Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1971_A7 S45.pdf [ 5.25MB ]
Metadata
JSON: 831-1.0050553.json
JSON-LD: 831-1.0050553-ld.json
RDF/XML (Pretty): 831-1.0050553-rdf.xml
RDF/JSON: 831-1.0050553-rdf.json
Turtle: 831-1.0050553-turtle.txt
N-Triples: 831-1.0050553-rdf-ntriples.txt
Original Record: 831-1.0050553-source.json
Full Text
831-1.0050553-fulltext.txt
Citation
831-1.0050553.ris

Full Text

FLUID-DYNAMIC EFFECTS ON THE RESPONSE OF OFFSHORE TOWERS TO WAVE AND EARTHQUAKE  FORCES  by ASOKE KUMAR SEN B. T e c h  (Hons.), I n d i a n I n s t i t u t e o f Technology, K h a r a g p u r , I n d i a , 1957  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in  t h e Department of  C I V I L ENGINEERING  We a c c e p t t h i s t h e s i s required  THE  as c o n f o r m i n g t o t h e  standard  UNIVERSITY OF BRITISH COLUMBIA J u l y , 1971  In p r e s e n t i n g  this  thesis  ments f o r  an  Columbia,  I agree that  able  in partial  advanced degree a t  for reference  for  extensive  may  be  and  the  g r a n t e d by  the  Library  study.  copying of  this  my  thesis  written  for financial  Civil  that  gain  Date J u l y f? ,  1971  for scholarly  Columbia  not  be  by  require-  British  agree that  copying or  shall  of  the  make i t f r e e l y  Department o r  Engineering  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  shall  thesis  permission.  Department of  University  I further  Head o f my  s e n t a t i v e s . . I t i s understood this  the  f u l f i l m e n t of  avail-  permission purposes  his  repre-  publication  allowed  of  without  (i) ABSTRACT The structures in  evaluation  i n a fluid  ted  flow  to f i n d  a general  cylinders  under c o n d i t i o n s  In this  study  f o r v a r i a b l e flow, v e l o c i t y flow,  motion.  The p a r a m e t e r s t h a t  using  uniformly  significance Accurate  of variable  knowsepara-  an a t t e m p t has b e e n made  method o f e v a l u a t i o n  constant  forces  on v i b r a t i n g f r a m e d  i n ocean e n g i n e e r i n g .  forces  i s lacking.  forces  environment i s o f c u r r e n t  view o f the a c t i v i t y  ledge o f the f l u i d  of f l u i d  of fluid  published accelerated  forces  on .  d a t a from t e s t s o f flow  and wave .  appear t o govern"the v a r i a b l e  a r e d i s c u s s e d , . a n d models f o r r e l a t i n g  force  flow  magnitudes  to these parameters are suggested.  The  dynamic r e s p o n s e o f f r a m e d s t r u c t u r e s  environment has n o t been i n v e s t i g a t e d e x c e p t s o i d a l wave m o t i o n i n deep w a t e r c o n d i t i o n s . shallow water s t r u c t u r e s well  on  for linear  taking  fluid  t y p e s o f wave f o r c e s , a s  motions.  The e f f e c t o f t h e mass and d r a g p a r a m e t e r s '  r e s p o n s e has been s t u d i e d .  f o r the design  structural  numerically  i n t o a c c o u n t t h e i n t e r a c t i o n between t h e s t r u c t u r e  the s t r u c t u r e  cases  sinu-  The r e s p o n s e o f  as t o e a r t h q u a k e e x c i t a t i o n , h a s b e e n a n a l y s e d  here, and  to various  i n an o c e a n  period  o f framed  and w a t e r  structures  depth.  Governing  load  have b e e n r e l a t e d t o  (ii)  TABLE OF  CONTENTS Page  ABSTRACT  ;(i)  TABLE OF CONTENTS  (ii)  L I S T OF TABLES  (vi)  L I S T OF FIGURES  (vii)  ACKNOWLEDGMENT  (ix)  CHAPTER I.-  II.  INTRODUCTION  1  1.1  Scope  1  1.2  Fluid  1.3  Earthquake problem  3  1.4  Wave p r o b l e m  3  1.5  Simplifications  3  1.6  O r g a n i s a t i o n o f the t h e s i s  4  interaction  1  FLUID FORCES  6  2.1  Cylindrical  2.2  B a s i c f l o w phenomena  2.3  Drag f l u c t u a t i o n s  2.4  Lift  2.5  Problems  2.6 2.7  C  : D  pile  - steady  f o r c e formula  8  - steady  flow  flow  i n unsteady  Coefficient  7  10 12  flow  o f drag  12 13  L i m i t s f o r p r e - s e p a r a t i o n s t a g e and associated  values  15  (iii) 2.8  Past.experimental data  15  2.9  Experiments f o r  16  i n o s c i l l a t o r y waves  2.10 R i g i d  cylinders  i n s t a n d i n g waves - mean  2.11 R i g i d  cylinders  i n s t a n d i n g waves -  varying  <_  19  D  2.12 L i m i t a t i o n s  of varying  2.13 T e s t s on r i g i d acceleration 2.14 C,.: M  values  model c y l i n d e r s  22  i n constant  flow  23  C o e f f i c i e n t o f mass  23  2.15 R i g i d  cylinders  i n s t a n d i n g waves - mean  2.16 R i g i d  cylinders  i n s t a n d i n g waves -  varying  C  model c y l i n d e r s  constant acceleration 2.18 O s c i l l a t e d f l e x i b l e  under  from r e s t  cylinders  27  - mean  resistance in flexible  25  26  M  2.17 E x p e r i m e n t s on r i g i d  2.19 L i f t  16  27 oscillating  cylinders  30  2.2 0 F r a m i n g o f r e l a t i o n s f o r i n s t a n t a n e o u s variable  d r a g and mass c o e f f i c i e n t s  2.21 C h o i c e o f p a r a m e t e r s 2.22 E m p i r i c a l  31  c o e f f i c i e n t o f mass i n a r b i t r a r y  motion 2.23 E m p i r i c a l motion  31  31 c o e f f i c i e n t of drag  2.24 F o r c e r e d u c t i o n s cylinders  in arbitrary 35  due t o n e i g h b o u r i n g 37  (iv) III.  WAVE THEORIES  38  3.1  Wave t h e o r i e s  38  3.2  Wave h e i g h t and p e r i o d  39  3.3  Ranges o f a p p l i c a b i l i t y  43  3.4  Characteristics  3.5  Additional  o f Stokes  limit  theory  of v a l i d i t y  of third  order Stokes theory  IV.  V.  46  51  3.6  B r e a k i n g waves - s o l i t a r y  wave t h e o r y  3.7  Impact t y p e o f b r e a k e r f o r c e s  54  3.8  Determinants  54  3.9  Earthquake  3.10  Comparative  of breakers -  motion ground  52  56 accelerations  DYNAMIC RESPONSE PROBLEM  56 58  4.1  Origin of nonlinear  terms  4.2  Assumptions  58  4.3  Basic  59  4.4  Earthquake  4.5  Further simplifications  61  4.6  Method o f s o l u t i o n  62  4.7  Wave f o r c e  62  4.8  Wave r e s p o n s e c o m p u t a t i o n s  formulation inputs  60  input  RESULTS OF COMPUTATIONS 5.1  Choice of structures earthquake  5.2  Earthquake  response  response  58  63 64  f o r evaluating 64 66  (v) 5.3  Effect  of s t r u c t u r a l  5.4  Effect  of C  5.5  Relevance  5.6  Dynamic r e s p o n s e  shape  72 72  D  of s u b c r i t i c a l  region  75  to f i n i t e - a m p l i t u d e  S t o k e s waves 5.7  Computed r e s p o n s e  5.8  F o r c e v a r i a t i o n s w i t h time  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  5.10 . S u p e r c r i t i c a l  t o S t o k e s waves  79 81  forces  83  flow conditions  85  5.11 K e u l e g a r i p a r a m e t e r  85  5.12 B r e a k i n g wave  85  5.13 C o m p a r a t i v e 5.14 B r o a d VI.  75  ( s o l i t a r y wave) r e s p o n s e  f o r c e s under  ranges  various excitations  of influence of load  types  86 94  CONCLUSIONS  96  6.1  Effects  o f mass c o e f f i c i e n t  6.2  Shallow water  6.3  Load  6.4  Other  96  waves  96  types governing design  96  conclusions  .  98  BIBLIOGRAPHY  100  APPENDIX  I  103  APPENDIX  II  104  APPENDIX.Ill  .  117  (vi)  LIST.OF  TABLES  TABLE  Page  1.  Wave F o r c e C o e f f i c i e n t s  ...........  2.  V a l u e s o f C,„ a n d C^ f o r S t a n d i n g Waves M D '  3.  Range o f A p p l i c a b i l i t y  4.  Periods  5.  Range o f P a r a m e t e r s  6.  E a r t h q u a k e Response  7.  Damping E q u i v a l e n t o f D r a g  73  8.  Water I n e r t i a a n d D r a g  74  9.  Parameters f o r F i n i t e - A m p l i t u d e  o f Wave T h e o r i e s  a n d Mode Shapes  .  . . .  17 20 48 68  .  70 .  to Finite-Amplitude  Wave Response  ...  Wave I n p u t  71  78  10.  Response  11.  P a r a m e t e r s f o r B r e a k i n g Wave F o r c e s  87  12.  L o a d i n g Due t o B r e a k i n g Waves  88  13.  Comparative  F o r c e s Under  14.  Comparative  Stresses  15.  G o v e r n i n g Load Cases  Various Excitations  f o r Water S t r u c t u r e s  80  ....  89  .  91 92  (vii)  L I S T OF FIGURES FIGURE  Page  1.  Flow diagram  2.  Separated  f l o w phenomena - s h e a r  3.  Separated  f l o w phenomena - I I  4.  Separated  f l o w phenomena - v o r t e x  5.  C i r c u l a t i o n around c y l i n d e r  11  6.  Variation of C  14  7.  Scatter of values  8.  Variation  9.  . ... - I  9 9  s t r e e t - I I I ..  D  of C  of C and M 1A  D r a g and l i f t  layers  2  D  9  18  D  w i t h |D  24  force ratios  28  10.  Predicted values of C  34  11.  Wave h e i g h t and p e r i o d  12.  Wave l e n g t h and p e r i o d . . .•  13.  Wave l e n g t h and w a t e r d e p t h  45  14.  Coefficient  47  15.  Ranges o f t h e o r i e s  49  16.  S o l i t a r y wave  53  17.  Impact m o d e l  53  18.  Breaker  55  19.  Structures analysed  20.  Key t o mode s h a p e s  21.  Structures analysed  f o r waves  22.  Structures analysed  f o rbreakers  M  41 45  c  ranges f o r earthquakes  65 .  .. . . i . . . .  67 76 77  (viii)  FIGURE  Page  23.  Wave f o r c e  24.  Pile  moment h i s t o r y  25.  Load  types  26.  Linear  wave  history  82 ......... ...  84 93 105  (ix)  ACKNOWLEDGMENT  The Dr.  Donald  tance  i s indebted to h i s thesis supervisor,  L. Anderson f o r h i s i n s p i r i n g  and g u i d a n c e  been b r o u g h t his  writer  a t every  to t h i s  and v a l u a b l e  s t a g e o f t h i s work.  assis-  The work has  stage o f c o m p i l a t i o n p r i m a r i l y  because o f  c o n s t a n t s u p p o r t and e n c o u r a g e m e n t .  Financial research  support  assistantship  from  f o rthe w r i t e r  i n t h e form  an N.R.C. g r a n t i s a l s o  of a  acknowledged.  CHAPTER I  INTRODUCTION 1.-1  Scope:  The structures  problem o f v i b r a t i o n s induced i n o f f s h o r e  by deep w a t e r waves h a s b e e n e x t e n s i v e l y  I n v i e w o f t h e i n c r e a s i n g numbers o f s u c h s t r u c t u r e s designed  and c o n s t r u c t e d  investigate  the forces  i t was c o n s i d e r e d  c a u s e d by o t h e r  namely, a n e a r t h q u a k e i n p u t which d e p a r t  significantly  the usual  the  only  problem o f e v a l u a t i o n  realistic 1.2  water flow  the response  range o f s t r u c t u r e s structures  are considered.  conditions  The m a g n i t u d e s o f t h e r e s p o n s e  o f e x c i t a t i o n a r e compared w i t h  Tower-supported p l a t f o r m tures  dynamic e x c i t a t i o n s ,  from t h e assumptions o f the l i n e a r  l i n e a r d e e p w a t e r waves and w i t h forces within  being  desirable to  and t h o s e o c e a n wave  s m a l l - a m p l i t u d e wave t h e o r i e s . to these kinds  studied.  the response t o to breaking  wave  i n shallow  water.  and s i m i l a r f r a m e d  struc-  Attention  has a l s o been g i v e n t o  o f the hydrodynamic  forces  under  conditions.  Fluid interaction: A  forces  flow  diagram o f the i n t e r a c t i o n between t h e water  and t h e s t r u c t u r a l r e s p o n s e  i s given  i n F i g . 1.  WAVE RESPONSE  Wave P e r i o d Wave H e i g h t Water D e p t h Bed S l o p e Roughness  FLOW  PROBLEM  OCEAN WATER  DIAGRAM  Water Velocities Water Accelerations  SYSTEM  Structure Response  FORCE GENERATION (Structure Geometry)  H y d r a u l i c Forces on — Structure —  STRUCTURAL SYSTEM  "7K  Final ->  (Displs., Velocities Forces)  Output  1 <-  -Hydrodynamic Mass and D r a g  EARTHQUAKE  Ground FORCE Motion  GENERATION  Effects-  PROBLEM  Equivalent Dynamic - F o r c e s on — Structure  Structure STRUCTURAL SYSTEM  Final Output  Response  Hydrodynamic Mass and D r a g  Fig.  1  Effects  to  3. 1.3  Earthquake In  force of  the case o f earthquake-caused v i b r a t i o n s , the  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 ,  recorded  This  problem:  ground  equivalent  a c c e l e r a t i o n values  dynamic  s y s t e m as an i n p u t .  force  Since  inertia  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  the s t r u c t u r a l motions  i n t e r a c t i o n w i t h t h e water and  t o d i s c r e t e masses.  i n the form o f hydrodynamic  f o r c e s , the f i n a l  o f the hydrodynamic  generated/influenced  by t h e r e s p o n s e .  active  themselves t o the other  Detailed  attach  expressions  given  later  drag e f f e c t s are nonlinear. inertia 1.4  e f f e c t s are also  Wave  inputs  i n that  relative  The h y d r o d y n a m i c  show t h a t  In general  inputted  forces.  t h e hydrodynamic  nonlinear.^  required  f o r t h e wave r e s p o n s e  the primary forces  to the s t r u c t u r e motion.  The w a t e r  o f wave p e r i o d ,  roughness  wave h e i g h t ,  and c o n f i g u r a t i o n  motion  v e l o c i t y and  using  inputs  problem  a r e c a u s e d by w a t e r  a r e c a l c u l a t e d f r o m one o f s e v e r a l wave  1.5  inter-  the hydrodynamic  acceleration  slope,  forces  problem: The  differ  drag  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  forces  c a u s e an  water  theories  d e p t h and t h e  of the bed.  Simplifications: Numerical studies  taking  the i n i t i a l  inputs  have b e e n  conducted i n both  as d e t e r m i n i s t i c .  The  problems,  hydrodynamic  4. interaction  f o r c e s were m o r e o v e r s i m p l i f i e d  dimensional  transverse flow past c i r c u l a r  a s s u m i n g two-  cylinders  t o be  applicable.  1.6  O r g a n i z a t i o n o f the t h e s i s :  The of  steady  second  chapter  and u n s t e a d y  fluid  i n p u t d a t a on t h e v e l o c i t i e s particles the  relative  fluid-induced  relating  experimental is  highlighted.  for  coefficient  The d i m e n s i o n l e s s  data a r e l a t i o n s h i p i s proposed  The  third  c r e a t e d by v a r i o u s t y p e s  quantitative velocities  discussed,  i n the unsteady evidence  and t h e need  area.  available f o r an  drag  and mass  coefficients  on a r e a n a l y s i s  f o r the instantaneous  of past mass  parameters:  d i s c u s s e s the flow c o n d i t i o n s  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 -  ground motion.  This chapter  presents  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 o f f l u i d  particle  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 .  applicability  Both  of-an a r b i t r a r y flow-history  i n terms o f t h e f l o w  chapter  o f the earthquake  is first  and e x p e r i m e n t a l  f o r the case  The n a t u r e o f  forces f o r progressively  t h e f o r c e s a r e i n t r o d u c e d and b a s e d  experimental  tion  o f such  forces are presented  approach  members when  and a c c e l e r a t i o n s o f t h e w a t e r  forces f o r steady.flow  (qualitative)  t o such  the determination  f o r c e s on c y l i n d r i c a l  complexities o f the flow  theoretical  with  t o t h e member a r e s u p p l i e d .  f o l l o w e d by an e x a m i n a t i o n increasing  i s concerned  The  o f t h e 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  t h e o c e a n - s t r u c t u r e g e o m e t r y , e t c . and t h e need f o r t a k i n g  into  account  t h e v a r i o u s k i n d s o f waves a r e s e t o u t .  The  f o u r t h chapter  formulates  the equations of  motion o f the s t r u c t u r e under earthquake force  and dynamic wave  inputs. The  results.of  response  computations  u n d e r a) e a r t h -  quake i n p u t s , b) s h a l l o w w a t e r n o n l i n e a r o s c i l l a t o r y inputs, fifth  and c) b r e a k i n g wave i n p u t s a r e s t a t e d i n d e t a i l  chapter.  The r e s p o n s e  v a r y i n g values o f drag, The  wave  o f the s e l e c t e d  i n the  structuresfor  mass and o t h e r p a r a m e t e r s a r e compared.  s t r u c t u r e f o r c e s u n d e r v a r i o u s k i n d s o f e x c i t a t i o n and  l o a d i n g a r e compared.  In the l a s t findings for  are given.  chapter,  t h e c o n c l u s i o n s and a summary o f  The d e s i g n c r i t e r i a  various ocean-structure  situations  which would  are indicated.  govern  CHAPTER I I  FLUID FORCES This  chapter  f o r c e s on c y l i n d r i c a l adjacent the  fluid.  These  ficients  inertia  o f drag  proportionality  For c y l i n d r i c a l drag  to the f l u i d  separation, that  for specific  and i n e r t i a  i n separated  the  experimental  as s t e a d y  c h a p t e r (Ch.  exist  i n both  the e a r t h -  with  occurrence of  I t i s pointed out i n the chapter  n o t s u s c e p t i b l e t o an a n a l y t i c a l  well  o f water  The m o t i o n has an a r b i t r a r y  and  flow a r e time-dependent  solution.  d e t e r m i n a t i o n o f the average  mass c h a r a c t e r i s t i c s  power)  flow conditions.  