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E X P E R I M E N T A L STUDY  OF LOGGING C A B L E SYSTEMS  b y  DANIEL YVES D i p l o m e  d ' I n g § n i e u r E N S A M  A  T H E S I S T H E  S U B M I T T E D  O F  e n  M e c a n i q u e  ( 1 9 7 5 )  I N  R E Q U I R E M E N T S M A S T E R  GUIMIER  P A R T I A L F O R  T H E  A P P L I E D  F U L F I L M E N T D E G R E E  O F  S C I E N C E  i n  T H E  We  a c c e p t t o  T H E  F A C U L T Y  t h i s  t h e  O F  M a y ,  D a n i e l  F O R E S T R Y  t h e s i s  r e q u i r e d  U N I V E R S I T Y  ©  O F  a s  c o n f o r m i n g  s t a n d a r d  B R I T I S H  C O L U M B I A  1 9 7 7  Y v e s  G u i m i e r  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e -  quirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference  and study.  I f u r t h e r agree  t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood  that  copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n  Department of  Forestry  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date  Columbia  permission.  ABSTRACT Two  t h e o r e t i c a l formulations of logging cable  sys-  tem p r o b l e m s , t h e c a t e n a r y m o d e l and t h e p a r a b o l i c m o d e l a r e i n v e s t i g a t e d and compared w i t h t h e r e s u l t s o f e x p e r i m e n t s executed  on a g r a v i t y s y s t e m f i e l d  The s t u d y  model.  shows t h a t a l t h o u g h  t h e shape o f a f r e e  hanging c a b l e i s b e t t e r d e s c r i b e d as a c a t e n a r y bola, both  t h e o r e t i c a l models a r e a c c u r a t e  than a  para-  enough t o s o l v e  p r a c t i c a l c a b l e system problems.  The f e w d y n a m i c t e s t s t r i e d on t h e f i e l d  model  show t h e g r e a t i m p o r t a n c e o f t h e d y n a m i c f o r c e s i n a l o g g i n g c a b l e s y s t e m and t h e need f o r f u r t h e r r e s e a r c h i n t h i s field.  iii  T A B L E OF  CONTENTS  Page Abstract Acknowledgements CHAPTER 1: INTRODUCTION  1  CHAPTER 2: INTRODUCTION TO CABLE MECHANICS  8  2. .1 2. 2 2. 3 2. 4  General D e s c r i p t i o n of the, System M o d e l l i n g Assumptions Catenary Model P a r a b o l i c Model  CHAPTER 3: DESCRIPTION OF 3. 3. 3. 3. 3. 3.  1 2 3 4 5 6  3. 7 3. 8 3. 9  THE  FIELD MODEL  S i t e Dimensions and C h a r a c t e r i s t i c s Cables Carriage Winches Dynamometers Surveying of Cable Shape and C a r r i a g e Position Other Measurements A c c u r a c i e s of Instruments and Expected E r r o r s i n the Measurements Dimensional S i m i l i t u d e between the Model and A Real Yarding System  CHAPTER 4: FREE HANGING CABLE  8 11 14 16 19 19 20 23 23 29 32 36 39 43 46  D e s c r i p t i o n of the Experiment A n a l y s i s of the R e s u l t s Error Analysis R e s u l t s and C o n c l u s i o n s  46 47 53 57  CHAPTER 5: CLAMPED LOAD ON A SINGLE LINE  80  D e s c r i p t i o n of the Experiment A n a l y s i s of the R e s u l t s R e s u l t s and C o n c l u s i o n s  80 81 87  4. 4. 4. 4.  1 2 3 4  5. 1 5. 2 5. 3  CHAPTER 6: GRAVITY SYSTEM 6. 1 6. 2 6. 3  D e s c r i p t i o n of the Experiment A n a l y s i s of the R e s u l t s R e s u l t s and C o n c l u s i o n s  95 95 96 98  iv  Page CHAPTER 7: DYNAMIC TESTS  111  7.1 7.2  Equipment Tests  111 111  7.3  Conclusion  124  CHAPTER 8: DISCUSSION AND CONCLUSION  125  Literature Cited  129  Appendix 1  131  Appendix 2  18 3  Appendix 3  186  Appendix 4  205  V  LIST  OF F I G U R E S  Figure 1  Page The three and  cable  systems  experimented  analyzed  7  2  Characteristics  3  Loading assumption for the derivation  and reference frames o f t h e catenary model  Loading  and reference  4  for  of a cable  assumption  the derivation  yarding  system  10  13  frames  of the parabolic  model  13  5  Sketch  of plan  view  of the f i e l d  model  22  6  Sketch  of  view  of the f i e l d  model  22  7  Plan view layout  and dimensions  Side view layout  and dimensions  8  9  10  11  Y-position Experiment  side  of the  surveying 34  of the  surveying 34  of points of the cable: versus models. T e s t #4  Y-position of points of the cable: Experiment versus catenary model Influence of t h e error on T . Test  50  #4  56  Y-position of points of the cable: Experiment versus p a r a b o l i c model Influence of t h e error on T . T e s t #4  56  Y-position Experiment T e s t #4  59  n  12  13  14  15  of points of the cable: versus model Error-zone  Y-position of points of the cable: Experiment versus model,- Average and maximum d i f f e r e n c e f o r t h e n i n e free hanging cable tests  59  Cable shape: Experiment versus b e s t - f i t models - Average d i f f e r e n c e f o r t h e nine free hanging cable tests  62  Percent d i f f e r e n c e between T and parameter c a l c u l a t e d f o r b e s t - f i r curves, for the nine free hanging cable tests  62  vi  Figure 16  Page Y - p o s i t i o n of p o i n t s o f the c a b l e : Catenary versus p a r a b o l i c model. Test  #4  17  Sketches of p a r a b o l i c and catenary hanging cable shapes  18  D e f l e c t i o n a t mid-span: Catenary p a r a b o l i c model  19  Tension a t the lower support: versus models  Experiment  20  Tension a t the lower support: versus p a r a b o l i c model  Catenary  21  free  versus  65. 65 67 70 70  Angle of the l i n e with the h o r i z o n t a l at the upper support: Experiment versus models  73  Angle o f the l i n e w i t h the h o r i z o n t a l a t the lower support: Experiment versus models  73  Angle of the l i n e w i t h the h o r i z o n t a l a t the upper support: Catenary versus p a r a b o l i c model  75  Angle of the l i n e w i t h the h o r i z o n t a l at the lower support: Catenary versus p a r a b o l i c model  75  25  Cable l e n g t h : Experiment  78  26  Cable l e n g t h : Catenary versus model  22  23  24  27  28  versus models parabolic  78  S k y l i n e w i t h a s i n g l e concentrated load f o r t h r e e d i f f e r e n t p o s i t i o n s of the clamped load  86  Y - p o s i t i o n of the l o a d : versus models  86  Experiment  29  Y - p o s i t i o n of the l o a d : Catenary p a r a b o l i c model  versus  30  Sketches of catenary and p a r a b o l i c clamped load load-paths  89 89  vii  Figure 31  Page Force balance at the clamped load u s i n g catenary and p a r a b o l i c model  91  32  Tension a t the lower support: versus models  Experiment  94  33  Tension a t the lower support: versus p a r a b o l i c model  Catenary  34 35  36  Y - p o s i t i o n of the c a r r i a g e : versus models  94  Experiment 100  M a i n l i n e shapes as p r e d i c t e d by the models f o r t h r e e of the g r a v i t y system tests  100  Y - p o s i t i o n of the c a r r i a g e : versus p a r a b o l i c model  104  Catenary  37  Skethces of the catenary and p a r a b o l i c g r a v i t y system load paths  104  38  Tension a t the lower support: versus models  Experiment 106  Tension at the lower support: versus p a r a b o l i c model  Catenary  39 40 41 42  43 44 45 46  106  Tension i n the m a i n l i n e at the upper support: Experiment versus models  10 9  Tension i n the m a i n l i n e at the upper support: Catenary versus p a r a b o l i c model  109  Chart r e c o r d i n g of the t e n s i o n i n the s k y l i n e a t the upper support d u r i n g v e r t i c a l o s c i l l a t i o n s of the load  116  Sketch of a dynamic t e s t . stopped w i t h the m a i n l i n e  Carriage 118  Sketch of a dynamic t e s t . stopped w i t h a clamp  Carriage 118  Chart r e c o r d i n g s of the s k y l i n e and m a i n l i n e t e n s i o n s d u r i n g a dynamic t e s t  121  Chart r e c o r d i n g s of s k y l i n e t e n s i o n d u r i n g dynamic t e s t s  123  viii  Figure  Page  47  P a r a b o l a i n the c o o r d i n a t e system  48  B a s i c p r i n c i p l e of the tensiometer  188  49  Copy of the b l u e - p r i n t of the tensiometer  191  50  Sketch of the equipment s e t up f o r the c a l i b r a t i o n of the tensiometer  19 6  51  Tensiometer r e a d i n g versus t e n s i o n i n the l i n e b e f o r e c a l i b r a t i o n  196  I n f l u e n c e of the gauge f a c t o r of the i n d i c a t o r , on the tensiometer r e a d i n g  199  I n f l u e n c e of the zero d i a l adjustment on the tensiometer r e a d i n g  199  Tensiometer r e a d i n g versus t e n s i o n i n the l i n e a f t e r c a l i b r a t i o n  202  52 53 54  (x,y)  183  ix  L I S T OF  PLATES  Plate 1  Page F i e l d model seen from the lower support  spar at  the 25  2  V i e w o f t h e c a r r i a g e and l o a d , towards the upper support  3  M a i n l i n e d i r e c t e d t o t h e G e a r m a t i c 19 winch w i t h a b l o c k a t the upper support  28  S k y l i n e passing the top of the spar at t h e l o w e r s u p p o r t and c o n n e c t e d t o t h e load c e l l  28  S k y l i n e and m a i n l i n e the upper support  31  4  5 6  looking  tensiometers  Reading of the t e n s i o n s at the upper support  i n the  25  at  two  lines 31  7  S u r v e y i n g o f t h e c a b l e and c a r r i a g e p o s i t i o n s w i t h the t h e o d o l i t e  38  8  Measurement o f t h e f r a c t i o n o f metre between lower s u p p o r t r e f e r e n c e p o i n t and t h e f i r s t p a i n t mark on t h e c a b l e  38  S t r i p c h a r t r e c o r d e r , g e n e r a t o r and transformer r e g u l a t o r used f o r the r e c o r d i n g of the tensions at the upper support  113  Manual i n i t i a t i o n of the v e r t i c a l o s c i l l a t o r y motion of the clamped l o a d  113  9  10  X  L I S T OF  TABLES  Table  Page  I  Cable  II  Requirements  III  Experimental errors  IV  Two  V  characteristics  23  f o r the winches  systems t h a t  26 42  the model can  F i e l d and computed r e s u l t s . . h a n g i n g t e s t #4  simulate  Free 51  VI  Experimental errors.  Free hanging  VII  Discrepancies i n free characteristics  hanging  VIII  Experimental errors.  Clamped  IX  Precision  of f i e l d  45  cable  53  segment 82 load  test.  84  model, r e a l y a r d i n g  system  126  X  Requirements  XI  Load  XII  Gauge f a c t o r  XIII  Experimental r e s u l t s .  Free hanging  test  208  XIV  Experimental r e s u l t s .  Clamped  load  test  209  XV  Experimental r e s u l t s .  Gravity  system  cells  f o r the tensiometers  19 2  characteristics  193  adjustment  200  test  210  xt  ACKNOWLEDGEMENTS I wish t o express my g r a t i t u d e t o Mr. G.G. my  s u p e r v i s o r , who  dent program  and guided me  I would of  i n the development  l i k e t o thank Mr. G.V.  graduate s t u of t h i s  thesis,  Wellburn, manager  the F o r e s t E n g i n e e r i n g Research I n s t i t u t e of Canada  (FERIC)  f o r the f i n a n c i a l support t h a t made t h i s study  possible. tor  a s s i s t e d me throughout my  Young,  I would  a l s o l i k e t o thank Mr. J . W a l t e r s , D i r e c -  of the U n i v e r s i t y of B r i t i s h Columbia Research F o r e s t  (UBCRF) f o r the use of f a c i l i t i e s on the F o r e s t .  I am a l s o t h a n k f u l to the f o l l o w i n g persons f o r their help: the  members of FERIC and the members of the UBCRF  who  a s s i s t e d me  i n the f i e l d work.  Mr. D. Myhrman, mechanical engineer at FERIC, f o r his  guidance and c o n s t r u c t i v e  Mr. D. Anderson  criticism.  f o r h i s a s s i s t a n c e i n the  field.  Mr. K. Vatsag f o r h i s e x c e l l e n t machining work. Messrs. H. J o l l i f f e and J . Walters f o r r e v i e w i n g my and  thesis.  Mrs. C. van Beusekom f o r her f a s t and a c c u r a t e typing.  E X P E R I M E N T A L STUDY  OF  CABLE LOGGING  CHAPTER  SYSTEMS  1  INTRODUCTION  Cable systems f o r h a n d l i n g and t r a n s p o r t i n g logs are widely u t i l i z e d by the f o r e s t i n d u s t r y i n the P a c i f i c Northwest.  E a r l y i n the h i s t o r y of l o g g i n g , c a b l e s were  used to h a r v e s t timber to  and c a b l e systems have been improved  i n c r e a s e the e f f i c i e n c y o f the o p e r a t i o n .  development of these  The major  systems was c e r t a i n l y the i n t r o d u c t i o n  of " h i g h - l e a d " at the t u r n o f the century. t e c h n o l o g i c a l advancements  Since then new  have been i n t r o d u c e d and the  e x i s t i n g systems have c o n s t a n t l y evolved  towards the new  requirements o f the l o g g i n g i n d u s t r y .  A look upon the present  s i t u a t i o n i n d i c a t e s a need  f o r y a r d i n g systems t h a t w i l l : i ) Reduce f o r e s t road d e n s i t y because of high road c o n s t r u c t i o n c o s t and because o f e n v i r onment c o n s t r a i n t s , i i ) Harvest  efficiently  timber  on s i t e s i n ^  a c c e s s i b l e with c o n v e n t i o n a l iii)  Meet the needs f o r improved practices.  systems, silvicultural  - 2 -  i v ) P r o t e c t the environment. Cable l o g g i n g systems can meet t h i s  challenge.  The most e f f i c i e n t use of e x i s t i n g methods and opportunity  of developing  new  ideas r e q u i r e s t h a t the  the  engin-  e e r i n g c h a r a c t e r i s t i c s of c a b l e systems be w e l l known. Those c h a r a c t e r i s t i c s are s t u d i e d i n cable mechanics.  S o l u t i o n s to problems i n cable mechanics f o r the p a r t i c u l a r case of l o g g i n g were attempted a long time Although the b a s i c problem i s easy to formulate  mathemati-  c a l l y , numerical s o l u t i o n s are d i f f i c u l t to o b t a i n . techniques were developed to circumvent the however, only the more recent and here.  Several  difficulty,  important ones are  Lysons and Mann(6) p u b l i s h e d  ago.  reported  a graphical-tabular  method to determine what payload a l o g g i n g system can  carry  over a given p r o f i l e .  the  Carson and Mann(2) reformulate  a n a l y s i s , d e s c r i b i n g the l i n e segment as a catenary, present catenary  and  an i t e r a t i v e technique f o r the s o l u t i o n of s k y l i n e equations.  Another p u b l i c a t i o n by Carson  and  Mann(3) proposes an a l g o r i t h m to determine the load path of a running  s k y l i n e u s i n g a s t r a i g h t l i n e approximation f o r the  load d i s t r i b u t i o n on the l i n e segments; t h i s  assumption  y i e l d s a p a r a b o l i c shape f o r the l i n e segment.  The  ment of the p a r a b o l i c model f o r the study of cable i s presented  i n Appendix 1 and  of t h i s t h e s i s .  represents  developsystems  a significant  part  - 3 -  Two major t h e o r i e s , the catenary model and the p a r a b o l i c model, are t h e r e f o r e a v a i l a b l e t o d e s c r i b e systems.  cable  However, even the most e l a b o r a t e f o r m u l a t i o n i s  based on c e r t a i n degrees o f assumptions, and a q u e s t i o n r e mains as t o know how w e l l the t h e o r i e s r e p r e s e n t the a c t u a l systems.  F i e l d measurements a r e r e p o r t e d t o have been made  on r e a l l o g g i n g systems(8)(9)(10) and p r a c t i c a l t a b l e s were proposed f o r some s p e c i f i c  To the author's  cases.  knowledge no other  experimentation  has been c a r r i e d out t o i n v e s t i g a t e thoroughly of l o g g i n g c a b l e systems.  The need f o r an  the mechanics  experimental  study t o c o n f i r m the t h e o r e t i c a l approaches would t h e r e f o r e seem  necessary.  The  l i m i t a t i o n s o f the mathematical  i s another p o i n t t o be c o n s i d e r e d .  formulations  Most o f the models  assume t h a t the c a b l e s and load are f r e e from the ground; only Carson(4) makes an attempt t o model the dragging l o g , but t h a t model should be t e s t e d i n the f i e l d .  of a  Another  l i m i t a t i o n , and c e r t a i n l y the most r e s t r i c t i v e one i s t h a t all  s t u d i e s a r e based on the f o r m u l a t i o n o f the s t a t i c  e q u i l i b r i u m o f the system when i t i s obvious t h a t the s k i d ding of logs i s a h i g h l y dynamic o p e r a t i o n .  No  simple  a n a l y t i c a l study can model a c c u r a t e l y the behaviour o f a c a b l e system i n s i t u a t i o n s such as l o g hangups,  dragging  - 4 -  logs, from  yarding other  plement  the  sources.  the  theory  This mechanics. carry  out  logs  Again and  thesis  The  to  validate  secondly  to  extend  iments the  and  study  second  more  phase  The gress  from  towards be  the  enumerated  as  i)  achieved  for  hanging  system  flyer  of  these  small  as  The  portion  to  the and  limits  f i r s t  to  experphase  of  of  the  thesis.  the  the  simplest  This  cable  Gravity  tion  system  study  was  to  cable  progression  prosystem  can  follows:  Free  system  Standing Running  and  the  for  sophisticated.  i i i )  Binkley  for  this  problems.  formulation, those  com-  primarily  situations.  a  of  cable  limitations  beyond  selected  load  V.W.  were  Only  Clamped  v)  study  loading  can  practical  the  results  ii)  iv)  the  same  shock  approach  of  dynamic  the  or  study  the  investigation  most  of  mathematical  complex of  ground  a  research  approach  the  of  completed.  was  some  within  the  analysis were  answer  the  the  empirical  describes  tests  theories  from an  objectives  f i e l d  investigate  free  D.D.  basic  on  a -  (Figure single  Live  (Figure  skyline  la).  line  skyline;  (Figure  lb).  shotgun  or  l c ) .  with  haulback  line,  skyline. Studier(l)  systems  and  present their  a  complete  numerous  descrip-  variations.  - 5 -  The running  skyline  experiment  hanging  studied  regarded suspended step  on  cable  as  the  of  the  u t i l i z i n g  a  the  should  component  i t  of  of  a  load  be  of  gravity  yarding  carried  of  any  on  a  and  on  f i r s t can  be  free  cable  single  system  the  system line  with  and  can  a  be  f u l l y  primarily  as  a  system.  high  system  out  the  The  considered  problems  and  assumptions  highlead  the  in  system.  should  the  cable be  modelling  clamped  line  considered  basic  study  real  haulback  gravity  The  but  not  the  simulation  Because  tests  were  with  simpler  the  f i r s t .  load,  towards  most  is  a  skyline  systems  since  investigated  was  standing  i t  an  cost  was  involved  realized  experimental  in  that  physical  model.  After parabolic  a  brief  models,  f i e l d  model  f i e l d  measurements  i s t i c s line bolic with  of  and  free the  models this  and  the  thesis  presents and  hanging  gravity are  are  the  to  describes  the the  comparative  catenary  and  experiment  on  analysis  theoretical values  for  cable,  load  system.  also  analysis.  Recommendations  introduction  compared The  the  clamped  The  catenary  to  dynamic  stated  in  the  each tests  are  the on  model  other  of  in  the  charactera  single  and  para-  p a r a l l e l  presented.  conclusion.  the  - 6 -  Figure 1  -  The three cable systems experimented and analysed: a) b) c)  free hanging cable clamped load on a single l i n e gravity system  - ,7  Sky!ine  S k y l i ne mped load  Skyline Ma in 1 i ne Carr i a  CHAPTER  2  INTRODUCTION TO CABLE MECHANICS  T h i s chapter  presents  t e r i s t i c s o f a c a b l e system. c a b l e mechanics are then  2.1  General  and d e f i n e s the b a s i c  The t h e o r e t i c a l approaches t o  introduced.  D e s c r i p t i o n of the System.  F i g u r e 2 i l l u s t r a t e s the important c a b l e y a r d i n g system. classified  charac-  f e a t u r e s of a  The c h a r a c t e r i s t i c s shown can be  i n two groups; the f i r s t group d e f i n e s the dimen-  s i o n s and geometry o f the system; the second group d e s c r i b e s the f o r c e s a c t i n g on i t .  The nomenclature i n t r o d u c e d  i n the  p r e s e n t a t i o n o f these c h a r a c t e r i s t i c s w i l l be used throughout the remainder of t h i s  Geometrical  thesis.  characteristics:  ;C:  Carriage  A:  Lower support  o f the c a b l e /  B:  Upper support  L:  Span; h o r i z o n t a l d i s t a n c e between the supports  E:  D i f f e r e n c e i n e l e v a t i o n between the supports  AB:  o f the c a b l e  _ J  Chord  0:  Angle of the chord with the h o r i z o n t a l  D:  D e f l e c t i o n ; v e r t i c a l d i s t a n c e between the  - 9 -  Figure 2  -  Characteristics of a cable yarding system  c a b l e and  t h e c h o r d a t any  point along  the  cable X:  H o r i z o n t a l p o s i t i o n o f a p o i n t on t h e i n the coordinate  Y:  system  cable  (X,Y)  V e r t i c a l p o s i t i o n o f a p o i n t on t h e c a b l e i n the coordinate  system  (X,Y)  S:  Cable  a:  Angle of the cable w i t h the h o r i z o n t a l a t point.  length any  - 11--  Force  characteristics:  co:  Weight o f the c a b l e per u n i t l e n g t h  R:  Weight o f the c a r r i a g e and l o a d  T:  Tension  H:  H o r i z o n t a l t e n s i o n i n the c a b l e  i n the c a b l e  R e l a t i o n s h i p s between the above c h a r a c t e r i s t i c s can be d e r i v e d u s i n g b a s i c mechanics p r i n c i p l e s .  V a r i o u s mathema-  t i c a l models have been proposed depending on the u n d e r l y i n g assumptions made.  2.2  Modelling  Assumptions.  The g e n e r a l d e r i v a t i o n s o f the e x i s t i n g  formula-  t i o n s , the catenary model and the p a r a b o l i c model, a r e c l a s s i c a l a p p l i e d mechanics problems and have been by I n g l i s ( 5 ) .  Both these models a r e based on the assumption  t h a t the c a b l e i s an i n f i n i t e l y  f l e x i b l e body which  t h a t no bending r e s i s t a n c e i s considered of the f o r c e s .  described  implies  i n the accounting  Another assumption i s made as t o how the  u n i f o r m l y d i s t r i b u t e d weight, w, a c t s on the system. catenary model c o n s i d e r s co as u n i f o r m l y d i s t r i b u t e d  The along  the c a b l e l e n g t h whereas the p a r a b o l i c model s i m p l i f i e s the problem and assumes w d i s t r i b u t e d on the chord o f the system (Figure 3 and F i g u r e 4 ) . distinct  T h i s b a s i c d i f f e r e n c e leads t o  formulations.  