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EXPERIMENTAL STUDY OF LOGGING CABLE SYSTEMS b y DANIEL YVES GUIMIER D i p l o m e d ' I n g § n i e u r e n M e c a n i q u e E N S A M ( 1 9 7 5 ) A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E i n T H E F A C U L T Y O F F O R E S T R Y W e a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A M a y , 1 9 7 7 © D a n i e l Y v e s G u i m i e r In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the re-quirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t fr e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Depart-ment or by his representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Forestry  The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT Two t h e o r e t i c a l f o r m u l a t i o n s o f l o g g i n g c a b l e s y s -tem problems, t h e c a t e n a r y model and t h e p a r a b o l i c model 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 e x e c u t e d on a g r a v i t y system f i e l d model. The s t u d y 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 t h a n a p a r a -b o l a , b o t h t h e o r e t i c a l models a r e a c c u r a t e 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 few dynamic 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 the dynamic f o r c e s i n a l o g g i n g c a b l e system 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 f i e l d . i i i TABLE OF CONTENTS Page Abstract Acknowledgements CHAPTER 1: INTRODUCTION 1 CHAPTER 2: INTRODUCTION TO CABLE MECHANICS 8 2. .1 General Description of the, System 8 2. 2 Modelling Assumptions 11 2. 3 Catenary Model 14 2. 4 Parabolic Model 16 CHAPTER 3: DESCRIPTION OF THE FIELD MODEL 19 3. 1 Site Dimensions and Characteristics 19 3. 2 Cables 20 3. 3 Carriage 23 3. 4 Winches 23 3. 5 Dynamometers 29 3. 6 Surveying of Cable Shape and Carriage Position 32 3. 7 Other Measurements 36 3. 8 Accuracies of Instruments and Expected Errors i n the Measurements 39 3. 9 Dimensional Similitude between the Model and A Real Yarding System 43 CHAPTER 4: FREE HANGING CABLE 46 4. 1 Description of the Experiment 46 4. 2 Analysis of the Results 47 4. 3 Error Analysis 53 4. 4 Results and Conclusions 57 CHAPTER 5: CLAMPED LOAD ON A SINGLE LINE 80 5. 1 Description of the Experiment 80 5. 2 Analysis of the Results 81 5. 3 Results and Conclusions 87 CHAPTER 6: GRAVITY SYSTEM 95 6. 1 Description of the Experiment 95 6. 2 Analysis of the Results 96 6. 3 Results and Conclusions 98 i v Page CHAPTER 7: DYNAMIC TESTS 111 7.1 Equipment 111 7.2 T e s t s 111 7.3 C o n c l u s i o n 124 CHAPTER 8: DISCUSSION AND CONCLUSION 125 L i t e r a t u r e C i t e d 129 Appendix 1 131 Appendix 2 18 3 Appendix 3 186 Appendix 4 205 V L I ST OF F IGURES F i g u r e P a g e 1 T h e t h r e e c a b l e s y s t e m s e x p e r i m e n t e d a n d a n a l y z e d 7 2 C h a r a c t e r i s t i c s o f a c a b l e y a r d i n g s y s t e m 10 3 L o a d i n g a s s u m p t i o n a n d r e f e r e n c e f r a m e s f o r t h e d e r i v a t i o n o f t h e c a t e n a r y m o d e l 13 4 L o a d i n g a s s u m p t i o n a n d r e f e r e n c e f r a m e s f o r t h e d e r i v a t i o n o f t h e p a r a b o l i c m o d e l 13 5 S k e t c h o f p l a n v i e w o f t h e f i e l d m o d e l 22 6 S k e t c h o f s i d e v i e w o f t h e f i e l d m o d e l 22 7 P l a n v i e w a n d d i m e n s i o n s o f t h e s u r v e y i n g l a y o u t 34 8 S i d e v i e w a n d d i m e n s i o n s o f t h e s u r v e y i n g l a y o u t 34 9 Y - p o s i t i o n o f p o i n t s o f t h e c a b l e : E x p e r i m e n t v e r s u s m o d e l s . T e s t #4 50 10 Y - p o s i t i o n o f p o i n t s o f t h e c a b l e : E x p e r i m e n t v e r s u s c a t e n a r y m o d e l -I n f l u e n c e o f t h e e r r o r o n T . T e s t #4 56 1 1 Y - p o s i t i o n o f p o i n t s o f t h e c a b l e : E x p e r i m e n t v e r s u s p a r a b o l i c m o d e l -I n f l u e n c e o f t h e e r r o r o n T n . T e s t #4 56 12 Y - p o s i t i o n o f p o i n t s o f t h e c a b l e : E x p e r i m e n t v e r s u s m o d e l - E r r o r - z o n e T e s t #4 5 9 13 Y - p o s i t i o n o f p o i n t s o f t h e c a b l e : E x p e r i m e n t v e r s u s m o d e l , - A v e r a g e a n d m a x i m u m d i f f e r e n c e f o r t h e n i n e f r e e h a n g i n g c a b l e t e s t s 59 14 C a b l e s h a p e : E x p e r i m e n t v e r s u s b e s t - f i t m o d e l s - A v e r a g e d i f f e r e n c e f o r t h e n i n e f r e e h a n g i n g c a b l e t e s t s 62 15 P e r c e n t d i f f e r e n c e b e t w e e n T a n d p a r a -m e t e r c a l c u l a t e d f o r b e s t - f i r c u r v e s , f o r t h e n i n e f r e e h a n g i n g c a b l e t e s t s 62 v i Figure Page 16 Y-position of points of the cable: Catenary versus parabolic model. Test #4 65. 17 Sketches of parabolic and catenary free hanging cable shapes 65 18 Deflection at mid-span: Catenary versus parabolic model 67 19 Tension at the lower support: Experiment versus models 70 20 Tension at the lower support: Catenary versus parabolic model 7 0 21 Angle of the l i n e with the horizontal at the upper support: Experiment versus models 73 22 Angle of the l i n e with the horizontal at the lower support: Experiment versus models 73 23 Angle of the l i n e with the horizontal at the upper support: Catenary versus parabolic model 75 2 4 Angle of the l i n e with the horizontal at the lower support: Catenary versus parabolic model 75 25 Cable length: Experiment versus models 78 26 Cable length: Catenary versus parabolic model 78 27 Skyline with a single concentrated load for three d i f f e r e n t positions of the clamped load 86 2 8 Y-position of the load: Experiment versus models 8 6 29 Y-position of the load: Catenary versus parabolic model 89 30 Sketches of catenary and parabolic clamped load load-paths 8 9 v i i Figure Page 31 Force balance at the clamped load using catenary and parabolic model 91 32 Tension at the lower support: Experiment versus models 94 33 Tension at the lower support: Catenary versus parabolic model 94 34 Y-position of the carriage: Experiment versus models 100 35 Mainline shapes as predicted by the models for three of the gravity system tests 100 36 Y-position of the carriage: Catenary versus parabolic model 104 37 Skethces of the catenary and parabolic gravity system load paths 104 3 8 Tension at the lower support: Experiment versus models 106 39 Tension at the lower support: Catenary versus parabolic model 106 40 Tension i n the mainline at the upper support: Experiment versus models 10 9 41 Tension i n the mainline at the upper support: Catenary versus parabolic model 109 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 s of the load 116 43 Sketch of a dynamic t e s t . Carriage stopped with the mainline 118 44 Sketch of a dynamic t e s t . Carriage stopped with a clamp 118 45 Chart recordings of the skyline and mainline tensions during a dynamic te s t 121 46 Chart recordings of skyline tension during dynamic tests 123 v i i i Figure Page 47 Parabola i n the coordinate system (x,y) 183 48 Basic p r i n c i p l e of the tensiometer 188 49 Copy of the blue-print of the tensiometer 191 50 Sketch of the equipment set up for the c a l i b r a t i o n of the tensiometer 19 6 51 Tensiometer reading versus tension i n the l i n e before c a l i b r a t i o n 196 52 Influence of the gauge factor of the indicator, on the tensiometer reading 199 53 Influence of the zero d i a l adjustment on the tensiometer reading 199 54 Tensiometer reading versus tension i n the l i n e after c a l i b r a t i o n 202 i x LIST OF PLATES P l a t e Page 1 F i e l d model seen from t h e s p a r a t t h e l o wer s u p p o r t 25 2 View of t h e c a r r i a g e and l o a d , l o o k i n g towards t h e upper s u p p o r t 25 3 M a i n l i n e d i r e c t e d t o t h e Gearmatic 19 w i n c h w i t h a b l o c k a t t h e upper s u p p o r t 28 4 S k y l i n e p a s s i n g t h e t o p o f t h e s p a r a t t h e l ower s u p p o r t and c o n n e c t e d t o t h e l o a d c e l l 28 5 S k y l i n e and m a i n l i n e t e n s i o m e t e r s a t t h e upper s u p p o r t 31 6 Reading of t h e t e n s i o n s i n t h e two l i n e s a t t h e upper s u p p o r t 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 t h e 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 9 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 t r a n s f o r m e r r e g u l a t o r used f o r t h e r e c o r d i n g o f t h e t e n s i o n s a t t h e upper s u p p o r t 113 10 Manual i n i t i a t i o n o f t h e v e r t i c a l o s c i l l a t o r y m o t i o n o f t h e clamped l o a d 113 X LIST OF TABLES Table Page I Cable c h a r a c t e r i s t i c s 23 I I Requirements f o r the winches 26 I I I Experimental e r r o r s 42 IV Two systems t h a t the model can simulate 45 V F i e l d and computed r e s u l t s . Free . hanging t e s t #4 51 VI Experimental e r r o r s . Free hanging c a b l e 5 3 VII D i s c r e p a n c i e s i n f r e e hanging segment c h a r a c t e r i s t i c s 82 V I I I Experimental e r r o r s . Clamped load t e s t . 8 4 IX P r e c i s i o n of f i e l d model, r e a l y a r d i n g system 126 X Requirements f o r the tensiometers 19 2 XI Load c e l l s c h a r a c t e r i s t i c s 193 XII Gauge f a c t o r adjustment 200 XIII Experimental r e s u l t s . Free hanging t e s t 208 XIV Experimental r e s u l t s . Clamped load t e s t 209 XV Experimental r e s u l t s . G r a v i t y system t e s t 210 xt ACKNOWLEDGEMENTS I wish to express my gratitude to Mr. G.G. Young, my supervisor, who assisted me throughout my graduate stu-dent program and guided me i n the development of t h i s thesis, I would l i k e to thank Mr. G.V. Wellburn, manager of the Forest Engineering Research I n s t i t u t e of Canada (FERIC) for the f i n a n c i a l support that made t h i s study possible. I would also l i k e to thank Mr. J. Walters, Direc-tor of the University of B r i t i s h Columbia Research Forest (UBCRF) for the use of f a c i l i t i e s on the Forest. I am also thankful to the following persons for t h e i r help: the members of FERIC and the members of the UBCRF who assisted me i n the f i e l d work. Mr. D. Myhrman, mechanical engineer at FERIC, for his guidance and constructive c r i t i c i s m . Mr. D. Anderson for his assistance i n the f i e l d . Mr. K. Vatsag for his excellent machining work. Messrs. H. J o l l i f f e and J. Walters for reviewing my thesis. and Mrs. C. van Beusekom for her fast and accurate typing. EXPERIMENTAL STUDY OF CABLE LOGGING SYSTEMS CHAPTER 1 INTRODUCTION Cable systems for handling and transporting logs are widely u t i l i z e d by the forest industry i n the P a c i f i c Northwest. Early i n the history of logging, cables were used to harvest timber and cable systems have been improved to increase the e f f i c i e n c y of the operation. The major development of these systems was c e r t a i n l y the introduction of "high-lead" at the turn of the century. Since then new technological advancements have been introduced and the ex i s t i n g systems have constantly evolved towards the new requirements of the logging industry. A look upon the present s i t u a t i o n indicates a need for yarding systems that w i l l : i) Reduce forest road density because of high road construction cost and because of envir-onment constraints, i i ) Harvest e f f i c i e n t l y timber on s i t e s i n ^ accessible with conventional systems, i i i ) Meet the needs for improved s i l v i c u l t u r a l practices. - 2 -iv) Protect the environment. Cable logging 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 the opportunity of developing new ideas requires that the engin-eering c h a r a c t e r i s t i c s of cable systems be well known. Those c h a r a c t e r i s t i c s are studied i n cable mechanics. Solutions to problems in cable mechanics for the p a r t i c u l a r case of logging were attempted a long time ago. Although the basic problem i s easy to formulate mathemati-c a l l y , numerical solutions are d i f f i c u l t to obtain. Several techniques were developed to circumvent the d i f f i c u l t y , however, only the more recent and important ones are reported here. Lysons and Mann(6) published a graphical-tabular method to determine what payload a logging system can carry over a given p r o f i l e . Carson and Mann(2) reformulate the analysis, describing the l i n e segment as a catenary, and present an i t e r a t i v e technique for the solution of skyline catenary equations. Another publication by Carson and Mann(3) proposes an algorithm to determine the load path of a running skyline using a straight l i n e approximation for 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 yiel d s a parabolic shape for the l i n e segment. The develop-ment of the parabolic model for the study of cable systems i s presented i n Appendix 1 and represents a s i g n i f i c a n t part of t h i s thesis. - 3 -Two major theories, the catenary model and the parabolic model, are therefore available to describe cable systems. However, even the most elaborate formulation i s based on certa i n degrees of assumptions, and a question re-mains as to know how well the theories represent the actual systems. F i e l d measurements are reported to have been made on r e a l logging systems(8)(9)(10) and p r a c t i c a l tables were proposed for some s p e c i f i c cases. To the author's knowledge no other experimentation has been c a r r i e d out to investigate thoroughly the mechanics of logging cable systems. The need for an experimental study to confirm the th e o r e t i c a l approaches would therefore seem necessary. The l i m i t a t i o n s of the mathematical formulations i s another point to be considered. Most of the models assume that the cables and load are free from the ground; only Carson(4) makes an attempt to model the dragging of a log, but that model should be tested i n the f i e l d . 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 that a l l studies are based on the formulation of the s t a t i c equilibrium of the system when i t i s obvious that the skid-ding of logs i s a highly dynamic operation. No simple a n a l y t i c a l study can model accurately the behaviour of a cable system i n situations such as log hangups, dragging - 4 -l o g s , y a r d i n g t h e l o g s f r e e f r o m t h e g r o u n d o r s h o c k l o a d i n g f r o m o t h e r s o u r c e s . A g a i n a n e m p i r i c a l a p p r o a c h c a n c o m -p l e m e n t t h e t h e o r y a n d a n s w e r s o m e o f t h e p r a c t i c a l p r o b l e m s . T h i s t h e s i s d e s c r i b e s a s t u d y o f c a b l e s y s t e m m e c h a n i c s . T h e o b j e c t i v e s o f t h e s t u d y w e r e p r i m a r i l y t o c a r r y o u t f i e l d t e s t s w i t h i n t h e s a m e l i m i t a t i o n s a s t h e t h e o r i e s t o v a l i d a t e t h e m a t h e m a t i c a l f o r m u l a t i o n , a n d s e c o n d l y t o e x t e n d t h e r e s e a r c h b e y o n d t h o s e l i m i t s t o i n v e s t i g a t e m o r e c o m p l e x d y n a m i c s i t u a t i o n s . T h e e x p e r -i m e n t s a n d a n a l y s i s o f t h e r e s u l t s f o r t h e f i r s t p h a s e o f t h e s t u d y w e r e c o m p l e t e d . O n l y a s m a l l p o r t i o n o f t h e s e c o n d p h a s e w a s a c h i e v e d f o r t h i s t h e s i s . T h e a p p r o a c h s e l e c t e d f o r t h e s t u d y w a s t o p r o -g r e s s f r o m t h e i n v e s t i g a t i o n o f t h e s i m p l e s t c a b l e s y s t e m t o w a r d s t h e m o s t s o p h i s t i c a t e d . T h i s p r o g r e s s i o n c a n b e e n u m e r a t e d a s f o l l o w s : i ) F r e e h a n g i n g c a b l e ( F i g u r e l a ) . i i ) C l a m p e d l o a d o n a s i n g l e l i n e ( F i g u r e l b ) . i i i ) G r a v i t y s y s t e m - L i v e s k y l i n e ; s h o t g u n o r f l y e r s y s t e m ( F i g u r e l c ) . i v ) S t a n d i n g s k y l i n e w i t h h a u l b a c k l i n e , v ) R u n n i n g s k y l i n e . V . W . B i n k l e y a n d D . D . S t u d i e r ( l ) p r e s e n t a c o m p l e t e d e s c r i p -t i o n o f t h e s e b a s i c s y s t e m s a n d t h e i r n u m e r o u s v a r i a t i o n s . - 5 -T h e s t a n d i n g s k y l i n e w i t h h a u l b a c k l i n e a n d t h e r u n n i n g s k y l i n e s y s t e m s w e r e n o t c o n s i d e r e d i n t h e f i r s t e x p e r i m e n t s i n c e m o s t o f t h e m o d e l l i n g a s s u m p t i o n s c a n b e i n v e s t i g a t e d o n t h e s i m p l e r g r a v i t y s y s t e m . T h e f r e e h a n g i n g c a b l e i s t h e b a s i c c o m p o n e n t o f a n y c a b l e s y s t e m a n d w a s s t u d i e d f i r s t . T h e c l a m p e d l o a d o n a s i n g l e l i n e c a n b e r e g a r d e d a s a s i m u l a t i o n o f a h i g h l e a d s y s t e m w i t h a f u l l y s u s p e n d e d l o a d , b u t i t s h o u l d b e c o n s i d e r e d p r i m a r i l y a s a s t e p t o w a r d s t h e s t u d y o f t h e g r a v i t y s y s t e m . B e c a u s e o f t h e p r o b l e m s a n d h i g h c o s t i n v o l v e d i n u t i l i z i n g a r e a l c a b l e y a r d i n g s y s t e m i t w a s r e a l i z e d t h a t t h e t e s t s s h o u l d b e c a r r i e d o u t o n a n e x p e r i m e n t a l p h y s i c a l m o d e l . A f t e r a b r i e f i n t r o d u c t i o n t o t h e c a t e n a r y a n d p a r a b o l i c m o d e l s , t h e t h e s i s d e s c r i b e s t h e e x p e r i m e n t o n t h e f i e l d m o d e l a n d p r e s e n t s t h e c o m p a r a t i v e a n a l y s i s o f t h e f i e l d m e a s u r e m e n t s a n d t h e o r e t i c a l v a l u e s f o r t h e c h a r a c t e r -i s t i c s o f f r e e h a n g i n g c a b l e , t h e c l a m p e d l o a d o n a s i n g l e l i n e a n d t h e g r a v i t y s y s t e m . T h e c a t e n a r y m o d e l a n d p a r a -b o l i c m o d e l s a r e a l s o c o m p a r e d t o e a c h o t h e r i n p a r a l l e l w i t h t h i s a n a l y s i s . T h e d y n a m i c t e s t s a r e p r e s e n t e d . R e c o m m e n d a t i o n s a r e s t a t e d i n t h e c o n c l u s i o n . - 6 -Figure 1 - The three cable systems experimented and analysed: a) free hanging cable b) clamped load on a single line c) gravity system - ,7 Sky!ine Skyl i ne mped load Skyl ine Ma in 1 i ne Carr ia CHAPTER 2 INTRODUCTION TO CABLE MECHANICS This chapter presents and defines the basic charac-t e r i s t i c s of a cable system. The t h e o r e t i c a l approaches to cable mechanics are then introduced. 2.1 General Description of the System. Figure 2 i l l u s t r a t e s the important features of a cable yarding system. The c h a r a c t e r i s t i c s shown can be c l a s s i f i e d i n two groups; the f i r s t group defines the dimen-sions and geometry of the system; the second group describes the forces acting on i t . The nomenclature introduced i n the presentation of 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 thesis. Geometrical c h a r a c t e r i s t i c s : ;C: Carriage A: Lower support of the cable / _ J B: Upper support of the cable L: Span; horizontal distance between the supports E: Difference i n elevation between the supports AB: Chord 0 : Angle of the chord with the horizontal D: Deflection; v e r t i c a l distance 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 p o i n t a l o n g t h e c a b l e 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 c a b l e i n t h e c o o r d i n a t e system (X,Y) 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 t h e c o o r d i n a t e system (X,Y) S: C a b l e l e n g t h a: A n g l e o f t h e c a b l e w i t h t h e h o r i z o n t a l a t any p o i n t . - 11--Force c h a r a c t e r i s t i c s : co: Weight of the cable per unit length R: Weight of the carriage and load T: Tension i n the cable H: Horizontal tension i n the cable Relationships between the above c h a r a c t e r i s t i c s can be derived using basic mechanics p r i n c i p l e s . Various mathema-t i c a l models have been proposed depending on the underlying assumptions made. 2.2 Modelling Assumptions. The general derivations of the ex i s t i n g formula-tions, the catenary model and the parabolic model, are c l a s s i c a l applied mechanics problems and have been described by I n g l i s ( 5 ) . Both these models are based on the assumption that the cable i s an i n f i n i t e l y f l e x i b l e body which implies that no bending resistance i s considered i n the accounting of the forces. Another assumption i s made as to how the uniformly d i s t r i b u t e d weight, w, acts on the system. The catenary model considers co as uniformly d i s t r i b u t e d along the cable length whereas the parabolic 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 of the system (Figure 3 and Figure 4 ). This basic difference leads to d i s t i n c t formulations. I t i s one of the objectives of t h i s study to com-pare the re s u l t s of both theories applied to cable logging - 12 -Figure 3 - Loading assumption and reference frames for the derivation of the catenary model Figure 4 - Loading assumption and reference frames for the derivation of the parabolic model - 13 -- 14 -s y s t e m s a n d t o c o m p a r e b o t h w i t h t h e e x p e r i m e n t . T h e r e -m a i n d e r o f t h e c h a p t e r d e s c r i b e s t h e c a t e n a r y m o d e l a n d t h e p a r a b o l i c m o d e l . 