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Fluid-dynamic effects on the response of offshore towers to wave and earthquake forces Sen, Asoke Kumar 1971

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FLUID-DYNAMIC EFFECTS ON THE RESPONSE OF OFFSHORE TOWERS TO WAVE AND EARTHQUAKE FORCES by ASOKE KUMAR SEN B. Tech (Hons.), Indian I n s t i t u t e of Technology, Kharagpur, I n d i a , 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e -ments f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l -a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e -s e n t a t i v e s . . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date J u l y f? , 1971 (i) ABSTRACT The e v a l u a t i o n of f l u i d f o r c e s on v i b r a t i n g framed s t r u c t u r e s i n a f l u i d environment i s of c u r r e n t s i g n i f i c a n c e i n view of the a c t i v i t y i n ocean e n g i n e e r i n g . Accurate know-ledge of the f l u i d f o r c e s under c o n d i t i o n s of v a r i a b l e separa-ted flow i s l a c k i n g . In t h i s study an attempt has been made to f i n d a gen e r a l method of e v a l u a t i o n o f f l u i d f o r c e s on . c y l i n d e r s f o r v a r i a b l e flow, u s i n g p u b l i s h e d data from t e s t s o f constant v e l o c i t y flow, u n i f o r m l y a c c e l e r a t e d flow and wave . motion. The parameters t h a t appear t o govern"the v a r i a b l e flow f o r c e s are discussed,.and models f o r r e l a t i n g f o r c e magnitudes to these parameters are suggested. The dynamic response of framed s t r u c t u r e s i n an ocean environment has not been i n v e s t i g a t e d except f o r l i n e a r s i n u -s o i d a l wave motion i n deep water c o n d i t i o n s . The response of shallow water s t r u c t u r e s to v a r i o u s types of wave f o r c e s , as w e l l as t o earthquake e x c i t a t i o n , has been analysed n u m e r i c a l l y here, t a k i n g i n t o account the i n t e r a c t i o n between the s t r u c t u r e and f l u i d motions. The e f f e c t o f the mass and drag parameters ' on the s t r u c t u r e response has been s t u d i e d . Governing l o a d cases f o r the d e s i g n of framed s t r u c t u r e s have been r e l a t e d to s t r u c t u r a l p e r i o d and water depth. ( i i ) TABLE OF CONTENTS Page ABSTRACT ;(i) TABLE OF CONTENTS ( i i ) LIST OF TABLES (vi) LIST OF FIGURES ( v i i ) ACKNOWLEDGMENT (ix) CHAPTER I.- INTRODUCTION 1 1.1 Scope 1 1.2 F l u i d i n t e r a c t i o n 1 1.3 Earthquake problem 3 1.4 Wave problem 3 1.5 S i m p l i f i c a t i o n s 3 1.6 O r g a n i s a t i o n of the t h e s i s 4 I I . FLUID FORCES 6 2.1 C y l i n d r i c a l p i l e f o r c e formula 7 2.2 B a s i c flow phenomena 8 2.3 Drag f l u c t u a t i o n s - steady flow 10 2.4 L i f t - steady flow 12 2.5 Problems i n unsteady flow 12 2.6 C D : C o e f f i c i e n t o f drag 13 2.7 L i m i t s f o r p r e - s e p a r a t i o n stage and a s s o c i a t e d v a l u e s 15 ( i i i ) 2.8 Past.experimental data 15 2.9 Experiments f o r i n o s c i l l a t o r y waves 16 2.10 R i g i d c y l i n d e r s i n s t a n d i n g waves - mean 16 2.11 R i g i d c y l i n d e r s i n st a n d i n g waves -v a r y i n g <_D 19 2.12 L i m i t a t i o n s of v a r y i n g values 22 2.13 Te s t s on r i g i d model c y l i n d e r s i n constant a c c e l e r a t i o n flow 23 2.14 C,.: C o e f f i c i e n t o f mass 23 M 2.15 R i g i d c y l i n d e r s i n st a n d i n g waves - mean 25 2.16 R i g i d c y l i n d e r s i n st a n d i n g waves -v a r y i n g C M 26 2.17 Experiments on r i g i d model c y l i n d e r s under constant a c c e l e r a t i o n from r e s t 27 2.18 O s c i l l a t e d f l e x i b l e c y l i n d e r s - mean r e s i s t a n c e 27 2.19 L i f t i n f l e x i b l e o s c i l l a t i n g c y l i n d e r s 30 2.2 0 Framing of r e l a t i o n s f o r instantaneous v a r i a b l e drag and mass c o e f f i c i e n t s 31 2.21 Choice of parameters 31 2.22 E m p i r i c a l c o e f f i c i e n t of mass i n a r b i t r a r y motion 31 2.23 E m p i r i c a l c o e f f i c i e n t of drag i n a r b i t r a r y motion 35 2.24 Force r e d u c t i o n s due to neighbouring c y l i n d e r s 37 (iv) I I I . WAVE THEORIES 38 3.1 Wave t h e o r i e s 3 8 3.2 Wave h e i g h t and p e r i o d 39 3.3 Ranges of a p p l i c a b i l i t y 43 3.4 C h a r a c t e r i s t i c s of Stokes theory 46 3.5 A d d i t i o n a l l i m i t of v a l i d i t y of t h i r d o r d e r Stokes theory 51 3.6 Breaking waves - s o l i t a r y wave theory 52 3.7 Impact type of breaker f o r c e s 54 3.8 Determinants of breakers - 54 3.9 Earthquake motion 56 3.10 Comparative ground a c c e l e r a t i o n s 56 IV. DYNAMIC RESPONSE PROBLEM 58 4.1 O r i g i n of n o n l i n e a r terms 58 4.2 Assumptions 58 4.3 B a s i c f o r m u l a t i o n 59 4.4 Earthquake i n p u t s 60 4.5 F u r t h e r s i m p l i f i c a t i o n s 61 4.6 Method of s o l u t i o n 62 4.7 Wave f o r c e i n p u t 6 2 4.8 Wave response computations 63 V. RESULTS OF COMPUTATIONS 6 4 5.1 Choice of s t r u c t u r e s f o r e v a l u a t i n g earthquake response 6 4 5.2 Earthquake response 66 (v) 5.3 E f f e c t of s t r u c t u r a l shape 72 5.4 E f f e c t o f C D 72 5.5 Relevance of s u b c r i t i c a l r e g i o n 75 5.6 Dynamic response to f i n i t e - a m p l i t u d e Stokes waves 75 5.7 Computed response to Stokes waves 79 5.8 Force v a r i a t i o n s w i t h time 81 5.9 I n t e r a c t i o n e f f e c t s on i n e r t i a f o r c e s 83 5.10 . S u p e r c r i t i c a l flow c o n d i t i o n s 85 5.11 Keulegari parameter 85 5.12 Breaking wave ( s o l i t a r y wave) response 85 5.13 Comparative f o r c e s under v a r i o u s e x c i t a t i o n s 86 5.14 Broad ranges of i n f l u e n c e o f lo a d types 9 4 VI. CONCLUSIONS 96 6.1 E f f e c t s of mass c o e f f i c i e n t 96 6.2 Shallow water waves 96 6.3 Load types governing d e s i g n 96 6.4 Other c o n c l u s i o n s . 9 8 BIBLIOGRAPHY 100 APPENDIX I 103 APPENDIX I I 104 APPENDIX.Ill . 117 (vi) LIST.OF TABLES TABLE Page 1. Wave Force C o e f f i c i e n t s ........... 17 2. Values of C,„ and C^ f o r Standing Waves . 20 M D ' 3. Range of A p p l i c a b i l i t y o f Wave Th e o r i e s . . 48 4. P e r i o d s and Mode Shapes . . 68 5. Range of Parameters 70 6. Earthquake Response . 71 7. Damping E q u i v a l e n t of Drag 73 8. Water I n e r t i a and Drag 74 9. Parameters f o r F i n i t e - A m p l i t u d e Wave Response ... 78 10. Response to F i n i t e - A m p l i t u d e Wave Input 80 11. Parameters f o r Breaking Wave Forces 87 12. Loading Due to Breaking Waves 88 13. Comparative Forces Under Var i o u s E x c i t a t i o n s .... 89 14. Comparative S t r e s s e s . 91 15. Governing Load Cases f o r Water S t r u c t u r e s 92 ( v i i ) LIST OF FIGURES FIGURE Page 1. Flow diagram . ... 2 2. Separated flow phenomena - shear l a y e r s - I 9 3. Separated flow phenomena - I I 9 4. Separated flow phenomena - v o r t e x s t r e e t - I I I .. 9 5. C i r c u l a t i o n around c y l i n d e r 11 6. V a r i a t i o n of C D 14 7. S c a t t e r of values of C D 18 8. V a r i a t i o n of C 1 A and wit h |- 24 M D D 9. Drag and l i f t f o r c e r a t i o s 28 10. P r e d i c t e d values of C M 34 11. Wave h e i g h t and p e r i o d 41 12. Wave l e n g t h and p e r i o d . . .• 45 13. Wave l e n g t h and water depth 45 14. C o e f f i c i e n t c 47 15. Ranges of t h e o r i e s 49 16. S o l i t a r y wave 53 17. Impact model 53 18. Breaker ranges 55 19. S t r u c t u r e s analysed f o r earthquakes 65 20. Key to mode shapes . 67 21. S t r u c t u r e s analysed f o r waves . . . . i . . . . 76 22. S t r u c t u r e s analysed f o r breakers 77 ( v i i i ) FIGURE Page 23. Wave f o r c e h i s t o r y 82 24. P i l e moment h i s t o r y ......... 84 25. Load types ... 93 26. L i n e a r wave 105 (ix) ACKNOWLEDGMENT The w r i t e r i s indebted to h i s t h e s i s s u p e r v i s o r , Dr. Donald L. Anderson f o r h i s i n s p i r i n g and v a l u a b l e a s s i s -tance and guidance a t every stage of t h i s work. The work has been brought to t h i s stage o f c o m p i l a t i o n p r i m a r i l y because of h i s constant support and encouragement. F i n a n c i a l support f o r the w r i t e r i n the form of a re s e a r c h a s s i s t a n t s h i p from an N.R.C. grant i s a l s o acknowledged. CHAPTER I INTRODUCTION 1.-1 Scope: The problem of v i b r a t i o n s induced i n o f f s h o r e s t r u c t u r e s by deep water waves has been e x t e n s i v e l y s t u d i e d . In view o f the i n c r e a s i n g numbers o f such s t r u c t u r e s being designed and c o n s t r u c t e d i t was con s i d e r e d d e s i r a b l e to i n v e s t i g a t e the f o r c e s caused by other dynamic e x c i t a t i o n s , namely, an earthquake input and those ocean wave c o n d i t i o n s which depart s i g n i f i c a n t l y from the assumptions of the l i n e a r small-amplitude wave t h e o r i e s . The magnitudes of the response to these kinds o f e x c i t a t i o n are compared with the response to l i n e a r deep water waves and wit h the response t o br e a k i n g wave f o r c e s w i t h i n the u s u a l range of s t r u c t u r e s i n shallow water. Tower-supported p l a t f o r m s t r u c t u r e s and s i m i l a r framed s t r u c -t u r e s o n l y are c o n s i d e r e d . A t t e n t i o n has a l s o been gi v e n to the problem of e v a l u a t i o n of the hydrodynamic f o r c e s under r e a l i s t i c water flow c o n d i t i o n s . 1.2 F l u i d i n t e r a c t i o n : A flow diagram of the i n t e r a c t i o n between the water f o r c e s and the s t r u c t u r a l response i s g i v e n i n F i g . 1. WAVE RESPONSE PROBLEM F L O W D I A G R A M Wave P e r i o d Wave Height Water Depth Bed Slope Roughness Water OCEAN WATER V e l o c i t i e s SYSTEM Water A c c e l e r a t i o n s S t r u c t u r e Response F i n a l ( D i s p l s . , V e l o c i t i e s -> Output Forces) 1 <- -Hydrodynamic Mass and Drag E f f e c t s -FORCE GENERATION (St r u c t u r e Geometry) "7K H y d r a u l i c Forces on — S t r u c t u r e — STRUCTURAL SYSTEM EARTHQUAKE PROBLEM Ground Motion FORCE GENERATION E q u i v a l e n t Dynamic -Forces on — St r u c t u r e S t r u c t u r e STRUCTURAL SYSTEM F i n a l Output Response Hydrodynamic Mass and Drag E f f e c t s F i g . 1 to 3. 1.3 Earthquake problem: In the case of earthquake-caused v i b r a t i o n s , the f o r c e system c o n s i s t s p r i m a r i l y of the a p p l i c a t i o n , i n e f f e c t , of recorded ground a c c e l e r a t i o n values to d i s c r e t e masses. T h i s e q u i v a l e n t dynamic f o r c e i s f e d i n t o the l i n e a r s t r u c t u r a l system as an i n p u t . Since the s t r u c t u r a l motions cause an i n t e r a c t i o n w i t h the water i n the form of hydrodynamic drag and i n e r t i a f o r c e s , the f i n a l response i s not the l i n e a r response, but i s a f u n c t i o n again of the hydrodynamic f o r c e s g e n e r a t e d / i n f l u e n c e d by the response. The hydrodynamic i n t e r -a c t i v e f o r c e s a t t a c h themselves to the other i n p u t t e d f o r c e s . D e t a i l e d expressions g i v e n l a t e r show t h a t the hydrodynamic drag e f f e c t s are n o n l i n e a r . In g e n e r a l the hydrodynamic i n e r t i a e f f e c t s are a l s o n o n l i n e a r . ^ 1.4 Wave problem: The inputs r e q u i r e d f o r the wave response problem d i f f e r i n t h a t the primary f o r c e s are caused by water motion r e l a t i v e to the s t r u c t u r e motion. The water v e l o c i t y and a c c e l e r a t i o n are c a l c u l a t e d from one of s e v e r a l wave t h e o r i e s u s i n g i n p u t s of wave p e r i o d , wave h e i g h t , water depth and the sl o p e , roughness and c o n f i g u r a t i o n of the bed. 1.5 S i m p l i f i c a t i o n s : Numerical s t u d i e s have been conducted i n both problems, t a k i n g the i n i t i a l i n p u t s as d e t e r m i n i s t i c . The hydrodynamic 4. i n t e r a c t i o n f o r c e s were moreover s i m p l i f i e d assuming two-dimensional t r a n s v e r s e flow p a s t c i r c u l a r c y l i n d e r s to be a p p l i c a b l e . 1.6 O r g a n i z a t i o n of the t h e s i s : The second chapter i s concerned w i t h the d e t e r m i n a t i o n of steady and unsteady f l u i d f o r c e s on c y l i n d r i c a l members when i n p u t data on the v e l o c i t i e s and a c c e l e r a t i o n s of the water p a r t i c l e s r e l a t i v e to the member are s u p p l i e d . The nature of the f l u i d - i n d u c e d f o r c e s f o r steady.flow i s f i r s t d i s c u s s e d , f o l l o w e d by an examination of such f o r c e s f o r p r o g r e s s i v e l y i n c r e a s i n g c o m p l e x i t i e s o f the flow i n the unsteady area. Both t h e o r e t i c a l ( q u a l i t a t i v e ) and experimental evidence a v a i l a b l e r e l a t i n g to such f o r c e s are presented and the need f o r an experimental approach f o r the case of-an a r b i t r a r y f l o w - h i s t o r y i s h i g h l i g h t e d . The dime n s i o n l e s s drag and mass c o e f f i c i e n t s f o r the f o r c e s are i n t r o d u c e d and based on a r e a n a l y s i s of past experimental data a r e l a t i o n s h i p f o r the instantaneous mass c o e f f i c i e n t i s proposed i n terms of the flow parameters: The t h i r d chapter d i s c u s s e s the flow c o n d i t i o n s c r e a t e d by v a r i o u s types o f waves and i n c l u d e s a s h o r t d e s c r i p -t i o n of the earthquake ground motion. T h i s chapter p r e s e n t s q u a n t i t a t i v e i n f o r m a t i o n f o r the d e t e r m i n a t i o n of f l u i d p a r t i c l e v e l o c i t i e s and a c c e l e r a t i o n s f o r d i f f e r e n t wave t h e o r i e s . The a p p l i c a b i l i t y of the v a r i o u s wave t h e o r i e s t o v a r y i n g c o n d i t i o n s 5. of the o c e a n - s t r u c t u r e geometry, e t c . and the need f o r t a k i n g i n t o account the v a r i o u s kinds o f waves are s e t out. The f o u r t h chapter formulates the equations of motion of the s t r u c t u r e under earthquake and dynamic wave f o r c e i n p u t s . The r e s u l t s . o f response computations under a) e a r t h -quake i n p u t s , b) shallow water n o n l i n e a r o s c i l l a t o r y wave i n p u t s , and c) breaking wave inputs are s t a t e d i n d e t a i l i n the f i f t h c hapter. The response of the s e l e c t e d s t r u c t u r e s f o r v a r y i n g values o f drag, mass and other parameters are compared. The s t r u c t u r e f o r c e s under v a r i o u s kinds o f e x c i t a t i o n and l o a d i n g are compared. In the l a s t chapter, the c o n c l u s i o n s and a summary of f i n d i n g s are gi v e n . The de s i g n c r i t e r i a which would govern f o r v a r i o u s o c e a n - s t r u c t u r e s i t u a t i o n s are i n d i c a t e d . CHAPTER I I FLUID FORCES Th i s chapter i s devoted to the de t e r m i n a t i o n of the f o r c e s on c y l i n d r i c a l members due to r e l a t i v e motion of the adjacent f l u i d . These f o r c e s are to be used i n c a l c u l a t i n g the s t r u c t u r e response. For c y l i n d r i c a l members the f o r c e s c o n s i s t of a v e l o c i t y - d e p e n d e n t drag component and an a c c e l e r a -tion-dependent i n e r t i a f o r c e component. Dimensionless coef-f i c i e n t s o f drag and i n e r t i a appear w i t h i n the constants of p r o p o r t i o n a l i t y to the f l u i d v e l o c i t i e s (to the second power) and the a c c e l e r a t i o n s r e s p e c t i v e l y . The kinematics o f water motion being d e a l t w i t h s e p a r a t e l y i n the next chapter (Ch. I l l ) , the f o r c e problem reduces to the de t e r m i n a t i o n of drag and i n e r t i a c h a r a c t e r i s t i c s f o r s p e c i f i c flow c o n d i t i o n s . Unsteady flow c h a r a c t e r i s t i c s e x i s t i n both the e a r t h -quake and wave f o r c e s i t u a t i o n s . The motion has an a r b i t r a r y c h a r a c t e r i n the former and i s o s c i l l a t o r y , with occurrence of s e p a r a t i o n , i n the l a t t e r . I t i s p o i n t e d out i n the chapter t h a t the drag and i n e r t i a i n separated flow are time-dependent and not s u s c e p t i b l e to an a n a l y t i c a l s o l u t i o n . Methods used i n the experimental d e t e r m i n a t i o n of the average drag and average mass c h a r a c t e r i s t i c s under s p e c i f i c types of unsteady flow as w e l l as steady flow are i n d i c a t e d . The flow parameters used 7. i n the d e t e r m i n a t i o n of the c o e f f i c i e n t s of mass and drag are s e l e c t e d on the b a s i s of dimensional a n a l y s i s and r e g r e s s i o n of the e x i s t i n g experimental data. 2.1 C y l i n d r i c a l p i l e f o r c e formula: In computations f o r the f o r c e s on a c y l i n d e r due to waves and to o t h e r types of s t r u c t u r e - f l u i d i n t e r a c t i o n , the t o t a l f o r c e i s taken to be a s u p e r p o s i t i o n of drag and i n e r t i a f o r c e s such that"'" F ( t ) = F ] ; ( t ) + F D ( t ) the f o r c e s being r e s p e c t i v e l y 1) An i n e r t i a f o r c e F (t) a r i s i n g out of a c c e l e r a t i o n of the f l u i d and represented by an added mass. 2) "A drag F D ( t ) comprising v i s c o u s f r i c t i o n and the p o r t i o n of the pressure d i f f e r e n t i a l upstream and downstream due to the e x i s t e n c e of the wake. In the range of Reynolds numbers of i n t e r e s t , t h i s i s p r o p o r t i o n a l to the square of the v e l o c i t y . The above s u p e r p o s i t i o n concept i s however true o n l y f o r two-dimensional flow p a s t c y l i n d e r s . E x p r e s s i o n s f o r i n d i v i d u a l terms are: F i ( t ) = CM p vo TE ( 2 - 1 } F D ( t ) = | c D p A v|v| (2.2) 8. where V Q = Enclosed volume of the member A = P r o j e c t e d area of the member v = R e l a t i v e v e l o c i t y between the member and the f l u i d p a r t i c l e s assuming f l u i d p a r t i c l e v e l o c i t y to be t h a t of the u n d i s t u r b e d flow of the surrounding f l u i d . C»„ = A dimensionless c o e f f i c i e n t of mass M C D = A dimensionless c o e f f i c i e n t of drag p = Mass d e n s i t y of the f l u i d . 