I t i s one of the o b j e c t i v e s of t h i s study pare the r e s u l t s o f both t h e o r i e s a p p l i e d t o c a b l e  t o comlogging  - 12 -  Figure 3  -  Loading assumption and reference frames f o r the derivation of the catenary model  Figure 4  -  Loading assumption and reference frames f o r the derivation of the parabolic model  - 13 -  - 14 -  systems  and  mainder  of  parabolic  to  compare  both  with  the  experiment.  the  chapter  describes  the  catenary  Catenary This  (x',  the  y')  below  to  basic  the  point  cable  of  so  system,  from  the  expressions  ment  OP  A.  The  given  catenary  =  The  is  the  sion,  is  is  in  coordinate  that  origin  sag  the  (Figure of  the  static  0'  is  3).  cable  at In  Both system  a  distance  this  shape  equilibrium  co-  derived  of  the  seg-  catenary:  x' — m--  with  equation with  m cosh  i t s  determined of  same  -  m  distance,  the  i t s  the  Mann(2).  m =  can  H/u)  be  origin  (a)  translated at  the  to  lower  a  co-  support  becomes:  y-intercept H,  the  presented  equation  m cosh  (x,y)  is  and  the  The  y  and  theory  Carson  of  by  catenary  equation  =  equation  (x',y')  the  y'  system  the  maximum  ordinate  is  of  Inglis(5)  positioned  ordinate  and  Model.  summary  reference  define  model  re-  model.  2.3  with  The  a,  by  at  from  E,  the  m cosh m  L  (x,y)  and  m,  catenary.  any  —  point  on  (b)  to  the  original  where  m,  Since  horizontal  the  equal  cable,  m is  frame  to  H/w ten-  a  m  - 15 -  constant  and becomes a convenient  formulation. expressed  The  parameter f o r the  catenary  other system c h a r a c t e r i s t i c s can e a s i l y  i n terms of the parameter m.  The  t e n s i o n T^ and  and the angles of the l i n e with the h o r i z o n t a l a  n  given a t the lower and  upper supports  and  by:  (c) (d)  T„ = oom cosh B  (e) (f)  13  and  Tg,  rv , are  T, = com cosh — and tga,. = s i n h — A m A m  m  be  tga„ = s i n h — — ^ B m  I t can e a s i l y be shown t h a t the d i f f e r e n c e between the  ten-  s i o n s at the supports' i s d e f i n e d by the simple r e l a t i o n s h i p :  T  B  - T  A  =  ooE  (g)  T h i s i s a very u s e f u l e x p r e s s i o n  i n the  catenary  f o r m u l a t i o n of c a b l e mechanics.  The  Dm  And  d e f l e c t i o n a t mid-span can be c a l c u l a t e d from:  E  =  2  , L/2-a - m cosh — m  (h)  i t i s easy to express the c a b l e l e n g t h as:  S = m  (sinh —  m  + s i n h -J) m  (i)  T h e r e f o r e , most of the system c h a r a c t e r i s t i c s are expressed tions.  simply u s i n g the h y p e r b o l i c s i n e and c o s i n e func-  However, the t r a n s c e n d e n t a l p r o p e r t y of the  hyper-  b o l i c f u n c t i o n s render t h e i r use i m p r a c t i c a b l e f o r the of c a b l e systems without has been devoted  the a i d of a computer.  t o develop  study  Much work  i t e r a t i v e techniques and com-  puter programs t o p r o v i d e numerical s o l u t i o n s t o catenary models of s k y l i n e problems.  The  l a t e s t and most e l a b o r a t e  i s t h a t by Carson and Mann(2) who i t e r a t i v e technique adopted this  developed  the " r i g i d  link"  f o r the catenary a n a l y s i s i n  paper.  2.4  P a r a b o l i c Model. The p a r a b o l i c theory as i t a p p l i e s to c a b l e l o g -  ging systems i s developed  i n Appendix 1.  The  development  presented progresses from the b a s i c f r e e hanging  cable to  the more complex f i v e - l i n e system and emphasize each major r e s u l t w i t h numerical examples.  Only the b a s i c f e a t u r e s and  r e s u l t s of t h i s theory are summarized i n t h i s  The equation of the f r e e hanging developed  i n the c o o r d i n a t e system  (x', y ) 1  section.  c a b l e shape, u s i n g the equa-  t i o n s of s t a t i c e q u i l i b r i u m of the segment of c a b l e OP (Figure 4) i s  -  This (x,y)  defines  equation, the  co  where  H,  meter. system  the The  can  V  The  L  of  the  expressed  tensions  "  the  port  line  are  with  defined  T  A  =  A  T  the  —  "and A  _ J _ cosa  ^  and  the  the  coL P  >  ,, ..  considered  as  (b ) 1  J  be  and  T_,  characteristics  and  =  ^  A  tga  B  the  f-  £  of  the  a_  a  para-  of  the  H.  angles  ti  lower  and  upper  sup-  relationships:  -  _  2  L  =  and  terms  A  following  tgcv  in  a,,  H  at  system  as:  ) x  cbs9H Q  coordinate  cable  conveniently  horizontal,  by  cosa  =  B  the  of  other  A  of  o  2  to  tension, can  expressions be  shape  2 . ,E + (7-  x  horizontal  also  translated  parabolic  2 cosGH  J  17  a  )  L n  „  (c')  cosGH  (d') v  <e'>  +  '  <f >  D  X5  The  deflection  The  length  L/ Oy  of  at  the  mid-span  is  cable  derived  is  simply  expressed  from:  as:  -  The  -  s o l u t i o n of t h i s  i s g i v e n i n A p p e n d i x 2. approximate formula  s = — ^cose  The  18  I f the c a b l e i s t i g h t the  c a n be  (1  and  following  used:  + 3 fL ^ ) ) 2  p a r a b o l i c equations  simple to manipulate.  are r e l a t i v e l y easy  However, t h e e x p r e s s i o n s o f  a n g l e s w i t h t h e h o r i z o n t a l and cumbersome and  i n t e g r a l i s not t r i v i a l  and  line  the cable length formula  c o n s t i t u t e a drawback t o the  theory.  are  -  19  -  CHAPTER 3  DESCRIPTION  OF T H E F I E L D  The the  major  f i e l d  desired and  model  components  instrumentation  was designed  of a gravity  was i n t e g r a t e d  variables.  the various  MODEL  system.  In  t o t h e system  The g e n e r a l  pieces  t o incorporate a l l  layout  o f equipment  addition, t o measure  of the f i e l d  are described  the  model  i n  this  chapter.  3.1  Site A  at  location  the University  (UBCRF). forest done the  Dimensions  and had been  mid-span  could  recently  the choice  Research  logged.  end of the  The f i n a l  was i n s t a l l e d  found  Forest  i n the northern  surveying,  and adjusted,  Span  L =  131.95  i n elevation,  E =  23.05  deflection  be obtained  Other in  Columbia  was  gave  dimensions:  Difference  span)  f o r the experiment  was l o c a t e d  t h e equipment  following  A maximum  suitable  of B r i t i s h  The s i t e  after  and C h a r a c t e r i s t i c s .  elements  of this  Dm =  f o r a free  were  site.  also  metres metres.  1 0 . 5 m ( i . e . 8% o f t h e hanging  taken  Anchorings  cable.  into were  consideration available  a t  -  each  end  for  rocky  base  lower  and  lation path to  the  for to  of  under  walk.  the  The  conditions  of  upper  cable, ground  very  the  supports.  that  and  the  cleared p r o f i l e  similar  to  cables,  There  end,  equipment  the  -  rigging  the  the  20  as  was  well  easy  of  of  branches  allowed  those  of  the a  firm to  the  the  i n s t a l -  the  tests.  and  snags  tests  real  a  access,  f a c i l i t a t e d execution  as  to  was  be  yarding  The easy  run  in  oper-  ation.  Figure  5  A  plan  view  and  p r o f i l e  and  6.  Each  element  of  of  the  the  site  model  are  w i l l  shown  now  be  in des-  cribed.  3.2  Cables. The  widely  used  cables  in  the  were  6x19  logging  different  diametres  were  7/16-inch  diameters  cables  eye  at  one  end.  The  marked  precisely  ment.  The  7/16  the  carriage  are  summarized  2-metre from  the  long  cable  along in  cable  supplier's  135 Both  came  with  a  cable,  used  as  was  Table  type  industry.  metre  the  a  obtained.  5/8  every  IWRC  with  skyline. I.  samples  The  The are  not  information.  the  per  of  two  made  length  mainline  and  flemish  skyline,  for  cables  rope  5/8-inch  factory  the  weight  wire  metres the  paint,  u t i l i z e d as  of  was measure-  to  move  characteristics metre  significantly  measured different  on  - 21 -  Figure  5 - Sketch of plan view of the f i e l d  model  Figure  6 - Sketch of s i d e view of the f i e l d  model  -  22  -  Upper support  - 23 -  Table I .  Cable c h a r a c t e r i s t i c s .  w (kg/m)  #(inch) Measured  Catalogue  Difference %  5/8  1.071  1.027  4  7/16  0.521  0.506  3  JZ5  B.S. (N) (NxlOOO)  E N/m  2  io  6  50  3.1  188  76  10  7.6  188  J.C'a'b'le.-.diameter Weight per metre  BS  Breaking s t r e n g t h  Tm  Maximum s t a t i c t e n s i o n expected during  SF  Safety  i  SF  157  w  E  Tm (N) (NxlOOO)  i n newtons  the experiment BS f a c t o r f o r the experiment c o n d i t i o n s = ^  E l a s t i c modulus of the cable 3.3  Carriage The  (Plate 2)  c a r r i a g e had a s i n g l e b a l l bearing-mounted  sheave t h a t f i t s the s k y l i n e . mainline  w i t h a shackle  means o f a clamp.  I t c o u l d be connected t o the  o r immobilized  on the s k y l i n e by  A basket was attached  t o the c a r r i a g e t o  r e c e i v e up t o 40 l e a d weights t o c o n s t i t u t e the l o a d .  The  maximum l o a d weight i n c l u d i n g the c a r r i a g e and basket was 535  kilograms.  3.4  Winches ( P l a t e 3) The  requirements f o r the winches, shown i n t a b l e I I ,  were d i c t a t e d by the cables c h a r a c t e r i s t i c s .  -  24  -  Plate 1  —  F i e l d model seen from the spar at the lower support.  Plate 2  —  View of the carriage and load, looking toward the upper support.  - 26 -  Table  Winch  II.  Cable diameter (inch)  Requirements  for  the  winches.  Yes  1  5/8  5  50,000  Slow  2  7/6  130  10,000  0 to 2  After conditions of The  were  proportion f i n a l  between  having  is  not  a  A  everytime  Its  to  were  of  slow  resulted  and a  Gearmatic  accept  i t s  i t  19  the  load  The  6-ton  speed  was  on  6-ton  winch had t o  variable skidder. which be  Comelong  manual  to  the  speed  and the  However  on  a  line  pull  was:just  mainline.  created  was  than  lack  winch.  capacity  of  lowered  appreciated  laborious  drum  120 m e t r e s  a  more  the  tensions.  mounted  hand  these  of  and expected  Comelong  i t s  that  because  19 w i n c h  supplied  Desired  obvious  exactly  diameter  but  i n f i n i t e l y  reversible  in  the  mounted  became  Gearmatic  the experiment  sufficient assets  a  i t  t o meet  cable  were  skidder  The for  research  d i f f i c u l t  choices  rubber-tired  needed  some  Reversible  Pull Speed Tensions (m/s) (Newtons)  Pull length (metre)  Its  main  convenience  the Gearmatic  problems  and a  hazard  slowly.  connected set  handling.  to  precise  the  skyline.  tensions  19  but  -  27  -  Plate 3  -  Mainline directed to the Gearmatic 19 winch with a block at the upper support.  Plate 4  -  Skyline passing the top of the spar at the lower support and connected to the load-cell.  - 29 =  Dynamometers  3.5  The line  and  meters.  tensions  at  the  No  force  Rugged  portable  in  the  f i e l d .  er  for  the  those  on  d i g i t a l  recorder. described  the  layout  of  the  only  load-cell  off  the  introduce  no  to  to  carriage The  to  reference  support  then  as  the gave  load  section  i t  to  was  a  taken  of  to  the at  can  the way  the  top  of  read tape  are  point  the  simple  of  the to The mounted  load-cell allowed  relieved.  the of  so  the The  that  lower the  in  straight-  bearing  cable to  used  cable  hook  or  be or  key  a l l  being  tension.  the  swivel  record-  of  was  the  b a l l  a  satisfy  weight  and  applied  height a l l  the  shape  connect  run  was  keep  problems  chart  a  support  a  enough  A  lower  of  was  i n s t a l l a t i o n  to  with  c e l l a  dynamo-  avoid  advantage  on  sky-  carriage.  load-cells  on  use  to  load  run  tension  could  point  sheave.  The  a  the  cable  cable to  the  the  with  the  dynamometers  the  of  load-cells  recorded  The  end  compatible  from  of  was  in  at  required  present  hanging  the  stump.  spar and  l e t  and  a  spin  1.2-metre  at  problem free  done  be  or  each  mainline  Electronic  the  disturbance  f i r s t ,  to  output  skyline  the  was  equipment.  The  line  the  was  in  were  indicator  rigging  forward.  anchored  The  3.  of  at  had  also  Appendix  on  sheave  and  4  characterisitcs  load-cell  solution  also  used.  The  measured  support  tests.  gauge  in  5)  dynamometers They  The  were  and  measurement  requirements commonly  a  upper  dynamic  very  (Plates  the  end.  lower  - 30 -  Plate 5  Plate 6  —  —  Skyline and mainline tensiometers at the upper support.  Reading of the tensions i n the two l i n e s at the upper support on the strain-gage indicators connected to the tensiometer.  -31-  - 32 -  For meter,  referred  built. of  The  on  a  the  with  to  mainline steel  the  the  were  with  tension  (Plate The locating and  the  A a  surveyed  in  most  the  instrument mark  knoll  surveying  T'.  T The  of  top  on  to  f i r s t  They given middle  be  simply and  bolted  After  their  upper  Shape  and  Carriage  experiment  required  to  leaving  respective  sheave  the  a  the  type  foundations.  be  capable  skyline  this  to  are  is  can  the  of  had  3,  cable  lever  Both  the  dynamo-  lines.  action  directed  of  as  the  of  any  point  where  the  circumstances. layout  stood  at  are  the  horizontal  so  as  to  read  to  plane  of  the  of  theodolite  from  justed the  the  a  line.  were  Cable  Salmoragy  on  Appendix  running  the  of  of  reference  the  sky-  point  B.  Position  7)  position  easily.  and  concrete  taken  of  in  or  type  tensiometer",  tensiometers  The  was  11  principle:  lines  object  purpose,  of  to  blocks.  Surveying  3.6  fixed  the  through  the  as  l o a d - c e l l by in  secured  tensiometer  in  sheaves a  special  thesis  mechanical  to  a  described  tension  run  frame  winches  support  this  three  tensiometers  line  in  transmitted  related  a  to  simple  deflection sheave  upper  tensiometers,  measuring  work  the  zero cable.  for  the  Since  plan  in  a  the  view  to  serve  could and  7 and  theodolite TM  1  of  accurately this  be  p r o f i l e The  8.  pre-surveyed  direction this  up,  cable  Figures  of  of  cable  set  entire  v e r t i c a l vernier  was  The  shown  the  a method  bench-  was  ad-  perpendicular  d i r e c t i o n was  not  Figure 7  -  Plan view and dimensions of the surveying layout  Figure £  -  Side view and dimensions of the surveying layout  -  35  -  l o c a t a b l e on the t e r r a i n or on the c a b l e the a c t u a l a d j u s t ment of t h e h o r i z o n t a l v e r n i e r was done a t 41° 18.3' w i t h the t e l e s c o p e of the t h e o d o l i t e p o i n t i n g the lower ence A.  Simple  refer-  geometric d e r i v a t i o n y i e l d s the f o l l o w i n g  equation f o r the h o r i z o n t a l d i s t a n c e X from a p o i n t of the c a b l e t o the lower support A.  X = A'M' - TM  1  tg(Alpha) where tg(Alpha)  represents  the tangent of the angle read on the h o r i z o n t a l v e r n i e r of the instrument.  With the dimensions  i n metres the p r e v i o u s  equation becomes: X = 48.411 - 55.094 tg(Alpha)(metres) The o r i g i n of the v e r t i c a l v e r n i e r was a d j u s t e d a t zero w i t h the a x i s o f the t e l e s c o p e of the t h e o d o l i t e i n the h o r i z o n t a l direction.  The f o l l o w i n g r e l a t i o n s h i p g i v e s the v e r t i c a l  d i s t a n c e Y from a p o i n t of the c a b l e t o the lower support A.  Y = AA' - (TM'/cos(Alpha))  x tg(Beta) where  cos(Alpha)  i s the c o s i n e of the h o r i z o n t a l v e r n i e r r e a d i n g and tg(Beta) i s the tangent of the angle read on the v e r t i c a l With the dimensions Y =  vernier.  i n metres the p r e v i o u s equation becomes:  TT*+14.975-(55.094/cos(Alpha))xtg(Beta)(metres)  TT' the h e i g h t of the instrument a x i s t o the bench-mark T' was remeasured a f t e r every s e t t i n g of the instrument.  The  shape of the c a b l e s and the c a r r i a g e p o s i t i o n  -  were  were  using  this  For  large  deflections  taken  by  surveyed  v i s i b l e direct bench  3.7  from  anics  marks  on  taking  at  and  simply  line, the  clinometer  gave  The  upper  lower  measurements  noted  number that  a  the  the  angles  of  few  their  height  from  line  with  the  horizontal.  they  have  in  cable  the  with  supports  and  at  carriage  the  This  good  was  pre-,  cables  This  the  points  position  the  placing  was  body level  rather  the  performed  of  the  tube  to  unusual  horizontal level  with  instrument the  mech-  a  in  c l i n o -  d i r e c t l y  horizontal  u t i l i z a t i o n of  and a  results.  painted  supports, the  and  a  of  unstretched  of  cable  cable.  importance  mainline.  were  the  their  the  Unstretched  skyline.  the  the  reading.  Marks  to  the  of  adjusting  3.7.2  and  under  Angle  skyline  the  of  3.7.1  of  of  theodolite  measurements.  recorded  by  the  Other  theories,  meter  technique.  measurement  Because  the  -  surveyed  not  were  36  two  whole  negligible  line every  length BA,  was  fractions metres error  length metre  of  the  8)  on  untensioned  of  metres A  the  cable  obtained  between is  (Plate  and  introduced  by  between addition  at  the  B.  It  since  the of  the  extremes should the  dis-  be  -  Plate 7  —  37  -  S u r v e y i n g of the cable and c a r r i a g e posit i o n s w i t h the t h e o d o l i t e .  Plate S  Measurement of the f r a c t i o n o f metre between the lower support r e f e r e n c e  p o i n t and the  f i r s t p a i n t mark on the c a b l e .  -  tances  39  -  at the extremes were measured on the c a b l e under t e n -  sion.  3.8  A c c u r a c i e s of instruments  and expected e r r o r s i n  the measurements. The magnitudes of the e r r o r s a f f e c t i n g the  various  measurements have to be known i n order to make any  conclu-  s i o n i n the a n a l y s i s of the r e s u l t s .  errors  have v a r i o u s o r i g i n s , they may p e r s o n a l or n a t u r a l .  Experimental  be i n s t r u m e n t a l ,  Instrumental  procedural,  e r r o r s r e s u l t from i n s t r u -  ment i m p e r f e c t i o n s and non-adjustments.  The magnitude o f  the p r o c e d u r a l e r r o r i n c r e a s e s with the number of  steps  performed and number of pre-measurements needed f o r the determination  of a given v a r i a b l e .  from human l i m i t a t i o n s and were the most d i f f i c u l t  Personal errors r e s u l t  accidents.  to apprehend.  The  natural errors  The weather c o n d i -  t i o n s i n p a r t i c u l a r a f f e c t e d the r e s u l t s as v a r i a t i o n s of the t e n s i o n and  sag r e s u l t e d from the expansion or c o n t r a c t -  i o n of the c a b l e from changes i n temperature. wind had  rain  and  the e f f e c t of i n c r e a s i n g the c a b l e weight per metre  r e s u l t i n g i n an i n c r e a s e i n t e n s i o n .  Other n a t u r a l e r r o r s  r e s u l t e d from the y i e l d i n g of the anchorings ment of the t h e o d o l i t e t r i p o d . observed and to them.  The  recorded  The  and  the  settle-  n a t u r a l e r r o r s were  but no numerical  values were attached  - 40 -  The remainder  of t h i s s e c t i o n e v a l u a t e s the  ex-  pected e r r o r f o r each types of measurement.  3.8.1  E r r o r s i n the t e n s i o n s measurements.  A complete t e s t c a r r i e d out on the  tensiometers  used a t the upper support i s r e p o r t e d i n Appendix 3.  With  the recommendations formulated i n Appendix 3, l e s s than  one  percent e r r o r can be obtained f o r t e n s i o n s g r e a t e r than 5 000 newtons i n the s k y l i n e and f o r t e n s i o n s g r e a t e r than newtons i n the m a i n l i n e .  1000  Although no s p e c i f i c t e s t was  done  on the r i g g i n g a t the lower support the l o a d c e l l a t t h a t p o i n t i s a l s o expected  to be accurate to p l u s or minus  one  percent.  3.8.2  E r r o r i n the c a b l e p o s i t i o n .  Instrumental and p e r s o n a l e r r o r s are c l o s e l y r e l a t e d i n t h i s case.  The  angular accuracy of the Salmoragy  t h e o d o l i t e i s g i v e n i n the s u p p l i e r ' s catalogue as 1/10 a minute.  However the experience has proved t h a t 1/5  minute was  p r o b a b l y a more r e a l i s t i c  of  of a  l i m i t f o r the angular  d e f i n i t i o n of the instrument, because the v e r n i e r s were difficult  t o read and the l e v e l l i n g r e q u i r e d d e x t e r i t y .  The e r r o r i n the c a b l e p o s i t i o n c r e a t e d by  the  angular e r r o r depends on the d i s t a n c e L from the t h e o d o l i t e to  the cable and i s d e f i n e d by: error =  * ;li = 3  ^ n 5x60x180  .000058L  -  For  the  longest  and  equal  to  6 mm.  simplification cable  value  errors  error of  of  each  the  computation.  The  the  telescope  than  cedural the  points  is  but  error  is  analysis is  6+12  of =  3.8.3  the  18  The be  positions  of  the  the  line  more  error  the  a  The  the  instrumental  error;  depends  on  the  dimensions AA',  on  the  reasonable. mm.  of  The  cable  the  be  of  less  of  position  the mea-  total  error the  in  v e r t i c a l  to  The  of  used  height  effect  total  from  improper the  r e a l i s t i c .2  measurement  clinometer  positioning  is  the  of  pro-  adopted  cable  of  the  line  angle  horizontal.  instrument.  in  a  of  averaging  d i f f i c u l t y  measurement  the  error  the  and  in  accuracy  e l i m i n a t e d by  on  point  estimated  appraise  be 12  is  sake  theodolite.  dimension  v e r t i c a l position  Error  instrumental  this  maximum  mm.  with The  to  inaccuracies  therefore  the  is  the  any  reproduced  error  to  to  positions  the  d i r e c t l y  6 mm s e e m s  to  for  pre-measured in  This  to  error  is  analysis  cable the  the  error  added  d i f f i c u l t  dimensional  surements  in  It  be  error  is  model  distance  of  measurement.  