a l Gap ice E l a s t i c Spring With L inear Damper Node on Ice Floe c d k1ce NonlInear Dashpot (Rate Dependent) Fig 3.1 Mathematical Model of Ice Element (Powell et al, 1985) a 2 HI cr W Q LLt N —I < cr O 2 STRAIN RATE - 1 / SEC Fig 3.2 Normalized Uniaxial Compressive Strength Vs Strain rate (Adapted from Bohon and Weingarten, 1985) FORCE ICE SHEET. INDENTER Fig 3.3 Indentation Test (Schematic Sketch) — CROTEAU. 1983 + + BONON ET AL, 1985 \ + + 0.0 0.5 1 .0 1 .5 2.0 2.5 3.0 ASPECT RATIO (D/T) 3.5 4.0 4.5 5.0 3.4 Indentation Factor for Brittle Failure: Comparison Between Experimental and Theoretical Results massless r i g i d disk Fig 3.5 Mathematical Model of Foundation 81 H D -5-Fig 3.7 Impact Configuration Fig 3.8 Mathematical Model of Impact (Hydrodynamic Force not Shown) 82 o o . L U o (_) cr . _j CO°. — r r ' o o r s j ' o o " 1 - Contact-point added mass (firm s o i l ) 2 - f a r - f i e l d added mass (firm soil) 3 - Contact-point added mass (weak s o i l ) 4 - f a r - f i e l d added mass (weak soil) 0.0 0.2 0.4 0.6 0.8 T I M E (SECOND) 1.0 1.2 1 A Fig 3.9 Added Mass Effects: Displacement of Structure _ o X o ~ o • o Contact-point added mass (firm s o i l ) f a r - f i e l d added mass (firm s o i l ) Contact-point added mass (weak s o i l ) f a r - f i e l d added mass (weak soi l ) 0.0 0.2 0.4 0.6 0.8 1 .0 T I M E (SECOND) 1.2 1 .4 Fig 3.10 Added Mass Effects: Impact Force on Structure 83 6.0 MPa o.o 2.0 MPa -i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 0.2 0.4 0.6 0.B 1.0 1.2 1.4 1.6 1.8 2.0 TIME (SECOND) Fig 3.11 Variation of Ice Strength: Displacement of Structure 2.2 6.0 MPa 5.0 MPa 2.0 MPa Fig 3.12 Variation of Ice Strength: Impact Force on Structure 84 4.0 m/s Fig 3.13 Variation of Impact Velocity: Displacement of Structure X LJO x.— 4 . 0 m / s - 1 1 — 2.0 2.2 Fig 3.14 Variation of Impact Velocity: Impact Force on Structure 85 oo Fig 3.15 Variation of Iceberg Draft: Displacement of Structure O o 1 .8 Fig 3.16 Variation of Iceberg Draft: Impact Force on Structure 

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