2.2 B a s i c flow phenomena: In order to g a i n g r e a t e r i n s i g h t i n t o the drag and i n e r t i a f o r c e s , c e r t a i n p h y s i c a l c h a r a c t e r i s t i c s of f l u i d flow are examined-in d e t a i l i n t h i s and succeeding s e c t i o n s . D i s -continuous f e a t u r e s of the flow appear i n the case of motion of a r e a l v i s c o u s f l u i d p a s t a c y l i n d e r . F i r s t , i n t h i s s e c t i o n , steady flow i s c o n s i d e r e d . For r e a l f l u i d flow, s e p a r a t i o n of the l a y e r i n con-t a c t with the boundary leads to the formation of v o r t i c e s ( F i g s . 2 to 4). As v e l o c i t y of flow i s i n c r e a s e d , i t e v e n t u a l l y r e s u l t s i n t h e i r b e i n g detached, g i v i n g r i s e to a wake f o r a d i s t a n c e downstream of 1 to 4 diameters. The d i s c h a r g e of v o r t i c e s [which when e s t a b l i s h e d c o n s t i t u t e a Karman vo r t e x s t r e e t , F i g . 4] o c c u r s , a t a s u f f i c i e n t l y high v e l o c i t y , a l t e r -nately-.' from opposite edges of the c y l i n d e r a t a frequency determined by the S t r o u h a l number S given by SEPARATED FLOW PHENOMENA Fig 4 s = s 1 - 1 9 - 7 NR 10, ^ (2.3) where f = frequency of shedding of a p a i r of eddies D = diameter vD vDp _ , , ... N„ = — = — - = Reynolds number. R v u J v = — = kinematic v i s c o s i t y . Over the range of i n t e r e s t the number S i s 0.21. The d i s c o n -t i n u i t y r e p r e s e n t e d i n the wake downstream i s bounded by shear l a y e r s s t a r t i n g from the s e p a r a t i o n p o i n t s on the o p p o s i t e edges of the c y l i n d e r and extending f o r the a f o r e s a i d d i s t a n c e down-stream. The wake c o n t r i b u t e s to the pressure d i f f e r e n t i a l t h a t b r i n g s about the major p a r t of the drag; i t a l s o accounts f o r f l u c t u a t i o n s i n the drag from i t s mean v a l u e . 2.3 Drag f l u c t u a t i o n s — s t e a d y flow: In steady flow when the v e l o c i t y i s s u f f i c i e n t l y high to l e a d to vortex-shedding, the mechanism f o r f l u c t u a t i o n s i n drag (and i n l i f t ) i s i n d i c a t e d , w i t h r e f e r e n c e to the changing t r a n s i e n t flow c o n f i g u r a t i o n i n F i g . 5. F l u c t u a t i o n s i n c i r -c u l a t i o n and v e l o c i t y are, a c c o r d i n g to B e r n o u l l i ' s equation, accompanied by f l u c t u a t i o n s i n p r e s s u r e , and hence i n l o n g i t u -d i n a l drag. An i n d i v i d u a l v o r t e x causes a complete c y c l e i n 9 22 2 3 the h i s t o r y o f l o n g i t u d i n a l f o r c e s (drag). ' ' T h i s occurs because the r i g h t and l e f t v o r t i c e s d i s s i p a t e i n a l o n g i t u d i n a l l y i d e n t i c a l wake ( F i g . 5). The f l u c t u a t i o n s i n drag, which can 11. Clockwise circulation round cylinder CIRCULATION AROUND CYLINDER F i g 5 amount to as much as 60 percent of the mean drag, occur a t a J T 1 22 frequency of =— where T = time f o r d i s c h a r g e of one eddy. 2.4 L i f t — s t e a d y flow: A comment on l i f t f o r c e s i n steady flow i s i n order. In the realm of p o s t - s e p a r a t i o n v e l o c i t i e s , a r i s i n g out of the c i r c u l a t i o n around t h e . c y l i n d e r ( F i g . 5) l i f t f o r c e s are genera-ted t r a n s v e r s e to the flow, being p r o p o r t i o n a l i n magnitude to the square of the v e l o c i t y and being of the order o f the drag f o r c e s . The c y c l i c r e v e r s a l s of c i r c u l a t i o n d e s c r i b e d p r e v i o u s l y make the l i f t f o r c e s r e v e r s e c y c l i c a l l y a t a frequency of ^ e ("the S t r o u h a l f r e q u e n c y " ) . Two stages of t r a n s v e r s e asymmetry of the vo r t e x l a y o u t are needed to complete a c y c l e i n the l i f t f o r c e h i s t o r y . 2.5 Problems i n unsteady flow: . V a r i a b i l i t y o f the flow parameters"is a l s o found f o r flow w i t h a time-dependent v e l o c i t y . Observations are as f o l l o w s : a) The l i m i t i n g N_. f o r s e p a r a t i o n i s time-dependent. b) P o s i t i o n s along the boundary where s e p a r a t i o n . o c c u r s are time-dependent. c) The wake geometry i n f l u e n c e s the drag more d r a s t i c a l l y . I t i s a f u n c t i o n of v e l o c i t y , c y l i n d e r diameter, v i s -c o s i t y and degree of tu r b u l e n c e . d) F l u c t u a t i o n s of the instantaneous drag (and, i n t h i s type o f flow, i n e r t i a f o r c e s ) from i t s mean v a l u e are more i r r e g u l a r . The c o m p l i c a t i o n s i n v o l v e d i n an attempt a t a n a l y t i c a l study can be seen from the f a c t t h a t when the flow r e v e r s e s , the e r s t w h i l e wake becomes the upstream s i d e of the c y l i n d e r . Quan-t i t a t i v e knowledge r e g a r d i n g the flow and f o r c e s i s l a c k i n g f o r the g e n e r a l case of a r b i t r a r y a c c e l e r a t i o n (with s e p a r a t i o n ) . 2.6 C D_- C o e f f i c i e n t of drag: The f o l l o w i n g s e c t i o n s w i l l be concerned w i t h the c h a r a c t e r i s t i c s o f the c o e f f i c i e n t C^. For steady flow a c o r r e l a t i o n between the drag coef-f i c i e n t C and N i s w e l l - e s t a b l i s h e d ( F i g . 6). The charac-t e r i s t i c s . o f the experimental p l o t ( F i g . 6) are as f o l l o w s : The drag c o e f f i c i e n t i s n e a r l y constant a t 1.2 i n the p r a c t i c a l range of 10 4 < N < 5 x 10 5 R except f o r a drop to a minimum of 0.4 f o r s u p e r c r i t i c a l 5 flows (N > 2 x 10 approx.). I t r i s e s f o r low Reynolds numbers to a l i m i t of 10. 102 10* 106 107 VARIATION OF Cp NR > Fig 6 2.7 L i m i t s f o r the p r e - s e p a r a t i o n stage and a s s o c i a t e d v a l u e s : While vortex-shedding i n steady flow s t a r t s a t an N of the order of 50, the p o i n t of s e p a r a t i o n occurs a t K 4 N R - 1.2 x 10 i n the case of constant a c c e l e r a t i o n s t a r t i n g from r e s t . Thus the instantaneous value of N alone i s not an R adequate parameter f o r determining s e p a r a t i o n . The f o l l o w i n g two c o n d i t i o n s are proposed as a means of p r e d i c t i n g s e p a r a t i o n i n v a r i a b l e flow: v rricix T i ) o s c i l l a t o r y flow: parameter — ^ = 15, where D i s the diameter and T the o s c i l l a t i o n p e r i o d , i i ) o t h e r : N R = 1000 combined with an o v e r r i d i n g l i m i t g of — = 0.3, where s = d i s t a n c e t r a v e r s e d on the c u r r e n t s t r o k e . The value of C D p r i o r to s e p a r a t i o n i s t h a t due to f r i c t i o n drag alone and ranges from = 1 to 2, as found by Keulegan,"*"^ f o r waves. In g e n e r a l , f o r waves and earthquakes the v e l o c i t i e s i n t h i s range are low and so the f o r c e a s s o c i a t e d w i t h drag i s s m a l l compared to the i n e r t i a f o r c e ; thus a high degree of accuracy i s not r e q u i r e d i n t h i s range. 2.8 Past experimental data: An experimental approach has to be r e s o r t e d to f o r f o r c e s i n unsteady flow w i t h s e p a r a t i o n . In succeeding s e c t i o n s 16. flow phenomena i n s p e c i f i c types of unsteady flow as observed by p a s t i n v e s t i g a t o r s are d e s c r i b e d — i n order of i n c r e a s i n g i r r e g u l a r i t y of motion and d e c r e a s i n g member r i g i d i t y . These o b s e r v a t i o n s y i e l d an i n s i g h t i n t o the important flow para-meters t h a t i n f l u e n c e f o r c e s . Such a knowledge of parameters i s necessary so as to attempt to formulate a r e l a t i o n f o r i n e r t i a and drag f o r c e s i n the case of a v a r i a b l e flow. 2.9 Experiments f o r i n o s c i l l a t o r y waves: Turning to the work of past experimenters on the wave motion type o f unsteady flow, the drag c o e f f i c i e n t C D has g e n e r a l l y been ev a l u a t e d by measuring the t o t a l f o r c e on a c y l i n d e r , immersed i n the flow, at the i n s t a n t the wave c r e s t passes the c y l i n d e r . At t h i s i n s t a n t the water p a r t i c l e a c c e l e r a t i o n s are t h e o r e t i c a l l y zero and so the t o t a l f o r c e i s equal to the drag f o r c e . The observed v a l u e s of on t h i s b a s i s show wide s c a t t e r when p l o t t e d a g a i n s t N ( F i g . 7). The values of C Q w i t h the r e s p e c t i v e data sources are t a b u l a t e d i n Table 1. The d i s p a r i t i e s among these v a l u e s are due to f a c t o r s l i s t e d i n Appendix I. I t i s a l s o commented that roughness 5 3 of the c y l i n d e r s i n c r e a s e s i n the zone N > 2 x 10 . 2.10 R i g i d c y l i n d e r s i n s t a n d i n g waves—mean C D:-13 McNown has determined the i n f l u e n c e of v o r t e x -shedding on C f o r r i g i d model c y l i n d e r s under standing waves. 17. TABLE 1 WAVE FORCE COEFFICIENTS [From Ref. 3] Experimenter and Date Diameter of C y l i n d e r (in.) 'D 'M Type of Flow Crooke, 1955 Keulegan & Carpenter, 1956 Model 2 1 ± 3 , 2 2 / 2 • L 2 ' ± 4 Keim, 1956 1,-Dean, 1956 " 3 Wiegel et a l , 1956 Prototype 24 Reid, 1956 B r e t s c h n e i d e r , 1957. Wilson, 1957 Paape, 1966 Model 8 5  8 8 16 30 1.60 2.30 O s c i l l a -t o r y 1.34 1.46 1.52 1.51 (Standing Waves) 1.00 0.93 A c c e l e r a t e d , n o n - o s c i l -l a t o r y 1.10 1.46 1.00 0.95 Ocean waves C a l i f o r n i a 0.53 1.47 Ocean waves Gulf of Mexico 0.40 1.10 1.00 1.45 V a r i a b l e with H r a t i o D S C A T T E R OF VALUES OF 19. The t e s t s i n v o l v e d l a r g e amplitude water o s c i l l a t i o n s . Average values of C D have been g i v e n as a f u n c t i o n of the parameter 2T ' e where T = p e r i o d of sta n d i n g waves T e = eddy-shedding p e r i o d f o r the maximum v e l o c i t y v max T Average C D f a l l s s t e e p l y from 2.0 f o r - — i n the neighbourhood T 6 of 2 t o an u l t i m a t e value of 1.2 i f — i s much d i f f e r e n t from T e 2. — can be p h y s i c a l l y i n t e r p r e t e d i n terms of vortex-shedding e ip and S t r o u h a l number. T h i s parameter — alone however would not e pr o v i d e good c o r r e l a t i o n to C D f o r an a r b i t r a r y k i n d of unsteady flow. 2.11 R i g i d c y l i n d e r s i n st a n d i n g w a v e s — v a r y i n g C D : A s p e c i f i c a n a l y s i s of the v a r i a t i o n of drag and i n e r t i a f o r c e s a t v a r i o u s i n s t a n t s i n the c y c l e of o s c i l l a t i o n was c a r r i e d out by Keulegan and - C a r p e n t e r . ^ The r i g i d model c y l i n d e r was pl a c e d a t the node of s t a n d i n g waves, wit h flow c o n d i t i o n s a d j u s t e d to ensure uniform h o r i z o n t a l v e l o c i t y from the s u r f a c e to the bottom. The t e s t s i n v o l v e d large-amplitude water o s c i l l a t i o n s . Through a F o u r i e r a n a l y s i s of the measured f o r c e s , and assuming t h a t the c o e f f i c i e n t s of higher harmonics were n e g l i g i b l e , they e v a l u a t e d C D a t v a r i o u s i n s t a n t s through-out a c y c l e of o s c i l l a t i o n (Table 2). The s e p a r a t i o n of the instantaneous v a l u e s o f C_ and C,, was e f f e c t e d as f o l l o w s : D M TABLE 2 VALUES OF-C„ AND (V FOR STANDING WAVES M D (CYLINDERS) •VT R.M.S. Averaqe Instantaneous Values o f C,_ & C_ M n , ^ - M D — 0 v e r Q y c l e — 1 1 1 1 1~ ~ ^ ^ ~ ^ ° ' 2 ^ °* 4 ^ ° ' 5 ^ = ° ' 6 ^ = ° ' 8 ^ C C C C C C C C C C C C CM ^D M D . M D M D UM ^D CM LD 3.0 2.14 0.70- 2.05 1.6 2.. 15.6* 0.80. 2.05 1.2 2.1 -0. 44.7 1.76 1.54 1.9 1.5 2. V CM CD CM CD CM C  CM 0.9 1.9 0.4 2.1 0.9 2.05 1.6 2.0 1.9 -2.0 2.0 -0.3 1.9 1.2 2.1 -1.4 1.4 2.2 1.6 2.1 1.4 1.9 1.5 2.2 T *This corresponds to = 1 e t = time from passage of c r e s t . 21. L e t t i n g T = the p e r i o d of the flow o s c i l l a t i o n s F = t o t a l f l u i d f o r c e and then u s i n g the f a c t of p e r i o d i c i t y of F and the symmetry of the flow, • F ( y t ) = - F ( ^ t + TT) (2.4) F The non-dimensionalised f o r c e 7=- can a c c o r d i n g l y be pv ZD ^ 2 K m expressed as a F o u r i e r s e r i e s with r e s p e c t to the v a r i a b l e t . The c o e f f i c i e n t s of the Fourier, s e r i e s are determined from measured values o f the flow-induced f o r c e s . On the other hand the Morison e x p r e s s i o n f o r the f l u i d f o r c e s , namely, 2TT F _ V o T . 2irt , 1 _ —-2I = CM~DV- S i n — + 2 CD pv D m K m 2-rrt cos —=— cos ^ (2.5) (where v^ = max. v e l o c i t y ) can a l s o be expanded as a t r i g o n o m e t r i c s e r i e s . L i k e terms i n the two t r i g o n o m e t r i c s e r i e s are compared to y i e l d s e r i e s e x p r e s s i o n s f o r C D and as a f u n c t i o n of t . Though C D i s time-dependent, weighted average val u e s over, a wave c y c l e can be e v a l u a t e d from an i n t e g r a l f o r the mean v a l u e . Furthermore e x p r e s s i n g F = f(t, T , v m , D , p , v ' ) (2.6) by means of dimensional a n a l y s i s they obtained - . v T v D • F f , 2 i F t m m \ = f (-sr-, - - = r - , max. N n = — — ) (2.7) 2 T ' D ' R ~ v pv D K m These experimenters went on to eva l u a t e c o e f f i c i e n t C D a t v a r i o u s i n s t a n t s o f the c y c l e from the computed co e f -f i c i e n t s o f the s e r i e s e x p r e s s i o n s a l r e a d y d e r i v e d ; t h i s was done v T f o r a s e r i e s of flow regimes r e p r e s e n t e d by the parameter m ^ x . Table 2 shows t h a t over the range of time when the instantaneous v e l o c i t i e s were non-zero, ^ 0.25), instantaneous v a l u e s of C n d i d not vary s i g n i f i c a n t l y . F u r t h e r by u s i n g the concepts of fD Vm T S t r o u h a l number — and the parameter —=—, i t was e s t a b l i s h e d t h a t v T V when n was much s m a l l e r than 15, no eddies formed; t h a t a v T s i n g l e v o rtex was formed i n each st r o k e when •• reached 15 and v T t h a t numerous eddies per str o k e formed f o r l a r g e v a l u e s o f ^ D i s t i n c t v a r i a t i o n s i n (mean values as w e l l as c y c l i c f l u c t u a -t i o n s ) C occu r r e d i n these ranges. T h i s i s e x e m p l i f i e d by the v T values i n Table 2. Mean C rose s h a r p l y from s m a l l t o a v T v T maximum of 2.2 a t p = 15 and f e l l g r a d u a l l y f o r l a r g e r ^ . There was e x c e l l e n t c o r r e l a t i o n found between mean C and the v T . m parameter — — . 