6 mm.  other  axis,  the  this  measured  affecting  position  in  to  -  maximum  i t s  has  the  on  This  applied  independent  procedural the  distance  41  zero  reading However the  is  of  a  adjustment  from  two  instrument for  degree. could  inverted  experience  expectation  degree.  1/10  has  shown  correctly the  angle  on  -  3.8.4  Error  Because of points error the  on  the  of  as  high  the the as  r e s u l t s from i n a c c u r a c y  reasonable  length  be  cable  to think  that  27  stretched  mm  fraction 5 mm.  .02  percent this  reference  cable  the  of metre a t each  Another  the  of the  cable.  total  III.  e r r o r , y i e l d i n g an The  o r m i n u s 37  expected experimental  unmm.  errors i s  Experimental errors a f f e c t i n g the measured v a r i a b l e s .  Nomenclature  Variable  T  Tension Vertical Angle of w i t h the Cable  position the l i n e .horizontal  length  It  cable  shown i n t a b l e I I I .  Table  of  procedural  metres l e n g t h .  i s t h e n known t o p l u s  A summary o f t h e  the  i n marking of  f o r a maximum 135  length  of the  marks on  i s t h e maximum m a g n i t u d e o f  error of  length.  poor d e f i n i t i o n  s h e a v e s and  extremes can  -  i n the  i n t h e measurement o f  error, is  the  42  Magnitude of error  the  1%  Y  18  a  .2  S  37  mm  degree mm  -  Dimensional  3.9  a  real  The any  two-line  length  of  gravity For  the  line  slackline  economical  model  had  yarding that  to  be  representative done and  for  map r^  and  metres  elevation quently  of  were  the  this  ratio.  unit  of  for were  a  real scaled  lationship  and  f u l l  the  system. by  r^ one  by  and  i f  were  i f  or  operated.  than  one  scaling  metre  of  The  span  to  the  same  the  represented and  the  same  ratio true.  the i f  the  be  is  for  the  lengths  of  the  system.  scale  for  in  conse-  scaled r a t i o  of  a  represents  and  units  weight  w i l l  difference  ratio  force  shows  scaling  model  and  real  model  the  d e f l e c t i o n were by  a  the  analysis  the  of  the  shot-gun  scale  similar  load  be  small  the  the  scaled  model  to  a  smaller  ratios,  The  assumed  or  system  is  system.  are  and  simulation  system  dimensional  scale  down  a  winches  was  determine  length  for  with  that  scaled  model  p r a c t i c a l considerations  down  ratio  real  Forces  is  two  independent  means  down  the  the  cable  force  i f and  forces  the  constant;  obtained the  the  as  skyline  a  length  simply  considered  Nevertheless,  Two  the  The  kept  system  of  properly. one  was  scaled  results  be  standing  purposes  system.  the  a  between  system.  could  system;  -  similitude  yarding  model  43  down  One  r^. of  the  following  by  force cable re-  4.4  r  P • R  W 0)  X  payload  R  weight  of  W  weight  per  real  i  r  P  in  2  r  kilogram carriage unit  for and  the load  length  real in  system.  kilogram.  of  the  cable  in  the  length, of  the  cable  in  the  system.  weight  co  -  per  unit  model, r^  length force  Angles with  a  one  example, model  two  are  systems  to  r e a l  almost  slopes  ratio  in  any  and  without  systems  described  of  ratio.  and  one  ratio.  which  table size  by  unitless  values  are  any  distortion.  As  can  be  IV.  The  merely  represented model  can  changing  related an by  the  simulate  the  two  scale  ratios.  The also  part  of  ground the  u t i l i z e d  inside  could  of  study.  be  configuration  general the  prime  and  similitude.  l i m i t s  of  importance  this in  terrain Although  thesis,  future  aspect not  those  extensions  were  very  two of  much  points the  -45  Table IV.  -  Example o f two systems t h a t the model c a n s i m u l a t e .  . System  , Model  characteristics  132  Span (metres)  Slackline system  Long r e a c h standing skyline  600  1500  4.55  11.36  Length r a t i o ( r ^ )  -  D i f f e r e n c e s i n e l e v a t i o n (metres)  23  105  261  10.5  48  119  Maximum d e f l e c t i o n  (metres)  Maximum d e f l e c t i o n  (%)  Average ground s l o p e  (%)  S k y l i n e l e n g t h of maximum d e f l e c t i o n Load r a t i o ( r ^ S k y l i n e diameter  (inch)  S k y l i n e weight p e r metre M a i n l i n e diameter  (kilogram)  8  8  17  17  17  136  619  1545  -  14:19.  65.64  5/8  1 1/8  1 1/2  1  3.4  6  3/4  1  1.5  2.6  7/16  (inch)  M a i n l i n e weight per metre  (m)  8  (kilogram)  .5  CHAPTER  FREE  HANGING  4  CABLE  The purpose of t h i s experiment was  to investigate  the c h a r a c t e r i s t i c s of a f r e e hanging l i n e segment.  The  s i n g l e l i n e segment i s the b a s i c element of a c a b l e system s i n c e the most complex  system can always be c o n s i d e r e d as a  more or l e s s i n t r i c a t e arrangement  of c a b l e segments hanging  f r e e l y between the d i f f e r e n t p o i n t s of attachment.  This  chapter d e s c r i b e s the f r e e hanging c a b l e experiment, compares the f i e l d  and t h e o r e t i c a l r e s u l t s and a l s o compares  the c a t e n a r y and p a r a b o l i c  4.1  models.  D e s c r i p t i o n of the Experiment. For t h i s t e s t the s k y l i n e was  r i g g e d so as t o hang  f r e e l y , under i t s own weight, between the lower and the upper supports and was  t e n s i o n e d w i t h the Gearmatic 19  winch.  4.1.1  Procedure and Data  Collection.  Nine f r e e hanging c a b l e t e s t s were executed f o r a range of t e n s i o n s a t the upper support between 2700 t o  11000  newtons, and r e s u l t i n g d e f l e c t i o n s at mid-span between  7.1  and 1.6 p e r c e n t of the span l e n g t h . p o r t e d i n Appendix  4.  The r e s u l t s are r e -  Each t e s t l a s t e d one hour on average and planned t o be executed by t h r e e o p e r a t o r s : d o l i t e , one a t the upper port.  produced  sup-  recorded i n i n d i v i d u a l  Sample pages of these note books are r e -  i n Appendix  Readings  one a t the theo-  support and one a t the lower  The r e q u i r e d i n f o r m a t i o n was  f i e l d note books.  was  4.  of the v e r t i c a l p o s i t i o n of the c a b l e were  taken a t s t a t i o n s f i v e metres apart along the h o r i z o n t a l span.  The p o s i t i o n a t mid-span was  :..  measured a t the b e g i n -  n i n g and a t the end of the t e s t t o check the v a r i a t i o n w i t h time.  The t e n s i o n s , the angles of the l i n e w i t h the h o r i z o n t a l and the c a b l e l e n g t h were recorded every 15 a t the upper  minutes  and lower support, as they proved t o change  s l i g h t l y w i t h time.  Those v a r i a t i o n s were a t t r i b u t e d t o  n a t u r a l phenomena l i k e the y i e l d i n g of the anchorings and the changes i n atmospheric c o n d i t i o n s as sun, r a i n or wind. For most of the t e s t s the upper  and lower support measure-  ments were done by the same o p e r a t o r which r e s u l t e d i n the i m p o s s i b i l i t y of o b t a i n i n g t r u l y simultaneous r e a d i n g s ; the time l a g between the readings was  4.2  about 5 minutes.  A n a l y s i s o f the R e s u l t s . The a n a l y s i s of the f r e e hanging c a b l e i s based  on  - 48 -  the comparison of the f i e l d measurements and p r e d i c t e d by the catenary  the r e s u l t s  as  and p a r a b o l i c models f o r the  f o l l o w i n g s i x c h a r a c t e r i s t i c s of the system: Cable shape d e f i n e d by v e r t i c a l Y - p o s i t i o n s of p o i n t s of the Dm:  cable  d e f l e c t i o n at mid-span  T.. and T„: A  t e n s i o n s at the  CL. and A  OL,:- angles of the l i n e with is  h o r i z o n t a l a t the S:  supports  15  the  supports  skyline length.  Only one of these c h a r a c t e r i s t i c s has t o be g i v e n as a parameter to determine the system and  the other v a r i a b l e s com-  pletely.  T  as the Parameter.  The measured t e n s i o n at the upper support taken as a parameter.  The  mined u s i n g the catenary by the f o l l o w i n g c h a r t :  other v a r i a b l e s c o u l d be  T ,  was  deter-  and p a r a b o l i c models as d e s c r i b e d  - 49 -  Figure 9  -  D i f f e r e n c e s i n the Y - p o s i t i o n o f the c a b l e between experiment and c a t e n a r y model, and between experiment and p a r a b o l i c model v e r s u s X - p o s i t i o n s on span, f o r f r e e hangi n g t e s t number 4.  -  50  -  Computation u s i n g Catenary and P a r a b o l a theories  Measured Parameter T  B  Theoretical values f o r the V a r i a b l e s Y, Dm, T , a , a , S  Measured v a l u e s f o r the Variables Y, Dm, T , a , a , S A  A  A  B  A  B  Conclusion  OL  r  E E T3  O E >I  ^  20  O • LA]  a. x I a> >o o  o  LA!  Yexp.-Ycat. Yexp.-Ypar.  kO  ± 18 mm  60  80  100  1 2  ^X»  m  -  T a b l e V.  .51-  F i e l d and computed r e s u l t s . f o r f r e e h a n g i n g c a b l e T e s t #4.  FREE HANGING CABLE  TEST # 04  DATE:  WEATHER:  31/08/76  SUNNY HOT  CREW D. Guimier D. Anderson D. Anderson  - THEODOLITE - WINCH - SPAR TEST STARTED AT:  COMPLETED AT:  13:00  Y position  o f t h e c a b l e on the span.  X POSITION IN METRES  Y POSITION IN METRES  0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 -60.0 65.0 65.97 70.0 75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0 120.0 125.0 130.0 131.95  EXPERIM.  CATENARY  PARABOLA  -0.013 -0.332 -0.558 -0.696 -0.741 -0.686 -0.548 -0.305 0.026 0.466 0.978 1.579 2.282 3.079 3.251 3.975 4.964 6.049 7.231 8.499 9.873 11.342 12.903 14.567 16.323 18.192 20.145 22.200 23.024  -0.000 -0.318 -0.543 -0.675 -0.714 -0.661 -0.514 -0.275 0.057 0.483 1.001 1.613 2.319 3.118 3.285 4.012 5.000 6.083 7.261 8.534 9.904 11.370 12.933 14.593 16.352 18.209 20.166 22.223 23.052  0.000 -0.320 -0.546 -0.677 -0.715 -0.659 -0.509 -0.265 0.073 0.506 1.032 1.652 2.366 3.174 3.343 4.076 5.073 6.163 7.347 8.625 9.997 11.464 13.024 14.678 16.426 18.268 20.205 22.235 23.052  14:00  DIFFERENCES IN METRES EXP-CAT  EXP-PAR  CAT-PAR  -0.013 •^0.014 -0.015 -0.021 -0.027 -0.025 -0.034 -0.030 -0.031 -0.017 -0.023 -0.034 -0.037 -0.039 -0.034 -0.037 -0.036 -0.034 -0.030 -0.035 -0.031 -0.028 -0.030 -0.026 -0.029 -0.017 -0.021 -0.023 -0.028  -0.013 -0.012 -0.012 -0.019 -0.026 -0.027 -0.039 -0.040 -0.047 -0.040 -0.054 -0.073 -0.084 -0.095 -0.092 -0.101 -0.109 -0.114 -0.116 -0.126 -0.124 -0.122 -0.121 -0.111 -0.103 -0.076 -0.060 -0.035 -0.028  -0.000 0.002 0.003 0.003 0.001 -0.002 -0.005 -0.010 -0.016 -0.023 -0.031 -0.039 -0.047 -0.056 -0.058 -0.065 -0.073 -0.030 -0.086 -0.091 -0.094 -0.094 -0.091 -0.085 -0.074 -0.059 -0.039 -0.012 0.000  -  Table  52 -  V.  (continued)  DEFLECTION AT MIDSPAN AS A PERCENT OF THE SPAN DEFLECTION IN PERCENT EXPERIM. C  CATENARY  PARABOLA 6.202  6.246  6..2 7.1  DIFFERENCES IN PERCENT EXP-CAT  EXP-PAR  CAT-PAR  0.026  0.070  0.044  TENSIONS AT THE SUPPORTS TENSIONS IN NEWTONS EXPERIM.  DIFFERENCES IN NEWTONS  CATENARY. PARABOLA i EXP-CAT iEXP-PAR , CAT-PAR 0  0  UPPER SUP.  3080  3080  3080  0  LOWER SUP.  2815  2838  2844  -22  -29  -6  HORIZONTAL  *  2830  2837  *  *  -6  * HORIZONTAL TENSION WAS NOT MEASURED  ANGLES OF THE LINES WITH THE HORIZONTAL ANGLES IN DEGREES  DIFFERENCES IN DEGREES  EXPERIM.  CATENARY  PARABOLA  EXP-CAT  EXP-PAR  CAT-PAR  UPPER END  -23.000  -23.232  -22.917  0.232  0.083  0.314  LOWER END  —4.000  -4.169  -4.196  0.169  0.196  0.027  SKYLINE LENGTH LENGTH IN METRES  DIFFERENCES IN METRES  .CATENARY' • PARABOLA' EXP-CAT' EXP-PAR:. !'GAT-PAR ! EXPERIM'. ' 135.140  135.252  135.232  -0.017  -0.003  0.020  -  The  5 3.  -  comparison of the r e s u l t s c o n s i s t s i n the eva-  l u a t i o n o f the d i f f e r e n c e s between the measured and c a l c u l ated v a l u e s f o r each o f the v a r i a b l e s .  As an example, the  r e s u l t s of the computation f o r Test'#4 i s shown i n Table V.  The comparison of the measured and t h e o r e t i c a l Y-  p o s i t i o n s of the cable Figure cable  i s a l s o presented g r a p h i c a l l y i n  9 where the d i f f e r e n c e s  i n the Y - p o s i t i o n s  a r e p l o t t e d f o r the e n t i r e span l e n g t h .  none o f the t h e o r i e s  of the  Apparently  seem t o agree c l o s e l y w i t h the f i e l d  r e s u l t s s i n c e the maximum d i f f e r e n c e f o r each of the model i s much g r e a t e r  than the maximum expected e r r o r  the c a b l e p o s i t i o n s .  However no c o n c l u s i o n  can be drawn  b e f o r e the i n f l u e n c e o f t h e e r r o r i n t e n s i o n  4.3  Error The  (18 mm) i n  i s examined.  Analysis.  expected experimental e r r o r s a f f e c t i n g t h e  measured v a r i a b l e s and parameters a r e summarized i n Table V I .  Table VI.  Designation  Experimental e r r o r s a f f e c t i n g the measured v a r i a b l e s .  Nomenclature  Parameter  T  ± 1%  B  ± 18 mm  Y T  Variables  a  A  ± 1%  A  , a  S  Experimental e r r o r  B  ± 0.2 degree ±  37  mm  -  54  -  The e r r o r s on the parameter T  are a f f e c t i n g  B  the c a l c u l a t e d v a l u e s o f the v a r i a b l e s .  F i g u r e 10  and  F i g u r e 11 show the same curves as F i g u r e 9 f o r the catenary model and the p a r a b o l i c model u s i n g v a l u e s of T  one  percent  g r e a t e r and one percent' s m a l l e r than the measured v a l u e .  It  can be seen t h a t one percent e r r o r i n the t e n s i o n measurement has t o be c o n s i d e r e d i n the a n a l y s i s .  The Y - p o s i t i o n of the c a b l e at 100 metres from lower support i s used as an example t o e x p l a i n the of the experimental e r r o r s used,  the  treatment  i n the a n a l y s i s , f o r Y  and  a l l the o t h e r v a r i a b l e s .  The e r r o r i n the parameter T_, allows the c a l c u l a t e d and Y~ 2  p o s i t i o n of the c a b l e Y to vary between Y+e^, shown i n F i g u r e s 10 and  e  11 f o r the two models.  a  I f the  s  seg-  ment AB o v e r l a p s the e r r o r zone the theory agrees w i t h the experiment  w i t h i n the margin of experimental e r r o r s .  examples shown i n F i g u r e s 10 and w i t h the experiment  In the  11 the catenary model agrees  f o r the Y - p o s i t i o n of the c a b l e a t  100  metres from the lower support and the p a r a b o l a does not agree w i t h the experiment.  A more convenient.way t o r e p r e -  sent the same c o n d i t i o n i s to e n l a r g e the e r r o r zone as shown i n F i g u r e 12 where the new d e f i n e d by the a d d i t i o n o f e^ and limits.  The  boundaries t  o  t  n  e  D and E are previous error  theory agrees w i t h the experiment  i f point C  - 55 -  F i g u r e 10 —  D i f f e r e n c e s i n the Y - p o s i t i p n o f the cable between experiment and catenary model  versus  X - p o s i t i o n on span, f o r f r e e hanging t e s t number 4.  The t h r e e graphs a r e drawn:  u s i n g the measured t e n s i o n Tg a t the upper support  as a parameter; u s i n g Tg one percent  g r e a t e r than the measured v a l u e ; and u s i n g T  F i g u r e 11 -  R  one percent  smaller.  D i f f e r e n c e s i n the Y - p o s i t i o n o f the cable between experiment and p a r a b o l i c model  versus  X - p o s i t i o n on the span, f o r f r e e hanging t e s t number 4.  The t h r e e graphs are drawn:  u s i n g the measured t e n s i o n Tg a t the upper support  as a parameter; u s i n g Tg one percent  g r e a t e r than the measured v a l u e ; and u s i n g T  R  one percent  smaller.  -  56  -  - 57 -  (or C ) f a l l s i n s i d e the new e r r o r zone.  T h i s procedure  was used f o r the a n a l y s i s of a l l  the v a r i a b l e s f o r a l l the t e s t s .  The e r r o r - z o n e i s d e f i n e d  by d a s h - l i n e s i n the d i f f e r e n t f i g u r e s throughout The  the t h e s i s .  a n a l y s i s has shown t h a t the values e^ and e.^ are almost  identical,  showing t h a t the system i s l i n e a r f o r s m a l l v a r -  i a t i o n s o f the parameter. found  The values e^ and e'^ were a l s o  i d e n t i c a l i n the a n a l y s i s which allows t o d e f i n e a  common e r r o r - z o n e f o r the catenary and p a r a b o l a models.  4.4  R e s u l t s and C o n c l u s i o n s .  4.4.1  Y - p o s i t i o n o f P o i n t s o f the Cable Experiment versus Models.  As shown i n F i g u r e 12 f o r T e s t #4, the average v a l u e s of the a b s o l u t e d i f f e r e n c e s between Y measured along the e n t i r e span and Y c a l c u l a t e d w i t h the catenary and p a r a b o l i c models a r e r e s p e c t i v e l y 27 mm and 70 mm.  The maximum o f  those d i f f e r e n c e s i s 39 mm f o r the catenary model and 126 mm f o r the p a r a b o l i c model. i s 10 8 mm.  The maximum h a l f e r r o r - z o n e  width  Those c h a r a c t e r i s t i c values c a l c u l a t e d f o r t h e  9 t e s t s a r e shown i n F i g u r e 13.  The e r r o r s w i t h t h e catenary  model are g e n e r a l l y s m a l l e r than t h a t of the p a r a b o l i c model. Except  f o r T e s t #1 the average a b s o l u t e d i f f e r e n c e s a r e  s m a l l e r than the maximum e r r o r .  The maximum d i f f e r e n c e s  - 58 -  F i g u r e 12 —  D i f f e r e n c e s i n t h e Y - p o s i t i o n o f the cable between experiment and catenary model, and between experiment and p a r a b o l i c model versus X - p o s i t i o n s on span, f o r f r e e hangi n g t e s t number 4.  The graph shows t h e  e r r o r zone taken i n account t h e e r r o r i n the parameter T . R  F i g u r e 13  —  Average and maximum d i f f e r e n c e s i n t h e Y - p o s i t i o n s o f p o i n t s o f the cable between experiment and catenary model, and between experiment and p a r a b o l i c model f o r the nine f r e e hanging t e s t s . width f o r each t e s t .  Maximum e r r o r zone  -  points f a l l  60  -  i n s i d e the e r r o r boundaries f o r 50% of the  f o r both the catenary  and  the p a r a b o l i c model.  The  as to whether the c a b l e hangs c l o s e r to a catenary  test  question or a  p a r a b o l i c shape cannot be answered c l e a r l y at t h i s p o i n t because of the dependence of the a n a l y s i s on the e r r o r i n the t e n s i o n T^.  A d i f f e r e n t approach w i l l now  be used t o  i n v e s t i g a t e the shape of the f r e e hanging c a b l e .  4.4.2 The catenary  Cable Shape: Catenary or  f o l l o w i n g approach c o n s i s t s of f i n d i n g  curve and  experimental  Parabola. the  the p a r a b o l i c curve t h a t best f i t the  p o s i t i o n measurements of the p o i n t s of the f r e e  hanging c a b l e .  For example, the catenary  f i t s the experiment i s the curve  curve t h a t b e s t  f o r which the sum  of  the  squares of the d i s c r e p a n c i e s between the measured Y - p o s i t i o n s of the c a b l e and i s minimum. support was  t h a t d e f i n e d by the equation  of the  curve  T_.c r e p r e s e n t i n g the t e n s i o n at the upper used as the parameter and  the problem was  to f i n d what t e n s i o n s at the upper support would g i v e best agreement between the experimental and a catenary  shape, and between the experimental  a p a r a b o l i c shape. cases was  shape of the  The  the cable  shape and  search f o r the optimum T_,c f o r a l l  implemented using a b i n a r y chop technique  F i b o n a c c i golden s e c t i o n s . The imental shape and  then  based on  agreement between the exper-  the b e s t - f i t curve  i s c h a r a c t e r i z e d by  average a b s o l u t e value of the d i s c r e p a n c i e s between the  the two.  - 61 -  F i g u r e 14 —  Average d i s c r e p a n c i e s between the measured Y - p o s i t i o n s of p o i n t s of the c a b l e and t h a t p r e d i c t e d by the b e s t - f i t  catenary  curve and by the b e s t - f i t p a r a b o l i c c u r v e , f o r the nine f r e e hanging t e s t s .  F i g u r e 15  —  Percent d i f f e r e n c e between Tg, measured t e n s i o n a t the upper support, and parameter c a l c u l a t e d f o r the  Tg^  best-fit  catenary curve and the b e s t - f i t p a r a b o l i c curve, f o r the nine f r e e hanging t e s t s .  -" 63 The nary for is  and  for  average the  b e s t - f i t  a l l  the  tests.  less  for  the  b e s t - f i t  parabola  the  b e s t - f i t  For  a l l  nine  where  This the  is  bola,  interesting  computed  the  actual  between  measured  cent  the  a l l  of the  curve.  both  1%  and  are for  the  expected  the  As  curve  Except  computed the  measured  tests.  parabolic  and  models  for  the  for  tests (see  compare  both  the  results  computed  tensions  a  18  and The  3,  as  p l o t t e d on  expected,  the  tensions  computed  that  the  Test  #1,  error  in  the  the  differences  tension the  are  tension  experiment.  the para-  differences  expressed  for  5  4).  the  are  than  is  4,  of  tension  smaller  mm.  parabolic  values  Tg.  the  results  Appendix  the  for  cable  number  catenary  tension  the  than  by  14  discrepancy  hanging  than  cate-  Figure  zone,  free  largest  to  the  smaller  curve  so  in  average  error  always  b e s t - f i t  shown  the  measured  measured  and  the  catenary  the  are  the  Whereas  of  p a r t i c u l a r l y  is  T_,c,  and  a  are  shape  for  tests  from  d e f l e c t i o n was  It optimum  by  9  shape.  catenary the  parabola  a l l  diverge  tests  represented  curve.  For  catenary  for  best  discrepancies  Figure  a 15  perfor  for  the  catenary  between  the  s u f f i c i e n t l y close to  give  confidence  to in  - 64 -  F i g u r e 16 - D i f f e r e n c e s i n the Y - P o s i t i o n s of the c a b l e between catenary  and p a r a b o l i c models  versus  X - p o s i t i o n s on span, f o r f r e e hanging Test  #4.  F i g u r e 17 - Sketches of the f r e e hanging c a b l e d e r i v e d from the catenary models.  