E X P E R I M E N T A L STUDY OF LOGGING C A B L E SYSTEMS b y DANIEL YVES D i p l o m e d ' I n g § n i e u r E N S A M A T H E S I S T H E S U B M I T T E D O F e n M e c a n i q u e ( 1 9 7 5 ) I N R E Q U I R E M E N T S M A S T E R GUIMIER P A R T I A L F O R T H E A P P L I E D F U L F I L M E N T D E G R E E O F S C I E N C E i n T H E We a c c e p t t o T H E F A C U L T Y t h i s t h e O F M a y , D a n i e l F O R E S T R Y t h e s i s r e q u i r e d U N I V E R S I T Y © O F a s c o n f o r m i n g s t a n d a r d B R I T I S H C O L U M B I A 1 9 7 7 Y v e s G u i m i e r In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e - quirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n Department of Forestry The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date Columbia permission. ABSTRACT Two t h e o r e t i c a l formulations of logging cable sys- tem p r o b l e m s , t h e c a t e n a r y m o d e l and t h e p a r a b o l i c m o d e l a r e i n v e s t i g a t e d and compared w i t h t h e r e s u l t s o f e x p e r i m e n t s executed on a g r a v i t y s y s t e m f i e l d The s t u d y model. shows t h a t a l t h o u g h t h e shape o f a f r e e hanging c a b l e i s b e t t e r d e s c r i b e d as a c a t e n a r y bola, both t h e o r e t i c a l models a r e a c c u r a t e than a para- enough t o s o l v e p r a c t i c a l c a b l e system problems. The f e w d y n a m i c t e s t s t r i e d on t h e f i e l d model show t h e g r e a t i m p o r t a n c e o f t h e d y n a m i c f o r c e s i n a l o g g i n g c a b l e s y s t e m and t h e need f o r f u r t h e r r e s e a r c h i n t h i s field. iii T A B L E OF CONTENTS Page Abstract Acknowledgements CHAPTER 1: INTRODUCTION 1 CHAPTER 2: INTRODUCTION TO CABLE MECHANICS 8 2. .1 2. 2 2. 3 2. 4 General D e s c r i p t i o n of the, System M o d e l l i n g Assumptions Catenary Model P a r a b o l i c Model CHAPTER 3: DESCRIPTION OF 3. 3. 3. 3. 3. 3. 1 2 3 4 5 6 3. 7 3. 8 3. 9 THE FIELD MODEL S i t e Dimensions and C h a r a c t e r i s t i c s Cables Carriage Winches Dynamometers Surveying of Cable Shape and C a r r i a g e Position Other Measurements A c c u r a c i e s of Instruments and Expected E r r o r s i n the Measurements Dimensional S i m i l i t u d e between the Model and A Real Yarding System CHAPTER 4: FREE HANGING CABLE 8 11 14 16 19 19 20 23 23 29 32 36 39 43 46 D e s c r i p t i o n of the Experiment A n a l y s i s of the R e s u l t s Error Analysis R e s u l t s and C o n c l u s i o n s 46 47 53 57 CHAPTER 5: CLAMPED LOAD ON A SINGLE LINE 80 D e s c r i p t i o n of the Experiment A n a l y s i s of the R e s u l t s R e s u l t s and C o n c l u s i o n s 80 81 87 4. 4. 4. 4. 1 2 3 4 5. 1 5. 2 5. 3 CHAPTER 6: GRAVITY SYSTEM 6. 1 6. 2 6. 3 D e s c r i p t i o n of the Experiment A n a l y s i s of the R e s u l t s R e s u l t s and C o n c l u s i o n s 95 95 96 98 iv Page CHAPTER 7: DYNAMIC TESTS 111 7.1 7.2 Equipment Tests 111 111 7.3 Conclusion 124 CHAPTER 8: DISCUSSION AND CONCLUSION 125 Literature Cited 129 Appendix 1 131 Appendix 2 18 3 Appendix 3 186 Appendix 4 205 V LIST OF F I G U R E S Figure 1 Page The three and cable systems experimented analyzed 7 2 Characteristics 3 Loading assumption for the derivation and reference frames o f t h e catenary model Loading and reference 4 for of a cable assumption the derivation yarding system 10 13 frames of the