and i s o s c i l l a t o r y ,  i n the l a t t e r .  the drag  (to t h e second  The k i n e m a t i c s  flow c h a r a c t e r i s t i c s  i n the former  coef-  to the determination o f drag  quake and wave f o r c e s i t u a t i o n s . character  Dimensionless  s e p a r a t e l y i n the next  characteristics  Unsteady  component and an a c c e l e r a -  velocities  the f o r c e problem reduces  inertia  i n calculating  appear w i t h i n the c o n s t a n t s o f  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 .  Ill),  motion of the  members t h e f o r c e s  f o r c e component.  and i n e r t i a  motion being d e a l t w i t h  and  f o r c e s a r e t o be u s e d  of a velocity-dependent  tion-dependent  t o the d e t e r m i n a t i o n o f the  members due t o r e l a t i v e  s t r u c t u r e response.  consist  and  i s devoted  under s p e c i f i c  flow are i n d i c a t e d .  types  Methods u s e d i n drag  and  of unsteady  average f l o w as  The f l o w p a r a m e t e r s  used  7. in  the  determination  selected of  the  2.1  on  the  of  basis  the  of d i m e n s i o n a l a n a l y s i s  e x i s t i n g experimental  Cylindrical  pile  force  In computations waves and total forces  c o e f f i c i e n t s o f mass and  to o t h e r  force  types of  forces  on  a c y l i n d e r due  to  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 a superposition  of  d r a g and  inertia  s u c h that"'"  forces 1)  2)  being  inertia  of  the  "A  drag  due  the  =  F (t) ] ;  +  F (t) D  respectively  An  of  force F  fluid F D  (t)  and  represented  comprising  pressure  to the  (t) a r i s i n g  existence  of  an  square of  The  above s u p e r p o s i t i o n  ( t )  F (t) D  added mass. and  u p s t r e a m and wake.  In  the  the  portion  downstream range  of  i s proportional  past  c o n c e p t i s however t r u e  to  =  only  cylinders.  f o r i n d i v i d u a l terms  i  acceleration  the v e l o c i t y .  two-dimensional flow  F  of  friction  interest, this  the  Expressions  the  out  by  viscous  differential  R e y n o l d s numbers o f  for  regression  formula:  F(t) the  are  data.  f o r the  i s t a k e n t o be  and  drag  C  M  = |c  p  D  v  o  TE  p A v|v|  are:  (  2  -  1  }  (2.2)  8. where  VQ  =  Enclosed  A  =  P r o j e c t e d a r e a o f the  v  =  Relative velocity  volume o f  fluid  particles  to  t h a t of  be  surrounding  fluid  the u n d i s t u r b e d  particle flow of  C  =  A dimensionless  coefficient  of  =  Mass d e n s i t y o f t h e  D  flow  steady  flow  fluid.  to gain g r e a t e r i n s i g h t  fluid  into  the drag  results  fluid  4).  i n their  As  flow  flow,  succeeding  appear i n the First,  of f l u i d  sections. case  flow Dis-  of motion  of  i n this section,  being  s e p a r a t i o n o f the  leads to the  velocity  d i s t a n c e downstream o f vortices  the  and  past a cylinder.  the boundary  2 to  in this  and  i s considered.  For r e a l tact with  the  phenomena:  f e a t u r e s of  viscous  velocity  drag  forces, certain physical characteristics  a real  the  fluid. o f mass  continuous  street,  assuming  coefficient  examined-in d e t a i l  (Figs.  between t h e member and  A dimensionless  In order  are  member  =  Basic  inertia  member  C»„ M  p 2.2  the  of  detached,  flow  formation  of  layer  vortices  i s increased, i t eventually  giving  1 to 4 diameters.  rise The  t o a wake f o r a discharge  [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 F i g . 4] o c c u r s ,  n a t e l y - . ' from o p p o s i t e d e t e r m i n e d by  the  at a s u f f i c i e n t l y edges o f  the  i n con-  S t r o u h a l number S g i v e n  vortex  high v e l o c i t y ,  cylinder by  at a  of  alter-  frequency  SEPARATED  FLOW  Fig  4  PHENOMENA  10,  s where  = s  1  -  1  -  9  N  ^  7  (2.3)  R  f  =  frequency o f shedding o f a p a i r  D  =  diameter  N„ R  =  vD — v  v  =  — =  =  vDp — - = u  _ ,, Reynolds J  of eddies  ... number.  kinematic v i s c o s i t y .  O v e r t h e r a n g e o f i n t e r e s t t h e number S i s 0.21. tinuity layers of  represented  stream. brings  2.3  about t h e major p a r t  lead  drag  steady  (and i n l i f t )  culation  f o r the a f o r e s a i d  down-  t o the pressure d i f f e r e n t i a l i t also  edges  that  accounts f o r  flow: high  t h e mechanism f o r f l u c t u a t i o n s i n  i s indicated,  flow c o n f i g u r a t i o n  with reference  i n F i g . 5.  and v e l o c i t y a r e , a c c o r d i n g  drag.  distance  f l o w when t h e v e l o c i t y i s s u f f i c i e n t l y  a c c o m p a n i e d by f l u c t u a t i o n s dinal  on t h e o p p o s i t e  i n t h e d r a g f r o m i t s mean v a l u e .  to vortex-shedding,  transient  points  of the drag;  Drag f l u c t u a t i o n s — s t e a d y In  to  and e x t e n d i n g  The wake c o n t r i b u t e s  fluctuations  discon-  i n t h e wake downstream i s bounded by s h e a r  s t a r t i n g from t h e s e p a r a t i o n  the c y l i n d e r  The  Fluctuations  to B e r n o u l l i ' s  i n pressure,  An i n d i v i d u a l v o r t e x  t o the changing  and hence  in cirequation,  i n longitu-  causes a complete  cycle i n  9 22 2 3 the  history of longitudinal  b e c a u s e t h e r i g h t and l e f t i d e n t i c a l wake  (Fig. 5).  forces vortices  (drag). ' dissipate  The f l u c t u a t i o n s  '  This  occurs  i n a longitudinally  i n drag, which can  11.  Clockwise circulation round cylinder  CIRCULATION  AROUND  Fig  5  CYLINDER  amount t o a s much as 60 p e r c e n t  J1 o f =— T  frequency  2.4  where T  Lift—steady  = time  circulation  f o r discharge  occur  at a  o f one eddy.  22  flow:  A comment on l i f t In the realm  o f t h e mean d r a g ,  f o r c e s i n steady  of post-separation velocities, around  flow  i s i n order.  arising  t h e . c y l i n d e r ( F i g . 5) l i f t  out of the  f o r c e s are genera-  ted  t r a n s v e r s e t o the flow, b e i n g p r o p o r t i o n a l i n magnitude t o  the  square  forces. make  o f the v e l o c i t y  and b e i n g  o f the order o f the drag  The c y c l i c r e v e r s a l s o f c i r c u l a t i o n  the l i f t  forces reverse  described previously  c y c l i c a l l y a t a frequency  of  ^ e  ("the of  the v o r t e x  force 2.5  Strouhal frequency").  Two s t a g e s  l a y o u t a r e needed  of transverse  t o complete  asymmetry  a c y c l e i n the l i f t  history. P r o b l e m s i n unsteady f l o w : . Variability  flow with  o f the flow p a r a m e t e r s " i s  a time-dependent v e l o c i t y .  a l s o found f o r  Observations  a r e as  follows: a)  The l i m i t i n g  b)  Positions along are  c)  N_. f o r s e p a r a t i o n i s t i m e - d e p e n d e n t . t h e b o u n d a r y where  separation.occurs  time-dependent.  The wake g e o m e t r y i n f l u e n c e s t h e d r a g more It  i s a function of velocity,  cosity  cylinder  and d e g r e e o f t u r b u l e n c e .  drastically.  diameter,  vis-  d)  Fluctuations of the instantaneous type o f flow, a r e more  The study  inertia  drag  f o r c e s ) from  (and, i n t h i s  i t s mean  irregular.  c o m p l i c a t i o n s i n v o l v e d i n an a t t e m p t  c a n be s e e n  value  from  the f a c t  at analytical  t h a t when t h e f l o w r e v e r s e s , t h e  e r s t w h i l e wake becomes t h e u p s t r e a m s i d e o f t h e c y l i n d e r . titative the  2.6  knowledge r e g a r d i n g t h e f l o w and f o r c e s i s l a c k i n g f o r  general case  of arbitrary  C _- Coefficient  The  following sections w i l l  characteristics For  o f the c o e f f i c i e n t  steady  and N  teristics.of  (with s e p a r a t i o n ) .  be c o n c e r n e d  flow a c o r r e l a t i o n  the e x p e r i m e n t a l drag  coefficient  the p r a c t i c a l 10  4  plot  with the  C^. between t h e d r a g  i s well-established (Fig. 6).  The in  acceleration  of drag:  D  ficient C  Quan-  coef-  The c h a r a c -  ( F i g . 6) a r e a s f o l l o w s : i s nearly constant  a t 1.2  range o f  < N  < 5 x 10  5  R  except  f o r a drop  t o a minimum o f 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.).  R e y n o l d s numbers t o a l i m i t  I t rises  o f 10.  f o r low  10  10*  2  10 NR  VARIATION OF Cp  Fig  6  10  6  >  7  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 associated  values:  While N N  vortex-shedding  o f the o r d e r K  - 1.2  R  from  x 10  rest.  and  of  4  50,  flow s t a r t s  at  the p o i n t of s e p a r a t i o n occurs  at  i n the case  Thus t h e  i n steady  of constant  instantaneous  acceleration  value  of N  an  starting  alone  i s not  an  R adequate parameter f o r d e t e r m i n i n g two in  c o n d i t i o n s are proposed variable  separation.  The f o l l o w i n g  as a means o f p r e d i c t i n g  separation  flow: v rricix T  i)  ii)  oscillatory  flow:  parameter — ^  = 15,  the diameter  and  T the o s c i l l a t i o n  period,  other:  N  = 1000  R  combined w i t h  where D i s  an o v e r r i d i n g  limit  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 vn al curre t u es t or fo k Ce . p r i o r  t o s e p a r a t i o n i s t h a t due  D  friction  drag  alone  and  Keulegan,"*"^ f o r waves. the v e l o c i t i e s with drag  in this  Past  forces  range are  as  low  to the  and  so t h e  found  to by  earthquakes  force associated  force;  i s not r e q u i r e d i n t h i s  range.  thus  a high  data:  experimental  i n unsteady  = 1 t o 2,  inertia  experimental An  from  I n g e n e r a l , f o r waves and  i s s m a l l compared  degree of accuracy 2.8  ranges  the  a p p r o a c h has  flow with  t o be  separation.  r e s o r t e d to f o r  In succeeding  sections  16. f l o w phenomena i n s p e c i f i c by  p a s t i n v e s t i g a t o r s are o f m o t i o n and  observations  yield  necessary  inertia  2.9  and  so as drag  insight  into  Turning motion  type  to attempt  for  passes  immersed  flow,  the  cylinder.  accelerations  case  are  At  flow, this  basis  show w i d e s c a t t e r when p l o t t e d  in  Table  factors  with  Q  1.  The  listed  the r e s p e c t i v e data disparities  i n Appendix  I.  para-  parameters  a relation  for  waves: on  coefficient total  instant  among t h e s e  total  the  2.10  cylinders Rigid  increases  of  on  this  (Fig. 7). are  values  cylinders  zone N  i n standing waves—mean  The  tabulated  a r e due  I t i s a l s o commented  i n the  crest  force i s  to  that  5 of  a  particle  against N sources  has  D  t h e wave  so t h e  values  C  the wave  f o r c e on  the water  observed  These  a v a r i a b l e flow.  z e r o and  t o the drag  of C  The  a t the  instant  theoretically force.  flow  experimenters  the d r a g  equal  values  of  by m e a s u r i n g t h e  i n the  observed  of i n c r e a s i n g  important  to formulate  t o t h e work o f p a s t  g e n e r a l l y been e v a l u a t e d cylinder,  the  in oscillatory  o f unsteady  as  Such a knowledge of  f o r c e s i n the  Experiments  flow  d e c r e a s i n g member r i g i d i t y .  meters t h a t i n f l u e n c e f o r c e s . is  of unsteady  d e s c r i b e d — i n order  irregularity  an  types  > 2 x 10  roughness  3  .  C :D  13 McNown shedding  on C  has  determined  for rigid  the  i n f l u e n c e of  model c y l i n d e r s  vortex-  under s t a n d i n g  waves.  17. TABLE 1 WAVE FORCE  COEFFICIENTS  [From R e f . 3]  Experimenter and D a t e  Crooke,  Diameter of Cylinder (in.) Model  1955  Keulegan & Carpenter,  1956  'D  'M  2 1 ±  1.60  2.30  3 , 2 2 / 2  1.34  1.46  1.52  1.51  •  Keim,  1956  Dean,  1956  "  W i e g e l e t a l , 1956 Reid,  Wilson,  1957.  1957 1966  Model  ±  4  Oscillatory  (Standing Waves)  1,-  1.00  0.93  3  1.10  1.46  24  1.00  0.95  Ocean waves California  8 8 8  0.53  1.47  Ocean waves Gulf of Mexico  16  0.40  1.10  30  1.00  1.45  5  1956  Bretschneider,  Paape,  Prototype  2 '  L  Type o f Flow  Variable H ratio D  with  Accelerated, non-oscillatory  SCATTER  OF V A L U E S  OF  19.  The  tests  values 2T  i n v o l v e d l a r g e amplitude  of C  water o s c i l l a t i o n s .  Average  have b e e n g i v e n as a f u n c t i o n o f t h e p a r a m e t e r  D  ' e  where  T T  e  =  p e r i o d of standing  waves  =  e d d y - s h e d d i n g p e r i o d f o r t h e maximum  velocity  v max T 2.0 f o r - — i n t h e n e i g h b o u r h o o d T 2 t o an u l t i m a t e v a l u e o f 1.2 i f — i s much d i f f e r e n t f r o m T — c a n 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 o f v o r t e x - s h e d d i n g  Average C  falls  D  s t e e p l y from  6  of 2. and  e  e  ip  S t r o u h a l number.  provide flow. 2.11  good c o r r e l a t i o n  Rigid A  inertia  T h i s parameter —  cylinders specific  to C  alone  however w o u l d n o t  f o r an a r b i t r a r y  i n standing waves—varying a n a l y s i s of the v a r i a t i o n  forces at various  was c a r r i e d  D  e  k i n d of unsteady  C : D  of drag and  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  o u t by K e u l e g a n and - C a r p e n t e r . ^  The r i g i d  c y l i n d e r was p l a c e d a t t h e node o f s t a n d i n g waves, w i t h conditions the  adjusted  t o ensure uniform  s u r f a c e t o t h e bottom.  water o s c i l l a t i o n s . forces,  they  involved  values  from  large-amplitude  Through a F o u r i e r a n a l y s i s o f t h e measured  evaluated C  a cycle of o s c i l l a t i o n  instantaneous  flow  horizontal velocity  and assuming t h a t the c o e f f i c i e n t s  were n e g l i g i b l e , out  The t e s t s  model  (Table  D  of higher  a t various 2).  harmonics  instants  through-  The s e p a r a t i o n o f t h e  o f C_ a n d C,, was e f f e c t e d D M  as f o l l o w s :  TABLE 2 VALUES O F - C „ AND (V FOR STANDING WAVES M D (CYLINDERS)  •VT M —  R.M.S. A v e r a q e , ^ y —  -  n  0  v  e  r  Q  c  l  e  ^~  ~ ^  1  ^C ° C' C  M  I n s t a n t a n e o u s V a l u e s o f C,_ & C_ M D 1 1 1  ^°*  2  4  ^D  C C M  VD  .  ^°'  5  C C MM DD  ^ °' =  C  C  C C MM DD C  C  6  ^ C° '  C MM  U C  =  1~ 8  ^D D C  C  MM C  ^  C  2.14  0.70- 2.05  1.6  2..  0.9  1.9  0.4  2.1  0.9  2.05  1.6  2.0  15.6*  0.80.  2.05  1.2  2.1  -0.  1.9  -2.0  2.0  -0.3  1.9  1.2  2.1  -1.4  44.7  1.76  1.54  1.9  1.5  2.  1.4  2.2  1.6  2.1  1.4  1.9  1.5  2.2  3.0  T *This  corresponds to  =1 e  t  = time  from p a s s a g e  of crest.  L  C D  21. Letting  and of  then  T  =  the p e r i o d o f the flow  F  =  total  using  fluid  the f a c t  force  of periodicity  o f F and t h e symmetry  the flow, •  F The  oscillations  ( y t ) =-F  non-dimensionalised  force  ( ^ t + TT)  (2.4)  F 7=- c a n a c c o r d i n g l y be pv D ^ m Z  2  K  expressed The  as a F o u r i e r s e r i e s w i t h  coefficients  measured v a l u e s the Morison  respect to the v a r i a b l e t .  of the Fourier, s e r i e s o f the flow-induced  expression  2TT F _ o T —-2I = M~DVpv D m m  S  i  n  forces.  f o r the f l u i d  .  V  C  2irt —  +  , 1 _ 2 D  are determined  On t h e o t h e r  hand  f o r c e s , namely, 2-rrt —=—  cos  C  from  cos ^  (2.5)  K  (where v ^ = max.  velocity)  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 .  the  two t r i g o n o m e t r i c s e r i e s  expressions  forC  D  and  evaluated  to y i e l d  as a f u n c t i o n o f t .  time-dependent, weighted be  a r e compared  average values  f r o m an i n t e g r a l  L i k e terms i n series  Though C  D  is  over, a wave c y c l e c a n  f o r t h e mean v a l u e .  Furthermore  expressing  F  =  by means o f d i m e n s i o n a l  F  pv K  f  2  m  = f  D  f(t,T,v ,D,p,v') m  a n a l y s i s they  (2.6)  obtained  - . v T v D • m m \ (-sr-, - - = r - , max. N = — — ) T ' D ' R ~ v  ,2iFt  n  (2.7)  T h e s e e x p e r i m e n t e r s went on t o e v a l u a t e C  D  at various  ficients for  o f the s e r i e s expressions  n  already  regimes r e p r e s e n t e d  derived;  were n o n - z e r o ,  by t h e p a r a m e t e r  ^ 0.25), i n s t a n t a n e o u s  d i d not vary  coef-  t h i s was done v T  2 shows t h a t o v e r t h e r a n g e o f t i m e when t h e  velocities C  i n s t a n t s o f t h e c y c l e f r o m t h e computed  a s e r i e s o f flow  Table  coefficient  m  ^  x  .  instantaneous values of  significantly. F u r t h e r by u s i n g t h e c o n c e p t s o f fD m number — and t h e p a r a m e t e r —=—, i t was e s t a b l i s h e d t h a t V  Strouhal v T when was much s m a l l e r  T  V  n  single vortex  was formed  t h a t numerous e d d i e s Distinct tions)  C  t h a n 15, no e d d i e s  formed; t h a t a v T i n e a c h s t r o k e when •• r e a c h e d 15 and v T  per stroke  formed f o r l a r g e v a l u e s  v a r i a t i o n s i n (mean v a l u e s occurred  i n these  as w e l l as c y c l i c  ranges.  