shapes  and p a r a b o l i c  - 66 -  F i g u r e IS  — D i f f e r e n c e s i n Dm,  d e f l e c t i o n a t mid-  span between the c a t e n a r y model and the p a r a b o l i c model versus Tg, t e n s i o n a t the upper support, and versus Dm mid-span d e f l e c t i o n i n the f r e e hanging  cable.  4.4.3  Y - p o s i t i o n o f P o i n t s o f the Cable. Catenary  Model versus P a r a b o l i c Model.  A t y p i c a l graph of the d i f f e r e n c e s between the two models, o b t a i n e d with.the F i g u r e 16.  c o n d i t i o n s o f . T e s t #4, i s shown i n  The p a r a b o l a i s s l i g h t l y under the catenary a t  the lower end but i s p l a i n l y above towards.the upper end. T h i s p o i n t i s i l l u s t r a t e d on the s k e t c h o f the c a b l e shapes shown i n F i g u r e 17.  In the example, the p a r a b o l a i s 60 mm  above the catenary a t mid^span.  The d i f f e r e n c e a t mid-span  between the two t h e o r i e s i s p l o t t e d i n F i g u r e 18 versus values o f the t e n s i o n a t the upper support T .. The d i f f e r - ' D  ences decrease (i.e.  r a p i d l y when the t e n s i o n i n -the l i n e i n c r e a s e s  when the percent d e f l e c t i o n a t mid-span d e c r e a s e s ) .  o  o o  E Cvl E  ro Ci E o •M O  ro o u <—  oE  5 ' I -i  10  8  h  TR,  4  H X  6 1000 3  D e f l e c t i o n at mid-span, %  10  -  I f  t h e  d e f l e c t i o n  c a t e n a r y  b y  i s  o n l y  2  4.4.4  t a k e n  a s  d i s c r e p a n c y  t e n s i o n  m o d e l s  T ^  a n d  s h o w n  i n  i m e n t  a n d  w h i c h  a n  e r r o r  z o n e  h o w e v e r  e n c e  m i g h t  i n  a t  t h e  c o m p a r e d  b o t h  19  b e  t o  t h e  s h o w  a  m o r e  t h e  i s  a  was  l  was  a b o v e  t h e  c a n  t e s t s  b e  t h a t  b o t h  b e t w e e n  #1  e x p e r -  f o r  T h e  m a x i m u m  t h a n  2%,  p e r c e n t  t h e  T h e  r e s u l t s  e x c e p t  o n e  ,  s h o w n .  T h e  t a k e n .  n  n o  w i t h  l a r g e r  w i t h i n  T  t h e r e f o r e  a g r e e m e n t  t h e  s h o w i n g  t h a n  s u p p o r t ,  v a l u e .  s l i g h t l y  f a l l  M o d e l s .  e v a l u a t e d  p r o b a b l y  v e r y  p o i n t s  u p p e r  p o i n t  g o o d  l  v e r s u s  a n a l y s i s ,  t h a t  v e r y  m e a s u r e m e n t s  4.4.5  p a r a b o l a  t h e  m e a s u r e d  f o r  i s  a c c u r a t e  t h e  a t  r e a d i n g  h a l f - w i d t h  o f  i n  a t  s u p p o r t  t h e o r i e s  e r r o n e o u s  t h e  t e n s i o n  l o w e r  t h e  E x p e r i m e n t  t e n s i o n  t h e  3%  m i d - s p a n .  p a r a m e t e r  F i g u r e  f r o m  a t  m e a s u r e d  a  m o s t  t h a n  T e n s i o n s .  T h e  was  l e s s  mm  -  68  d i f f e r -  d y n a m o m e t e r s  e x p e c t e d .  T e n s i o n s :  C a t e n a r y  M o d e l  v e r s u s  P a r a b o l i c  M o d e l .  T h e  s u p p o r t  c o m p u t e d  m o d e l  a r e  u p p e r  s u p p o r t .  i s  a l w a y s  d e f l e c t i o n  t h e  t w o  d i f f e r e n c e s  w i t h  p l o t t e d  T h e  l a r g e r  o f  o n  t h e  t h e o r i e s  c a t e n a r y  F i g u r e  20  t e n s i o n  t h a n  t h a t  c a b l e  i s  b e t w e e n  a t  a b o u t  t h e  m o d e l  w i t h  t h e  m i d - s p a n  1%  a n d  v e r s u s  o b t a i n e d  a n d  t e n s i o n s  w i t h  t h e  w i t h  t h e  t e n s i o n  t h e  c a t e n a r y  t h e  a t  t h e  p a r a b o l i c  a t  t h e  p a r a b o l i c  m o d e l .  d i s c r e p a n c y  d e c r e a s e s  l o w e r  r a p i d l y  A t  m o d e l  1 0 %  b e t w e e n  t o  . 0 3 %  - 69 -  Figure 19 - Differences i n T , tension at the lower A  support, between experiment and catenarymodel, and between experiment and parabolic model, f o r the nine free hanging t e s t s .  Figure 20 — Differences i n T^, tension at the lower support, between the catenary model and parabolic model versus Tg, tension at the upper support, and versus Dm, mid-span d e f l e c t i o n i n the free hanging cable.  -70. -  ^-Catenary •fParabol i c  Mid-span d e f l e c t i o n , %  - 71 -  f o r a d e f l e c t i o n a t mid-span o f 3%.  The same a n a l y s i s c a r r i e d f o r the h o r i z o n t a l t e n ^ s i o n i n the l i n e y i e l d s the same c o n c l u s i o n s as f o r the t e n s i o n a t the lower  4.4.6  support.  Angles o f the Gable w i t h the H o r i z o n t a l . Experiment  versus Models.  The h i s t o r i g r a m s o f the d i f f e r e n c e s between the measured angles and the c a l c u l a t e d angles are shown i n F i g u r e s 21 and 2 2 f o r both model and both end o f the c a b l e . 85% of the p o i n t s are w i t h i n the e r r o r zone a t the upper support and o n l y 45% a t the lower support. of t h i s disagreement sheaves  An e x p l a n a t i o n  i s found c o n s i d e r i n g the r a d i u s of the  a t the upper and lower support not taken i n t o  con-  s i d e r a t i o n i n the t h e o r e t i c a l models.  4.4.7  Angles o f the Cable with the H o r i z o n t a l . Catenary Model versus P a r a b o l i c Model.  The d i s c r e p a n c y between the angles o f the l i n e a t the upper support c a l c u l a t e d with both model i s p l o t t e d i n F i g u r e 23 versus t e n s i o n a t the upper support.  The p a r a b o l a  i s always above the catenary a t the upper support but the d i f f e r e n c e gets very s m a l l as the t e n s i o n i n c r e a s e s .  The  e q u i v a l e n t curve i s shown i n F i g u r e 2 4 f o r the angle a t the lower support.  For l a r g e d e f l e c t i o n at mid-span  ( i . e . i f the  -  Figure 21 —  72  -  Differences i n « , angle of the l i n e with B  the horizontal at the upper support, between experiment and catenary model, and between experiment and parabolic model, f o r the nine free hanging t e s t s .  Figure 22 —  Differences i n o^, angle of the l i n e with the horizontal at the lower support, between experiment and catenary model, and between experiment and parabolic model, f o r the nine free hanging t e s t s .  - 74 -  F i g u r e 23 —  D i f f e r e n c e s i n dg, angle of the l i n e w i t h the h o r i z o n t a l a t the upper support,  between  catenary model and p a r a b o l i c model, versus Tg, t e n s i o n a t the upper support, and v e r s u s Dm,  mid-span d e f l e c t i o n i n the f r e e  cable.  hanging  Sketch of the r e l a t i v e p o s i t i o n o f  the c a t e n a r y and p a r a b o l a .  Figure 24 —  Differences i n a , A  angle of the l i n e w i t h  the h o r i z o n t a l a t the lower support, between c a t e n a r y model and p a r a b o l i c model, versus Tg, t e n s i o n a t the upper support, and versus Dm, mid-span d e f l e c t i o n i n the f r e e cable.  Sketches  of the r e l a t i v e  of the c a t e n a r y and p a r a b o l a .  hanging  positions  -,7 5  - 76  t e n s i o n i s l e s s than catenary  2750 n e w t o n s ) t h e p a r a b o l a  a l o n g the e n t i r e span.  the p a r a b o l a t h e two  curves  i n t e r c e p t a l o n g the span.  i s horizontal.  span d e f l e c t i o n i s  The  i s above  the  For d e f l e c t i o n l e s s than  i s under the c a t e n a r y a t the lower  f l e c t i o n at which the tangent support  -  7%  7%  support  and  i s not the  de-  t o the cable at the  lower  T h i s s i t u a t i o n o c c u r s when t h e  mid-  4.3%.  d i s c r e p a n c i e s between the angles o f the  lines  a s c o m p u t e d w i t h t h e c a t e n a r y m o d e l and w i t h t h e p a r a b o l i c m o d e l c a n be  c o n s i d e r e d as n e g l e c t a b l e f o r m i d - s p a n  t i o n s l e s s than  4.4.8 The  6%.  Cable  Length:  Experiment versus  l e n g t h measurement gave t h e t o t a l  length of the cable.  Models.  unstretched  To o b t a i n a v a l u e c o m p a r a b l e w i t h  t h e o r e t i c a l r e s u l t s a c o r r e c t i o n f o r e l o n g a t i o n has a p p l i e d t o the f i e l d the approximate S  =  deflec-  results.  The  to  be  correction i s defined  relationship: Sm  the  (1 + g | )  S  elongated  length  Sm  measured u n s t r e t c h e d  T_,  t e n s i o n i n the l i n e at the upper  E  e l a s t i c modulus o f t h e  A  c r o s s - s e c t i o n area  length  cable  support  by  -  Figure 25 —  77  -  Differences i n S, cable length, between experiment and catenary model, and between experiment and parabolic model, f o r the nine free hanging t e s t s .  Figure 26 — Difference i n S, cable length, between catenary model and parabolic model, versus Tg, tension at the upper support, and versus Dm, mid-span d e f l e c t i o n i n the free hanging cable.  - 79 -  t h e s k y l i n e , S i s given by:  For  S = Sm  (1 + T^/37000) w i t h T  D  e x p r e s s e d i n newtons.  The d i f f e r e n c e s b e t w e e n t h e m e a s u r e d  elongated  l e n g t h a n d t h e c o m p u t e d t h e o r e t i c a l l e n g t h a r e shown i n Figure  25 f o r t h e n i n e  f r e e hanging t e s t s .  I f T e s t #1. i s  ignored,  the r e s u l t s demonstrate the v a l i d i t y of both the  catenary  and p a r a b o l i c models l e n g t h f o r m u l a t i o n s  confirm  the value  and a l s o  chosen f o r t h e e l a s t i c modulus o f t h e  -  cable.  4.4.9  Cable Length:, Parabolic  Catenary Model  Model.  The d i f f e r e n c e s i n t h e c a b l e both models:are p l o t t e d i n Figure upper support. the catenary  versus  lengths  computed  with  26.versus t e n s i o n a t the  A r a p i d d e c r e a s e o f t h e d i f f e r e n c e between  model and t h e p a r a b o l i c model i s n o t e d as t h e  tension increases.  The p a r a b o l i c l e n g t h i s s h o r t e r b y 180  a t 10% m i d - s p a n d e f l e c t i o n . : a h d o n l y by l e s s t h a n 1 mm d e f l e c t i o n a t mid-span i s l e s s than  3%.  mm  i f the  - 80 -  CHAPTER 5  CLAMPED LOAD ON A S I N G L E The  LINE  f o l l o w i n g experiment was designed t o study the  e f f e c t o f a c o n c e n t r a t e d v e r t i c a l l o a d clamped a t a known distance  along the span o f a s i n g l e l i n e .  t i o n occurs i n a c t u a l y a r d i n g  This  configura-  systems when the c a r r i a g e i s  equipped with a s k y l i n e stop or when a c a r r i a g e bumper i s clamped on the s k y l i n e .  The same s i t u a t i o n i s a l s o found  when the chokers a r e attached  d i r e c t l y on the l i n e as i n  highlead.  5.1  D e s c r i p t i o n o f the Experiment. The  5/8-inch s k y l i n e was used f o r t h i s t e s t .  c a r r i a g e and l e a d weights c o n s t i t u t e d the v e r t i c a l  The  load and  a s m a l l clamp t h a t f i t the 5/8-inch s k y l i n e was manufactured to stop t h e c a r r i a g e from r o l l i n g .  A convenient way t o  execute the experiment was t o proceed as f o l l o w s : i ) choose a c a r r i a g e p o s i t i o n a t about 1/8 of the cable  span s t a r t i n g from the upper  support; ii)  choose a load from 100 Kg t o 500 Kg by steps of 100 Kg;  iii)  increase  the t e n s i o n  i n the s k y l i n e a t the  upper support from the minimum p o s s i b l e t o about 30,000 N i n 4 t o 6 steps;  - 81 -  i v ) lower the c a r r i a g e and  change the l o a d .  the e n t i r e range of load values has  If  been  i n v e s t i g a t e d change the c a r r i a g e p o s i t i o n .  While t h i s procedure i m p l i e s 175 only 29 were a c t u a l l y done and u l a t e d i n Appendix  5.2  different  tests,  the r e s u l t s of these are  tab-  4.  A n a l y s i s of the R e s u l t s .  5.2.1  Pre-considerations.  A s k y l i n e with a s i n g l e concentrated  load i s shown  i n F i g u r e 27 f o r three d i f f e r e n t p o s i t i o n s of the l o a d . Such a system can be c o n s i d e r e d  to be comprised of two  segments f r e e l y hanging between the load and.the two For the three p o s i t i o n s presented  cable supports.  the t o t a l d e f l e c t i o n a t  the load i s about 7 percent but the d e f l e c t i o n at the midp o i n t i n each of the f r e e hanging segments i s l e s s than percent.  Therefore  u s i n g the r e s u l t s of the a n a l y s i s of  f r e e hanging c a b l e presented  2.5 the  i n Chapter 4, the f o l l o w i n g  c o n c l u s i o n s can be drawn f o r the expected d i f f e r e n c e s b e t ween experiment catenary characteristics.  and parabola  f o r the two  segments  - 82' "  Table V I I .  Expected d i s c r e p a n c i e s i n the c h a r a c t e r i s t i c s of the f r e e hanging segments.  Differences between experiment and theories  Differences between catenary and parabola less than .02%  Tension Within the margin  Deflection at mid-point  33 mm  of Angles of the lines with the horizontal  .04° experimental errors 1 mm  Cable length  Table VII shows t h a t no d e t e c t a b l e introduced theories;  on the i n d i v i d u a l l i n e segments by the two thus the study can be r e s t r i c t e d only  a n a l y s i s of the load p o s i t i o n s tensions  d i f f e r e n c e s are  t o the  f o r d i f f e r e n t loads and  and a d e t a i l e d a n a l y s i s o f each l i n e segment i s  unnecessary.  5.2.2  Method of A n a l y s i s .  F i v e elements, shown i n F i g u r e consideration  27, are taken i n t o  i n the a n a l y s i s :  - R  load weight  - T_,  tension support  i n the s k y l i n e a t the upper  -  -  T  83  tension  A  -  in  the  skyline  at  the  lower  support -  X  horizontal  -  Y  vertical  The these  parameters.  T_. a n d This as  system  X  as  is  The  parameters  procedure  was  position  position  completely usual and  adopted  way  solve and  the  the  defined  of  load  load.  by  any  proceeding  for  the  of  of  the  is  variables  analysis  can  three to  take  Y  be  of  and  R, T  .  summarized  follows:  Measured Parameters R, T_ and X B  Computation u s i n g nary and p a r a b o l a  catetheories  Measured V a r i a b l e s Y and A  T h e o r e t i c a l v a l u e s f o r the v a r i a b l e s Yc and T.c A  Conclusion  5.2.3 The iables  and  Error  Analysis.  experimental  parameters  are  errors recalled  affecting in  Table  the  measured  VIII.  var-  - 84 -  Table  VIII.  Expected e r r o r s i n the measured v a r i a b l e s and parameters of the clamped l o a d on a s i n g l e l i n e system.  Experimental  Nomenclature  Designation  1%  R Parameters  Variables  the  same way  for  the  iations  treatment as  for  measured in  the in  the  Y  18 mm  the  variables  experimental  hanging T^  are  calculated values  of  Y  the  calculated variables  Y  parameters  upper  is  support  R  and  T  T_)  The  T^  The  with  is  done  error  enlarged  and .  obtained  (1.01  errors  cable.  and  the  the  1%  A  free  errors  at  Neglected  of  the  sion  X  T  The  1%  B  T  error  by  zones  the  resulting upper  the  combined  with  from  limit  largest the  var-  for ten-  small-  13  est  load  opposite extreme  had  The  extremes.  The  lower error  limit  is  obtained  zone  is  defined  with by  the  those  cases.  It load  R).  (.99  a  was  found  considerable  that  small  influence  errors on  the  in  the  tension  results.  and  - 85 -  Figure 27 —  Skyline with a single concentrated load f o r three d i f f e r e n t positions of the clamped load.  Figure 28 - Differences i n Y-position of the load, between experiment and catenary and parabolic models, f o r the 29 clamped load on a single l i n e t e s t s .  - 87 -  5.3  Results  5.3.1  and  Conclusions.  Y - p o s i t i o n o f the Load:  Experiment versus  Models. The  d i f f e r e n c e s between the measured Y and the Y  c a l c u l a t e d w i t h the catenary on F i g u r e  and p a r a b o l i c models are p l o t t e d  28 f o r the 29 t e s t s .  The d i f f e r e n c e between the  models i s so small t h a t i t does not appear on t h i s graph. Figure and  28 shows complete agreement between the experiment  the models f o r a l l the t e s t s but one.  5.3.2  Y - p o s i t i o n o f the Load - Catenary Model versus P a r a b o l i c Model.  The  discrepancies  between the two models w i l l be  maximum when the d e f l e c t i o n a t mid-point i n the l i n e segments i s maximum.  T h i s s i t u a t i o n occurs f o r the s m a l l e s t  (R = 100 Kg) and the minimum t e n s i o n clearance  f o r the complete load path.  load  (T_, = 5886N) t h a t  gives  The d i f f e r e n c e s , i n  those c o n d i t i o n s , between the two t h e o r i e s are p l o t t e d on Figure  29 versus h o r i z o n t a l p o s i t i o n s o f the load along the  e n t i r e span.  The p a r a b o l i c model over-estimates the d e f l e c t -  ion of the load r e l a t i v e t o the catenary p a r t of the load-path upper end.  model i n the lower  and under-estimates i t towards the  T h i s p o i n t i s i l l u s t r a t e d on a sketch  paths shown i n F i g u r e  30.  of the load  The maximum d i f f e r e n c e i s 15 mm.  - 88 -  Figure 29 —  Differences i n the Y-positi;bns of the load, between catenary and parabolic models versus X-positions on the span Tg R  tension at the upper support = 5885 N load = 100 Kg.  Figure 30 — Sketches of the catenary and parabolic model load paths f o r the clamped load on a single l i n e .  - 89 -  - 90 -  Figure  31 - a) Force balance a t the l e v e l i n a clamped load on a s i n g l e l i n e system, u s i n g catenary and p a r a b o l i c models. b) Unbalanced f o r c e s at the load l e v e l i n a clamped load on a s i n g l e l i n e ,  using  the p a r a b o l i c model and a Y - p o s i t i o n of the  load higher than t h a t a t e q u i l i b r i u m .  - 91 -  I t w o u l d be l o g i c a l  t o t h i n k t h a t because t h e l i n e .  weights are under-estimated with should  yield  cases.  a d e f l e c t i o n smaller  this  than t h e catenary  theory in a l l  H o w e v e r , t h e r e s u l t s show t h a t t h i s a f f i r m a t i o n i s  false for a large proportion  o f t h e span.  t i o n i s investigated i n Figure the  the parabola,  31.  Figure  forces a t the load f o r the catenary  equilibrium.  i s 107 mm a b o v e t h e p a r a b o l a .  parabola  has been a r t i f i c i a l l y  catenary;  contradic-  31a r e p r e s e n t s  and t h e p a r a b o l a a t  F o r t h e load p o s i t i o n chosen  catenary  This  (X = 65 m) t h e  In Figure  31b t h e  s e t a t t h e same l e v e l a s t h e  t h e f o r c e s c a l c u l a t i o n shows t h a t t h e l i f t i n g  c a p a c i t y o f t h e system i n t h i s p o s i t i o n i s s m a l l e r  than t h e  l o a d a n d t h e s y s t e m i s f o r c e d t o s a g more b e c a u s e o f t h e unbalanced  force.  a)  b) X - p o s i t i o n of R weight of T  R  the load on the span : 6 5 m  the load  t e n s i o n at  the upper support  : 100  Kg  : 5886 N  - 92 -  5.3.3  t e n s i o n a t the lower Experiment  support:  versus Models.  The d i f f e r e n c e between the measured T  and the  t e n s i o n c a l c u l a t e d w i t h the catenary and p a r a b o l i c models are p l o t t e d on F i g u r e 32 f o r the 29 t e s t s .  The d i f f e r e n c e  between catenary and parabola i s s m a l l and does not appear on the graph.  The r e s u l t s are between the upper and lower  e r r o r boundaries  f o r a l l the t e s t s but one.  T h e r e f o r e the  t h e o r i e s a r e c o n s i d e r e d t o agree w i t h the a c t u a l measurements .  5.3.4  Tension a t the lower support - Catenary Model versus P a r a b o l i c Model.  The d i s c r e p a n c i e s between the two t h e o r i e s are p l o t t e d on F i g u r e 33 f o r p o s i t i o n s o f the l o a d along the e n t i r e span.  The t e n s i o n at the lower support computed with  the p a r a b o l i c model are l a r g e r than t h a t with the catenary model f o r any p o s i t i o n o f the l o a d .  The d i f f e r e n c e i s  however always very s m a l l and l e s s than 3N (.05%) f o r the study case.  The agreement between the two t h e o r i e s i s best  when the load i s i n the mid-span area.  - 93 -  F i g u r e 32  -  Differences i n T , A  support,  t e n s i o n at the lower  between experiment and  p a r a b o l i c models, f o r the 29  and  l o a d on a s i n g l e l i n e  F i g u r e 33  -  catenary  Differences i n T , A  support,  clamped  tests.  t e n s i o n at the  between catenary  and  lower  parabolic  model versus X - p o s i t i o n s on the  span.  Tg  t e n s i o n at the upper support  = 586*5 N  R  load  = 100  Kg.  - 94 -  6  CHAPTER  GRAVITY SYSTEM  The the  experiment described  v e r y s i m p l e and  i n t h i s Chapter  commonly u s e d g r a v i t y  g r a v i t y system a c a r r i a g e  r u n n i n g on  to the  the  point  upper support w i t h o f maximum r e a c h by  6.1  Description The  and  the  of  stretched  length  the  skyline could  system.  The  be  tensioned with on  the  Set  at the ii)  upper  Choose a l o a d of  iii)  length  100  Let go  the  The  r e s u l t s of  the  be  the  The  un-  kept constant the  in a live  tension  skyline  c a r r i e d out  as  to in  gravity  follows:  the  skyline  support, f r o m 100  Kg  t o 500  Kg  by  steps  i t  can  Kg. carriage  run  down as  f r e e l y under i t s w e i g h t .  iv) P u l l  6-ton Comelong  (or t e n s i o n ) of  p o s i t i o n i s thus  of  the  system or  m o n i t o r e d as  the  pulled to i t s  G e a r m a t i c 19.  skyline could  t e s t s were t h e r e f o r e i)  returns  the  Experiment.  similate a fixed skyline gravity the  In  skyline i s  m a i n l i n e and  the  stored  of  system.  gravity.  s k y l i n e was  m a i n l i n e was  the  tested  t e s t s are  of  The  starting  determined,  carriage  a b o u t 1/8  f a r as  along the  the  span  reported  skyline in  length.  i n Appendix  4.  steps  - 96 -  6.2  A n a l y s i s o f the R e s u l t s .  6.2.1 The  Pre-considerations.  system can be (very s i m i l a r l y t o the clamped  load on a s i n g l e l i n e s t u d i e d three  i n Chapter 5) considered  as  s i n g l e l i n e segments f r e e l y hanging between the c a r -  r i a g e and each o f the supports.  