2 . 3 C a t e n a r y M o d e l . T h i s s u m m a r y o f t h e c a t e n a r y t h e o r y i s p r e s e n t e d w i t h r e f e r e n c e t o I n g l i s ( 5 ) a n d C a r s o n a n d M a n n ( 2 ) . B o t h d e f i n e t h e b a s i c c a b l e e q u a t i o n i n t h e c o o r d i n a t e s y s t e m ( x ' , y ' ) p o s i t i o n e d s o t h a t i t s o r i g i n 0 ' i s a t a d i s t a n c e m b e l o w t h e p o i n t o f m a x i m u m s a g ( F i g u r e 3 ) . I n t h i s c o -o r d i n a t e s y s t e m , t h e e q u a t i o n o f t h e c a b l e s h a p e d e r i v e d f r o m t h e e x p r e s s i o n s o f t h e s t a t i c e q u i l i b r i u m o f t h e s e g -m e n t OP i s g i v e n b y t h e c a t e n a r y : x ' y ' = m c o s h — w i t h m = H/u) ( a ) m--T h e c a t e n a r y e q u a t i o n c a n b e t r a n s l a t e d t o a c o -o r d i n a t e s y s t e m ( x , y ) w i t h i t s o r i g i n a t t h e l o w e r s u p p o r t A . T h e e q u a t i o n b e c o m e s : y = m c o s h - m c o s h — ( b ) m m T h e d i s t a n c e , a , f r o m ( x , y ) t o t h e o r i g i n a l f r a m e ( x ' , y ' ) i s d e t e r m i n e d b y E , L a n d m, w h e r e m, e q u a l t o H/w i s t h e y - i n t e r c e p t o f t h e c a t e n a r y . S i n c e h o r i z o n t a l t e n -s i o n , H , i s t h e s a m e a t a n y p o i n t o n t h e c a b l e , m i s a - 15 -constant and becomes a convenient parameter for the catenary formulation. The other system c h a r a c t e r i s t i c s can e a s i l y be expressed i n terms of the parameter m. The tension T^ and Tg, and the angles of the l i n e with the horizontal an and rv , are given at the lower and upper supports by: T, = com cosh — and tga,. = sinh — (c) (d) A m 13 A m T„ = oom cosh and tga„ = sinh — — (e) (f) B m ^ B m I t can e a s i l y be shown that the difference between the ten-sions at the supports' i s defined by the simple relationship: T B - T A = ooE (g) This i s a very useful expression i n the catenary formulation of cable mechanics. The d e f l e c t i o n at mid-span can be calculated from: E , L/ 2-a Dm = - m cosh — (h) 2 m And i t i s easy to express the cable length as: S = m (sinh — + sinh -J) (i) m m Therefore, most of the system c h a r a c t e r i s t i c s are expressed simply using the hyperbolic sine and cosine func-tions. However, the transcendental property of the hyper-b o l i c functions render t h e i r use impracticable for the study of cable systems without the aid of a computer. Much work has been devoted to develop i t e r a t i v e techniques and com-puter programs to provide numerical solutions to catenary models of skyline problems. The l a t e s t and most elaborate i s that by Carson and Mann(2) who developed the " r i g i d l i n k " i t e r a t i v e technique adopted for the catenary analysis i n t h i s paper. 2.4 Parabolic Model. The parabolic theory as i t applies to cable log-ging systems i s developed i n Appendix 1. The development presented progresses from the basic free hanging cable to the more complex f i v e - l i n e system and emphasize each major res u l t with numerical examples. Only the basic features and res u l t s of t h i s theory are summarized i n t h i s section. The equation of the free hanging cable shape, developed i n the coordinate system (x', y 1) using the equa-tions of s t a t i c equilibrium of the segment of cable OP (Figure 4) i s - 1 7 T h i s e q u a t i o n , t r a n s l a t e d t o t h e c o o r d i n a t e s y s t e m ( x , y ) d e f i n e s t h e p a r a b o l i c s h a p e o f t h e c a b l e a s : co 2 . ,E coL > ,, .. x + (7- " o ^ Q P ) x ( b 1 ) J 2 c o s G H VL 2 cbs9H J w h e r e H , t h e h o r i z o n t a l t e n s i o n , c a n b e c o n s i d e r e d a s a p a r a -m e t e r . T h e e x p r e s s i o n s o f t h e o t h e r c h a r a c t e r i s t i c s o f t h e s y s t e m c a n a l s o b e e x p r e s s e d c o n v e n i e n t l y i n t e r m s o f H . T h e t e n s i o n s T a n d T _ , a n d a,, a n d a_ t h e a n g l e s A H A ti o f t h e l i n e w i t h t h e h o r i z o n t a l , a t t h e l o w e r a n d u p p e r s u p -p o r t a r e d e f i n e d b y t h e f o l l o w i n g r e l a t i o n s h i p s : T A = — " a n d t g c v = f- - _ a ) L n „ ( c ' ) ( d ' ) A c o s a A ^ A L 2 c o s G H v ' = _ J _ and t g a B = £ + <e'> < f > B c o s a D X5 T h e d e f l e c t i o n a t m i d - s p a n i s s i m p l y e x p r e s s e d a s : T h e l e n g t h o f t h e c a b l e i s d e r i v e d f r o m : L/ Oy - 18 -The s o l u t i o n o f t h i s i n t e g r a l i s not t r i v i a l and i s g i v e n i n Appendix 2. I f the c a b l e i s t i g h t t h e f o l l o w i n g a p proximate f o r m u l a can be used: s = — ^ - ( 1 + f ^ ) 2 ) cose 3 L The p a r a b o l i c e q u a t i o n s a r e r e l a t i v e l y easy and s i m p l e t o m a n i p u l a t e . However, the e x p r e s s i o n s o f l i n e a n g l e s w i t h t h e h o r i z o n t a l and t h e c a b l e l e n g t h f o r m u l a a r e cumbersome and c o n s t i t u t e a drawback t o t h e t h e o r y . - 19 -CHAPTER 3 DESCRIPTION OF THE F IELD MODEL T h e f i e l d m o d e l w a s d e s i g n e d t o i n c o r p o r a t e a l l t h e m a j o r c o m p o n e n t s o f a g r a v i t y s y s t e m . I n a d d i t i o n , i n s t r u m e n t a t i o n w a s i n t e g r a t e d t o t h e s y s t e m t o m e a s u r e t h e d e s i r e d v a r i a b l e s . T h e g e n e r a l l a y o u t o f t h e f i e l d m o d e l a n d t h e v a r i o u s p i e c e s o f e q u i p m e n t a r e d e s c r i b e d i n t h i s c h a p t e r . 3 . 1 S i t e D i m e n s i o n s a n d C h a r a c t e r i s t i c s . A l o c a t i o n s u i t a b l e f o r t h e e x p e r i m e n t w a s f o u n d a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a R e s e a r c h F o r e s t ( U B C R F ) . T h e s i t e w a s l o c a t e d i n t h e n o r t h e r n e n d o f t h e f o r e s t a n d h a d b e e n r e c e n t l y l o g g e d . T h e f i n a l s u r v e y i n g , d o n e a f t e r t h e e q u i p m e n t w a s i n s t a l l e d a n d a d j u s t e d , g a v e t h e f o l l o w i n g d i m e n s i o n s : S p a n L = 1 3 1 . 9 5 m e t r e s D i f f e r e n c e i n e l e v a t i o n , E = 2 3 . 0 5 m e t r e s . A m a x i m u m m i d - s p a n d e f l e c t i o n Dm = 1 0 . 5 m ( i . e . 8% o f t h e s p a n ) c o u l d b e o b t a i n e d f o r a f r e e h a n g i n g c a b l e . O t h e r e l e m e n t s w e r e a l s o t a k e n i n t o c o n s i d e r a t i o n i n t h e c h o i c e o f t h i s s i t e . A n c h o r i n g s w e r e a v a i l a b l e a t - 2 0 -e a c h e n d f o r t h e r i g g i n g o f t h e c a b l e s , a s w e l l a s a f i r m r o c k y b a s e f o r t h e s u p p o r t s . T h e r e w a s e a s y a c c e s s , t o t h e l o w e r a n d t o t h e u p p e r e n d , t h a t f a c i l i t a t e d t h e i n s t a l -l a t i o n o f t h e e q u i p m e n t a n d t h e e x e c u t i o n o f t h e t e s t s . T h e p a t h u n d e r t h e c a b l e , c l e a r e d o f b r a n c h e s a n d s n a g s w a s e a s y t o w a l k . T h e g r o u n d p r o f i l e a l l o w e d t h e t e s t s t o b e r u n i n c o n d i t i o n s v e r y s i m i l a r t o t h o s e o f a r e a l y a r d i n g o p e r -a t i o n . A p l a n v i e w a n d p r o f i l e o f t h e s i t e a r e s h o w n i n F i g u r e 5 a n d 6 . E a c h e l e m e n t o f t h e m o d e l w i l l n o w b e d e s -c r i b e d . 3 . 2 C a b l e s . T h e c a b l e s w e r e 6 x 1 9 IWRC a t y p e o f w i r e r o p e w i d e l y u s e d i n t h e l o g g i n g i n d u s t r y . 1 3 5 m e t r e s o f t w o d i f f e r e n t d i a m e t r e s w e r e o b t a i n e d . B o t h t h e 5 / 8 - i n c h a n d 7 / 1 6 - i n c h d i a m e t e r s c a b l e s c a m e w i t h a f a c t o r y m a d e f l e m i s h e y e a t o n e e n d . T h e 5/8 c a b l e , u s e d a s t h e s k y l i n e , w a s m a r k e d p r e c i s e l y e v e r y m e t r e w i t h p a i n t , f o r l e n g t h m e a s u r e -m e n t . T h e 7 / 1 6 c a b l e w a s u t i l i z e d a s t h e m a i n l i n e t o m o v e t h e c a r r i a g e a l o n g t h e s k y l i n e . T h e c a b l e s c h a r a c t e r i s t i c s a r e s u m m a r i z e d i n T a b l e I. T h e w e i g h t p e r m e t r e m e a s u r e d o n 2 - m e t r e l o n g c a b l e s a m p l e s a r e 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 f r o m t h e s u p p l i e r ' s i n f o r m a t i o n . - 21 -F i g u r e 5 - S k e t c h o f p l a n view o f t h e f i e l d model F i g u r e 6 - S k e t c h o f s i d e view o f t h e 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 . #(inch) w (kg/m) B.S. (N) (NxlOOO) Tm (N) (NxlOOO) SF E N/m2 i o 6 Measured Catalogue D i f f e r -ence % 5/8 1.071 1.027 4 157 50 3.1 188 7/16 0.521 0.506 3 76 10 7.6 188 JZ5 J.C'a'b'le.-.diameter w Weight per metre BS Breaking strength i n newtons Tm Maximum s t a t i c tension expected during the experiment BS SF Safety factor for the experiment conditions = ^ i E E l a s t i c modulus of the cable 3.3 Carriage (Plate 2) The carriage had a single b a l l bearing-mounted sheave that f i t s the skyline. I t could be connected to the mainline with a shackle or immobilized on the skyline by means of a clamp. A basket was attached to the carriage to receive up to 40 lead weights to constitute the load. The maximum load weight including the carriage and basket was 535 kilograms. 3.4 Winches (Plate 3) The requirements for the winches, shown i n table I I , were dictated by the cables c h a r a c t e r i s t i c s . - 24 -Plate 1 — Field model seen from the spar at the lower support. Plate 2 — View of the carriage and load, looking toward the upper support. - 26 -T a b l e I I . R e q u i r e m e n t s f o r t h e w i n c h e s . Winch Cable dia-meter (inch) Pull length (metre) Tensions (Newtons) Pull Speed (m/s) Reversible 1 5/8 5 50,000 Slow Yes 2 7/6 130 10,000 0 to 2 Desired A f t e r s o m e r e s e a r c h i t b e c a m e o b v i o u s t h a t t h e s e c o n d i t i o n s w e r e d i f f i c u l t t o m e e t e x a c t l y b e c a u s e o f t h e l a c k o f p r o p o r t i o n b e t w e e n c a b l e d i a m e t e r a n d e x p e c t e d t e n s i o n s . T h e f i n a l c h o i c e s w e r e a G e a r m a t i c 19 w i n c h m o u n t e d o n a r u b b e r - t i r e d s k i d d e r a n d a 6 - t o n C o m e l o n g h a n d w i n c h . T h e G e a r m a t i c 19 s u p p l i e d m o r e t h a n t h e l i n e p u l l n e e d e d f o r t h e e x p e r i m e n t b u t i t s d r u m c a p a c i t y w a s : j u s t s u f f i c i e n t t o a c c e p t t h e 1 2 0 m e t r e s o f m a i n l i n e . I t s m a i n a s s e t s w e r e i t s i n f i n i t e l y v a r i a b l e s p e e d a n d t h e c o n v e n i e n c e o f h a v i n g i t m o u n t e d o n a s k i d d e r . H o w e v e r t h e G e a r m a t i c 19 i s n o t a A r e v e r s i b l e w i n c h w h i c h c r e a t e d p r o b l e m s a n d a h a z a r d e v e r y t i m e t h e l o a d h a d t o b e l o w e r e d s l o w l y . T h e 6 - t o n C o m e l o n g w a s c o n n e c t e d t o t h e s k y l i n e . I t s s l o w s p e e d w a s a p p r e c i a t e d t o s e t p r e c i s e t e n s i o n s b u t r e s u l t e d i n l a b o r i o u s m a n u a l h a n d l i n g . - 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 = 3 . 5 D y n a m o m e t e r s ( P l a t e s 4 a n d 5 ) T h e t e n s i o n s w e r e m e a s u r e d a t e a c h e n d o f t h e s k y -l i n e a n d a t t h e u p p e r s u p p o r t i n t h e m a i n l i n e w i t h d y n a m o -m e t e r s . N o f o r c e m e a s u r e m e n t w a s d o n e a t t h e c a r r i a g e . R u g g e d p o r t a b l e d y n a m o m e t e r s w e r e r e q u i r e d t o a v o i d p r o b l e m s i n t h e f i e l d . T h e y a l s o h a d t o b e c o m p a t i b l e w i t h a r e c o r d -e r f o r t h e d y n a m i c t e s t s . E l e c t r o n i c l o a d - c e l l s s a t i s f y a l l t h o s e r e q u i r e m e n t s a n d a l s o p r e s e n t t h e a d v a n t a g e o f b e i n g v e r y c o m m o n l y u s e d . T h e o u t p u t f r o m a l o a d c e l l c a n b e r e a d o n a d i g i t a l g a u g e i n d i c a t o r o r r e c o r d e d o n a c h a r t o r t a p e r e c o r d e r . T h e c h a r a c t e r i s i t c s o f t h e l o a d - c e l l s u s e d a r e d e s c r i b e d i n A p p e n d i x 3 . T h e r i g g i n g o f t h e d y n a m o m e t e r s w a s a k e y p o i n t i n t h e l a y o u t o f t h e e q u i p m e n t . T h e i n s t a l l a t i o n o f t h e s i m p l e l o a d - c e l l o n t h e s k y l i n e a t t h e l o w e r s u p p o r t w a s s t r a i g h t -f o r w a r d . T h e o n l y p r o b l e m w a s t o k e e p t h e w e i g h t o f t h e l o a d - c e l l o f f t h e f r e e h a n g i n g s e c t i o n o f t h e c a b l e t o i n t r o d u c e n o d i s t u r b a n c e i n c a b l e s h a p e a n d t e n s i o n . T h e s o l u t i o n w a s t o l e t t h e c a b l e r u n o n a b a l l b e a r i n g m o u n t e d s h e a v e f i r s t , a n d t h e n t o c o n n e c t i t t o t h e l o a d - c e l l a n c h o r e d t o a s t u m p . T h e u s e o f a s w i v e l h o o k a l l o w e d t h e l i n e t o s p i n a s t h e t e n s i o n w a s a p p l i e d o r r e l i e v e d . T h e 1 . 2 - m e t r e s p a r g a v e e n o u g h h e i g h t t o t h e c a b l e s o t h a t t h e c a r r i a g e a n d l o a d c o u l d r u n a l l t h e w a y t o t h e l o w e r e n d . T h e r e f e r e n c e p o i n t A w a s t a k e n a t t h e t o p o f t h e l o w e r s u p p o r t s h e a v e . - 30 -Plate 5 — Skyline and mainline tensiometers at the upper support. Plate 6 — Reading of the tensions in the two lines at the upper support on the strain-gage indicators connected to the tensiometer. -31-- 32 -F o r t h e u p p e r s u p p o r t a s p e c i a l t y p e o f d y n a m o -m e t e r , r e f e r r e d t o i n t h i s t h e s i s a s 1 1 t e n s i o m e t e r " , h a d t o b e b u i l t . T h e t e n s i o m e t e r s , d e s c r i b e d i n A p p e n d i x 3, a r e c a p a b l e o f m e a s u r i n g t h e t e n s i o n i n f i x e d o r r u n n i n g l i n e s . T h e y w o r k o n a s i m p l e m e c h a n i c a l p r i n c i p l e : t h 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 s h e a v e s a n d t h e a c t i o n o n t h e m i d d l e s h e a v e t r a n s m i t t e d t o a l o a d - c e l l b y a l e v e r c a n b e s i m p l y r e l a t e d t o t h e t e n s i o n i n t h e l i n e . B o t h t h e s k y l i n e a n d m a i n l i n e w e r e r u n t h r o u g h t e n s i o m e t e r s o f t h i s t y p e b o l t e d t o a s t e e l f r a m e s e c u r e d t o c o n c r e t e f o u n d a t i o n s . A f t e r l e a v i n g t h e t e n s i o m e t e r s t h e l i n e s w e r e d i r e c t e d t o t h e i r r e s p e c t i v e w i n c h e s w i t h b l o c k s . T h e t o p o f t h e f i r s t s h e a v e o f t h e s k y -l i n e t e n s i o m e t e r w a s t a k e n a s t h e u p p e r r e f e r e n c e p o i n t B . 3.6 S u r v e y i n g o f C a b l e S h a p e a n d C a r r i a g e P o s i t i o n ( P l a t e 7) T h e o b j e c t o f t h e e x p e r i m e n t r e q u i r e d a m e t h o d o f l o c a t i n g t h e p o s i t i o n o f a n y p o i n t o f t h e c a b l e a c c u r a t e l y a n d e a s i l y . A S a l m o r a g y t h e o d o l i t e w a s s e t u p , t o s e r v e t h i s p u r p o s e , o n a k n o l l f r o m w h e r e t h e e n t i r e c a b l e c o u l d b e s u r v e y e d i n m o s t c i r c u m s t a n c e s . T h e p l a n v i e w a n d p r o f i l e o f t h e s u r v e y i n g l a y o u t a r e s h o w n i n F i g u r e s 7 a n d 8. T h e i n s t r u m e n t T s t o o d a t t h e v e r t i c a l o f a p r e - s u r v e y e d b e n c h -m a r k T ' . T h e h o r i z o n t a l v e r n i e r o f t h e t h e o d o l i t e w a s a d -j u s t e d s o a s t o r e a d z e r o f o r t h e d i r e c t i o n T M 1 p e r p e n d i c u l a r t o t h e p l a n e o f t h e c a b l e . S i n c e t h i s d i r e c t i o n w a s n o t Figure 7 - Plan view and dimensions of the surveying layout Figure £ - Side view and dimensions of the surveying layout - 35 -locatable on the t e r r a i n or on the cable the actual adjust-ment of the horizontal vernier was done at 41° 18.3' with the telescope of the theodolite pointing the lower r e f e r -ence A. Simple geometric derivation y i e l d s the following equation for the horizontal distance X from a point of the cable to the lower support A. X = A'M' - TM1 tg(Alpha) where tg(Alpha) represents the tangent of the angle read on the horizontal vernier of the instrument. With the dimensions i n metres the previous 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 vernier was adjusted at zero with the axis of the telescope of the theodolite i n the horizontal d i r e c t i o n . The following r e l a t i o n s h i p gives the v e r t i c a l distance Y from a point of the cable to the lower support A. Y = AA' - (TM'/cos(Alpha)) x tg(Beta) where cos(Alpha) i s the cosine of the horizontal vernier reading and tg(Beta) i s the tangent of the angle read on the v e r t i c a l vernier. With the dimensions i n metres the previous equation becomes: Y = TT*+14.975-(55.094/cos(Alpha))xtg(Beta)(metres) TT' the height of the instrument axis to the bench-mark T' was remeasured afte r every setting of the instrument. The shape of the cables and the carriage p o s i t i o n - 36 -w e r e s u r v e y e d u s i n g t h i s t e c h n i q u e . F o r l a r g e d e f l e c t i o n s o f t h e c a b l e a f e w p o i n t s w e r e n o t v i s i b l e f r o m t h e t h e o d o l i t e a n d t h e i r p o s i t i o n w a s t a k e n b y d i r e c t m e a s u r e m e n t o f t h e i r h e i g h t f r o m p r e - , s u r v e y e d b e n c h m a r k s u n d e r t h e c a b l e . 3 . 7 O t h e r m e a s u r e m e n t s . 3 . 7 . 1 A n g l e o f t h e l i n e w i t h t h e h o r i z o n t a l . B e c a u s e o f t h e i m p o r t a n c e t h e y h a v e i n c a b l e m e c h -a n i c s t h e o r i e s , t h e a n g l e s o f t h e c a b l e s w i t h t h e h o r i z o n t a l w e r e r e c o r d e d a t t h e s u p p o r t s a n d a t t h e c a r r i a g e l e v e l i n t h e s k y l i n e a n d m a i n l i n e . T h i s w a s p e r f o r m e d w i t h a c l i n o -m e t e r b y s i m p l y p l a c i n g t h e b o d y o f t h e i n s t r u m e n t d i r e c t l y o n t h e l i n e , a d j u s t i n g t h e l e v e l t u b e t o t h e h o r i z o n t a l a n d t a k i n g t h e r e a d i n g . T h i s r a t h e r u n u s u a l u t i l i z a t i o n o f a c l i n o m e t e r g a v e g o o d r e s u l t s . 3 . 7 . 2 U n s t r e t c h e d l i n e l e n g t h ( P l a t e 8) M a r k s w e r e p a i n t e d e v e r y m e t r e o n t h e u n t e n s i o n e d s k y l i n e . T h e u n s t r e t c h e d l e n g t h o f t h e c a b l e b e t w e e n t h e u p p e r a n d l o w e r s u p p o r t s , B A , w a s o b t a i n e d b y a d d i t i o n o f t h e m e a s u r e m e n t s o f t h e t w o f r a c t i o n s o f m e t r e s a t t h e e x t r e m e s t o t h e n u m b e r o f w h o l e m e t r e s b e t w e e n A a n d B . I t s h o u l d b e n o t e d t h a t a n e g l i g i b l e e r r o r i s i n t r o d u c e d s i n c e t h e d i s -- 37 -Plate 7 — Surveying of the cable and carriage posi-tions with the theodolite. Plate S Measurement of the f r a c t i o n of metre between the lower support reference point and the f i r s t paint mark on the cable. - 39 -tances at the extremes were measured on the cable under ten-sion. 3.8 Accuracies of instruments and expected errors i n the measurements. The magnitudes of the errors a f f e c t i n g the various measurements have to be known in order to make any conclu-sion i n the analysis of the r e s u l t s . Experimental errors have various o r i g i n s , they may be instrumental, procedural, personal or natural. Instrumental errors r e s u l t from i n s t r u -ment imperfections and non-adjustments. The magnitude of the procedural error increases with the number of steps performed and number of pre-measurements needed for the determination of a given variable. Personal errors r e s u l t from human li m i t a t i o n s and accidents. The natural errors were the most d i f f i c u l t to apprehend. The weather condi-tions i n p a r t i c u l a r affected the results as variations of the tension and sag resulted from the expansion or contract-ion of the cable from changes i n temperature. The r a i n and wind had the e f f e c t of increasing the cable weight per metre re s u l t i n g i n an increase i n tension. Other natural errors resulted from the y i e l d i n g of the anchorings and the s e t t l e -ment of the theodolite tripod. The natural errors were observed and recorded but no numerical values were attached to them. - 40 -The remainder of t h i s section evaluates the ex-pected error for each types of measurement. 