of  ^  fluctua-  This  i s e x e m p l i f i e d by t h e v T v a l u e s i n T a b l e 2. Mean C r o s e s h a r p l y from s m a l l to a v T v T maximum o f 2.2 a t = 15 and f e l l g r a d u a l l y f o r l a r g e r ^ . T h e r e was e x c e l l e n t c o r r e l a t i o n f o u n d b e t w e e n mean C and t h e v T . m parameter — — . p  2.12  Limitations of varying Factors  obtained 1)  D  values:  that i n v a l i d a t e the a p p l y i n g o f these  i n S e c t i o n 2.11 t o an o s c i l l a t i o n Deviation  values  problem a r e :  o f the p a t t e r n o f water o s c i l l a t i o n  that of a standing 2)  C  from  wave.  Geometric s i m i l a r i t y  (ratio  w a  Y  e  height^ diameter t o be t h e same i n t h e p r o t o t y p e . J  ^  s  un^-y^giy J  2.13  T e s t s on r i g i d  model c y l i n d e r s  in  constant a c c e l e r a t i o n flow:  For  a non-reversing  unsteady  flow  s i t u a t i o n , 'N  again R  is  not  an  on  velocity  from the to  be  adequate parameter, s i n c e s e p a r a t i o n i s not alone.  This  i s due  to the  fact  that i t takes  s t a r t of motion f o r s e p a r a t i o n to occur  formed.  For  and  u n i f o r m l y a c c e l e r a t e d motion from  p l o t s o f Cjj a g a i n s t t h e p a r a m e t e r — were g i v e n by Garrison"'""'" traversed  (reproduced from r e s t .  considerations.  p a r a m e t e r — was  plot  ( F i g . 8)  s  a maximum a t — - 2.5  reaches  vortex  configuration.  rest,  Sarpkaya  s e l e c t e d on  shows t h a t C  and  dimensional  i s low  Q  first  vortex  the v a r i a t i o n s tices.  in C  at  small  fCo„u n-d . C o e f f i c i e n t M  experimental w i l ' l be tional  and  1.2  time  taken  shedding  at  of i t  large — - 6 to first  two  f o r t r a v e r s i n g an  c o r r e l a t i o n of C  symmetric  for C with  N  adequate  t o assume v T or  succeeding  summarised.  The  M  s e c t i o n s the p a s t data  for particular theoretical  cases  value  —™—  on  of unsteady  for inviscid  F o r wave i n p u t s on m o d e l and  7,  vor-  o f mass:  of C  i s 2.0.  the  No  values  flow  a value of  a  a r o u n d — = 4.8;  f o r t h e wake t o f o r m and  flow v a l u e s .  In t h i s  a t the  o c c u r r i n g o n l y d u r i n g the  D  i n a stroke  separated  t o 1.0  (asymmetric v o r t e x p a t t e r n )  This highlights  distance  a t w h i c h t h e r e was  I t decreases  thereafter eventually attains  was 2.14  vortices  s  — and  the  time  a t F i g . 8 ) , where s = c u r r e n t d i s t a n c e The  The  dependent  flow  irrota-  prototype  Fig  8  25. piles t o 2.3  (rigid  piles),  Table  (Appendix I d e t a i l s  c o e f f i c i e n t was e v a l u a t e d At is  the reasons therein using  the i n s t a n t  a t the s t i l l  theoretically force.  1 indicates values  of C  from  0.95  f o r the s c a t t e r ) .  The  M  the f o l l o w i n g approach:  t h e l e v e l o f t h e wave  water  level,  surface  the v e l o c i t i e s are  z e r o and t h e f o r c e i s p u r e l y an  The m e a s u r e d  force at this i s then  instant yields  value  o f C... M  stant  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  wave  This value  a  assumed t o be c o n of  force.  In view of the s c a t t e r  of the data  judgment must be e x e r c i s e d i n s e l e c t i n g sideration  inertia  the s i m i l a r i t y  C , M  available  so f a r ,  taking into  con-  of conditions i n a given s i t u a t i o n to  t h o s e w h i c h p r e v a i l e d i n an e x p e r i m e n t . 2.15  Rigid  cylinders  i n s t a n d i n g w a v e s — m e a n C^:  E x p e r i m e n t s on r i g i d  model c y l i n d e r s  under  waves t o examine t h e i n f l u e n c e o f t h e v o r t e x - s h e d d i n g on  C I M  these  Paralleling experiments  s e c t i o n 2.10, t h e r e s u l t s  show t h a t a v e r a g e C  M  falls  standing frequency  o f McNown f r o m from a v a l u e  T T low 5 ^ — t o a minimum o f 1 a t ^ — = 2 t o 3. I t increases e e with large — t o 2, i . e . , t h e r e i s a; d e f i n i t e c o r r e l a t i o n at  of 2 again  T  parameter  — . e  with  2.16  Rigid cylinders i n standing waves—varying Experiments f o r instantaneous  c y l i n d e r s i n s t a n d i n g waves (K.eulegan  C  M  C : M  for rigid  model  and C a r p e n t e r ) : 10  S e c t i o n 2.11 has i n d i c a t e d t h a t instantaneous values o f C», were M segregated  i n a s e r i e s form, w i t h r e s p e c t to the time v a r i a b l e  2 t The e x p r e s s i o n f o r the instantaneous C was found v T to be d i r e c t l y p r o p o r t i o n a l t o . The computed v a l u e s of the time-dependent a t v a r i o u s c y c l e p o i n t s are g i v e n i n =  0  — m — •  M  Table 2; they show t h a t C C  D  values f l u c t u a t e more markedly than v T  v a l u e s , s p e c i a l l y when  = 15.  A l s o , weighted average  v a l u e s o f C^ over a wave c y c l e were found c  = b ^ M * * M TT o M  s  6  M  i  n  2  e e d  =  4¥ TT3  D  ^ o  from the e x p r e s s i o n F  s i  f  d e  2_ pv D m  (2.8)  M  where, i n framing the e x p r e s s i o n f o r C higher order terms have been n e g l e c t e d . seen t o i n f l u e n c e C at  by dimensional  M  as a  0 - s e r i e s , the v T The parameter is  d i r e c t l y , a c o n c l u s i o n which i s a l s o a r r i v e d  a n a l y s i s ( S e c t i o n 2.11).  Mean v a l u e s o f C v T d u r i n g a cycle,were c o r r e l a t e d s t r o n g l y w i t h —™— and d i s t i n c t v T v a r i a t i o n s i n C o c c u r r e d f o r s p e c i f i c bands of values o f —™— . v T Mean C s h a r p l y f a l l s from 2.1 a t low —=— to a minimum o f 0.8 a t v T v T — — =15.. Tt then r i s e s g r a d u a l l y f o r l a r g e r — . The parameter v T M  M  M  p  was i n d i c a t e d t o be important  i n any r e g r e s s i o n .  The values  o b t a i n e d should be i n t e r p r e t e d w i t h c a u t i o n as some computed values of C shows.  M  are negative  ( p h y s i c a l l y impossible) as Table 2  27. 2.17  Experiments  on r i g i d  model  cylinders  under c o n s t a n t a c c e l e r a t i o n  from  rest:  Section  2.13 i n d i c a t e d t h a t f o r C t h e r e was no v T c o r r e l a t i o n w i t h N_. o r — — i n t h i s f l o w . As e x p e c t e d t h i s a l s o h o l d s t r u e f o r C... M —  (due t o S a r p k a y a  tion.  o f C.. a g a i n s t t h e p a r a m e t e r M ^ r  and G a r r i s o n : F i g . 8) show s t r o n g  This highlights  distance  Plots  the time  taken  i n traversing  i n a s t r o k e f o r C„„ v a l u e s t o d r o p M  correlaan  adequate  from  2.0 a t r e s t t o  A t the s t a r t o f motion  when t h e r e l a -  c  a lower tive  ultimate value.  flow i s v i r t u a l l y  C„, assumes t h e v a l u e 2. M w h i c h d e c r e a s e s t h e r e a f t e r w i t h i n c r e a s i n g ^ t o 1.2. The C „ ^ D M c u r v e r i s e s a g a i n , r e a c h i n g a.ri a s y m p t o t i c v a l u e o f 1.3. 2.18 Oscillated flexible cylinders—mean resistance: The  results  d r a g and i n e r t i a The flexible water. of  irrotational,  of a test  forces  are mentioned  t e s t s were c o n d u c t e d  model c y l i n d e r s Laird  f o r t h e measurement o f combined  longitudinal  i n the l a b o r a t o r y with  oscillated  9 21 22 ' ' investigated oscillations  next.  at large  the f o r c e s  -(^ > > 1) w i t h i n  single  amplitudes  in still  f o r large  amplitude  t h e r e g i o n 2 x 10"^  4 N  < 4 x 10  which f e l l  found  large  force  over  times  (Fig. 9).  within  the p r a c t i c a l  i n c r e a s e s i n the f l u c t u a t i n g t h o s e on r i g i d  cylinders  Combined r e s i s t a n c e  range  intensity  of flow.  He  o f the t o t a l  o f - t h e o r d e r o f up t o 5 (drag + i n e r t i a )  m e a s u r e d and r e p o r t e d , b u t t h e d r a g p r e d o m i n a t e d .  The  was average  28,  A  A  D D  'n 1.5  1  o  7  a  1.72  Old  1.9  -JlQ  i s 5 10  4  t> v.  a  o oO  7 9  A  I 3 52  V  0 4  D a  10 10.8  o.i DRAG  AND  LIFT  FORCE  Fig  RATIOS  9  N  c  • •  resistance the  f o r c e s were f o u n d  steady-state  drag  D.  Laird  0  forces  i n terms o f t h e flow f p a r a m e t e r s ^ — and f_ n where  f_ . =  t o i n c r e a s e by a b o u t 3 t o 4 t i m e s i n t e r p r e t e d the unsteady  phenomena a t v a r i o u s  forced o s c i l l a t i o n (This  frequency  values  f^  =  n a t u r a l frequency  f  =  Strouhal  of the  of the c y l i n d e r  i n f l u e n c e s mean o s c i l l a t i o n  hence t h e S t r o u h a l  flow  s p e e d and  frequency f ^ ) . of the f l e x i b l e  model i n a i r .  frequency.  Fig. total the  drag  3 oscillators  Laird f  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 f o r v a l u e s o f ^— n e a r u n i t y . B a s e d on t h e d a t a f o r n f  = 1.51, 1.72 and 1.97 r e s p e c t i v e l y ,  has s t a t e d t h a t as f  , so long  drag  with  these  slower runs  there  i s greater.time,for each  (smaller f  offered i s that i n  corresponding  the s t r u c t u r a l  force increases  over  oscillations  was r e l a t i v e l y  high,  those  oscillations  f o r c e s and t o t h e i n c r e a s e  directly  equal to  i s s m a l l , t h e maximum v a r i a b l e  An e x p l a n a t i o n  were a t t r i b u t e d t o l a t e r a l  lateral  from a value  to smaller f ) ,  amplitude  to increase  for rigid  cylinders  stroke.  The  lift  i s reduced  as t h e r e d u c t i o n  does n o t d e c r e a s e .  during  g  caused. the t o t a l  induced  by  fluctuating  i n t h e wake w i d t h s w h i c h t h e  When t h e maximum resisting  fluctuating  lift  f o r c e was c o r r e l a t e d  t o t h e s q u a r e o f t h e wake w i d t h s .  Drag  predominated  30. over  inertia  i n these t e s t s ,  d r a g and i n e r t i a  forces  Structures lower  magnify.C  than  0.3.  D  no s e p a r a t e v a l u e s o f t h e  are a v a i l a b l e .  s h o u l d be d e s i g n e d  t h a n what w o u l d s i g n i f i c a n t l y  (i.e., less  though  t o have a  flexibility-  the f l u i d f 1.2); from F i g . 9 the ^— r a t i o n  ^>  Practical  piles  fall  within  this  2.19  Lift  i n flexible  raise  structures with braced  forces s h o u l d be  cylindrical  category. oscillating  cylinders: 9 21 2 2  The oscillations  preceding experiments with large amplitudes  of Laird  '  show l i f t  '  forces  concerning t o be  sig-  n i f i c a n t when t h e S t r o u h a l f r e q u e n c y i s c l o s e t o . o r lower than f (= 0.6 f t o f ) . A p o s s i b l e c a u s e f o r t h e a b o v e e f f e c t n n n c  when f was l e s s e n the secondary  than  f  w o u l d be t h e t r a n s f e r  drag o s c i l l a t i o n s  o f energy ^  from  J  a t a frequency of 2 f . g  From  v the  relationship  f  = 0.21 — , i t i s s e e n  wave f l o w v e l o c i t i e s above l i f t diameters  of less  than  that f o r the p r a c t i c a l  12 f t . / s e c .  e f f e c t s w o u l d n o t be s i g n i f i c a n t o f 1 t o 3 f t . and s t r u c t u r a l  (r.m.s.), the  f o r the u s u a l  f r e q u e n c i e s o f 0.3 t o  1 c y c l e s / s e c . . The extreme c a s e s w a r r a n t i n g e x a m i n a t i o n o f lift  w o u l d be p i l e s  heights.  of small diameters  Furthermore  lift  effects  are ruled  quake c a s e  since the distance t r a v e l l e d  sufficient  t o cause  p r o l o n g e d eddy  u n d e r waves o f l a r g e out i n the earth-  i n each  shedding.  stroke i s not  The  magnitudes o f l i f t  small-diameter  2.20  Framing drag  piles  are about  of r e l a t i o n s  i n the s h a l l o w water case w i t  those  f o r the instantaneous  and mass c o e f f i c i e n t s For  cases  of drag.  of a r b i t r a r y  (separated  t o r e c o g n i s e t h e most i m p o r t a n t  influence  the value o f C  n  experimental data into  the important  mental 2.21  parameters  i t is that  and C„„, and t o have r e c o u r s e t o M  to determine  the c o r r e l a t i o n .  An  insight  v a r i a b l e s h a s b e e n o f f e r e d by t h e e x p e r i -  work d e s c r i b e d i n S e c t i o n s 2.9 t o 2.18. Choice  of  The  parameters:  technique  of dynamical  to s e l e c t d i m e n s i o n l e s s parameters mental  flow):  a c c e l e r a t i v e motion,  necessary  D  variable  values of the drag  similarity  has b e e n  t h a t would c o r r e l a t e  and i n e r t i a  coefficients.  used experi  Deriva-  .4 tions  g i v e n by M o r i s o n  2.22  Empirical related  and C r o o k e may be r e f e r r e d t o .  coefficient  o f mass i n a r b i t r a r y  to dimensionless  Some v a r i a b l e s  motion  parameters:  i n f l u e n c i n g C,, a f t e r ^ M  the onset o f  separation are: 1)  Acceleration  o f t h e body  2)  Acceleration  i n the surrounding  presence tion  fluid  due t o t h e  o f t h e body--depends on boundary c o n f i g u r a -  3)  D u r a t i o n of  4)  Rate o f  5)  Interaction i)  the  acceleration  change o f of  the  the  the  acceleration  velocity  distance  and  traversed  on  the  current  stroke ii) 6)  the  Residual  time elapsed  vorticity  on  the  current  from p r e v i o u s c y c l e s  stroke of  oscilla-  tion 7)  Symmetric or related  non-symmetric n a t u r e o f v o r t e x  to Strouhal  I t e m number 4) account i n the meters, the  the  value of  number.  could  not  parameters chosen.  most i m p o r t a n t o f C„ M  are:  L  =  length  v  =  v e l o c i t y of  A  =  local  T  =  time  the  be  basic  the  relevant  para-  influencing  acceleration parameter  C„: M  Physical  into  body  being found ^  to i n f l u e n c e ,  -  variables  taken  parameter  c a r r i e d out,  M  explicitly  Examining the  D i m e n s i o n a l a n a l y s i s was  C  formation-  C  M  s i g n i f i c a n c e of  ? >  parameters:  leading  t o two  parameters  (  2  -  9  »  AT  2  D  a measure  ;  of  (Local i n e r t i a ) x ( V i s c o u s force)  2  3  (Convective vT —  AT taken i n c o n j u n c t i o n with — , represents ., .. Convective i n e r t i a the r a t i o — Local inertia vT The b r o a d e f f e c t o f 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  ,  is  l a r g e and  A r e a n a l y s i s of and  2.17  disclosed  adequately  inertia)  the  t h a t the  determined  as  v is  low.  experimental  instantaneous  data  i n Section  value  of C„ M  a q u a d r a t i c s u r f a c e i n the  2.16  was  £-n-C  M  space,  where 2  S  =  Log-^l^! -!  0.125  + 0.985 |  1  | -  4.11  2 n T being The  =  the  -Log  time  1 Q  l  1 0  elapsed  p  T  1 + 0 - 1 2 4 |^|  from the  best regressive r e l a t i o n C  =  M  1.35  Fig.  10  shows t h e v a r i a t i o n 2 iOAT vT  Log  — - —  values  and  of C  M  —  s t a r t of  found  + 0.026 £  2  current stroke.  - 0.152? + 0 . 6 2 n  of C  the  shown f o r s p e c i f i c  the  M  (2.10)  was:  with  i n the range covered  as p r e d i c t e d by  + 0.903  the  by  the  equation data.  two  The  and  (2.11)  2  parameters  tests. as  test  The  experimentally  observed  are  results  constant  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  for only  a few  of  the o s c i l l a t o r y  appreciable  |  T  being  choice of  and  1OAT ^ in — - — and rotation  |^-|  vT — .  £ and  enabled  The  £-n  n as  diverged  space  of the o r t h o g o n a l  independent  elimination  to  an  variables instead  o f - c r o s s - p r o d u c t terms  representation involved a 1OAT ^ vT  axes  |—-—|  s m a l l , i . e . , s i n "*"(0.125).  is valid  results  extent.  The of- | ^  flow t e s t  i n t h e r a n g e bounded by  C  and  / the  rotation  g i v e n by e q u a t i o n  M  the  (2.11)  following inequalities: 2  1.088  <  (0.99  Log  |  1 Q  1 0  p  | -0.15  T  |^|)  < 2.1 ' (2.12)  2 0.8  <  0.7  <  Log  of the  | < 3  (2.13)  < 14  in practice the ranges  1 p XU  1 0  (2.14)  f l o w p a r a m e t e r s w o u l d u s u a l l y be  e x p r e s s i o n s g i v e n by  (2.12),  (2.13)  within and  (2.14). 2.23  Empirical coefficient  of drag  in arbitrary  2-D  motion  (with s e p a r a t i o n ) : Although that  for  was  a dimensional  f o l l o w e d , i t was  analysis not  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 correlated  the e x p e r i m e n t a l  influencing  the  coefficient  found that  values.  