2.12 L i m i t a t i o n s of v a r y i n g C D v a l u e s : F a c t o r s t h a t i n v a l i d a t e the a p p l y i n g of these v a l u e s obtained i n S e c t i o n 2.11 to an o s c i l l a t i o n problem a r e : 1) D e v i a t i o n o f the p a t t e r n of water o s c i l l a t i o n from t h a t o f a st a n d i n g wave. 2) Geometric s i m i l a r i t y ( r a t i o w aY e height^ ^ s un^ -y^ giy J diameter J t o be the same i n the pro t o t y p e . 2.13 T e s t s on r i g i d model c y l i n d e r s i n c o n s t a n t a c c e l e r a t i o n flow: For a n o n - r e v e r s i n g unsteady flow s i t u a t i o n , 'N a g a i n R i s not an adequate parameter, s i n c e s e p a r a t i o n i s not dependent on v e l o c i t y alone. T h i s i s due to the f a c t t h a t i t takes time from the s t a r t of motion f o r s e p a r a t i o n to occur and v o r t i c e s to be formed. For u n i f o r m l y a c c e l e r a t e d motion from r e s t , p l o t s of Cjj a g a i n s t the parameter — were gi v e n by Sarpkaya and Garrison"'""'" (reproduced a t F i g . 8), where s = c u r r e n t d i s t a n c e t r a v e r s e d from r e s t . The parameter — was s e l e c t e d on dimensional c o n s i d e r a t i o n s . The p l o t ( F i g . 8) shows t h a t C Q i s low a t s m a l l s s — and reaches a maximum a t — - 2.5 a t which there was a symmetric v o r t e x c o n f i g u r a t i o n . I t decreases to 1.0 a t the shedding of the f i r s t v o r t e x (asymmetric v o r t e x pattern) around — = 4.8; i t t h e r e a f t e r e v e n t u a l l y a t t a i n s a v a l u e of 1.2 at l a r g e — - 6 to 7, the v a r i a t i o n s i n C D o c c u r r i n g o n l y d u r i n g the f i r s t two vor-t i c e s . T h i s h i g h l i g h t s the time taken f o r t r a v e r s i n g an adequate d i s t a n c e i n a s t r o k e f o r the wake to form and f o r C to assume v T separated flow v a l u e s . No c o r r e l a t i o n of C with N or —™— was found. 2.14 C„ - C o e f f i c i e n t of mass: M In t h i s and succeeding s e c t i o n s the past data on experimental v a l u e s of C M f o r p a r t i c u l a r cases of unsteady flow wil'l be summarised. The t h e o r e t i c a l value f o r i n v i s c i d i r r o t a -t i o n a l flow i s 2.0. For wave in p u t s on model and prototype Fig 8 25. p i l e s ( r i g i d p i l e s ) , Table 1 i n d i c a t e s v a l u e s of C M from 0.95 to 2.3 (Appendix I d e t a i l s the reasons f o r the s c a t t e r ) . The c o e f f i c i e n t was ev a l u a t e d t h e r e i n u s i n g the f o l l o w i n g approach: A t the i n s t a n t the l e v e l of the wave s u r f a c e i s a t the s t i l l water l e v e l , the v e l o c i t i e s are t h e o r e t i c a l l y zero and the f o r c e i s p u r e l y an i n e r t i a f o r c e . The measured f o r c e a t t h i s i n s t a n t y i e l d s a value of C... T h i s value i s then assumed to be con-M s t a n t f o r subsequent p r e d i c t i o n s / c o m p u t a t i o n s of wave f o r c e . In view of the s c a t t e r of the data a v a i l a b l e so f a r , judgment must be e x e r c i s e d i n s e l e c t i n g C M, t a k i n g i n t o con-s i d e r a t i o n the s i m i l a r i t y of c o n d i t i o n s i n a giv e n s i t u a t i o n to those which p r e v a i l e d i n an experiment. 2.15 R i g i d c y l i n d e r s i n st a n d i n g waves—mean C^: Experiments on r i g i d model c y l i n d e r s under s t a n d i n g waves to examine the i n f l u e n c e o f the vortex-shedding frequency on CMI P a r a l l e l i n g s e c t i o n 2.10, the r e s u l t s of McNown from these experiments show t h a t average C M f a l l s from a value of 2 T T at low 5 ^ — to a minimum of 1 a t ^ — = 2 to 3. I t i n c r e a s e s a g a i n e T e with l a r g e — to 2, i . e . , there i s a; d e f i n i t e c o r r e l a t i o n w i t h parameter — . e 2.16 Rigid cylinders i n standing waves—varying C M: Experiments for instantaneous C M for r i g i d model cylinders i n standing waves (K.eulegan and Carpenter 1 0) : Section 2.11 has indicated that instantaneous values of C», were M segregated i n a series form, with respect to the time variable 2 t 0 = — m — • The expression for the instantaneous C was found v T M to be directly proportional to . The computed values of the time-dependent at various cycle points are given i n Table 2; they show that C values fluctuate more markedly than v T C D values, s p e c i a l l y when = 15. Also, weighted average values of C^ over a wave cycle were found from the expression c M = b ^ M * 6 * s i n 2 e d e = 4 ¥ ^ F s i f d e (2.8) M TT o M TT3 D o 2_ pv D M m where, i n framing the expression for C as a 0 - s e r i e s , the v T higher order terms have been neglected. The parameter i s seen to influence C M d i r e c t l y , a conclusion which i s also arrived at by dimensional analysis (Section 2.11). Mean values of C v T M during a cycle,were correlated strongly with —™— and d i s t i n c t v T variations i n C M occurred for s p e c i f i c bands of values of —™— . v T Mean C M sharply f a l l s from 2.1 at low —=— to a minimum of 0.8 at v T v T — — =15.. Tt then r i s e s gradually for larger — . The parameter v T p was indicated to be important i n any regression. The values obtained should be interpreted with caution as some computed values of C M are negative (physically impossible) as Table 2 shows. 27. 2.17 Experiments on r i g i d model c y l i n d e r s under constant a c c e l e r a t i o n from r e s t : S e c t i o n 2.13 i n d i c a t e d t hat f o r C there was no v T c o r r e l a t i o n w i t h N_. or — — i n t h i s flow. As expected t h i s a l s o holds t r u e f o r C... P l o t s o f C.. a g a i n s t the parameter M M ^ r — (due to Sarpkaya and G a r r i s o n : F i g . 8) show s t r o n g c o r r e l a -t i o n . T h i s h i g h l i g h t s the time taken i n t r a v e r s i n g an adequate d i s t a n c e i n a st r o k e f o r C„„ v a l u e s to drop from 2.0 a t r e s t to M c a lower u l t i m a t e v a l u e . A t the s t a r t of motion when the r e l a -t i v e flow i s v i r t u a l l y i r r o t a t i o n a l , C„, assumes the va l u e 2. M which decreases t h e r e a f t e r w i t h i n c r e a s i n g ^ to 1.2. The C„ ^ D M curve r i s e s again, r e a c h i n g a.ri asymptotic value of 1.3. 2.18 O s c i l l a t e d f l e x i b l e c y l i n d e r s — m e a n r e s i s t a n c e : The r e s u l t s o f a t e s t f o r the measurement of combined drag and i n e r t i a f o r c e s are mentioned next. The t e s t s were conducted i n the l a b o r a t o r y w i t h s i n g l e f l e x i b l e model c y l i n d e r s o s c i l l a t e d a t l a r g e amplitudes i n s t i l l 9 21 22 water. L a i r d ' ' i n v e s t i g a t e d the f o r c e s f o r l a r g e amplitude of l o n g i t u d i n a l o s c i l l a t i o n s -(^ > > 1) w i t h i n the r e g i o n 2 x 10"^  4 N < 4 x 10 which f e l l w i t h i n the p r a c t i c a l range of flow. He found l a r g e i n c r e a s e s i n the f l u c t u a t i n g i n t e n s i t y of the t o t a l f o r c e over those on r i g i d c y l i n d e r s of - the order of up to 5 times ( F i g . 9). Combined r e s i s t a n c e (drag + i n e r t i a ) was measured and r e p o r t e d , but the drag predominated. The average A A O l d - J l Q 4 t> v. a o oO i s 5 10 10 'n 1 . 5 1 1 . 7 2 1 .9 7 7 9 I 3 5 2 0 4 1 0 . 8 D Dc o a A V D a 28, • • o.i D R A G A N D L I F T F O R C E R A T I O S N F ig 9 r e s i s t a n c e f o r c e s were found to i n c r e a s e by about 3 to 4 times the s t e a d y - s t a t e drag D0. L a i r d i n t e r p r e t e d the unsteady flow f o r c e s i n terms of the flow phenomena a t v a r i o u s v a l u e s of the f parameters ^— and f_ n where f_ . = f o r c e d o s c i l l a t i o n frequency of the c y l i n d e r (This i n f l u e n c e s mean o s c i l l a t i o n speed and hence the S t r o u h a l frequency f ^ ) . f ^ = n a t u r a l frequency of the f l e x i b l e model i n a i r . f = S t r o u h a l frequency. F i g . 9 h i g h l i g h t s the a m p l i f i c a t i o n i n f l u c t u a t i n g f t o t a l drag f o r value s of ^— near u n i t y . Based on the data f o r n the 3 o s c i l l a t o r s with f = 1.51, 1.72 and 1.97 r e s p e c t i v e l y , L a i r d has s t a t e d t h a t as f g i s reduced from a value equal to f , so long as the r e d u c t i o n i s s m a l l , the maximum v a r i a b l e drag does not decrease. An e x p l a n a t i o n o f f e r e d i s t h a t i n these slower runs (smaller f corresponding to s m a l l e r f ) , there i s g r e a t e r . t i m e , f o r the s t r u c t u r a l amplitude t o i n c r e a s e d u r i n g each s t r o k e . The f o r c e i n c r e a s e s over those f o r r i g i d c y l i n d e r s were a t t r i b u t e d to l a t e r a l o s c i l l a t i o n s induced by f l u c t u a t i n g l i f t f o r c e s and to the i n c r e a s e i n the wake widths which the l a t e r a l o s c i l l a t i o n s caused. When the maximum f l u c t u a t i n g l i f t was r e l a t i v e l y h i g h , the t o t a l r e s i s t i n g f o r c e was c o r r e l a t e d d i r e c t l y to the square of the wake widths. Drag predominated 30. over i n e r t i a i n these t e s t s , though no separate values of the drag and i n e r t i a f o r c e s are a v a i l a b l e . S t r u c t u r e s should be designed to have a f l e x i b i l i t y -lower than what would s i g n i f i c a n t l y r a i s e the f l u i d f o r c e s f ( i . e . , magnify.C D ^> 1.2); from F i g . 9 the ^ — r a t i o should be n l e s s than 0.3. P r a c t i c a l s t r u c t u r e s with braced c y l i n d r i c a l p i l e s f a l l w i t h i n t h i s category. 2.19 L i f t i n f l e x i b l e o s c i l l a t i n g c y l i n d e r s : 9 21 2 2 The preceding experiments o f L a i r d ' ' concerning o s c i l l a t i o n s with l a r g e amplitudes show l i f t f o r c e s to be s i g -n i f i c a n t when the S t r o u h a l frequency i s c l o s e t o . or lower than f (= 0.6 f to f ). A p o s s i b l e cause f o r the above e f f e c t n n n c when f was l e s s than f would be the t r a n s f e r of energy from e n ^ J the secondary drag o s c i l l a t i o n s a t a frequency of 2 f g . From v the r e l a t i o n s h i p f = 0.21 —, i t i s seen t h a t f o r the p r a c t i c a l wave flow v e l o c i t i e s o f l e s s than 12 f t . / s e c . (r.m.s.), the above l i f t e f f e c t s would not be s i g n i f i c a n t f o r the u s u a l diameters of 1 to 3 f t . and s t r u c t u r a l f r e q u e n c i e s of 0.3 to 1 c y c l e s / s e c . . The extreme cases w a r r a n t i n g examination of l i f t would be p i l e s of smal l diameters under waves of l a r g e h e i g h t s . Furthermore l i f t e f f e c t s are r u l e d out i n the e a r t h -quake case s i n c e the d i s t a n c e t r a v e l l e d i n each s t r o k e i s not s u f f i c i e n t to cause prolonged eddy shedding. The magnitudes of l i f t i n the shallow water case w i t small-diameter p i l e s are about those of drag. 2.20 Framing of r e l a t i o n s f o r the instantaneous v a r i a b l e drag and mass c o e f f i c i e n t s (separated f l o w ) : For cases of a r b i t r a r y a c c e l e r a t i v e motion, i t i s necessary to r e c o g n i s e the most important parameters t h a t i n f l u e n c e the value of C n and C„„, and to have recourse to D M experimental data to determine the c o r r e l a t i o n . An i n s i g h t i n t o the important v a r i a b l e s has been o f f e r e d by the e x p e r i -mental work d e s c r i b e d i n S e c t i o n s 2.9 to 2.18. 2.21 Choice of parameters: The technique of dynamical s i m i l a r i t y has been used to s e l e c t d i mensionless parameters t h a t would c o r r e l a t e e x p e r i mental v a l u e s of the drag and i n e r t i a c o e f f i c i e n t s . D e r i v a -. 4 t i o n s g i v e n by Morison and Crooke may be r e f e r r e d t o . 2.22 E m p i r i c a l c o e f f i c i e n t of mass i n a r b i t r a r y motion r e l a t e d to dimensionless parameters: Some v a r i a b l e s i n f l u e n c i n g C,, a f t e r the onset of ^ M s e p a r a t i o n are: 1) A c c e l e r a t i o n of the body 2) A c c e l e r a t i o n i n the surrounding f l u i d due to the presence of the body--depends on boundary c o n f i g u r a -t i o n 3) D u r a t i o n of the a c c e l e r a t i o n 4) Rate of change of the a c c e l e r a t i o n 5) I n t e r a c t i o n of the v e l o c i t y and i ) the d i s t a n c e t r a v e r s e d on the c u r r e n t s t r o k e i i ) the time e l a p s e d on the c u r r e n t s t r o k e 6) R e s i d u a l v o r t i c i t y from p r e v i o u s c y c l e s of o s c i l l a -t i o n 7) Symmetric or non-symmetric nature of v o r t e x formation-r e l a t e d to S t r o u h a l number. Item number 4) c o u l d not be e x p l i c i t l y taken i n t o account i n the parameters chosen. Examining the r e l e v a n t para-meters, the most important of the b a s i c v a r i a b l e s i n f l u e n c i n g the value of C„ are: M L = l e n g t h parameter v = v e l o c i t y of the body A = l o c a l a c c e l e r a t i o n T = time parameter Dimensional a n a l y s i s was c a r r i e d out, l e a d i n g to two parameters being found to i n f l u e n c e C„: ^ , M C M - C M ? > ( 2 - 9 » P h y s i c a l s i g n i f i c a n c e of parameters: AT 2 D ; a measure of 2 (L o c a l i n e r t i a ) x ( V i s c o u s f o r c e ) 3 (Convective i n e r t i a ) vT AT — , taken i n c o n j u n c t i o n w i t h — , r e p r e s e n t s ., .. Convective i n e r t i a the r a t i o — L o c a l i n e r t i a vT The broad e f f e c t of v a r i a t i o n s i n -=- on C,. D M vT . i s t h a t an i n c r e a s e i n -=r- i n c r e a s e s C„ when T D M i s l a r g e and v i s low. A r e a n a l y s i s of the experimental d a t a i n S e c t i o n 2.16 and 2.17 d i s c l o s e d t h a t the instantaneous value of C„ was M adequately determined as a q u a d r a t i c s u r f a c e i n the £-n-C M space, where 2 S = 0.125 L o g - ^ l ^ ! 1 - ! + 0.985 | | - 4.11 2 n = - L o g 1 Q l 1 0 p T 1 + 0 - 1 2 4 | ^ | + 0.903 (2.10) T being the time e l a p s e d from the s t a r t of the c u r r e n t s t r o k e . The b e s t r e g r e s s i v e r e l a t i o n found was: C M = 1.35 + 0.026 £ 2 - 0.152? + 0.62n 2 (2.11) F i g . 10 shows the v a r i a t i o n of C M with the two parameters 2 iOAT vT Log — - — and — i n the range covered by the t e s t s . The v a l u e s of C M as p r e d i c t e d by the equation and as e x p e r i m e n t a l l y observed are shown f o r s p e c i f i c data. The t e s t r e s u l t s f o r c o n s t a n t a c c e l e r a t i o n a l l showed good c o r r e l a t i o n , w h i l e only a few of the o s c i l l a t o r y flow t e s t r e s u l t s d i v e r g e d to an a p p r e c i a b l e extent. The c h o i c e of £ and n as independent v a r i a b l e s i n s t e a d of- | ^ T | and |^-| enabled e l i m i n a t i o n o f - c r o s s - p r o d u c t terms 1OAT ^ vT i n — - — and — . The £-n space r e p r e s e n t a t i o n i n v o l v e d a 1OAT ^ vT r o t a t i o n of the o r t h o g o n a l axes | — - — | and / the r o t a t i o n being s m a l l , i . e . , s i n "*"(0.125). C M g i v e n by e q u a t i o n (2.11) i s v a l i d i n the range bounded by the f o l l o w i n g i n e q u a l i t i e s : 2 1.088 < (0.99 L o g 1 Q | 1 0 p T | -0.15 |^|) < 2.1 ' (2.12) 2 0.8 < L o g 1 0 1XUp | < 3 (2.13) 0.7 < < 14 (2.14) i n p r a c t i c e flow parameters would u s u a l l y be w i t h i n the ranges of the e x p r e s s i o n s g i v e n by (2.12), (2.13) and (2.14). 2.23 E m p i r i c a l c o e f f i c i e n t of drag i n a r b i t r a r y 2-D motion (with s e p a r a t i o n ) : Although a dimensional a n a l y s i s approach s i m i l a r to t h a t f o r was f o l l o w e d , i t was not found p o s s i b l e t o formu-l a t e an e m p i r i c a l e x p r e s s i o n f o r t h a t s a t i s f a c t o r i l y , c o r r e l a t e d the experimental v a l u e s . The s i g n i f i c a n t v a r i a b l e s i n f l u e n c i n g the c o e f f i c i e n t of drag a f t e r s e p a r a t i o n occurs are: 1) Degree to which the wake has been e s t a b l i s h e d 2) Symmetry of v o r t i c e s 36. 