3.8.1 Errors i n the tensions measurements. A complete test c a r r i e d out on the tensiometers used at the upper support i s reported i n Appendix 3. With the recommendations formulated i n Appendix 3, less than one percent error can be obtained for tensions greater than 5 000 newtons i n the skyline and for tensions greater than 1000 newtons i n the mainline. Although no s p e c i f i c test was done on the rigging at the lower support the load c e l l at that point i s also expected to be accurate to plus or minus one percent. 3.8.2 Error i n the cable position. Instrumental and personal errors are c l o s e l y re-lated i n t h i s case. The angular accuracy of the Salmoragy theodolite i s given i n the supplier's catalogue as 1/10 of a minute. However the experience has proved that 1/5 of a minute was probably a more r e a l i s t i c l i m i t for the angular d e f i n i t i o n of the instrument, because the verniers were d i f f i c u l t to read and the l e v e l l i n g required dexterity. The error i n the cable position created by the angular error depends on the distance L from the theodolite to the cable and i s defined by: error = ^ *n3;li = .000058L 5x60x180 - 4 1 -F o r t h e l o n g e s t d i s t a n c e o n t h e m o d e l t h e e r r o r i s m a x i m u m a n d e q u a l t o 6 mm. T h i s m a x i m u m e r r o r i s f o r t h e s a k e o f s i m p l i f i c a t i o n a p p l i e d i n t h i s a n a l y s i s t o a n y p o i n t o f t h e c a b l e i n d e p e n d e n t o f i t s d i s t a n c e t o t h e t h e o d o l i t e . T h e p r o c e d u r a l e r r o r h a s t o b e a d d e d t o t h i s i n s t r u m e n t a l e r r o r ; t h e v a l u e o f t h e m e a s u r e d c a b l e p o s i t i o n s d e p e n d s o n t h e e r r o r s a f f e c t i n g e a c h o f t h e p r e - m e a s u r e d d i m e n s i o n s u s e d i n t h e c o m p u t a t i o n . T h e e r r o r i n t h e d i m e n s i o n A A ' , h e i g h t o f t h e t e l e s c o p e a x i s , i s d i r e c t l y r e p r o d u c e d o n t h e v e r t i c a l p o s i t i o n m e a s u r e m e n t . T h i s e r r o r i s e s t i m a t e d t o b e l e s s t h a n 6 mm. I t i s d i f f i c u l t t o a p p r a i s e t h e e f f e c t o f t h e o t h e r d i m e n s i o n a l i n a c c u r a c i e s o n t h e c a b l e p o s i t i o n m e a -s u r e m e n t s b u t 6 mm s e e m s t o b e r e a s o n a b l e . T h e t o t a l p r o -c e d u r a l e r r o r i s t h e r e f o r e 12 mm. T h e t o t a l e r r o r a d o p t e d i n t h e a n a l y s i s o f t h e v e r t i c a l p o s i t i o n o f t h e c a b l e p o i n t s i s 6+12 = 18 mm. 3 . 8 . 3 E r r o r i n t h e m e a s u r e m e n t o f t h e l i n e a n g l e w i t h t h e h o r i z o n t a l . T h e a c c u r a c y o f a c l i n o m e t e r i s 1 / 1 0 o f a d e g r e e . T h e i n s t r u m e n t a l e r r o r f r o m i m p r o p e r z e r o a d j u s t m e n t c o u l d b e e l i m i n a t e d b y a v e r a g i n g t h e r e a d i n g f r o m t w o i n v e r t e d p o s i t i o n s o f t h e i n s t r u m e n t . H o w e v e r e x p e r i e n c e h a s s h o w n t h e d i f f i c u l t y i n p o s i t i o n i n g t h e i n s t r u m e n t c o r r e c t l y o n t h e l i n e a n d a m o r e r e a l i s t i c e x p e c t a t i o n f o r t h e a n g l e m e a s u r e m e n t e r r o r i s . 2 d e g r e e . - 42 -3.8.4 E r r o r i n the c a b l e l e n g t h . Because of the poor d e f i n i t i o n of the r e f e r e n c e p o i n t s on the sheaves and of the marks on the c a b l e the e r r o r i n the measurement of the f r a c t i o n of metre at each of the extremes can be as h i g h as 5 mm. Another p r o c e d u r a l e r r o r , r e s u l t s from i n a c c u r a c y i n marking of the c a b l e . I t i s reasonable to t h i n k t h a t .02 percent of the t o t a l c a b l e l e n g t h i s the maximum magnitude of t h i s e r r o r , y i e l d i n g an e r r o r of 27 mm f o r a maximum 135 metres l e n g t h . The un-s t r e t c h e d l e n g t h i s then known to p l u s or minus 37 mm. A summary of the expected experimental e r r o r s i s shown i n t a b l e I I I . Table I I I . 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 . Variable Nomenclature Magnitude of the error Tension T 1% V e r t i c a l p o s i t i o n Y 18 mm Angle of the l i n e with the .horizontal a .2 degree Cable length S 37 mm - 43 -3.9 D i m e n s i o n a l s i m i l i t u d e b e t w e e n t h e m o d e l a n d a r e a l y a r d i n g s y s t e m . T h e m o d e l c o u l d b e c o n s i d e r e d a s a s i m u l a t i o n o f a n y t w o - l i n e s y s t e m ; a s t a n d i n g s k y l i n e s y s t e m i f t h e l e n g t h o f t h e l i n e w a s k e p t c o n s t a n t ; o r a s h o t - g u n o r g r a v i t y s l a c k l i n e s y s t e m i f t h e t w o w i n c h e s w e r e o p e r a t e d . F o r e c o n o m i c a l p u r p o s e s a n d p r a c t i c a l c o n s i d e r a t i o n s t h e m o d e l h a d t o b e s c a l e d d o w n a n d w a s s m a l l e r t h a n a r e a l y a r d i n g s y s t e m . N e v e r t h e l e s s , a d i m e n s i o n a l a n a l y s i s s h o w s t h a t t h e r e s u l t s o b t a i n e d w i t h t h e s m a l l s c a l e m o d e l w i l l b e r e p r e s e n t a t i v e o f t h e f u l l s c a l e s y s t e m i f t h e s c a l i n g i s d o n e p r o p e r l y . Two i n d e p e n d e n t r a t i o s , o n e f o r t h e l e n g t h s a n d o n e f o r t h e f o r c e s d e t e r m i n e t h e s c a l i n g o f t h e s y s t e m . T h e l e n g t h r a t i o r ^ i s s i m i l a r t o t h e s c a l e f o r a m a p a n d s i m p l y m e a n s t h a t o n e m e t r e o f t h e m o d e l r e p r e s e n t s r ^ m e t r e s o f t h e r e a l s y s t e m . T h e s p a n a n d d i f f e r e n c e i n e l e v a t i o n w e r e s c a l e d d o w n b y t h e s a m e r a t i o a n d c o n s e -q u e n t l y t h e c a b l e l e n g t h a n d d e f l e c t i o n w e r e s c a l e d d o w n b y t h i s r a t i o . F o r c e s a r e s c a l e d b y t h e f o r c e r a t i o r^. O n e u n i t o f f o r c e f o r t h e m o d e l r e p r e s e n t e d u n i t s o f f o r c e f o r a r e a l s y s t e m . T h e l o a d a n d t h e w e i g h t o f t h e c a b l e w e r e s c a l e d d o w n b y t h e s a m e r a t i o i f t h e f o l l o w i n g r e -l a t i o n s h i p i s a s s u m e d t o b e t r u e . r 4.4 -P • W R 0) X r i r 2 P p a y l o a d i n k i l o g r a m f o r t h e r e a l s y s t e m . R w e i g h t o f c a r r i a g e a n d l o a d i n k i l o g r a m . W w e i g h t p e r u n i t l e n g t h o f t h e c a b l e i n t h e r e a l s y s t e m . co w e i g h t p e r u n i t l e n g t h , o f t h e c a b l e i n t h e m o d e l , r ^ l e n g t h r a t i o . f o r c e r a t i o . A n g l e s a n d s l o p e s a n d u n i t l e s s v a l u e s a r e r e l a t e d w i t h a o n e t o o n e r a t i o w i t h o u t a n y d i s t o r t i o n . A s a n e x a m p l e , t w o r e a l s y s t e m s w h i c h c a n b e r e p r e s e n t e d b y t h e m o d e l a r e d e s c r i b e d i n t a b l e I V . T h e m o d e l c a n s i m u l a t e s y s t e m s o f a l m o s t a n y s i z e b y m e r e l y c h a n g i n g t h e t w o s c a l e r a t i o s . T h e g r o u n d c o n f i g u r a t i o n a n d t e r r a i n a s p e c t w e r e a l s o p a r t o f t h e g e n e r a l s i m i l i t u d e . A l t h o u g h n o t v e r y m u c h u t i l i z e d i n s i d e t h e l i m i t s o f t h i s t h e s i s , t h o s e t w o p o i n t s c o u l d b e o f p r i m e i m p o r t a n c e i n f u t u r e e x t e n s i o n s o f t h e s t u d y . - 4 5 -T a b l e IV. Example o f two systems t h a t the model can s i m u l a t e . . , System c h a r a c t e r i s t i c s Model Slack-l i n e system Long reach standing skyline Span (metres) 132 600 1500 Length r a t i o (r^) - 4.55 11.36 Differences i n elevation (metres) 23 105 261 Maximum d e f l e c t i o n (metres) 10.5 48 119 Maximum d e f l e c t i o n (%) 8 8 8 Average ground slope (%) 17 17 17 Skyline length of maximum d e f l e c t i o n (m) 136 619 1545 Load r a t i o ( r ^ - 14:19. 65.64 Skyline diameter (inch) 5/8 1 1/8 1 1/2 Skyline weight per metre (kilogram) 1 3.4 6 Mainline diameter (inch) 7/16 3/4 1 Mainline weight per metre (kilogram) .5 1.5 2.6 CHAPTER 4 FREE HANGING 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 free hanging l i n e segment. The single l i n e segment i s the basic element of a cable system since the most complex system can always be considered as a more or less i n t r i c a t e arrangement of cable segments hanging f r e e l y between the d i f f e r e n t points of attachment. This chapter describes the free hanging cable experiment, com-pares 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 also compares the catenary and parabolic models. 4.1 Description of the Experiment. For t h i s test the skyline was rigged so as to hang fr e e l y , under i t s own weight, between the lower and the upper supports and was tensioned with the Gearmatic 19 winch. 4.1.1 Procedure and Data C o l l e c t i o n . Nine free hanging cable tests were executed for a range of tensions at the upper support between 2700 to 11000 newtons, and r e s u l t i n g deflections at mid-span between 7.1 and 1.6 percent of the span length. The re s u l t s are re-ported i n Appendix 4. Each te s t lasted one hour on average and was planned to be executed by three operators: one at the theo-d o l i t e , one at the upper support and one at the lower sup-port. The required information was recorded i n i n d i v i d u a l f i e l d note books. Sample pages of these note books are re-produced i n Appendix 4. Readings of the v e r t i c a l p o sition of the cable were taken at stations f i v e metres apart along the horizontal :.. span. The position at mid-span was measured at the begin-ning and at the end of the te s t to check the v a r i a t i o n with time. The tensions, the angles of the l i n e with the h o r i -zontal and the cable length were recorded every 15 minutes at the upper and lower support, as they proved to change s l i g h t l y with time. Those variations were attributed to natural phenomena l i k e the y i e l d i n g of the anchorings and the changes i n atmospheric conditions as sun, r a i n or wind. For most of the tests the upper and lower support measure-ments were done by the same operator which resulted i n the i m p o s s i b i l i t y of obtaining t r u l y simultaneous readings; the time lag between the readings was about 5 minutes. 4.2 Analysis of the Results. The analysis of the free hanging cable i s based on - 48 -the comparison of the f i e l d measurements and the r e s u l t s as predicted by the catenary and parabolic models for the following six c h a r a c t e r i s t i c s of the system: Cable shape defined by v e r t i c a l Y-positions of points of the cable Dm: d e f l e c t i o n at mid-span T.. and T„: tensions at the supports A 15 CL. and OL,:- angles of the l i n e with the A is horizontal at the supports S: skyline length. Only one of these c h a r a c t e r i s t i c s has to be given as a para-meter to determine the system and the other variables com-pl e t e l y . T as the Parameter. The measured tension at the upper support T , was taken as a parameter. The other variables could be deter-mined using the catenary and parabolic models as described by the following chart: - 49 -Figure 9 - Differences i n the Y-position of the cable between experiment and catenary model, and between experiment and parabolic model versus X-positions on span, f o r free hang-ing test number 4. - 50 -Measured Parameter TB Measured values f o r the Variables Y, Dm, T A, a A , a B , S Computation using Catenary and Parabola theories Theoretical values f o r the Variables Y, Dm, T A, a A , a B , S Conclusion O L E E T3 O E >-I O • LA] a. x I a> >-o o o LA! r ± 18 mm ^ 20 kO 60 80 100 1 2 ^ X » m Yexp.-Ycat. Yexp.-Ypar. - . 5 1 -T a b l e V. F i e l d and computed r e s u l t s . f o r f r e e h a n ging c a b l e T e s t #4. FREE HANGING CABLE DATE: 31/08/76 CREW - THEODOLITE - WINCH - SPAR TEST # 04 WEATHER: SUNNY HOT D. D. D. Guimier Anderson Anderson TEST STARTED AT: 13:00 COMPLETED AT: 14:00 Y p o s i t i o n of the cable on the span. X POSITION Y POSITION IN METRES DIFFERENCES IN METRES IN METRES EXPERIM. CATENARY PARABOLA EXP-CAT EXP-PAR CAT-PAR 0.0 -0.013 -0.000 0.000 -0.013 -0.013 -0.000 5.0 -0.332 -0.318 -0.320 •^0.014 -0.012 0.002 10.0 -0.558 -0.543 -0.546 -0.015 -0.012 0.003 15.0 -0.696 -0.675 -0.677 -0.021 -0.019 0.003 20.0 -0.741 -0.714 -0.715 -0.027 -0.026 0.001 25.0 -0.686 -0.661 -0.659 -0.025 -0.027 -0.002 30.0 -0.548 -0.514 -0.509 -0.034 -0.039 -0.005 35.0 -0.305 -0.275 -0.265 -0.030 -0.040 -0.010 40.0 0.026 0.057 0.073 -0.031 -0.047 -0.016 45.0 0.466 0.483 0.506 -0.017 -0.040 -0.023 50.0 0.978 1.001 1.032 -0.023 -0.054 -0.031 55.0 1.579 1.613 1.652 -0.034 -0.073 -0.039 -60.0 2.282 2.319 2.366 -0.037 -0.084 -0.047 65.0 3.079 3.118 3.174 -0.039 -0.095 -0.056 65.97 3.251 3.285 3.343 -0.034 -0.092 -0.058 70.0 3.975 4.012 4.076 -0.037 -0.101 -0.065 75.0 4.964 5.000 5.073 -0.036 -0.109 -0.073 80.0 6.049 6.083 6.163 -0.034 -0.114 -0.030 85.0 7.231 7.261 7.347 -0.030 -0.116 -0.086 90.0 8.499 8.534 8.625 -0.035 -0.126 -0.091 95.0 9.873 9.904 9.997 -0.031 -0.124 -0.094 100.0 11.342 11.370 11.464 -0.028 -0.122 -0.094 105.0 12.903 12.933 13.024 -0.030 -0.121 -0.091 110.0 14.567 14.593 14.678 -0.026 -0.111 -0.085 115.0 16.323 16.352 16.426 -0.029 -0.103 -0.074 120.0 18.192 18.209 18.268 -0.017 -0.076 -0.059 125.0 20.145 20.166 20.205 -0.021 -0.060 -0.039 130.0 22.200 22.223 22.235 -0.023 -0.035 -0.012 131.95 23.024 23.052 23.052 -0.028 -0.028 0.000 - 52 -T a b l e V. ( c o n t i n u e d ) DEFLECTION AT MIDSPAN AS A PERCENT OF THE SPAN DEFLECTION IN PERCENT DIFFERENCES IN PERCENT EXPERIM. CATENARY PARABOLA EXP-CAT EXP-PAR CAT-PAR C 6..2 7.1 6.246 6.202 0.026 0.070 0.044 TENSIONS AT THE SUPPORTS TENSIONS IN NEWTONS DIFFERENCES IN NEWTONS EXPERIM. CATENARY. PARABOLA i EXP-CAT iEXP-PAR , CAT-PAR UPPER SUP. 3080 3080 3080 0 0 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 ! EXPERIM'. ' .CATENARY' • PARABOLA' EXP-CAT' EXP-PAR:. !'GAT-PAR 135.140 135.252 135.232 -0.017 -0.003 0.020 - 5 3. -The comparison of the re s u l t s consists i n the eva-luation of the differences between the measured and c a l c u l -ated values for each of the variables. As an example, the resu l t s of the computation for 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-positions of the cable i s also presented graphically i n Figure 9 where the differences i n the Y-positions of the cable are plotted for the entire span length. Apparently none of the theories seem to agree c l o s e l y with the f i e l d r e s u l t s since the maximum difference for each of the model i s much greater than the maximum expected error (18 mm) i n the cable positions. However no conclusion can be drawn before the influence of the error i n tension i s examined. 4 .3 Error Analysis. The expected experimental errors a f f e c t i n g the measured variables and parameters are summarized i n Table VI. Table VI. Experimental errors a f f e c t i n g the measured variables. Designation Nomenclature Experimental error Parameter TB ± 1% Variables Y TA a A , a B S ± 18 mm ± 1% ± 0.2 degree ± 37 mm - 54 -The errors on the parameter T B are a f f e c t i n g the calculated values of the variables. Figure 10 and Figure 11 show the same curves as Figure 9 for the catenary model and the parabolic model using values of T one percent greater and one percent' smaller than the measured value. I t can be seen that one percent error i n the tension measurement has to be considered i n the analysis. The Y-position of the cable at 100 metres from the lower support i s used as an example to explain the treatment of the experimental errors used, i n the analysis, for Y and a l l the other variables. The error i n the parameter T_, allows the calculated p o s i t i o n of the cable Y to vary between Y+e^, and Y~e2 a s shown i n Figures 10 and 11 for the two models. If the seg-ment AB overlaps the error zone the theory agrees with the experiment within the margin of experimental errors. In the examples shown i n Figures 10 and 11 the catenary model agrees with the experiment for the Y-position of the cable at 100 metres from the lower support and the parabola does not agree with the experiment. A more convenient.way to repre-sent the same condition i s to enlarge the error zone as shown i n Figure 12 where the new boundaries D and E are defined by the addition of e^ and t o t n e previous error l i m i t s . The theory agrees with the experiment i f point C - 55 -Figure 10 — Differences i n the Y-positipn of the cable between experiment and catenary model versus X-position on span, f o r free hanging test number 4. The three graphs are drawn: using the measured tension Tg at the upper support as a parameter; using Tg one percent greater than the measured value; and using T R one percent smaller. Figure 11 - Differences i n the Y-position of the cable between experiment and parabolic model versus X-position on the span, f o r free hanging te s t number 4. The three graphs are drawn: using the measured tension Tg at the upper support as a parameter; using Tg one percent greater than the measured value; and using T R one percent smaller. - 56 -- 57 -(or C ) f a l l s inside the new error zone. This procedure was used for the analysis of a l l the variables for a l l the tes t s . The error-zone i s defined by dash-lines i n the d i f f e r e n t figures throughout the thesis. The analysis has shown that the values e^ and e.^ are almost i d e n t i c a l , showing that the system i s li n e a r for small var-iatio n s of the parameter. The values e^ and e'^ were also found i d e n t i c a l i n the analysis which allows to define a common error-zone for the catenary and parabola models. 4.4 Results and Conclusions. 4.4.1 Y-position of Points of the Cable -Experiment versus Models. As shown in Figure 12 for Test #4, the average values of the absolute differences between Y measured along the entire span and Y calculated with the catenary and parabolic models are respectively 27 mm and 70 mm. The maximum of those differences i s 39 mm for the catenary model and 126 mm for the parabolic model. The maximum half error-zone width i s 10 8 mm. Those c h a r a c t e r i s t i c values calculated for the 9 tests are shown i n Figure 13. The errors with the catenary model are generally smaller than that of the parabolic model. Except for Test #1 the average absolute differences are smaller than the maximum error. The maximum differences - 58 -Figure 12 — Differences i n the Y-position of the cable between experiment and catenary model, and between experiment and parabolic model versus X-positions on span, f o r free hang-ing test number 4. The graph shows the error zone taken i n account the error i n the parameter T R. Figure 13 — Average and maximum differences i n the Y-positions of points of the cable between experiment and catenary model, and between experiment and parabolic model f o r the nine free hanging t e s t s . Maximum error zone width f o r each t e s t . - 60 -points f a l l inside the error boundaries for 50% of the t e s t for both the catenary and the parabolic model. The question as to whether the cable hangs closer to a catenary or a parabolic shape cannot be answered c l e a r l y at t h i s point because of the dependence of the analysis on the error i n the tension T^. A d i f f e r e n t approach w i l l now be used to investigate the shape of the free hanging cable. 