The  of drag  after  1)  D e g r e e t o w h i c h t h e wake has  2)  Symmetry o f  vortices  approach s i m i l a r possible to  to  formu-  satisfactorily, significant  variables  separation occurs  been e s t a b l i s h e d  are:  36. 3)  Instantaneous related  of c i r c u l a t i o n — t h i s i s  to v i s c o s i t y ,  shear  velocity  4)  Fluid  5)  Nearness of the frequency Strouhal  6)  function  of c y l i n d e r motion t o  frequency from  previous  cycles—related  3)  Treating L,v,A,T,u and  and d e n s i t y  a t t h e b o u n d a r y o f t h e body  Residual v o r t i c i t y to  p, C  t h e s e as b e i n g D  i s found  represented  by d i m e n s i o n a l  i n the v a r i a b l e s  analysis  t o be a  o f t h e f o l l o w i n g ( f a c t o r no. 5 above c o u l d n o t be  explicitly  taken  onto „  where  value  w  R  AL .2 'v v'T — Li  account) AL R' ~2  , vT a  n  T "  d  i s a measure o f . xs  Convective i n e r t i a Viscous forces  j . a measure o f  (ref.  section  Local inertia Convective i n e r t i a  2.22)  i s an i n d i r e c t measure o f  Convective i n e r t i a Local inertia I n an a l t e r n a t i v e  choice of dimensionless  parameters,  is a  function of: 2 2 y'v'T VT',; , AT ^-3 , 3- and — pD  , where  2 uv'T j pD  . _ (Viscous force)x (Convective i s a measure o f — ^ (Local i n e r t i a ) 5  inertia)  37. AT D  2 is  a measure o f  (Local  inertia)x(Viscous forces)^  3 (Convective  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 mental data  Force  t h e d a t a b e i n g meagre.  the presence  cent  Further  r e d u c t i o n s due t o n e i g h b o u r i n g  Reduction to  o u t on t h e p a s t e x p e r i -  d i d n o t g i v e c l o s e a g r e e m e n t f o r t h e many  relationships,  2.24  inertia)  f o r a clear  experimental  cylinders:  o f wave f o r c e s on a t r a i l i n g  o f a l e a d i n g neighbour spacing of 3 diameters  proposed  amounted  cylinder to only  and 45 p e r c e n t  owing  15 p e r fora  20 21 2 3 clear  spacing of 1 diameter  ( v i d e t e s t s by L a i r d ,  No r e d u c t i o n s o r a l t e r a t i o n s justified  i n practical  i n most c a s e s  i n drag  4  '  ).  forces are therefore  t o w e r s t r u c t u r e s where  g r e a t e r than  '  diameters.  the spacing i s  CHAPTER I I I WAVE .THEORIES In for  t h e p r e v i o u s c h a p t e r methods have b e e n  the d e t e r m i n a t i o n  relative and  velocity  and  of the f l u i d  chapter  information  In order  i s concerned  for determining  the  Wave  fluid  as T s u n a m i s . predicting  many v a r i a b l e s  with of  waves, and  fluid  i s the  lateral  occasionally  the  frequency  a reasonable particle  resulting  1 2 '  Because o f water  amount o f e f f o r t , velocity  and  fairly  acceleration  In order accurate  have :been f o r m u l a t e d .  Some o f t h e  wave t h e o r y a p p r o p r i a t e f o r use  factors  are:  and the  arbi-  to obtain, estimates  u n d e r some o f  more common r e g i m e s o f f l o w , v a r i o u s s i m p l i f i e d  of  kinematics,  foo? t h e m e c h a n i c s o f w a t e r waves i n an very complicated.  such  the problem  o f wave m o t i o n s 12 3  i n f l u e n c i n g wave g e o m e t r y and  theory  force  wharves  s i n g l e waves  particle, velocities. ' '  s i t u a t i o n w o u l d be  fluid  necessary  i n waves, o r  Many i n v e s t i g a t o r s have worked on  t h e m a g n i t u d e and  resulting  trary  the  tower s t r u c t u r e s i n t h e o c e a n s as w e l l as  from wind-generated  a general  motion  latter require-  situation.  o f the major d e s i g n c r i t e r i a  the  l a y i n g out  particles  theories: For  one  fluid  t o meet t h e  with  g r o u n d m o t i o n i n an e a r t h q u a k e  3.1  a member i f t h e  a c c e l e r a t i o n between t h e  t h e member a r e known.  ment, t h i s  f o r c e s on  presented  the  wave t h e o r i e s  t h a t determine  the  - depth of water - f e t c h , i . e . , exposed - wind  o f beach.  any one l o c a t i o n w h i c h d e t e r m i n e s d e p t h , f e t c h and b e a c h  slope,  d i f f e r e n t wind v e l o c i t i e s  heights  and f r e q u e n c i e s .  structure, ponse  taking  levels  In particle  ledge are  p r o d u c e waves o f d i f f e r e n t "  The s t r u c t u r a l d e s i g n e r  determine which c o n d i t i o n  for  o f water  conditions  - slope  For  length  must t h e n  i s most s e v e r e f o r t h e p r o p o s e d  into account the d i f f e r i n g  for different  f o r c e and r e s -  s t r u c t u r a l frequencies.  t h e f o l l o w i n g p r o p o s e d wave t h e o r i e s motion w i l l  of a t l e a s t  be p r e s e n t e d  two p a r a m e t e r s .  t h e wave h e i g h t  H  (measured  p e r i o d T, t h e t i m e between  expressions  which r e q u i r e  prior  The most common ones  from t r o u g h t o c r e s t )  the passage o f s u c c e s s i v e  know-  used  and t h e waves.  Thus we must be a b l e  t o d e t e r m i n e H a n d T f r o m knowledge o f  the  o f wind  local  and w i n d 3.2  conditions  speed, f e t c h , depth, beach  duration.  Wave h e i g h t  and p e r i o d :  Methods f o r t h e e v a l u a t i o n the  period  T i n t h e deep w a t e r  Relationships in  slope  shallow  f o r wave h e i g h t v  o f t h e wave h e i g h t  situation will  and p e r i o d  considered.  f o r waves,generated  w a t e r a r e a v a i l a b l e i n r e f . 6.  d 1 — > (d b e i n g  now be  H and•  I n deep w a t e r , where  t h e w a t e r d e p t h and L t h e wave l e n g t h ) , t h e  4 0. principal  parameters that  influence  t h e wave h e i g h t  H and t h e  wave p e r i o d T o f w i n d - g e n e r a t e d waves a r e t h e mean w i n d U,  the f e t c h length  analysis  F and.the wind d u r a t i o n  i t c a n be shown t h a t , n e g l e c t i n g  m e t e r s , t h e f o l l o w i n g r e l a t i o n s must  3l  =  U  1  31 U The  Fig.  11)  example, in height  2  f (31 2  and o f i n f i n i t e forinfinite  F  v  U  dimensional  l e s s important  para-  hold  3±\ u  £iv U * ;  cases of i n f i n i t e  [expressed  duration  and p e r i o d w i t h  2 '  By  f ^ and f ^ have been  f o r the l i m i t i n g  determined t  (depicted i n  i n eqns. 3 . 4 ] .  t , F i g . 11  shows an  For  increase  w i n d v e l o c i t y and f e t c h as w o u l d  p h y s i c a l l y expected.  For are  =  2  nature of the f u n c t i o n s  empirically  be  f (3l u '  t.  speed  In effect  l a r g e time d u r a t i o n reached  t , steady-state  and t h e wave h e i g h t  conditions  H and p e r i o d  T depend  crF upon t h e F r o u d e number  This i s the f e t c h - l i m i t e d case. U gH R e g r e s s i o n on e x p e r i m e n t a l d a t a g i v e s t h e c u r v e s o f — ^ — and l / 3 aF — - j - — versus shown i n F i g . 1 1 . u ^ g  T  U  Here H-jy^  =  s i g n i f i c a n t wave h e i g h t , the  T-jy-j  =  u p p e r 1/3  significant  values  period,  i . e . , average o f  o f H. similarly  defined.  ..2  WAVE  HEIGHT  AND  Fig II  PERIOD  qF I n t h e zone  4 < 10  t h e r e l a t i o n s f o r H and T a r e n e a r l y U l i n e a r o n l o g - l o g p l o t s and so r e d u c e t o H  l/3  T  1  /  °-  =  =  3  0 4 5  U  F  ° '  (3.1)  5  0.6 U ° ' F ° ' 4  (3.2)  3  where t h e u n i t s a r e U  =  surface  F  =  fetch length  Hjy., = T  l/3  height P  =  e r : L O C  qF l/3 -a— > 10 , Y-— U U  ^  wind  s p e e d i n m.p.h. i n miles  i n feet i  n  seconds T t o a b o u t 0.35 and —-f-^-  g H  For  5  becomes a s y m p t o t i c very  large  increase  g  t o 9.  This  max.  max.  [valid  points  =  wave h e i g h t H  the other  3  =  t o the case o f a  i n the p r a c t i c a l  o f f e t c h b e y o n d 100 m i l e s  H  For  corresponds  f e t c h and shows t h a t  maximum d e s i g n  /  U  From t h e a p p r o x i m a t e e m p i r i c a l  the  1  becomes a s y m p t o t i c  r a n g e , an  has no i n f l u e n c e  relation^  (p.3)  1. 87 H. 1/3 becomes 0.084  UF  f o r 2|J < i U  4 0  0 , 5  ]  c a s e o f a d u r a t i o n - l i m i t e d w i n d wave,  to the f o l l o w i n g :  on t h e waves.  (3.3) regression  l/3  H  and  H  a  '  u 1  5  t  a t * 0  1 / 3  4  o  '  u 1  to t  6  0  ,  5  (3.4) l/3  A  and  l/3  T  a  U  t  a  (Ref.  Approximate are  0  '  3  1)  expressions v a l i d  i n the range  20000 < ^ j -  as f o l l o w s :  H  l/3  =  °-  0 0 5  t 0  '  T  l/3  =  °-  0 0 6  : t  °'  Here t  =  d u r a t i o n i n seconds  U  =  s u r f a c e wind  l/3 for large t, — ^ — g H  Again  U '  4  1  3 u  ° '  speed  in  l/3 —-^f—  tend  6  <-)  7  3  5  ft./sec.  g T  and  to c o n s t a n t  values  u  independent  o f SL-.  wind v e l o c i t y  i/3  E  a  U in this  n  d  T  i / 3 increase faster with  case  compared w i t h  the  the  fetch-limited  case. 3.3  Ranges o f The  applicability: v a r i o u s wave t h e o r i e s t o be made use  a p p r o p r i a t e ranges a)  of  flow c o n d i t i o n s are  Linear small-amplitude t h e s i t u a t i o n j- >  of i n the  as f o l l o w s :  theory, which i s l i m i t e d (i.e. , ~ T  > 2.5  ft./sec. ) 2  to  (vide Appendix theory  II).  A plot  of L versus  i s g i v e n i n F i g . 12.  The r a t i o  T for this of L to the  2 length L  forinfinite  Q  depth  (L  Q  = gT /2TT) i s p l o t t e d  cl  against — i n F i g . 13. T h i r d o r d e r Stokes'? e q u a t i o n s 0  is  a nonlinear theory.  equations improve  1 d 1 f o r -^=- < 7- <  There are other  o f h i g h e r o r d e r , b u t they  the accuracy  This  Z  II  2D  Stokes"  g e n e r a l l y do n o t  commensurate w i t h  the increased  computations Cnoidal  i n v o l v e d and s o w i l l n o t be d i s c u s s e d . d 1 theory f o r — < - - n o t made u s e o f due t o  excessive results  ZD  Li  computations  i n this  being  involved.  r a n g e may be o b t a i n e d  Approximate from  the theory  i n b) . Solitary when ^  wave t h e o r y > 0.78.  f o r b r e a k i n g waves, b r o a d l y  For a f i r s t  check, the v a l u e of H  g i v e n by deep w a t e r e x p r e s s i o n s may be u s e d this  the above, d = depth water L  and g  T  2TT  gT  °°  = wave  o f water below t h e o r i g i n a l level length  L i s obtained  from  the equation:  . . 2^d . . t a n h —=— f o r c a s e a) L  2  27T  i n checking  inequality. In  —~—  valid  2 7 r d  tanh ~ — L f o r case  2 14+4 c o s h (^-) ( . L T ^ - I - 4 16 s i n h  4ird  2  [1 + d) .  „ ,)]for 27Td —=—  case  b)  JJ  (3.6)  T  =  time p e r i o d  o f t h e wave, w i t h u s u a l  < 16 s e e s , o b t a i n e d  values  as o u t l i n e d i n S e c t i o n  3.2. H i s a variable limit of  selected  of the range o f the C n o i d a l  the c o e f f i c i e n t  Fig.  as i n d i c a t e d i n S e c t i o n theory,  c i n the.following  3.2.  cl  1  Li  AD  namely =- =  expression  For the '  a plot  i s given i n  14: 2 L where  (Wave l e n g t h  T = wave p e r i o d  Appendix a),  i n feet)  II gives  b ) , c ) , and d ) .  =  cT  in. seconds.  a brief  o u t l i n e o f the t h e o r i e s f o r  O n l y d e t e r m i n i s t i c m o d e l s o f waves a r e  considered. In T a b l e which the v a r i o u s  3 the ranges o f the r e l e v a n t wave t h e o r i e s  ted.  A graphical representation  these  theories  each theory 3.4  i s embodied  are discussed  The (finite  over  indica-  o f the ranges o f v a l i d i t y o f The r a n g e s p e r t i n e n t t o  i n the succeeding  paragraphs.  theory:  mathematical basis  amplitude)  a r e a p p l i c a b l e have b e e n  i n F i g . 15.  C h a r a c t e r i s t i c s of Stokes  parameters  and s o l i t a r y  f o r the Stokes wave t h e o r i e s  s h a l l o w water  are given i n  Appendix I I . The gives  third  order  Stokes o s c i l l a t o r y  shallow water  a s u b s t a n t i a l improvement i n a c c u r a c y o v e r  small-amplitude theory  mainly  i n the f o l l o w i n g  that  ranges:  theory  i n the  TABLE 3 RANGE OF APPLICABILITY OF THE WAVE THEORIES  Linear Airy Theory  Parameter 1  2  ^ T  (ft./sec. ) 2  Equivalent'^'  Stokes" Shallow Water Finite-Amplitude  2  4  Equivalent  5  Side c o n d i t i o n f o r range o f H T  ^  Interpolation  0.2< -^-<2.5 T  <0.20  <0. 08  0.08<- <2.5 T  >0.5  0.04< -<0.5  <0. 04  <0.016  0.016<^<0.5  d  d  JL  (ft./sec. )  No T h e o r y -  >2.5  Li  3  Cnoidal  Solitary Wave (Breaking)  <<0.3  <0.3 [limited  range]  d  5  Large  v.small  -  <0.78  -  -  -  = • 0.78  | < 0.78  -  ^<0. 08 T  2  N.B. 1. 2.  F i g u r e s are approximate. C o n d i t i o n s 1,2,3 a n d 4 a r e t a k e n t o g e t h e r  >0.3  (>9.8)  forclassifying  each  case.  RANGES  io  ' 0  OF  49.  THEORIES  <  io"'  d  ,  i  o  ( F t / sec?)  ^.  ( Deep water)  ^  1 0  ° 2  0.02  0.2  2  M  LEGEND ( a ) .. ( b) ( c ) ..  B r e a k i n g waves r e p r e s e n t e d by s o l i t a r y waves " " " " i n t e r p o l a t e d Stokes theory " " " " deep water l i m i t H/L = 0 . 1 4 2 Fig  15  50.  — L or  , i . e . , — ? r < 1.8 T  < ->  0.08 <  < 0.3  ft./sec.  ft./sec.  2  T The most amplitude  important  expressions f o r the t h i r d  t h e o r y , as summarised  order  finite-  from Appendix I I , a r e as  follows: 14+4cosh ^ C =/ | ^ t a n h ^ [ 1 + ( f f i ( % )] 16sinh — 2  Wave v e l o c i t y  4  „2 L = ^ t a n h  Wave l e n g t h  o o ^ [1 ( f f i  2  d  ^ - J Q  l+8cosh ^ — sinh ' 6 2  Depth o f t r o u g h =  a  Horizontal  u = C [F.cosh  L  r d  )]  (3.9)  (3.10)  *2±2*L  c  o  s  2  (  K  X  2 T r 1  - ^  t  f  : S + d  ^ cos ( K X - ^ t ) + F c o s h 0  )  +  F  c o s h ^ t  3  d  l  cos3 ( K x ~ t ) ] a c c e l e r a t i o n |£'= o "Cl  [F, c o s h 1  X  cosh % 4  cosh \ 6  K  =  2 i r  (3.11)  ( Z + d ) T  s i n ( x - ^ t ) +2F K  t e + d )  sin2( x-ilLt) 3F, K  J-J ( z + d )  +  1  cos3( x-|lt)] K  F  a n <  ^ 3 F  =  f  u  n  c  t  i  o  n  s  ^ stipulated  -J  (3.12)  1  (3.13)  J_i  l' 2  0  A  1  J_j  J-l  F  (3.8)  62ird  2  velocity  14+4cosh ^ % )] 16sihh L 2  (  +  2 . H = 2a+- -^ ^ L  Wave h e i g h t  Local  (3.7)  2  i n Appendix I I .  51. 3.5  Additional of  third  order  Apart transition the  limit  ( o f wave s t e e p n e s s )  Stokes  theory:  from t h e f o l l o w i n g  t o breaking waves.in  solitary  tional break.  l i m i t o f the v a l i d  s h a l l o w w a t e r r e p r e s e n t e d by  b  < "  0.75  limiting  condition  f o r t h e wave h e i g h t s  =  g r e a t e r h e i g h t s , the t h i r d represent For  and  the water  of breaking  velocities  lengths obtained  such  (3.15)  t o b r e a k i n g waves  order Stokes  theory  cases,  are evaluated  by i n t e r p o l a t i o n .  i s employed  does n o t  (3.14) and  f r o m wave  The S t o k e s  t o compute p a r t i c l e  the range o f t h i s  0.08 <  directly).  kinematics.  c o n d i t i o n s i n t e r m e d i a t e between  w a t e r wave t h e o r y in  a t w h i c h waves  Li  ( L i n e a r deep w a t e r waves t r a n s f o r m  values  i s an a d d i - -.  0.142 t a n h ^  Li  correctly  there  i s as u n d e r : ^  For  (3.14)  t h e w a t e r d e p t h below t h e t r o u g h ,  This  range ( f o r  wave t h e o r y ) , n a m e l y ,  d  d^ being  of v a l i d i t y  modified  < 2.5  (3.15),  heights  shallow velocities  approach  being  ft./sec.  T (3.16) 0.3  <  < 0.78  ft./sec.  2  T^ These l i m i t s  apply  to f r i c t i o n l e s s  flat  ocean  beds.  The considerably Plots  h e i g h t o f b r e a k i n g waves w o u l d limited  of experimental  transformation  along  by t h e e f f e c t  actually  o f the s l o p e o f the beach.  heightH with i t s 1 8 i 19 the bed i n s h o r e a r e a v a i l a b l e as guidev a l u e s o f the breaker  H  H  lines.  The p l o t s  are available  as f u n c t i o n s o f - — , L  and  be  0  —  ,  d  bed s l o p e i  3.6  HQ  =  wave h e i g h t o f i n c o m i n g  LQ  =  incoming  d  =  water depth  Breaking  waves:  Solitary  wave  This surface  almost  wave  wave i n deep w a t e r  l e n g t h i n deep water  i n shallow  water.  theory:  represents a symmetrical  wave w i t h  wholly  as q u a l i t a t i v e l y  above  the trough,  in  Fig. 1 6 .  at  S e c t i o n 3 . 3 d/, which a p p l i e s t o non-viscous  beds.  the water  For sloping  beaches,  modified variables  relatively friction cyele from  shown  The g e o m e t r y o f t h e wave i s s u b j e c t t o t h e l i m i t flat  ocean  the experimentally obtained 1_8-  of  0  such  as H and d a r e a v a i l a b l e .  small reduction of breaking  i s obtained  from  computation  values  2 0  The  h e i g h t s due t o b o t t o m o f the energy  t h a t i s d i s s i p a t e d by l a m i n a r d a m p i n g .  p e r wave  This i s obtained  the.empirical expression e, x  (-—) H  =  HQ  e  L  ( 3 . 