The s k y l i n e segments are  very t i g h t , w i t h d e f l e c t i o n a t mid-span l e s s than 2.5%, i n most of the cases p e r m i t t e d by the l i m i t s o f the study.  As  concluded i n the a n a l y s i s o f the clamped load on a s i n g l e l i n e , u n d e t e c t a b l e d i f f e r e n c e s w i l l be i n t r o d u c e d  by the  p a r a b o l i c and catenary models on the c h a r a c t e r i s t i c s of the s k y l i n e f r e e hanging segments.  The t e n s i o n  i n the m a i n l i n e  i s u s u a l l y l a r g e enough t o keep the m a i n l i n e segment t i g h t when the c a r r i a g e approaches the upper support; however as the c a r r i a g e runs out the t e n s i o n and  i n the m a i n l i n e decreases  i s very small when the c a r r i a g e e v e n t u a l l y  stops.  The  d e f l e c t i o n i n the m a i n l i n e can then be l a r g e enough t o result i n noticeable  discrepancies  The  e f f e c t o f those d i s c r e p a n c i e s  and  on the t e n s i o n s  6.2.2  between the two t h e o r i e s . on the c a r r i a g e p o s i t i o n  i s i n v e s t i g a t e d i n the a n a l y s i s .  Method of A n a l y s i s .  S i x c h a r a c t e r i s t i c s o f the system are taken i n t o consideration  i n the a n a l y s i s :  - R - T  c a r r i a g e and load weight D  tension  i n the s k y l i n e a t the upper  support "  tension  A  T  i n the s k y l i n e a t the lower  support "  T  t e n s i o n i n the m a i n l i n e  B3  a t the upper  support - X  h o r i z o n t a l p o s i t i o n o f the c a r r i a g e  - Y  v e r t i c a l p o s i t i o n o f the c a r r i a g e  Any  combination o f three o f these c h a r a c t e r i s t i c s  t h a t does not i n c l u d e the p a i r completely.  R,X,T  D  ( T , T ) d e f i n e s the system g  were taken as parameters, i n the a n a l -  y s i s , t o determine the v a r i a b l e s T_, T catenary  and p a r a b o l i c f o r m u l a t i o n s .  y s i s procedure i s given  D O  and Y u s i n g the  A c h a r t o f the a n a l -  as f o l l o w s :  Measured Parameters R,X,T  Computation u s i n g catenary and p a r a b o l i c theories  Measured V a r i a b l e s T T , Y  T h e o r e t i c a l values f o r the v a r i a b l e s Y  0  r  3  Conclusion  - 98 -  6.2.3 The  Analysis.  computation of the experimental  determination  of the error-zone  as d e s c r i b e d line.  Error  e r r o r s and t h e  b o u n d a r i e s w e r e done e x a c t l y  i n C h a p t e r 5 f o r t h e clamped l o a d on a s i n g l e  The same i n f l u e n c e o f t h e e r r o r s i n t h e t e n s i o n a n d  l o a d on t h e t h e o r e t i c a l v a l u e s  6.3  Results  6.3.1  and  o f t h e v a r i a b l e s was f o u n d .  Conclusions.  Y-position of the Carriage  - Experiment  versus Models. The  d i f f e r e n c e s between t h e measured Y and t h e Y  calculated with Figure  34.  the catenary  and p a r a b o l a  F o r 13 c a r r i a g e p o s i t i o n s f r o m t h e l o w e r t o t h e  upper support.  T e s t #1 was a t t h e p o i n t o f maximum r e a c h o f  the  system f o r t h e given  tension  the  ground f o r t h e f i r s t  6 tests.  The  and t h e m a i n l i n e  t h e o r i e s agree w i t h  margin o f experimental The  are plotted i n  touched  the experiment w i t h i n the  e r r o r f o r t h e l a s t 2/3 o f t h e s e r i e s .  f a c t that the theories diverge  boundaries f o rthe f i r s t  g r e a t l y from t h e e r r o r  t e s t s c o u l d be e x p e c t e d s i n c e no  p r o v i s i o n i s made i n t h e a s s u m p t i o n s o f t h e t h e o r i e s f o r t h e n a t u r a l l i m i t a t i o n s o f t h e ground p r o f i l e . mainline  as c a l c u l a t e d by t h e p a r a b o l a  The s h a p e o f t h e  and c a t e n a r y  i s purely  - 99 -  Figure  34 - D i f f e r e n c e s i n Y - p o s i t i o n o f t h e c a r r i a g e , between e x p e r i m e n t and c a t e n a r y  and p a r a -  b o l i c m o d e l s , f o r t h e 13 g r a v i t y s y s t e m tests.  Figure  35 - M a i n l i n e s h a p e s a s p r e d i c t e d by t h e t h e o r e t i c a l models f o r t h r e e o f t h e g r a v i t y s y s t e m tests„showing i n t e r s e c t i o n w i t h ground p r o f i l e .  - 100 -  - 101  -  t h e o r e t i c a l f o r t h e f i r s t 6 t e s t s a s shown i n F i g u r e 3 5 . The m a i n l i n e i s c a l c u l a t e d t o be f r e e f r o m t h e g r o u n d f o r t e s t 7 compatibly  with the a c t u a l s i t u a t i o n i n the f i e l d  during the test.  Another deduction  the parabola catenary  does n o t d i v e r g e  does.  34 i s t h a t  from F i g u r e  f r o m t h e t e s t s a s much a s t h e  T h i s c a n be e x p l a i n e d c o n s i d e r i n g t h e f o r c e s  a c t i n g on t h e c a r r i a g e i n t h e a r e a o f maximum r e a c h system.  As t h e c a r r i a g e a p p r o a c h e s t h e l o w e r  h o r i z o n t a l t e n s i o n i n the mainline decreases component o f t h e t e n s i o n i n t h i s  support  l i n e at the c a r r i a g e  This v e r t i c a l  the v e r t i c a l  component a d d i n g  formulation the v e r t i c a l  in the mainline  stops  With the  component o f t h e t e n s i o n  i n c r e a s e s r a p i d l y towards i n f i n i t y  r e s u l t of the combination  to the  In the  component o f t h e m a i n l i n e t e n s i o n  i n c r e a s i n g when t h e m a i n l i n e t o u c h e s t h e g r o u n d . catenary  level  of the l i n e  w e i g h t o f t h e l o a d c a u s e s t h e s y s t e m t o s a g more. field  the  but the v e r t i c a l  i n c r e a s e s because o f the i n c r e a s e o f the angle with the h o r i z o n t a l .  of the  of the l i n e angle  as a  and l i n e  weight.  For the p a r a b o l i c model, the t o t a l weight o f the m a i n l i n e i s assumed t o be d i s t r i b u t e d on t h e c h o r d constant  as t h e l i n e  and t h e r e f o r e r e m a i n s  s a g s more a n d c o n s e q u e n t l y  the v e r t i c a l  component o f t h e t e n s i o n i n t h e m a i n l i n e c o n v e r g e s t o w a r d s a f i n i t e value. reality better.  As a r e s u l t t h e p a r a b o l i c model  represents  - 102  6.3.2  -  Y - p o s i t i o n of the C a r r i a g e Model v e r s u s P a r a b o l i c  S i m i l a r l y t o the  - Catenary-  Model.  a n a l y s i s of the  clamped l o a d  a s i n g l e l i n e i n C h a p t e r 5,  the  c o m p a r i s o n was  conditions  Kg)  and  support  of load  (T-£  (R = 100  = 6867N) t h a t r e s u l t s i n t h e  ment b e t w e e n t h e  two  shows d i s c r e p a n c i e s the  theories.  f o r the  The  as  l a s t s t a t i o n s of the  s e c t i o n of the  load path.  l e v e l was two  load  as  the  upper disagreeanalysis  s e v e r a l metres i n 36 up  The  to  parabola  carriage for a  short  T h i s phenomenon r e s u l t i n g f r o m  lines with  explained  largest  load path.  the d e f l e c t i o n at the  angles of the  at the  shown i n F i g u r e  over-estimates  the  done w i t h  r e s u l t s of t h i s  t h a t r a n g e as h i g h  a r e a o f maximum r e a c h , and  10 mm  tension  on  the h o r i z o n t a l at the  i n C h a p t e r 5.  Figure  37  carriage  sketches  the  paths.  6.3.3  Tension i n the Support:  The  S k y l i n e at the  Lower  Experiment versus Models.  d i f f e r e n c e s b e t w e e n t h e m e a s u r e d T,  and  the  the parabola  are  A  tension calculated with shown i n F i g u r e catenary on  the  and  f o r the  13  and  tests.  p a r a b o l i c models i s s m a l l  graph.  w i t h i n the  38  the catenary  The  two  and  t h e o r i e s agree w i t h  l i m i t s of experimental  errors.  noted f o r the v e r t i c a l p o s i t i o n of the present f o r the  The  tension, T . &  This  d i f f e r e n c e between does not the  experiment  The  divergence  carriage i s  should  appear  be  not  expected  con-  -  F i g u r e 36  -  103  -  D i f f e r e n c e s i n the Y - p o s i t i o n s of the c a r r i a g e , between catenary and  parabolic  models versus X - p o s i t i o n s on the Tg R  Figure  37  —  t e n s i o n at the upper  span.  support  6867  i n the s k y l i n e  =  Carriage plus load  = 100  Sketches of the catenary and  parabolic  models l o a d paths f o r the g r a v i t y system.  N Kg.  Or  104 -  - 105 -  F i g u r e 38  -  D i f f e r e n c e s i n T^, support  t e n s i o n at the lower  between experiment and  catenary  and p a r a b o l i c models, f o r the 13  gravity  system t e s t s .  F i g u r e 39  -  D i f f e r e n c e s i n T^, support,  t e n s i o n a t the lower  between catenary and  parabolic  models versus X - p o s i t i o n s of the c a r r i a g e on the Tg R  span. t e n s i o n i n the s k y l i n e at the  6867  upper support  =  Carriage plus load  = 100  N Kg.  - 106 -  X on the span, m  - 107  -  s i d e r i n g the catenary theory f o r which the d i f f e r e n c e between  and  depends on to and E o n l y .  6.3.4  Tension i n the S k y l i n e a t the Lower Support:  Catenary Model versus P a r a b o l i c  Model. The graph of the d i f f e r e n c e s between the v a l u e s of T  A  c a l c u l a t e d w i t h the two models f o r p o s i t i o n s of the c a r -  r i a g e along the e n t i r e l o a d path i s shown i n F i g u r e 39.  The  catenary y i e l d s the l a r g e s t v a l u e s f o r the t e n s i o n s f o r a l l the p o s i t i o n s of the c a r r i a g e .  The maximum d i f f e r e n c e i s  obtained a t each end of the l o a d path and i s 7 newtons or (.1%).  The two  t h e o r i e s agree almost p e r f e c t l y when the  c a r r i a g e i s a t mid-span.  6.3.5  Tg^ Tension i n the M a i n l i n e a t the Upper Support:  Experiment  Since the t e n s i o n was when the l i n e was  versus Models.  not recorded i n the m a i n l i n e  touching the ground, F i g u r e 40 shows the  h i s t o r i g r a m of the d i f f e r e n c e s between experiment i e s f o r the l a s t 7 t e s t s . the e r r o r boundaries;  and  H a l f of the t e s t s d i v e r g e  however, the tensiometer was  theorfrom  utilized  to measure t e n s i o n s l e s s than 1000N f o r those t e s t s and f o r t h i s range of t e n s i o n a one percent expected optimistic.  e r r o r i s too  -  Figure  40  -  Differences upper  gravity  41  -  and  the  upper  the  carriage  T  ,  3  the  between  0  0  ,  mainline  at  experiment  models,  tension  support, model, on  tension upper  R  B  in  for  the  and  the  13  tests.  in  parabolic  0  T  parabolic  and  T  -  tension  system  Differences at  in  support,  catenary  Figure  108  the in  between versus  the  mainline  catenary  X-positions  model of  span. the  skyline  support  Carriage  in  plus  load  at  the =  6867  =  100  N Kg.  109  GO  -  -  6.3.6  n o  -  Tension i n the M a i n l i n e Support:  a t the Upper  Catenary Model versus  P a r a b o l i c Model. The mainline  p l o t of the d i f f e r e n c e s f o r the t e n s i o n i n the  c a l c u l a t e d w i t h both t h e o r i e s and shown i n F i g u r e  41 demonstrates, as expected, t h a t the disagreement between the two t h e o r i e s i s small f o r p o s i t i o n o f the c a r r i a g e towards the upper support.  The maximum value  crepancy i s 8N a t the p o i n t o f maximum  reach.  of the d i s -  7  CHAPTER  D Y N A M I C  T E S T S  The tems few  of  was n o t one o f dynamic  potential idea and  study  of  tests  of  7.1  the prime  the field  model  and equipment  of  results  are presented  f i r s t  sys-  the thesis.  approach  to  The  A  show t h e  and o b t a i n  t o be expected.  i n this  cable  an  equipment  chapter.  Equipment. input  and output  were  changed.  volts  DC.  The output  of variable  speed  devices  f o r the  tensiometer,  The l o a d - c e l l s  were  excited  was c o n n e c t e d  to a  strip-chart  and s e n s i t i v i t y  (Plate  by a  10 record-  9).  Tests. Various  presented  a  of  of  as  only,  7.2  objectives  tried  The  er  behaviour  were  the type  tests  t h e dynamic  types  of  tests  were  carried  o u t ; two are  here: i)  V e r t i c a l  ii)  Instantaneous by  7.2.1 The  o s c i l l a t i o n stop  of  of  the  load  the carriage  running  gravity.  V e r t i c a l  position  of  Oscillation the load  of  being  the  Load.  surveyed  by the  - 112 -  Plate 9  —  Strip-chart recorder, generator and transformer-regulator  used f o r the recording of  the tensions at the upper support.  Plate 10  —  Manual i n i t i a t i o n of the v e r t i c a l tory motion of the clamped load.  oscilla-  -113-  - 114  usual technique manually  the  system i s brought i n t o v e r t i c a l  ( P l a t e 10).  r e a c h e s a b o u t 150%  -  motion  When t h e t e n s i o n i n t h e s k y l i n e  of i t s s t a t i c value  the e x c i t a t i o n i s  s t o p p e d and  the t e n s i o n a t the upper support  i s recorded  u n t i l damping b r i n g s i t back t o i t s o r i g i n a l  s t a t i c value.  A t y p i c a l c h a r t i s shown i n F i g u r e  Conclusion The e x a m p l e has  ii)  of the  42.  Test.  v i b r a t i o n recorded  the i)  i n the s k y l i n e  f o r t h e t e s t g i v e n , as  an  following characteristics: frequency  .475  damping r a t i o  Hertz  ( p e r i o d 2.10  seconds)  .05.  No m a j o r d i f f e r e n c e s w e r e n o t i c e d w h e t h e r t h e c a r r i a g e clamped or hooked t o the  7.2.2  mainline.  Instantaneous R u n n i n g by  For  was  Stop of the  Carriage  Gravity.  t h i s t e s t the c a r r i a g e i s f r e e d t o run  from a  known p o s i t i o n , t h e n s t o p p e d s u d d e n l y w i t h t h e m a i n l i n e w i t h a c l a m p on t h e recorded skyline.  skyline.  a t the upper support The  The  43.  The  the t e n s i o n s i s  on b o t h t h e m a i n l i n e  f i r s t e x a m p l e was  d e s c r i b e d on F i g u r e  e f f e c t on  performed f o r the  r e s u l t s are  or  and  the  conditions  shown i n F i g u r e  45.  - 115  Figure 42  -  -  Chart-recording of the tension i n the skyline at the upper support during v e r t i c a l o s c i l l a t i o n of the load of the gravity system. Recorder: - input - output - chartspeed  = 10 V = 20 mV = 10 sec/inch  Carriage and load:  - 495  Kg. - located at mid-span - deflection 6.7$  - 117 -  F i g u r e 43 -  Sketch of dynamic t e s t . C a r r i a g e stopped by the main l i n e . S t a r t i n g p o s i t i o n o f the c a r r i a g e .  X = 106.09 m Y = 11.OS m  Figure 4 4 -  S k e t c h of dynamic t e s t . C a r r i a g e stopped w i t h a clamp on the s k y l i n e . S t a r t i n g p o s i t i o n o f the c a r r i a g e .  X = 72.02 Y = 2.96  - 119 -  C o n c l u s i o n of the T e s t .  Tension i n the s k y l i n e .  J u s t a f t e r the shock, a drop t o 60% of the o r i g i n a l s t a t i c tension rapidly  f o l l o w s a s m a l l peak a t 135%.  The maximum t e n s i o n reached  i s 150% of the s t a t i c t e n s i o n  and a complex v i b r a t i o n  phenomenon takes p l a c e i n the l i n e  with a dominant low frequency of .67 Hertz  (period 1.5  seconds).  Tension i n the m a i n l i n e .  An i n c r e a s e o f more than 700% from the o r i g i n a l s t a t i c t e n s i o n i s recorded a f t e r the shock. e f f e c t i s repeated p e r i o d i c a l l y  This  snapping  every 1.5 seconds w i t h a  r a p i d l y d e c r e a s i n g magnitude (about 300% f o r the t h i r d peak).  Other v i b r a t i o n s of h i g h e r f r e q u e n c i e s  interfere  with t h i s b a s i c pattern.  The  second  example was performed  d e s c r i b e d i n F i g u r e 44.  The r e s u l t s  i n the c o n d i t i o n s  are shown i n F i g u r e 46.  C o n c l u s i o n of the t e s t .  The example i s presented i n p a r a l l e l w i t h the r e s u l t o b t a i n e d f o r a t e s t o f the f i r s t type.  The b a s i c  -.120  Figure 45 -  -  Chart-recordings of tensions i n the l i n e s of a gravity system during dynamic t e s t s . a) i n the skyline b) i n the mainline Carriage stopped with the mainline.  Time,  seconds  -  Figure 4 6 —  122  -  Chart-recordings of tensions i n the skyline at the upper support. a) carriage stopped with a clamp b) carriage stopped with the mainline  -  123 -  - 124 -  difference following  i s t h e s h a r p p e a k o f 155% o f t h e s t a t i c t e n s i o n t h e s h o c k on t h e c l a m p .  d i s t u r b e d i n t h i s second example  The s k y l i n e and complex  i s much more  vibration  phenomena o c c u r i n t h e s y s t e m l o n g a f t e r t h e s h o c k .  7.3  Conclusion. D y n a m i c t e s t s w e r e e a s i l y p e r f o r m e d on t h e f i e l d  model.  The f i r s t  results  show t h a t c o n s i d e r a b l e  dynamic  t e n s i o n s c a n be d e v e l o p e d i n t h e l i n e s a n d d e m o n s t r a t e c l e a r l y t h e need f o r f u r t h e r  research  i n this  area.  - 125 -  CHAPTER  D I S C U S S I O N  A N D  8  C O N C L U S I O N  Both the r e s u l t s o f the a n a l y s i s and the p r a c t i c a l aspect o f c a b l e l o g g i n g problems should be taken i n t o cons i d e r a t i o n to propose an o p e r a t i o n a l t h e o r e t i c a l model f o r c a b l e l o g g i n g systems.  The  catenary  w i t h an a c t u a l f i e l d configurations.  and the p a r a b o l i c model were compared  t e s t f o r three t y p i c a l c a b l e system  The f a c t , shown i n Chapter 4, t h a t the shape  of a f r e e hanging c a b l e segments i s c l o s e r t o a catenary a parabola ing  i s probably  not a s u r p r i s e t o an engineer  than  trust-  the b a s i c laws o f mechanics; the catenary model des-  c r i b e s a c a b l e y a r d i n g system with b e t t e r p r e c i s i o n than the p a r a b o l i c model does, but i s t h i s p r e c i s i o n needed f o r pract i c a l applications?  Cable y a r d i n g systems do not operate  i n i d e a l experim-  e n t a l c o n d i t i o n s and much u n c e r t a i n t y i s attached  i n the  field  to t h e i r various c h a r a c t e r i s t i c s .  cases  the ground p r o f i l e i s known with a p r e c i s i o n of about  300 mm  (1 foot) .  In the best o f the  The u n c e r t a i n t y on the v a l u e s of the  a t i n g t e n s i o n s i s as l a r g e as 10% on the e x i s t i n g  oper-.j.  yarders  even equipped with the most s o p h i s t i c a t e d t e n s i o n c o n t r o l  - 126 -  systems and the i n a c c u r a c y i n the s c a l i n g of the l o g s can r e s u l t i n an i n p r e c i s i o n of 10% i n the payload v a l u e . Table IX compares these a c t u a l p r e c i s i o n s w i t h the p r e c i s i o n s i n the c h a r a c t e r i s t i c s of the f i e l d model, and w i t h the differences  between catenary and p a r a b o l i c  Table IX.  System Characteristics  models.  P r e c i s i o n i n the knowledge of the c h a r a c t e r i s t i c s o f : the f i e l d model a r e a l y a r d i n g system D i s c r e p a n c i e s between catenary model and p a r a b o l i c model f o r the same characteristics.  Nomenclature  F i e l d model precision  R e a l system precision  Discrepancies Cat. - Par.  Ground p r o f i l e Vertical position  Y  Operating tension  T  1%  10%  l e s s than  Payload  R  1%  10%  l e s s than 1%  18  mm  300  15  mm  mm  1%  The f i e l d model c h a r a c t e r i s t i c s are ten times more a c c u r a t e than a r e a l y a r d i n g system; t h i s i s j u s t i f i e d by the s c i e n t i f i c aspect o f the experimental approach. the catenary and p a r a b o l i c  Both  model compute the system charac-  t e r i s t i c s w i t h a p r e c i s i o n f a r beyond what i s known and  - 127  needed i n t h e f i e l d , of  -  t h e r e f o r e any m o d e l o r any  combination  t h e m a i n a s s e t s o f b o t h m o d e l c a n be u s e d t o  investigate  the s t a t i c  c h a r a c t e r i s t i c s of a cable logging  B r i e f D e s c r i p t i o n of the Model  The  model proposed  system.  Proposed.  uses equation g  (Chapter  2)  from the c a t e n a r y f o r m u l a t i o n , t o t r a n s f e r a l l the f o r c e s known i n t h e s y s t e m , bolic  to the c a r r i a g e l e v e l .  Then, t h e  model i s used t o f o r m u l a t e the e q u a t i o n s of  brium of the c a r r i a g e . i m p l e m e n t e d by C a r s o n  This procedure  was  para-  equili-  successfully  and Mann(3) f o r a r u n n i n g  skyline  system.  A s i m p l e and  sufficiently  a c c u r a t e model i s a v a i l -  a b l e to d e s c r i b e a c a b l e y a r d i n g system tions.  T h i s model would s a t i s f y  d e s i g n e r s and  i m p l i e d by t h e m o d e l .  i s not e x a c t l y the case, c a b l e system  u s e r s have t r a d i t i o n a l l y static The  r e s u l t s to account  system i n the  Conscious designers  that and  a p p l i e d a f a c t o r of s a f e t y to  the  f o r the v a r i o u s u n c e r t a i n t i e s .  f a c t o r 5 recommended by t h e Workmen's C o m p e n s a t i o n  B o a r d ( 1 1 ) i s more " i g n o r a n c e " t h a n  " s a f e t y " and  more r e s e a r c h s h o u l d be done i n t h e s t u d y o f t h e of  condi-  the needs of c a b l e  u s e r s i f c a b l e systems were o p e r a t e d  conditions of s t a t i c this  in static  cable logging  systems.  proves  that  behaviour  -  128  -  The c o n c l u s i o n s from the dynamic t e s t s d e s c r i b e d i n Chapter limited.  7 are t e n t a t i v e s i n c e the experiment was I t was  noted t h a t the t e n s i o n s were very much  d i s t u r b e d by the c a r r i a g e motion and 700%  of the s t a t i c value was  a t e n s i o n as h i g h as  recorded i n the m a i n l i n e .  should be noted, however, t h a t the m a i n l i n e was initially.  very  It  understressed  The dynamic t e s t s y i e l d more q u e s t i o n s  than  answers to the problem at t h a t p o i n t of the study but  clearly  show the need f o r f u r t h e r i n v e s t i g a t i o n s i n t h i s f i e l d  and  a l s o demonstrate the p o t e n t i a l u s e f u l n e s s of the f i e l d model to  c a r r y those  investigations.  