4.4.2 Cable Shape: Catenary or Parabola. The following approach consists of finding the catenary curve and the parabolic curve that best f i t the experimental p o s i t i o n measurements of the points of the free hanging cable. For example, the catenary curve that best f i t s the experiment i s the curve for which the sum of the squares of the discrepancies between the measured Y-positions of the cable and that defined by the equation of the curve i s minimum. T_.c representing the tension at the upper support was used as the parameter and the problem was then to f i n d what tensions at the upper support would give the best agreement between the experimental shape of the cable and a catenary shape, and between the experimental shape and a parabolic shape. The search for the optimum T_,c for a l l cases was implemented using a binary chop technique based on Fibonacci golden sections. The agreement between the exper-imental shape and the b e s t - f i t curve i s characterized by the average absolute value of the discrepancies between the two. - 61 -Figure 14 — Average discrepancies between the measured Y-positions of points of the cable and that predicted by the b e s t - f i t catenary curve and by the b e s t - f i t parabolic curve, f o r the nine free hanging t e s t s . Figure 15 — Percent difference between Tg, measured tension at the upper support, and Tg^ parameter calculated f o r the b e s t - f i t catenary curve and the b e s t - f i t parabolic curve, f o r the nine free hanging t e s t s . -" 63 -T h e a v e r a g e d i s c r e p a n c i e s f o r t h e b e s t - f i t c a t e -n a r y a n d f o r t h e b e s t - f i t p a r a b o l a a r e s h o w n i n F i g u r e 1 4 f o r a l l t h e t e s t s . F o r a l l 9 t e s t s t h e a v e r a g e d i s c r e p a n c y i s l e s s f o r t h e c a t e n a r y s h a p e . W h e r e a s t h e r e s u l t s f o r t h e b e s t - f i t p a r a b o l a d i v e r g e f r o m t h e e r r o r z o n e , t h e r e s u l t s f o r t h e b e s t - f i t c a t e n a r y a r e a l w a y s s m a l l e r t h a n 18 mm. F o r a l l n i n e t e s t s t h e s h a p e o f t h e f r e e h a n g i n g c a b l e i s b e s t r e p r e s e n t e d b y a c a t e n a r y c u r v e t h a n b y a p a r a b o l i c c u r v e . T h i s i s p a r t i c u l a r l y s o f o r t e s t s n u m b e r 3 , 4 , 5 w h e r e t h e d e f l e c t i o n w a s t h e l a r g e s t ( s e e A p p e n d i x 4 ) . I t i s i n t e r e s t i n g t o c o m p a r e t h e v a l u e s o f t h e o p t i m u m T _ , c , c o m p u t e d f o r b o t h t h e c a t e n a r y a n d t h e p a r a -b o l a , a n d t h e a c t u a l m e a s u r e d t e n s i o n T g . T h e d i f f e r e n c e s b e t w e e n m e a s u r e d a n d c o m p u t e d t e n s i o n s e x p r e s s e d a s a p e r -c e n t o f t h e m e a s u r e d t e n s i o n a r e p l o t t e d o n F i g u r e 1 5 f o r a l l t h e t e s t s . A s e x p e c t e d , t h e t e n s i o n s c o m p u t e d f o r t h e p a r a b o l i c c u r v e a r e s m a l l e r t h a n t h a t f o r t h e c a t e n a r y c u r v e . E x c e p t f o r T e s t # 1 , t h e d i f f e r e n c e s b e t w e e n t h e c o m p u t e d a n d t h e m e a s u r e d t e n s i o n a r e s u f f i c i e n t l y c l o s e t o t h e 1% e x p e c t e d e r r o r i n t h e t e n s i o n t o g i v e c o n f i d e n c e i n b o t h t h e m o d e l s a n d t h e e x p e r i m e n t . - 64 -Figure 16 - Differences i n the Y-Positions of the cable between catenary and parabolic models versus X-positions on span, for free hanging Test #4. Figure 17 - Sketches of the free hanging cable shapes derived from the catenary and parabolic models. - 66 -Figure IS — Differences i n Dm, de f l e c t i o n at mid-span between the catenary model and the parabolic model versus Tg, tension at the upper support, and versus Dm mid-span de f l e c t i o n i n the free hanging cable. 4.4.3 Y-position of Points of the Cable. Catenary Model versus Parabolic Model. A t y p i c a l graph of the differences between the two models, obtained with.the conditions of.Test #4, i s shown i n Figure 16. The parabola i s s l i g h t l y under the catenary at the lower end but i s p l a i n l y above towards.the upper end. This point i s i l l u s t r a t e d on the sketch of the cable shapes shown i n Figure 17. In the example, the parabola i s 60 mm above the catenary at mid^span. The difference at mid-span between the two theories i s plotted i n Figure 18 versus values of the tension at the upper support TD.. The d i f f e r - ' ences decrease rapidly when the tension i n -the l i n e increases ( i . e . when the percent d e f l e c t i o n at mid-span decreases). o o o E Cvl E ro Ci E o •M O ro o u <— E o ' I - i h 5 6 T R , H X 1000 10 8 4 3 Def lect ion at mid-span, % 10 - 68 -I f t h e d e f l e c t i o n i s l e s s t h a n 3 % t h e p a r a b o l a i s a b o v e t h e c a t e n a r y b y o n l y 2 mm a t m i d - s p a n . 4.4.4 T e n s i o n s . E x p e r i m e n t v e r s u s M o d e l s . T h e m e a s u r e d t e n s i o n a t t h e u p p e r s u p p o r t , T n , w a s t a k e n a s a p a r a m e t e r i n t h e a n a l y s i s , t h e r e f o r e n o d i s c r e p a n c y i n t h e t e n s i o n a t t h a t p o i n t c a n b e s h o w n . T h e t e n s i o n T ^ a t t h e l o w e r s u p p o r t w a s e v a l u a t e d w i t h b o t h m o d e l s a n d c o m p a r e d t o t h e m e a s u r e d v a l u e . T h e r e s u l t s s h o w n i n F i g u r e 1 9 s h o w a v e r y g o o d a g r e e m e n t b e t w e e n e x p e r -i m e n t a n d b o t h t h e o r i e s f o r a l l t h e t e s t s e x c e p t # 1 f o r w h i c h a n e r r o n e o u s r e a d i n g w a s p r o b a b l y t a k e n . T h e m a x i m u m e r r o r z o n e h a l f - w i d t h i s v e r y s l i g h t l y l a r g e r t h a n 2 % , h o w e v e r m o s t o f t h e p o i n t s f a l l w i t h i n o n e p e r c e n t d i f f e r -e n c e f r o m t h e m e a s u r e m e n t s s h o w i n g t h a t t h e d y n a m o m e t e r s m i g h t b e m o r e a c c u r a t e t h a n e x p e c t e d . 4.4.5 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 d i f f e r e n c e s b e t w e e n t h e t e n s i o n s a t t h e l o w e r s u p p o r t c o m p u t e d w i t h c a t e n a r y m o d e l a n d w i t h t h e p a r a b o l i c m o d e l a r e p l o t t e d o n F i g u r e 2 0 v e r s u s t h e t e n s i o n a t t h e u p p e r s u p p o r t . T h e t e n s i o n o b t a i n e d w i t h t h e p a r a b o l i c m o d e l i s a l w a y s l a r g e r t h a n t h a t w i t h t h e c a t e n a r y m o d e l . A t 1 0 % d e f l e c t i o n o f t h e c a b l e a t m i d - s p a n t h e d i s c r e p a n c y b e t w e e n t h e t w o t h e o r i e s i s a b o u t 1 % a n d d e c r e a s e s r a p i d l y t o . 0 3 % - 69 -Figure 19 - Differences in T A, tension at the lower support, between experiment and catenary-model, and between experiment and parabolic model, for the nine free hanging tests. Figure 20 — Differences in 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 deflection in the free hanging cable. - 7 0 . -^-Catenary •fParabol ic Mid-span d e f l e c t i o n , % - 71 -for a de f l e c t i o n at mid-span of 3%. The same analysis carried for the horizontal ten^ sion i n the l i n e y i e l d s the same conclusions as for the tension at the lower support. 4.4.6 Angles of the Gable with the Horizontal. Experiment versus Models. The historigrams of the differences between the measured angles and the calculated angles are shown i n Figures 21 and 2 2 for both model and both end of the cable. 85% of the points are within the error zone at the upper support and only 45% at the lower support. An explanation of t h i s disagreement i s found considering the radius of the sheaves at the upper and lower support not taken into con-sideration in the t h e o r e t i c a l models. 4.4.7 Angles of the Cable with the Horizontal. Catenary Model versus Parabolic Model. The discrepancy between the angles of the l i n e at the upper support calculated with both model i s plotted i n Figure 23 versus tension at the upper support. The parabola i s always above the catenary at the upper support but the difference gets very small as the tension increases. The equivalent curve i s shown i n Figure 2 4 for the angle at the lower support. For large d e f l e c t i o n at mid-span ( i . e . i f the - 7 2 -Figure 21 — Differences in « B, angle of the line with the horizontal at the upper support, between experiment and catenary model, and between experiment and parabolic model, for the nine free hanging tests. Figure 22 — Differences in o^ , angle of the line with the horizontal at the lower support, between experiment and catenary model, and between experiment and parabolic model, for the nine free hanging tests. - 74 -Figure 23 — Differences i n dg, angle of the l i n e with the horizontal at the upper support, 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. Sketch of the r e l a t i v e p o s i t i o n of the catenary and parabola. Figure 24 — Differences i n a A , angle of the l i n e with the horizontal at the lower support, 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. Sketches of the r e l a t i v e positions of the catenary and parabola. -,7 5 - 76 -t e n s i o n i s l e s s t h a n 2750 newtons) th e p a r a b o l a i s above t h e c a t e n a r y a l o n g the e n t i r e span. For d e f l e c t i o n l e s s t h a n 7% the p a r a b o l a i s under t h e c a t e n a r y a t t h e lower s u p p o r t and t h e two c u r v e s i n t e r c e p t a l o n g t h e span. 7% i s not t h e de-f l e c t i o n a t w h i c h t h e t a n g e n t t o t h e c a b l e a t t h e lower s u p p o r t i s h o r i z o n t a l . T h i s s i t u a t i o n o c c u r s when t h e mid-span d e f l e c t i o n i s 4.3%. The d i s c r e p a n c i e s between t h e a n g l e s o f t h e l i n e s as computed w i t h t h e c a t e n a r y model and w i t h t h e p a r a b o l i c model can 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 mid-span d e f l e c -t i o n s l e s s t h a n 6%. 4.4.8 C a b l e Length: E x p e r i m e n t v e r s u s Models. The l e n g t h measurement gave t h e t o t a l u n s t r e t c h e d l e n g t h o f t h e c a b l e . To o b t a i n a v a l u e comparable w i t h t h e 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 t o be a p p l i e d t o t h e f i e l d r e s u l t s . The c o r r e c t i o n i s d e f i n e d by the a p p r o ximate r e l a t i o n s h i p : S = Sm (1 + g | ) S e l o n g a t e d l e n g t h Sm measured u n s t r e t c h e d l e n g t h T_, t e n s i o n i n t h e l i n e a t t h e upper s u p p o r t E e l a s t i c modulus o f t h e c a b l e A c r o s s - s e c t i o n a r e a - 7 7 -Figure 25 — Differences in S, cable length, between experiment and catenary model, and between experiment and parabolic model, for the nine free hanging tests. Figure 26 — Difference in S, cable length, between catenary model and parabolic model, versus Tg, tension at the upper support, and versus Dm, mid-span deflection in the free hanging cable. - 79 -For t h e s k y l i n e , S i s g i v e n by: 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 between t h e measured e l o n g a t e d l e n g t h and t h e computed t h e o r e t i c a l l e n g t h a r e shown i n F i g u r e 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 i g n o r e d , t h e r e s u l t s demonstrate t h e v a l i d i t y o f b o t h t h e c a t e n a r y 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 and a l s o c o n f i r m t h e v a l u e chosen f o r t h e e l a s t i c modulus o f t h e -c a b l e . 4.4.9 C a b l e Length:, C a t e n a r y Model v e r s u s P a r a b o l i c Model. The d i f f e r e n c e s i n t h e c a b l e l e n g t h s computed w i t h b o t h models:are p l o t t e d i n F i g u r e 26.versus t e n s i o n a t t h e upper s u p p o r t . 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 t h e c a t e n a r y model and t h e p a r a b o l i c model i s noted as t h e t e n s i o n i n c r e a s e s . 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 by 180 mm a t 10% mid-span def 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 i f t h e d e f l e c t i o n a t mid-span i s l e s s t h a n 3%. - 80 -CHAPTER 5 CLAMPED LOAD ON A SINGLE L INE The following experiment was designed to study the ef f e c t of a concentrated v e r t i c a l load clamped at a known distance along the span of a single l i n e . This configura-t i o n occurs i n actual yarding systems when the carriage i s equipped with a skyline stop or when a carriage bumper i s clamped on the skyline. The same s i t u a t i o n i s also found when the chokers are attached d i r e c t l y on the l i n e as i n highlead. 5.1 Description of the Experiment. The 5/8-inch skyline was used for t h i s t e s t . The carriage and lead weights constituted the v e r t i c a l load and a small clamp that f i t the 5/8-inch skyline was manufactured to stop the carriage from r o l l i n g . A convenient way to execute the experiment was to proceed as follows: i) choose a carriage p o s i t i o n at about 1/8 of the cable span s t a r t i n g from the upper support; i i ) choose a load from 100 Kg to 500 Kg by steps of 100 Kg; i i i ) increase the tension i n the skyline at the upper support from the minimum possible to about 30,000 N i n 4 to 6 steps; - 81 -iv) lower the carriage and change the load. I f the entire range of load values has been investigated change the carriage position. While t h i s procedure implies 175 d i f f e r e n t t e s t s , only 29 were ac t u a l l y done and the r e s u l t s of these are tab-ulated i n Appendix 4. 5.2 Analysis of the Results. 5.2.1 Pre-considerations. A skyline with a single concentrated load i s shown in Figure 27 for three d i f f e r e n t positions of the load. Such a system can be considered to be comprised of two cable segments f r e e l y hanging between the load and.the two supports. For the three positions presented the t o t a l d e f l e c t i o n at the load i s about 7 percent but the d e f l e c t i o n at the mid-point i n each of the free hanging segments i s less than 2.5 percent. Therefore using the r e s u l t s of the analysis of the free hanging cable presented i n Chapter 4, the following conclusions can be drawn for the expected differences bet-ween experiment catenary and parabola for the two segments c h a r a c t e r i s t i c s . - 82' " Table VII. Expected discrepancies i n the c h a r a c t e r i s t i c s of the free hanging segments. Differences between experiment and theories Differences between catenary and parabola less than Tension .02% Deflection at mid-point Within the margin 33 mm Angles of the lines with the horizontal of experimental errors .04° Cable length 1 mm Table VII shows that no detectable differences are introduced on the i n d i v i d u a l l i n e segments by the two theories; thus the study can be r e s t r i c t e d only to the analysis of the load positions for d i f f e r e n t loads and tensions and a detailed analysis of each l i n e segment i s unnecessary. 5.2.2 Method of Analysis. Five elements, shown i n Figure 27, are taken into consideration i n the analysis: - R load weight - T_, tension i n the skyline at the upper support - 8 3 -- T t e n s i o n i n t h e s k y l i n e a t t h e l o w e r A s u p p o r t - X h o r i z o n t a l p o s i t i o n o f t h e l o a d - Y v e r t i c a l p o s i t i o n o f t h e l o a d . T h e s y s t e m i s c o m p l e t e l y d e f i n e d b y a n y t h r e e o f t h e s e p a r a m e t e r s . T h e u s u a l w a y o f p r o c e e d i n g i s t o t a k e R, T_. a n d X a s p a r a m e t e r s a n d s o l v e f o r t h e v a r i a b l e s Y a n d T . T h i s p r o c e d u r e w a s a d o p t e d a n d t h e a n a l y s i s c a n b e s u m m a r i z e d a s f o l l o w s : Measured Parameters R, T_ and X B Measured Variables Y and A Computation using cate-nary and parabola theories Theoretical values for the variables Yc and T.c A Conclusion 5 . 2 . 3 E r r o r A n a l y s i s . T h e e x p e r i m e n t a l e r r o r s a f f e c t i n g t h e m e a s u r e d v a r -i a b l e s a n d p a r a m e t e r s a r e r e c a l l e d i n T a b l e V I I I . - 84 -T a b l e V I I I . E x p e c t e d e r r o r s i n t h e m e a s u r e d v a r i a b l e s a n d p a r a m e t e r s o f t h e c l a m p e d l o a d o n a s i n g l e l i n e s y s t e m . D e s i g n a t i o n N o m e n c l a t u r e E x p e r i m e n t a l e r r o r P a r a m e t e r s R T B X 1% 1% N e g l e c t e d V a r i a b l e s Y T A 18 mm 1% T h e t r e a t m e n t o f t h e e x p e r i m e n t a l e r r o r s i s d o n e t h e s a m e w a y a s f o r t h e f r e e h a n g i n g c a b l e . T h e e r r o r z o n e s f o r t h e m e a s u r e d v a r i a b l e s Y a n d T ^ a r e e n l a r g e d b y t h e v a r -i a t i o n s i n t h e c a l c u l a t e d v a l u e s o f Y a n d T ^ r e s u l t i n g f r o m t h e e r r o r s i n t h e p a r a m e t e r s R a n d T . T h e u p p e r l i m i t f o r t h e c a l c u l a t e d v a r i a b l e s i s o b t a i n e d w i t h t h e l a r g e s t t e n -s i o n a t t h e u p p e r s u p p o r t (1.01 T _ ) c o m b i n e d w i t h t h e s m a l l -13 e s t l o a d (.99 R ) . T h e l o w e r l i m i t i s o b t a i n e d w i t h t h e o p p o s i t e e x t r e m e s . T h e e r r o r z o n e i s d e f i n e d b y t h o s e e x t r e m e c a s e s . I t w a s f o u n d t h a t s m a l l 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 h a d a c o n s i d e r a b l e i n f l u e n c e o n t h e r e s u l t s . - 85 -Figure 27 — Skyline with a single concentrated load for three different positions of the clamped load. Figure 28 - Differences in Y-position of the load, between experiment and catenary and para-bolic models, for the 29 clamped load on a single line tests. - 87 -5.3 Results and Conclusions. 5.3.1 Y-position of the Load: Experiment versus Models. The differences between the measured Y and the Y calculated with the catenary and parabolic models are plotted on Figure 28 for the 29 tests. The difference between the models i s so small that i t does not appear on t h i s graph. Figure 28 shows complete agreement between the experiment and the models for a l l the tests but one. 5.3.2 Y-position of the Load - Catenary Model versus Parabolic Model. The discrepancies between the two models w i l l be maximum when the de f l e c t i o n at mid-point i n the l i n e segments i s maximum. This s i t u a t i o n occurs for the smallest load (R = 100 Kg) and the minimum tension (T_, = 5886N) that gives clearance for the complete load path. The differences, i n those conditions, between the two theories are plotted on Figure 29 versus horizontal positions of the load along the entire span. The parabolic model over-estimates the d e f l e c t -ion of the load r e l a t i v e to the catenary model i n the lower part of the load-path and under-estimates i t towards the upper end. This point i s i l l u s t r a t e d on a sketch of the load paths shown i n Figure 30. The maximum difference i s 15 mm. - 88 -Figure 29 — Differences in the Y-positi;bns of the load, between catenary and parabolic models versus X-positions on the span -Tg tension at the upper support = 5885 N R load = 100 Kg. Figure 30 — Sketches of the catenary and parabolic model load paths for the clamped load on a single l i n e . - 89 -- 90 -Figure 31 - a) Force balance at the l e v e l i n a clamped load on a single l i n e system, using catenary and parabolic models. b) Unbalanced forces at the load l e v e l i n a clamped load on a single l i n e , using the parabolic model and a Y-position of the load higher than that at equilibrium. - 91 -I t would 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 . w e i g h t s a r e u n d e r - e s t i m a t e d w i t h t h e p a r a b o l a , t h i s t h e o r y s h o u l d y i e l d a d e f l e c t i o n s m a l l e r than t h e c a t e n a r y i n a l l c a s e s . However, 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 f a l s e f o r a l a r g e p r o p o r t i o n o f t h e span. T h i s c o n t r a d i c -t i o n i s i n v e s t i g a t e d i n F i g u r e 31. F i g u r e 31a r e p r e s e n t s t h e f o r c e s a t t h e l o a d f o r t h e c a t e n a r y and t h e p a r a b o l a a t e q u i l i b r i u m . F o r t h e l o a d p o s i t i o n chosen (X = 65 m) t h e c a t e n a r y i s 107 mm above t h e p a r a b o l a . I n F i g u r e 31b t h e p a r a b o l a has been a r t i f i c i a l l y s e t a t t h e same l e v e l as t h e c a t e n a r y ; 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 t h a n t h e l o a d and t h e system i s f o r c e d t o sag more because o f t h e unbalan c e d f o r c e . a) b) X -pos i t i on of the load on the span : 6 5 m R weight of the load : 100 Kg T R tension at the upper support : 5 8 8 6 N - 92 -5.3.3 tension at the lower support: Experiment versus Models. The difference between the measured T and the tension calculated with the catenary and parabolic models are plotted on Figure 32 for the 29 tests. The difference between catenary and parabola i s small and does not appear on the graph. The results are between the upper and lower error boundaries for a l l the tests but one. Therefore the theories are considered to agree with the actual measure-ments . 5.3.4 Tension at the lower support - Catenary Model versus Parabolic Model. The discrepancies between the two theories are plotted on Figure 33 for positions of the load along the entire span. The tension at the lower support computed with the parabolic model are larger than that with the catenary model for any position of the load. The difference i s however always very small and less than 3N (.05%) for the study case. The agreement between the two theories i s best when the load i s i n the mid-span area. - 93 -Figure 32 - Differences i n T A, tension at the lower support, 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 . Figure 33 - Differences i n T A, tension at the lower support, between catenary and parabolic model versus X-positions on the span. Tg tension at the upper support = 586*5 N R load = 100 Kg. - 94 -CHAPTER 6 GRAVITY SYSTEM The e x p e r i m e n t d e s c r i b e d i n t h i s Chapter t e s t e d the v e r y s i m p l e and commonly used g r a v i t y system. I n t h e 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 t h e s k y l i n e i s p u l l e d t o t h e upper s u p p o r t w i t h t h e m a i n l i n e and r e t u r n s t o i t s p o i n t o f maximum r e a c h by g r a v i t y . 