1 7 )  Ay  c  IMPACT MODEL  Fig  17  53.  where  e ^ i s a b o t t o m damping  f a c t o r which i n c r e a s e s with  increasing kinematic decreases  with  viscosity  v.  i n c r e a s i n g w a t e r d e p t h s and  wave l e n g t h  _3 and to  i n c r e a s e s w i t h wave p e r i o d ;  values  from  10  2 x 10~ . . 2  x = d i s t a n c e from These v a l u e s solitary 3.7  wave  such  among t h e s p i l l i n g waves. expressions  In p a r t i c u l a r  3.8  t o be  slope.  f e d back i n t o  cases  of  impulse  for piles.  the  f o r asymmetric  The  the  p o r t i o n of the  impulse  contact i s comparatively  those  steeper  f o r e a c h wave i s o b t a i n e d  t h e computed i m p a c t - t y p e  s m a l l e r than  Determinants  i s accurate  as p l u n g i n g waves and  The  t o change o f momentum on  found  the beach  forces:  ( F i g . 17)  b r e a k i n g wave p r o f i l e s  drag  toe o f  relations.  Impact t y p e o f b r e a k e r  from  the  f o r b r e a k i n g wave h e i g h t s a r e  T h i s model  due  t a k e s on  small.  excitations  were  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 .  breakers:  V Fig. (i being different  18  i n d i c a t e s .4 r e g i o n s  i n the  i - -g- p l a n e  the b e a c h s l o p e , H Q t h e deep w a t e r wave h e i g h t ) types  of-waves r e a c h  the  shore—spilling,  when  plunging,  7 .and . n o n - b r e a k i n g . tical The  figure of  plot  breaker  h e i g h t s exceeded the  0.78d i n model t e s t s w i t h  a t F i g . 18  b r e a k i n g ones.  The  a p p r e c i a b l e bed  demarcates b r e a k i n g regimes  P l u n g i n g waves g e n e r a t e  theore-  from  the h i g h e s t  slope.  nonvelocities  OJOI  d BREAKER  Fig  RANGES  18  and wave f o r c e s o u t o f t h e v a r i o u s  t y p e s o f b r e a k i n g waves.  The  the p l u n g i n g type o f  solitary  breaker for  wave model r e p r e s e n t s  fairly  accurately  and h e n c e wave f o r c e  calculations  t h e s o l i t a r y wave, w i t h o u t any s e a l i n g , would be c o n s e r v a -  tive  f o r other breakers also.  3.9  Earthquake  motion:  The p r i m a r y i n p u t u s e d acceleration problem  t o the ground,  the s u p e r p o s i t i o n  motion.  of the motion  The d y n a m i c a l of the s t r u c t u r e  of the r e l a t i v e motion  S i n c e the ground motion  motion  i s also  the r e l a t i v e  and t h e g r o u n d  o c c u r s w i t h o u t i m p a r t i n g any  t o t h e main body o f t h e w a t e r ,  the absolute  s t r u c t u r e - - f l u i d motion  structure ( i n the  o f o t h e r waves).  However t h e r a n g e o f t h e t o t a l m o t i o n it  i s the ground  the absolute motion of the s t r u c t u r e  motion  absence  thesis  o f an a c t u a l e a r t h q u a k e .  i s f o r m u l a t e d i n terms  relative being  record  i n this  i s such  that  w o u l d n o t c a u s e s e p a r a t i o n and v o r t e x - s h e d d i n g ; t h e r e f o r e  the v a l u e s o f C effective. as w e l l 3.10  and C  M  D  i n the p r e - s e p a r a t i o n  range  s h o u l d be  T h i s was c h e c k e d by n u m e r i c a l c o m p u t a t i o n s f o r s t i f f  as f l e x i b l e  Comparative The  structures.  ground  accelerations:  characteristics  o f t h e E l C e n t r o , 1940, N.-S. o  ground  record  and t h e T a f t ,  w h i c h were u s e d  July  as t h e i n p u t s  21, 1952 S.21-W g r o u n d  t o the s t r u c t u r e s  record,  analysed, are  now s t a t e d . for  The E l C e n t r o g r o u n d m o t i o n p e r s i s t s  the f i r s t . 1 0  second r e a c h i n g dominant  seconds  f r e q u e n c y i s 2.05 c y c l e s / s e c .  acceleration 3.0  till  t h e 30th  a maximum a c c e l e r a t i o n o f 0.3g a t 2.5 s e c .  extends a p p r e c i a b l y  is  and l e s s p e r c e p t i b l y  strongly  motion  f o r 30 s e c o n d s and r e a c h e s a maximum  o f 0.144g a t t = 4.1 s e e s .  cycles/sec.  The T a f t g r o u n d  The  The d o m i n a n t  frequency  CHAPTER IV  DYNAMIC This equations  RESPONSE  PROBLEM  c h a p t e r p r e s e n t s the systems  of motion  f o r framed  structures  of differential  u s e d , t o compute t h e  dynamic r e s p o n s e t o t h e e a r t h q u a k e e x c i t a t i o n inputs discussed 4.1  Origin  applied drag  terms:  s t r u c t u r a l .analysis portion  dynamic p r o b l e m u s i n g lation.  force  i n the p r e v i o u s c h a p t e r s .  of nonlinear  The  and wave  i s a linear  the standard s t i f f n e s s  N o n l i n e a r terms  are introduced  to the s t r u c t u r e which  f o r c e s o f the f l u i d  arise  damped  method o f f o r m u -  through the f o r c e s  t h r o u g h t h e i n e r t i a and  on t h e s t r u c t u r e  due t o t h e r e l a t i v e  displacement.  4.2  Assumptions: The  dynamic 1)  following  assumptions  formulation: The f r a m e d  structure  is elastic,  members and i s s y m m e t r i c the 2)  have b e e n made i n t h e  ground/wave  an assumption  tions .  to the d i r e c t i o n of  motion.  R o t a t o r y and v e r t i c a l ted,  normal  has c y l i n d r i c a l  translatory  checked  inertia  are neglec-  s u b s e q u e n t l y by computa-  59, 3)  Fluid  f o r c e s and r e s i s t a n c e s c a n be d i s c r e t i s e d a t  nodes 4)  The e f f e c t cylinder  5)  Fluid in  4.3  Basic  i n t e r a c t i o n was  fluid-  neglected  f o r c e s on t h e c y l i n d e r a r e t w o - d i m e n s i o n a l  nature.  formulation: The  the  o f t h e change o f s e c t i o n shape on  basic equation  f o r the dynamical problem  i s of  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 this  case represents  horizontal [m] = mass [C  of generalised coordinates,  nodal  the reduced  column m a t r i x  displacements,  viscous  damping m a t r i x  representing  damping r e l a t e d t o t h e r e d u c e d v e c t o r  by  The i n d i v i d u a l  setting C. . D 1  so  that  first  of  matrix.  = relative  {u}.  which i n  constants  =  a  of v e l o c i t i e s  terms o f t h e m a t r i x  are obtained  a and g i n t h e e x p r e s s i o n  m. . + i j  g  k 1 .• .  D  t h e p e r c e n t a g e o f c r i t i c a l damping  two modes  internal  i s a preselected  value.  i n the  60. [k]  = reduced s t i f f n e s s matrix with translations  {H(U ,U )} = vector r  f o r c e s due  which i s a f u n c t i o n and (P(t)}  horizontal  t o the of  the  hydrodynamic relative  of  be  other  f o r c e s on  either physical  s u c h as  the  imparted  the  system.  forces  inertia  fluid.  These  or c o n c e p t u a l i n the  effect/  velocities  a c c e l e r a t i o n s ' b e t w e e n s t r u c t u r e and  = vector may  to  only.  of  r  respect  forces forces  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  The on  the  inputs:  exact  assumptions  equations  are  form of stated  written  the  i n Section  s  equivalent  m  =  where  {U" } r  = n x  s t r  fC  In m a t r i x form  {0}  =  r  [ [m J + r K V j ] { U } + [ C  4.2.  i s established the  as:  + [k] {U } An  equations of motion  equation  (4.2)  of dynamic e q u i l i b r i u m i s  ] {U}+rK A|U|J{U}+[k]{U}  s t r  D  ]{U }+[k]{U }  1 vector  g  (4.3)  g  of  generalised  displacements r e l a t i v e  to  coordinates,  the  ground  in  i.e., the  horizontal direction {U  9J  } = n x  1 vector  f u n c t i o n of is  the  of  the  time.  ground d i s p l a c e m e n t , a Every element of  same f u n c t i o n  of  time.  the  given  vector  { g} U  61. {U}={U +U } = v e c t o r o f a b s o l u t e r g that relative [m J  structure  w h i c h was a l s o  t o water.  = diagonal matrix  s  displacements ^  o f d i s c r e t i s e d masses i n t h e  ( o f f - d i a g o n a l terms a p p e a r i n c a s e o f  coupling) [~K Vj = d i a g o n a l m a t r i x m  coefficient enclosed  o f added mass, c o n t a i n i n g t h e  o f mass, w a t e r d e n s i t y and t h e  volume c o r r e s p o n d i n g  |^K A(i|.U+U |.)Hu+U } = n x 1 v e c t o r o f f l u i d g  D  [ j- ]  ~  c  S  g  r  [k]  a  s  Further  differential  (4.3) r e p r e s e n t s  equations of C  M  allows  and  i n the o v e r a l l response  at a particular  a separate  constant  coefficients  value  To e v a l u a t e they  throughout  were the  c o m p u t a t i o n b e i n g made f o r e v e r y  c h o i c e o f C,„ and C^. M D the equations  a system o f n o n l i n e a r  with v a r i a b l e c o e f f i c i e n t s .  assumed t o be c o n s t a n t motion, with  time-differentiation.  simplifications:  importance  ferent  forces.  = as p r e v i o u s l y d e f i n e d .  Equation  the  drag  node.  previously defined,  Dots r e p r e s e n t 4.5  t o each  A s s u m i n g C„„ and M D  dif-  as c o n s t a n t s  3  o f m o t i o n t o be r e d u c e d  t o a system  with  as u n d e r :  [m . . J { U +U }+[C . ]{U" }+rK_A(;|U +U |)J{U +U }+[k]{U } virtual r g str r D r g' r g r 1  n  :  = {0}  1  J  (4.4)  or  equivalently  virtualJ ^ > [ u  +  c  ] s  t  "  {  r  fK A|u|g {U}+[k] {U}  } +  D  t  =  where  4.6  [m  virtual  Method o f The  integration studies  equation  For  since  resorted  (4.4)  to.  t o 1/10  3rd  The  the  structure  The  steps  for  structures with  ^  {  U  g  (4.5)  }  values  of  ranged from  small  the  by  of  K. m  time-step  the  to  inputs. the  Previous  non-linear  advantage i s  and  formulae  the  of  for this  the  degrees of the  used  are  given  t o be  kept  of  the  of  solution.  0.0005 s e e s . , t h e  latter  t h e s e were  f l u c t u a t i o n s i n the  used f o r  was  natural periods  f r e e d o m , and  C h a p t e r V d e t a i l s the  thereby  linearisation  t i m e . s t e p s had  smallest  follow  numerical  R u n g e - K u t t a method was  0.005 s e c o n d s t o  10  no  to m a i n t a i n s t a b i l i t y  ground r e c o r d  Wave f o r c e  solved  involved,  order  s i z e of  i n order  ground r e c o r d .  structures  irregular  analysed  and-  input.  input:  Besides the sary  +  techniques f o r l i n e a r i s i n g  The  down t o 1/4  4.7  }  constant  was  i t e r a t i o n s are  III.  types of  g  U  a deterministic input  i n Appendix  the  {  ground motion r e c o r d  f o r n u m e r i c a l s o l u t i o n , and  sufficiently  ]  solution:  have l e d t o  secured,  s t r  incorporates  for specific  drag terms.  not  J  C  to s i m p l i f y the  assumptions  i n Section  e x c i t a t i o n which though  4.2  i t was  neces-  deterministic  in direction amplitudes sis of  i n shallow  and f r e q u e n c y .  only  negligible equations  slenderness. and t h e f l o w  o f motion  Cm J { U } + [ C s  s t r  K  {V } = v e c t o r w T7  ture and  The  equations  V j {  V  U } + r K  height)  t o be s u b - c r i t i c a l .  D  A (  IV l &  ) J { V  o f water p a r t i c l e  symbols  W-U  The,  }  ( 4  velocities  a r e as p r e v i o u s l y  and r e s u l t i n g  '  6 )  a t the s t r u c -  defined.  give.rise high  drag  o f m o t i o n become h i g h l y n o n l i n e a r . time-step  the amplitude value.  While  selecting  as e l a b o r a t e d  corresponding  forces.  chosen.  Then  They were over  wave c y c l e s the inputs  several  converged f o r res-  i n C h a p t e r V, a p e r i o d o f  t o the s t r u c t u r a l  amplitude  i n e a c h c a s e were  t o high  i n t e g r a t i o n extending  i n successive  n o n l i n e a r wave r e s o n a n t  greatest  t o be  computations:  ponse computations, the  f o r c e s were assumed  presumed  velocities  a steady-state  noting  become:  other  s o l v e d by n u m e r i c a l until  Lift  l a r g e r o f t h e wave h e i g h t s  water p a r t i c l e  to  f o r the c o n t r i b u t i o n  nodes,  Wave r e s p o n s e  cycles  i n the analy-  ] {U}+[k] {U}  ^ m  =  the  i n allowing  respect to  one s e t o f s w a y - b r a c i n g s t o t h e s t i f f n e s s ,  large  where  i s stochastic with  A f u r t h e r assumption  of structures consisted  their  4.8  water,  of excitation  period (i.e.,  and t h e wave  CHAPTER V  RESULTS OF COMPUTATIONS Calculations structures  o f t h e dynamic r e s p o n s e o f s e l e c t e d  t o e a r t h q u a k e e x c i t a t i o n and s h a l l o w water  l i n e a r wave a c t i o n a r e p r e s e n t e d h e r e i n . ponse a n d s t r e s s e s along with selected  Displacement  res-  u n d e r t h e above two t y p e s o f e x c i t a t i o n  t h o s e u n d e r b r e a k i n g waves have b e e n compared f o r  structure  of varying  non-  geometries and water depths.  t h e v a l u e s o f t h e p a r a m e t e r s C^ a n d C  The e f f e c t D  on e a r t h -  quake r e s p o n s e h a s a l s o b e e n e x a m i n e d . 5.1  Choice of structures earthquake The  matically  f o r evaluating  response:  structures  chosen f o r a n a l y s i s  shown i n F i g .  19.  Structures  resemblance t o b e l l - t y p e well-head submerged.  The d i s p l a c e d  with the s t i f f n e s s  and n a t u r a l  a d d e d masses a r e a p p r e c i a b l e , light  the influence  o f C^.  frequency  f o r b o t h A and B low f o r A.  Structure  C represents  small  The n a t u r a l  As t h e  enclosed periods  high-  the other  v o l u m e s and s t r u c t u r a l  I t i s a tower-supported deck p l a t f o r m  with a r e l a t i v e l y frequency.  and a r e t o t a l l y  the responses o f A and B  extreme o f the range o f d i s p l a c e d stiffness.  A a n d B have a  structures  volume i s l a r g e  are diagram-  volume and a h i g h  structure natural  a n d mode s h a p e s f o r t h e t h r e e  STRUCTURES  ANALYSED  FOR  EARTHQUAKES  65.  66. structures  a r e g i v e n i n T a b l e 4 where t h e nodes r e p r e s e n t  horizontal  degrees o f freedom  matrix)  (as u s e d i n t h e r e d u c e d  and a r e numbered as shown i n F i g . 20.  merged b e l l s  A and B, i t i s s e e n t h a t  reases the fundamental p e r i o d arriving tions tics  stiffness  F o r the f u l l y  the influence  a t the t a b u l a t e d values o f the periods  sub-  of C„ i n c -  by as much as 25 p e r c e n t .  o f the response, the d i s t r i b u t e d  only  In  and i n c a l c u l a -  mass/inertia  characteris-  o f t h e . u p p e r member o f s t r u c t u r e s A a n d B were t a k e n  into  a c c o u n t . • Beam members o f t h e p l a n e frame t y p e were u s e d i n modelling  the s t r u c t u r e s ,  a member h a v i n g s i x d e g r e e s o f f r e e d o m .  5.2 E a r t h q u a k e r e s p o n s e : The (structural) in  T a b l e 5.  range o f parameters  damping f o r w h i c h In addition  applied  to the three  used-as  the input  ments and b a s e  C , C M  D  and t h e p e r c e n t c r i t i c a l  c o m p u t a t i o n s were made, a r e g i v e n  t o t h e E l C e n t r o ground  structures,  the T a f t  f o r s t r u c t u r e A.  shears—to  r e c o r d was  The response-maximum  earthquake  i n c r e a s i n g v a l u e s o f the parameter  ground  shock w h i c h was  inputs  For  l a r g e r v a l u e s o f C^, w h i c h  cause  the  maximum d i s p l a c e m e n t u n d e r  the E l Centro input  s t r u c t u r e A as e x p e c t e d  The  1.0 < C < 1.5 1.5 < C „ < 2 M•  26% o v e r t h e r a n g e  1  42 p e r c e n t a s C  i n T a b l e 6.  fundamental  periods,  increased f o r  (of the order of the f o l l o w i n g ) :  20% o v e r t h e r a n g e 6% o v e r t h e r a n g e ^  maximum b a s e  displace-  f o r progressively  C„, a r e t a b u l a t e d M longer  also  M  < C  shear f o r s t r u c t u r e increased  from  M  < 2). A also  1 t o 2.  i n c r e a s e d by a b o u t  For structure  B the  KEY .TO MODE SHAPES o Degrees  of  freedom  2  I  -3> J T  L  Shear  deformations  not  considered  r & A  STRUCTURE A  A  A  STRUCTURE B  STRUCTURE C Fin  9n  PLANE  BEAM  ELEMENT  TABLE 4 PERIODS AND MODE SHAPES  Structure  No. o f Nodes Mode  p  M  Natural Periods (sees)  1  1+0  2.3  1  1+1  2.814  2  1+0  0.383  2  1+1  0.463  3  1+0  0.038 .  3  1+1  0.042  1  1+0  1. 217  1  1+1  1.49  2  1+0  0.153  2  1+1  0.186  3  1+0  0. 008  3  1+1  0.009  Participation Factor (For L i n e a r Behaviour)  Mode Shape Node Ampl.  1 0.129 1 0.129 1 -0.691 1 -0.992 1 0.992 1 0.994  2 0.259 2 0.259 2 -0.593 2 -0.595 2 -0.110 2 -0.090  3 0.957 3 0.957 3 0.412 3 0.411 3 0.066 3 0.054  1.485  1 0.067 1 0.067 1 -0.660 1 -0.660 1 0.997 1 0.998  2 0.129 2 0.129 2 -0.646 2 -0.646 2 -0.068 2 -0.054  3 0.989 3 0.989 3 0.385 3 0.384 3 0.037 3 0.029  1.488  1.49 1.010 1.04 0.080 0.077  1.495 1.234 1.255 0.0805 0.078  TABLE 4  No. Struc- of ture Nodes Mode 10  Natural Period 'M (sees)  (Cont'd.) Participation Factor (For L i n e a r Behaviour)  Mode Shape  1  1+1 0. 