3) Instantaneous value of c i r c u l a t i o n — t h i s i s r e l a t e d to v i s c o s i t y , v e l o c i t y and d e n s i t y 4) F l u i d shear a t the boundary of the body 5) Nearness of the frequency of c y l i n d e r motion t o S t r o u h a l frequency 6) R e s i d u a l v o r t i c i t y from p r e v i o u s c y c l e s — r e l a t e d to 3) T r e a t i n g these as being r e p r e s e n t e d i n the v a r i a b l e s L,v,A,T,u and p , C D i s found by dimensional a n a l y s i s t o be a f u n c t i o n of the f o l l o w i n g ( f a c t o r no. 5 above c o u l d not be e x p l i c i t l y taken onto account) „ AL , vT R' ~2 a n d T " where w i s a measure o f Convective i n e r t i a R Viscous f o r c e s AL . j . L o c a l i n e r t i a xs a measure of .2 Convective i n e r t i a ' v v'T — ( r e f . s e c t i o n 2.22) i s an i n d i r e c t measure of Li Convective i n e r t i a L o c a l i n e r t i a In an a l t e r n a t i v e choice of dimensionless parameters, i s a f u n c t i o n o f : 2 2 y'v'T VT',; , AT ^ - 3 , 3- and — pD 2 , uv'T . _ (Viscous f o r c e ) x (Convective i n e r t i a ) where j is a measure of — 5 ^ p D (Local i n e r t i a ) 37. AT D 2 (Loc a l i n e r t i a ) x ( V i s c o u s f o r c e s ) ^ 3 (Convective i n e r t i a ) i s a measure of As s t a t e d p r e v i o u s l y , r e g r e s s i o n c a r r i e d out on the p a s t e x p e r i -mental data d i d not give c l o s e agreement f o r the many proposed r e l a t i o n s h i p s , the data being meagre. F u r t h e r experimental to the presence of a l e a d i n g neighbour amounted to onl y 15 per-cent f o r a c l e a r spacing of 3 diameters and 45 per c e n t f o r a 20 21 2 3 c l e a r spacing of 1 diameter (vide t e s t s by L a i r d , ' ' ). No r e d u c t i o n s or a l t e r a t i o n s i n drag f o r c e s are t h e r e f o r e j u s t i f i e d i n p r a c t i c a l tower s t r u c t u r e s where the spacing i s i n most cases g r e a t e r than 4 diameters. 2.24 Force r e d u c t i o n s due to neighbouring c y l i n d e r s : Reduction of wave f o r c e s on a t r a i l i n g c y l i n d e r owing CHAPTER I I I WAVE .THEORIES In the previous chapter methods have been presented f o r the d e t e r m i n a t i o n of the f l u i d f o r c e s on a member i f the r e l a t i v e v e l o c i t y and a c c e l e r a t i o n between the f l u i d p a r t i c l e s and the member are known. In order to meet the l a t t e r r e q u i r e -ment, t h i s chapter i s concerned w i t h l a y i n g out the necessary i n f o r m a t i o n f o r determining the f l u i d motion i n waves, or the ground motion i n an earthquake s i t u a t i o n . 3.1 Wave t h e o r i e s : For tower s t r u c t u r e s i n the oceans as w e l l as wharves one of the major d e s i g n c r i t e r i a i s the l a t e r a l f o r c e r e s u l t i n g from wind-generated waves, and o c c a s i o n a l l y s i n g l e waves such as Tsunamis. Many i n v e s t i g a t o r s have worked on the problem of 1 2 p r e d i c t i n g the magnitude and frequency of wave motions ' and 1 2 3 the r e s u l t i n g f l u i d p a r t i c l e , v e l o c i t i e s . ' ' Because of the many v a r i a b l e s i n f l u e n c i n g wave geometry and water k i n e m a t i c s , a g e n e r a l theory foo? the mechanics of water waves i n an a r b i -t r a r y s i t u a t i o n would be very complicated. In order to o b t a i n , with a reasonable amount of e f f o r t , f a i r l y a c c urate estimates of f l u i d p a r t i c l e v e l o c i t y and a c c e l e r a t i o n under some of the more common regimes of flow, v a r i o u s s i m p l i f i e d wave t h e o r i e s have :been formulated. Some of the f a c t o r s t h a t determine the wave theory a p p r o p r i a t e f o r use are: - depth of water - f e t c h , i . e . , exposed l e n g t h o f water - wind c o n d i t i o n s - slope of beach. For any one l o c a t i o n which determines depth, f e t c h and beach s l o p e , d i f f e r e n t wind v e l o c i t i e s produce waves of d i f f e r e n t " h e i g h t s and f r e q u e n c i e s . The s t r u c t u r a l d e s i g n e r must then determine which c o n d i t i o n i s most severe f o r the proposed s t r u c t u r e , t a k i n g i n t o account the d i f f e r i n g f o r c e and r e s -ponse l e v e l s f o r d i f f e r e n t s t r u c t u r a l f r e q u e n c i e s . In the f o l l o w i n g proposed wave t h e o r i e s e x p r e s s i o n s f o r p a r t i c l e motion w i l l be presented which r e q u i r e p r i o r know-ledge of a t l e a s t two parameters. The most common ones used are the wave h e i g h t H (measured from trough to c r e s t ) and the p e r i o d T, the time between the passage of s u c c e s s i v e waves. Thus we must be able to determine H and T from knowledge of the l o c a l c o n d i t i o n s of wind speed, f e t c h , depth, beach s l o p e and wind d u r a t i o n . 3.2 Wave h e i g h t and p e r i o d : Methods f o r the e v a l u a t i o n of the wave h e i g h t H and• the p e r i o d T i n the deep water s i t u a t i o n w i l l now be c o n s i d e r e d . R e l a t i o n s h i p s f o r wave vheight and p e r i o d f o r waves,generated i n shallow water are a v a i l a b l e i n r e f . 6. In deep water, where d 1 — > (d being the water depth and L the wave length) , the 4 0. p r i n c i p a l parameters t h a t i n f l u e n c e the wave h e i g h t H and the wave p e r i o d T of wind-generated waves are the mean wind speed U , the f e t c h l e n g t h F and.the wind d u r a t i o n t . By dim e n s i o n a l a n a l y s i s i t can be shown t h a t , n e g l e c t i n g l e s s important para-meters, the f o l l o w i n g r e l a t i o n s must hold 3l = f (3l 3±\ U 1 u 2 ' u 31 = f (31 £iv 2 2 v 2 ' U ; * U U The nature of the f u n c t i o n s f ^ and f ^ have been determined e m p i r i c a l l y f o r the l i m i t i n g cases of i n f i n i t e t (depicted i n F i g . 11) and of i n f i n i t e F [expressed i n eqns. 3 . 4 ] . For example, f o r i n f i n i t e d u r a t i o n t , F i g . 11 shows an i n c r e a s e i n h e i g h t and p e r i o d with wind v e l o c i t y and f e t c h as would be p h y s i c a l l y expected. For l a r g e time d u r a t i o n t , s t e a d y - s t a t e c o n d i t i o n s are I n e f f e c t reached and the wave hei g h t H and p e r i o d T depend crF upon the Froude number T h i s i s the f e t c h - l i m i t e d case. U gH Regression on experimental data g i v e s the curves of — ^ — and g T l / 3 aF U — - j - — versus shown i n F i g . 1 1 . u ^ Here H-jy^ = s i g n i f i c a n t wave h e i g h t , i . e . , average of the upper 1/3 v a l u e s o f H. T-jy-j = s i g n i f i c a n t p e r i o d , s i m i l a r l y d e f i n e d . . .2 W A V E HE IGHT AND PERIOD Fig II qF 4 In the zone < 10 the r e l a t i o n s f o r H and T are n e a r l y U l i n e a r on l o g - l o g p l o t s and so reduce t o H l / 3 = ° - 0 4 5 U F ° ' 5 (3.1) T 1 / 3 = 0.6 U ° ' 4 F ° ' 3 (3.2) where the u n i t s are U = s u r f a c e wind speed i n m.p.h. F = f e t c h l e n g t h i n mi l e s Hjy., = h e i g h t i n f e e t T l / 3 = P e r : L O C^ i n seconds qF 5 g H l / 3 g T 1 / 3 For -a— > 10 , Y-— becomes asymptotic to about 0.35 and —-f-^-U U U becomes asymptotic to 9. T h i s corresponds to the case of a very l a r g e f e t c h and shows t h a t i n the p r a c t i c a l range, an i n c r e a s e of f e t c h beyond 100 m i l e s has no i n f l u e n c e on the waves. From the approximate e m p i r i c a l r e l a t i o n ^ (p.3) H = 1. 87 H. max. 1/3 the maximum des i g n wave h e i g h t becomes H = 0.084 U F 0 , 5 max. [ v a l i d f o r 2|J < i 0 4 ] (3.3) U For the other case of a d u r a t i o n - l i m i t e d wind wave, r e g r e s s i o n p o i n t s to the f o l l o w i n g : H l / 3 a u 1 ' 5 t o u 1 ' 6 and H 1 / 3 a t 0 * 4 to t 0 , 5 A l / 3 a U and T l / 3 a t 0 ' 3 (Ref. 1) Approximate expressions v a l i d i n the range 20000 < ^ j -are as f o l l o w s : (3.4) H l / 3 = ° - 0 0 5 t 0 ' 4 U 1 ' 6 T l / 3 = ° - 0 0 6 : t ° ' 3 u ° ' 7 < 3- 5) Here t = d u r a t i o n i n seconds U = s u r f a c e wind speed i n f t . / s e c . g H l / 3 g T l / 3 Again f o r l a r g e t , — ^ — and —-^f— tend to c o n s t a n t v a l u e s u independent of SL-. Ei/3 a n d T i / 3 i n c r e a s e f a s t e r with the wind v e l o c i t y U i n t h i s case compared with the f e t c h - l i m i t e d case. 3.3 Ranges of a p p l i c a b i l i t y : The v a r i o u s wave t h e o r i e s to be made use of i n the a p p r o p r i a t e ranges of flow c o n d i t i o n s are as f o l l o w s : a) L i n e a r small-amplitude theory, which i s l i m i t e d to the s i t u a t i o n j- > ( i . e . , ~ > 2.5 f t . / s e c . 2 ) T (vide Appendix I I ) . A p l o t of L versus T f o r t h i s theory i s g i v e n i n F i g . 12. The r a t i o of L to the 2 leng t h L Q f o r i n f i n i t e depth ( L Q = gT /2TT) i s p l o t t e d cl a g a i n s t — i n F i g . 13. 0 1 d 1 T h i r d order Stokes'? equations f o r -^=- < 7- < T h i s 2 D I I Z i s a n o n l i n e a r theory. There are other Stokes" equations of h i g h e r order, but they g e n e r a l l y do not improve the accuracy commensurate wi t h the i n c r e a s e d computations i n v o l v e d and so w i l l not be d i s c u s s e d . d 1 C n o i d a l theory f o r — < --not made use of due to Li Z D e x c e s s i v e computations being i n v o l v e d . Approximate r e s u l t s i n t h i s range may be obtained from the theory i n b) . S o l i t a r y wave theory f o r breaking waves, br o a d l y v a l i d when ^ > 0.78. For a f i r s t check, the value of H give n by deep water e x p r e s s i o n s may be used i n checking t h i s i n e q u a l i t y . In the above, d = depth of water below the o r i g i n a l water l e v e l L = wave l e n g t h and L i s obtained from the equation: g T . . 2^d . . —~— tanh —=— f o r case a) 2TT L 2 2 4ird gT 2 7 r d 2 14+4 cosh tanh ~ — [1 + (^-) ( . „ , ) ] f o r case b) 27T L L T ^ - I - 4 27Td 16 s i n h —=— JJ °° f o r case d) . (3.6) T = time p e r i o d of the wave, wit h u s u a l v a l u e s < 16 sees, o b t a i n e d as o u t l i n e d i n S e c t i o n 3.2. H i s a v a r i a b l e s e l e c t e d as i n d i c a t e d i n S e c t i o n 3.2. For the cl 1 l i m i t of the range of the C n o i d a l theory, namely =- = a p l o t Li AD ' of the c o e f f i c i e n t c i n t h e . f o l l o w i n g e x p r e s s i o n i s giv e n i n F i g . 14: 2 L (Wave l e n g t h i n fe e t ) = cT where T = wave p e r i o d in. seconds. Appendix I I g i v e s a b r i e f o u t l i n e of the t h e o r i e s f o r a ) , b ) , c ) , and d ) . Only d e t e r m i n i s t i c models of waves are co n s i d e r e d . In Table 3 the ranges of the r e l e v a n t parameters over which the v a r i o u s wave t h e o r i e s are a p p l i c a b l e have been i n d i c a -t e d . A g r a p h i c a l r e p r e s e n t a t i o n o f the ranges o f v a l i d i t y of these t h e o r i e s i s embodied i n F i g . 15. The ranges p e r t i n e n t to each theory are d i s c u s s e d i n the succeeding paragraphs. 3.4 C h a r a c t e r i s t i c s of Stokes theory: The mathematical b a s i s f o r the Stokes shallow water ( f i n i t e amplitude) and s o l i t a r y wave t h e o r i e s are giv e n i n Appendix I I . The t h i r d order Stokes o s c i l l a t o r y shallow water theory g i v e s a s u b s t a n t i a l improvement i n accuracy over t h a t i n the small-amplitude theory mainly i n the f o l l o w i n g ranges: TABLE 3 RANGE OF APPLICABILITY OF THE WAVE THEORIES Stokes" L i n e a r Shallow S o l i t a r y No Theory-A i r y Water Wave Parameter Theory F i n i t e - A m p l i t u d e C n o i d a l (Breaking) I n t e r p o l a t i o n 1 ^ ( f t . / s e c . 2 ) T >2.5 0.2<d-^-<2.5 T <0.20 <0. 08 0.08<-d5<2.5 T 2 E q u i v a l e n t ' ^ ' Li >0.5 0.04<d-<0.5 JL <0. 04 <0.016 0.016<^<0.5 3 ( f t . / s e c . 2 ) <<0.3 <0.3 [ l i m i t e d range] -(>9.8) Large >0.3 4 E q u i v a l e n t ^ v.s m a l l - <0.78 = • 0.78 | < 0.78 5 Side c o n d i t i o n f o r range of H T 2 - - - ^<0. 0 8 T -N.B. 1. F i g u r e s are approximate. 2. C o n d i t i o n s 1,2,3 and 4 are taken together f o r c l a s s i f y i n g each case. RANGES OF THEOR IES 49. io io"' < d , i o ( Ft / sec?) ^. ' 1 0 0 ° 2 0.02 0.2 M 2 LEGEND ( Deep water) ^ ( a ) .. Breaking waves represented by s o l i t a r y waves ( b) " " " " i n t e r p o l a t e d Stokes theory ( c ) .. " " " " deep water l i m i t H/L = 0 . 1 4 2 F i g 15 50. — < , i . e . , — ? r < 1.8 f t . / s e c . L -> T or 0.08 < < 0.3 f t . / s e c . 2 T The most important expressions f o r the t h i r d order f i n i t e -amplitude theory, as summarised from Appendix I I , are as f o l l o w s : 1 4 + 4 c o s h 2 ^ Wave v e l o c i t y C =/ | ^ t a n h ^ [ 1 + ( f f i 2 ( 4 % )] (3.7) 16sinh — „2 o o 14+4cosh 2^ d-Wave length L = ^ t a n h 2 ^ [ 1 + ( f f i ( % )] (3.8) 16sihh 2 . l+8cosh L 6 2 i r d Wave h e i g h t H = 2a+-2-^ ^ ^ - J Q ^ — 6 2 - r d L )] (3.9) L s i n h ' Depth of trough = a (3.10) H o r i z o n t a l v e l o c i t y u = C [ F . c o s h 2 T r 1 f : S + d ^ cos ( K X - ^ t ) +F 0cosh *2±2*L c o s 2 ( K X - ^ t ) + F 3 c o s h ^ t d l cos3 ( K x ~ t ) ] (3.11) L o c a l a c c e l e r a t i o n |£'= [F, c o s h 2 i r T ( Z + d ) s i n ( K x - ^ t ) +2F 0 o "Cl 1 X J_j 1 A c o s h 4 % t e + d ) s i n 2 ( K x - i l L t ) + 3 F , J-J 1 -J c o s h 6 \ ( z + d ) c o s 3 ( K x - | l t ) ] (3.12) J-l 1 K = ^ (3.13) J_i F l ' F 2 a n < ^ F 3 = f u n c t i o n s s t i p u l a t e d i n Appendix I I . 51. 3.5 A d d i t i o n a l l i m i t (of wave steepness) of v a l i d i t y of t h i r d order Stokes theory: Apart from the f o l l o w i n g l i m i t of the v a l i d range ( f o r t r a n s i t i o n t o bre a k i n g waves.in shallow water re p r e s e n t e d by the s o l i t a r y wave theory) , namely, < 0.75 (3.14) d b " d^ being the water depth below the trough, there i s an a d d i - -. t i o n a l l i m i t i n g c o n d i t i o n f o r the wave h e i g h t s a t which waves break. T h i s i s as under: ^ = 0.142 tanh ^ (3.15) L i L i (Linear deep water waves transform to brea k i n g waves d i r e c t l y ) . For g r e a t e r h e i g h t s , the t h i r d o r d e r Stokes theory does not c o r r e c t l y r e p r e s e n t the water k i n e m a t i c s . For c o n d i t i o n s i n t e r m e d i a t e between (3.14) and (3.15), values o f brea k i n g v e l o c i t i e s are ev a l u a t e d from wave he i g h t s and lengths obtained by i n t e r p o l a t i o n . The Stokes shallow water wave theory i s employed to compute p a r t i c l e v e l o c i t i e s i n such cases, the range o f t h i s m o d i f i e d approach being 0.08 < < 2.5 f t . / s e c . T (3.16) 0.3 < < 0.78 f t . / s e c . 2 T^ These l i m i t s apply to f r i c t i o n l e s s f l a t ocean beds. The h e i g h t of b r e a k i n g waves would a c t u a l l y be c o n s i d e r a b l y l i m i t e d by the e f f e c t of the slope o f the beach. P l o t s of experimental v a l u e s o f the breaker h e i g h t H w i t h i t s 1 8 i 1 9 t r a n s f o r m a t i o n along the bed inshore are a v a i l a b l e as guide-H H 0 l i n e s . The p l o t s are a v a i l a b l e as f u n c t i o n s o f -— , — , L 0 d and bed sl o p e i H Q = wave h e i g h t o f incoming wave i n deep water L Q = incoming wave l e n g t h i n deep water d = water depth i n shallow water. 3 . 6 Breaking waves: S o l i t a r y wave theory: T h i s r e p r e s e n t s a symmetrical wave wit h the water s u r f a c e almost wholly above the trough, as q u a l i t a t i v e l y shown i n F i g . 1 6 . The geometry of the wave i s s u b j e c t to the l i m i t a t S e c t i o n 3 . 3 d/, which a p p l i e s to non-viscous f l a t ocean beds. For s l o p i n g beaches, the e x p e r i m e n t a l l y obtained v a l u e s 1_8- 2 0 of m o d i f i e d v a r i a b l e s such as H and d are a v a i l a b l e . The r e l a t i v e l y small r e d u c t i o n of breaking h e i g h t s due to bottom f r i c t i o n i s ob t a i n e d from computation of the energy per wave c y e l e t h a t i s d i s s i p a t e d by laminar damping. T h i s i s ob t a i n e d from t h e . e m p i r i c a l e x p r e s s i o n e, x (-—) H = H Q e L ( 3 . 1 7 ) A y c 53. IMPACT MODEL F i g 17 where e ^ i s a bottom damping f a c t o r which i n c r e a s e s w i t h i n c r e a s i n g k i n e m a t i c v i s c o s i t y v. decreases w i t h i n c r e a s i n g water depths and wave l e n g t h _ 3 and i n c r e a s e s with wave p e r i o d ; takes on v a l u e s from 10 to 2 x 10~ 2. . x = d i s t a n c e from the toe of the beach s l o p e . These values f o r b r e a k i n g wave h e i g h t s are fed back i n t o the s o l i t a r y wave r e l a t i o n s . 