Conclusion.  The  study i s s u c c e s s f u l i n s e l e c t i n g a simple  t h e o r e t i c a l model f o r the d e t e r m i n a t i o n of c a b l e y a r d i n g system s t a t i c c h a r a c t e r i s t i c s . i s based on the comparative field  The  s e l e c t i o n of the model  a n a l y s i s of the r e s u l t s of the  t e s t s and the t h e o r i e s , showing t h a t although  the  shape of a f r e e hanging c a b l e i s b e t t e r d e s c r i b e d as a c a t e nary than a p a r a b o l a both t h e o r e t i c a l models are  accurate  enough to s o l v e p r a c t i c a l c a b l e system problems.  The  t h e s i s a l s o shows the importance of the  dynamic f o r c e s a c t i n g on the system. t h i s area i s r e q u i r e d .  Further research i n  - 129 -  L I T E R A T U R E  C I T E D  B i n k l e y , V.W. and S t u d i e r , D.D. 1974. Cable Logging Systems. USDA F o r e s t S e r v i c e , P o r t l a n d , Oregon, 190 p. Carson, W.W. and Mann, C.N. 1970. A technique f o r t h e s o l u t i o n of s k y l i n e catenary equations. PNW-110, 18 p., i l l u s . P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n , P o r t l a n d , Oregon. Carson, W.W. and .'.Mann, C.N. 1971. An a n a l y s i s of runn i n g s k y l i n e l o a d path. PNW-120, 9 p., i l l u s . P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n , P o r t l a n d , Oregon. Carson, W.W. 1975. A n a l y s i s o f running s k y l i n e with . :: drag. PNW-193, 8 p., i l l u s . P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n , P o r t l a n d , Oregon. I n g l i s , S i r Charles Edward. 1963. A p p l i e d Mechanics f o r Engineers. 404 p., i l l u s . New York Dover P u b l i c a t i o n s , Inc. 6.  Lysons, H.H. and Mann, C.N. 1967. S k y l i n e Tension and D e f l e c t i o n Handbook. U.S. F o r e s t Serv. Res. Pap. PNW-39, 44 p.,. i l l u s . P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n , P o r t l a n d , Oregon.  7.  Selby, S.M. 1964. Standard Mathematical T a b l e s . The Chemical Rubber Co., C l e v e l a n d , Ohio.  632 p.  Veda, M., S a i t o , T., Tominaga, M.,and S h i b a t a , J . 1962. S t u d i e s on the Main Cable i n S k y l i n e Logging. F i r s t Report. B u l l e t i n of the Government F o r e s t Experimental S t a t i o n no.188. Tokyo, Japan. Veda, M. and S a i t o , T. 1965. S t u d i e s on the Main Cable i n S k y l i n e Logging. Second Report. B u l l e t i n o f the Government F o r e s t Experiment S t a t i o n no.188. Tokyo, Japan. 10  Veda, M. 1966. Studies on the Main Cable i n S k y l i n e Logging. T h i r d Report. B u l l e t i n o f the Government F o r e s t Experiment S t a t i o n no.18 8. Tokyo, Japan.  - 130 -  11.  Workmen's C o m p e n s a t i o n B o a r d . 1972. A c c i d e n t P r e v e n tion Regulations. 298 p. Workmen's C o m p e n s a t i o n Board o f B r i t i s h Columbia.  12.  W i r e r o p e Handbook. 1959. U n i t e d S t a t e s S t e e l C o r p o r a t i o n , San F r a n c i s c o , C a l i f o r n i a . 193 p.  - 131 -  APPENDIX  1  Appendix 1 p r e s e n t s the p a r a b o l i c model developed by G.G. Young"*" and the author, f o r the study o f c a b l e l o g ging system problems.  Appendix 1 was p r i m a r i l y produced  as  a t e a c h i n g a i d f o r a u n i v e r s i t y course i n f o r e s t h a r v e s t i n g . I t s t u d i e s c a b l e mechanics from the very b a s i c f r e e c a b l e t o the more complex f i v e l i n e system.  hanging  Because the  course can be attended by students w i t h a l i m i t e d knowledge i n mechanics the development i s very d e t a i l e d i n the f i r s t c h a p t e r s , t o become reasonably s u c c i n c t towards the end as the student p r o g r e s s e s .  Numerical  examples a r e presented  a f t e r every major development.  Although  intended p r i m a r i l y f o r f o r e s t r y  students,  we b e l i e v e t h a t t h i s paper can be o f use t o anyone d e a l i n g w i t h problems i n c a b l e mechanics.  Assistant Professor H a r v e s t i n g and Operations Research F a c u l t y of F o r e s t r y U n i v e r s i t y of B r i t i s h Columbia  -  132  -  CONTENTS  Introduction  A.  B.  Free hanging  cable  1.  General d e s c r i p t i o n of  2.  Models of  3.  Equation of  4.  T e n s i o n s and d e f l e c t i o n  system  2  the c a b l e shape  3 7  concentrated  load  Governing equations  for  The c a b l e w e i g h t  assumed t o  on the  1  the system  Cable with a s i n g l e 1.  the  is  the  system act  subchords  16  2.  Sag and d e f l e c t i o n  3.  Problem 1 -  19  Given the  sag  •  ( o r d e f l e c t i o n D^) a t x = x^ and t h e clamped l o a d R what the 4.  t e n s i o n s T ^ and T ^ a t  Problem 2 -  Given the  are  the  supports?  22  sag  ( o r d e f l e c t i o n D^) a t x = x^ a n d the  tension T at the upper support B what i s the l i f t i n g c a p a c i t y R o f t h e 5.  Problem 3 -  Given  the  u p p e r s u p p o r t and the what i s at  tension T  B  at  system?  27  the  l o a d R a t x = x^  t h e s a g S ( o r d e f l e c t i o n D)  the p o s i t i v e  of  the  load?  31  133  -  C.  Standing s k y l i n e .  Gravity  1.  Description of  2.  Governing equations  3.  Problem 1 (or  the  Given  the d e f l e c t i o n  what i s  the  -  system  system  34  f o r the system  the  •  35  sag S  D) and t h e  t e n s i o n T^  at  l o a d R-  B in  the  skyline? 4.  37  Problem 2 (or in  Given  the d e f l e c t i o n the s k y l i n e a t  c a p a c i t y R of 5.  the  what i s at  sag  S•  D) a n d t h e B what i s  tension  the  lift  the system?  Problem 3 - G i v e n the in  D.  the  40  tension  T  f i 2  s k y l i n e a t B and the l o a d R the  sag  S (or  the  deflection)  the c a r r i a g e ?  Five  line  1.  D e s c r i p t i o n of  2.  Governing  3.  Mathematical  43  system  skyline  the system  equations for  the  44 system  s i m i l i t u d e between  and t h e f i v e  l i n e system  the  45 standing ;  47  CABLE MECHANICS  Free hanging 1.  cable  General d e s c r i p t i o n of  ;  the  system  V •y  •••  :  B  c. ^  \  E  B  '  VA'  y'  A  H  ^ \ C  = cable  .  '." *  o  :  x'  -  L  ^ • .. .  1.1  9  : -  -—'  -•- '  '-' -----  • •:• -- - -  Geometry C = cable B = upper support of  the  cable  A = lower  the  cable  L = span;  support of  horizontal distance  E = difference  in- e l e v a t i o n  between the  between t h e  supports  supports  AB = c h o r d 6 = angle  between t h e  P = any p o i n t  on t h e  c h o r d and t h e  1.2  x',  y'  = coordinate  [6 = a r c t g  (E/L)]  cable  0 = p o i n t of h o r i z o n t a l tangent 0,  horizontal  of  the  cable  system  Forces The t e n s i o n of  the  at  cable.  any p o i n t o n a c a b l e ' a c t s  a l o n g the  tangent  at  that  point  - 135 -  T  P  H  P  V  P  T  A  T  B  tension  = horizontal = vertical  Models  of  the  cable  at  component o f  component o f  P the  tha  at  the  lower  tension  at  the  upper support  of  the  cable  tension  tension  tension  = weight 2.  i n the  at  at P P  support  per u n i t  length  system  D i f f e r e n t e q u a t i o n s c a n b e d e v e l o p e d d e p e n d i n g o n t h e a s s u m p t i o n s made a s how t h e u n i f o r m l y d i s t r i b u t e d c a b l e w e i g h t a c t s o n t h e c a b l e . Three d i f f e r e n t  models  c o u l d be  to  used:-  model 1  : cable weight i s of the system  u n i f o r m l y d i s t r i b u t e d over the h o r i z o n t a l span  model 2  : cable weight system  u n i f o r m l y d i s t r i b u t e d over  model 3  : cable weight i s cable i t s e l f  model  1  is  the  u n i f o r m l y d i s t r i b u t e d over the  model  2  chord of  length of  model  the  the  3  T h e b e s t a c c u r a c y i s o b t a i n e d w i t h m o d e l 3 , b u t on t h e o t h e r h a n d m o d e l s and 2 p r o v i d e a s i m p l e r m a t h e m a t i c a l f o r m u l a t i o n f o r the s y s t e m .  1  The d e v e l o p m e n t o f t h e e q u a t i o n s f o r t h e s y s t e m i n t h i s p a p e r i s b a s e d on m o d e l 2. I t g i v e s a v e r y good d e s c r i p t i o n o f l o g g i n g c a b l e s i n most c a s e s . T h e e q u a t i o n s a r e most a c c u r a t e f o r t i g h t c a b l e s .  - 136 -  3.  Equation of  3.1  the  cable  Equation of  shape  the  cable  The o r i g i n 0 i s  shape  •  a c t i n g on t h e  *  H  tension  *  **P  *  W = x'u/cosG  +  ^P  c o o r d i n a t e system 0,  x',  y'  t h e p o i n t o f maximum s a g .  The s y s t e m c o n s i d e r e d i s Forces  i n the  =  ^P  the  c a b l e b e t w e e n 0 and P .  system: at  0,  horizontal  tension  at P  assumed w e i g h t  of  the  c a b l e between  0 and P  - 137 -  Equations of e q u i l i b r i u m *  Sum o f  v e r t i c a l forces V  *  Sum o f  -  p  x'w/cose  = 0 = 0  horizontal forces  = 0  - H + H p = 0 of *  Sum o f  the  cable  moments a b o u t Hy' cos6  The l a s t y'  tension  3_  component  constant  throughout  the  cable  0  to  This  x' 2  horizontal  2  c a n be a r r a n g e d t o  2cos6H  is  i.e.  P = 0  equation =  .Hp=H  is  the  yield equation of  the  cable  in  c o o r d i n a t e system O x ' y ' . Conclusion: c a b l e hangs l i k e a p a r a b o l a .  the the  The above e q u a t i o n i s r e l a t i v e t o a c o o r d i n a t e s y s t e m w i t h o r i g i n a t t h e p o i n t o f maximum s a g i n t h e c a b l e . S i n c e t h i s p o i n t moves w i t h v a r y i n g t e n s i o n , i t i s not a very u s e f u l e x p r e s s i o n . N o r m a l l y one r e q u i r e s a c o o r d i n a t e s y s t e m w i t h o r i g i n a t one o f t h e s u p p o r t s .  Equation of  the  cable  shape  i n the  coordinate  system A , x,  y.  - 138 -  The t r a n s l a t i o n  of  rx = x '  the  coordinate  + a o  Substituting  system i s  defined  by:  rx' = x - a iy' = y + b  r  f o r x' and y' i n y'  x'  yields,  2  the  new  equation  2cos9H of  the  cable  i n the  system A ,  x,  y + b =  (x -  oi  a)  2  2cos6H ^ Now t o  determine  Limit  the  values  conditions,  of  a.and  b:  x= 0 when y = 0 and x = L when y = E  o)  give:  and  a'  2cos9H  E  +  b  =  <" ( 2cos6H  L  " a)  S u b t r a c t i n g t h o s e two  2  from which  b =  to 2cos6H  y +  to • ( L 2cos9H L 2  and  y +  to (L 2cos6H  2  + a  -  2La)  2  -  2La)  cos6HE uL  L - COSSHE! aiL  2  to 2cosGH  and b i n t h e  equation  of  fL - COS9HE!  2  2  2  equations  E =  Substituting"a  =  the  x -  toL  cable: L 2  2cos6H  cos9HE uL  x + fL - cs*efl|T - 2x 2  2  to 2cos8H  toL L -  2cos6H  y =  T h i s i s the p a r a b o l i c e q u a t i o n at the l e f t h a n d s u p p o r t .  u x 2cos6H  of  the  cos6H  2  +  E  L  cable  cosBHE toL  cos9HE toL  -2 c o s 6toL H  with  L 2  the  X  coordinate  syste  - 139 -  T h e above e q u a t i o n can be r e a r r a n g e d s o t h a t i t s components m e a n i n g f u l from a p h y s i c a l s t a n d p o i n t y i e l d i n g :  - u x ( L - x) 2cos8H  From t h e  above  Ex L  -MX(L -  x)  2cos6H  figure  it  c a n be  seen  that:  NP  =  y  position  of  the  cable  at  x  NM  =  y  position  of  the  chord at  x  NP -  NM = NP + MN = MP  referred  of  the  at to  point as  cable  at  the  x.  x = L/2  Tangents  the  to  cable  and  vertical This  x = D -  distance  quantity  DEFLECTION o f  x  At midspan  more  + E_x L  and c a b l e  Deflection  are  is  the  cable  = MX(L - x) 2cos6H  Dm = t o L / 2 ( L - L / 2 ) 2cos6H  =  coL 8cos6H 2  between chord  classically at  x.  - 140 -  Since the necessary  t e n s i o n a c t s on t h e t a n g e n t o f t h e c a b l e i t i s v e r y o f t e n t o know t h e s l o p e o f t h i s t a n g e n t a t a g i v e n p o i n t P .  The t a n g e n t a i s g i v e n b y t h e v a l u e c a b l e e q u a t i o n w i t h r e s p e c t t o x:  the  2cosGH  (ii  L  x +  -  cos8H  Example:  the  the  slope  at  uiL ] 2cos6H  A is:  = tga  = Slope  = E -  tga  slope  at  B is:  tga  = D  Tensions  and  derivative  of  the  2cosGHj  L  4.  first  01  y =  dx  of  of  the  cable at  point  x.  oiL 2cos9H  oiL  + E  cos6H  —  L  E  2cos9H  uL  +  L  2cos6H  deflection  G i v e n the geometry o f the system ( E and L ) and the type of c a b l e ( u ) e q u a t i o n s o f t h e c a b l e s 1 and 2 o n l y d i f f e r b e c a u s e o f t h e v a l u e s o f horizontal tensions and H ^ .  the the  The h o r i z o n t a l t e n s i o n i s a v e r y h a n d y p a r a m e t e r f o r t h e d e r i v a t i o n o f t h e e q u a t i o n s b u t i t i s n o t o f r e a l i n t e r e s t when s o l v i n g a p r a c t i c a l p r o b l e m . In p r a c t i c e  the  important f a c t o r s  are:  -  deflection  at  -  tension  the  at  a given  point  supports  - 141 -  4.1  P r o b l e m 1:  Given  tensions  and  4.1.1  the  deflection  at  Horizontal  the  tension  The e q u a t i o n  for  D =  at  point  is  D  x =  in  the  the  X  are  the  cable  deflection  =  1  • Then H = "  what  supports?  1_  2cos6H  1 l> 2D cos9 (  L  x  1  If in  4.1.2  the the  e q u a t i o n of the c a b l e i s needed equation d e r i v e d at 3.2.  Tensions  T  f  i  at  = _ H _ cosa o D  the  and  plug this  value  supports  T  A  =  H cosa. A  and f r o m 3 . 3 ,  t  with  H =  U  X  1  " l 2D,cos9 ±  & r>  ~ j± + L  • IOL 2cos6H  tga.  = F, L  oiL 2cos6H  0L  (  L  X  )  of H  - 142 -  The d e f l e c t i o n  D = Dm i s  In  H =  this  case  OJL  known a t m i d s p a n .  x^ = L / 2  D = Dm  2  8Dmcos8 and  tga  tga.  The t e n s i o n s a t T  f i  =  = _E +  ooL  L  2cos6  (j,  = E L  h)L 2cos9  8Dmcos6 2  the  8Dmcos9 = E + 4Dm  supports  wL . 8 c o s a „ Dmcos9  and  2  =  w L  c a n be  T^ =  K  NOTE:  L  2  E - 4Dm L  obtained  from  o)L 8 c o s a . Dmcos9 2  A  A n o t h e r way o f d e r i v i n g t h e s l o p e o f t h e c a b l e a t i s to use the f o l l o w i n g p r o p e r t y of p a r a b o l a s : The t a n g e n t s a t A and B c r o s s t h r o u g h m i d s p a n and PQ = MP  at  Q on the  Then: tga.  =  JEL AN  tga  =  -2Dm + E / 2 = E - 4Dm L/2 L  = J £ _ = -(2Dm + E / 2 ) T5R  -(V2)  = 4Dm + E L  N  A and B  vertical  - 143 -  o  If the cable i s very different. Assuming the chord With t h i s  Then T = B  B  1.5  the  the  angle  and 8 a r e  t e n s i o n at  B acts  not  along  assumption  H  =  a  cose  D = Dm i s T =  tight  = 8 i s assuming that of the system.  uix, ( L - x , ) 1 1_ 2DjCose  H =  if  sufficiently  )  , and  X  l  (  L  "  X  ' c o s a ^ = cos8  recall  tg8  = E_ L  l>  aD^cose)  2  g i v e n at inidspan  coL 8Dm(cos8) 2  2  Example Problem A f r e e h a n g i n g c a b l e has a 500-meter span w i t h a d i f f e r e n c e of 200 m e t e r s b e t w e e n t h e s u p p o r t s . The w e i g h t p e r m e t e r o f t h e 1" c a b l e i s a b o u t 6 l b s . The m i d s p a n d e f l e c t i o n i s 2%. a)  Assuming t h a t the  b)  tensions  Check t h a t is  close  to  the at  the the  tensions  both  act  a l o n g the  cable,  what  are  ends?  difference difference  i n t e n s i o n between in elevation  the  t i m e s oi.  two  supports  - 144 -  c)  Assume now t h a t tension  at  the  the  cable  is  tight  enough and c a l c u l a t e  the  upper s u p p o r t .  Solution: a)  from  4.1.3  Horizontal  tension <oL 8Dmcose  u) = 6  L  lbs/meter  Dm = .02 x 500  = 500 m  9 = Arctg  = 10  (200/500)  .2 6 x (500) 8 x 10 x c o s 2 1 . 8  = 21.8°  20194  lbs  Tens i o n s Lower T.  support =  tga  H cosaA  = 200 -  4-x 500  10 =  .32  10  .48  a . = 17.74' A 20194 cosl7.74  Upper  = 21203  lbs  support tga_  H cosa. B  = 200 + 4 x 500 25.64  20194 cos25.64  b)  22400 E x (o  200 x 6  22400  v  lbs  21203 = 1197  lbs  1200  lbs  NOTE: One o f t h e c o n c l u s i o n s o f t h e c a t e n a r y f o r m u l a t i o n f o r t h e c a b l e ( b a s e d on m o d e l 3 , w e i g h t d i s t r i b u t e d o v e r t h e l e n g t h o f the c a b l e i t s e l f ) i s t h a t t h e d i f f e r e n c e i n t e n s i o n b e t w e e n two p o i n t s of a f r e e h a n g i n g c a b l e i s e q u a l t o the w e i g h t p e r u n i t l e n g t h o f the c a b l e t i m e s the d i f f e r e n c e i n e l e v a t i o n between the two p o i n t s . Because in  this  any m o d e l .  characteristic is  s i m p l e and exact  it  c a n be  used  - 145 -  c)  from  4.1.A  8Dm(cosO)  B  T' = B  This  6 x  .  Given the  point  cable?  the  2  (cos21.8)  2  = 21749  assumption underestimates  P r o b l e m 2: of  (500)  8 x 10 x  s u p p o r t by 4.2  2  S i n c e the e q u a t i o n of  (22400 tension  21749) T„ at  lbs  the  = 650 B what  tension  lbs is  at  the  upper  o r 3%. the  deflection  of  each  " the  deflection  is  D = tox(L - x ) 2cos6H  for a  given  ' i  g e o m e t r y o f t h e s y s t e m and t y p e H only. The p r o b l e m i s t h e n t o 4.2.1  Horizontal  H = T cosa B  and t g a  Recall:  R  B  tension  i n the  cable  B  = E_ + uiL L 2cos6H  (from  3.3)  cosa. B V1  Then:  of cable the d e f l e c t i o n find H given T .  H = T  V  +  tg a 2  B  D  1 +  E + coL L 2cos6H  at  x depends  on  - 146 -  H yi +  T_  =  E + - uL L 2cos6H  H (1 + ]E_ + L  Squaring both s i d e s :  H  2  1  oil 2cos6H  'E' 2  +  +  •  u>L 2cosG  L  H  2  'l  E L  +  2i  +  H  J  This  quadratic equation aH  c a n be  + bH + c = 0  with  is  given  H  +  Eto cosf3  Ed) cos6H  oiL  2  -  T  B  = 0  2cos6^  a = 1 +  Eoi cosO oiL 2cos6  solution  + 2  written  b =  The  1  2  -<  2  H = -b V7"  by  +  4ac  2a 4.2.2  Deflection H is  t h e n known and t h e  deflection  at  point  x  can be  D = oix(L - x ) 2cos6H 4.2.3  To c a l c u l a t e H = T coso B  tga  i  deflection  at  midspan given T ^  B  = E + _ u L _ L 2cos9H  H c a n be and  the  d e r i v e d as .2 Dm = <oL 8cos0H  (from  in  3.3)  4.2.1 (from  3.2)  calculated  from  - 147 -  2.4  Tight  T  D  cable  c a n be assumed t o a c t  a l o n g AB  Tgcosctg = T c o s e  Then:  B  The d e f l e c t i o n D =  at  any. p o i n t x i s  uix(L -  recall  t£  = E L  given by  x)  2(cos6) T 2  2.5  B  Example Problem A f r e e h a n g i n g c a b l e h a s a 500 m e t e r s p a n w i t h a d i f f e r e n c e o f 200 meters between the s u p p o r t s . The w e i g h t p e r m e t e r o f t h e 1" c a b l e i s a b o u t 6 l b s and t h e b r e a k i n g s t r e n g t h 4 4 . 8 t o n s . Use a s a f e t y f a c t o r o f 4. a)  Assuming t h a t the t e n s i o n s act midspan d e f l e c t i o n i f the l i n e  b)  Assume now t h a t t h e c a b l e i s t i g h t enough a n d c a l c u l a t e m i d s p a n d e f l e c t i o n f o r t h e same c o n d i t i o n s a s b e f o r e .  Solution a)  from  4.2  Horizontal  tension  a l o n g t h e c a b l e what i s t h e i s tensioned to c a p a c i t y . the  - 148 -  H is  a solution 1 +  + H  E = 200 T  of:  L = 500  That  „ 2 B  8 = Arctg  (200/500)  = 21.8  lbs  :  yields  H fl 2  + f200l 500  1.16H  2  + 1292H -  5 x 10 2  from  (22400)  x 5 x  10  8  =  1.16  4.2.3 6 x (500) = 10 8 c o s 2 1 . 8 x 20212  Deflection  a t midspan =  10 500  meters  = 2%  4.2.4  Horizontal  = 0  = 0  lbs  Dm = a) x L 8cos9H or  8  + 4 x 1.16 2 x  H = 20212  6 x 500 2cos21.8  + H 200 x 6 + cos21.8  H = -1292 + Vl292  from  ~  to = 6  = 4 4 . 8 x 2000 = 22400 a  2  uL 2cos6  Eh) + cos8  tension  H = T^cosE  H = 22400 x c o s 2 1 . 8 = 20800 l b s =9.7 6 x (500)' 8 x c o s 2 1 . 8 x 20800  Dm =  oiL 8cos6H  or  Deflection  a t m i d s p a n = 9 . 7 = 1.9% 500  This  assumption  underestimates  the  meters  deflection.  - 1 2 9 2 + 48184 2.32  - 149 -  B.  Cable with a s i n g l e 1.  concentrated  Governing equations the  1.1  f o r the  load system.  The c a b l e w e i g h t  is  assumed t o a c t  subchords  Additional  notation  C = point of R = weight  attachment  of  the  of  S = sag of  the  l o a d on the  cable  load  x = h o r i z o n t a l p o s i t i o n of cable  6^ = a n g l e b e t w e e n  the  at  the  load  point  C (Distance  ' from  x  axis)  s u b c h o r d CA and t h e h o r i z o n t a l  9^^ = A r c t g [ ( S / x ) ] &2 = a n g l e b e t w e e n e  OL„  a  r 0  2  the  s u b c h o r d CB and t h e h o r i z o n t a l  = A r c t g [(S + E)/(L  -  x)]  = a n g l e between  cable  1 and t h e h o r i z o n t a l a t C  = a n g l e between  cable  2 and t h e h o r i z o n t a l a t C  on  - 150 -  The p o s i t i o n o f w i t h two o f t h e  Two e q u a t i o n s 1.2  Equation  -  deflection  -  load R  -  tension  static  at  one o f  the  supports  relate  these v a r i a b l e s :  e q u i l i b r i u m of C.  Sum o f h o r i z o n t a l  forces  The e q u i l i b r i u m o f  H (tga  the  x components o f  component o f  Sum o f v e r t i c a l R -  or sag at C  can be w r i t t e n t o  horizontal *  t y p e o f c a b l e (oi) a l o n g the system c o m p l e t e l y .  (1) :  Express the *  the- s u p p o r t s ( E , L ) and t h e following variables define  r 1  the  forces + tga  )  =0  cable  the  tension  forces is  shows t h a t  a constant H.  the  - 151  -  The sections of cable 1 and 2 are two f r e e hanging cables. Then t g a can be derived from the formula i n Part A, Section 3.3. r  E and L have to be s u b s t i t u t e d by t h e i r corresponding each s e c t i o n .  x t  g  a r  values f o r  2cos6 H  9 = S + E - oi(L'- x) L - x 2cos9 H 2  This y i e l d s R = H  (S  x  1 x  R =  Equation  (2):  2COS6JH + S + E L - x  + S + E - oi(L - x) L - x 2cosG H 2  2cos9,  HL S + HE L - x x(L - x)  oi (L - x) 2cos8„  oix 2cos6,  oi(L - x) 2cos8„  (equation  1)  - 152 -  Express the t e n s i o n at B i n terms of H.  Tg = H/cosa  B  T. = H/cosa  ='H V l + (tgctg)  2  = H \]l + (tga )  2  From Part A, Section 3.3, we can c a l c u l a t e t g a ^ f o r the f r e e hanging s e c t i o n • of cable 2, and t g a ^ f o r s e c t i o n 1. ^ x,  t  =  a  'S * 01  =  E + S •+ L - x .§ x  A  2  aix 2cos9 H  +  1  E + S L - x  Tg = H | / l +  T  - x) 2cos8 H  OJ(L  _S x  = H|/l +  io(L - x) 2cos9 H  (equation 2)  2  (equation 2 )  MX  2cos9 H  Squaring both s i d e s of equation 2 2 2 Tg - H 2  2  1  E + S  +  \2  L - x  0 = H'  (L - x) 2cos6 H  + f + ) + (E + S)m [ L - xj J cos9  1  E  s  E + S L - x  2  j(L - x) cos9 H 2  [ai(L - x)] _ T [ 2cos9 )  2  2  H  2  2  2  H can be e a s i l y obtained i f needed by s o l v i n g t h i s quadratic equation. Sag and d e f l e c t i o n Further development of equation 1 y i e l d s : S = Rx(L - x) - xE + HL i n which cos9^ and c o s f l  2  2  (L - x) + o)x(L - x) 2HLcos9, 2HL-COS0,  MX  are dependent on S.  d e r i v e d without f u r t h e r assumptions.  