99  2  1+1 0. 263  3  1+1 0. 145  4  1+1 0. 114  5  1+1 0. 078  6  1+1 0. 075  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  Node Ampl.  1&2* 3 4&5* 6 7&8* 9&10* 0.128 0. 145 0.475 0.490 0.78 1.0 A n t i - s y m m e t r i c mode 1&2 3 4&5 6 7&8 9&10 0.525 0. 594 1.0 0.772 0.27 -•0. 387 A n t i - s y m m e t r i c mode 1&2 3 4&5 6 7&8 9&10 1.0 0. 909 -0.345 -0.368 -0.313 0.135 A n t i - s y m m e t r i c mode 1&2 3 4&5 6 7&8 9&10 0.17 1.0 -•0. 249 0. 134 -0.518 -0.313 A n t i - s y m m e t r i c , mode 1&2 3 4&5 6 7&8 9&10 0. 088 0 -1.0 0 0.003 0 A n t i - s y m m e t r i c mode Symmetric mode  *Ampls. e q u a l i n a n t i - s y m m e t r i c modes, b u t e q u a l and o p p o s i t e  i n symmetric  1...192 0. 636 0. 503 0. 056 0. 0005 0. 0005  modes.  TABLE 5  RANGE OF PARAMETERS Q.  Structure  A  \  B ) (For E l Centro)  C (For E l C e n t r o ) A (For  Taft)  C  M  C  D  Critical Damping  T  (sees.) B 1.49 1.42 1.35 1.22 1.49 1.49 1.49  A 2.81 2.68 2.56 2.3 2.81 2.81 2.81 only)2.81  1+1 1+0. 75 1+0. 5 1+0 1+1 1+1 1+1 1+1  1.2 1.2 1.2 1.2 0 0 0 0  2 2 2 2 2.5 3 4 5(A  1+1 1+0  1.2 1.2  2 2  0.99 0.988  1+1 1+0. 5 1+1 1+1  1.2 1.2 0 0  2 2 3 4  2.81 2.56 2.81 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 w i t h o u t  water.  -  TABLE 6 EARTHQUAKE  RESPONSE  Parameters "  STR. A  Earthquake Record El  Centro 1940  Natural Period (sees.)  Maximum Displacement (ins.)  D  1+0  1.2  0.02  2.30  17.95  28.91  1+0.5 1.2 1+0.75 1.2 1.2 1+1  0.02 0.02 0.02  2.56 • 2.68 2.81  21.42 22.51 22.60  36.12 37.91 51.2  1+0 1+0.5 1+0.75 1+1  1.2 1.2 1.2 1.2  0.02 0.02 0.02 0.02  1.22 1.35 1.42 1.49  6.44 6.26 6.81 7.32  36.10 41.84 46.25 48.70 9.92 11.57 14.18  El  A  Taft,1952  1+0 1+0.5 1+1  1.2 1.2 1.2  0.02 0.02 0.02  2.30 2.56 2.81  3.79 3.92 4.22  C  El  1+0 1+1  1.2 1.2  0.02 0.02  0.988 0.99  6.78 6.71  M  Centro, 1940  Maximum Base Shear (Kips)  M  B  N.B. C  Centro 1940  Ratio of Critical Damping £  = 1+0 r e p r e s e n t s s t r u c t u r e w i t h o u t  water.  603.9 641.5  response  was  less  the range 1 < C ment and was  shear  negligible  sensitive  respectively. difference  effect  volume, s t i f f n e s s illustrated  by  sensitive  i s the response  5.4  Effect A  response  of  the  2  percent  shape, i . e . , d i s p l a c e d on  the response  chosen.  The  smaller  is the  s m a l l e r the n a t u r a l p e r i o d , the t o hydrodynamic  less  effects.  D  is insensitive  t h a t term  to C  Q  study v a r y i n g C f o r earthquake  for structure A with  b e i n g r e p l a c e d by 0.02  t h e maximum r e s p o n s e  damping e f f e c t s due 3 percent  with.or without  shape:  similar parametric  Comparing  there  comparisons.  v i s c o u s damping r a t i o o f 0.01, 7.  of s t r u c t u r e C  C :  t i o n s of the response and w i t h  the  for displace-  and  £ =  t h r e e examples  volume and  over  of  mass d i s t r i b u t i o n  displaced  increases  35 p e r c e n t  1.2  of s t r u c t u r a l  and  the  =  D  were assumed i n making t h e  The  values  damping  E f f e c t of s t r u c t u r a l  and  In the case  Constant  critical  percent  i n the response  C  5.3  The  < 2 were 14 p e r c e n t  M  added mass e f f e c t .  Percent  t o C^.  critical  to water drag  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 ,  inputs.  the  Computa-  the n o n l i n e a r drag  0.03  in either  v i s c o u s damping  showed t h a t  term  an a d d i t i o n a l e q u i v a l e n t and  do  Q  are g i v e n i n Table case,  additional  not e v i d e n t l y exceed  f o r such from  structures.  Table  8,  drag  2 to Further-  (and  also  TABLE 7 DAMPING EQUIVALENT OF DRAG  STR.  Ratio of Critical Damping  Earthquake Record El  Natural Period (sees.)  Maximum Displacement (ins.)  Maximum Base Shear (Kips)  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 El  C.. = 1+1 M  C e n t r o Ground  C^ = 1.2 D Record  Node  (Max. Water „ .. I n e r t i a Force) R a t i o -rrr = —r(Max. D r a g F o r c e )  or-> ^ 853.0  r-> _ 67.0  13.7  75. lift)  5.5  effects  are  Relevance  of  Checks for  very  motion  short  between  region.  5.6  node  well  as  6  between  the  s t r u c t u r e was  flow  only  member  with  sizes  natural  the  periods  response and  was  heights  high  sizes  computed were  pile  (period of  only  waves  covering  (over  of  waves:  4.4  2  are  selected  a  to  sees.)  periods  shallowthe  struc-  reasonably member  chosen  keep  so  standard  sizes  and  as  induce  range.  the  as  structures  While  practical  to  The  flexibility  and  sees.).  parameters  s e l e c t e d wave given so  The  are  were  designed  the  and  i n F i g . 21  periods  the  fundamental  encountered.  c o n s t r u c t i o n , the  therefore  values  structural  shown  relative  structures.  platforms  be  structural  were  The  single  would  practical  ocean  at  f t . , i . e . , where  c o n f i g u r a t i o n s as  the  occur  except  subcritical  100  tural  to  i n the  f t . to  wave  inertia.  the  reported.  40  conditions  node,  are  water  resonance  from  top  seconds  i n depth  consequently  a  added  that  f i n i t e - a m p l i t u d e Stokes  for  3.45  showed  extreme  flexible  pile-supported  and  amenable  most  to  results  the  N  s e p a r a t i o n would  the  with  region:  instantaneous  and  range  and  compared  fluid  other  2.11  small  at  response  The  be  durations  of  Dynamic  to  subcritical  of  Further,  topmost  seen  C  one  and  data  i n Table  that  M  of  C  D  used,  f o r which  9.  The  wave  the  harmonics  some  of  vibration periods of  the  2  STRUCTURES  VI *, VII  STRUCTURE  X  STRUCTURE IX  For Structure VIII, see Str. C STRUCTURES  ANALYSED  Fig  22  FOR  of Fig 19 BREAKERS  TABLE 9 STRUCTURAL AND OTHER PARAMETERS FOR FINITE-AMPLITUDE WAVE RESPONSE  Str. No.  I  Natural Base Period of F i x i t y Structure (sees.) 1st 2nd Mode Mode  Height Dia. of of S t r u c - Main ture Piles Ft. D Ft.  Total Projected Area Sq.Ft.  Total Ratio Depth P e r i o d Enclosed (Proj.Area) of of E n c l o s e d V o l . Water Volume Wave Cub.ft. d T _i Ft. sec. Ft. 1  H e i g h t Wave of Length Wave L Ft. H Ft.  d L  2.11 0.23 R e s t rained -EI L  60  1.5  179.6  167  1.07  40  4.2  12  90  2.65 0.28 R e s t rained  60  1.5  179.4  155.2  1.16  40  5.3  19  139  . 29  3.45 0.40 F i x e d  60  1.5  166.2  161.8  1.03  40  6.9  25  244  .16  4.4  0.50 F i x e d  90  3  405  954.2  0. 42  60  6.0  25  184  . 33  1.44  0.17 R e s t rained  82  2  336  490  0.69  75  2.87  128  4  997.6  2466  0.41  100  5.0  128 2.84 0.41 R e s t trained v-EI • L C =12* .0 D ' M  2  467.6  0.75  100  5.65  .44  k  II  k=^ 4L K  III IV V  3L 2.55 0.52 R e s t rained  42  1.79  17  128  .78  21  164  .61  5.8  K  VI  k* 2L E I  VII  U  612.8  k = rotational IFL  stiffness  a r e member i n e r t i a  o f base j o i n t .  and l e n g t h t o n e x t j t . t o r  bottom  section of p i l e .  n o n l i n e a r waves w o u l d be vibration  of the s t r u c t u r e .  by m a t c h i n g pondingly of  i n resonance  the  first  This c r i t e r i o n  The  i n p u t , o r one particular  a r e t h e ones r e p o r t e d h e r e . these depths  5.7  the  second  fundamental  harmonics  of non-breaking  t h e maximum s t e a d y - s t a t e  p e r i o d i n every  displacements  Despite the f a c t  the  case  i n T a b l e 10 do  change i n . d y n a m i c  second  accordingly in  e x c e p t S t r . IV,  the  the  not i n c r e a s e i n a r e g u l a r  Although  Consequently o f the  structure  I I was  causes  the water  depths  because of the  second  and  to  synchro-  greater;  g r e a t e r , c a u s i n g an i n c r e a s e comparatively  h a r m o n i c o f -the wave, t h e  i s much g r e a t e r f o r S t r . I I t h a n  s t r u c t u r e s VI  A  II,  I I , the v a l u e o f T s e l e c t e d  t h e wave s i z e was  g r e a t e r amplitude  cases of  response.  harmonic w i t h  the r a t i o  displacement  that  i n c r e a s i n g h e i g h t d o r such o t h e r parameter.  a r e t h e same f o r I and nise  waves  f o r b r e a k i n g waves  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 b e t w e e n I and a large  the  h a r m o n i c o f t h e wave e x c i t a t i o n e q u a l l e d  maximum d i s p l a c e m e n t s manner w i t h  corres-  waves: .  o v e r t u r n i n g moments a t t h e b a s e . of the  satisfied  later.  to Stokes  T a b l e 10 l i s t s  period  c a n be  mode o f  s i t u a t i o n s which generate  Computations  are p r e s e n t e d  Computed r e s p o n s e  and  first  of the h i g h e r  l a r g e s t dynamic f o r c e s under the a c t i o n  in  the  h a r m o n i c o f a s m a l l wave, w i t h a  s m a l l energy  l a r g e r waves.  with  VII are  similar.  for S t r . I.  peak The  TABLE 10 RESPONSE VALUES FOR FINITE-AMPLITUDE WAVE INPUT  Str.  T n sec.  D Ft.  Ratio Projected Area E n c l o s e d Volume Ft."  d Ft.  H T Sec. F t .  Max. D i s p l a c e m e n t (Time A f t e r C r e s t ) X (Wave P e r i o d ) max In.  Max. Overturning Moment K.In.  4.63  0.4  24100  1  I  2.11  1.5  1.07  40  4.2  12  II  2.65  1.5  1.16  40  5.3  19  16.3  0.1  53200  III  3.45  1.5  1.03  40  6.9  25  41t  0.2  82100  IV  4.4*  3  0.42  60  6.0  25  28t  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  •  C  D  =  1 .2  *Simple p i l e tFlexibility.high,  nonlinear 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  displace-  ments a r e : a)  G r e a t e r wave h e i g h t s H s e l e c t e d greater T d i r e c t l y excitation.  b)  i n c r e a s e s the amplitude  T h i s accounts  high displacements  c)  The e x t r e m e l y  d)  the average  influences Similarly  for VI.  low s t r u c t u r a l  stiffness  "static  n o r do t h e y  displacements. structures  5.8  h y d r o d y n a m i c damping r a t i o , particle  resonance  velocity,  differs  the degree  structure  the s t r u c t u r e  a function w i d e l y and  o v e r t u r n i n g moments do n o t o r i n p u t wave  i n c r e a s e i n t h e same manner as t h e peak due t o t h e f a c t  of participation  Force v a r i a t i o n s with  typical  I l l  amplification.  t h e maximum  This i s partly  A plot  of Str.  d e f l e c t i o n " and  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 height;  t p k e e p down  i n t h e l a r g e peak d i s p l a c e m e n t shown.  The e f f e c t i v e of  f o r the comparatively  t o drag i s seen  would engender a l a r g e results  of the  f o r S t r . I I , I I I and V I I .  The s m a l l a r e a e x p o s e d the displacements  t o accompany t h e  that i n different  by the second  mode v a r i e s .  time:  o f t h e t o t a l wave f o r c e ( S t r . I) t a k i n g  into  i s shown i n F i g . 23.  on t h e p i l e s  account  The t o t a l  v a r i a b l e water s u r f a c e t o the base o f the p i l e  of a  the motion of force  from the  has been  plotted.  83. The  plot indicates  1)  the f o l l o w i n g :  drag predominates over i n e r t i a in question, relative  2)  where t h e wave d i m e n s i o n s a r e l a r g e  t o the water  depth.  the f o r c e - h i s t o r y p l o t i s not symmetrical time o f passage o f t h e c r e s t . inertia  force being  and  higher order  the i n e r t i a  force  still  water l e v e l  order  terms.  The the p i l e  is  p l o t t e d i n F i g . 24.  ond  owing t o s t r u c t u r e  p l o t i s not symmetrical  motion  about t h e  t i m e , t h i s b e i n g due t o h i g h e r  adjacent t o the platform The moment  bending  moment  f o r structure  fluctuates  o f t h e wave, w h i c h i s e x p l a i n e d  I  a t twice the  by t h e f a c t t h a t  coincided  the sec-  with the struc-  period. 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 The  is  to the  terms.  h a r m o n i c a t h a l f t h e wave p e r i o d  tural 5.9  section  about the  i s due t o t h e  time v a r i a t i o n o f t h e s t e a d y - s t a t e  in  frequency  This  a t 90° p h a s e and a l s o  change i n t h e d r a g p a t t e r n  3)  f o r the structure  inertia  significantly  compared w i t h structure particle  portion  forces:  o f t h e wave f o r c e  different fora flexible  a corresponding r i g i d  accelerations accelerations  pile.  a r e comparable  pile This  on p i l e  members  s t r u c t u r e , as i s because the  i n magnitude t o the water  even though t h e v e l o c i t i e s d i f f e r  widely.  85.  Thus the t a k i n g in  arriving  into  account of the feedback o f s t r u c t u r e  at inertia  ment o f t h e r e s p o n s e  5.10  Supercritical  Although  flow  >  2 x 10^)  exceeded  This  5.11  Keulegan  N  approached  2 x  topmost  t o t h e h i g h e s t waves.  6 x 10^  10  For  ( b a s e d on r.m.s. v a l u e s ) . A t o t h e r nodes  parameter:  a  x  values  >  i . e . , greater  be d i s c h a r g e d . natural  f o r the  dura-  T m  5.12  i t approached  a t the  values f o r longer  situation.  subjected  adopted  values prevailed.  v  30,  critical  f e a t u r e w o u l d r e d u c e t h e wave f o r c e s .  subcritical  (1.2) were  D  b a s e d on r.m.s. v e l o c i t i e s  node f o r some s t r u c t u r e s 4 f t . dia. piles  values of C  i n t h e wave p r o b l e m , v e l o c i t i e s  than i n the earthquake  (i.e.,  refine-  conditions:  subcritical  s u r f a c e of the water  to a  solution.  f o r most c o m p u t a t i o n s  tions  force values contributes  motion  t h a n 15,  B r e a k i n g wave  ruling  2.18)  vortex to  f r e q u e n c i e s were much l o w e r  out l i f t  than  resonance tendencies.  ( s o l i t a r y wave) r e s p o n s e :  take i n t o  account the e f f e c t  i n c r e a s i n g wave h e i g h t s g i v e n by  of s h o a l i n g i n  at breaking, plots 18  depth r e l a t i o n s  r a n g e d f r o m 20 t o  t h e v a l u e f o r a t l e a s t one  Eddy-shedding  frequencies,  To  (Ref. S e c t i o n  experimenters  of the breaker h e i g h t -  19 '  20 '  were  adopted.  86. To make u s e o f t h e a f o r e s a i d  s l o p e was t a k e n a t t h e p o i n t where — =  f o r s h a l l o w water) The s o l i t a r y  and was u s e d  depths.  water  k i n e m a t i c s a t - t h e passage  force  levels.  tabulated  The p a r a m e t e r s i n T a b l e 11.  T h e summarised  computed  forces  from t h e t r u e  t h e computed a)  initial  t o f i n d the  force  forces  computations  particulars  The d e v i a t i o n s  o f the  of these  o c c u r due t o t h e f o l l o w i n g  i s to warrant a s l i g h t  decrease  forces:  Increase of the s t a t i c a l l y and  b)  effect  A  o f t h e c r e s t and t h u s t h e  f o r the v a r i o u s  l o a d s a r e i n T a b l e 12.  f a c t o r s , whose o v e r a l l  to define  wave t h e o r y was u s e d  computed  in  X  J-j  water  are  t h e p o i n t o f commence-  cl *  ment o f t h e . b e a c h (the l i m i t  relations,  stresses  Decrease: portions tions,  due t o d y n a m i c  .Supercritical N of the p i l e s  this  computed member  forces  amplification. v a l u e s a t the upper  D  reduce C  b e i n g by a f a c t o r  D  i n steady flow  situa-  o f 3 i n t h e upper  portions. c)  Decrease:  For s p i l l i n g  b r e a k e r s and f o r waves  i n g b u t n o t b r e a k i n g under ities  would  the p a r t i c u l a r  deform-  slope,  be l o w e r t h a n f o r a t h e o r e t i c a l  veloc-  solitary  wave o f t r a n s l a t i o n . 5.13  Comparative  forces  Comparative  under v a r i o u s  values of forces  t h e v a r i o u s wave and e a r t h q u a k e The m o m e n t s / f o r c e s  inputs  excitations: and moments p r o d u c e d by are given  i n T a b l e 13.  f o r e a r t h q u a k e i n p u t s h a v e - b e e n . s c a l e d down  TABLE 11 STRUCTURAL AND OTHER PARAMETERS  i t r . No.  Height of Structure Ft.  Dia. of Main P i l e s D Ft.  FOR BREAKING WAVE  Total  Projected Area Sq. F t .  (SOLITARY WAVE) FORCES  Depth Below Trough Ft.  H e i g h t o f Wave F l a t Smooth Sloping Bed Bed Ft. Ft.  I  60  1.5  180  40  II  60  1.5  180  40  IV  90  3  405  60  -  V  82  2  336  75  -  VI  128  4  998  100  VII  128  2  468  100  -  IX  135  2  1013  95  50  X  165  3  1450  135  30  31 31 40 45 50 50  —  TABLE 12 LOADING DUE TO BREAKING WAVES Str.  No.  ,11,III  Type of Breaker  Pile Total Depth Beach Wave C h a r a c t e r i s t i c s Dia. Height of Slope T, H W 0 D Of S t r . Water i Ft. Sec. Ft. Ft. Ft. d Ft. .. . .  Spilling  1.5  (Proj.Area) (Proj.Area of P i l e )  Total Force on Piles K  Total Overturning Moment K.In.  60  40  10  33  31  1.11  128  63600  .05  IV  Plunging 3 (.05) Spilling (.02,.01)  90  60 .05,.02,10 .01  40  40  1. 