3.7 Impact type of breaker f o r c e s : T h i s model ( F i g . 17) i s accurate f o r asymmetric breaking wave p r o f i l e s such as p l u n g i n g waves and the steeper among the s p i l l i n g waves. The impulse f o r each wave i s obtained from drag e x p r e s s i o n s f o r p i l e s . The p o r t i o n of the impulse due to change of momentum on c o n t a c t i s comparatively s m a l l . In p a r t i c u l a r cases the computed impact-type e x c i t a t i o n s were found to be s m a l l e r than those f o r s o l i t a r y wave r e p r e s e n t a t i o n s . 3.8 Determinants of b r e a k e r s : V F i g . 18 i n d i c a t e s .4 r e g i o n s i n the i - -g- plane ( i being the beach s l o p e , HQ the deep water wave height) when d i f f e r e n t types of-waves reach the s h o r e — s p i l l i n g , p l u n g i n g , 7 .and . non-breaking. The breaker h e i g h t s exceeded the theore-t i c a l f i g u r e of 0.78d i n model t e s t s with a p p r e c i a b l e bed s l o p e . The p l o t at F i g . 18 demarcates b r e a k i n g regimes from non-brea k i n g ones. P l u n g i n g waves generate the h i g h e s t v e l o c i t i e s OJOI d B R E A K E R RANGES F i g 18 and wave f o r c e s out of the v a r i o u s types of breaking waves. The s o l i t a r y wave model r e p r e s e n t s the pl u n g i n g type of breaker f a i r l y a c c u r a t e l y and hence wave f o r c e c a l c u l a t i o n s f o r the s o l i t a r y wave, without any s e a l i n g , would be conserva-t i v e f o r other breakers a l s o . 3.9 Earthquake motion: The primary i n p u t used i n t h i s t h e s i s i s the ground a c c e l e r a t i o n r e c o r d of an a c t u a l earthquake. The dynamical problem i s formulated i n terms of the motion of the s t r u c t u r e r e l a t i v e to the ground, the absolute motion of the s t r u c t u r e being the s u p e r p o s i t i o n of the r e l a t i v e motion and the ground motion. Since the ground motion occurs without imparting any motion to the main body of the water, the ab s o l u t e s t r u c t u r e motion i s a l s o the r e l a t i v e s t r u c t u r e - - f l u i d motion ( i n the absence of other waves). However the range of the t o t a l motion i s such t h a t i t would not cause s e p a r a t i o n and vortex-shedding; t h e r e f o r e the values of C M and C D i n the p r e - s e p a r a t i o n range should be e f f e c t i v e . T h i s was checked by numerical computations f o r s t i f f as w e l l as f l e x i b l e s t r u c t u r e s . 3.10 Comparative ground a c c e l e r a t i o n s : The c h a r a c t e r i s t i c s of the E l Centro, 1940, N.-S. o ground r e c o r d and the T a f t , J u l y 21, 1952 S.21-W ground r e c o r d , which were used as the in p u t s t o the s t r u c t u r e s analysed, are now s t a t e d . The E l Centro ground motion p e r s i s t s s t r o n g l y f o r the f i r s t . 1 0 seconds and l e s s p e r c e p t i b l y t i l l the 30th second r e a c h i n g a maximum a c c e l e r a t i o n of 0.3g a t 2.5 sec. The dominant frequency i s 2.05 c y c l e s / s e c . The T a f t ground motion extends a p p r e c i a b l y f o r 30 seconds and reaches a maximum a c c e l e r a t i o n of 0.144g a t t = 4.1 sees. The dominant frequency i s 3.0 c y c l e s / s e c . CHAPTER IV DYNAMIC RESPONSE PROBLEM Th i s chapter p r e s e n t s the systems of d i f f e r e n t i a l equations of motion f o r framed s t r u c t u r e s used, to compute the dynamic response t o the earthquake e x c i t a t i o n and wave f o r c e i n p u t s d i s c u s s e d i n the pr e v i o u s cha p t e r s . 4.1 O r i g i n of n o n l i n e a r terms: The s t r u c t u r a l .analysis p o r t i o n i s a l i n e a r damped dynamic problem u s i n g the standard s t i f f n e s s method of formu-l a t i o n . N o n l i n e a r terms are i n t r o d u c e d through the f o r c e s a p p l i e d to the s t r u c t u r e which a r i s e through the i n e r t i a and drag f o r c e s of the f l u i d on the s t r u c t u r e due to the r e l a t i v e displacement. 4.2 Assumptions: The f o l l o w i n g assumptions have been made i n the dynamic f o r m u l a t i o n : 1) The framed s t r u c t u r e i s e l a s t i c , has c y l i n d r i c a l members and i s symmetric normal to the d i r e c t i o n of the ground/wave motion. 2) Rotatory and v e r t i c a l t r a n s l a t o r y i n e r t i a are ne g l e c -ted, an assumption checked subsequently by computa-t i o n s . 59, 3) F l u i d f o r c e s and r e s i s t a n c e s can be d i s c r e t i s e d a t nodes 4) The e f f e c t of the change of s e c t i o n shape on f l u i d -c y l i n d e r i n t e r a c t i o n was n e g l e c t e d 5) F l u i d f o r c e s on the c y l i n d e r are two-dimensional i n nature. 4.3 B a s i c f o r m u l a t i o n : The b a s i c equation f o r the dynamical problem i s of the form [m]{U> •+ [C . ]{U} + [k]{U> + {H(U ,U ) } = P(t) (4.1) where {u} =.n x 1 v e c t o r of g e n e r a l i s e d c o o r d i n a t e s , which i n t h i s case r e p r e s e n t s the reduced column matrix of h o r i z o n t a l nodal displacements, [m] = mass matrix. [C = r e l a t i v e v i s c o u s damping matrix r e p r e s e n t i n g i n t e r n a l damping r e l a t e d to the reduced v e c t o r of v e l o c i t i e s {u}. The i n d i v i d u a l terms of the matrix are ob t a i n e d by s e t t i n g constants a and g i n the e x p r e s s i o n C. . = a m. . + g k .• . 1D i j 1D so that the percentage of c r i t i c a l damping i n the f i r s t two modes i s a p r e s e l e c t e d v a l u e . 60. [k] = reduced s t i f f n e s s m atrix w i t h r e s p e c t to h o r i z o n t a l t r a n s l a t i o n s o n l y . {H(U r,U r)} = v e c t o r of f o r c e s due to the hydrodynamic e f f e c t / which i s a f u n c t i o n of the r e l a t i v e v e l o c i t i e s and a c c e l e r a t i o n s ' between s t r u c t u r e and f l u i d . ( P ( t ) } = v e c t o r of other f o r c e s on the system. These f o r c e s may be e i t h e r p h y s i c a l f o r c e s or c o n c e p t u a l f o r c e s such as the imparted i n e r t i a i n the earthquake.case. Dots r e p r e s e n t t i m e - d i f f e r e n t i a t i o n . 4.4 Earthquake i n p u t s : The exact form of the equations o f motion i s e s t a b l i s h e d on the assumptions s t a t e d i n S e c t i o n 4.2. In matrix form the equations are w r i t t e n as: + [k] {U r} = {0} (4.2) An e q u i v a l e n t equation of dynamic e q u i l i b r i u m i s [ [msJ + r K m V j ] { U } + [ C s t r ] {U}+rK DA|U|J{U}+[k]{U} = f C s t r ] { U g } + [ k ] { U g } (4.3) where {U"r} = n x 1 v e c t o r of g e n e r a l i s e d c o o r d i n a t e s , i . e . , displacements r e l a t i v e to the ground i n the h o r i z o n t a l d i r e c t i o n {U } = n x 1 v e c t o r of the ground displacement, a g i v e n 9J f u n c t i o n of time. Every element of the v e c t o r { Ug} i s the same f u n c t i o n of time. 61. {U}={U +U } = v e c t o r of ab s o l u t e displacements which was a l s o r g ^ t h a t r e l a t i v e t o water. [msJ = d i a g o n a l matrix of d i s c r e t i s e d masses i n the s t r u c t u r e ( o f f - d i a g o n a l terms appear i n case of coupling) [~KmVj = d i a g o n a l matrix of added mass, c o n t a i n i n g the c o e f f i c i e n t of mass, water d e n s i t y and the enclosed volume corresponding to each node. |^KDA(i|.U+Ug|.)Hu+Ug} = n x 1 v e c t o r of f l u i d drag f o r c e s . [ c S j - r ] ~ a s p r e v i o u s l y d e f i n e d , [k] = as p r e v i o u s l y d e f i n e d . Dots r e p r e s e n t t i m e - d i f f e r e n t i a t i o n . 4.5 F u r t h e r s i m p l i f i c a t i o n s : Equation (4.3) r e p r e s e n t s a system of n o n l i n e a r d i f f e r e n t i a l equations with v a r i a b l e c o e f f i c i e n t s . To eva l u a t e the importance of C M and i n the o v e r a l l response they were assumed to be constant a t a p a r t i c u l a r value throughout the motion, with a separate computation being made f o r every d i f -f e r e n t c h o ice of C,„ and C^. Assuming C„„ and as constants M D 3 M D allows the equations o f motion to be reduced to a system with c o n s t a n t c o e f f i c i e n t s as under: [m . . nJ{U +U }+[C . ]{U": }+rK_A(;|U +U |)J{U +U }+[k]{U } 1 v i r t u a l r g s t r r D 1 r g' r g J r = {0} (4.4) or e q u i v a l e n t l y v i r t u a l J ^ u > + [ c s t r ] { " } + fK DA|u|g {U}+[k] {U} = t C s t r ] { U g } + ^ { U g } (4.5) where [m v i r t u a l J i n c o r p o r a t e s constant v a l u e s of K. m 4.6 Method of s o l u t i o n : The equation (4.4) was s o l v e d by time-step numerical i n t e g r a t i o n f o r s p e c i f i c ground motion r e c o r d i n p u t s . Previous s t u d i e s have l e d to techniques f o r l i n e a r i s i n g the n o n - l i n e a r drag terms. For a d e t e r m i n i s t i c i n p u t no advantage i s thereby secured, s i n c e i t e r a t i o n s are i n v o l v e d , and l i n e a r i s a t i o n was not r e s o r t e d t o . The 3rd order Runge-Kutta method was used f o r n umerical s o l u t i o n , and the formulae f o r t h i s are g i v e n i n Appendix I I I . The s i z e of the time.steps had to be kept down to 1/4 to 1/10 of the s m a l l e s t of the n a t u r a l p e r i o d s of the s t r u c t u r e i n order to m a i ntain s t a b i l i t y of the s o l u t i o n . The steps ranged from 0.005 seconds to 0.0005 sees., the l a t t e r f o r s t r u c t u r e s w i t h 10 degrees of freedom, and these were s u f f i c i e n t l y s m a l l to f o l l o w the f l u c t u a t i o n s i n the i r r e g u l a r ground r e c o r d . Chapter V d e t a i l s the s t r u c t u r e s analysed and-the types of ground r e c o r d used f o r i n p u t . 4.7 Wave f o r c e i n p u t : Besides the assumptions i n S e c t i o n 4.2 i t was neces-sary to s i m p l i f y the e x c i t a t i o n which though d e t e r m i n i s t i c i n d i r e c t i o n i n shallow water, i s s t o c h a s t i c w i t h r e s p e c t to amplitudes and frequency. A f u r t h e r assumption i n the analy-s i s of s t r u c t u r e s c o n s i s t e d i n a l l o w i n g f o r the c o n t r i b u t i o n of o n l y one s e t of sway-bracings t o the s t i f f n e s s , n o t i n g t h e i r l a r g e s l e n d e r n e s s . L i f t f o r c e s were assumed to be n e g l i g i b l e and the flow presumed to be s u b - c r i t i c a l . The, equations of motion become: CmsJ {U}+[C s t r] {U}+[k] {U} = ^ K m V j { V U } + r K D A ( I V & l ) J { V W - U } ( 4 ' 6 ) where {VT7} = v e c t o r of water p a r t i c l e v e l o c i t i e s a t the s t r u c -w ture nodes, and other symbols are as p r e v i o u s l y d e f i n e d . 4.8 Wave response computations: The l a r g e r of the wave h e i g h t s g i v e . r i s e to high water p a r t i c l e v e l o c i t i e s and r e s u l t i n g high drag f o r c e s . Then the equations of motion become h i g h l y n o n l i n e a r . They were s o l v e d by numerical time-step i n t e g r a t i o n extending over s e v e r a l c y c l e s u n t i l the amplitude i n s u c c e s s i v e wave c y c l e s converged to a s t e a d y - s t a t e v a l u e . While s e l e c t i n g the in p u t s f o r r e s -ponse computations, as e l a b o r a t e d i n Chapter V, a p e r i o d of the n o n l i n e a r wave resonant to the s t r u c t u r a l p e r i o d and the g r e a t e s t corresponding amplitude of e x c i t a t i o n ( i . e . , wave height) i n each case were chosen. CHAPTER V RESULTS OF COMPUTATIONS C a l c u l a t i o n s o f the dynamic response of s e l e c t e d s t r u c t u r e s to earthquake e x c i t a t i o n and shallow water non-l i n e a r wave a c t i o n are presented h e r e i n . Displacement r e s -ponse and s t r e s s e s under the above two types of e x c i t a t i o n along w i t h those under b r e a k i n g waves have been compared f o r s e l e c t e d s t r u c t u r e geometries and water depths. The e f f e c t of v a r y i n g the va l u e s of the parameters C^ and C D on e a r t h -quake response has a l s o been examined. 5.1 Choice of s t r u c t u r e s f o r e v a l u a t i n g earthquake response: The s t r u c t u r e s chosen f o r a n a l y s i s are diagram-m a t i c a l l y shown i n F i g . 1 9 . S t r u c t u r e s A and B have a resemblance t o b e l l - t y p e well-head s t r u c t u r e s and are t o t a l l y submerged. The d i s p l a c e d volume i s l a r g e f o r both A and B with the s t i f f n e s s and n a t u r a l frequency low f o r A. As the added masses are a p p r e c i a b l e , the responses o f A and B h i g h -l i g h t the i n f l u e n c e of C^. S t r u c t u r e C r e p r e s e n t s the other extreme of the range of d i s p l a c e d volumes and s t r u c t u r a l s t i f f n e s s . I t i s a tower-supported deck p l a t f o r m s t r u c t u r e w i t h a r e l a t i v e l y s m a l l enclosed volume and a hig h n a t u r a l frequency. The n a t u r a l p e r i o d s and mode shapes f o r the three S T R U C T U R E S A N A L Y S E D F O R E A R T H Q U A K E S 65. 66. s t r u c t u r e s are giv e n i n Table 4 where the nodes r e p r e s e n t only h o r i z o n t a l degrees o f freedom (as used i n the reduced s t i f f n e s s matrix) and are numbered as shown i n F i g . 20. For the f u l l y sub-merged b e l l s A and B, i t i s seen t h a t the i n f l u e n c e of C„ i n c -reases the fundamental p e r i o d by as much as 25 percent. In a r r i v i n g a t the t a b u l a t e d values of the pe r i o d s and i n c a l c u l a -t i o n s of the response, the d i s t r i b u t e d m a s s / i n e r t i a c h a r a c t e r i s -t i c s o f the.upper member of s t r u c t u r e s A and B were taken i n t o account. • Beam members o f the plane frame type were used i n mo d e l l i n g the s t r u c t u r e s , a member having s i x degrees of freedom. 5.2 Earthquake response: The range of parameters C M, C D and the per c e n t c r i t i c a l ( s t r u c t u r a l ) damping f o r which computations were made, are g i v e n i n Table 5. In a d d i t i o n to the E l Centro ground shock which was a p p l i e d to the three s t r u c t u r e s , the T a f t ground r e c o r d was a l s o used-as the i n p u t f o r s t r u c t u r e A. The response-maximum d i s p l a c e -ments and base s h e a r s — t o earthquake i n p u t s f o r p r o g r e s s i v e l y i n c r e a s i n g values of the parameter C„, are t a b u l a t e d i n Table 6. M For l a r g e r values of C^, which cause longer fundamental p e r i o d s , the maximum displacement under the E l Centro input i n c r e a s e d f o r s t r u c t u r e A as expected (of the order of the f o l l o w i n g ) : 20% over the range 1.0 < C M < 1.5 6% over the range 1.5 < C„ < 2 ^ M • 26% over the range 1 < C M < 2). The maximum base shear f o r s t r u c t u r e A a l s o i n c r e a s e d by about 42 p e r c e n t as C i n c r e a s e d from 1 to 2. For s t r u c t u r e B the KEY .TO MODE SHAPES r & A A A STRUCTURE A STRUCTURE B STRUCTURE C F i n 9 n o D e g r e e s of f r e e d o m 2 I -3> J T L S h e a r d e f o r m a t i o n s no t c o n s i d e r e d PLANE BEAM ELEMENT TABLE 4 PERIODS AND MODE SHAPES P a r t i c i p a t i o n N a t u r a l F a c t o r No. of p Periods (For L i n e a r S t r u c t u r e Nodes Mode M (sees) Mode Shape Behaviour) 1 1+0 2.3 Node 1 Ampl. 0.129 1 1+1 2.814 1 0.129 2 1+0 0.383 1 -0.691 2 1+1 0.463 1 -0.992 3 1+0 0.038 . 1 0.992 3 1+1 0.042 1 0.994 1 1+0 1. 217 1 0.067 1 1+1 1.49 1 0.067 2 1+0 0.153 1 -0.660 2 1+1 0.186 1 -0.660 3 1+0 0. 008 1 0.997 3 1+1 0.009 1 0.998 2 3 1.485 0.259 0.957 2 3 1.49 0.259 0.957 2 3 1.010 -0.593 0.412 2 3 1.04 -0.595 0.411 2 3 0.080 -0.110 0.066 2 3 0.077 -0.090 0.054 2 3 1.488 0.129 0.989 2 3 1.495 0.129 0.989 2 3 1.234 -0.646 0.385 2 3 1.255 -0.646 0.384 2 3 0.0805 -0.068 0.037 2 3 0.078 -0.054 0.029 TABLE 4 (Cont'd.) No. S t r u c - o f ture Nodes Mode Natu-r a l Per-i o d 'M (sees) Mode Shape P a r t i c i p a -t i o n F a c t o r (For L i n e a r Behaviour) 10 1 1+1 0. 99 Node 1&2* 3 4&5* 6 7&8* 9&10* 1.. .192 Ampl. 0.128 0. 145 0.475 0.490 0.78 1.0 Anti-symmetric mode 2 1+1 0. 263 1&2 3 4&5 6 7&8 9&10 0. 636 0.525 0. 594 1.0 0.772 0.27 -•0. 387 Anti-symmetric mode 3 1+1 0. 145 1&2 3 4&5 6 7&8 9&10 0. 503 1.0 0. 909 -0.345 -0.368 -0.313 0.135 Anti-symmetric mode 4 1+1 0. 114 1&2 3 4&5 6 7&8 9&10 0. 056 0.17 0. 134 -0.518 -0.313 1.0 -•0. 249 Anti-symmetric, mode 5 1+1 0. 078 1&2 3 4&5 6 7&8 9&10 0. 0005 0. 