2  No simple expression f o r S can be  - 153 -  C a b l e weight  assumed  A s s e e n on t h e  figure  cose^ = c o s 6  The p r e v i o u s S  to  2  act  above  on t h e  this  c h o r d AB  s i m p l i f i c a t i o n assumes  = cos9  expression  for S  becomes:  = R x ( L - x) - x E + a ) x a - x ) + iox(L - x )  2  2  HL =  foC -  =  Kx(L HL  1  L  ~ *) HL x)  S = R x ( L - x) HL  that:  2HLcos9  2 - x E + cox L L -  3 aix  x E + mxL(L -  "3  o  + aixL + aix 2HLcos6  o  -  2mx  I  x)  2HLcos6  + ti)x(L -  x)  2Hcos0  -  xE L  Sag a t at  concentrated  a distance  x from  load R lower  support  - 154 -  This  e q u a t i o n c a n be b r o k e n i n t o 1) O J X ( L - x) 2cos0H  ii)  iii)  Rx(L HL  x)  three  = MP = d e f l e c t i o n  components:  due t o c a b l e w e i g h t  = PC = a d d i t i o n a l d e f l e c t i o n  due t o  - x E = NM  (Part A, Section  3.2)  load  -  L  s a g = 1TC = P 1 : + M P + ' N M The d e f l e c t i o n MC p r o d u c e d a t t o be Deflection  = Rx(L HL  At midspan x = L / 2 2 Dm = RL + mL 4H 8cos9H  x)  the  concentrated  + oix(L - x ) 2cos6H  = MC  l o a d can thus  easily  be  seen  - 155 -  3.  Problem 1 - Given load  3.1  the  R what a r e t h e  Horizontal Solving  ( o r d e f l e c t i o n D^) a t x = x ^ and t h e  sag  tensions  T  and  A  t e n s i o n i n the  at  cable  f o r H i n e q u a t i o n (1)  yields  LS,  R +  x (L -  1 x (L  +  x  E  l  x  co(L -  +  2cos6,  =  R  ux  +  x )  x  M  <  ~ l> 2cos6, L  x  x ) ±  +  x  1  +  cox.  1 ] =R - x )J E X  x (L -  +  2cos0,  + Ex /L)-|  and S  ^ l  x^  1  L S  the s u p p o r t s ?  2cos9,  x  +  co(L -  2cos6,  x  Deflection  X ; L  2cos9  ) r  x = x^  at  L H =  x  l ^  L  ~ LD,  x  p  R  +  u x  ]  ^  ~  L  x  i ^ + ^ ( L  2LD cos6 1  1  2LD  -  x  1  ) '  cos8  2  clamped  - 156 -  Recall  from 1.1  tg6  = S /x 1  then  1  1  V  1  cosO,  tg6  = (S  2  + E)/(L -  1  X;L  then  )  ^i'*!*  +  __1  1 +  cos9. H is  3.2  then  completely  T e n s i o n s at  the  defined.  supports t  From 1 . 3  (equation  2 and 2 )  T  and T  A  T  A  = H  1+  s *  T  B  = Hl/1 +  where H h a s  2003  1  E + S  3.3  If  R is  g i v e n and the  1 +  U  x^  the  by:  D  ^  1 +  L -  are given  sag  6^  (L - x) ±  2cos9 H 2  value  defined  Sm i s  in  3.1.  known a t  midspan, i . e .  x^ = L / 2 S = Sm  3.3.1  Horizontal  tension  (from  3.1)  Dm = Sm + E / 2 = D e f l e c t i o n a t m i d s p a n or  Sm = Dm -  H = LR + 4Dm  E/2  toL  +  2  16Dmcos9  and  coL  2  16Dmcos9  1 +  2Sm  1 +  2(Sm + E)  = LR + tuL  1 +  2  f  1  16Dm I c o s 9  4Dm  +  2Dm - E  /l  +  f2Dm +  1 cos9.  1  cos9.  c o s 6,  S  l  L -  Y)  1  +  E  x.  ]  Introducing and  = E = s l o p e of the c h o r d L p = Dm = p e r c e n t deflection/100 L s = tg6  1 +  H = _K_ + toL 4p 16p  3.3.2  Tensions  T  A  T  at  the  s)  2  +  \ / l +  |2Sm + oiL L 4cos6 H  = H HI +  ( 2 ( E + Sm) + L 1 cos  1 +  s)  2  and 2  oiL 4cos6„H 1  and i n t r o d u c i n g p and s  2 p  yields:  cos8^  A= l/ ( - Hl/l (2p - s) H  (2p +  supports  = H Wl +  Substituting  T  (2p -  S +  :  +  T  B  = Hl/1 +  where H has  '2  P +  B  +  g y  i  +  the v a l u e d e f i n e d  Assumption t h a t the t e n s i o n s chords of the system  at  (  2  p  in  +  s )  :  3.3.1.  the s u p p o r t s  A and B a c t  a l o n g the  sub-  - 158.-  3.4.1  Horizontal  tension  The e x p r e s s i o n •  =  X  l  ( L -  of  x )R 1  +  H i s u n c h a n g e d and i s as d e f i n e d 2 2 uxj (L - x ) o) (L - x ) x  LD^  3.4.2  Tensions  T  at  the  1  = H ^1 +  (S /x )'  Tg = H / c o s 8  2  = H \jl +  f(S  the  the v a l u e  sag S is  Introducing s H = _R_ + 4p  1  1 +  1  + E)/(L -  1  defined  known a t  and p  oiL 16p  2  supports  1  If  3.1.  t  2LD cos6  = H/cos6  A  X l  2LD^cos0^  where H h a s  3.4.3  +  in  in  x^'  3.4.1  m i d s p a n (x.^ = L / 2 and  (recall (2p -  s)  s = Sm)  3.3.1)  2  + \jl  + (2p +  s)  2  and  V 3.5  H \jl +  (2p -  s)'  H \ | l +  (2p +  s)'  Example A c a b l e system has 200 m e t e r s b e t w e e n i s about 6 l b s . A d e f l e c t i o n at that a)  Assuming t h a t tensions  b)  a 5 0 0 - m e t e r s p a n w i t h an e l e v a t i o n d i f f e r e n c e the s u p p o r t s . The w e i g h t p e r m e t e r o f t h e 1" 5 , 8 8 5 l b l o a d i s c l a m p e d a t m i d s p a n and the p o i n t i s 10%. the  at both  Assume now t h a t the  tensions  act  tangent  to  the  c a b l e , what  are  of cable  the  ends? the  upper s u p p o r t .  cable  is  tight  and c a l c u l a t e  the  tension  at  - 159 -  Solution a)  Tensions  act along  Horizontal  the  tension  cable  (3.3.1)  H = _R_ + jJL_ (\[ 1 + (2p 4p '16p ' R = 5,885  lbs  s)  \jl +  +  2  (2p  +'s)  L = 500 m  p = .1  2  s = E / L = 200/500  u = 6 lbs/m  Then  H = 14712 + 4099 = 18811  lbs  Tensions  (3.3.2)  a t the supports  Tg = H1 / 1 +  = 18811  2p + s + u L j / l + (2p + s ) ' 4H  . 2 + . 4 + _ 6 x 500 y  1 +  i  +  (  .  2  +  .  4 )  2  4 x 18811 Tg = 18811 V l . 4 2 = 22400  lbs  Similarly T . = 19048 l b s A b)  Tensions  a c t on t h e s u b c h o r d s  Horizontal  tension  Same as b e f o r e  Tg =  T  A  =  H\/I H^l  (3.4.3)  H = 18811 l b s  + (2p + s )  + (2p -  s)  2  =  2  =  18811  188.11  Vl736  VI704  = 21937 l b s  = 19183  lbs  160  -  Problem 2 at 4.1  the  Given  the  sag  (or x i s the  Horizontal tension Recall  H  equation  1 +  +  This  equation  2  The  lifting  capacity  x = x , and t h e t e n s i o n J. "~ • R o f the system?  T •' B  cable  ( E + S)toH .+ cos9„  x,  l/cos9„  the  at  JL  2i  L -  aH  in  D )  (2)  E +  with  deflection  S  u p p e r s u p p o r t what  -  oi (L -  x)  2cos8„  2 ~  „, 2 B  1 +  is  a quadratic of  the  form:  + bH + c = 0 solution  for H is  Wb  H =  2 -  4ac  2a  4.2  Lifting  capacity  Recall  equation HLS, -  x^(L  where H h a s 4.3  If  Tg i s  4.3.1  R f o r l o a d clamped at  x ) 1  the  value and t h e  Equation 2(E  oix^  HE L - x^  Horizontal  1 +  x^  (1)  +  given  point  (2)  u(L -  20086^  defined sag  at  Sm i s  Xj)  2cos9, 4.1.  R can then e a s i l y  known a t  midspan  be  = L/2  found. S = Sm,  tension is  reduced  + Sm)j'  H  +  to:  (E + Sm)oiH + cos9„  w i t h Dm = Sm + E / 2 = D e f l e c t i o n  at  oiL 4cos9,  midspan  (from cos9.  -  T  B  (Sm = Dm -  3.3.1)  = 0  E/2)  - 161 -  Introducing s qH  and p ,  the  + / q L a (2p + s ) 2  UJLJ I H + q fwLi  where q = 1 + (2p + s) H is  A.3.2  the p o s i t i v e  Lifting  (2)  is  reduced  L  4  by a n a l o g y w i t h  P  With 4.4.1  the  =0  2  the  quadratic.  R  " f  1 cos 9  (Vl+ (2pbe  assumption that  Horizontal  to:  + 1  1 ^ cos9 | 2  3.3.1  R can t h e n e a s i l y  4.4  1/cos  T„ B  coL toL 4cos9^ 4cos02  R = 2H (2Sm + E) - u L  4  =  s o l u t i o n of  R = 4HSm + 2HE -  =  2  -  becomes:  capacity R  Equation  R  quadratic equation  tension  the  s )  (2P +  2 +  s)  2 ,  found.  tension  acts  on t h e  s u b c h o r d CB  - 162 -  H = T cos0 B  with  2  cos6  =  2  1 1 + (S  4.4.2  Lifting R is  H L S  x^L  4.4.3  If  the  x  = L/2  x  from e q u a t i o n  1  +  x^  2  1  sag  Sm i s  and f r o m  -  known a t  2  " ^ " V  ^1  2cos6  1  2  midspan  S = Sm  H = T cos9 B  _EH  1  ( L - -x.^)' 2 c o s 6  - x )  Introducing s  Dm = Sm + E / 2  and p  = T  B  / V l +  (2p +  s)  :  4.3.2  R = 4pH - u L 4  4.5  + E/L -  capacity R  obtained  R -  L  Vl +  - s)  (2p  2  + Vl +  (2p  + s)  2  Example A c a b l e s y s t e m h a s a 5 0 0 - m e t e r s p a n w i t h an e l e v a t i o n d i f f e r e n c e o f 200 meters between the s u p p o r t s . T h e w e i g h t p e r m e t e r o f t h e 1" c a b l e i s a b o u t 6 l b s and t h e b r e a k i n g s t r e n g t h 4 4 . 8 t o n s . The d e f l e c t i o n a t m i d s p a n i m p o s e d b y t h e g r o u n d p r o f i l e i s 10%. What i s t h e l o a d t h e s y s t e m can l i f t a t m i d s p a n i n t h i s c o n d i t i o n ? Use a s a f e t y f a c t o r o f 4. a)  Assuming t h a t  the  tensions  at  the  supports  act  on t h e  cable  b)  Assuming that  the  tensions  at  the  supports  act  on t h e  subchord  Solution a)  Horizontal tension H is qH  Z  a solution +  (4.3)  of:  co (2p + s ) H + q (wL 2  [4] 2  where with oi = 6  q = l + ( 2 p + s ) L = 500,  , s = E / L = 200/500 = .4  T_ — 4 4 . 8  x 2000 = 22400  lbs  p =  .1  itself  163 -  -  Then and  q = 1 + (.2 the e q u a t i o n  + .4)  2  =  1.36  becomes:  1.36H  2  + 1050H + 765000 -  1.36H  2  + 1050H -  501760000 = 0  500995000 = 0  H = - 1 0 5 0 + V ( 1 0 5 0 ) + 4 . x 1.36 2 x 1.36 2  Lifting  capacity  R = 4pH - wL j ^ | l + (2p - s )  R = 4 x  . 1 x 18811 -  = 7524 The  'b)  lift  R is  +  (  __  2 )  2  s) ' 2  +  \j  ±+  {  f  >  )  2  5885 l b s .  on t h e s u b c h o r d  (4.4.3  of:  V l + (2p  +  s)  H = 22400/\/l + (-6) Lifting  + (2p +  tension  solution  H = T /  +Vl  1639 = 5885 l b s  capacity  Horizontal  2  6 x 500 4  Assuming the t e n s i o n  H is  x 50099500 = 18811 l b s  2  2  = 19208  lbs  capacity  R = 4 x . 1 x 19208 -  1639 = 6044  The l i f t i n g  is  capacity  lbs  overestimated  by t h i s  method.  - 164 -  To s o l v e t h i s p r o b l e m f o r a l l p o s i t i o n s o f t h e d e s c r i b e d by p o i n t C w h i l e the l o a d p r o g r e s s e s r e f e r r e d t o as " l o a d p a t h " a t c o n s t a n t t e n s i o n  l o a d i s to f i n d the curve from A to B . This curve i s T . B  U n f o r t u n a t e l y p r o b l e m 3 d o e s n o t h a v e any e a s y s o l u t i o n b e i s o l a t e d i n any o f t h e e q u a t i o n s . D e p e n d i n g on t h e a c c u r a c y d e s i r e d and o n t h e means from d i f f e r e n t methods. Two o f  5.1  t h o s e methods  Graphical  are presented  since  used,S  S cannot  can b e  obtained  here:  -  graphical solution  -  iterative  technique  solution  T h i s s o l u t i o n c a n be w o r k e d o u t w i t h a s i m p l e i s as f o l l o w s :  calculator.  The p r o c e d u r e  - 165 -  l)  make a g u e s s  2)  now assume explained plot  3)  S  1  at  that you are  if T ,  versus  t o 2) ,  3.  and R and d e t e r m i n e  I  TL  BI  as  T B1  T - is is  given  for problem 1 i n  i A)  S:  smaller  greater to  than y o u r given  T  choose a s m a l l e r  t h a n T„ c h o o s e a l a r g e r  determine  a new p o i n t  on the  S:  S and  if  and go b a c k  curve S versus  When y o u e s t i m a t e t h a t y o u h a v e d e f i n e d t h e c u r v e S v e r s u s accuracy f i n d the v a l u e of S f o r your g i v e n T . . B  T  with  T B enough  Tension at B Curve T e n s i o n at  B versus  Sag  G i v e n T,  (solution  for  the  Sag)  T h i s method c a n be v e r y l a b o r i o u s i f the. f i r s t g u e s s e s f o r S d i f f e r g r o s s l y f r o m the f i n a l s o l u t i o n . To a v o i d u n n e c e s s a r y c a l c u l a t i o n t h e u s e r o f t h i s method i s a d v i s e d t o use t h e a s s u m p t i o n 3 . 4 f o r t h e f i r s t g u e s s e s .  -  5.2  Iterative  166 -  technique  Can be w o r k e d w i t h The p r o c e d u r e i s  a pocket  as  c a l c u l a t o r but  make a g u e s s  2)  assume now t h a t  at  S:  explained  determine  a computer.  S =  you are  given  S  1  and T  I 3)  more a p p r o p r i a t e f o r  follows:  .1)  as  is  f o r problem' 2 i n  t h e new v a l u e  c  .  new b = p r e v i o u s  „  and d e t e r m i n e H and R  1  B  4  f o r S g i v e n by  . (R - R.)  S-^ +  1  x, 1  (L  -  x,) 1_  HL where R i s 4)  go t o is  your given  load  1 u n t i l R^ i s n o t s i g n i f i c a n t l y d i f f e r e n t then the v a l u e you are l o o k i n g f o r .  T h i s method c o n v e r g e s c h o i c e s o f S.  towards  the  right  answer  f r o m R.  The s a g  for S for reasonable  first  - 167 -  Standing s k y l i n e . 1.  Gravity  D e s c r i p t i o n of  1.1  the  Additional  system  system  notation  C = carriage  a  T  , a  C1'  T  C2  ,  to = w e i g h t p e r u n i t  length  cf  u>2 = w e i g h t p e r u n i t  length  of main  a_.  T  C3  = a n g l e s o f the the c a r r i a g e =  t  e  n  s  i  o  n  s  a  t  c  The s u b s c r i p t s application.of  respective  1"  t  for the  h  e  skyline  cables  respective  line with  t h e a n g l e s and t e n s i o n s f o r c e and the l i n e .  = angles of  subchords  1  = Arctg  ((S/x))  2  = Arctg  ((S + E ) / ( L -  Q 6  the  horizontal  at  lines  Recall 6^ and  the  x))  define  the  point  of  -  1.2  168 -  Variables In p r a c t i c a l a p p l i c a t i o n s 4 v a r i a b l e s d e f i n e the system i n a d d i t i o n t o the u s u a l g e o m e t r i c a l p a r a m e t e r s ( E , L ) and t h e w e i g h t s p e r u n i t l e n g t h . S T  _ -  2.  (or d e f l e c t i o n )  tension  upper support  -  load,  x  -  horizontal position  lifting  equations  c a n be w r i t t e n  Nine equations  for  the  that  Sum o f h o r i z o n t a l + H  2  3  -  U  load those v a r i a b l e s  1  c l  + H tga  C o n t i n u i t y of  the whole  l  and  system.  c  R  forces  = 0  ±  (1)  Sum o f v e r t i c a l H tga  2.2  system  relate  describe  --.--i-----------  *  the  line  carriage  H  H  the  load  system  are developed  E q u i l i b r i u m of  *  to  the  i n main  capacity of of  of  system.  Governing equations  2.1  at  R  Various the  ( o r D) s a g  2  the  forces c 2  + ^ t g a ^  tension  in  = R the  skyline  (2) through the  carriage  describe  - 169 -  The t e n s i o n through the T  T  T  H  2.3  Cl  = T  lV  1  =  +  H /cosa  c l  H /cosa  c 2  2  cl  the  = H ^ l  +  (tga  )  2  c l  = H \/l  +  (tga  )  2  c 2  =  +  (tga  2  (tga )  Angles at  ends  H \/l 2  of  tga  t  g  a  c l  A l  tga  +  of  cable  (4)  " 21^0036^  (5)  x  2  = S + E L - x  tga  ai(L - x ) 2H cos9 2  (6) 2  = S + E + ai(L - x) L - x 2H cos8 2  (7)  2  Section 3 tga--  tga  -  S + E L - x  M  3  = S + E -r 3 L -  x  (L  2H cos6 X  '  (L  X  2  (9)  )  2H cos9 3  (8)  )  3  M  p 3  passing  (3)  2  1  x  Section  )  o f t g a h a s b e e n d e r i v e d i n P a r t A , 3.3. by t h e i r c o r r e s p o n d i n g v a l u e s f o r each  = S aix x ' 2H^cos9^  =  c 2  each s e c t i o n  The g e n e r a l e x p r e s s i o n h a v e t o be s u b s t i t u t e d Section  i s o n l y d i s t u r b e d i n d i r e c t i o n when carriage. Then:  C2 1  Cl  C2  i n the s k y l i n e s h e a v e s of t h e  2  ..  E and L section.  -  Problem 1 tension  3.1  Given  T„„ at BZ  B in  Horizontal 3.1.1  the  sag  the  S (or  170  the  -  deflection  D) and t h e  l o a d R , what  is  the  skyline?  tensions  H, is "tga  obtained " are  (1)  H  defined  + H  2  from the  -  3  H  by  x  (4),  (6)  + H„ x  2H cos6 1  -  or  and  (2)  i n which  the  (8).  uix 2cos6  + E L - x  2  S + E L - x  + H,  co(L - x ) 2H cos6  2  W  3 " 2H cos9 (  L  X  3  )  2  as: -  co(L - x ) 2cos8  1  -  "3 " 2cos8 ( L  2  S + S + E x L - x  H,  and  fs  1  can be r e a r r a n g e d  x|  (1)  = 0  (2)  (2)  equations  2cos8  x  o,(L - x ) 2cos8  1  +  )  (H  2  S + E  + H ) 3  L  2  -  "3 " 2cos6, ( L  2  X  )  -  x  = R  finally  l =  H  R +  cox 2cos6,  is  known.  Therefore  3.1.2  H  - x) SL + x E  + "3 2cos6„ ( L  2  Equation  H  " lx(L X )  + co(L - x ) 2cos8«  n  Let  is  (3):  H ^ l +  known and t g a  us  Then H  call  2  is  .  thf> f i r s t the  (tga^)  = H ^ l +  2  can be  calculated  term of  equation  solution  of  1 +  11  S q u a r i n g b o t h s i d e s and r e p l a c i n g yields:  1 +  S + E L - x  io(L - x ) 2H cos8 2  2  (tga  )'  from equation (3)  (4).  " a " , which i s  (tg<* )  tga^  c 2  =  known.  a  c2  ^  y  i  t  s  e x  pression  equation  (6)  - 171 -  further  development  H  3.2  2  is  the  T e n s i o n i n the  H (S + E)oo +  S + E L - x  1 +  gives:  cose  solution  skyline:  of  this  - x)  J(L  2  2  a  2  = 0  2cos0,  2  q u a d r a t i c e q u a t i o n as  on p a g e  13.  T B2  skyline  T  B2  =  H  2^  equation  c o s a  B2  (7)  =  H  2  tRa , n ;  +  ^  t g c l  B2^  2  a  3.3  tension  i n the  d  f  r  o  m  = S + E + ui(L - x ) L - x 2H cos9 2  The  n  skyline  at  the  2  upper support B i s  then completely  defined.  Example Problem:  A g r a v i t y c a b l e s y s t e m h a s a 500 m e t e r s p a n w i t h a n e l e v a t i o n d i f f e r e n c e o f 200 m e t e r s b e t w e e n t h e s u p p o r t s . The w e i g h t p e r p e r m e t e r o f t h e 1" s k y l i n e i s 6 l b s and t h a t o f t h e 3 / 4 " m a i n l i n e 3.4 l b s . T h e 6 2 3 4 . 5 l b l o a d e d c a r r i a g e i s a t m i d s p a n and t h e d e f l e c t i o n a t t h a t p o i n t i s 10%. What i s  the  tension  in  the  skyline  at  the upper  Solution: .. 3 . 3 . 1  H  r R  +  oix  2cos8,  +  OJ(L  - x'  2cos8„  w.j(L - x) 2cos8„ 2  J  x(L - x) SL + xE  support?  - 172 -  R = 6234.5 l b s  oj = 6 lbs/tn  L = 500 m  x = 250 m  S = -50 m  tg6_ = -50 * 200 = - . 6 500 - 250  l/cos6  2  = 1.1662  tge, =  l/cos6  1  = 1.0198  ^  3.3.2  3  E = 200 m  -50 500 - 250  = 20924 l b s  H  2  1 +  a  o> = 3.4 lbs/m  2  =  u  S + E L - x  2  1 +  H (S 2  +  E)to  cos9  +  2  o>(L - x) 2cos9,  a  = 0  (tga )' c l  tga . = ^50 - 6 x 250 x 1.0198 = -.2365 250 2 x 20924 a = 21501 l b s ( t e n s i o n i n the s k y l i n e a t the c a r r i a g e ) H 3.3.3  2  = 18812 l b s  T. B2 T  B 2  = H /cosa 2  B 2  tga , = -50 + 200 + 6(250)(1.1662) 250 2 x 18812 T  B 2  = 18812/.8397 = 22401 l b s  = .6465  - 173 -  Problem 2 at  4.1  B what  is  Given the  lift  Horizontal  T  sag  S (or  capacity  the  R of  deflection  the  and the  tension  in  the  skyline  system?  2  B2  =  H  2  / c O S O l  equation  B2  =  2  H  (7)  t  V  1  (  +  8 B2 A  =  s  t  g  a  B 2  +  E  Solving  H  2  and  is is  for H  the  is  2  identical  solution  therefore  of  the  )  2  a  n  d  f  l  ">(L - x) 2H cos6  +  L -  4.1.2  D)  tensions  I  4.1.1  the  x  2  as what has  resulting  2  been  quadratic  d e r i v e d i n P a r t B,  equation,  known.  «1 Equation  H  2  is  (3)  is  us  1 +  known and t g  everything is Let  "xV  call  0 1  ^  c  a  (tga  n  ^  e  c  c l  a  l  i + (tga  )'  c  u  l  a  t  e  <  the  y  from e q u a t i o n  (6)  where  known.  "a" the  known q u a n t i t y  i n the  equation. H^ i s  i  c 2  solution  of  -xV1  +  (tga  c l  )  = a  second  term of  the  1.3  - 174 -  S q u a r i n g b o t h s i d e s and r e p l a c i n g e q u a t i o n (4) yields:  tg  0 1  ^ by i  t  s  expression  1 + x  further  2H^cos8^  development  1 +  _S  gives  2  2  4.1.3  H  the  solution  4.3  Lift  2  - =n0  this  quadratic  equation.  3  From e q u a t i o n  4.2  of  a  2cos8,  cos8.  is  -  (1)  H+H2  3  capacity R  H-  n  x  = 0  Equation  (2)  The  tga  are g i v e n by equations  (4),  the  lift  capacity  defined.  Example  R = ^ t g a ^  = H,  is  + H tga  completely  2  c 2  + H-jtga^  (6),  (8).  Therefore,  %  Problem:  A g r a v i t y c a b l e s y s t e m h a s a 500 m e t e r s p a n w i t h a n e l e v a t i o n d i f f e r e n c e o f 200 m e t e r s b e t w e e n t h e s u p p o r t s . The weight p e r m e t e r o f t h e 1" s k y l i n e i s 6 l b s and t h e b r e a k i n g s t r e n g t h 4 4 . 8 tons. The w e i g h t p e r meter o f the 3/4" m a i n l i n e i s 3.4 l b s . T h e d e f l e c t i o n a t m i d s p a n i s 10%. What i s t h e conditions?  l o a d the system can l i f t a t midspan i n Use a s a f e t y f a c t o r o f 4.  these  Solution: T, B2  22400  L  500 m -50 m  lbs  oo = 6 l b s / m x = 250 m l/cos8  2  = 1.1662  co = 3 . 4 3  lbs/m  200 m 1/cose^^ = 1 . 0 1 9 8  1 7 5  -  4.3.1  H„ Solving in  for H  I  2  s  ^  n  a  X  t h e e x a m p l e page 29  Then H  4.3.2  -  2  1  respects similar  to  the s o l u t i o n f o r H  (4.5).  = 18811 l b s  H  1 +  H So)  2  V X  cos8,  with  a  tga  = - 5 0 + 200 250  c 2  H \ / 1 + 2  +  2  =n0  2  -  a  2cos9,  (tgapj)'  6(250) x 2 x 18811  1.1662  .5535  a = 1 8 8 1 l \ / l + ( . 5 5 3 5 ) " = 21500 l b s Then ^  4.3.3  H H  4.3.4  = 20921 l b s  3  = 20921 -  3  Lift  capacity R  R = H  tga  C2  tga  l  t  ga  c  + H tga  l  2  = -50 250  C  t g a  18811 = 2110 l b s  =  "  5  5  3  + ^ t g a ^  6 x 250 x 1 . 0 1 9 8 2 x 20921  .2365  5  _ = - 5 0 + 200 250  R = 6234.5  c 2  lbs  3.4(250) 2 x  x 1.1662 2110  =  .3651  - 176 -  Problem 3 the sag This  Given  S (or  the  tension T^^ i n  the d e f l e c t i o n )  problem i s  similar  to  at  the  the  the  s k y l i n e at  S = f(S)  5.1  can be s o l v e d  of  one d e s c r i b e d on p a g e  (5.)  form S = f ( S )  where  f  is  an  f(S)  expression for  S = x(L -  S appears i n  o n p a g e 37 c a n b e r e a r r a n g e d t o  x) R +  x) R +  LH,  5.2  the  31  f o r S on a c o m p u t e r b y i t e r a t i o n .  S =  SL + x E = x ( L H.  or  is  function.  Derivation The  l o a d R , what  carriage?  S c a n o n l y be o b t a i n e d f r o m a n e x p r e s s i o n o f intricate  B and t h e  oix 2cos9,  cox  + io(L - x ) 2cos9,  + co(L - x )  2cos9^ the  second  +  +  term i n  3 2cos9„  3 2cos0„  2cos82  yield:  Ex L  the e x p r e s s i o n of H ^ , cos8^,  cos92.  Iteration No m a t t e r what i t e r a t i v e remains I d e n t i c a l :  technique  is  used the  1)  make a g u e s s a t  2)  compute t h e v a l u e s  of  The  d e r i v e d as  value of  3)  Find  4)  Check on Go  to  general procedure  S =  is  cos9^,  cos92 and d e s c r i b e d f o r Problem 2  (4.1).  t h e new v a l u e f o r S convergence  2 until  the sag  converges  inside  given  tolerances.  - 177 -  D.  Five  line  1.  D e s c r i p t i o n of  1.1  system the  Additional to  system  notation  weight per u n i t  length  of  haulback  u> w e i g h t p e r u n i t  length  of  mainline  3  u^i a  weight per u n i t a n g l e of  length  a cable with  of  the  slackpuller horizontal  T h e s u b s c r i p t s f o r t h e a n g l e s and t e n s i o n s of the f o r c e and the l i n e .  