0  230  171100  V  Plunging 2 (.05) Spilling (.02,.01)  82  75 .05,.02,12 . 01  50  45  1.06  266  214800  VI  Spilling  4  128  100 .02,.01 10  55  50  1.02  735  783000  VII  Spilling  2  128  100 .02,.01 10  55  50  1.05  367  391000  IX  Spilling  2  135  120  0  10  50  50  1.13  950  960000  X  Unspecified  3  165  150  0  16  50  -  1.13  a) *303  270000  b) +950  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 c a l c u l a t i o n s f o r moments. 2. Beach slopes are i n d i c a t e d i n parentheses. 3. C = 1.2 D  t Upper l i m i t :  as f o r IX 00. CO  89. TABLE 13  COMPARATIVE VALUES OF MOMENTS AND AXIAL FORCES UNDER VARIOUS EXCITATIONS  B r e a k i n g Waves Oscillatory T Str. n No. • S e e s .  Cl  <°n  Ft.  W o r s t Mom. W o r s t A x i a l K" Force K  I 2.11  2.98  40  302*  2.65  2.37  40  -  II  S h a l l o w Water Waves  L i n e a r Waves S t o k e s  1  1  -  W o r s t Mom. W o r s t A x i a l K" Force K  Beach Slope  -  S c a l e d by 0 . 7 5 W o r s t Mom. W o r s t A x i a l K" j Force K  E l C e n t r o Quake Response Spectrum Scaled f o r Yielding W o r s t Mom. W o r s t A x i a l K" Force K  950 940  240 270  0.05  2770  13  1185  143  3200 3160  230 640  0.05  4500 .  10  1850 1550  140 40  0.05  2340 ;  25  890*  2410 • 2910 '  18 95  787 415  92 330  290 5020  5400 1030  215 2050  1210 2300 .  720 17  800 250  100 450  ;  3.45  1.82  40  -  -  2180  240  V 1.44  4.36  75  -  -  396 156  55 1309  0.05 0 . 0 2 , 0 . 01  2.55  2.46  100  13000  250  23100 3500  1480 2170  0 . 0 2 , 0 . 01  VII  2. 84  2.21  100  3500*  10  5550 1540  0 1030  0 . 0 2 , 0 . 01  VIII  0.99  6.29  200  1231  304  1250  300  o.  4830 1  1360  2720@  760@  IX 1.18  5.33  120  -  -  -  -  0  3400  1480  1090  625  X 0.80  7.97  150  1680  500  -  -  0  3400  1480*4  2110  670  XI  0.99  6.29  250  1900  510  -  0  3350  3200  735  XII  4. 48  1.40  800  -  -  XIII  6.28  1.00  1200  -  -  -  III  VI  31800000+ %  3  * S t a t i c a p p l i c a t i o n o f s h a l l o w w a t e r wave f o r c e s . * Estimated. *3 C o n c e n t r a t i o n o f l o a d a t a p o i n t assumed. * Estimated. @ E x a c t e l a s t i c v a l u e s by t i m e s t e p i n t e g r a t i o n : Mom.: 6500 K"; Axial force: 1800K. t 3 s t a n d a r d d e v i a t i o n s . R e f . : F o s t e r , E.T.: Model f o r n o n l i n e a r d y n a m i c s o f o f f s h o r e t o w e r s ( J . A . S . C . E . V o l . 9 6 , No.EMI, F e b . 1 9 7 0 ) . L  z  _  %  4  +  11500 2450  !  610 _  280000  85  2  _  3 standard deviations o f displacement a t top = 2 . 7 f t . R e f . : M a l h o t r a A, and P e n z i e n J . : R e s p o n s e o f o f f s h o r e s t r u c t u r e s t o random wave f o r c e s ( J . A . S . C . E . V o l . 9 6 , No. S T . 1 0 , O c t o b e r , 1 9 7 0 ) . D i s p l a c e m e n t a t t o p i n e l a s t i c b e h a v i o u r = 1.33 f t .  90,  by  a half to allow  f o r the reduction  i n design  forces  that  w o u l d r e s u l t f r o m d u c t i l e y i e l d i n g o f t h e s t r u c t u r a l members. This  i s conservative  when compared  to current  earthquake  design  16 philosophy the by  of structures.  computed b r e a k i n g 25 p e r c e n t .  i n Table  F o r reasons quoted  i n Section  wave m o m e n t s / f o r c e s have b e e n s c a l e d  The c o m p a r a t i v e v a l u e s  of stresses  14.  1)  natural  2)  water  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 .  period  depth  o f the beach s l o p e ,  relevant,  i i ) b e d r o u g h n e s s , e t c . w o u l d a l s o be  each s t r u c t u r e  d e p t h d and p e r i o d  considerations  R  wave  forces.  the c r i t i c a l  l o a d case, as w e l l  T , have b e e n n o t e d i n T a b l e  discussed  natural period T the  s u c h as i ) d e p t h o f w a t e r a t t h e  e s p e c i a l l y f o r breaking For  as  on compara-  responses a r e :  Other parameters toe  down  are given  Some o f t h e p a r a m e t e r s t h a t have a b e a r i n g tive  5.12,  i n previous  sections  15.  From  and c h a p t e r s ,  the  and t h e w a t e r d e p t h d were a d j u d g e d t o be  p a r a m e t e r s most m a t e r i a l l y a f f e c t i n g t h e c o m p a r a t i v e  response.  TABLE .14 COMPARATIVE  STRESSES  Stresses Str. No.  n  X-section  Osc. i . d Linear I n . F t . Waves  Stokes'  Area  Water Waves  1  2. 11  31  861  40  ±3. 1+0. 03  II  2. 65  31  861  40  -  III  3. 45  287  40  IV V VI  7. 1  4. 4  56  800  60  1. 44  28  1200  75  2. 55 112  (K/in. )  Shallow Breaking Waves  E l C e n t r o Quake Response Spectrum M a s s e s a t Top Stresses  ±9.9+ 7.7 ±9.8+ 8.7  ±38.7+  0.5  2x150  ±9. 4+ 3. 9  ±33.5+ 7.4 ±32.9+20.6  ±47.1+  0.3  2x150  ±19. 4 + 1 . 3  4.5  2x200  ± 7 . 9+ 3. 3 ± 4 . 2+11. 7  Unrepresentative  Structure  II  -  ±4.0+ 2.0 ±1.6+46.8  ±38.8+  21000 100 ± 1 4 . 8+2. 2  ±26.4+13. 2 ± 4.0+19.4  ±17.5+3.4 ±3.7+59.8  2x500  ± 6 . 2+ 1. 9 ± 1 . 2+18. 3  1360 100 ± 3 1 . 0+0. 4  ±49.1+ 0 ±13.6+36.8  ±14.4+34.1 ± 2 7 . 1 + 0.8  2x300  ± 7 . 1+ 3. 6 ± 2 . 2+16. 1  ± 1 . 8+3. 6  ±1.8+3.6  ±6.9+16.0  2x400  ± 3 . 9 + 8 . 9*  VII  2. 84  28  VIII  0. 99  85  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  12600 200  - • 18900 250 . ± 2.4+4. 0 XI 0. 99 126 * E x a c t v a l u e s (by t i m e s t e p i n t e g r a t i o n ) f o r e l a s t i c  ±4.3+4.8 2x320 , b e h a v i o u r : ±5.2+11.9 K / i n .  ±  4  - 1+ 5. 8  TABLE 15  Structure  GOVERNING  LOAD CASES FOR OFFSHORE TOWERS  T  d  n  Sees.  Governing Load  Ft.  I  2. 11  40  Breaking  (Solitary)  wave  II  2. 65  40  Breaking  (Solitary)  wave  III  3. 45  40  Breaking  (Solitary)  wave  V  1. 44  75  Breaking  (Solitary)  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  VIII  0. 99  200  1. B r e a k i n g  IX  1. 18  120  X  0. 80  150  XI  0. 99  250  wave 2. E a r t h q u a k e  Breaking  wave  1. B r e a k i n g wave 2. E a r t h q u a k e Earthquake  LOAD  TYPES  93.  The  fundamental p e r i o d T  characteristic ance w i t h the  excitations with  offshore  similar,  that p r i n c i p a l l y  and  the  increase  In the  the in  the  relative  these type  i n masses was  magnitudes of higher  to the  B since practical region.  ferent  load  i n the  5.14  d-T  n  contact  govern design  the v a l u e s  were f o u n d  the  in-  r  For  to  the is was  bounding l i n e s fall  o f d and  to govern, these  T  A  types  i n f l u e n c e of plot  are  and  outside  , four  being  i)  difoscilla-  waves, i i i ) e a r t h Between  load case governs, t r a n s i t i o n  adjacent  Broad ranges of  area.  attention  likely  and waves;  b e e n made and  plot  water, i i ) breaking  individual  depth  height'and  contact  has  two  s t r u c t u r e s would not  shown where two  following:  to  area,  space c o r r e s p o n d i n g  r e g i o n between the  In overview the the  the  i v ) o s c i l l a t o r y waves i n deep w a t e r .  zones where an are  period T  the  In c o n s t r u c t i n g t h i s  t o r y waves i n s h a l l o w q u a k e s and  d,  harmonics of n o n l i n e a r  o f t h e wave and  D e p e n d i n g on types  almost,  in relation  waves i t d e t e r m i n e s  l o a d i n g t h a t may  shown i n F i g . 25.  this  for a given  Since  graded  maximum wave d i m e n s i o n s , t h e  distribution  restricted  geometrically  c a s e o f o s c i l l a t o r y waves, t h e w a t e r  reasons a p l o t of  degree of reson- .  member c r o s s - s e c t i o n a l s i z e s .  the c a s e o f b r e a k i n g  velocity  system  dominant f r e q u e n c i e s .  s t r u c t u r e s were t a k e n as  reflects  influences  determines the  various  depth, i t also follows that directly  i s t h e most i m p o r t a n t  n  equally  likely  to  the zones govern.  load types: -  i n F i g . 25  i s seen to  highlight  95..  i)  t h e d o m i n a n t i n f l u e n c e o f 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  a certain  e x t e n t on  than  90  f t . and  those with depths l e s s  to  than  160 f t . ii)  iii)  the dominant i n f l u e n c e of earthquake  loads  structures with natural periods less  than  the  importance  o f d e s i g n i n g on  falling  o u t s i d e i ) or  ii).  2 sec.  the b a s i s o f  d e e p w a t e r waves f o r s t r u c t u r e s w i t h tion  on  a d-T  periodic combina-  CHAPTER V I CONCLUSIONS 6.1 E f f e c t s  o f mass  coefficient:  V i r t u a l mass e f f e c t s i n determining response structures with  have t o be e x a m i n e d i n d e t a i l  t o earthquake  large periods.  tionship of C although has  M  been s u g g e s t e d .  structures increased those  6.2  flow phases.  by a b o u t  structures coefficients  A conclusive rela-  that  experimental  data,  f o r some o f t h e displacements  25 p e r c e n t f o r t h e h i g h e s t v a l u e s o f C  M  over  f o r z e r o added C,,.  Shallow  w a t e r waves: r e g a r d s w a t e r wave i n p u t s i n s h a l l o w w a t e r ,  dynamic d i s p l a c e m e n t s  w o u l d be s u s t a i n e d o n l y by  s t r u c t u r e s with periods w e l l over f o r c e s occur a t o r near Load  types  such  t h e time  governing  A graphical type,  submerged  was n o t e s t a b l i s h e d ,  on l i m i t e d  I t was o b s e r v e d  mass  c o n s i d e r e d , t h e peak e a r t h q u a k e - i n d u c e d  As  6.3  the v i r t u a l  t o the flow parameters  a p o s s i b l e one, based  f o r bulky  In the case o f such  t h i s would n e c e s s i t a t e d e t e r m i n i n g corresponding to the v a r i a b l e  motion  2 seconds.  of o f f s h o r e s t r u c t u r e s .  flexible  The g r e a t e s t wave  o f the passage o f the c r e s t .  design:  relationship  as earthquake  large  i s presented  o r wave f o r c e s ,  showing t h e l o a d  t h a t governs  The two p a r a m e t e r s  the d e s i g n  t h a t govern  the load  type are the n a t u r a l p e r i o d The  o f the s t r u c t u r e  c h o i c e o f t h e s e as t h e b a s i c  being the p r i n c i p a l under v a r i o u s  and t h e w a t e r  independent parameters  determinants of the comparative  depth.  and as  response  t y p e s o f e x c i t a t i o n , was b a s e d on t h e f o l l o w i n g  considerations:  a)  b)  the fundamental p e r i o d T mainly  influences  fering  frequencies.  the water  n  i s the c h a r a c t e r i s t i c  dynamic r e s p o n s e t o i n p u t s o f d i f -  d e p t h d d e t e r m i n e s t h e maximum wave  and t h e w a t e r  contact  area.  influences  of h i g h e r harmonics  o f n o n l i n e a r waves t h a t  the r e l a t i v e  r e s o n a n c e and m o r e o v e r ,  t h e wave.  overall  c)  structural  size  depth a l s o  resulting  induce  distribution  i n f l u e n c e s the  and hence t h e n a t u r a l f r e q u e n c y .  o t h e r m a t e r i a l parameters in  The  The w a t e r  magnitudes  i n the case of  b r e a k i n g waves i t d e t e r m i n e s t h e v e l o c i t y in  dimensions  In the case of o s c i l l a -  t o r y waves, i t a l s o  structural  are r e f l e c t e d  i n some f o r m  t h e s e two p a r a m e t e r s . plot  o f t h e d-T s p a c e i n F i g . 25 d e l i n e a t e s t h e  r e g i o n s where v a r i o u s  t y p e s o f o c e a n waves and e a r t h q u a k e  would  The v a r i o u s  pair tical  that  govern design. of bounding  l i n e s which  structural  geometries.  regions  constitute  loading  are located within  a restriction  a  on p r a c -  98.  From an o v e r a l l p o i n t o f v i e w t h e f o l l o w i n g b r o a d trends  appear i n the v a r i o u s areas  i)  of the p l o t :  t h e d o m i n a n t i n f l u e n c e o f b r e a k i n g wave  forces i n  the d e s i g n o f s t r u c t u r e s w i t h water depths l e s s 90 f e e t ,  and a l s o  depths l e s s ii)  than  to a lesser 160  e x t e n t , on t h o s e  than with  feet,  the a p p r o p r i a t e n e s s o f c o n s i d e r i n g earthquake  loads  in  than  the design o f s t r u c t u r e s with  periods less  2  seconds iii)  t h e d o m i n a n t i n f l u e n c e o f p e r i o d i c deep w a t e r waves on  o f f s h o r e s t r u c t u r e s i n t h e r e s t o f t h e d-T r e g i o n .  T h e - e f f e c t s o f o t h e r k i n d s o f l o a d i n g such w a t e r c u r r e n t s c a n be s u p e r p o s e d preponderance of the e f f e c t This plot  without  as dead  affecting  l o a d s and the r e l a t i v e  o f one o f t h e above l o a d  i s a useful  types.  a i d i n preparing a f i r s t  design  o f a s h a l l o w o r deep w a t e r s t r u c t u r e o f t h e p l a t f o r m deck t y p e . 6.4  Other a)  conclusions: I n the shallow water range,  manipulation  s p a c i n g would n o t s i g n i f i c a n t l y but  structural  wave  response,  g e o m e t r y and t h e d e s i g n o f 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 cies  reduce  of p i l e  separated  from  o f t h e h i g h e r waves w o u l d do s o .  the frequen-  99. b)  The s t e a d y - s t a t e considerably and is  c)  r e s p o n s e t o waves as computed i s  l e s s when t h e i n t e r a c t i o n between w a t e r  structure velocities  i s considered  ignored.  Wave f o r c e s  i n the l a r g e . w a v e - h e i g h t range a r e p r e -  dominantly drag  f o r c e s whereas f l u i d  earthquake e x c i t a t i o n are mainly d)  forces  inertia  The r a n g e s o f w a t e r v e l o c i t y and p i l e the  m a g n i t u d e and f r e q u e n c y o f l i f t  have been s p e c i f i e d . and  t h a n when i t  lateral  forces greater  be s t u d i e d  than 3 seconds.  here the l i f t  forces.  d i a m e t e r where  are important  Combined r e s p o n s e i n l o n g i t u d i n a l  directions taking  should  u n d e r an  into consideration  f o r structures with F o r the s t r u c t u r e s  f o r c e s were n e g l i g i b l e .  lift  periods considered  BIBLIOGRAPHY  1.  W i e g e l , R.L. Oceanographical engineering: Prentice-Hall, I n c . , Englewood C l i f f s , N.J.: 1964. ,  2.  K i n s m a n , B. Wind waves: Prentice Hall, C l i f f s , N.J.: 1965.  3.  I p p e n , A.T. E s t u a r y and c o a s t l i n e h y d r o d y n a m i c s : E n g i n e e r i n g S o c i e t i e s Monographs: McGraw-Hill Co., I n c . : 1966.  Inc.,  Englewood  Book  4.  M o r i s o n , 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 : P e t r o l e u m T r a n s a c t i o n s , A.I.M.M.E. V o l . 189, 1950.  5.  H i n o , M. A t h e o r y on t h e f e t c h g r a p h , t h e r o u g h n e s s o f t h e s e a and, t h e e n e r g y t r a n s f e r b e t w e e n w i n d and wave: P r o c . 1 0 t h C o n f e r e n c e 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 R e i d , R.O. S u r f a c e waves and offshore 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 T e x a s A. & M. R e s e a r c h F o u n d a t i o n .  7.  C a m f i e l d , F . E . and S t r e e t R.L. O b s e r v a t i o n s and e x p e r i m e n t s on s o l i t a r y wave d e f o r m a t i o n : 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.  Hall,  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 H a r b o r s D i v i s i o n , A.S.C.E., V o l . 88, NO. WW3, August 1962.  10.  K e u l e g a n , G.H. and C a r p e n t e r , L.H. F o r c e s 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 B u r e a u o f S t a n d a r d s , V o l . 60, No. 5, May 1958.  11.  S a r p k a y a , T. and G a r r i s o n C . J . V o r t e x f o r m a t i o n 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, T r a n s . A.S.M.E., V o l . 85, S e r i e s E , M a r c h , 1963.  12.  A g e r s c h o u , H.A. and E d e n s , J . J . 5 t h and 1 s t o r d e r wave f o r c e coefficients for cylindrical piles: Coastal Engineering, Santa Barbara S p e c i a l t y Conference, October, 1965.  M.A. L a b o r a t o r y s t u d y o f b r e a k i n g wave f o r c e s piles: B e a c h E r o s i o n B o a r d T. Memo. No. 106, A u g u s t , 1958.  on  101.  13.  McNown, J . S . D r a g i n u n s t e a d y f l o w : n a t i o n a l de M e c a n i q u e A p p l i q u e e , 1957.  IX Congres I n t e r A c t e s , Tome I I I ,  14.  McNown, J . S . and K e u l e g a n , G.H. V o r t e x f o r m a t i o n 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. D i v . , A.S.C.E., V o l . 85, EM 1, P a r t 1, J a n u a r y , 1959.  15.  P a a p e , A. and B r e u s e r s , H. The i n f l u e n c e o f p i l e dimension on f o r c e s e x e r t e d by waves: Proc.. 1 0 t h C o n f e r e n c e o n 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 . , C o r n i n g , L.H. and Newmark, N.M. Design of multistory r e i n f o r c e d concrete b u i l d i n g s f o r earthquake m o t i o n s : 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. T a x t r a n s f o r m a t i o n o f waves i n s h a l l o w w a t e r : P r o c e e d i n g s o f 1 s t C o n f e r e n c e 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 o n Wave R e s e a r c h , 1950.  