088 0 -1.0 0 0.003 0 Anti-symmetric mode 6 1+1 0. 075 Symmetric mode 0. 0005 7 1+1 0. 039 Symmetric mode 0. 0002 8 1+1 0. 036 Symmetric mode 0. 0002 9 1+1 0. 023 Anti-symmetric mode 0. 016 10 1+1 0. 017 Anti-symmetric mode 0. 001 *Ampls. equal i n anti-symmetric modes, but equal and opp o s i t e i n symmetric modes. TABLE 5 RANGE OF PARAMETERS Q. S t r u c t u r e CM CD C r i t i c a l T (sees.) Damping A B A \ 1+1 1.2 2 2.81 1.49 B ) 1+0. 75 1.2 2 2.68 1.42 (For E l Centro) 1+0. 5 1.2 2 2.56 1.35 1+0 1.2 2 2.3 1.22 1+1 0 2.5 2.81 1.49 1+1 0 3 2.81 1.49 1+1 0 4 2.81 1.49 1+1 0 5(A only)2.81 -C 1+1 1.2 2 0.99 (For E l Centro) 1+0 1.2 2 0.988 A 1+1 1.2 2 2.81 (For T a f t ) 1+0. 5 1.2 2 2.56 1+1 0 3 2.81 1+1 0 4 2.81 N.B. C„, = 1+0 i n d i c a t e s s t r u c t u r e without water. TABLE 6 EARTHQUAKE RESPONSE Parameters " R a t i o of C r i t i c a l N a t u r a l Maximum Maximum Earthquake Damping P e r i o d Displacement Base Shear STR. Record M D £ (sees.) (ins.) (Kips) A E l Centro 1+0 1.2 0.02 2.30 17.95 28.91 1940 1+0.5 1.2 0.02 2.56 • 21.42 36.12 1+0.75 1.2 0.02 2.68 22.51 37.91 1+1 1.2 0.02 2.81 22.60 51.2 B E l Centro 1+0 1.2 0.02 1.22 6.44 36.10 1940 1+0.5 1.2 0.02 1.35 6.26 41.84 1+0.75 1.2 0.02 1.42 6.81 46.25 1+1 1.2 0.02 1.49 7.32 48.70 A Taft,1952 1+0 1.2 0.02 2.30 3.79 9.92 1+0.5 1.2 0.02 2.56 3.92 11.57 1+1 1.2 0.02 2.81 4.22 14.18 C E l Centro, 1+0 1.2 0.02 0.988 6.78 603.9 1940 1+1 1.2 0.02 0.99 6.71 641.5 N.B. C M = 1+0 represents s t r u c t u r e without water. response was l e s s s e n s i t i v e to C^. The percent i n c r e a s e s over the range 1 < C M < 2 were 14 percent and 35 p e r c e n t f o r d i s p l a c e -ment and shear r e s p e c t i v e l y . In the case of s t r u c t u r e C there was n e g l i g i b l e d i f f e r e n c e i n the response with.or without the added mass e f f e c t . Constant v a l u e s of C D = 1.2 and Percent c r i t i c a l damping £ = 2 p e r c e n t were assumed i n making the comparisons. 5.3 E f f e c t of s t r u c t u r a l shape: The e f f e c t of s t r u c t u r a l shape, i . e . , d i s p l a c e d volume, s t i f f n e s s and mass d i s t r i b u t i o n on the response i s i l l u s t r a t e d by the three examples chosen. The s m a l l e r the d i s p l a c e d volume and the s m a l l e r the n a t u r a l p e r i o d , the l e s s s e n s i t i v e i s the response to hydrodynamic e f f e c t s . 5.4 E f f e c t of C D: A s i m i l a r p a r a m e t r i c study v a r y i n g C Q showed t h a t the response i s i n s e n s i t i v e to C Q f o r earthquake i n p u t s . Computa-t i o n s of the response f o r s t r u c t u r e A w i t h the n o n l i n e a r drag term and w i t h t h a t term being r e p l a c e d by an a d d i t i o n a l e q u i v a l e n t v i s c o u s damping r a t i o o f 0.01, 0.02 and 0.03 are g i v e n i n Table 7. Comparing the maximum response i n e i t h e r case, a d d i t i o n a l damping e f f e c t s due to water drag do not e v i d e n t l y exceed 2 to 3 p e r c e n t c r i t i c a l v i s c o u s damping f o r such s t r u c t u r e s . F u r t h e r -more, f o r l a r g e - d i a m e t e r c y l i n d e r s , from Table 8, drag (and a l s o TABLE 7 DAMPING EQUIVALENT OF DRAG Earthquake STR. Record R a t i o of C r i t i c a l Damping N a t u r a l P e r i o d (sees.) Maximum Displacement (ins.) Maximum Base Shear (Kips) E l Centro, 1940 1+1 1+1 1+1 1+1 1+1 1.2 0 0 0 0 0.02 0.025 0.03 0.04 0.05 2.81 2.81 2.81 2.81 2.81 22.60 26.04 25.32 23.95 22.70 51.21 58.74 55.06 48.88 44.23 T a f t , 1952 1+1 1+1 1+1 1.2 0 0 0.02 0.03 0.04 2.81 2.81 2.81 4.22 4.13 4.00 14.18 13.39 12.25 74. TABLE 8 WATER INERTIA AND DRAG FORCES FOR STRUCTURE B T = 1.49 C.. = 1+1 C^ = 1.2 M D E l Centro Ground Record Node (Max. Water „ . . I n e r t i a Force) o r - > ^ r-> _ R a t i o -rrr - = — r - 853.0 67.0 13.7 (Max. Drag Force) 75. l i f t ) e f f e c t s a r e s e e n t o be s m a l l compared w i t h added i n e r t i a . 5.5 R e l e v a n c e o f s u b c r i t i c a l r e g i o n : C h e c k s o f t h e i n s t a n t a n e o u s N showed t h a t e x c e p t f o r v e r y s h o r t d u r a t i o n s a t t h e e x t r e m e t o p node, t h e r e l a t i v e m o t i o n b e t w e e n f l u i d and s t r u c t u r e was i n t h e s u b c r i t i c a l r e g i o n . F u r t h e r , f l o w s e p a r a t i o n w o u l d o c c u r a t o n l y t h e t o p m o s t node o f o n l y t h e most f l e x i b l e s t r u c t u r e s . 5.6 Dynamic r e s p o n s e t o f i n i t e - a m p l i t u d e S t o k e s waves: The r e s u l t s f o r a s i n g l e p i l e ( p e r i o d 4.4 s e e s . ) as w e l l as 6 o t h e r p i l e - s u p p o r t e d p l a t f o r m s o f f u n d a m e n t a l p e r i o d s between 2.11 and 3.45 s e c o n d s a r e r e p o r t e d . The s t r u c t u r e s r a n g e i n d e p t h f r o m 40 f t . t o 100 f t . , i . e . , where s h a l l o w -w a t e r wave c o n d i t i o n s w o u l d be e n c o u n t e r e d . W h i l e t h e s t r u c -t u r a l c o n f i g u r a t i o n s as shown i n F i g . 21 a r e r e a s o n a b l y s t a n d a r d and amenable t o p r a c t i c a l c o n s t r u c t i o n , t h e member s i z e s and c o n s e q u e n t l y t h e s t r u c t u r a l p e r i o d s were c h o s e n so as t o i n d u c e r e s o n a n c e w i t h o c e a n waves c o v e r i n g a p r a c t i c a l r a n g e . The member s i z e s were t h e r e f o r e d e s i g n e d t o keep t h e f l e x i b i l i t y and n a t u r a l p e r i o d s h i g h ( o v e r 2 s e e s . ) . The v a l u e s o f t h e p a r a m e t e r s C M and C D u s e d , some o f t h e s t r u c t u r a l s i z e s and s e l e c t e d wave d a t a f o r w h i c h v i b r a t i o n r e s p o n s e was computed a r e g i v e n i n T a b l e 9. The wave p e r i o d s and h e i g h t s were s e l e c t e d so t h a t one o f t h e h a r m o n i c s o f t h e 2 S T R U C T U R E S VI *, VII STRUCTURE X STRUCTURE IX For Structure VIII, see Str. C of Fig 19 S T R U C T U R E S A N A L Y S E D FOR B R E A K E R S Fig 22 TABLE 9 STRUCTURAL AND OTHER PARAMETERS FOR FINITE-AMPLITUDE WAVE RESPONSE S t r . N a t u r a l Base Height D i a . T o t a l T o t a l R a t i o Depth P e r i o d Height Wave No. P e r i o d of F i x i t y of of Pro- Enclosed (Proj.Area) of of of Length d S t r u c t u r e S t r u c - Main j e c t e d Volume En c l o s e d V o l . Water Wave Wave L L (sees.) t u r e P i l e s Area Cub.ft. _ i d T H F t . 1st 2nd F t . D Sq.Ft. F t . 1 F t . sec. F t . Mode Mode F t . I 2.11 0.23 Rest-r a i n e d k - E I L 60 1.5 179.6 167 1.07 40 4.2 12 90 .44 II 2.65 0.28 Rest-r a i n e d k = ^ K 4L 60 1.5 179.4 155.2 1.16 40 5.3 19 139 . 29 I I I 3.45 0.40 F i x e d 60 1.5 166.2 161.8 1.03 40 6.9 25 244 .16 IV 4.4 0.50 F i x e d 90 3 405 954.2 0. 42 60 6.0 25 184 . 33 V 1.44 0.17 Rest-r a i n e d K 3L 82 2 336 490 0.69 75 2.87 5.8 42 1.79 VI 2.55 0.52 Rest-r a i n e d k - E I * 2L 128 4 997.6 2466 0.41 100 5.0 17 128 .78 VII 2.84 0.41 Rest- 128 2 467.6 612.8 0.75 100 5.65 21 164 .61 t r a i n e d v-EI • L k = r o t a t i o n a l s t i f f n e s s of base j o i n t . C = 1 2 * UD ' M .0 I F L are member i n e r t i a and l e n g t h to next j t . to r bottom s e c t i o n of p i l e . n o n l i n e a r waves would be i n resonance w i t h the f i r s t mode of v i b r a t i o n of the s t r u c t u r e . T h i s c r i t e r i o n can be s a t i s f i e d by matching the f i r s t harmonic of a s m a l l wave, wit h a c o r r e s -pondingly s m a l l energy i n p u t , or one of the h i g h e r harmonics o f l a r g e r waves. The p a r t i c u l a r s i t u a t i o n s which generate the l a r g e s t dynamic f o r c e s under the a c t i o n of non-breaking waves are the ones r e p o r t e d here. Computations f o r b r e a k i n g waves i n these depths are presented l a t e r . 5.7 Computed response to Stokes waves: . T a b l e 10 l i s t s the maximum s t e a d y - s t a t e displacements and o v e r t u r n i n g moments a t the base. D e s p i t e the f a c t t h a t the p e r i o d of the second harmonic of the wave e x c i t a t i o n e q u a l l e d the fundamental p e r i o d i n every case except S t r . IV, the maximum displacements i n T a b l e 10 do not i n c r e a s e i n a r e g u l a r manner wit h i n c r e a s i n g h e i g h t d or such other parameter. A s m a l l change i n s t r u c t u r a l p e r i o d , as between I and I I , causes a l a r g e change in.dynamic response. Although the water depths are the same f o r I and I I , the value of T s e l e c t e d to synchro-n i s e the second harmonic w i t h s t r u c t u r e I I was g r e a t e r ; a c c o r d i n g l y the wave s i z e was g r e a t e r , causing an i n c r e a s e i n the r a t i o Consequently because of the comparatively g r e a t e r amplitude of the second harmonic of -the wave, the peak displacement i s much g r e a t e r f o r S t r . I I than f o r S t r . I . The cases of s t r u c t u r e s VI and VII are s i m i l a r . TABLE 10 RESPONSE VALUES FOR FINITE-AMPLITUDE WAVE INPUT S t r . T n sec. D F t . Ratio P r o j e c t e d Area d F t . T Sec. H F t . Max. X max In. Displacement (Time A f t e r C r e s t ) Max. Ov e r t u r n i n g E n c l o s e d Volume F t . " 1 (Wave Period) Moment K.In. I 2.11 1.5 1.07 40 4.2 12 4.63 0.4 24100 I I 2.65 1.5 1.16 40 5.3 19 16.3 0.1 53200 I I I 3.45 1.5 1.03 40 6.9 25 41t 0.2 82100 IV 4.4* 3 0.42 60 6.0 25 2 8 t 0.4 38700 V 1.44 2 0.69 75 2.87 5.8 1.25 0.5 38500 VI 2.55 4 0.41 100 5.0 17 3.8 0.4 152000 VII 2.84 2 0.75 100 5.65 21 6.7 0.1 148600 N.B • CD = 1 .2 *Simple p i l e t F l e x i b i l i t y . h i g h , n o n l i n e a r a n a l y s i s warranted. 81. Other causes f o r the v a r i a t i o n s i n maximum d i s p l a c e -ments a r e : a) Greater wave h e i g h t s H s e l e c t e d to accompany the g r e a t e r T d i r e c t l y i n c r e a s e s the amplitude of the e x c i t a t i o n . T h i s accounts f o r the comparatively h i g h displacements f o r S t r . I I , I I I and V I I . b) The s m a l l a r e a exposed to drag i s seen tp keep down the displacements f o r VI. c) The extremely low s t r u c t u r a l s t i f f n e s s of S t r . I l l would engender a l a r g e " s t a t i c d e f l e c t i o n " and r e s u l t s i n the l a r g e peak displacement shown. d) The e f f e c t i v e hydrodynamic damping r a t i o , a f u n c t i o n of the average p a r t i c l e v e l o c i t y , d i f f e r s w i d e l y and i n f l u e n c e s resonance a m p l i f i c a t i o n . S i m i l a r l y the maximum o v e r t u r n i n g moments do not i n c r e a s e m o n o t o n i c a l l y w i t h e i t h e r water depth or i n p u t wave hei g h t ; nor do they i n c r e a s e i n the same manner as the peak displacements. T h i s i s p a r t l y due to the f a c t t h a t i n d i f f e r e n t s t r u c t u r e s the degree of p a r t i c i p a t i o n by the second mode v a r i e s . 5.8 Force v a r i a t i o n s w i t h time: A p l o t of the t o t a l wave f o r c e on the p i l e s of a t y p i c a l s t r u c t u r e ( S t r . I) t a k i n g i n t o account the motion of the s t r u c t u r e i s shown i n F i g . 23. The t o t a l f o r c e from the v a r i a b l e water s u r f a c e t o the base of the p i l e has been p l o t t e d . 83. The p l o t i n d i c a t e s the f o l l o w i n g : 1) drag predominates over i n e r t i a f o r the s t r u c t u r e i n q u e s t i o n , where the wave dimensions are l a r g e r e l a t i v e to the water depth. 2) the f o r c e - h i s t o r y p l o t i s not symmetrical about the time of passage of the c r e s t . T h i s i s due to the i n e r t i a f o r c e being a t 90° phase and a l s o to the change i n the drag p a t t e r n owing to s t r u c t u r e motion and h i g h e r o r d e r terms. 3) the i n e r t i a f o r c e p l o t i s not symmetrical about the s t i l l water l e v e l time, t h i s being due to h i g h e r order terms. The time v a r i a t i o n of the s t e a d y - s t a t e bending moment i n the p i l e s e c t i o n a djacent t o the p l a t f o r m f o r s t r u c t u r e I i s p l o t t e d i n F i g . 24. The moment f l u c t u a t e s a t twice the frequency of the wave, which i s e x p l a i n e d by the f a c t t h a t the sec-ond harmonic a t h a l f the wave p e r i o d c o i n c i d e d w i t h the s t r u c -t u r a l p e r i o d . 5.9 I n t e r a c t i o n e f f e c t s on i n e r t i a f o r c e s : The i n e r t i a p o r t i o n of the wave f o r c e on p i l e members i s s i g n i f i c a n t l y d i f f e r e n t f o r a f l e x i b l e p i l e s t r u c t u r e , as compared with a co r r e s p o n d i n g r i g i d p i l e . T h i s i s because the s t r u c t u r e a c c e l e r a t i o n s are comparable i n magnitude to the water p a r t i c l e a c c e l e r a t i o n s even though the v e l o c i t i e s d i f f e r w i d e l y . 85. Thus the t a k i n g i n t o account of the feedback o f s t r u c t u r e motion i n a r r i v i n g a t i n e r t i a f o r c e v a l u e s c o n t r i b u t e s to a r e f i n e -ment of the response s o l u t i o n . 5.10 S u p e r c r i t i c a l flow c o n d i t i o n s : Although s u b c r i t i c a l v alues o f C D (1.2) were adopted f o r most computations i n the wave problem, v e l o c i t i e s a t the s u r f a c e of the water exceeded c r i t i c a l v a l u e s f o r longer dura-t i o n s than i n the earthquake s i t u a t i o n . N approached 2 x 10 ( i . e . , > 2 x 10^) based on r.m.s. v e l o c i t i e s f o r the topmost node f o r some s t r u c t u r e s s u b j e c t e d t o the h i g h e s t waves. For 4 f t . d i a . p i l e s i t approached 6 x 10^ (based on r.m.s. v a l u e s ) . T h i s f e a t u r e would reduce the wave f o r c e s . At other nodes s u b c r i t i c a l v a l u e s p r e v a i l e d . 5.11 Keulegan parameter: v T m a x > v a l u e s (Ref. S e c t i o n 2.18) ranged from 20 to 30, i . e . , g r e a t e r than 15, the value f o r a t l e a s t one v o r t e x to be d i s c h a r g e d . Eddy-shedding f r e q u e n c i e s were much lower than n a t u r a l f r e q u e n c i e s , r u l i n g out l i f t resonance t e n d e n c i e s . 5.12 Breaking wave ( s o l i t a r y wave) response: To take i n t o account the e f f e c t of s h o a l i n g i n i n c r e a s i n g wave h e i g h t s a t b r e a k i n g , p l o t s of the breaker h e i g h t -18 19 20 depth r e l a t i o n s g i v e n by experimenters ' ' were adopted. 86. To make use o f the a f o r e s a i d r e l a t i o n s , the p o i n t of commence-cl * X ment of the. beach slope was taken a t the p o i n t where — = J-j A (the l i m i t f o r shallow water) and was used to d e f i n e i n i t i a l water depths. The s o l i t a r y wave theory was used to f i n d the water kinematics at - t h e passage of the c r e s t and thus the f o r c e l e v e l s . The parameters f o r the v a r i o u s f o r c e computations are t a b u l a t e d i n Table 11. The summarised p a r t i c u l a r s of the computed loads are i n Table 12. The d e v i a t i o n s of these computed f o r c e s from the tr u e f o r c e s occur due to the f o l l o w i n g f a c t o r s , whose o v e r a l l e f f e c t i s to warrant a s l i g h t decrease i n the computed f o r c e s : a) Increase of the s t a t i c a l l y computed member f o r c e s and s t r e s s e s due to dynamic a m p l i f i c a t i o n . b) Decrease: . S u p e r c r i t i c a l N D v a l u e s a t the upper p o r t i o n s of the p i l e s reduce C D i n steady flow s i t u a -t i o n s , t h i s being by a f a c t o r o f 3 i n the upper p o r t i o n s . c) Decrease: For s p i l l i n g breakers and f o r waves deform-i n g but not bre a k i n g under the p a r t i c u l a r s l o p e , v e l o c -i t i e s would be lower than f o r a t h e o r e t i c a l s o l i t a r y wave of t r a n s l a t i o n . 5.13 Comparative f o r c e s under v a r i o u s e x c i t a t i o n s : Comparative v a l u e s of f o r c e s and moments produced by the v a r i o u s wave and earthquake i n p u t s are g i v e n i n Table 13. The moments/forces f o r earthquake i n p u t s have-been.scaled down TABLE 11 STRUCTURAL AND OTHER PARAMETERS FOR BREAKING WAVE (SOLITARY WAVE) FORCES i t r . No. Height of Dia. of T o t a l P r o j e c t e d Depth Height of Wave St r u c t u r e Main P i l e s Area Below F l a t Smooth S l o p i n g F t . D Sq. F t . Trough Bed Bed F t . F t . F t . F t . I 60 1.5 180 40 - 31 II 60 1.5 180 40 - 31 IV 90 3 405 60 - 40 V 82 2 336 75 - 45 VI 128 4 998 100 - 50 VII 128 2 468 100 - 50 IX 135 2 1013 95 50 -X 165 3 1450 135 30 — TABLE 12 LOADING DUE TO BREAKING WAVES Str. No. Type of Breaker P i l e Total Depth Beach Wave Cha r a c t e r i s t i c s (Proj.Area) Dia. Height of Slope T, D Of Str. Water i Ft. Ft. d Ft. . . . . W Sec. H Ft. 0 Ft. (Proj.Area of Pile) Total Total Force Over-on turning P i l e s Moment K K.In. ,11,III S p i l l i n g 1.5 60 40 .05 10 3 3 31 1.11 128 63600 IV Plunging (.05) S p i l l i n g (.02,.01) 3 90 60 .05,.02,10 .01 40 40 1. 