parabolic model 13 5 Sketch of plan view of the f i e l d model 22 6 Sketch of view of the f i e l d model 22 7 Plan view layout and dimensions Side view layout and dimensions 8 9 10 11 Y-position Experiment side of the surveying 34 of the surveying 34 of points of the cable: versus models. T e s t #4 Y-position of points of the cable: Experiment versus catenary model Influence of t h e error on T . Test 50 #4 56 Y-position of points of the cable: Experiment versus p a r a b o l i c model Influence of t h e error on T . T e s t #4 56 Y-position Experiment T e s t #4 59 n 12 13 14 15 of points of the cable: versus model Error-zone Y-position of points of the cable: Experiment versus model,- Average and maximum d i f f e r e n c e f o r t h e n i n e free hanging cable tests 59 Cable shape: Experiment versus b e s t - f i t models - Average d i f f e r e n c e f o r t h e nine free hanging cable tests 62 Percent d i f f e r e n c e between T and parameter c a l c u l a t e d f o r b e s t - f i r curves, for the nine free hanging cable tests 62 vi Figure 16 Page Y - p o s i t i o n of p o i n t s o f the c a b l e : Catenary versus p a r a b o l i c model. Test #4 17 Sketches of p a r a b o l i c and catenary hanging cable shapes 18 D e f l e c t i o n a t mid-span: Catenary p a r a b o l i c model 19 Tension a t the lower support: versus models Experiment 20 Tension a t the lower support: versus p a r a b o l i c model Catenary 21 free versus 65. 65 67 70 70 Angle of the l i n e with the h o r i z o n t a l at the upper support: Experiment versus models 73 Angle o f the l i n e w i t h the h o r i z o n t a l a t the lower support: Experiment versus models 73 Angle of the l i n e w i t h the h o r i z o n t a l a t the upper support: Catenary versus p a r a b o l i c model 75 Angle of the l i n e w i t h the h o r i z o n t a l at the lower support: Catenary versus p a r a b o l i c model 75 25 Cable l e n g t h : Experiment 78 26 Cable l e n g t h : Catenary versus model 22 23 24 27 28 versus models parabolic 78 S k y l i n e w i t h a s i n g l e concentrated load f o r t h r e e d i f f e r e n t p o s i t i o n s of the clamped load 86 Y - p o s i t i o n of the l o a d : versus models 86 Experiment 29 Y - p o s i t i o n of the l o a d : Catenary p a r a b o l i c model versus 30 Sketches of catenary and p a r a b o l i c clamped load load-paths 89 89 vii Figure 31 Page Force balance at the clamped load u s i n g catenary and p a r a b o l i c model 91 32 Tension a t the lower support: versus models Experiment 94 33 Tension a t the lower support: versus p a r a b o l i c model Catenary 34 35 36 Y - p o s i t i o n of the c a r r i a g e : versus models 94 Experiment 100 M a i n l i n e shapes as p r e d i c t e d by the models f o r t h r e e of the g r a v i t y system tests 100 Y - p o s i t i o n of the c a r r i a g e : versus p a r a b o l i c model 104 Catenary 37 Skethces of the catenary and p a r a b o l i c g r a v i t y system load paths 104 38 Tension a t the lower support: versus models Experiment 106 Tension at the lower support: versus p a r a b o l i c model Catenary 39 40 41 42 43 44 45 46 106 Tension i n the m a i n l i n e at the upper support: Experiment versus models 10 9 Tension i n the m a i n l i n e at the upper support: Catenary versus p a r a b o l i c model 109 Chart r e c o r d i n g of the t e n s i o n i n the s k y l i n e a t the upper support d