6.1 D e s c r i p t i o n o f t h e E x p e r i m e n t . The s k y l i n e was t e n s i o n e d w i t h the 6-ton Comelong and t h e m a i n l i n e was s t o r e d on t h e G e a r m a t i c 19. The un-s t r e t c h e d l e n g t h o f t h e s k y l i n e c o u l d be k e p t c o n s t a n t t o s i m i l a t e a f i x e d s k y l i n e g r a v i t y system o r t h e t e n s i o n i n t h e s k y l i n e c o u l d be m o n i t o r e d as i n a l i v e s k y l i n e g r a v i t y system. The t e s t s were t h e r e f o r e c a r r i e d o u t as f o l l o w s : i ) S et t h e l e n g t h (or t e n s i o n ) o f t h e s k y l i n e a t t h e upper s u p p o r t , i i ) Choose a l o a d from 100 Kg t o 500 Kg by s t e p s o f 100 Kg. i i i ) L e t t h e c a r r i a g e r u n down as f a r as i t can go f r e e l y under i t s w e i g h t . The s t a r t i n g p o s i t i o n i s t h u s d e t e r m i n e d , i v ) P u l l t h e c a r r i a g e a l o n g th e s k y l i n e i n s t e p s of about 1/8 o f the span l e n g t h . The r e s u l t s o f t h e t e s t s a r e r e p o r t e d i n Appendix 4. - 96 -6.2 Analysis of the Results. 6.2.1 Pre-considerations. The system can be (very s i m i l a r l y to the clamped load on a single l i n e studied i n Chapter 5) considered as three single l i n e segments f r e e l y hanging between the car-riage and each of the supports. The skyline segments are very t i g h t , with d e f l e c t i o n at mid-span less than 2.5%, i n most of the cases permitted by the l i m i t s of the study. As concluded i n the analysis of the clamped load on a single l i n e , undetectable differences w i l l be introduced by the parabolic and catenary models on the c h a r a c t e r i s t i c s of the skyline free hanging segments. The tension i n the mainline i s usually large enough to keep the mainline segment t i g h t when the carriage approaches the upper support; however as the carriage runs out the tension i n the mainline decreases and i s very small when the carriage eventually stops. The defl e c t i o n i n the mainline can then be large enough to re s u l t i n noticeable discrepancies between the two theories. The e f f e c t of those discrepancies on the carriage p o s i t i o n and on the tensions i s investigated i n the analysis. 6.2.2 Method of Analysis. Six c h a r a c t e r i s t i c s of the system are taken into consideration i n the analysis: - R carriage and load weight - T D tension i n the skyline at the upper support tension i n the skyline at the lower support tension i n the mainline at the upper support horizontal p o s i t i o n of the carriage v e r t i c a l p o sition of the carriage Any combination of three of these c h a r a c t e r i s t i c s that does not include the pair (T g, T ) defines the system completely. R,X,TD were taken as parameters, i n the anal-y s i s , to determine the variables T_, T D O and Y using the catenary and parabolic formulations. A chart of the anal-y s i s procedure i s given as follows: " TA " T B 3 - X - Y Measured Parameters R,X,T0 Measured Variables T r T 3, Y Computation using cate-nary and parabolic theories T h e o r e t i c a l values for the v a r i a b l e s Y Conclusion - 98 -6.2.3 E r r o r A n a l y s i s . The c o m p u t a t i o n o f t h e e x p e r i m e n t a l e r r o r s and t h e d e t e r m i n a t i o n o f the e r r o r - z o n e b o u n d a r i e s were done e x a c t l y as d e s c r i b e d i n Chapter 5 f o r t h e clamped l o a d on a s i n g l e l i n 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 and 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 o f the v a r i a b l e s was found. 6.3 R e s u l t s and C o n c l u s i o n s . 6.3.1 Y - p o s i t i o n o f t h e C a r r i a g e - Experiment v e r s u s Models. The d i f f e r e n c e s between the measured Y and t h e Y c a l c u l a t e d w i t h t h e c a t e n a r y and p a r a b o l a a r e p l o t t e d i n F i g u r e 34. For 13 c a r r i a g e p o s i t i o n s from t h e lower t o t h e upper s u p p o r t . 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 g i v e n t e n s i o n and t h e m a i n l i n e touched t h e ground f o r t h e f i r s t 6 t e s t s . The t h e o r i e s agree w i t h t h e exp e r i m e n t w i t h i n t h e margin o f e x p e r i m e n t a l e r r o r f o r the l a s t 2/3 o f t h e s e r i e s . The f a c t t h a t t h e t h e o r i e s d i v e r g e g r e a t l y from t h e e r r o r b o u n d a r i e s f o r t h e f i r s t 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 assumptions 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 . The shape o f t h e m a i n l i n e as c a l c u l a t e d by t h e p a r a b o l a and c a t e n a r y i s p u r e l y - 99 -F i g u r e 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 ex 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 models, f o r t h e 13 g r a v i t y system t e s t s . F i g u r e 35 - M a i n l i n e shapes as 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 system 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 as 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 from t h e ground f o r t e s t 7 c o m p a t i b l y w i t h t h e a c t u a l s i t u a t i o n i n t h e f i e l d d u r i n g t h e t e s t . A nother d e d u c t i o n from F i g u r e 34 i s t h a t t h e p a r a b o l a does not d i v e r g e from the t e s t s as much as t h e c a t e n a r y does. T h i s can 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 o f t h e system. As t h e c a r r i a g e approaches t h e lower s u p p o r t t h e h o r i z o n t a l t e n s i o n i n t h e m a i n l i n e d e c r e a s e s b u t t h e 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 i s l i n e a t the c a r r i a g e l e v e l i n c r e a s e s because o f t h e i n c r e a s e o f the a n g l e o f t h e l i n e w i t h t h e h o r i z o n t a l . T h i s v e r t i c a l component ad d i n g t o t h e w e i g h t o f t h e l o a d causes the system t o sag more. I n t h e f i e l d t h e v e r t i c a l component o f t h e m a i n l i n e t e n s i o n s t o p s i n c r e a s i n g when t h e m a i n l i n e t o uches the ground. W i t h t h e c a t e n a r y f o r m u l a t i o n t h e 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 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 as a r e s u l t o f t h e c o m b i n a t i o n o f t h e l i n e a n g l e and l i n e w e i g h t . For t h e p a r a b o l i c model, t h e t o t a l w e i g h t o f t h e 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 and t h e r e f o r e remains c o n s t a n t as the l i n e sags more and c o n s e q u e n t l y t h e 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 converges towards a f i n i t e v a l u e . As a r e s u l t the p a r a b o l i c model r e p r e s e n t s r e a l i t y b e t t e r . - 102 -6.3.2 Y - p o s i t i o n o f t h e C a r r i a g e - Catenary-Model v e r s u s P a r a b o l i c Model. S i m i l a r l y t o the a n a l y s i s o f the clamped l o a d on a s i n g l e l i n e i n Chapter 5, the comparison was done w i t h t h e c o n d i t i o n s o f l o a d (R = 100 Kg) and t e n s i o n a t t h e upper s u p p o r t (T-£ = 6867N) t h a t r e s u l t s i n t h e l a r g e s t d i s a g r e e -ment between t h e two t h e o r i e s . The r e s u l t s o f t h i s a n a l y s i s shows d i s c r e p a n c i e s t h a t range as h i g h as s e v e r a l metres i n t h e a r e a o f maximum reach, and as shown i n F i g u r e 36 up t o 10 mm f o r t h e l a s t s t a t i o n s o f t h e l o a d p a t h . The p a r a b o l a o v e r - e s t i m a t e s t h e d e f l e c t i o n a t t h e c a r r i a g e f o r a s h o r t s e c t i o n o f t h e l o a d p a t h . T h i s phenomenon r e s u l t i n g from the a n g l e s o f t h e l i n e s w i t h t h e h o r i z o n t a l a t t h e c a r r i a g e l e v e l was e x p l a i n e d i n C h apter 5. F i g u r e 37 s k e t c h e s th e two l o a d p a t h s . 6.3.3 T e n s i o n i n the S k y l i n e a t t h e Lower Su p p o r t : Experiment v e r s u s Models. The d i f f e r e n c e s between the measured T, and t h e A t e n s i o n c a l c u l a t e d w i t h t h e c a t e n a r y and t h e p a r a b o l a a r e shown i n F i g u r e 38 f o r t h e 13 t e s t s . The d i f f e r e n c e between c a t e n a r y and p a r a b o l i c models i s s m a l l and does not appear on the graph. The two t h e o r i e s agree w i t h t h e e x p e r i m e n t w i t h i n t h e l i m i t s o f e x p e r i m e n t a l e r r o r s . The d i v e r g e n c e n o t ed f o r t h e v e r t i c a l p o s i t i o n o f t h e c a r r i a g e i s not p r e s e n t f o r t h e t e n s i o n , T &. T h i s s h o u l d be e x p e c t e d con-- 103 -Figure 36 - Differences i n the Y-positions of the carriage, between catenary and parabolic models versus X-positions on the span. Tg tension at the upper support i n the skyline = 6867 N R Carriage plus load = 100 Kg. Figure 37 — Sketches of the catenary and parabolic models load paths f o r the gravity system. - 104 -O r - 105 -Figure 38 - Differences i n T^, tension at the lower support between experiment and catenary and parabolic models, f o r the 13 gravity system t e s t s . Figure 39 - Differences i n T^, tension at the lower support, between catenary and parabolic models versus X-positions of the carriage on the span. Tg tension i n the skyline at the upper support = 6867 N R Carriage plus load = 100 Kg. - 106 -X on the span, m - 107 -sidering the catenary theory for which the difference between and depends on to and E only. 6.3.4 Tension i n the Skyline at the Lower Support: Catenary Model versus Parabolic Model. The graph of the differences between the values of T A calculated with the two models for positions of the car-riage along the entire load path i s shown i n Figure 39. The catenary y i e l d s the largest values for the tensions for a l l the positions of the carriage. The maximum difference i s obtained at each end of the load path and i s 7 newtons or (.1%). The two theories agree almost p e r f e c t l y when the carriage i s at mid-span. 6.3.5 Tg^ Tension i n the Mainline at the Upper Support: Experiment versus Models. Since the tension was not recorded i n the mainline when the l i n e was touching the ground, Figure 40 shows the historigram of the differences between experiment and theor-ies for the l a s t 7 tests. Half of the tests diverge from the error boundaries; however, the tensiometer was u t i l i z e d to measure tensions less than 1000N for those tests and for th i s range of tension a one percent expected error i s too optim i s t i c . - 1 0 8 -F i g u r e 40 - D i f f e r e n c e s i n t e n s i o n i n t h e m a i n l i n e a t t h e u p p e r s u p p o r t , T B 3 , b e t w e e n e x p e r i m e n t a n d c a t e n a r y a n d p a r a b o l i c m o d e l s , f o r t h e 1 3 g r a v i t y s y s t e m t e s t s . F i g u r e 4 1 - D i f f e r e n c e s i n T 0 0 , t e n s i o n i n t h e m a i n l i n e a t t h e u p p e r s u p p o r t , b e t w e e n c a t e n a r y m o d e l a n d p a r a b o l i c m o d e l , v e r s u s X - p o s i t i o n s o f t h e c a r r i a g e o n t h e s p a n . T 0 t e n s i o n i n t h e s k y l i n e a t t h e u p p e r s u p p o r t = 6 8 6 7 N R C a r r i a g e p l u s l o a d = 1 0 0 K g . 109 -GO - n o -6.3.6 Tension i n the Mainline at the Upper Support: Catenary Model versus Parabolic Model. The p l o t of the differences for the tension i n the mainline calculated with both theories and shown i n Figure 41 demonstrates, as expected, that the disagreement between the two theories i s small for position of the carriage towards the upper support. The maximum value of the d i s -crepancy i s 8N at the point of maximum reach. CHAPTER 7 D Y N A M I C T E S T S T h e s t u d y o f t h e d y n a m i c b e h a v i o u r o f c a b l e s y s -t e m s w a s n o t o n e o f t h e p r i m e o b j e c t i v e s o f t h e t h e s i s . A f e w d y n a m i c t e s t s w e r e t r i e d a s a f i r s t a p p r o a c h t o s h o w t h e p o t e n t i a l o f t h e f i e l d m o d e l a n d e q u i p m e n t a n d o b t a i n a n i d e a o f t h e t y p e o f r e s u l t s t o b e e x p e c t e d . T h e e q u i p m e n t a n d t e s t s a r e p r e s e n t e d i n t h i s c h a p t e r . 7 . 1 E q u i p m e n t . T h e i n p u t a n d o u t p u t d e v i c e s f o r t h e t e n s i o m e t e r , o n l y , w e r e c h a n g e d . T h e l o a d - c e l l s w e r e e x c i t e d b y a 10 v o l t s D C . T h e o u t p u t w a s c o n n e c t e d t o a s t r i p - c h a r t r e c o r d -e r o f v a r i a b l e s p e e d a n d s e n s i t i v i t y ( P l a t e 9 ) . 7 . 2 T e s t s . V a r i o u s t y p e s o f t e s t s w e r e c a r r i e d o u t ; t w o a r e p r e s e n t e d h e r e : i ) V e r t i c a l o s c i l l a t i o n o f t h e l o a d i i ) I n s t a n t a n e o u s s t o p o f t h e c a r r i a g e r u n n i n g b y g r a v i t y . 7 . 2 . 1 V e r t i c a l O s c i l l a t i o n o f t h e L o a d . T h e p o s i t i o n o f t h e l o a d b e i n g s u r v e y e d b y t h e - 112 -Plate 9 — Strip-chart recorder, generator and trans-former-regulator used for the recording of the tensions at the upper support. Plate 10 — Manual i n i t i a t i o n of the vertical o s c i l l a -tory motion of the clamped load. - 1 1 3 -- 114 -u s u a l t e c h n i q u e t h e system i s b r o u g h t i n t o v e r t i c a l m o tion m a n u a l l y ( P l a t e 1 0 ) . When the t e n s i o n i n t h e s k y l i n e r e a c h e s about 150% o f i t s s t a t i c v a l u e the e x c i t a t i o n i s stopped and t h e t e n s i o n a t the upper s u p p o r t i n t h e s k y l i n e i s r e c o r d e d 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 v a l u e . A t y p i c a l c h a r t i s shown i n F i g u r e 42. C o n c l u s i o n o f t h e T e s t . The v i b r a t i o n r e c o r d e d f o r the t e s t g i v e n , as an example has t h e f o l l o w i n g c h a r a c t e r i s t i c s : i ) f r e q u e n c y .475 H e r t z ( p e r i o d 2.10 seconds) i i ) damping r a t i o .05. No major d i f f e r e n c e s were n o t i c e d whether the c a r r i a g e was clamped o r hooked t o t h e m a i n l i n e . 7.2.2 I n s t a n t a n e o u s Stop o f t h e C a r r i a g e Running by G r a v i t y . F o r 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 r u n from a known p o s i t i o n , t h e n stopped s u d d e n l y w i t h t h e m a i n l i n e o r w i t h a clamp on t h e s k y l i n e . The e f f e c t on the t e n s i o n s i s r e c o r d e d a t t h e upper s u p p o r t on b o t h the m a i n l i n e and t h e s k y l i n e . The f i r s t example was performed f o r the c o n d i t i o n s d e s c r i b e d on F i g u r e 43. The r e s u l t s a r e shown i n F i g u r e 45. - 115 -Figure 42 - Chart-recording of the tension in the skyline at the upper support during vertical o s c i l -lation of the load of the gravity system. Recorder: - input = 10 V - output = 20 mV - chartspeed = 10 sec/inch Carriage and load: - 495 Kg. - located at mid-span - deflection 6 .7$ - 117 -Figure 43 - Sketch of dynamic t e s t . Carriage stopped by the main l i n e . Starting p o s i t i o n of the carriage. X = 106.09 m Y = 11.OS m Figure 4 4 - Sketch of dynamic t e s t . Carriage stopped with a clamp on the skyline. S t a r t i n g p o s i t i o n of the carriage. X = 72.02 Y = 2.96 - 119 -Conclusion of the Test. Tension i n the skyline. Just after the shock, a drop to 60% of the o r i g -i n a l s t a t i c tension rapidly follows a small peak at 135%. The maximum tension reached i s 150% of the s t a t i c tension and a complex vi b r a t i o n phenomenon takes place i n the l i n e with a dominant low frequency of .67 Hertz (period 1.5 seconds). Tension i n the mainline. An increase of more than 700% from the o r i g i n a l s t a t i c tension i s recorded aft e r the shock. This snapping e f f e c t i s repeated p e r i o d i c a l l y every 1.5 seconds with a rap i d l y decreasing magnitude (about 300% for the t h i r d peak). Other vibrations of higher frequencies i n t e r f e r e with t h i s basic pattern. The second example was performed i n the conditions described i n Figure 44. The res u l t s are shown i n Figure 46. Conclusion of the te s t . The example i s presented i n p a r a l l e l with the re-s u l t obtained for a test of the f i r s t type. The basic - . 1 2 0 -Figure 45 - Chart-recordings of tensions i n the lines of a gravity system during dynamic tests. a) in the skyline b) in the mainline Carriage stopped with the mainline. Time, seconds - 122 -Figure 4 6 — Chart-recordings of tensions in the skyline at the upper support. a) carriage stopped with a clamp b) carriage stopped with the mainline - 123 -- 124 -d i f f e r e n c e i s the sharp peak o f 155% o f t h e s t a t i c t e n s i o n f o l l o w i n g t h e shock on t h e clamp. The s k y l i n e i s much more d i s t u r b e d i n t h i s second example and complex v i b r a t i o n phenomena o c c u r i n t h e system l o n g a f t e r t h e shock. 7.3 C o n c l u s i o n . Dynamic t e s t s were e a s i l y performed on t h e f i e l d model. The f i r s t r e s u l t s 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 can be d e v e l o p e d i n t h e l i n e s and demonstrate c l e a r l y 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 a r e a . - 125 -CHAPTER 8 D I S C U S S I O N A N D C O N C L U S I O N Both the res u l t s of the analysis and the p r a c t i c a l aspect of cable logging problems should be taken into con-sideration to propose an operational t h e o r e t i c a l model for cable logging systems. The catenary and the parabolic model were compared with an actual f i e l d t e s t for three t y p i c a l cable system configurations. The fact, shown i n Chapter 4, that the shape of a free hanging cable segments i s closer to a catenary than a parabola i s probably not a surprise to an engineer t r u s t -ing the basic laws of mechanics; the catenary model des-cribes a cable yarding system with better precision than the parabolic model does, but i s t h i s p r ecision needed for prac-t i c a l applications? Cable yarding systems do not operate i n i d e a l experim-ental conditions and much uncertainty i s attached i n the f i e l d to t h e i r various c h a r a c t e r i s t i c s . In the best of the cases the ground p r o f i l e i s known with a precision of about 300 mm (1 foot) . The uncertainty on the values of the oper-.j. ating tensions i s as large as 10% on the ex i s t i n g yarders even equipped with the most sophisticated tension control - 126 -systems and the inaccuracy i n the scaling of the logs can re s u l t i n an inp r e c i s i o n of 10% i n the payload value. Table IX compares these actual precisions with the precisions 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 with the differences between catenary and parabolic models. Table IX. Precision i n the knowledge of the c h a r a c t e r i s t i c s of: the f i e l d model a r e a l yarding system Discrepancies between catenary model and parabolic model for the same c h a r a c t e r i s t i c s . System C h a r a c t e r i s t i c s Nomen-clature F i e l d model p r e c i s i o n Real system p r e c i s i o n Discrepancies Cat. - Par. Ground p r o f i l e V e r t i c a l p o s i t i o n Y 18 mm 300 mm 15 mm Operating tension T 1% 10% less than 1% Payload R 1% 10% less than 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 accurate than a r e a l yarding 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 of the experimental approach. Both the catenary and parabolic model compute the system charac-t e r i s t i c s with a precision far beyond what i s known and - 127 -needed i n t h e f i e l d , t h e r e f o r e any model o r any c o m b i n a t i o n o f t h e main a s s e t s o f b o t h model can be used t o i n v e s t i g a t e t h e s t a t i c c h a r a c t e r i s t i c s o f a c a b l e l o g g i n g system. B r i e f D e s c r i p t i o n o f t h e Model Proposed. The model proposed uses e q u a t i o n g (Chapter 2) from t h e 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 t h e f o r c e s known i n t h e system, t o t h e c a r r i a g e l e v e l . Then, t h e p a r a -b o l i c model i s used t o f o r m u l a t e t h e e q u a t i o n s o f e q u i l i -b r i u m o f the c a r r i a g e . T h i s p r o c e d u r e was s u c c e s s f u l l y implemented by Carson and Mann(3) f o r a r u n n i n g s k y l i n e system. A s i m p l e and s u f f i c i e n t l y a c c u r a t e model i s a v a i l -a b l e t o d e s c r i b e a c a b l e y a r d i n g system i n s t a t i c c o n d i -t i o n s . T h i s model would s a t i s f y t h e needs o f c a b l e system d e s i g n e r s and u s e r s i f c a b l e systems were o p e r a t e d i n t h e c o n d i t i o n s o f s t a t i c i m p l i e d by the model. C o n s c i o u s t h a t t h i s i s n o t e x a c t l y t h e c a s e , c a b l e system d e s i g n e r s and u s e r s have t r a d i t i o n a l l y a p p l i e d a f a c t o r o f s a f e t y t o t h e s t a t i c r e s u l t s t o account f o r t h e v a r i o u s u n c e r t a i n t i e s . The f a c t o r 5 recommended by t h e Workmen's Compensation Board(11) i s more " i g n o r a n c e " t h a n " s a f e t y " and p r o v e s t h a t 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 b e h a v i o u r of c a b l e l o g g i n g systems. - 1 2 8 -The conclusions from the dynamic tests described in Chapter 7 are tentative since the experiment was very limited. I t was noted that the tensions were very much disturbed by the carriage motion and a tension as high as 700% of the s t a t i c value was recorded i n the mainline. I t should be noted, however, that the mainline was understressed i n i t i a l l y . The dynamic tests y i e l d more questions than answers to the problem at that point of the study but c l e a r l y show the need for further investigations i n t h i s f i e l d and also demonstrate the potential usefulness of the f i e l d model to carry those investigations. Conclusion. The study i s successful i n selecting a simple t h e o r e t i c a l model for the determination of cable yarding system s t a t i c c h a r a c t e r i s t i c s . The selection of the model i s based on the comparative analysis of the results of the f i e l d tests and the theories, showing that although the shape of a free hanging cable i s better described as a cate-nary than a parabola both t h e o r e t i c a l models are accurate enough to solve p r a c t i c a l cable system problems. The thesis also shows the importance of the dynamic forces acting on the system. Further research i n t h i s area i s required. - 129 -L I T E R A T U R E C I T E D 6. Binkley, V.W. and Studier, D.D. 1974. Cable Logging Systems. USDA Forest Service, Portland, Oregon, 190 p. Carson, W.W. and Mann, C.N. 1970. A technique for the solution of skyline catenary equations. PNW-110, 18 p., i l l u s . P a c i f i c Northwest Forest and Range Experiment Station, Portland, Oregon. Carson, W.W. and .'