0 230 171100 V Plunging (.05) S p i l l i n g (.02,.01) 2 82 75 .05,.02,12 . 01 50 45 1.06 266 214800 VI S p i l l i n g 4 128 100 .02,.01 10 55 50 1.02 735 783000 VII S p i l l i n g 2 128 100 .02,.01 10 55 50 1.05 367 391000 IX S p i l l i n g 2 135 120 0 10 50 50 1.13 950 960000 X Unspeci-f i e d 3 165 150 0 16 50 - 1.13 a) *303 b) +950 270000 960000 N.B. *Lower l i m i t (under l i n e a r o s c i l l a t o r y waves) 1. S t a t i c calculations for moments. 2. Beach slopes are indicated i n parentheses. 3. C D = 1.2 t Upper l i m i t : as for IX 00. CO TABLE 13 COMPARATIVE VALUES OF MOMENTS AND AXIAL FORCES UNDER VARIOUS EXCITATIONS O s c i l l a t o r y L i n e a r Waves Stokes Shallow Water Waves S t r . No. • T n Sees. <°n Cl F t . Worst Mom. K" Worst A x i a l Force K Worst Mom. Worst A x i a l K" Force K I 2.11 2.98 40 3 0 2 * 1 1 950 940 240 270 II 2.65 2.37 40 - - 3200 3160 230 640 I I I 3.45 1.82 40 - - 2180 240 V 1.44 4.36 75 - - 396 156 55 1309 VI 2.55 2.46 100 13000 250 23100 3500 1480 2170 VII 2. 84 2.21 100 3 5 0 0 * 3 10 5550 1540 0 1030 V I I I 0.99 6.29 200 1231 304 1250 300 IX 1.18 5.33 120 - - - -X 0.80 7.97 150 1680 500 - -XI 0.99 6.29 250 1900 510 -XII 4. 48 1.40 800 31800000+ - - -X I I I 6.28 1.00 1200 % - - -* L S t a t i c a p p l i c a t i o n of shallow water wave f o r c e s . * z Estimated. *3 C o n c e n t r a t i o n of l o a d a t a p o i n t assumed. * 4 Estimated. @ E x a c t e l a s t i c v a l u e s by time step i n t e g r a t i o n : Mom.: 6500 K"; A x i a l f o r c e : 1800K. t 3 standard d e v i a t i o n s . Ref.: F o s t e r , E.T.: Model f o r n o n l i n e a r dynamics of o f f s h o r e towers (J.A.S.C.E. V o l . 96, No.EMI, Feb.1970). 89. E l Centro Quake Breaking Waves - Response Spectrum Scaled by 0.75 S c a l e d f o r Y i e l d i n g Worst Mom. Worst A x i a l Worst Mom. Worst A x i a l Beach K" j Force K" Force Slope K K 0.05 2770 13 1185 143 0.05 4500 . 10 1850 140 ; 1550 40 0.05 2340 ; 25 8 9 0 * 2 85 0.05 2410 • 18 787 92 0 . 0 2 , 0 . 01 2910 ' 95 415 330 0 . 0 2 , 0 . 01 11500 290 5400 215 2450 5020 1030 2050 0 . 0 2 , 0 . 01 1210 720 800 100 2300 . 17 250 450 o. 4830 1 1360 2720@ 760@ 0 3400 ! 1480 1090 625 0 3400 1480*4 2110 670 0 3350 610 3200 735 _ _ 280000 _ % 3 standard d e v i a t i o n s o f displa c e m e n t a t top = 2 . 7 f t . Ref.: Malhotra A, and Penzien J . : Response o f o f f s h o r e s t r u c t u r e s to random wave f o r c e s (J.A.S.C.E. V o l . 96, No. ST.10, October, 1970). + Displacement a t top i n e l a s t i c behaviour = 1.33 f t . 90, by a h a l f to al l o w f o r the r e d u c t i o n i n d e s i g n f o r c e s t h a t would r e s u l t from d u c t i l e y i e l d i n g of the s t r u c t u r a l members. T h i s i s c o n s e r v a t i v e when compared to c u r r e n t earthquake d e s i g n 16 p h i l o s o p h y of s t r u c t u r e s . For reasons quoted i n S e c t i o n 5.12, the computed b r e a k i n g wave moments/forces have been s c a l e d down by 25 p e r c e n t . The comparative val u e s of s t r e s s e s are giv e n i n Table 14. Some of the parameters t h a t have a b e a r i n g on compara-t i v e responses are: 1) n a t u r a l p e r i o d 2) water depth 3) diameter of p i l e s 4) mass/(member s t i f f n e s s ) r a t i o 5) bed s l o p e . Other parameters such as i ) depth of water a t the toe of the beach s l o p e , i i ) bed roughness, e t c . would a l s o be r e l e v a n t , e s p e c i a l l y f o r brea k i n g wave f o r c e s . For each s t r u c t u r e the c r i t i c a l l o a d case, as w e l l as depth d and p e r i o d T , have been noted i n Table 15. From c o n s i d e r a t i o n s d i s c u s s e d i n pr e v i o u s s e c t i o n s and ch a p t e r s , the n a t u r a l p e r i o d T R and the water depth d were adjudged to be the parameters most m a t e r i a l l y a f f e c t i n g the comparative response. TABLE .14 COMPARATIVE STRESSES S t r e s s e s (K/in. ) X - s e c t i o n Osc. Stokes' Shallow E l Centro Quake S t r . i . d L i n e a r Breaking Response Spectrum No. n Area In. F t . Waves Water Waves Waves Masses a t Top S t r e s s e s 1 2. 11 31 861 40 ±3. 1+0. 03 ±9.9+ 7.7 ±9.8+ 8.7 ±38.7+ 0.5 2x150 ±9. 4+ 3. 9 II 2. 65 31 861 40 - ±33.5+ 7.4 ±32.9+20.6 ±47.1+ 0.3 2x150 ±19. 4+1. 3 I I I 3. 45 7. 1 287 40 Unrepresentative S t r u c t u r e IV 4. 4 56 800 60 II V 1. 44 28 1200 75 - ±4.0+ 2.0 ±1.6+46.8 ±38.8+ 4.5 2x200 ±7. ±4. 9+ 3. 2+11. 3 7 VI 2. 55 112 21000 100 ±14. 8+2. 2 ±26.4+13. 2 ± 4.0+19.4 ± 1 7 . 5 + 3 . 4 ±3.7+59.8 2x500 ±6. ±1. 2+ 1. 2+18. 9 3 VII 2. 84 28 1360 100 ±31. 0+0. 4 ±49.1+ 0 ±13.6+36.8 ±14.4+34.1 ±27.1+ 0.8 2x300 ±7. ±2. 1+ 3. 2+16. 6 1 VII I 0. 99 85 12600 200 ±1. 8+3. 6 ± 1 . 8 + 3 . 6 ±6.9+16.0 2x400 ±3. 9+8. 9* IX 1. 18 37. 7 2720 120 - - ±15.0+39.2 2x400 ±4. 0+13. 8 X 0. 80 56. 5 9170 150 ±3. 3 + 8. 9 - ±6.7+26.2 2x400 ±3. 5+. 9. 9 XI 0. 99 126 18900 250 . ± 2. 4+4. 0 - • ± 4 . 3 + 4 . 8 2x320 , ± 4 - 1+ 5. 8 *Exact values (by time step i n t e g r a t i o n ) f o r e l a s t i c behaviour: ±5.2+11.9 K / i n . TABLE 15 GOVERNING LOAD CASES FOR OFFSHORE TOWERS S t r u c t u r e T n d Governing Load Sees. F t . I 2. 11 40 Breaking ( S o l i t a r y ) wave I I 2. 65 40 Breaking ( S o l i t a r y ) wave I I I 3. 45 40 Breaking ( S o l i t a r y ) wave V 1. 44 75 Breaking ( S o l i t a r y ) wave VI 2. 55 100 Shallow water o s c i l l a t o r y wave VII 2. 84 100 Shallow water o s c i l l a t o r y wave V I I I 0. 99 200 1. Breaking wave 2. Earthquake IX 1. 18 120 Breaking wave X 0. 80 150 1. Breaking wave 2. Earthquake XI 0. 99 250 Earthquake L O A D T Y P E S 93. The fundamental p e r i o d T n i s the most important system c h a r a c t e r i s t i c t h a t p r i n c i p a l l y determines the degree of reson- . ance w i t h e x c i t a t i o n s w i t h v a r i o u s dominant f r e q u e n c i e s . Since the o f f s h o r e s t r u c t u r e s were taken as g e o m e t r i c a l l y almost, s i m i l a r , and the i n c r e a s e i n masses was graded i n r e l a t i o n to depth, i t a l s o f o l l o w s t h a t f o r a g i v e n d, the p e r i o d T r i n -d i r e c t l y r e f l e c t s member c r o s s - s e c t i o n a l s i z e s . In the case of o s c i l l a t o r y waves, the water depth i n f l u e n c e s the maximum wave dimensions, the c o n t a c t area, and the r e l a t i v e magnitudes of higher harmonics of n o n l i n e a r waves; i n the case of b r e a k i n g waves i t determines the height'and v e l o c i t y d i s t r i b u t i o n of the wave and the c o n t a c t area. For these reasons a p l o t i n the d-T n space corre s p o n d i n g to the type of l o a d i n g t h a t may govern d e s i g n has been made and i s shown i n F i g . 25. In c o n s t r u c t i n g t h i s p l o t a t t e n t i o n was r e s t r i c t e d t o the r e g i o n between the two bounding l i n e s A and B s i n c e p r a c t i c a l s t r u c t u r e s would not l i k e l y f a l l o u t s i d e t h i s r e g i o n . Depending on the v a l u e s of d and T , f o u r d i f -f e r e n t l o a d types were found to govern, these being i ) o s c i l l a -t o r y waves i n shallow water, i i ) b r e a k i n g waves, i i i ) e a r t h -quakes and i v ) o s c i l l a t o r y waves i n deep water. Between the zones where an i n d i v i d u a l l o a d case governs, t r a n s i t i o n zones are shown where two a d j a c e n t types are e q u a l l y l i k e l y t o govern. 5.14 Broad ranges of i n f l u e n c e of l o a d types: -In overview the p l o t i n F i g . 25 i s seen to h i g h l i g h t the f o l l o w i n g : 95.. i ) the dominant i n f l u e n c e of b r e a k i n g wave f o r c e s on s t r u c t u r e s w i t h depths l e s s than 90 f t . and to a c e r t a i n extent on those w i t h depths l e s s than 160 f t . i i ) the dominant i n f l u e n c e of earthquake loads on s t r u c t u r e s w i t h n a t u r a l p e r i o d s l e s s than 2 sec. i i i ) the importance of d e s i g n i n g on the b a s i s of p e r i o d i c deep water waves f o r s t r u c t u r e s with a d-T combina-t i o n f a l l i n g o u t s i d e i ) or i i ) . CHAPTER VI CONCLUSIONS 6.1 E f f e c t s of mass c o e f f i c i e n t : V i r t u a l mass e f f e c t s have to be examined i n d e t a i l i n determining response t o earthquake motion f o r bulky submerged s t r u c t u r e s w i t h l a r g e p e r i o d s . In the case o f such s t r u c t u r e s t h i s would n e c e s s i t a t e determining the v i r t u a l mass c o e f f i c i e n t s c o rresponding to the v a r i a b l e flow phases. A c o n c l u s i v e r e l a -t i o n s h i p of C M to the flow parameters was not e s t a b l i s h e d , although a p o s s i b l e one, based on l i m i t e d experimental data, has been suggested. I t was observed t h a t f o r some of the s t r u c t u r e s c o n s i d e r e d , the peak earthquake-induced displacements i n c r e a s e d by about 25 per c e n t f o r the h i g h e s t v a l u e s of C M over those f o r zero added C,,. 6.2 Shallow water waves: As regards water wave i n p u t s i n shallow water, l a r g e dynamic displacements would be s u s t a i n e d o n l y by f l e x i b l e s t r u c t u r e s with p e r i o d s w e l l over 2 seconds. The g r e a t e s t wave f o r c e s occur at or near the time of the passage of the c r e s t . 6.3 Load types governing d e s i g n : A g r a p h i c a l r e l a t i o n s h i p i s presented showing the l o a d type, such as earthquake o r wave f o r c e s , t h a t governs the de s i g n of o f f s h o r e s t r u c t u r e s . The two parameters t h a t govern the l o a d type are the n a t u r a l p e r i o d of the s t r u c t u r e and the water depth. The c h o i c e o f these as the b a s i c independent parameters and as being the p r i n c i p a l determinants of the comparative response under v a r i o u s types of e x c i t a t i o n , was based on the f o l l o w i n g c o n s i d e r a t i o n s : a) the fundamental p e r i o d T n i s the c h a r a c t e r i s t i c t h a t mainly i n f l u e n c e s dynamic response t o i n p u t s of d i f -f e r i n g f r e q u e n c i e s . b) the water depth d determines the maximum wave dimensions and the water co n t a c t area. In the case of o s c i l l a -t o r y waves, i t a l s o i n f l u e n c e s the r e l a t i v e magnitudes of h i g h e r harmonics of n o n l i n e a r waves t h a t induce s t r u c t u r a l resonance and moreover, i n the case of brea k i n g waves i t determines the v e l o c i t y d i s t r i b u t i o n i n the wave. The water depth a l s o i n f l u e n c e s the o v e r a l l s t r u c t u r a l s i z e and hence the n a t u r a l frequency. c) o t h e r m a t e r i a l parameters are r e f l e c t e d i n some form i n these two parameters. The r e s u l t i n g p l o t of the d-T space i n F i g . 25 d e l i n e a t e s the reg i o n s where v a r i o u s types of ocean waves and earthquake l o a d i n g would govern d e s i g n . The v a r i o u s r e g i o n s are l o c a t e d w i t h i n a p a i r of bounding l i n e s which c o n s t i t u t e a r e s t r i c t i o n on p r a c -t i c a l s t r u c t u r a l geometries. 98. From an o v e r a l l p o i n t of view the f o l l o w i n g broad trends appear i n the v a r i o u s areas of the p l o t : i ) the dominant i n f l u e n c e of bre a k i n g wave f o r c e s i n the d e s i g n of s t r u c t u r e s w i t h water depths l e s s than 90 f e e t , and a l s o to a l e s s e r e x t e n t , on those w i t h depths l e s s than 160 f e e t , i i ) the a p p r o p r i a t e n e s s of c o n s i d e r i n g earthquake loads i n the d e s i g n of s t r u c t u r e s with p e r i o d s l e s s than 2 seconds i i i ) the dominant i n f l u e n c e of p e r i o d i c deep water waves on o f f s h o r e s t r u c t u r e s i n the r e s t of the d-T r e g i o n . T h e - e f f e c t s o f other kinds of l o a d i n g such as dead loads and water c u r r e n t s can be superposed without a f f e c t i n g the r e l a t i v e preponderance of the e f f e c t of one of the above lo a d types. T h i s p l o t i s a u s e f u l a i d i n p r e p a r i n g a f i r s t d esign of a shallow o r deep water s t r u c t u r e of the p l a t f o r m deck type. -6.4 Other c o n c l u s i o n s : a) In the shallow water range, m a n i p u l a t i o n of p i l e s p acing would not s i g n i f i c a n t l y reduce wave response, but s t r u c t u r a l geometry and the de s i g n of s t r u c t u r a l modal f r e q u e n c i e s w i d e l y separated from the frequen-c i e s of the hig h e r waves would do so. 99. b) The s t e a d y - s t a t e response to waves as computed i s c o n s i d e r a b l y l e s s when the i n t e r a c t i o n between water and s t r u c t u r e v e l o c i t i e s i s c o n s i d e r e d than when i t i s i g n o r e d . c) Wave f o r c e s i n the large.wave-height range are pre-dominantly drag f o r c e s whereas f l u i d f o r c e s under an earthquake e x c i t a t i o n are mainly i n e r t i a f o r c e s . d) The ranges of water v e l o c i t y and p i l e diameter where the magnitude and frequency of l i f t are important have been s p e c i f i e d . Combined response i n l o n g i t u d i n a l and l a t e r a l d i r e c t i o n s t a k i n g i n t o c o n s i d e r a t i o n l i f t f o r c e s should be s t u d i e d f o r s t r u c t u r e s w i t h p e r i o d s g r e a t e r than 3 seconds. For the s t r u c t u r e s c o n s i d e r e d here the l i f t f o r c e s were n e g l i g i b l e . BIBLIOGRAPHY 1. Wiegel, R.L. Oceanographical e n g i n e e r i n g : P r e n t i c e - H a l l , Inc., Englewood C l i f f s , N.J.: 1964. , 2. Kinsman, B. Wind waves: P r e n t i c e H a l l , Inc., Englewood C l i f f s , N.J.: 1965. 3. Ippen, A.T. E s t u a r y and c o a s t l i n e hydrodynamics: E n g i n e e r i n g S o c i e t i e s Monographs: McGraw-Hill Book Co., Inc.: 1966. 4. Morison, J.R.. e t a l . The f o r c e e x e r t e d by s u r f a c e waves on p i l e s : Petroleum T r a n s a c t i o n s , A.I.M.M.E. V o l . 189, 1950. 5. Hino, M. A theory on the f e t c h graph, the roughness of the sea and, the energy t r a n s f e r between wind and wave: Proc. 10th Conference on C o a s t a l E n g i n e e r i n g , 1966. 6. B r e t s c h n e i d e r , C L . and Reid, R.O. Surface waves and o f f s h o r e s t r u c t u r e , e t c . : T e c h n i c a l r e p o r t , October, 1953, The Texas A. & M. Research Foundation. 7. C a m f i e l d , F.E. and S t r e e t R.L. Observations and experiments on s o l i t a r y wave deformation: 10th Conference on C o a s t a l E n g i n e e r i n g , V o l . I , 1966. 8. H a l l , M.A. L a b o r a t o r y study of b r e a k i n g wave f o r c e s on p i l e s : Beach E r o s i o n Board T. Memo. No. 106, August, 1958. 9. L a i r d , A. Water f o r c e s on f l e x i b l e o s c i l l a t i n g c y l i n d e r s : J o u r n a l Waterways and Harbors D i v i s i o n , A.S.C.E., V o l . 88, NO. WW3, August 1962. 10. Keulegan, G.H. and Carpenter, L.H. Forces on c y l i n d e r s and p l a t e s i n an o s c i l l a t o r y f l u i d : J o u r n a l N a t i o n a l Bureau of Standards, V o l . 60, No. 5, May 1958. 11. Sarpkaya, T. and G a r r i s o n C.J. Vortex formation and r e s i s -tance i n unsteady flow: J o u r n a l A p p l i e d Mechanics, Trans. A.S.M.E., V o l . 85, S e r i e s E, March, 1963. 12. Agerschou, H.A. and Edens, J . J . 5th and 1st order wave f o r c e c o e f f i c i e n t s f o r c y l i n d r i c a l p i l e s : C o a s t a l E n g i n e e r i n g , Santa Barbara S p e c i a l t y Conference, October, 1965. 101. 13. McNown, J.S. Drag i n unsteady flow: IX Congres I n t e r -n a t i o n a l de Mecanique Appliquee, A c t e s , Tome I I I , 1957. 14. McNown, J.S. and Keulegan, G.H. Vortex formation and r e s i s t a n c e i n p e r i o d i c motion: J o u r n a l E n g i n e e r i n g Mech. Div., A.S.C.E., V o l . 85, EM 1, P a r t 1, January, 1959. 15. Paape, A. and Breusers, H. The i n f l u e n c e of p i l e dimension on f o r c e s e x e r t e d by waves: Proc.. 10th Conference on C o a s t a l E n g i n e e r i n g , V o l . I I , A.S.C.E., 1967. 