define  the  point  of  application  - 178 -  1.2  Variables I n p r a c t i c a l a p p l i c a t i o n s A v a r i a b l e s d e f i n e the system i n a d d i t i o n to t h e u s u a l g e o m e t r i c a l p a r a m e t e r s ( E , L ) and t h e w e i g h t s p e r u n i t l e n g t h . -  S ( o r D)  -  x horizontal position  -  R load,  -  T .  Governing 2.1  at  of  capacity  of  the of  load)  load the  upper support  equations  the  system  i n main  can be w r i t t e n  to  line  relate  those v a r i a b l e s  and  describe  system.  equations  Tensions  that  for  in lines  The t e n s i o n at  (or d e f l e c t i o n  lifting  tension  Various the  sag  in  point.  the  1 and  1'  the haulback r u n n i n g through a b l o c k at A remains T h e r e f o r e the  b e t w e e n A and C a r e In p a r t i c u l a r the and a l s o H  system  =H',.  sections  of  the  free  hanging  lines  unchanged 1 and  identical.  tensions at  the  carriage T  c  l  and T  c  l  , are  the  same  1'  - 179 -  2.2  Equilibrium  Sum o f  Sum o f  2.3  the  + H  2  l  H  t  g  a  C l  H\Jl  H  the  2.2  2  2H  (1)  = 0  X  forces C2  t g a  +  H  3  t g a  C3  +  H  '3  t g a  C3  t e n s i o n i n the haulback  page  =  (2)  R  through  the  carriage  35)  = T C2  H /cosa , 1  to  +  forces  -  3  the v e r t i c a l  (similar  Cl  + H '  3  Continuity of  T  carriage  the h o r i z o n t a l H  2  of  (  1  =  + (tga  H /cosa 2  c l  )  2  c 2  = H  2  \ / l+  (tga  c 2  )'  (3)  -  2.4  Angles at  the ends of  180  -  each s e c t i o n  of  cable  The g e n e r a l e x p r e s s i o n of t g a has b e e n d e r i v e d i n p a r t A ( 3 . 3 ) . E and L and to have t o be s u b s t i t u t e d b y t h e i r c o r r e s p o n d i n g v a l u e s for  each  Section tga  c l  section. 1 (and  iox.  = S  g  a  A l  =  Section tga  tga  =  x  +  — 2H cose ] |  tga  „ = S + E L - x  2  „ = S + E + to(L - x ) L - x 2H cos6  2  section  2  (5)  (5')  (6)  (7)  3 for  2 and to^ f o r to i n  the  expressions  of  tga  written  2.  ^  3'  S u b s t i t u t e 3 ' f o r 2 and to,, f o r to i n t h e o f t g a w r i t t e n f o r s e c t i o n 2.  Mathematical  (4')  3  Substitute  Section  Al'  (4)  i  co(L - x ) 2H cos6  Section  Cl'  2  2  for  tga  2^0080^^  X  t  1')  s i m i l i t u d e between the  standing  expressions (8')  s k y l i n e and the  The g o v e r n i n g e q u a t i o n s d e r i v e d f o r the s t a n d i n g i d e n t i c a l to t h a t o f t h e f i v e l i n e s y s t e m .  five  line  (9')  system  s k y l i n e a r e i n many ways  F o r most o f the c a s e s the s o l v i n g p r o c e d u r e s d e v e l o p e d f o r t h e t h r e e p r o b l e m s s t u d i e d f o r the s t a n d i n g s k y l i n e w i l l be a p p l i c a b l e f o r the f i v e l i n e system w i t h v e r y few c h a n g e s . When t h e i n d i v i d u a l t e n s i o n s i n l i n e s 3 and 3 ' a r e n o t n e e d e d we c a n d e f i n e to^* = to^ + to^,, c o m b i n e d w e i g h t p e r u n i t l e n g t h o f t h e m a i n l i n e a n d s l a c k puller,  and co* = 2to.  - 1 8 1  With  t h e s e new n o t a t i o n s  1  the e x p r e s s i o n  +  + co(L - x ) 2cosG  2  -  of  c a n be d e r i v e d f r o m  io *(L -  x)  3  2cos6  2  w h i c h compares w i t h the e x p r e s s i o n o f  2  on page  (1)  and  (2)  x ( L - x) SL + x  37  ( 3 . 1 . 1 )  and „  .  •  ,  v  co*x + oi(L - x) S = x ( L - x) R + 2H L 2cos9^ 2cos6 T  1  w h i c h compares w i t h This  similitude is  a l m o s t any k i n d o f  ,  +  OJ_*(L  2  the e x p r e s s i o n of used  to  S on page  develop v e r s a t i l e  cable yarding  system.  -  3 2cosG,  4 3  x)  xE L  ( 5 . 1 ) .  computer programs t h a t  can handle  - 182 -  Figure 1+7 -  Parabola i n the coordinate system. (x,y)  -  C A B L E  L E N G T H  P R O M  T H E  18 3  -  APPENDIX  2  P A R A B O L I C  The p a r a b o l i c e q u a t i o n i n a t e system (x,y)  Y =  T H E O R Y  o f t h e c a b l e i n t h e coord:  i s :  to 2cos9H  2  . E k  L  _  o)L  ,  2cos0H  ;  (b')  The l e n g t h o f a s m a l l e l e m e n t o f c a b l e i s e v a l u a t e d as:  ds  2 2 dx- +dy  1  +  dx;  y  n—TI  . B  - 184  Therefore  the t o t a l l e n g t h of the cable i s :  S =  The  gjx l+( cosGH  Ids =  E L  2cos9H  )  dx =  square r o o t i n the i n t e g r a l can be expressed  P(x)  =  where  The  -  and  ,E _ cuL , L 2cos0H  A  =  B  =  2u ,E cosGH L* T  C  =  (  2Cx + B 4C  dx  T h i s exact formula cations.  as:  1/(1 + A) + Bx + Cx l  2  ;  V  "  0)L  2cos0H  )  ^COS0rT  s o l u t i o n to t h i s i n t e g r a l  • (  /P(x)dx  i s g i v e n by Selby(7)  4AC-B'  P(x) o  (i  log (  8C  as dx  '  J o  • x^/T  l/p(x) '  +  -J=.)J  i s cumbersome to use f o r p r a c t i c a l  appli-  An approximate e x p r e s s i o n of the l e n g t h of a t i g h t  - 185 -  cable i n the case of l e v e l Inglis(5)  s u p p o r t s (E = o) i s g i v e n by  and t h e W i r e Rope H a n d b o o k ( 1 2 ) a s :  s - L  ( l + §-'<22> ) 2  I f t h e two s u p p o r t s a r e n o t l e v e l t h e f o l l o w i n g e x p r e s s i o n c a n be  used: 2 b  cose  U  +  3  '  - 186  -  APPENDIX 3  T E N S I O M E T E R  A3.1 I n t r o d u c t i o n . A tensiometer i s d e f i n e d here as an capable of measuring  instrument  the t e n s i o n i n f i x e d or running  lines.  The need f o r a tensiometer a r i s e s when a load c e l l or any other s o r t of t e n s i o n measuring  d e v i c e cannot be p l a c e d a t  the dead end of the rope i n which the t e n s i o n i s t o be measured.  Two  types of tensiometers have been developed  by  d i f f e r e n t manufacturers  mainly to meet the needs f o r crane  i n d i c a t i n g and warning  systems.  The f i r s t  type manufactured  by Rucker Company  1  operates on the f i x e d r e l a t i o n s h i p between the t e n s i o n and the n a t u r a l frequency of a wirerope.  In t h i s system,  an  e x c i t o r causes the c a b l e to v i b r a t e .  The t e n s i o n i n the  line  of a g i v e n weight per u n i t l e n g t h i s d e r i v e d from the reading of the frequency w i t h the sensor.  2 D i l l o n and Company  produces  a tensiometer of the  second  "'"Rucker C o n t r o l Systems, 47 00 San Pablo Avenue, Oakland, C a l i f o r n i a 94608. 2 D i l l o n and Company, Inc., 14620 Keswick S t r e e t , Van Nuys, C a l i f o r n i a 91407.  - 187 -  Figure A-8 -  Basic P r i n c i p l e of the  tensiometer.  - 188 -  type.  T h i s type works on a simpler mechanic p r i n c i p l e .  c a b l e i s g i v e n a d e f l e c t i o n w i t h t h r e e sheaves. i n the l i n e i s deduced middle sheave.  Two  The  The  tension  from the f o r c e thus c r e a t e d on the  tensiometers of the second type were  built.  A3.2  D e s c r i p t i o n o f the Tensiometers. A3.2.1  Principle.  The b a s i c mechanical p r i n c i p l e of the tensiometer i s i l l u s t r a t e d i n F i g u r e 48. three sheaves  1, 2 and 3.  a l e v e r p i v o t i n g about 0.  The c a b l e winds through the  The c e n t r e sheave  2 i s mounted on  The a c t i o n of the c a b l e on the  middle sheave i s thus t r a n s m i t t e d t o the load c e l l to an i n d i c a t o r which measures the t e n s i o n .  connected  A study of the  - 189  f o r c e balance on the cable  -  l e v e r shows t h a t the t e n s i o n T i n the  i s r e l a t e d to the f o r c e F a p p l i e d to the  T  =  load c e l l  by:  F /(2 x sinG)  where 0 i s the angle of the t a l between the  l i n e w i t h the  horizon-  sheaves.  I f the geometry of the machine remains the same when the load i s a p p l i e d , sin0 i s a constant and  the  previous  r e l a t i o n s h i p becomes:  T  =  k x F  where k i s the constant of the machine.  A3 .2.2  Design.  A scaled reproduction  of the blue p r i n t of the  siometer designed f o r t h i s study i s shown i n F i g u r e p a r a l l e l channel bars c o n s t i t u t e the g e n e r a l machine.  The  outer sheaves and  l e v e r i s composed of two  b o l t e d on the  by f r i c t i o n i n the t r a n s m i s s i o n r o l l e r bearing l e v e r and  the  frame.  The and  s h a f t of  taken so as to avoid of the f o r c e s .  the losses  A special  i s a l s o used f o r the connection between load c e l l .  The  the  rotate  s t e e l p l a t e s welded t o g e t h e r  S p e c i a l care was  Two  s t r u c t u r e of  machined to support the b a l l bearings f o r the middle sheave.  49.  the t r i a n g u l a r l e v e r  on s h a f t s guided by p i l l o w b l o c k s  ten-  frame i s over designed to  the  - 190 -  Figure 49  - Copy of the blue-print of the tensiometer.  -  minimize the at  the flexion  deflection maximum  i n  192  and very  the  load  -  l i t t l e  c e l l  error  which  is  does  not  requirements  two tensiometers shown  Table  X.  i n  Table  were  manufactured  L i n e diameter  Requirements  for  (inches)  Maximum l i n e t e n s i o n  A3.2.3 The  (newtons)  Load  heart  of  electronic  the  transducers  changes  in  of  the  load  c e l l  fed into  is  calibrated  c e l l .  Very  gauges  bonded  force  d i r e c t l y  i n  changes  the  a  i n  strong steel  Tensiometer 2 7/16  50000  5000  an  terms a  is  the  that  voltage.  schematically, to  the  tensiometers.  5/8  the machine  into  is  for  Cells.  forces  deflection  . 3 mm  X.  Tensiometer 1  are  exceed  by  load.  The  cells  introduced  The  load  the c e l l  the resistance  of  due t o the  or  load is  a  output  applied  to of  element.  the application  strain-gauges  i n signal  recorder  composed  steel  Load  changes  electronic  but e l a s t i c  element  c e l l .  translate  indicator of  load  which the  strainThe of  a  connected  -  to  form  a balanced  The in  wheatsbone  major  the tensiometers  Table  Load c e l l  193  -  bridge.  characteristics a r e shown  XI.  Load  i n  c e l l s  Table  the load-cells  used  XI.  characteristics.  characteristics  Brand name  of  Tensiometer 1  Tensiometer 2  BLH  BLH  U3G1  U3G1  Capacity ( l b s )  10,000  11,000  Safe working l o a d ( l b s )  15,000  1,500  Weight ( l b s )  10  6  Recommended e x c i t a t i o n ( v o l t s AC o r DC)  12  12  1  Designation  Output/input millivolt/volt  3 ( a t maximum c a p a c i t y )  3 ( a t maximum c a p a c i t y )  2 P r e c i s i o n f o r the t e n s i o n %  BLH  This  -  Electronics, Massachusett figure atory  .3  I n c . , 42 F o u r t h 02154.  was d e t e r m i n e d conditions.  Avenue,  experimentally  .3  Waltham,  i n  ideal  labor-  -  A3.3  Tensiometer The  relate  the  indicator  relationship  and  the  reading  F  =  and force  F  on  a'R in  the  b  to  T  =  k  T  =  ka'R  +  T  =  aR  b  to  tensiometer the  force  F  tension applied  indicator  is  is in  to  necessary the  the  to  line. load  c e l l  linear:  1  theory  x  a  the  +  transmitted  of  reading  between R  -  Calibration.  calibration  The  194  the the  tension load  T  c e l l  in  the  are  line  and  related  the  by:  F  Therefore:  or  the and  the  graph  T  versus  tensiometer  sketched hooked and  at  gave  gradually at  the  on  the  and  between  the  tension  R  in  shown  set-up  end  ka'  R  Figure  one  =  reading  1 before  The  with  relationship  indicator  of  +  kb'  "true"  for  The  the line  other  end  of  the  indicator  and  had  was  to  line. consists  T. be  The of  kb T  in  the  example,  determined to  be  the  tension  the for  the  is  c e l l  1  calibrated was  calibrated, with calibration  of  setting  gauge  the  line  linear.  load  laboratory  The  1  calibration  and  been  =  an  proved  indicator  tension  tensiometer  51  b  As  tensiometer  line  the  and  linear.  Figure  the  to  siometer  is  calibration  50. of  a  applied a  the  winch tenfactor  - 195 -  Figure  50 -  Sketch o f the equipment set-up f o r the c a l i b r a t i o n o f the t e n s i o m e t e r .  F i g u r e 51 -  Graph o f t e n s i o n read by the tensiometer versus t e n s i o n i n the l i n e before bration.  cali-  - 196 -  -  and  the  equal the  zero  to  the  value  scale  so  on  the  indicator  tension  of  the  that  T  =  T  aR  +  cuted  i f  the  A  proper  setting  tained  when  dition  is  t r i a l s  slope  the  are  other  R  words, of  be find  the  tension  relation:  =  R  (i.e.  a  =  can  be  quickly  explained  the  gauge  factor  the  slope  and  line  gauge of  i f  x  T  versus  factor  the  line  =  for  s u f f i c i e n t l y before  y  far this  and  hereafter  of  the  1  b  =  0).  Procedure.  Factor.  required  reading  origin  Gauge  satisfied taken  the  the  of  of  and  the  In  Set  change  the  readings  T  procedure  varies  that  line.  factor  calibration  A3.4.1  so  b  Calibration The  -  2  the  previous  becomes:  A3.4  in  gauge  the  197  on T  on R  accurately  is  followed:  the  indicator  (Figure  the  R  and  the  apart. value  2  In of  52).  indicator  versus R  exe-  is  is  1.  values  gauge  ob-  This  practice,  the  The  of  contwo  several  factor  is  found.  The in  Table  XII.  different  t r i a l s  for  tensiometer  1  are  reported  - 198 -  Figure 52 -  Influence of the gage factor of the i n d i cator on the graph of tension read by the tensiometer versus tension i n the l i n e .  Figure 53 —  Influence of the zero knob adjustment on the graph of tension read by the tensiometer versus tension i n the l i n e .  - 199 -  -  Table X I I .  T r i a l number Gauge f a c t o r  =  2.02  2 =  2.07  3 2.08  20620  T  R  ±  5200  T  15420  y  GF  =  2.09  2  20040  T  R  ±  5250  T  14790  y  5350  20180  14670  T  R  ±  5350  T  14620  y  2  ±  20120 5560 14560  R  2  20250  T  R  ±  5080  T  15170  y  R + b  20100  5510  ±  19970  R of equation  =  2  2  The r e s u l t of t h i s f i r s t  T  ±  R  x  2  14750  R  x  4  Tension T (newtons)  2  x  GF ••==  R e s u l t s o f the gauge f a c t o r adjustment f o r the s k y l i n e 'i tensiometer.  R  x  GF  -  Tensiometer r e a d i n g R (newtons)  1 GF  200  ±  2  20480 5300 15180  s e t t i n g i s a l i n e T versus  - 201 -  Figure 54 -  a) Graph of tension read by the skyline tensiometer versus tension i n the l i n e after f i n a l calibration. b) Graph of the discrepancies between tension read by the skyline tensiometer and the tension i n the l i n e , f o r increasing tensions and f o r decreasing tensions.  -  202  -  R tensiometer  reading,  NX  1000  - 203 -  A3.4.2  S e t t i n g the O r i g i n of the Tension to  Scale  Zero.  The a d j u s t m e n t o f t h e o r i g i n k n o b o n t h e i n d i c a t o r translates the l i n e T versus R  ( F i g u r e 5 3).  The  proper  s e t t i n g o f t h e o r i g i n knob i s o b t a i n e d when T^ = R 2 / w i t h .. T^ a t e n s i o n c h o s e n i n t h e m i d d l e For tensiometer in  of the tensiometer  range.  1, T^ was r e a d t o be 2006 on t h e l o a d c e l l  1  t h e l i n e a n d t h e o r i g i n k n o b was s e t t o 5.038 o n t h e t e n -  siometer  i n d i c a t o r t o r e a d t h e same v a l u e f o r R^.  A3.4.3  Checking  the C a l i b r a t i o n .  To v e r i f y t h e c a l i b r a t i o n t h e c u r v e T v e r s u s R was determined  f o r t e n s i o n s f r o m z e r o t o t h e maximum c a p a c i t y o f  t h e i n s t r u m e n t and back t o z e r o . for  tensiometer  The r e s u l t s o f t h i s  test  1 i s shown i n F i g u r e 54.  A3.5 C o n c l u s i o n . The f i n a l t e s t shows h y s t e r e s i s f o r t h e c u r v e v e r s u s R. the l i n e  The t e n s i o m e t e r o v e r - e s t i m a t e s t h e t e n s i o n when i s slackened.  To a v o i d t h i s e r r o r t h e t e n s i o n mea-  s u r e m e n t s w e r e made o n t h e a s c e n d i n g e s i s curve. then  T  lifted  branch  of the hyster-  T h i s n e c e s s i t a t e d t h a t t h e l i n e be d r o p p e d and e v e r y t i m e t h e t e n s i o n had t o be a d j u s t e d t o a  v a l u e s m a l l e r t h a n t h e p r e v i o u s one.  - 204  The  -  e x p e c t e d e r r o r c a n be d e d u c e d f r o m t h e  final  t e s t t o be l e s s t h a n 1% f o r t e n s i o n s g r e a t e r t h a n 5000N.  -  205  -  APPENDIX 4 Note book sample page.  Record of the data mea-  sured w i t h the t h e o d o l i t e f o r the f r e e hanging t e s t .  1 H o r i z . angle Verti. degree minute degree  Station 0  ,  angle minute  2.57  45 .0  2.S7  _J« •^  2  41 38  _.1.8_.3._ 14.2  4-  34  53.0  6  31  14.0  8  27  _16.8  . 37-i  10  . ; 2>  „°1'3  47-»  12  18  28.7  __i..3__. _ 4 0 . 9 . _  14 16  ••  zsy  ^•7  0 -1  18  3  32.6  2?3  20  358  20.9  2£0  eS"'S  Ul  IV - 0  _3_>JL_. _10_.8_ 348 343  07.3 14.6  28  338  ..1C.3  30  ...334 ..  32  330  34 36 38 40 42 44  322  10.3  ..3_26__ ....24.7 ,  319 316  2  66  IS-3  263  0S--I  52.9  2-53  s-*,-«*  46  309  36.2  48 50  307 '305  34.9 43.8  no 27i ^7^  . W e i l lher  i-l  Tempie.ra... ure. time sltirti:  -  400 41 .1 0 4 .6  U IV  '.c'  ip  I !  |l I  M  '  i 1  Lite. Start  ncJrikiJleVel:  yd W i>kL  Endj-  .U?>d_!_!_LLl Ja/C  I ! ! i I II!II!I•I 1 _ M i d | a t a t i d n ! horiz. angle i 4 2 19.1 . v e r t i la'ngle :sta_}.t:.L_L. . ZQl: 2 7 . }  I ! ! i. i ! i  rerti.  I ! ; ' ..2.6$  j  angle end 1 Heigh't of. i n s  j :  U.t'. :  : ,  U i l l j i i 111 i j 1 i i  Comments,  L THeod  Statlijoh 52  -3  1  Theodolite! check  plummet;  /  1  02-  Coll.  IS-5  46.9  _3_L4._. ._14„._0_ 48.8 311  fl.w..[2!7l]JM  J : I!  _ie - o l_..<t  .57.1.  Date.  _L-zJrb. s e t t  I i  '6 -1  14.3.  FH  ife'rjtlca'll l e v e l  V  2. f>L  26i_  Experiment if  IjtjUr \H*<\  LS2__  40.8  22  -  ._2.se  8  24 26  Comments  53.  u,  olite  man  '  304  d^ejgrfej [minute  303  Vert degre:  50.4  01.8  Z4-3  angleL plnute  27  4  "02.9  Coaaeiit!  jiiT  -  206  -  N o t e book s a m p l e p a g e .  Record o f t h edata  at t h e lower support, f o r t h e free hanging  Time  zero  Indie. reading  - J X P '.:" i me  Tensior i Angle degree  •  IV. 14. 10 1? -35  11  lit,JO  -Si  II 0 l  -81  1031  10 IS"  IS, 00 _IDJ5:„ -S^  01  test.  \l >j  if  11  + 6.5" ' -1a t *. i  1"  + 6.3,  Jae a  _L J  .116 r  n  r a t i 11  11 i n  a t i i '_t  -{e  fi i Ini  -C a  t 3  L!  (3 1  ,h  .tn..lnJer  r e f er e n c e :  - L in 16 in t  p a i n t ni a r k ...OS  oidz&b  S  JJumbei ..of. ..pai n t  marl s_beiwe en t h e  Z7  j |  ok : i  -  !  w u. wo  A  sP u •_. mrt  1  \  1  0X  i -  -  i i -  i  !  ovxcj  /  ,..  ..two„su pp.or.ts.:  --  1  r h 111  - A b i >i'- c 1 !C k losest  1. s  1,  J  ATI IT f- Vl  i  R M  pc  b a t .6 r v  I  taken  J  P  1 i i 1 1 i  |  j i i |  i i  |  i  i ;  i i •!  ! !  i !I  i ri i i !I !i 1! 1 i i !i J  \  ! 1  j• ij j|  11 1 >  -  207  -  N o t e book s a m p l e p a g e .  Record o f t h e data  taken  at t h e upper support, f o r t h e f r e e hanging t e s t .  i  ;  Time - i n d i e , Xensioi  Comments  rsadinf  I /, -15 J.13.CL  -IU.SL. CUj H i 3 .ttockcahr. J4 1 5-,oo JJJ.5" 1 S\ 15 l.l II  1  )  •  ExperiitaeiVtl  L DktA "Wei.  \r>3\ m i l !  Temperature Ifjinis'h,  OMidafcoiJchecM  ' ' 'J I I I I I I ' ' '  Gil  ! i  !  i i i_i I  !  I  I ! I  battery.  ! ! !  AbneyLbiieckj: length -from-c l o s e s t p a i n t r to_upp e r ref« rencej.  Cpmme.ntS.  '/4 Winih  1  i! i ^2.1 '<?r6MC/bta I Ii iM il-f!  -  Table X I I I .  Test #  Mid-span deflection  %  208  -  Summary o f t h e e x p e r i m e n t a l o f t;he f r e e h a n g i n g t e s t .  Tension upper support (N)  results  Tension lower support (N)  Angle upper support (deg)  Angle lower support (deg)  length (m)  3.7  5200  4758  17.8  1.5  134.2  2  1.7  11000  10800  13.2  6.5  133.5  3  7.0  2727  2501  25.0  -5.8  135.5  4  6.3  3080  2815  23.0  -4.0  135.1  5  4.9  3885  3541  20.0  -1.0  134.6  6  2.9  6327  6121  16.1  3.1  134.0  7  2.4  7789  7514  15.0  5.5  133.9  8  2.1  8878  8613  14.2  5.1  133.8  9  1.9  9633  9349  14.0  5.7  133.7  - 209 -  Table XIV.  Test  Summary o f e x p e r i m e n t a l r e s u l t s . Clamped l o a d on a s i n g l e l i n e .  Load position (m) Y (m)  X  Tension upper support (N)  Load (Kg)  Tension lower support (N)  1  126.57  19.26  3747  100  3099  2  127.13  19.58  5886  200  4738  3  127.34  19.76  8103  300  6651  4  127.85  21.51  19168  300  18089  5  127.98  21.14  19914  495  18050  6  118.45  16.58  18442  495  16706  7  118.42  17.23  21788  495  20081  8  118.58  17.83  25309  495  23671  9  118.55  18.19  28939  495  27399  10  93.24  5.77  7710  200  6965  11  93.05  10.83  14224  200  13577  12  93.15  12.67  21169  200  20512  13  93.06  13.52  27546  200  26957  14  93.84  5.14  15244  495  14185  15  93.48  8.70  21562  495  20581  16  93.51  10.78  29037  495  28184  17  67.92  1.77  19620  495  18472  18  67.96  2.77  21719  495  20512  19  67.96  4.25  25662  495  24515  20  68.00  5.11  28704  495  27624  21  67.81  1.79  12841  300  11997  22  67.78  3.32  14950  300  14185  23  67.84  5.98  21395  300  20620  24  67.83  7.41  27978  300  27203  25  50.03  0.84  23073  500  22376  26  50.07  2.38  28537  500  27781  27  49.90  0.90  15195  300  14548  28  49.92  3.13  21042  300  20394  29  49.92  4.52  27909  300  27124  - 210 -  T a b l e XV.  Test #  Carriage location, , Y (m) "•X (m)  Summary o f e x p e r i m e n t a l of t h e g r a v i t y system.  results  T e n s i o n skyl i n e upper support (N)  Load  T e n s i o n skyl i n e lower support (N)  Tension i n the mainline N  1  11.94  -1.42  11870:  300  11409  *  2  18.47  -1.24  13714  300  *  3  29.87  -0.39  15745  300  13272 *  4  44.64  1.27  17432  300  ft  5  56.50  2.98  18403  300  17118 *  6  65.07  4.38  18540  300  17913  *  7  72.22  5.66  18383  300  ft  313  8  82.92  7.75  17775  300  372  9  96.00  10.64  16510  300  17216 *  10  104.78  12.81  14930  300  529  11  115.32  15.81  12517  300  14616 *  12  125.03  19.26  9545  300  814  13  128.85  21.11  8112  300  9388 *  14  11.28  -1.49  17677  495  17147  ft  15  12.93  -1.51  18266  495  18001  ft  16  35.68  oo:o8  25162  495  24701  *  17  49.94  1.65  26143  495  *  ft  18  65.51  3.84  25721  495  627  19  84.97  7.42  24054  495  25378 *  20  106.26  12.49  19982  495  1030  21  123.41  18.08  13380  495  18560 *  1486  22  129.13  20.73  8544  495  *  2020  Not measured.  * *  441 637 922  794