.Mann, C.N. 1971. An analysis of run-ning skyline load path. PNW-120, 9 p., i l l u s . P a c i f i c Northwest Forest and Range Experiment Station, Portland, Oregon. Carson, W.W. 1975. Analysis of running skyline with . :: drag. PNW-193, 8 p., i l l u s . P a c i f i c Northwest Forest and Range Experiment Station, Portland, Oregon. I n g l i s , S i r Charles Edward. 1963. Applied Mechanics for Engineers. 404 p., i l l u s . New York Dover Publications, Inc. Lysons, H.H. and Mann, C.N. 1967. Skyline Tension and Deflection Handbook. U.S. Forest Serv. Res. Pap. PNW-39, 44 p.,. i l l u s . P a c i f i c Northwest Forest and Range Experiment Station, Portland, Oregon. 7. Selby, S.M. 1964. Standard Mathematical Tables. 632 p. The Chemical Rubber Co., Cleveland, Ohio. 10 Veda, M., Saito, T., Tominaga, M.,and Shibata, J. 1962. Studies on the Main Cable i n Skyline Logging. F i r s t Report. B u l l e t i n of the Government Forest Experimental Station no.188. Tokyo, Japan. Veda, M. and Saito, T. 1965. Studies on the Main Cable i n Skyline Logging. Second Report. B u l l e t i n of the Government Forest Experiment Station no.188. Tokyo, Japan. Veda, M. 1966. Studies on the Main Cable i n Skyline Logging. Third Report. B u l l e t i n of the Government Forest Experiment Station no.18 8. Tokyo, Japan. - 130 -11. Workmen's Compensation Board. 1972. A c c i d e n t P r e v e n -t i o n R e g u l a t i o n s . 298 p. Workmen's Compensation 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 Corp-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 presents the parabolic model developed by G.G. Young"*" and the author, for the study of cable log-ging system problems. Appendix 1 was primarily produced as a teaching a i d for a university course i n forest harvesting. It studies cable mechanics from the very basic free hanging cable to the more complex f i v e l i n e system. Because the course can be attended by students with a lim i t e d knowledge i n mechanics the development i s very detailed i n the f i r s t chapters, to become reasonably succinct towards the end as the student progresses. Numerical examples are presented after every major development. Although intended primarily for forestry students, we believe that t h i s paper can be of use to anyone dealing with problems i n cable mechanics. Assistant Professor Harvesting and Operations Research Faculty of Forestry University of B r i t i s h Columbia - 1 3 2 -CONTENTS I n t r o d u c t i o n A . F r e e hang ing c a b l e 1. G e n e r a l d e s c r i p t i o n o f the system 1 2. Mode l s o f the system 2 3. E q u a t i o n o f the c a b l e shape 3 4. T e n s i o n s and d e f l e c t i o n 7 B . C a b l e w i t h a s i n g l e c o n c e n t r a t e d l o a d 1. G o v e r n i n g e q u a t i o n s f o r the sys tem The c a b l e we igh t i s assumed to a c t on the subchords 16 2 . Sag and d e f l e c t i o n 19 3 . Prob lem 1 - G i v e n the sag • ( o r d e f l e c t i o n D^) a t x = x^ and the c lamped l o a d R what a r e the t e n s i o n s T ^ and T ^ a t the s u p p o r t s ? 22 4 . Prob lem 2 - G i v e n the sag ( o r d e f l e c t i o n D^) a t x = x^ and the t e n s i o n T a t the upper s u p p o r t B what i s the l i f t i n g c a p a c i t y R o f the system? 27 5 . Prob lem 3 - G i v e n the t e n s i o n T a t t h e B upper s u p p o r t and the l o a d R a t x = x^ what i s the sag S ( o r d e f l e c t i o n D) a t the p o s i t i v e o f the load? 31 - 133 -C . S t a n d i n g s k y l i n e . G r a v i t y system 1. D e s c r i p t i o n o f the system 34 2. G o v e r n i n g e q u a t i o n s f o r the system • 35 3 . Problem 1 - G i v e n the sag S ( o r the d e f l e c t i o n D) and the l o a d R-what i s the t e n s i o n T ^  a t B i n the s k y l i n e ? 37 4 . Prob lem 2 - G i v e n the sag S • (or the d e f l e c t i o n D) and the t e n s i o n i n the s k y l i n e a t B what i s the l i f t c a p a c i t y R o f the system? 40 5. Problem 3 - G i v e n the t e n s i o n T f i 2 i n the s k y l i n e a t B and the l o a d R what i s the sag S ( o r the d e f l e c t i o n ) a t the c a r r i a g e ? 43 D . F i v e l i n e system 1. D e s c r i p t i o n o f the system 44 2. G o v e r n i n g e q u a t i o n s f o r the system 45 3. M a t h e m a t i c a l s i m i l i t u d e between the s t a n d i n g s k y l i n e and the f i v e l i n e system ; 47 CABLE MECHANICS F r e e h a n g i n g c a b l e 1. G e n e r a l d e s c r i p t i o n o f the system ; : ••• V B • y c. ^ VA' \ E ' y' B HA ^ \ C = c a b l e '." * . o : x ' -L 9 • .. . ^ : - -—' -•- ' '-' ----- • •:• -- - -1.1 Geometry C = c a b l e B = upper s u p p o r t o f the c a b l e A = lower s u p p o r t o f the c a b l e L = s p a n ; h o r i z o n t a l d i s t a n c e between the s u p p o r t s E = d i f f e r e n c e in- e l e v a t i o n between t h e s u p p o r t s AB = c h o r d 6 = a n g l e between the c h o r d and the h o r i z o n t a l [6 = a r c t g ( E / L ) ] P = any p o i n t on the c a b l e 0 = p o i n t o f h o r i z o n t a l tangent o f the c a b l e 0, x ' , y ' = c o o r d i n a t e system 1.2 F o r c e s The t e n s i o n at any p o i n t on a c a b l e ' a c t s a l o n g the tangent at t h a t p o i n t o f the c a b l e . - 135 -T P t e n s i o n i n the c a b l e at P H P = h o r i z o n t a l component o f the t e n s i o n at P V P = v e r t i c a l component o f t h a t e n s i o n at P T A t e n s i o n a t the lower s u p p o r t T B t e n s i o n a t the upper s u p p o r t = weight o f the c a b l e per u n i t l e n g t h 2 . Mode l s o f the system D i f f e r e n t e q u a t i o n s can be d e v e l o p e d depending on the as sumpt ions 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 c a b l e we ight a c t s on the c a b l e . T h r e e d i f f e r e n t models c o u l d be u s e d : -model 1 : c a b l e we ight i s 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 o f the system model 2 : c a b l e we ight i s u n i f o r m l y d i s t r i b u t e d over the c h o r d o f the sys tem model 3 : c a b l e we ight i s u n i f o r m l y d i s t r i b u t e d over the l e n g t h o f the c a b l e i t s e l f model 1 model 2 model 3 The 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 model 3 , but on the o t h e r hand models 1 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 sys tem. The development of the e q u a t i o n s f o r the system i n t h i s paper i s based on model 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 . The 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. E q u a t i o n o f the c a b l e shape 3 .1 E q u a t i o n o f the c a b l e shape i n the c o o r d i n a t e system 0, x ' , y ' The o r i g i n 0 i s the p o i n t o f maximum s a g . The system c o n s i d e r e d i s the c a b l e between 0 and P . F o r c e s a c t i n g on the sys tem: * H t e n s i o n at 0 , h o r i z o n t a l * **P + ^P = ^P t e n s i o n at P • * W = x ' u / c o s G assumed weight of the c a b l e between 0 and P - 137 -E q u a t i o n s of e q u i l i b r i u m * Sum of v e r t i c a l f o r c e s = 0 V p - x 'w/cose = 0 * Sum of h o r i z o n t a l f o r c e s = 0 - H + H p = 0 . H p = H i . e . h o r i z o n t a l component o f c a b l e t e n s i o n i s c o n s t a n t throughout the c a b l e * Sum of the moments about P = 0 H y ' cos6 3_ 2 0 The l a s t e q u a t i o n can be a r r a n g e d t o y i e l d y' = to x' 2 T h i s i s the e q u a t i o n o f the c a b l e i n the 2cos6H c o o r d i n a t e system O x ' y ' . C o n c l u s i o n : the c a b l e hangs l i k e a p a r a b o l a . 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 system w i t h o r i g i n a t the p o i n t of maximum sag i n the 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 v e r y 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 sys tem w i t h o r i g i n a t one o f the s u p p o r t s . E q u a t i o n o f the c a b l e shape i n the c o o r d i n a t e sys tem A , x , y . - 138 -The t r a n s l a t i o n o f the c o o r d i n a t e system i s d e f i n e d b y : rx = x' + a rx ' = x - a o r i y ' = y + b S u b s t i t u t i n g f o r x' and y' i n y' of the c a b l e i n the system A , x , x' 2 y i e l d s , the new e q u a t i o n 2cos9H y + b = oi (x - a ) 2 2cos6H ^ Now to determine the v a l u e s o f a . a n d b : L i m i t c o n d i t i o n s , x= 0 when y = 0 and x = L when y = E g i v e : o) a' and 2cos9H E + b = <" ( L " a ) 2 = to ( L 2 + a 2 - 2La) 2cos6H 2cos6H S u b t r a c t i n g t h o s e two e q u a t i o n s E = to • ( L 2 - 2La) 2cos9H from which and L - cos6HE 2 b = to uL L - COSSHE!2 2cosGH aiL S u b s t i t u t i n g " a and b i n the e q u a t i o n o f the c a b l e : y + to 2cos6H y + to fL - COS9HE! 22 toL 2cos6H 2cos8H x - L - cos9HE 2 uL x 2 + fL - cs*efl|T2 - 2x 2cos6H cos6H toL L - cos9HE L - cosBHE 2 toL toL y = u x 2 + E - toL X 2cos6H L 2cos6H T h i s i s the p a r a b o l i c e q u a t i o n o f the c a b l e w i t h the c o o r d i n a t e syste a t the l e f t h a n d s u p p o r t . - 139 -The above e q u a t i o n can be r e a r r a n g e d so t h a t i t s components a r e more m e a n i n g f u l from a p h y s i c a l s tand p o i n t y i e l d i n g : - u x ( L - x) + E_x 2cos8H L From the above f i g u r e i t can be seen t h a t : E x L NP = y p o s i t i o n of the c a b l e a t x NM = y p o s i t i o n of the c h o r d at x -MX(L - x) 2cos6H NP - NM = NP + MN = MP v e r t i c a l d i s t a n c e between chord and c a b l e at p o i n t x . T h i s q u a n t i t y i s c l a s s i c a l l y r e f e r r e d t o as the DEFLECTION o f the c a b l e a t x. D e f l e c t i o n o f the c a b l e a t x = D = MX(L - x) x - 2cos6H A t midspan x = L / 2 and Dm = toL/2(L - L / 2 ) = coL2 2cos6H 8cos6H Tangents to the c a b l e - 140 -S i n c e the t e n s i o n a c t s on the tangent o f the c a b l e i t i s v e r y o f t e n n e c e s s a r y to know the s l o p e o f t h i s tangent at a g i v e n p o i n t P . The tangent a i s g i v e n by the v a l u e of the f i r s t d e r i v a t i v e o f the 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: y = 01 2cosGH (ii x + dx cos8H L 2cosGHj - uiL ] = t g a = S lope o f the c a b l e a t p o i n t x . 2cos6H Example: the s l o p e at A i s : tga = E - oiL L 2cos9H the s l o p e a t B i s : tga = oiL + E D — E + u L cos6H L 2cos9H L 2cos6H 4. T e n s i o n s and d e f l e c t i o n G i v e n the geometry o f the system ( E and L ) and the type of c a b l e ( u ) the e q u a t i o n s o f the c a b l e s 1 and 2 o n l y d i f f e r because o f the v a l u e s o f the h o r i z o n t a l t e n s i o n s and H ^ . 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 handy parameter f o r the d e r i v a t i o n o f the e q u a t i o n s but i t i s not 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 i m p o r t a n t f a c t o r s a r e : - d e f l e c t i o n a t a g i v e n p o i n t - t e n s i o n a t the s u p p o r t s - 141 -4.1 Problem 1: G i v e n the d e f l e c t i o n D = at p o i n t x = what are the  t e n s i o n s and at the s u p p o r t s ? 4 . 1 . 1 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 The e q u a t i o n f o r the d e f l e c t i o n i s D = 1 1_ • 2cos6H Then H = " X 1 ( L - x l > 2D 1 cos9 I f the e q u a t i o n o f the c a b l e i s needed p l u g t h i s v a l u e o f H i n the e q u a t i o n d e r i v e d a t 3 . 2 . 4 . 1 . 2 T e n s i o n s a t the s u p p o r t s T f i = _ H _ and T A = H w i t h H = U X 1 ( L " X l ) c o s a D c o s a . 2D,cos9 o A ± and from 3 . 3 , t&0Lr> ~ j± + • IOL L 2cos6H t g a . = F, - oiL L 2cos6H - 142 -The d e f l e c t i o n D = Dm i s known a t m i d s p a n . x^ = L / 2 D = Dm In t h i s case H = O J L 2 8Dmcos8 and tga = _E + ooL 8Dmcos9 = E + 4Dm L 2cos6 ( j , 2 L t g a . = E - h)L 8Dmcos6 = E - 4Dm L 2cos9 w L 2 L The t e n s i o n s a t the s u p p o r t s can be o b t a i n e d from T f i = wL2. and T ^ = o)L2  8 c o s a „ Dmcos9 8 c o s a . Dmcos9 K A NOTE: A n o t h e r way o f d e r i v i n g the s l o p e o f the c a b l e a t A and B i s to use the f o l l o w i n g p r o p e r t y o f p a r a b o l a s : The tangent s a t A and B c r o s s at Q on the v e r t i c a l through midspan and PQ = MP Then: t g a . = JEL = -2Dm + E / 2 = E - 4Dm N AN L / 2 L t g a = J £ _ = -(2Dm + E / 2 ) = 4Dm + E T5R - ( V 2 ) L - 143 -o I f the c a b l e i s s u f f i c i e n t l y t i g h t the a n g l e and 8 a r e n o t v e r y d i f f e r e n t . Assuming = 8 i s assuming t h a t the t e n s i o n a t B a c t s a l o n g the chord o f the sys tem. W i t h t h i s a s s u m p t i o n uix, (L - x , ) , ' H = 1 1_ and c o s a ^ = cos8 2 D j C o s e Then T = H = a ) X l ( L " X l > B r e c a l l tg8 = E_ L cose aD^cose) 2 i f D = Dm i s g i v e n a t inidspan T B = coL2 8Dm(cos8 ) 2 1.5 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 meters between the s u p p o r t s . The weight per meter o f the 1" c a b l e i s about 6 l b s . The midspan d e f l e c t i o n i s 2%. a) Assuming t h a t the t e n s i o n s a c t a l o n g the c a b l e , what a r e the t e n s i o n s a t b o t h ends? b) Check t h a t the d i f f e r e n c e i n t e n s i o n between the two s u p p o r t s i s c l o s e to the d i f f e r e n c e i n e l e v a t i o n t imes oi. - 144 -c) Assume now t h a t the c a b l e i s t i g h t enough and c a l c u l a t e the t e n s i o n at the upper s u p p o r t . S o l u t i o n : a) from 4 . 1 . 3 H o r i z o n t a l t e n s i o n u) = 6 l b s / m e t e r Dm = .02 x 500 = 10 <oL 8Dmcose L = 500 m 9 = A r c t g (200/500) = 2 1 . 8 ° .2 6 x (500) 20194 l b s 8 x 10 x c o s 2 1 . 8 Tens i o n s Lower s u p p o r t T . = H cosaA tga = 200 - 4-x 10 = .32 500 a . = 17 .74 ' A 20194 = 21203 l b s c o s l 7 . 7 4 Upper s u p p o r t H cosa. B tga_ = 200 + 4 x 10 .48 500 2 5 . 6 4 v b) 20194 cos25 .64 22400 l b s 22400 21203 = 1197 l b s E x (o 200 x 6 1200 l b s NOTE: One o f the c o n c l u s i o n s o f the c a t e n a r y f o r m u l a t i o n f o r the c a b l e (based on model 3 , weight d i s t r i b u t e d o v e r the 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 the d i f f e r e n c e i n t e n s i o n between 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 to the we ight per u n i t l e n g t h o f the c a b l e t imes 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 t h i s c h a r a c t e r i s t i c i s s i m p l e and e x a c t i t can be used i n any m o d e l . - 145 -c) from 4 . 1 . A B 8Dm(cosO) 2 T' = 6 x ( 5 0 0 ) 2 = 21749 l b s B 2 8 x 10 x (cos21 .8 ) T h i s as sumpt ion u n d e r e s t i m a t e s the t e n s i o n at the upper s u p p o r t by (22400 - 21749) = 650 l b s or 3%. 4.2 Problem 2: G i v e n the t e n s i o n T„ at B what i s the d e f l e c t i o n of each  p o i n t o f the c a b l e ? " . S i n c e the e q u a t i o n o f the d e f l e c t i o n i s D = tox(L - x) f o r a g i v e n 2cos6H ' i geometry of the system and type o f c a b l e the d e f l e c t i o n a t x depends on H o n l y . The prob lem i s t h e n to f i n d H g i v e n T . 4 . 2 . 1 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 H = T B c o s a B and t g a R = E_ + uiL ( from 3 .3) L 2cos6H B R e c a l l : cosa. B V1 + t g 2 a B Then: H = T D V 1 + E + coL L 2cos6H - 146 -H yi + E + - u L L 2 c o s 6 H = T_ S q u a r i n g b o t h s i d e s : H (1 + ]E_ + oil L 2 c o s 6 H H 2 1 + 'E' 2 + • u>L 2 1 + Ed) L 2 c o s G H 2 c o s 6 H H 2 ' l + E 2 i + H Eto + oiL 2 LJ c o s f 3 2 c o s 6 ^ - T B = 0 T h i s q u a d r a t i c e q u a t i o n can be w r i t t e n aH + bH + c = 0 w i t h a = 1 + b = Eoi cosO oiL 2cos6 2-< The s o l u t i o n i s g i v e n by 4 . 2 . 2 D e f l e c t i o n H = - b + V 7 " 4ac 2a H i s then known and the d e f l e c t i o n a t p o i n t x can be c a l c u l a t e d from D = o i x ( L - x ) 2cos6H 4 . 2 . 3 To c a l c u l a t e the d e f l e c t i o n at midspan g i v e n T ^ H = T B c o s o B tga = E + _ u L _ ( from 3 .3) L 2cos9H i H can be d e r i v e d as i n 4 . 2 . 1 .2 and Dm = <oL ( from 3 .2) 8cos0H - 147 -2.4 T i g h t c a b l e T can be assumed to a c t a l o n g AB D Then: Tgcosctg = T B c o s e r e c a l l t £ = E L The d e f l e c t i o n at any. p o i n t x i s g i v e n by D = uix(L - x) 2 ( c o s 6 ) 2 T B 2.5 Example Prob lem 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 o f 200 meters between the s u p p o r t s . The weight per meter o f the 1" c a b l e i s about 6 l b s and the b r e a k i n g s t r e n g t h 44 .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 a l o n g the c a b l e what i s the midspan d e f l e c t i o n i f the l i n e i s t e n s i o n e d to c a p a c i t y . b) Assume now t h a t the c a b l e i s t i g h t enough and c a l c u l a t e the midspan d e f l e c t i o n f o r the same c o n d i t i o n s as b e f o r e . S o l u t i o n a) from 4 .2 H o r i z o n t a l t e n s i o n - 148 -H i s a s o l u t i o n o f : 1 + + H Eh) + cos8 uL 2cos6 2 „ 2 ~ B E = 200 L = 500 to = 6 8 = A r c t g (200/500) = 21.8 T = 44 .8 x 2000 = 22400 l b s a : That y i e l d s H 2 f l + f 2 0 0 l 500 + H 200 x 6 + cos21 .8 6 x 5 0 0 2cos21 .8 (22400) = 0 1 . 1 6 H 2 + 1292H - 5 x 1 0 8 = 0 H = -1292 + Vl292 2 + 4 x 1.16 x 5 x 1 0 8 = -1292 + 48184 2 x 1.16 2.32 H = 20212 l b s from 4 . 2 . 3 Dm = a) x L 6 x (500) = 10 meters 8cos9H 8cos21 .8 x 20212 o r D e f l e c t i o n a t midspan = 10 = 2% 5 0 0 from 4 . 2 . 4 H o r i z o n t a l t e n s i o n H = T^cosE H = 22400 x c o s 2 1 . 8 = 20800 l b s Dm = oiL 6 x (500)' = 9 . 7 meters 8cos6H 8 x c o s 2 1 . 8 x 20800 o r D e f l e c t i o n a t midspan = 9 .7 = 1.9% 500 T h i s a s sumpt ion u n d e r e s t i m a t e s the d e f l e c t i o n . - 149 -B. Cab le w i t h a s i n g l e c o n c e n t r a t e d l o a d 1. G o v e r n i n g e q u a t i o n s f o r the sys tem. The c a b l e weight i s assumed to a c t on the subchords 1 .1 A d d i t i o n a l n o t a t i o n C = p o i n t o f a t tachment o f the l o a d on the c a b l e R = weight o f the l o a d x = h o r i z o n t a l p o s i t i o n o f l o a d ' S = sag o f the c a b l e at p o i n t C ( D i s t a n c e f r o m x a x i s ) 6^ = ang le between the subchord CA and the h o r i z o n t a l 9^ ^ = A r c t g [(S/x)] &2 = a n g l e between the subchord CB and the h o r i z o n t a l e 2 = A r c t g [(S + E)/(L - x)] OL„ = a n g l e between c a b l e 1 and the h o r i z o n t a l a t C a r 0 = ang le between c a b l e 2 and the h o r i z o n t a l at C - 150 -The p o s i t i o n o f the- s u p p o r t s ( E , L ) and the type of c a b l e (oi) a l o n g w i t h two of the f o l l o w i n g v a r i a b l e s d e f i n e the system c o m p l e t e l y . - d e f l e c t i o n or sag a t C - l o a d R - t e n s i o n at one o f the s u p p o r t s Two e q u a t i o n s can be w r i t t e n t o r e l a t e these v a r i a b l e s : 1.2 E q u a t i o n (1) : E x p r e s s the s t a t i c e q u i l i b r i u m o f C . * Sum o f h o r i z o n t a l f o r c e s The e q u i l i b r i u m o f the x components o f the f o r c e s shows t h a t the h o r i z o n t a l component of the c a b l e t e n s i o n i s a c o n s t a n t H . * Sum o f v e r t i c a l f o r c e s R - H ( t g a r 1 + tga ) = 0 - 151 -The sections of cable 1 and 2 are two free hanging cables. Then tga r can be derived from the formula in Part A, Section 3.3. E and L have to be substituted by their corresponding values for each section. x 2cos6 H t g a r 9 = S + E - oi(L'- x) L - x 2cos9 2H This yields R = H (S x + S + E - oi(L - x) 2COS6JH L - x 2cosG2H 1 + S + E x L - x 2cos9, oi (L - x) 2cos8„ R = HL S + HE oix x(L - x) L - x 2cos6, oi(L - x) (equation 1) 2cos8„ Equation (2): - 152 -Express the tension at B in terms of H. Tg = H/cosa B ='H V l + (tgctg) 2 T. = H/cosa = H \]l + (tga ) 2 From Part A, Section 3.3, we can calculate tga^ for the free hanging section • of cable 2, and tga^ for section 1. t^ax, = E + S •+ OJ(L - x) L - x 2cos8 2H 'S01* = .