16. Blume, J.A., Corning, L.H. and Newmark, N.M. Design of m u l t i s t o r y r e i n f o r c e d concrete b u i l d i n g s f o r e a r t h -quake motions: Pub.'s P o r t l a n d Cement A s s o c i a t i o n . 17. Mason, M.A. Tax t r a n s f o r m a t i o n of waves i n shallow water: Proceedings of 1 s t Conference on C o a s t a l E n g i n e e r i n g , C o u n c i l on Wave Research, 1950. 18. I v e r s e n , H.W. Waves and breakers i n s h o a l i n g water: -Proceedings of 3rd Conference on C o a s t a l E n g i n e e r i n g , 1952. 19. Nakamura, M., S h i r a i s h i , H. and S a s a k i , Y. Wave decaying due to brea k i n g : Proceedings of 10th Conference on C o a s t a l E n g i n e e r i n g , A.S.C.E., 1966. 20. K i s h i , T. and S a e k i , H. The s h o a l i n g , breaking and run-up of the s o l i t a r y wave on impermeable rough s l o p e s : Proc. 10th Conference on C o a s t a l E n g i n e e r i n g , A.S.C.E., 1966. 21. L a i r d , A.D.K., Johnson, C.A. and Walker, R.W. Water f o r c e s on a c c e l e r a t e d c y l i n d e r s : J o u r n a l Waterways and Harbors D i v i s i o n , A.S.C.E., Proc. V o l . 85, No. W.W.I, 1959. 22. L a i r d , A.D.K., Johnson, C.A. and Walker, R.W; Water eddy f o r c e s on o s c i l l a t i n g c y l i n d e r s : J o u r n a l o f the H y d r a u l i c s D i v i s i o n , A.S.C.E., V o l . 86, No. HY9, November, 1960. 23. L a i r d , A.D.K. Eddy f o r c e s on r i g i d c y l i n d e r s : J o u r n a l of Waterways and Harbors D i v i s i o n , A.S.C.E., V o l . 87, No. W.W.4, November, 1961. 24. L a i r d , A.D.K. and Warren R.P. Groups of v e r t i c a l c y l i n d e r s o s c i l l a t i n g i n water: J o u r n a l of E n g i n e e r i n g Mechanics D i v i s i o n , A.S.C.E., V o l . 89, No. EM 1, February, 1963. 102. 25. L a i r d , A.D.K. Forces on a f l e x i b l e p i l e : A.S.C.E. S p e c i a l t y Conference on C o a s t a l E n g i n e e r i n g , Santa Barbara, C a l i f o r n i a , October, 1965. 26. Wiegel, R.L. Earthquake e n g i n e e r i n g : P r e n t i c e H a l l , 1970. 103. APPENDIX I CAUSES OF DISPARITIES BETWEEN WAVE FORCE COEFFICIENT DATA 1) Most experimenters used val u e s f o r water v e l o c i t i e s which were computed i n s t e a d of being measured d u r i n g the wave f o r c e experiments. The t r u e v e l o c i t i e s , . i t i s con-clu d e d , d i f f e r e d from the computed valu e s to v a r y i n g e x t e n t s . Another source of divergence was t h a t some experimenters measured the v e l o c i t i e s a t the c r e s t or a t some other p o i n t and r e l a t e d the drag c o e f f i c i e n t s to the measured ones. 2) Some experimenters used the l i n e a r theory i n computing v e l o c i t y values whereas o t h e r s used n o n l i n e a r t h e o r i e s . 3) N e g l e c t of the random v a r i a t i o n o f the wave form i n p r o t o t y p e t e s t s . 4) The e f f e c t of parameters, which c o u l d not be p i n -p o i n t e d , other than N . 5) U n c e r t a i n knowledge of the d i f f u s i o n of t u r b u l e n c e . 6) N e g l e c t of the c o n v e c t i o n a l terms of the a c c e l e r a t i o n l^jr'in the f o r c e e x p r e s s i o n s . 7) V a r y i n g roughness and f l e x i b i l i t y of the models and pr o t o t y p e s . 8) V i b r a t i o n s of t e s t p i l e s . 9) Turbulence around the s t r u c t u r e s by which the prototype t e s t p i l e s were supported. 104. APPENDIX I I WAVE THEORIES The time p e r i o d of ocean waves and t h e i r h e i g h t s have been c o r r e l a t e d e x p e r i m e n t a l l y with wind i n p u t s ( f e t c h e s , wind speeds and d u r a t i o n ) . For computations of the c h a r a c t e r i s t i c s of d e t e r m i n i s t i c wave f o r c e s , the p e r i o d and h e i g h t would be known independent data. L i n e a r Theory For a simple harmonic wave p r o g r e s s i n g i n the x-d i r e c t i o n w i t h phase v e l o c i t y C ^ as shown i n F i g . 26, the 2 d i f f e r e n t i a l equation t o be s a t i s f i e d f o r a l l x and w i t h i n - d < z < r i i s r i f t + i f i = o ( i i - D 8x 8x where <(> i s the v e l o c i t y p o t e n t i a l f u n c t i o n such t h a t H o r i z o n t a l v e l o c i t y u = — V e r t i c a l v e l o c i t y w = — r ^ - (II-2) oX o Z The boundary c o n d i t i o n a t the bottom i s : W = -|i = 0 on z = -d 8z (II-3) u = —1^ - = 0 on z = -d 3x The c o n d i t i o n on the upper boundary i s a mixed boundary c o n d i -t i o n : 105. z=-d  LINEAR WAVE Fig 26 106. - '|||' + \ ( u 2 + w 2) + || + gz| = 0 z=n p z=n z=n where p = f l u i d p r e ssure (zero a t f r e e surface) (p' = mass d e n s i t y of the f l u i d S ince 2 3cb • 2 ^ 3d) U 3t ; W K < 9t t h i s becomes n = — o n 2 = n g In the small-amplitude l i n e a r theory t h i s i s s i m p l i f i e d to rj = I l i on z = 0 (H-4) g 9t The g e n e r a l s o l u t i o n of equation ( I I - l ) i s of the form C 8 z -C 8z d> = C,+C„x+G z+(C.cosCoX+C-sinCoX)(C,e +C_e ) T X Z S 4 O D O D / or e q u i v a l e n t l y <J> = C 1+C 2x+C 3z+(C 4cosCgX+C 5sinC 8x) {C gcoshC 8 (z+C i : L) + C 1 ( ) s i n h C 8 (z+C 1 2) } Using the s p a t i a l p e r i o d i c i t y o f <j> a t i n t e r v a l s of l e n g t h L, c2 = 0 107. Having chosen the moving o r i g i n of c o o r d i n a t e s a t the c r e s t means C c = 0 c 4 t. o The zero net t r a n s p o r t of f l u i d i n the v e r t i c a l d i r e c t i o n y i e l d s c 3 = 0. Use of the c o n d i t i o n II-3 makes C10 = 0 C12 = d cj) = C/•C Qcosh{- 2^ (z+d)} s i n 4^ x Changing the moving o r i g i n of c o o r d i n a t e s to a f i x e d one, v e l o c i t y b e i n g ^ * ej) = C 4.C 9cosh{- 2^(z+d) }sin ( 4 £ x- ^-t) L H Use of II-4 and a v e l o c i t y u = ^ a t z = ^ y i e l d s i /-% H cosh 2TT (z+d)/L . ,2ir . 2m, x , T T * " C p h 2 s i n h 27rd/L S i n ( - J 7 X - T" t ) ( I I _ 5 ) where C p h = |. The v e l o c i t i e s " ~ are then d e r i v e d : 108. H o r i z o n t a l V e l o c i t y component 2Tr(z+d) c 2Tr(z+d) ° ' f l i ^ 2 W L c o s ( ^ K - ^ t ) ( I I - 6 , v e r t i c a l component w = f S ^ h 2 ^ d / L sin(^ x - ^ t ) ( I I - 7 ) H 2 2 Surface e l e v a t i o n n = ^ cos (-^ x - t) (II-8) cl 1 S p e c i a l i s i n g to the case of deep water (— > •=•) , 2 aT Length o f wave L^ = (II-9). Wave phase . ; C . = ^ v e l o c i t y ^ (11-10) i gT V a r i a t i o n s of p r e s s u r e w i t h depth are n e g l i g i b l e . F i n i t e - A m p l i t u d e Stokes Theory 2 3 T h i s theory ' s t a r t s with the assumptions t h a t the motion i s i r r o t a t i o n a l and both p o t e n t i a l <f> and stream f u n c t i o n T/J e x i s t . The f r e e s u r f a c e boundary c o n d i t i o n i s however d i f -f e r e n t from the l i n e a r case. The s o l u t i o n of V2. = 0 (11-11) s u b j e c t to the boundary c o n d i t i o n s as under, w i t h a moving system o f c o o r d i n a t e axes, . 109. z=n * l z = - d = k i g n + 1 [ ( | i , 2 + ( | i ) 2 ] . = k 3 Z=T] i s found to be T|;(X , Z) = C^z + C 2 +(C 3 cos C 7x + s i n C 7x) C z " C 7 z : 7 + C e (C 5e ' + u 6 e ) (H-12) Using the c o n d i t i o n s of 1) s p a t i a l p e r i o d i c i t y a t i n t e r v a l s of l e n g t h L 2) v e r t i c a l v e l o c i t y at bottom f o r d->°° being zero 3) h o r i z o n t a l v e l o c i t y a t bottom f o r d-*°° being zero C, = 0 6 C, = C , 1 ph where C ^ = speed of motion of the c o o r d i n a t e axes, T|;(X , Z) = ~ c p h z + C 2 + C 5 e K Z * C3 C O S K X + c 4 s i n K x) , 2 TT where = -^r-Since ip| = 0 f o r a l l n , z=n c 2 = 0 110. r ' — - — = -z + 3e C O S K X i s a p a r t i c u l a r s o l u t i o n i f c p h C 3 C 5 3 = (II-13) Ph and C 4 = ^ Using Cauchy-Riemann r e l a t i o n s , the p o t e n t i a l f u n c t i o n <j) i s found <b(x,z) , . K Z . = -x + £e sin<x. Ph From the e x p r e s s i o n f o r ^ , p u t t i n g z = n, n = 3 e K r i c o s K X . . . . 1 2 1 3 n = 3 t l+KTl+2" (KT|) +g-(KTl) + ]CQSKX (11-14) 2 3 Stokes' T h i r d Order Theory ' Approximate val u e s f o r n,ip and $ are g i v e n by t h i s theory c o r r e c t t o the t h i r d order i n 3. E x p r e s s i n g n as n = 3 n 0 + 3 n-L + 3 n 2 and s u b s t i t u t i n g 2 3 3 n 0 + 3 n-L + 3 n 2 = F ( 3 / n 0 ' n 1 f n 2 ' K ' x ) 2 R e t a i n i n g terms o n l y up to 3 , 111. 2 2 1 2 2 n Q + gn-L + 3 n 2 = [ c o s K x ] + 3 [ K n 0 c o s K x ] + 3 [ ( K O ^ ^ K n Q ) C O S K X ] 0 1 2 e q u a l i n g c o e f f i c i e n t s o f 3 , 3 a n d 3 r e s p e c t i v e l y , n 0 = C O S ' K X 1 ^ 1 „ ^1 = K n Q - c o s K X = 2"^  2K c o s 2 | < x 1 2 9 2 3 2 n 2 = ^ T 1 l + 2 K T 1 o ^ C O S k X = 8"K C O S K X + {j* cos 3 K X . Therefore n = \<$2 + 3 (1+-|K2 3 2 ) C O S K X + ^ K 3 2 cos 2 K X + | - K 2 3 3 C O S 3 K X (11-15) 9 2 2 The c o e f f i c i e n t of cos K X i s rearranged, l e t t i n g a = 3(l+-g1-K 3 ) 2 3 S o l v i n g f o r 3, 3 = e^a + e 2 a + e^a +... r E 2 F. 3 9K2,F 2 3,3 or a = ^ a + b 2 a + ^ a + ^ K ( ^ a + e ^ +e3a ) R e t a i n i n g terms upto 3rd power of a and equating c o e f f i c i e n t s of l i k e powers, 3, s^, e 2 and a r e o b t a i n e d , and then 1 2 1 2 3 2 3 h. = + A C O S K X + -^a C O S 2 K X + g-K a C O S 3 K X (11-16) 1 2 Choosing a new moving o r i g i n o f c o o r d i n a t e s a t z = ^ K 3 1 2 3 2 3 n = a C O S K X + C O S 2 K X + -g-K a cos 3K X (11-17) 2 3 S u b s t i t u t i o n of g = e^a + e 2 a + e 3 a 9 2 3 = a — g- K a 112. i n the expre s s i o n s f o r <j>, y i e l d <(>, and by d i f f e r e n t i a t i o n , the components of v e l o c i t y . The e x p r e s s i o n changes f o r f i x e d c o o r d i n a t e s to , 2TT,.,1 2 „, 2ir,, ,3 2 3 2 IT. » n = a C O S ( K X — — t ) + - ^ K a C O S 2 ( K X - — t ) + g - K a C O S 3 ( K X — ^ - t ) (11-18) Stokes 1 T h i r d Order T h e o r y — F i n i t e Depth S i m i l a r t o the - f o r e g o i n g d e r i v a t i o n f o r the case when d->-°°, a p e r t u r b a t i o n technique a p p l i e d to the s o l u t i o n s c{>, n and c e l e r i t y C ^ and a p p l i c a t i o n of the s u r f a c e and 1 3 bottom boundary c o n d i t i o n s y i e l d ' the g e n e r a l t h i r d order r e l a t i o n s : Wave p r o f i l e n = a c o s ( K X - ^ t ) + ^ f 2 c o s 2 ( K X - 2 ^ t ) + ^ - a - f 3 c o s 3 ( K x - ^ t ) (11-19) L ^ c - ^  /d x (2+cosh4-rrd/L)cosh27Td/L where ± 2 = f 2 (j-) = ^ 2. s m n —=— XJ f = f (^) = _ L l+8cosh 6 2fTd/L 3 _ 3 L 16 . ,6 n j /T sxnh 2Trd/L 2 Wave h e i g h t H = 2a + a 3 f (£) (11-20) XJ H o r i z o n t a l v e l o c i t y u = C . [F n c o s h 2 7 T T ( z + d ) cos ( K x - ^ t ) + F 0 c o s h 4 7 r T ( Z + d ) ph 1 L T 2 Li cos2 ( K X - ^ t ) + F 3 c o s h 6 7 T ] f Z + d ) cos3 ( K x - ^ t ) ] (H-21) 113. " 3u and h o r i z o n t a l l o c a l a c c e l e r a t i o n ' — d t |H = ! ! ^ F i C O S h ^ I s i n C ^ t j ^ F ^ o s h i l i ^ I s i n 2 ( K X - ^ t ) + i ^ £ h F 3 c o s h ^ ± d ) s i n 3 ( K X - ^ t ) (11-22) where F n = 2 ™ 1 L s i n h ; ^ £ o • A * T ' 2 4 L . ,4 2Trd s i n h —=— _ _3_ iZ-rra. 3 pll-^cosh-^n^--, *3 64 L . . 7 2ud J s i n h —=— " 0 , . _ 14+4cosh 2 c =/ 2 ± l t a n h ^ [ l + ( ^ ) 2 V % ] (11-23) 1 6 s i n h 4 ^ i i 2 4ird m 2 _ . _ _ 14+4cosh .212. L = 2 T _ t a n h 2 r f £ 1 + ,2™, 2 L , , ( I I . 2 4 ) 16smh - 7 = — The above development presupposes t h a t the wave l e n g t h L i s known. L i s to be found from the n o n l i n e a r r e l a t i o n a t (11-24) above. S o l i t a r y Wave Theory T h i s r e p r e s e n t s a wave w i t h the e n t i r e water body l y i n g above the o r i g i n a l water l e v e l , and mathematically the water p a r t i c l e s move o n l y i n the d i r e c t i o n of wave advance/.^ As F i g . 16- ' shows the wave l e n g t h i s i n f i n i t e . The equations f o r the 114, water p r o f i l e and wave v e l o c i t y a re: y s = d+H s e c h 2 [ / | -3 (x-Ct)] (11-25) d C = VgS ( 1 + f §) = Vd+f) (11-26) These are c o r r e c t to the 1st order and along w i t h a d d i t i o n a l e x p r e s s i o n s f o r the p a r t i c l e v e l o c i t y u are not adequate i n the v i c i n i t y of the c r e s t f o r l a r g e v a l u e s of the H r a t i o -r d JL_ = | s e c h 2 [ ^ (x-Ct)] (11-27) /glT d" Munk-McCowan S o l i t a r y Wave Theory T h i s theory''" i s more r e l i a b l e p a r t i c u l a r l y i n the v i c i n i t y o f the c r e s t of the wave f o r l a r g e v a l u e s o f ^ and prov i d e a b e t t e r f i t to the scanty experimental data. I t i s however more d i f f i c u l t i n computation and the s u r f a c e pressure i s not con s t a n t . C = ^dd+l) (11-28) 1 .MyN , ,MxN 1- cos (.—4r) cosh (-5-) H = n[~ —J± — - 5 — _ ] (11-29) C r /Myi , , ,Mx. n 2 [cos (-^-)+cosh (-^ -) ] ^  where M and N are found from the f o l l o w i n g : 115. | = | tan l l M U + l ) ] N = | s i n 2 [ M ( 1 + | | ) ] T h i s theory y i e l d s lower val u e s of the dimensionless v e l o c i t y ^ than the pre v i o u s one and i t s g e n e r a l i s e d t h i r d order form. C n o i d a l Theory T h i s i s a n o n l i n e a r theory"'' f o r permanent p e r i o d i c cl 1 1 waves i n shallow water where — < -=-?- to J a c o b i a n e l l i p t i c L -LU Zo f u n c t i o n s K ( k ) , E ( k ) , c n u and sn u appear i n the e x p r e s s i o n s , which are i n v o l v e d . The wave p e r i o d T and h e i g h t H are indepen-dent i n p u t s . Wave l e n g t h L i s g i v e n by k = — K(k) ( 2 L + 1 - - ^ ) " 1 / 2 (11-30) /I a i n which L and k are d e f i n e d by 2 equations as f o l l o w s : k 2 = <** y A / d ) - ( V d ) 2L+1-( Yt/d) y y y ( 2 L + 1 — ^ ) E ( k ) = ( 2 L + 2 ~ ' - -^)K(k) (Ref.l) where K(k) = complete e l l i p t i c i n t e g r a l of the 1st k i n d of modulus k y t = d i s t a n c e from the ocean bottom t o the trough y = d i s t a n c e from the ocean bottom to the c r e s t E(k) = complete e l l i p t i c i n t e g r a l o f the 2nd k i n d of modulus k. ' 116. An approximation t o L i s / 16d 3 L = v kK(k) (II-31) The wave p r o f i l e i n terms of y measured from the bed i s given s by Y s = y t + H cn 2[2K(k) - |) ,k] Y t , 2 r / 3 ( 2 L + l - - e r ) , L j = y t + H cn [/| -(x-|t),k] (11-32) i The wave v e l o c i t y C = / g d [ l + | (^ - f-S] (11-33) d k 2 v2 K(k) Water p a r t i c l e v e l o c i t y u: u - r l , 3 y t ^ f 3 H y t H . 2, . / i d * u 4d" ^ 2d' H 2 4. , 8HK 2(k) ,d y 2 w .2 2. . —j cn ( ) '- ( o ~ = ^ ) i - k sn ( ) 4d^ IT J ^ a c n 2 ( ) + c n 2 ( ) d n 2 ( ) - s n 2 ( ) d n 2 ( )}] where sn ( ) = s n [ 2 K ( k ) ( x - - h] e t c . (11-34) •Li x 117. APPENDIX I I I RELATIONS FOR THE THIRD ORDER RUNGE—KUTTA METHOD The system of equations to be s o l v e d are r e w r i t t e n i n the form of 1st order equations: i=l,....n ( I H - 1 ) i=l,.....n ( I H - 2 ) i s a v a i l a b l e from the computed valu e s of v a r i a b l e s of the preceding s t e p , and through the use of the equations of motion. For a succeeding time s t e p , the dependent v a r i a b l e u and i t s f i r s t d e r i v a t i v e are found as f o l l o w s : z ± ( t + A t ) = z ± ( t ) + + | K I 3 ( I H - 3 ) (At) z . (z. , z_, . . . z , u,,u_,...u ,t) 1 1 2. n 1 2 n ( A t ) z i ( z 1 + | K l l , Z 2 + | K 2 i - - - V l K n l ' U l + 3 q l l ' U 2 + I q 2 1 ' • • • u n + 3 q n l ' t + I ( A t > ) (At) z. ( 2 1 + § K 1 2 , Z 2 + f K 2 2 ' ' ' • Z n + § K n 2 ' U l + l q 1 2 ' U 2 + l < 3 2 2 ' , . .u n+|q n 2,t+|(At)) (III-4) U i ( t + A t ) = u ± ( t ) + \ q±1.+ | q i 3 (III-5) u. u, z . 1 dz, dt where K i l i 2 K i 3 118 where q i l = ( A t ) z i q i 2 = (At) (z. + l K i l ) q i 3 = (At) (z± + § K ± 2 ) ( I I I - 6 ) S u b s t i t u t i n g (ITI - 6 ) i n t o (III- 4 ) and ( I I I - 5 ) , ( I I I - 5 ) • i s r e w r i t t e n u ± ( t + A t ) = u ± ( t ) + ( A t ) z ± ( t ) + | ( A t ) K ± 2 ( I H - 7 ) and K ± 1 = ( A t ) z ± [ z 1 ( t ) , z 2 ( t ) , z 3 ( t ) . . . z n ( t ) , u x ( t ) , u 2 ( t ) , . . . u n ( t ) , t ] K i 2 = ( A t ) Z i [ Z l 4 K l l ' Z 2 + I K 2 1 ' ' • • V k l ' V l ( A t ) V u 2 +| (A t ) z 2 . , . . .u n+i (At ) z n , t + | ( A t ) ] K ± 3 = ( A t ) z i [ z 1 + | K l 2 , z 2 + § K 2 2 , . . . z n ; V . . X : ^ + | c n 2 , 2 2 2 2 (At) z 1+^-(At) K 1 1,U 2 + - 3 (At) z 2+-g(At) K 2 1 , . . .u +|(At) z +f (At) K -j , t+| (At ) ] ( I I I - 8 ) n 3 n 9 n i 3 ( I I I - 3 ) , ( I I I - 7 ) , and ( I I I - 8 ) a re e x p l i c i t f o r m u l a e . 119. The above s e t of equations are p a r t of one form of the t h i r d order Runge^Kutta r e l a t i o n s f o r a p a r t i c u l a r c h o i ce of parameters. • 

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