18.  I v e r s e n , H.W. Waves and b r e a k e r s i n s h o a l i n g w a t e r : Proceedings o f 3 r d 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. a n d S a s a k i , Y. Wave d e c a y i n g due t o b r e a k i n g : P r o c e e d i n g s o f 1 0 t h C o n f e r e n c e 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 , b r e a k i n g and r u n - u p o f t h e s o l i t a r y wave o n i m p e r m e a b l e r o u g h s l o p e s : P r o c . 1 0 t h C o n f e r e n c e 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., J o h n s o n , C.A. and W a l k e r , 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 H a r b o r s D i v i s i o n , A.S.C.E., P r o c . V o l . 85, No. W.W.I, 1959.  22.  L a i r d , A.D.K., J o h n s o n , C.A. and W a l k e r , 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 t h e 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 o n r i g i d c y l i n d e r s : J o u r n a l o f Waterways and H a r b o r s 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 W a r r e n R.P. G r o u p s o f 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 o f 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, F e b r u a r y , 1963.  102.  25.  L a i r d , A.D.K. F o r c e s 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 C o n f e r e n c e on C o a s t a l E n g i n e e r i n g , S a n t a B a r b a r a , C a l i f o r n i a , O c t o b e r , 1965.  26.  W i e g e l , R.L. 1970.  Earthquake  engineering:  Prentice  Hall,  103.  APPENDIX I CAUSES OF DISPARITIES BETWEEN WAVE FORCE COEFFICIENT DATA 1)  Most e x p e r i m e n t e r s w h i c h were computed  used v a l u e s  extents.  differed  experimenters  The t r u e v e l o c i t i e s , . i t  f r o m t h e computed v a l u e s  Another  source  measured  of divergence  the v e l o c i t i e s  some o t h e r p o i n t and r e l a t e d measured 2)  3)  prototype  4)  was t h a t  some  a t the c r e s t or a t  the drag  used  coefficients  the l i n e a r  o f t h e random v a r i a t i o n  to the  theory  i n computing  o f t h e wave  pointed, other  o f p a r a m e t e r s , w h i c h c o u l d n o t be p i n than N .  5)  U n c e r t a i n knowledge o f t h e d i f f u s i o n  6)  Neglect  of turbulence.  o f t h e c o n v e c t i o n a l terms o f t h e a c c e l e r a t i o n  l^jr'in t h e f o r c e Varying  form  tests.  The e f f e c t  7)  to varying  v a l u e s whereas o t h e r s u s e d n o n l i n e a r t h e o r i e s .  Neglect in  i s con-  ones.  Some e x p e r i m e n t e r s velocity  velocities  i n s t e a d o f b e i n g measured d u r i n g the  wave f o r c e e x p e r i m e n t s . cluded,  f o r water  expressions.  r o u g h n e s s and f l e x i b i l i t y  o f t h e m o d e l s and  prototypes. 8)  Vibrations of test  9)  Turbulence test  p i l e s were  piles.  a r o u n d t h e s t r u c t u r e s by w h i c h t h e p r o t o t y p e supported.  104.  APPENDIX I I WAVE THEORIES The  time p e r i o d  o f o c e a n waves and t h e i r  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 s p e e d s and d u r a t i o n ) .  w i t h wind  Linear  (fetches,  have wind  For computations o f the c h a r a c t e r i s t i c s  o f d e t e r m i n i s t i c wave f o r c e s , t h e p e r i o d known i n d e p e n d e n t  inputs  heights  and h e i g h t  w o u l d be  data.  Theory For  a s i m p l e h a r m o n i c wave p r o g r e s s i n g  i n t h e x-  d i r e c t i o n w i t h p h a s e v e l o c i t y C ^ a s shown i n F i g . 26, t h e 2 differential - d < z  e q u a t i o n 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  < riis r  ift+  ifi = o  8x  8x  (ii-D  where <(> i s t h e 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 Horizontal  velocity u = — V e r t i c a l  such  velocity w = —r^-  boundary  condition W u  The  condition  tion:  (II-2)  o Z  oX  The  that  a t the bottom i s :  =  -|i = 0  =  8z —1^=0 3x  on z = -d  (II-3)  on z = - d  on t h e u p p e r b o u n d a r y  i s a mixed boundary  condi-  105.  z=-d  LINEAR WAVE  Fig  26  106.  - '|||'  + \  (u  + w)  2  + ||  2  z=n where  p = fluid  p  pressure  (zero a t free  (p' = mass d e n s i t y o f t h e  + gz|  z=n  = 0  z=n  surface)  fluid  Since 2  3cb • 3t  U  n =  t h i s becomes  K  o  linear  <  n  theory  Ig l9 ti  rj =  general  W  — g  In the small-amplitude  The  2 ^ ;  solution  3d) 9t  2  =  n  this  i s simplified  on z = 0  (H-4)  of equation  (II-l)  i s o f t h e form  C z  -C z  8  d>  =  C,+C„x+G X  T  Z  z+(C.cosCoX+C-sinCoX)(C,e 4  S  O  D  O  to  8  +C_e  D  )  /  or e q u i v a l e n t l y <J> =  C +C x+C z+(C cosCgX+C sinC x) {C coshC 1  2  3  4  +C  Using  1 ( )  5  sinhC  8  the s p a t i a l p e r i o d i c i t y  8  g  (z+C  i:L  )  (z+C ) } 1 2  o f <j> a t i n t e r v a l s  c = 2  8  0  o f l e n g t h L,  107. Having chosen t h e moving  o r i g i n of coordinates a t the c r e s t  means  The  C  c  =  0  c  4  t.  o  zero net transport of f l u i d  c Use o f t h e c o n d i t i o n  i n the v e r t i c a l d i r e c t i o n  =  3  0.  I I - 3 makes  C  10  =  0  C  12  =  d  cj) = C / • C c o s h { - ^ 2  Q  Changing the moving velocity  *  being  o r i g i n of coordinates  where C  p h  t o a f i x e d one,  ^  2  4  9  Use o f I I - 4 and a v e l o c i t y  "  (z+d)} s i n 4^ x  C . C c o s h { - ^ ( z + d ) } s i n ( 4 £ x- ^-t)  ej) =  i *  yields  C  L H u = ^ at z = ^  /-% H c o s h 2TT (z+d)/L ph 2 s i n h 27rd/L  S i n (  . ,2ir . -J7 X  yields  2m, x T" t )  = |.  The v e l o c i t i e s  " ~  are then derived:  , (  I  I  _  T T 5  )  108. Horizontal Velocity  component 2Tr(z+d)  °'f  c o s c  l i ^ 2 W L  (  ^ K - ^ t )  (II-6,  2Tr(z+d) vertical  Surface  component w  n  elevation  Specialising  = f  S  ^  h  2  H 2 = ^ c o s (-^  ^  d  /  sin(^  L  x  - ^ t )  (II-7)  2  x -  t)  t o t h e c a s e o f deep w a t e r  (II-8)  cl 1 (— > •=•) ,  2 aT L e n g t h o f wave L ^ = Wave p h a s e . velocity  (II-9).  ; C . = ^ i  Variations  ^ gT  (11-10)  of pressure with depth are  Finite-Amplitude  Stokes  negligible.  Theory  2 3 This motion  i s irrotational  T/J e x i s t . ferent  theory '  The  free  and b o t h p o t e n t i a l  s u r f a c e boundary  from the l i n e a r The  s t a r t s w i t h the assumptions  solution  of coordinate  axes,  .  function  i s however  dif-  of  2  to the boundary  condition  stream  the  case.  V. subject  <f> and  that  =  conditions  0  (11-11) as u n d e r , w i t h a m o v i n g  system  109.  z=n  *lz=-d  g  n  +  1  [  | i , 2  (  +  i  = k  | i  (  )  2  ]  .  =  k  3  Z=T]  is  found  t o be  T|;(X,Z)  =  C^z  +  C  +(C  2  cos  3  C x  +  7  sin  C x) 7  C z " 7 7 + C e (C e ' 6 ) C  +  u  z :  (H-12)  e  5  Using  the c o n d i t i o n s o f 1)  spatial periodicity  2)  vertical velocity  3)  horizontal velocity  at intervals  of length L  a t b o t t o m f o r d->°°  being  a t b o t t o m f o r d-*°° b e i n g  C, 6  =  0  C, 1  =  C , ph  where C ^ = s p e e d o f m o t i o n o f t h e c o o r d i n a t e axes,  =  T|;(X,Z)  , where  2 TT = -^r-  Since  ip| =0 z=n  c  2  =  ~ ph c  z  +  C  2  +  C  5  e  K  Z  * 3 C  for a l l , n  0  C  O  S  K  X  +  c  4  sinK  x)  zero zero  110. r  ' — - — = - z + 3e ph  i s a particular solution i f  COSKX  c  C  3  3 5 C  =  (II-13) Ph  and  C  Using  4  ^  =  Cauchy-Riemann r e l a t i o n s ,  the p o t e n t i a l  function  <j) i s f o u n d  <b(x,z)  , . KZ . = -x + £e sin<x.  Ph From t h e e x p r e s s i o n  f o r ^, p u t t i n g  n  =  3e  1  Third  correct  Expressing  ....  3 ]CQSKX  (11-14)  2 3 '  values  f o r n,ip and $  t o the t h i r d  order  a r e g i v e n by t h i s  i n 3.  n as  n  and  2 1  Order Theory  Approximate theory  cosKX  3 t l+KTl+2" (KT|) +g-(KTl) +  n =  Stokes'  K r i  z = n,  =  3n  0  + 3 n-L + 3 n  2  substituting  3n Retaining  + 0  2 3 n-L +  3 3 n  2 terms o n l y up t o 3 ,  = 2  F  (3/n ' 0  n  fn 'K' ) x  1  2  111.  n  Q  +  gn-L  3  +  2  n  c o e f f i c i e n t s  3  o f  , 3  n  ^1  =  2  =  n  Therefore  n  KnQ-  ^  l 2  T 1  +  \<$  =  1  K T 1  2 o^ C  =  O  S  X  )COSKX]  r e s p e c t i v e l y ,  K  k  2 Q  C O S ' K X  8"  =  2  2  c  o  s  2  O  S  K  COSKX  X  „  |  9 2  K C  + 3 (1+-|K 3 )  2  3  1 ^ 1 2"^ 2  =  n  2 a n d  0  KX  c o s  1  1 2  [ ( K O ^ ^ K  0  0 e q u a l i n g  2  [cosKx] + 3[Kn cosKx]+3  = 2  <  x  ^K3  +  3 2 cos 3 K X .  {j*  +  2  cos  2KX+|-K 3 COS3KX 2  3  (11-15)  The  coefficient  Solving  9 2 2 l e t t i n g a = 3(l+-g -K 3 )  o f cos K X i s rearranged,  f o r 3, 3  1  2 3 e^a + e a + e^a +...  =  2  r E 2 F. 3 9K2,F 2 or a = ^ a + a + ^ a + ^ ( ^ a + e ^ +e a b  Retaining like  h. =  Choosing  3  terms upto  powers,  3,3 )  K  2  3 r d power o f a and e q u a t i n g  3, s^, e 1  2  +  and  2  a  e  obtained,  1 2 + -^a COS2KX  COSKX  A  r  +  c o e f f i c i e n t s of  and t h e n 3 2 3 a COS3KX  g-K  (11-16)  1 2 a new m o v i n g o r i g i n o f c o o r d i n a t e s a t z = ^ K 3  n = a  Substitution  COSKX  of g = =  1  +  2  COS2KX  2 e^a + e a + e 2  a  —  9 2 3 g- K a  + 3 a 3  3 2 3  -g-K a  cos  3KX  (11-17)  112.  in  the expressions  f o r <j>, y i e l d  components o f v e l o c i t y . coordinates  <(>, a n d by d i f f e r e n t i a t i o n , t h e  The e x p r e s s i o n  changes f o r f i x e d  to  n  =  ,  2TT,.,1  COS(KX——t)+-^Ka  a  2  „,  2ir,, ,3 2 3  COS2(KX-—t)+g-K  a  2 IT. »  COS3(KX—^-t)  (11-18)  Stokes  T h i r d Order T h e o r y — F i n i t e  1  Similar  t o the - f o r e g o i n g  when d->-°°, a p e r t u r b a t i o n c{>,  Depth  n and c e l e r i t y  technique  d e r i v a t i o n f o r the case a p p l i e d t o the s o l u t i o n s  C ^ and a p p l i c a t i o n o f t h e s u r f a c e a n d 1 3  bottom boundary  conditions  yield  '  the general  third  order  relations: Wave p r o f i l e n = a c o s ( K X - ^ t ) + ^ f  ^  c  where ±  - ^ /d  = f (j-) =  2  x  cos2(KX- ^t)+^- -f L 2  2  a  3  cos3( x-^t) K  (11-19)  (2+cosh4-rrd/L)cosh27Td/L ^  2  2.  smn  —=— XJ  f  = f (^) 3 3 L _  Wave h e i g h t  = _ L l+8cosh 2fTd/L 16 . ,6 j / sxnh 2Trd/L 2 H = 2a + a f (£) 6  n  T  (11-20)  3  XJ  Horizontal  velocity  u = C . [F c o s h ph 1 n  2 7 T  ( z + d ) T  cos ( x - ^ t ) + F c o s h T 2 K  L  cos2 ( K X - ^ t ) + F c o s h 3  6 7 T ]  f  4 7 r  0  Z + d )  ( Z + d ) T  Li  cos3 ( x - ^ t ) ] K  (H-21)  113.  and h o r i z o n t a l  |H  ! ! ^  =  F  " 3u acceleration'—  local  i  C  O  dt  h ^ I  S  s i n C ^ t j ^ F ^ o s h i l i ^ I  sin2 ( K X - ^ t ) i ^ £ h +  where F  =  n  £  *3  sinh  L  o 2  c o s h ^ ±  •A  *  4  LT  '  sin3 ( K X - ^ t )  (11-22)  . ,4 2Trd sinh —=—  J  " ,. _ 2±ltanh^[l+(^)  2  14+4cosh V 16sinh  2  4  =  2 2T_ m  t a n h  _ . 2rf  £  1  +  ii  % ] ^  (11-23)  2 4ird _ _ 14+4cosh .212. ,2™,2 L , , 16smh - 7 = —  The above d e v e l o p m e n t p r e s u p p o s e s t h a t known.  )  _ _3_ iZ-rra. 3 pll-^cosh-^n^--, 64 L . . 7 2ud sinh —=—  =/  L  d  ^  ;  0  c  3  ™  2  1  F  (  I  I  .  2  4  )  t h e wave l e n g t h L i s  L i s t o be f o u n d f r o m t h e n o n l i n e a r  relation at  (11-24)  above. S o l i t a r y Wave T h e o r y This above  r e p r e s e n t s a wave w i t h t h e e n t i r e w a t e r body  the o r i g i n a l water l e v e l ,  p a r t i c l e s move o n l y  and m a t h e m a t i c a l l y  the water  i n t h e d i r e c t i o n o f wave advance/.^  16- ' shows t h e wave l e n g t h  i s infinite.  lying  As F i g .  The e q u a t i o n s f o r t h e  114,  water p r o f i l e  and wave v e l o c i t y a r e :  y  = d+H s e c h [ | -3 d 2  s  (x-Ct)]  /  (11-25)  = VgS ( 1 + f §)  C  = Vd+f) These a r e c o r r e c t additional adequate  expressions  (11-26)  to the 1st order  f o r the p a r t i c l e  i n the v i c i n i t y  of the c r e s t  and a l o n g  velocity  with  u are not  f o r large values  of the  H r a t i o -r d  JL_ = | s e c h [ ^ /glT d"  (x-Ct)]  2  Munk-McCowan  S o l i t a r y Wave T h e o r y  This vicinity provide  theory''" i s more r e l i a b l e  of the c r e s t  particularly  o f t h e wave f o r l a r g e v a l u e s  a b e t t e r f i t to the scanty  however more d i f f i c u l t is  (11-27)  experimental  i n computation  i n the  o f ^ and  data.  and t h e s u r f a c e  I ti s pressure  not constant.  C =  ^dd+l)  (11-28)  1 .My , ,Mx 1- c o s (.—4r) c o s h (-5-) H = n[~ —J± —-5—_] C /Myi , , ,Mx. 2 [cos (-^-)+cosh (-^-) ] ^ N  r  where M a n d N a r e f o u n d  N  n  from the f o l l o w i n g :  (11-29)  115.  | = |  tan  llMU+l)]  N = | sin [M(1+| |)] 2  This velocity  theory  yields  ^ than the previous  lower v a l u e s  of the  dimensionless  one a n d i t s g e n e r a l i s e d  third  order  form.  Cnoidal  Theory  This  i s a nonlinear  cl 1 1 w a t e r where — < -=-?- t o L -LU Zo  waves i n s h a l l o w functions  K(k),E(k),cn  which a r e i n v o l v e d . dent i n p u t s .  theory"'' f o r p e r m a n e n t Jacobian  periodic elliptic  u and s n u a p p e a r i n t h e e x p r e s s i o n s ,  T h e wave p e r i o d T and h e i g h t  Wave l e n g t h  H a r e indepen-  L i s g i v e n by  k = —  K(k) ( 2 L + 1 - - ^ ) "  /I  1 / 2  (11-30)  a  i n w h i c h L and k a r e d e f i n e d  by 2 e q u a t i o n s a s f o l l o w s :  y /d)-(V ) d  A  k  2  =  <** 2L+1-( t/d) Y  y (2L+1—^)E(k)  =  y y ( 2 L + 2 ~ ' - -^)K(k)  where K ( k ) = c o m p l e t e e l l i p t i c  integral  (Ref.l)  of the 1 s t k i n d o f  modulus k y  t  y E(k)  = distance  from the ocean bottom t o t h e t r o u g h  = distance  from the ocean bottom t o the c r e s t  = complete e l l i p t i c modulus k. '  i n t e g r a l o f t h e 2nd k i n d o f  116.  An a p p r o x i m a t i o n  to L i s  L  =  v  / 16d  3  kK(k)  The wave p r o f i l e i n t e r m s o f y  (II-31)  measured from the bed i s g i v e n s  by Y  s  = y  + H cn [2K(k)  - |) ,k]  2  t  Y  = y  2  r /  3  + H c n [/|  t  t,  (2L l--er)  ,  +  L j  -(x-|t),k]  (11-32)  i  The wave v e l o c i t y  C =  /gd[l+| d 2  v  k  (^ - f-S] 2 K(k)  (11-33)  Water p a r t i c l e v e l o c i t y u:  u  -  r l ,  /id  * H  3  y  u  2  —j  cn  t  ^  f  3 H  4d" 4. (  y  ^  2  where s n (  H  . 2,  .  2d'  ,  8HK (k)  ,d y  2  )  4d^ cn (  t  ) + cn ( 2  w  .2  (o~=^) i - k  IT  J  ^  ) dn ( 2  ) = s n [ 2 K ( k ) ( - - h] x  •Li  2  '-  x  etc.  sn  2. (  .  )  a  ) -sn ( 2  ) dn ( 2  )}]  (11-34)  117.  APPENDIX I I I RELATIONS FOR  The the  form o f  THE  THIRD ORDER RUNGE—KUTTA METHOD  system o f equations  1st order  t o be  solved  are r e w r i t t e n i n  equations:  u.  z.  i=l,....n  (IH-1)  i=l,.....n  (IH-2)  1  dz, u,  is  a v a i l a b l e from the  preceding For  dt  step,  and  a succeeding  first  computed v a l u e s  through the  use  time s t e p , the  d e r i v a t i v e are  f o u n d as  o f the  where  K  il i2  + |  K  I  ,t)  (At)z (z +|  U  1  u  +  K l l  (At) z. (  q  2 1  +§  t +  K  ,Z2+|  I  K 2  i---  Vl nl' l 3 ll' 2 I 21' K  U i  (  t +  At)  +  K  Z  +  K  q  U  +  q  = u (t)  + \  U  +  q  U  +  <  (III-4)  n 2  ±  +  2 , 2 f 2 2 ''' • n § n2' l l 12' 2 l 322'  , . .u +|q ,t+|(At)) n  its  ( A t > )  Z  1  motion.  (IH-3)  3  (At) z . ( z . , z_, . . . z , u,,u_,...u 1 1 2. n 1 2 n  ••• n 3 nl' i3  of  the  follows:  ±  i  K  equations  d e p e n d e n t v a r i a b l e u and  z (t+At) = z ( t ) + ±  of v a r i a b l e s of  q .+ ±1  | q  i 3  (III-5)  118  where i l  q  Substituting • is  =  (At)z  i  q  i 2  =  (At)  (z. +  q  i 3  =  (At)  (z±  (ITI-6)  into  l  +  (III-4)  K  § K  i  l  ±  2  ) (III-6)  )  and ( I I I - 5 ) ,  (III-5)  rewritten  u (t+At) ±  = u (t)  1  =  +  ±  (At)z (t)  +|  ±  (At)K  ±  (IH-7)  2  and K  ±  (At)z [z (t) ,z (t) ,z (t) ...z (t) ,u (t) ,u (t) , ±  1  2  3  n  x  2  . . . u (t) , t ] n  K  i2  =  ( A t ) Z  i l4 ll' 2 I 21' ' • • V k l ' V l [ Z  K  Z  +  K  V  ( A t )  u +|(At) z ., . . .u +i(At) z , t + | ( A t ) ] 2  K  ±  3  =  2  (At)z [z +| i  2  1  n  n  K l 2  ,z +§K 2  2 2  ,...z ;V..X:^+|c n  2 2 2 (At) z +^-(At) K , U + - ( A t ) z +-g(At) K 1  1 1  2  3  2  . . . u + | ( A t ) z + f (At) K -j , t + | ( A t ) ] n 3 n 9 ni 3 ( I I I - 3 ) , ( I I I - 7 ) , and  (III-8)  are e x p l i c i t  formulae.  2  1  n 2  ,  , (III-8)  119. The third  a b o v e s e t o f e q u a t i o n s a r e p a r t o f one  o r d e r Runge^Kutta  parameters. •  relations  for a particular  form of the  c h o i c e of  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0050553/manifest

Comment

Related Items