§ + aix x 2cos9 1H Tg = H|/l + E + S io(L - x) L - x 2cos9 2H (equation 2) T A = H|/l + _S M X x 2cos9 H (equation 2 ) Squaring both sides of equation 2 \2 2 2 Tg 2 - H 2 0 = H' 1 + E + S L - x (L - x) 2cos6 2H E + S L - x j(L - x) cos9 2H 1 + f E + s) 2 + (E + S)m H [ai(L - x)] 2 _ T 2 [L - xj J cos9 2 [ 2cos9 2 ) H can be ea s i l y obtained i f needed by solving this quadratic equation. Sag and deflection Further development of equation 1 y i e l d s : 2 2 S = Rx(L - x) - xE + M X (L - x) + o)x(L - x) HL 2HLcos9, 2HL-COS0, i n which cos9^ and cosfl 2 are dependent on S. No simple expression for S can be derived without further assumptions. - 153 -C a b l e weight assumed to a c t on the c h o r d AB As seen on the f i g u r e above t h i s s i m p l i f i c a t i o n assumes t h a t : cose^ = c o s 6 2 = c o s 9 The p r e v i o u s e x p r e s s i o n f o r S becomes: S = Rx(L - x) - x E + a )x 2a - x) + iox(L - x ) 2 HL L 2HLcos9 2 3 o "3 o = foC1- ~ *) - x E + cox L - aix + aixL + aix - 2mx I HL L 2HLcos6 = K x ( L - x) - xE + mxL(L - x) HL 2HLcos6 S = Rx(L - x) + ti)x(L - x) - x E HL 2Hcos0 L Sag at c o n c e n t r a t e d l o a d R a t a d i s t a n c e x from lower s u p p o r t - 154 -T h i s e q u a t i o n can be broken i n t o t h r e e components: 1) OJX(L - x) = MP = d e f l e c t i o n due t o c a b l e weight ( P a r t A , S e c t i o n 3 .2) 2cos0H i i ) Rx(L - x) = PC = a d d i t i o n a l d e f l e c t i o n due to l o a d HL i i i ) - x E = NM -L sag = 1TC = P1:+MP+'NM The d e f l e c t i o n MC produced a t the c o n c e n t r a t e d l o a d can thus e a s i l y be seen to be D e f l e c t i o n = R x ( L - x) + oix(L - x) = MC HL 2cos6H A t midspan x = L / 2 2 Dm = RL + mL 4H 8cos9H - 155 -3. Problem 1 - G i v e n the sag (or d e f l e c t i o n D^) at x = x ^ and the clamped  l o a d R what are the t e n s i o n s T A and at the s u p p o r t s ? 3 .1 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 S o l v i n g f o r H i n e q u a t i o n (1) y i e l d s L S , x 1 ( L - x ^ R + ^ l + M < L ~ x l > 2cos0 , 2cos6 , L S 1 + E X 1 ] = R + x x ( L - x x ) J cox. + co(L - x ± ) 2cos6 , 2cos9 , + E x 1 / L ) - | = R + uxx + co(L - X ; L ) x x ( L - x x ) 2cos6 , 2 c o s 9 r and S + E x l L D e f l e c t i o n at x = x ^ H = x l ^ L ~ x p R + u x ] ^ L ~ x i ^ + ^ ( L - x 1 ) ' L D , 2 L D 1 c o s 6 1 2LD c o s 8 2 - 156 -R e c a l l from 1.1 tg6 = S 1 / x 1 then 1 cosO, V1 + ^ i ' * ! * t g 6 2 = ( S 1 + E)/(L - X ; L) then __1 1 + c o s 9 . S l + E ] L - x. H i s then c o m p l e t e l y d e f i n e d . 3 .2 T e n s i o n s at the s u p p o r t s t From 1.3 ( e q u a t i o n 2 and 2 ) T and T a r e g i v e n b y : A D T A = H 1 + s 1 + ^ * 1 2 0 0 3 6 ^ T B = H l / 1 + E + S 1 + U ( L - x±) L - x^ 2 c o s 9 2 H where H has the v a l u e d e f i n e d i n 3 . 1 . 3 . 3 I f R i s g i v e n and the sag Sm i s known a t m i d s p a n , i . e . x^ = L / 2 S = Sm 3 . 3 . 1 H o r i z o n t a l t e n s i o n ( from 3 .1) Dm = Sm + E/2 = D e f l e c t i o n at midspan o r Sm = Dm - E/2 H = LR + toL2 + coL2 = LR + tuL2 f 1 + 1 4Dm 16Dmcos9 16Dmcos9 4Dm 16Dm Icos9 1 co s9 . and c o s 9 . 1 + 2Sm 1 + 2Dm - E cos 6, 1 + 2(Sm + E) / l + f2Dm + Y)1 I n t r o d u c i n g s = tg6 = E = s l o p e of the c h o r d L and p = Dm = per cent d e f l e c t i o n / 1 0 0 L H = _K_ + toL 4p 16p 1 + (2p - s ) 2 + \ / l + (2p + s ) 2 3 . 3 . 2 Tens ions at the s u p p o r t s T A = H W l + |2Sm + oiL L 4cos6 H T = H H I + (2(E + Sm) + oiL L 4 c o s 6 „ H S u b s t i t u t i n g 1 and 1 and i n t r o d u c i n g p and s y i e l d s : cos 2 cos8^ T A = H l / 1 + ( 2 p - S + H l / l + ( 2 p - s ) : T B = H l / 1 + '2 P + B + g y i + ( 2 p + s ) : where H has the v a l u e d e f i n e d i n 3 . 3 . 1 . Assumpt ion t h a t the t e n s i o n s a t the s u p p o r t s A and B a c t a l o n g the s u b - chords of the sys tem - 158.-3 . 4 . 1 H o r i z o n t a l t e n s i o n The e x p r e s s i o n o f H i s unchanged and i s as d e f i n e d i n 3 . 1 . 2 2 • = X l ( L - x 1 ) R + uxj (L - x x ) + o ) X l ( L - x t ) L D ^ 2LD^cos0^ 2 L D 1 c o s 6 2 3 . 4 . 2 T e n s i o n s a t the s u p p o r t s T A = H / c o s 6 1 = H ^1 + ( S 1 / x 1 ) ' Tg = H / c o s 8 2 = H \ j l + f ( S 1 + E ) / ( L - x ^ ' where H has the v a l u e d e f i n e d i n 3 . 4 . 1 3 . 4 . 3 I f the s a g S i s known a t midspan (x.^ = L / 2 and s = Sm) I n t r o d u c i n g s and p ( r e c a l l 3 . 3 . 1 ) H = _R_ + oiL 4p 16p 1 + (2p - s ) 2 + \jl + (2p + s ) 2 and V H \ j l + (2p - s ) ' H \ | l + (2p + s ) ' 3 .5 Example A c a b l e sys tem has a 500-meter span w i t h an e l e v a t i o n d i f f e r e n c e of 200 meters between the s u p p o r t s . The w e i g h t p e r meter o f the 1" c a b l e i s about 6 l b s . A 5,885 l b l o a d i s c lamped a t midspan and the d e f l e c t i o n a t t h a t p o i n t i s 10%. a) Assuming t h a t the t e n s i o n s a c t tangent t o the c a b l e , what are the t e n s i o n s at b o t h ends? b) Assume now t h a t the c a b l e i s t i g h t and c a l c u l a t e the t e n s i o n a t the upper s u p p o r t . - 159 -S o l u t i o n a) Tens ions ac t a l o n g the c a b l e H o r i z o n t a l t e n s i o n ( 3 . 3 . 1 ) H = _R_ + jJL_ (\[ 4p '16p ' 1 + (2p - s )2 + \jl + (2p + ' s ) 2 s = E / L = 200/500 R = 5,885 l b s L = 500 m p = .1 u = 6 l b s / m Then H = 14712 + 4099 = 18811 l b s T e n s i o n s a t the s u p p o r t s ( 3 . 3 . 2 ) Tg = H1 / 1 + 2p + s + uL j / l + (2p + s ) ' 4H = 18811 1 + . 2+ . 4 +_6 x 500 y i + ( . 2 + . 4 ) 2 4 x 18811 Tg = 18811 V l . 4 2 = 22400 l b s S i m i l a r l y T . = 19048 l b s A b) T e n s i o n s a c t on the subchords ( 3 . 4 . 3 ) H o r i z o n t a l t e n s i o n Same as b e f o r e H = 18811 l b s Tg = H\/I + (2p + s ) 2 = 18811 Vl736 = 21937 l b s T A = H^ l + (2p - s ) 2 = 188.11 VI704 = 19183 lb s - 1 6 0 -Problem 2 - Given the sag S (or d e f l e c t i o n D ) at x = x, and the t e n s i o n T x JL J. "~ • •' B a t the upper s u p p o r t what i s the l i f t i n g c a p a c i t y R o f the system? 4 .1 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 R e c a l l e q u a t i o n (2) 2i H 1 + E + L - x, + (E + S)toH .+ c o s 9 „ oi (L - x) 2 c o s 8 „ 2 „, 2 ~ B w i t h l / c o s 9 „ 1 + T h i s e q u a t i o n i s a q u a d r a t i c o f the form: a H 2 + bH + c = 0 The s o l u t i o n f o r H i s H = Wb 2 - 4ac 2a 4.2 L i f t i n g c a p a c i t y R f o r l o a d clamped a t p o i n t x^ R e c a l l e q u a t i o n (1) H L S , + HE oix^ u ( L - Xj) x^(L - x 1 ) L - x^ 20086^ 2cos9, where H has the v a l u e d e f i n e d a t 4 . 1 . R can then e a s i l y be f o u n d . 4 .3 I f Tg i s g i v e n and the sag Sm i s known a t midspan = L / 2 S = Sm, 4 . 3 . 1 H o r i z o n t a l t e n s i o n E q u a t i o n (2) i s r e d u c e d t o : 1 + 2(E + Sm)j' H + (E + Sm)oiH + c o s 9 „ oiL 4cos9 , - T B = 0 w i t h Dm = Sm + E / 2 = D e f l e c t i o n a t midspan (Sm = Dm - E / 2 ) ( f rom 3 . 3 . 1 ) cos9 . - 161 -I n t r o d u c i n g s and p , the q u a d r a t i c e q u a t i o n becomes: qH + / q L a (2p + s) H + q fwLi - T„ = 0 2 U J J I B 2 2 where q = 1 + (2p + s) = 1 /cos H i s the p o s i t i v e s o l u t i o n o f the q u a d r a t i c . A . 3 . 2 L i f t i n g c a p a c i t y R E q u a t i o n (2) i s reduced t o : R = 4HSm + 2HE - coL - toL 4cos9^ 4cos02 R = 2H (2Sm + E) - u L L 4 by ana logy w i t h 3 . 3 . 1 1 + 1 ^ cos 9 1 c o s 9 2 | R = 4 P R " f ( V l + ( 2 p - s ) 2 + (2P + s ) 2 , R can then e a s i l y be f o u n d . 4 .4 W i t h the assumpt ion t h a t the t e n s i o n a c t s on the subchord CB 4 . 4 . 1 H o r i z o n t a l t e n s i o n - 162 -H = T B c o s 0 2 w i t h c o s 6 2 = 1  1 + ( S L + E / L - x^2 4 . 4 . 2 L i f t i n g c a p a c i t y R R i s o b t a i n e d from e q u a t i o n 1 R - H L S 1 + _ E H - ^ 1 " ^ " V x ^ L - x 1 ) (L - -x.^)' 2 c o s 6 1 2 c o s 6 2 4 . 4 . 3 I f the sag Sm i s known at midspan x x = L / 2 S = Sm Dm = Sm + E / 2 I n t r o d u c i n g s and p H = T B c o s 9 2 = T B / V l + (2p + s ) : and from 4 . 3 . 2 R = 4pH - uL 4 V l + (2p - s )2 + V l + (2p + s ) 2 4 .5 Example A c a b l e system has a 500-meter span 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 . The w e i gh t p e r meter o f the 1" c a b l e i s about 6 l b s and the b r e a k i n g s t r e n g t h 44 .8 t o n s . The d e f l e c t i o n a t m i d -span imposed by the ground p r o f i l e i s 10%. What i s the l o a d the system can l i f t a t midspan 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 of 4. a) Assuming t h a t the t e n s i o n s a t the s u p p o r t s a c t on the c a b l e i t s e l f b) Assuming t h a t the t e n s i o n s a t the s u p p o r t s a c t on the s u b c h o r d S o l u t i o n a) H o r i z o n t a l t e n s i o n (4 .3 ) H i s a s o l u t i o n o f : q H Z + co (2p + s ) H + q (wL 2 [ 4 ] 2 where q = l + ( 2 p + s ) , w i t h L = 500, s = E / L = 200/500 = .4 p = .1 oi = 6 T_ — 44 .8 x 2000 = 22400 l b s - 163 -Then q = 1 + ( .2 + . 4 ) 2 = 1.36 and the e q u a t i o n becomes: 1 . 3 6 H 2 + 1050H + 765000 - 501760000 = 0 1 . 3 6 H 2 + 1050H - 500995000 = 0 H = -1050 + V ( 1 0 5 0 ) 2 + 4 . x 1.36 x 50099500 = 18811 lbs 2 x 1.36 L i f t i n g c a p a c i t y R = 4pH - wL j ^ | l + (2p - s ) 2 + V l + (2p + s ) 2 ' R = 4 x .1 x 18811 - 6 x 500 + ( _ _ 2 ) 2 + \j± + { f > ) 2 4 = 7524 - 1639 = 5885 l b s The l i f t c a p a c i t y R i s 5885 l b s . 'b) Assuming the t e n s i o n on the subchord ( 4 . 4 . 3 H o r i z o n t a l t e n s i o n H i s s o l u t i o n o f : H = T / V l + (2p + s ) 2 H = 2 2 4 0 0 / \ / l + ( - 6 ) 2 = 19208 l b s L i f t i n g c a p a c i t y R = 4 x .1 x 19208 - 1639 = 6044 l b s The l i f t i n g c a p a c i t y i s o v e r e s t i m a t e d 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 the l o a d i s to f i n d the curve 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 from A to B . T h i s curve i s r e f e r r e d to 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 T . B U n f o r t u n a t e l y p r o b l e m 3 does n o t have any easy s o l u t i o n s i n c e S cannot be i s o l a t e d i n any o f the e q u a t i o n s . Depending on the a c c u r a c y d e s i r e d and on the means u s e d , S can be o b t a i n e d f rom d i f f e r e n t methods. Two o f those methods a r e p r e s e n t e d h e r e : - g r a p h i c a l s o l u t i o n - i t e r a t i v e t e c h n i q u e 5 . 1 G r a p h i c a l s o l u t i o n T h i s s o l u t i o n can be worked out w i t h a s i m p l e c a l c u l a t o r . The p r o c e d u r e i s as f o l l o w s : - 165 -l ) 2) 3) A) make a guess at S: now assume t h a t you are g i v e n and R and determine TL as I BI e x p l a i n e d f o r p r o b l e m 1 i n 3. p l o t S 1 v e r s u s T i B1 i f T - i s s m a l l e r than y o u r g i v e n T choose a s m a l l e r S and i f T , i s g r e a t e r than T„ choose a l a r g e r S: and go back to 2) , to de termine a new p o i n t on the curve S v e r s u s T B When you e s t i m a t e t h a t you have d e f i n e d the c u r v e S v e r s u s T w i t h enough a c c u r a c y f i n d the v a l u e o f S f o r y o u r g i v e n T . . B T e n s i o n a t B Curve T e n s i o n a t B v e r s u s Sag G i v e n T, ( s o l u t i o n f o r the Sag) T h i s method can be v e r y l a b o r i o u s i f the. f i r s t guesses f o r S d i f f e r g r o s s l y from the f i n a l s o l u t i o n . To a v o i d unnecessary c a l c u l a t i o n the u s e r o f t h i s method i s a d v i s e d to use the assumpt ion 3.4 f o r the f i r s t guesses . - 166 -5.2 I t e r a t i v e t echn ique Can be worked w i t h a pocket c a l c u l a t o r b u t i s more a p p r o p r i a t e f o r a computer. The p r o c e d u r e i s as f o l l o w s : .1) make a guess a t S: S = 2) assume now t h a t you are g i v e n S 1 and T and de termine H and R I B 1 as e x p l a i n e d f o r problem' 2 i n 4 3) de termine the new v a l u e f o r S g i v e n by c . „ . (R - R.) x, (L - x , ) new b = p r e v i o u s S-^  + 1 1 1_ HL where R i s y o u r g i v e n l o a d 4) go to 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 from R. The sag i s then the v a l u e you are l o o k i n g f o r . T h i s method converges towards the r i g h t answer f o r S f o r r e a s o n a b l e f i r s t c h o i c e s of S. - 167 -S t a n d i n g s k y l i n e . G r a v i t y sys tem 1. D e s c r i p t i o n o f the sys tem 1.1 A d d i t i o n a l n o t a t i o n C = c a r r i a g e to = w e i g h t p e r u n i t l e n g t h c f s k y l i n e u>2 = w e i g h t p e r u n i t l e n g t h o f main l i n e a , a , a _ . = ang le s o f the r e s p e c t i v e c a b l e s w i t h the h o r i z o n t a l a t the c a r r i a g e T C 1 ' T C 2 T C 3 = t e n s i o n s a t c 1" t h e r e s p e c t i v e l i n e s The s u b s c r i p t s f o r the a n g l e s and t e n s i o n s d e f i n e the p o i n t o f a p p l i c a t i o n . o f the f o r c e and the l i n e . R e c a l l 6^ and = a n g l e s o f the subchords Q1 = A r c t g ( ( S / x ) ) 6 2 = A r c t g ((S + E ) / ( L - x ) ) - 168 -1 . 2 V a r i a b l e s 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 to the u s u a l g e o m e t r i c a l parameters ( E , L ) and the we ights p e r u n i t l e n g t h . S - (or D) sag (or d e f l e c t i o n ) o f the l o a d T _ - t e n s i o n at upper s u p p o r t i n main l i n e R - l o a d , l i f t i n g c a p a c i t y o f the system x - h o r i z o n t a l p o s i t i o n of the l o a d V a r i o u s e q u a t i o n s can be w r i t t e n to r e l a t e those v a r i a b l e s and d e s c r i b e the sys tem. 2. G o v e r n i n g e q u a t i o n s f o r the sy s t em N i n e e q u a t i o n s a r e d e v e l o p e d t h a t d e s c r i b e the whole sys tem. 2 . 1 E q u i l i b r i u m o f c a r r i a g e H l c - - . - - i - - - - - - - - - - - R * Sum of h o r i z o n t a l f o r c e s H 2 + H 3 - U± = 0 (1) * Sum of v e r t i c a l f o r c e s H 1 t g a c l + H 2 t g a c 2 + ^ t g a ^ = R (2) 2 .2 C o n t i n u i t y o f the t e n s i o n i n the s k y l i n e through the c a r r i a g e - 169 -The t e n s i o n i n the s k y l i n e 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 p a s s i n g through the sheaves of the c a r r i a g e . Then: T = C l T C2 T C l H 1 / c o s a c l = H ^ l + ( t g a c l ) 2 T C 2 = H 2 / c o s a c 2 = H 2 \ / l + ( t g a c 2 ) 2 H l V 1 + (tga c l) = H 2 \ / l + ( t g a c 2 ) 2 (3) 2.3 A n g l e s a t the ends o f each s e c t i o n o f c a b l e The g e n e r a l e x p r e s s i o n o f tga has been d e r i v e d i n P a r t A , 3.3. E and L have to be s u b s t i t u t e d 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 e c t i o n . S e c t i o n 1 t g a c l = S - aix (4) x ' 2H^cos9^ t g a A l = - + " x (5) x 21^0036^ S e c t i o n 2 tga = S + E - ai(L - x) (6) L - x 2 H 2 c o s 9 2 tga = S + E + ai(L - x) (7) L - x 2 H 2 c o s 8 2 S e c t i o n 3 t g a - - - S + E - M3 ( L - X ) (8) L - x 2 H 3 c o s 6 2 t g a p 3 = S + E -r M3 ( L ' X ) (9) L - x 2 H 3 c o s 9 2 .. - 1 7 0 -Problem 1 - G i v e n the sag S (or the d e f l e c t i o n D) and the l o a d R, what i s the t e n s i o n T „ „ a t B i n the s k y l i n e ? BZ 3 .1 H o r i z o n t a l t e n s i o n s 3 . 1 . 1 H , i s o b t a i n e d from the e q u a t i o n s (1) and (2) i n which the "tga " a r e d e f i n e d by ( 4 ) , (6) and ( 8 ) . (1) H 2 + H 3 - H x = 0 (2) x 2 H 1 c o s 6 1 + H„ fs + E - co(L - x) L - x 2 H 2 c o s 6 2 + H , S + E - W 3 ( L " X ) L - x 2 H 3 c o s 9 2 (2) can be r e a r r a n g e d a s : - uix - co(L - x) - " 3 ( L " x ) + ( H 2 + H 3 ) x | 2 c o s 6 1 2 c o s 8 2 2 c o s 8 2 S + E L - x o r H , f i n a l l y S + S + E x L - x o,(L - x) - " 3 ( L " X ) = R 2 c o s 8 1 2 c o s 8 2 2cos6 , H l = R + cox + co(L - x) + " 3 ( L " X ) l 2cos6 , 2 c o s 8 « 2 c o s 6 „ x ( L - x) SL + x E T h e r e f o r e i s known. 3 . 1 . 2 H 2 E q u a t i o n ( 3 ) : H ^ l + ( t g a ^ ) 2 = H ^ l + ( t g a c 2 ) ' H n i s known and tga . can be c a l c u l a t e d from e q u a t i o n ( 4 ) . L e t us c a l l thf> f i r s t term of e q u a t i o n (3) " a " , which i s known. Then H 2 i s the s o l u t i o n o f 11 1 + (tg<* c 2) = a 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 t g a ^ ^ y i t s e x p r e s s i o n e q u a t i o n (6) y i e l d s : 1 + S + E - io(L - x) L - x 2 H 2 c o s 8 2 - 171 -f u r t h e r development g i v e s : 1 + S + E L - x H 2(S + E)oo + c o s e 2 J ( L - x) 2cos0 , 2 2 a = 0 H 2 i s the s o l u t i o n o f t h i s q u a d r a t i c e q u a t i o n as on page 13. 3 .2 T e n s i o n i n the s k y l i n e : T B2 s k y l i n e T B 2 = H 2 ^ c o s a B 2 = H 2 + ^ t g c l B 2 ^ 2 a n d f r o m e q u a t i o n (7) t R a n ; , = S + E + ui(L - x) L - x 2 H 2 c o s 9 2 The t e n s i o n i n the s k y l i n e a t the upper s u p p o r t B i s then c o m p l e t e l y d e f i n e d . 3 . 3 Example P r o b l e m : A g r a v i t y c a b l e system has a 500 meter span 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 . The we ight per p e r meter o f the 1" s k y l i n e i s 6 l b s and t h a t o f the 3/4" m a i n l i n e 3 .4 l b s . The 6234.5 l b l oaded c a r r i a g e i s a t midspan and the 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 t e n s i o n i n the s k y l i n e a t the upper s u p p o r t ? S o l u t i o n : .. 3 . 3 . 1 H r R + oix + OJ(L - x' 2cos8 , 2 c o s 8 „ w.j(L - x) x(L - x) 2 c o s 8 „ SL + xE 2 J - 172 -R = 6234.5 lbs L = 500 m oj = 6 lbs/tn x = 250 m o>3 = 3.4 lbs/m E = 200 m S = -50 m tg6_ = -50 * 200 = -.6 500 - 250 l /cos6 2 = 1.1662 t g e , = -50 500 - 250 l / co s6 1 = 1.0198 ^ = 20924 lbs 3.3.2 H 2 1 + S + E L - x H 2(S + E)to + cos9 2 o>(L - x) 2cos9, a = 0 2 u 2 a = 1 + ( t g a c l ) ' tga . = ^50 - 6 x 250 x 1.0198 = -.2365 250 2 x 20924 a = 21501 lbs (tension i n the skyl ine at the carriage) H 2 = 18812 lbs 3.3.3 T. B2 T B 2 = H 2 / c o s a B 2 tga , = -50 + 200 + 6(250)(1.1662) = .6465 250 2 x 18812 T B 2 = 18812/.8397 = 22401 lbs - 173 -Problem 2 - G i v e n the sag S (or the d e f l e c t i o n D) and the t e n s i o n i n the s k y l i n e  a t B what i s the l i f t c a p a c i t y R o f the system? 4 .1 H o r i z o n t a l t e n s i o n s 4 . 1 . 1 I 2 T B 2 = H 2 / c O S O l B 2 = H 2 V1 + ( t g a B 2 ) 2 a n d f l e q u a t i o n (7) t 8 A B 2 = s + E + ">(L - x) L - x 2 H 2 c o s 6 2 S o l v i n g f o r H 2 i s i d e n t i c a l as what has been d e r i v e d i n P a r t B , 1.3 H 2 i s the s o l u t i o n o f the r e s u l t i n g q u a d r a t i c e q u a t i o n , and i s t h e r e f o r e known. 4 .1 .2 « 1 E q u a t i o n (3) i s "xV 1 + ( t g a c l ) ' i + ( t g a c 2 y H2 i s known and t g 0 1 ^ c a n ^ e c a l c u l a t e < i from e q u a t i o n (6) where e v e r y t h i n g i s known. L e t us c a l l " a " the known q u a n t i t y i n the second term of the e q u a t i o n . H^ i s the s o l u t i o n o f -xV 1 + ( t g a c l ) = a - 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 t g 0 1 ^ by i t s e x p r e s s i o n e q u a t i o n (4) y i e l d s : 1 + x 2H^cos8^ f u r t h e r development g i v e s 2 1 + _S cos8 . 2cos8 , 2 2 - n - a = 0 i s the s o l u t i o n o f t h i s q u a d r a t i c e q u a t i o n . 4 . 1 . 3 H 3 From e q u a t i o n (1) H2 + H3 - nx = 0 H- = H , 4 .2 L i f t c a p a c i t y R E q u a t i o n (2) R = ^ t g a ^ + H 2 t g a c 2 + H - j t g a ^ The tga a r e g i v e n by e q u a t i o n s ( 4 ) , ( 6 ) , ( 8 ) . T h e r e f o r e , t h e l i f t c a p a c i t y i s c o m p l e t e l y d e f i n e d . 4 . 3 Example % P r o b l e m : A g r a v i t y c a b l e system has a 500 meter span 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 . The we ight p e r meter o f the 1" s k y l i n e i s 6 l b s and the b r e a k i n g s t r e n g t h 44 .8 t o n s . The w e i g h t per meter o f the 3/4" m a i n l i n e i s 3.4 l b s . The d e f l e c t i o n a t midspan i s 10%. What i s the l o a d the system can l i f t a t midspan i n these c o n d i t i o n s ? Use a s a f e t y f a c t o r o f 4 . S o l u t i o n : T, B2 L 22400 l b s 500 m -50 m oo = 6 l b s / m x = 250 m l / c o s 8 2 = 1.1662 co3 = 3 .4 l b s / m 200 m 1/cose^^ = 1.0198 - 1 7 5 -4 . 3 . 1 H„ S o l v i n g f o r H 2 I s ^ n a X 1 r e s p e c t s s i m i l a r to the s o l u t i o n f o r H i n the example page 29 ( 4 . 5 ) . Then H 2 = 18811 l b s 4 . 3 . 2 H 1 + V 2 X H So) + c o s 8 , 2cos9 , 2 2 n - a = 0 w i t h a H 2 \ / 1 + ( t g a p j ) ' t g a c 2 = -50 + 200 - 6(250) x 1.1662 .5535 250 2 x 18811 a = 1 8 8 1 l \ / l + ( .5535)" = 21500 l b s Then ^ = 20921 l b s 4 . 3 . 3 H 3 H 3 = 20921 - 18811 = 2110 l b s 4 . 3 . 4 L i f t c a p a c i t y R R = H l t g a c l + H 2 t g a c 2 + ^ t g a ^ tga = -50 - 6 x 250 x 1.0198  C 250 2 x 20921 .2365 t g a C 2 = " 5 5 3 5 tga _ = -50 + 200 - 3 .4 (250) x 1.1662 = .3651 250 2 x 2110 R = 6234.5 l b s - 176 -Problem 3 - G i v e n the t e n s i o n T ^ ^ i n the s k y l i n e a t B and the l o a d R, what i s  the sag S (or the d e f l e c t i o n ) a t the c a r r i a g e ? T h i s problem i s s i m i l a r to the one d e s c r i b e d on page 31 ( 5 . ) S can o n l y be o b t a i n e d from an e x p r e s s i o n o f the form S = f ( S ) where f i s an i n t r i c a t e f u n c t i o n . S = f ( S ) can be s o l v e d f o r S on a computer by i t e r a t i o n . 5 .1 D e r i v a t i o n of S = f ( S ) The e x p r e s s i o n f o r on page 37 can be r e a r r a n g e d to y i e l d : SL + x E = x ( L - x) H . R + oix + io(L - x) + 3  2cos9 , 2cos9 , 2 c o s 9 „ o r S = x ( L - x) L H , R + cox + co(L - x) + 3  2cos9^ 2cos82 2 c o s 0 „ Ex L S appears i n the second term i n the e x p r e s s i o n o f H ^ , cos8^ , cos92 . 5.2 I t e r a t i o n No m a t t e r what i t e r a t i v e t e c h n i q u e i s used the g e n e r a l p r o c e d u r e remains I d e n t i c a l : 1) make a guess a t S = 2) compute the v a l u e s o f cos9^ , cos92 and The v a l u e o f i s d e r i v e d as d e s c r i b e d f o r Problem 2 ( 4 . 1 ) . 3) F i n d the new v a l u e f o r S 4) Check on convergence Go to 2 u n t i l the sag converges i n s i d e g i v e n t o l e r a n c e s . - 177 -D . F i v e l i n e system 1. D e s c r i p t i o n o f the system 1.1 A d d i t i o n a l n o t a t i o n to we ight per u n i t l e n g t h o f h a u l b a c k u>3 we ight per u n i t l e n g t h o f m a i n l i n e u ^ i we ight p e r u n i t l e n g t h o f s l a c k p u l l e r a a n g l e of a c a b l e w i t h the h o r i z o n t a l The s u b s c r i p t s f o r the a n g l e s and t e n s i o n s d e f i n e the p o i n t o f a p p l i c a t i o n o f the f o r c e and the l i n e . - 178 -1.2 V a r i a b l e s 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 the u s u a l g e o m e t r i c a l parameters ( E , L ) and the we ights per u n i t l e n g t h . - S (or D) sag ( o r d e f l e c t i o n o f the l o a d ) - x h o r i z o n t a l p o s i t i o n o f the l o a d - R l o a d , l i f t i n g c a p a c i t y o f the system - T . t e n s i o n a t upper s u p p o r t i n main l i n e V a r i o u s e q u a t i o n s can be w r i t t e n to r e l a t e those v a r i a b l e s and d e s c r i b e the sys tem. G o v e r n i n g e q u a t i o n s f o r the system 2 .1 T e n s i o n s i n l i n e s 1 and 1' The t e n s i o n i n the h a u l b a c k r u n n i n g t h r o u g h a b l o c k a t A remains unchanged a t t h a t p o i n t . T h e r e f o r e the s e c t i o n s o f the f r e e hang ing l i n e s 1 and 1' between A and C a r e i d e n t i c a l . I n p a r t i c u l a r the t e n s i o n s a t the c a r r i a g e T c l and T c l , a r e the same and a l s o H = H ' , . - 179 -2.2 E q u i l i b r i u m o f the c a r r i a g e Sum of the h o r i z o n t a l f o r c e s H 2 + H 3 + H ' 3 - 2 H X = 0 Sum o f the v e r t i c a l f o r c e s 2 H l t g a C l + H 2 t g a C 2 + H 3 t g a C 3 + H ' 3 t g a C 3 = R (1) (2) 2 .3 C o n t i n u i t y o f the t e n s i o n i n the h a u l b a c k t h r o u g h the c a r r i a g e ( s i m i l a r to 2 .2 page 35) T = T C l C2 H 1 / c o s a ( , 1 = H 2 / c o s a c 2 H\Jl + ( t g a c l ) 2 = H 2 \ / l + ( t g a c 2 ) ' (3) - 180 -2.4 A n g l e s a t the ends of each s e c t i o n o f c a b l e The g e n e r a l e x p r e s s i o n of tga has been d e r i v e d i n p a r t A ( 3 . 3 ) . E and L and to have to be s u b s t i t u t e d 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 e c t i o n . S e c t i o n 1 (and 1') t g a c l = S iox. = tga X 2 ^ 0 0 8 0 ^ ^ t g a A l = - + — x 2 H ] | c o s e i C l ' t ga A l ' (4) (4 ' ) (5) (5 ' ) S e c t i o n 2 tga „ = S + E - co(L - x) (6) L - x 2 H 2 c o s 6 2 t g a „ = S + E + to(L - x) (7) L - x 2 H 2 c o s 6 2 S e c t i o n 3 S u b s t i t u t e 3 f o r 2 and to^ f o r to i n the e x p r e s s i o n s o f t g a w r i t t e n f o r s e c t i o n 2 . ^ S e c t i o n 3' S u b s t i t u t e 3 ' f o r 2 and to,, f o r to i n the e x p r e s s i o n s o f tga w r i t t e n f o r s e c t i o n 2. (8 ' ) ( 9 ' ) M a t h e m a t i c a l s i m i l i t u d e between the s t a n d i n g s k y l i n e and the f i v e l i n e system 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 s k y l i n e a r e i n many ways 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 sys t em. F o r most o f the case 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 the three problems 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 changes . When the 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 needed we can d e f i n e to^* = to^  + to^,, combined we igh t per u n i t l e n g t h o f the m a i n l i n e and s l a c k p u l l e r , and co* = 2to. - 1 8 1 -W i t h these new n o t a t i o n s the e x p r e s s i o n of can be d e r i v e d from ( 1 ) and (2) 1 2 + co(L - x) + i o 3 * ( L - x) x ( L - x) 2 c o s G 2 2 c o s 6 2 SL + x w h i c h compares w i t h the e x p r e s s i o n o f on page 3 7 ( 3 . 1 . 1 ) and S = x ( L - x) 2 H 1 L „ . • , T v , O J _ * ( L - x) R + co*x + o i(L - x) + 3  2cos9^ 2 c o s 6 2 2cosG, x E L w h i c h compares w i t h the e x p r e s s i o n o f S on page 4 3 ( 5 . 1 ) . T h i s s i m i l i t u d e i s used to deve lop v e r s a t i l e computer programs t h a t can h a n d l e a l m o s t any k i n d o f c a b l e y a r d i n g s y s t e m . - 182 -Figure 1+7 - Parabola in the coordinate system. (x,y) - 18 3 -A P P E N D I X 2 C A B L E L E N G T H P R O M T H E P A R A B O L I C T H E O R Y The p a r a b o l i c e q u a t i o n o f the c a b l e i n t h e coord: i n a t e system (x,y) i s : Y = 2cos9H to 2 . E _ o)L , k L 2cos0H; (b') The l e n g t h o f a s m a l l element 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 . B n—TI - 184 -Therefore the t o t a l length of the cable i s : S = Ids = l+( gjx E cosGH L 2cos9H ) dx = /P(x)dx The square root i n the i n t e g r a l can be expressed as: P(x) = 1/(1 + A) + Bx + Cx 2 where A = ,E _ cuL , lL 2cos0H; B = 2u ,E * T 0)L cosGH VL 2cos0H C = ( " ) ^COS0rT The solution to t h i s i n t e g r a l i s given by Selby(7) as • ( and 2Cx + B 4C dx P(x) 4AC-B' dx 8C ' J l/p(x) o o ' (i log ( • x^/T + -J=.)J This exact formula i s cumbersome to use for p r a c t i c a l a p p l i -cations. An approximate expression of the length of a t i g h t - 185 -c a b l e i n the case o f l e v e l s u p p o r t s (E = o) i s g i v e n by I n g l i s ( 5 ) and t h e Wire Rope Handbook(12) a s : s - L ( l + §-'<22> 2) I f t h e two s u p p o r t s are not 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 can be used: 2 b cose U + 3 ' - 186 -APPENDIX 3 T E N S I O M E T E R A3.1 Introduction. A tensiometer i s defined here as an instrument capable of measuring the tension i n fixed or running l i n e s . The need for a tensiometer arises when a load c e l l or any other sort of tension measuring device cannot be placed at the dead end of the rope i n which the tension i s to be mea-sured. Two types of tensiometers have been developed by d i f f e r e n t manufacturers mainly to meet the needs for crane indicating and warning systems. The f i r s t type manufactured by Rucker Company1 operates on the fixed r e l a t i o n s h i p between the tension and the natural frequency of a wirerope. In t h i s system, an excitor causes the cable to vibrate. The tension i n the l i n e of a given weight per unit length i s derived from the read-ing of the frequency with the sensor. 2 D i l l o n and Company produces a tensiometer of the second "'"Rucker Control 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 Street, Van Nuys, C a l i f o r n i a 91407. - 187 -Figure A-8 - Basic Principle of the tensiometer. - 188 -type. This type works on a simpler mechanic p r i n c i p l e . The cable i s given a d e f l e c t i o n with three sheaves. The tension in the l i n e i s deduced from the force thus created on the middle sheave. Two tensiometers of the second type were b u i l t . A3.2 Description of the Tensiometers. A3.2.1 P r i n c i p l e . The basic 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 Figure 48. The cable winds through the three sheaves 1, 2 and 3. The centre sheave 2 i s mounted on a lever pivoting about 0. The action of the cable on the middle sheave i s thus transmitted to the load c e l l connected to an indicator which measures the tension. A study of the - 189 -force balance on the lever shows that the tension T i n the cable i s related to the force F applied to the load c e l l by: T = F /(2 x sinG) where 0 i s the angle of the l i n e with the horizon-t a l between the sheaves. If the geometry of the machine remains the same when the load i s applied, sin0 i s a constant and the previous re 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 ten-siometer designed for t h i s study i s shown i n Figure 49. Two p a r a l l e l channel bars constitute the general structure of the machine. The outer sheaves and the triangular lever rotate on shafts guided by pillow blocks bolted on the frame. The lever i s composed of two s t e e l plates welded together and machined to support the b a l l bearings for the shaft of the middle sheave. Special care was taken so as to avoid losses by f r i c t i o n i n the transmission of the forces. A special r o l l e r bearing i s also used for the connection between the lever and the load c e l l . The frame i s over designed to - 190 -Figure 49 - Copy of the blue-print of the tensiometer. - 1 9 2 -m i n i m i z e t h e f l e x i o n a n d v e r y l i t t l e e r r o r i s i n t r o d u c e d b y t h e d e f l e c t i o n i n t h e l o a d c e l l w h i c h d o e s n o t e x c e e d . 3 mm a t m a x i m u m l o a d . T h e t w o t e n s i o m e t e r s w e r e m a n u f a c t u r e d f o r t h e r e q u i r e m e n t s s h o w n i n T a b l e X . T a b l e X . R e q u i r e m e n t s f o r t h e t e n s i o m e t e r s . Line diameter (inches) Tensiometer 1 5/8 Tensiometer 2 7/16 Maximum l i n e tension (newtons) 50000 5000 A 3 . 2 . 3 L o a d C e l l s . T h e h e a r t o f t h e m a c h i n e i s t h e l o a d c e l l . L o a d c e l l s a r e e l e c t r o n i c t r a n s d u c e r s t h a t t r a n s l a t e c h a n g e s i n f o r c e s i n t o c h a n g e s i n v o l t a g e . T h e e l e c t r o n i c o u t p u t s i g n a l o f t h e l o a d c e l l i s f e d i n t o a n i n d i c a t o r o r a r e c o r d e r w h i c h i s c a l i b r a t e d d i r e c t l y i n t e r m s o f t h e l o a d a p p l i e d t o t h e c e l l . V e r y s c h e m a t i c a l l y , a l o a d c e l l i s c o m p o s e d o f s t r a i n -g a u g e s b o n d e d t o a s t r o n g b u t e l a s t i c s t e e l e l e m e n t . T h e d e f l e c t i o n i n t h e s t e e l e l e m e n t d u e t o t h e a p p l i c a t i o n o f a f o r c e c h a n g e s t h e r e s i s t a n c e o f t h e s t r a i n - g a u g e s c o n n e c t e d - 1 9 3 -t o f o r m a b a l a n c e d w h e a t s b o n e b r i d g e . T h e m a j o r c h a r a c t e r i s t i c s o f t h e l o a d - c e l l s u s e d i n t h e t e n s i o m e t e r s a r e s h o w n i n T a b l e X I . T a b l e X I . L o a d c e l l s c h a r a c t e r i s t i c s . Load c e l l c h a r a c t e r i s t i c s Tensiometer 1 Tensiometer 2 Brand name1 BLH BLH Designation U3G1 U3G1 Capacity (lbs) 10,000 11,000 Safe working load (lbs) 15,000 1,500 Weight (lbs) 10 6 Recommended e x c i t a t i o n (volts AC or DC) 12 12 Output/input m i l l i v o l t / v o l t 3 (at maxi-mum capacity) 3 (at maxi-mum capacity) 2 Pr e c i s i o n f o r the tension % .3 .3 B L H - E l e c t r o n i c s , I n c . , 42 F o u r t h A v e n u e , W a l t h a m , M a s s a c h u s e t t 0 2 1 5 4 . T h i s f i g u r e w a s d e t e r m i n e d e x p e r i m e n t a l l y i n i d e a l l a b o r -a t o r y c o n d i t i o n s . - 1 9 4 -A 3 . 3 T e n s i o m e t e r C a l i b r a t i o n . T h e c a l i b r a t i o n o f t h e t e n s i o m e t e r i s n e c e s s a r y t o r e l a t e t h e i n d i c a t o r r e a d i n g t o t h e t e n s i o n i n t h e l i n e . T h e r e l a t i o n s h i p b e t w e e n a f o r c e F a p p l i e d t o t h e l o a d c e l l a n d t h e r e a d i n g R o n t h e i n d i c a t o r i s l i n e a r : F = a ' R + b 1 a n d i n t h e o r y t h e t e n s i o n T i n t h e l i n e a n d t h e f o r c e F t r a n s m i t t e d t o t h e l o a d c e l l a r e r e l a t e d b y : T = k x F T h e r e f o r e : T = k a ' R + k b ' o r T = a R + b w i t h a = k a ' a n d b = k b 1 t h e r e l a t i o n s h i p b e t w e e n t h e t e n s i o n T i n t h e l i n e a n d t h e i n d i c a t o r r e a d i n g R i s l i n e a r . A s a n e x a m p l e , t h e g r a p h o f T v e r s u s R s h o w n i n F i g u r e 5 1 w a s d e t e r m i n e d f o r t h e t e n s i o m e t e r 1 b e f o r e c a l i b r a t i o n a n d p r o v e d t o b e l i n e a r . T h e s e t - u p f o r t h e t e n s i o m e t e r c a l i b r a t i o n i s s k e t c h e d o n F i g u r e 5 0 . T h e i n d i c a t o r a n d t h e l o a d c e l l 1 h o o k e d a t o n e e n d o f t h e l i n e h a d b e e n l a b o r a t o r y c a l i b r a t e d a n d g a v e t h e " t r u e " l i n e t e n s i o n T . T h e t e n s i o n w a s a p p l i e d g r a d u a l l y t o t h e t e n s i o m e t e r t o b e c a l i b r a t e d , w i t h a w i n c h a t t h e o t h e r e n d o f t h e l i n e . T h e c a l i b r a t i o n o f t h e t e n -s i o m e t e r a n d i n d i c a t o r c o n s i s t s o f s e t t i n g t h e g a u g e f a c t o r - 195 -Figure 50 - Sketch of the equipment set-up f o r the c a l i b r a t i o n of the tensiometer. Figure 51 - Graph of tension read by the tensiometer versus tension i n the l i n e before c a l i -bration. - 196 -- 1 9 7 -a n d t h e z e r o o n t h e i n d i c a t o r 2 s o t h a t t h e r e a d i n g R b e e q u a l t o t h e t e n s i o n T i n t h e l i n e . I n o t h e r w o r d s , f i n d t h e v a l u e o f t h e g a u g e f a c t o r a n d t h e o r i g i n o f t h e t e n s i o n s c a l e s o t h a t t h e p r e v i o u s r e l a t i o n : T = a R + b b e c o m e s : T = R ( i . e . a = 1 a n d b = 0 ) . A 3 . 4 C a l i b r a t i o n P r o c e d u r e . T h e c a l i b r a t i o n c a n b e q u i c k l y a n d a c c u r a t e l y e x e -c u t e d i f t h e p r o c e d u r e e x p l a i n e d h e r e a f t e r i s f o l l o w e d : A 3 . 4 . 1 S e t t h e G a u g e F a c t o r . A c h a n g e o f t h e g a u g e f a c t o r o n t h e i n d i c a t o r v a r i e s t h e s l o p e o f t h e l i n e T v e r s u s R ( F i g u r e 5 2 ) . T h e p r o p e r s e t t i n g o f t h e g a u g e f a c t o r o n t h e i n d i c a t o r i s o b -t a i n e d w h e n t h e s l o p e o f t h e l i n e T v e r s u s R i s 1 . T h i s c o n -d i t i o n i s s a t i s f i e d i f x = y f o r a n d R 2 t h e v a l u e s o f t w o r e a d i n g s t a k e n s u f f i c i e n t l y f a r a p a r t . I n p r a c t i c e , s e v e r a l t r i a l s a r e r e q u i r e d b e f o r e t h i s v a l u e o f t h e g a u g e f a c t o r i s f o u n d . T h e d i f f e r e n t t r i a l s f o r t e n s i o m e t e r 1 a r e r e p o r t e d i n T a b l e X I I . - 198 -Figure 52 - Influence of the gage factor of the in d i -cator on the graph of tension read by the tensiometer versus tension in the li n e . Figure 53 — Influence of the zero knob adjustment on the graph of tension read by the tensiometer versus tension in the li n e . - 199 -- 200 -Table XII. Results of the gauge factor adjustment for the skyline 'i tensiometer. T r i a l number Gauge factor Tensiometer reading R (newtons) Tension T (newtons) 1 GF = 2.02 R 2 20620 R ± 5200 x 15420 T 2 20100 T± 5350 y 14750 2 GF = 2.07 R 2 20040 R ± 5250 x 14790 T 2 20180 T ± 5510 y 14670 3 GF ••== 2.08 R 2 19970 R ± 5350 x 14620 T 2 20120 T± 5560 y 14560 4 GF = 2.09 R 2 20250 R ± 5080 x 15170 T 2 20480 T± 5300 y 15180 The r e s u l t of t h i s f i r s t setting i s a l i n e T versus R of equation T = R + b - 201 -Figure 54 - a) Graph of tension read by the skyline tensiometer versus tension in the line after f i n a l calibration. b) Graph of the discrepancies between ten-sion read by the skyline tensiometer and the tension in the l i n e , for increasing tensions and for decreasing tensions. - 202 -R tensiometer reading, N X 1000 - 203 -A3.4.2 S e t t i n g t h e O r i g i n o f t h e T e n s i o n S c a l e t o Zero. The a d j u s t m e n t o f the o r i g i n knob on t h e i n d i c a t o r t r a n s l a t e s t h e l i n e T v e r s u s R ( F i g u r e 5 3 ) . The p r o p e r 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 chosen i n t h e m i d d l e o f the t e n s i o m e t e r range. F o r t e n s i o m e t e r 1, T^ was r e a d t o be 2006 on t h e l o a d c e l l 1 i n t h e l i n e and t h e o r i g i n knob was s e t t o 5.038 on t h e t e n -s i o m e t e r i n d i c a t o r t o r e a d the same v a l u e f o r R^. A3.4.3 C h e c k i n g t h e 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 d e t e r m i n e d f o r t e n s i o n s from z e r o t o t h e maximum c a p a c i t y o f the i n s t r u m e n t and back t o z e r o . The r e s u l t s o f t h i s t e s t f o r t e n s i o m e t e r 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 T v e r s u s R. 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 th e l i n e i s s l a c k e n e d . 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-surements were made on the a s c e n d i n g b r a n c h o f t h e h y s t e r -e s i s c u r v e . 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 dropped and the n l i f t e d 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 can be deduced from t h e f i n a l 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 with the theodolite for the free hanging t e s t . S t a t i o n H o r i z . degree angle minute _.1.8_.3._ 14 .2 V e r t i . degree 2.57 angle minute 1 Comments 0 4 1 45 .0 _ J « • ^ 2 38 2.S7 4 - 34 5 3 . 0 •• , 6 31 1 4 . 0 zsy ._2.se LS2__ 2?3 ^ • 7 8 27 . ; 2> 18 _ 1 6 . 8 „ ° 1 ' 3 28.7 _40 .9._ 4 0 . 8 32.6 2 0 . 9 . 3 7 - i 47-» 0 -1 -10 12 14 16 __i..3__. 8 IjtjUr \H*<\ I 18 3 i 20 358 2£0 eS"'S / 1 22 _3_>JL_. 348 343 _10_.8_ 07.3 14.6 Ul 2. f>L IV - 0 IS-3 J : 24 V 26 26i_ 2 6  -28 30 338 ...334 .. 330 . .1C.3 14.3. 10 .3 '6 -1 _ie - o IS-5 32 34 36 ..3_26__ 322 ....24.7 .57 .1. 4 6 . 9 2 6 3 l_..<t 38 , 319 0S--I 40 316 5 2 . 9 ._14„._0_ 4 8 . 8 2-53 s-*,-«* 42 _3_L4._. 311 n o 4 0 0 44 41 .1 46 309 3 6 . 2 2 7 i 0 4 .6 48 307 3 4 . 9 ^ 7 ^ U -3 IV '.c' 50 '305 4 3 . 8 Experiment if Date. .Weil l Tempi time her Theodolite! check ncJrikiJleVel: ure. e.ra... sltirti: _L-zJrb. ife'rjtlca'll l e v e l sett I ! plummet; Heigh't Statlijoh _ M i d | a t a t i d n ! .verti I ! ! i. r e r t i . horiz. la'ngle i ! i I ! ; ' . angle end of. Comments, j 1 i n s L THeod 52 53. u, o l i t e ' d^ ejgrfej 304 303 man [minute 01.8 Z4-3 F H 02-fl.w..[2!7l]JM i - l 1 i p M | l I Coll. ' Lite. S t a r t yd W i>kL I ! ! i 1 I ! Endj-.U?>d_!_!_LLl Ja/C i I I I ! I I ! I • I 1 angle i42 19.1 :sta_}.t:.L_L. . ZQl: 2 7 . } j : .2.6$ U.t'. : : , U i l l j i i 111 i j 1 i i V e r t degre: 27 4 angleL plnute 50.4 "02.9 Coaaeiit! j i i T - 206 -Note book sample page. Record o f t h e d a t a t a k e n a t t h e low e r s u p p o r t , f o r t h e f r e e hanging t e s t . Time I n d i e . r e a d i n g z e r o • Ten s i o r i A n g l e -J X P t a n n '.: " i m e 11 if \l >j deg ree IV. 14. 10 1? -35 11 01 + 6.5" ' --1 Ja { 11 I a e e i n *. .1 16 r a f pc _ L R i M 1. s - -l i t , J O 10 IS" - S i II 0 l i r a t i t a t i i i i 3 t J -81 1031 1 " 11 '_t L ! .6 ( 3 i IS, 00 _IDJ5:„ -S^  + 6.3, 1, i ,h r h 1 1 \ i -C a b 111 -1 | r v 0 X - A b i J j | j i i i >i ' - c 1 !C k - ok : i •! ! ! i ! ! i | i ! I I ATI IT f- Vl l o s e s t r e n c e : p a i n t n ...OS i a rk - L in 16 in t S A w u. wo ovxcj P i i r i . t n . . l n J e r r e f e oidz&b - i / i i , . . 1 | i J i i i ; ! I ! i JJumbei ..two„su ..of. ..pai pp.or.ts.: n t mar l Z 7 s_beiwe en t h e - i 1 1! 1 i s P u •_. m rt 1 J 1 i i i \ ! ! 1 j • i j -j | 11 1 > - 2 0 7 -Note book sample page. Record o f t h e d a t a t a k e n a t t h e upper s u p p o r t , f o r t h e f r e e hanging t e s t . Time - i n d i e , rsadinf Xensioi i ;  Comments I /, -15 J.13.CL -IU.SL. H i 3 CUj .ttockcahr. J 4 1 ) • 1 5-,oo JJJ.5" l.l II 1 S\ 15 length -from-c er ref« lo s e s t rencej. paint r to_upp ExperiitaeiVtl L DktA "Wei. Temperature Ifjinis'h, OMidafcoiJchecM ' ' ' J I I I I I I ' ' ' battery. Cpmme.ntS. '/4 AbneyLbiieckj: Winih1 \r>3\ m i l ! Gil ! i ! ! i i i _ i I I I ! I ! ! ! i ! i 2^.1 '<?r6MC/bta I I i i M i l - f ! - 208 -T a b l e X I I I . Summary o f t h e e x p e r i m e n t a l r e s u l t s o f t ;he f r e e h a n g i n g t e s t . Test # Mid-span d e f l e c t i o n % Tension upper support (N) 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 -T a b l e XIV. 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 . Test Load position Tension upper Load Tension lower X (m) Y (m) support (N) (Kg) 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. Summary o f e x p e r i m e n t a l r e s u l t s o f t h e g r a v i t y system. Test Carriage Tension sky- Load Tension sky- Tension i n # l o c a t i o n , , l i n e upper l i n e lower the main-"•X (m) Y (m) support (N) support (N) l i n e N 1 11.94 -1.42 11870: 300 11409 * 2 18.47 -1.24 13714 300 13272 * 3 29.87 -0.39 15745 300 * * 4 44.64 1.27 17432 300 17118 ft 5 56.50 2.98 18403 300 * * 6 65.07 4.38 18540 300 17913 * 7 72.22 5.66 18383 300 ft 313 8 82.92 7.75 17775 300 17216 372 9 96.00 10.64 16510 300 * 441 10 104.78 12.81 14930 300 14616 529 11 115.32 15.81 12517 300 * 637 12 125.03 19.26 9545 300 9388 814 13 128.85 21.11 8112 300 * 922 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 25378 627 19 84.97 7.42 24054 495 * 794 20 106.26 12.49 19982 495 18560 1030 21 123.41 18.08 13380 495 * 1486 22 129.13 20.73 8544 495 * 2020 Not measured.