u r i n g v e r t i c a l o s c i l l a t i o n s of the load 116 Sketch of a dynamic t e s t . stopped w i t h the m a i n l i n e Carriage 118 Sketch of a dynamic t e s t . stopped w i t h a clamp Carriage 118 Chart r e c o r d i n g s of the s k y l i n e and m a i n l i n e t e n s i o n s d u r i n g a dynamic t e s t 121 Chart r e c o r d i n g s of s k y l i n e t e n s i o n d u r i n g dynamic t e s t s 123 viii Figure Page 47 P a r a b o l a i n the c o o r d i n a t e system 48 B a s i c p r i n c i p l e of the tensiometer 188 49 Copy of the b l u e - p r i n t of the tensiometer 191 50 Sketch of the equipment s e t up f o r the c a l i b r a t i o n of the tensiometer 19 6 51 Tensiometer r e a d i n g versus t e n s i o n i n the l i n e b e f o r e c a l i b r a t i o n 196 I n f l u e n c e of the gauge f a c t o r of the i n d i c a t o r , on the tensiometer r e a d i n g 199 I n f l u e n c e of the zero d i a l adjustment on the tensiometer r e a d i n g 199 Tensiometer r e a d i n g versus t e n s i o n i n the l i n e a f t e r c a l i b r a t i o n 202 52 53 54 (x,y) 183 ix L I S T OF PLATES Plate 1 Page F i e l d model seen from the lower support spar at the 25 2 V i e w o f t h e c a r r i a g e and l o a d , towards the upper support 3 M a i n l i n e d i r e c t e d t o t h e G e a r m a t i c 19 winch w i t h a b l o c k a t the upper support 28 S k y l i n e passing the top of the spar at t h e l o w e r s u p p o r t and c o n n e c t e d t o t h e load c e l l 28 S k y l i n e and m a i n l i n e the upper support 31 4 5 6 looking tensiometers Reading of the t e n s i o n s at the upper support i n the 25 at two lines 31 7 S u r v e y i n g o f t h e c a b l e and c a r r i a g e p o s i t i o n s w i t h the t h e o d o l i t e 38 8 Measurement o f t h e f r a c t i o n o f metre between lower s u p p o r t r e f e r e n c e p o i n t and t h e f i r s t p a i n t mark on t h e c a b l e 38 S t r i p c h a r t r e c o r d e r , g e n e r a t o r and transformer r e g u l a t o r used f o r the r e c o r d i n g of the tensions at the upper support 113 Manual i n i t i a t i o n of the v e r t i c a l o s c i l l a t o r y motion of the clamped l o a d 113 9 10 X L I S T OF TABLES Table Page I Cable II Requirements III Experimental errors IV Two V characteristics 23 f o r the winches systems t h a t 26 42 the model can F i e l d and computed r e s u l t s . . h a n g i n g t e s t #4 simulate Free 51 VI Experimental errors. Free hanging VII Discrepancies i n free characteristics hanging VIII Experimental errors. Clamped IX Precision of f i e l d 45 cable 53 segment 82 load test. 84 model, r e a l y a r d i n g system 126 X Requirements XI Load XII Gauge f a c t o r XIII Experimental r e s u l t s . Free hanging test 208 XIV Experimental r e s u l t s . Clamped load test 209 XV Experimental r e s u l t s . Gravity system cells f o r the tensiometers 19 2 characteristics 193 adjustment 200 test 210 xt ACKNOWLEDGEMENTS I wish t o express my g r a t i t u d e t o Mr. G.G. my s u p e r v i s o r , who dent program and guided me I would of i n the development l i k e t o thank Mr. G.V. graduate s t u of t h i s thesis, Wellburn, manager the F o r e s t E n g i n e e r i n g Research I n s t i t u t e of Canada (FERIC) f o r the f i n a n c i a l support t h a t made t h i s study possible. tor a s s i s t e d me throughout my Young, I would a l s o l i k e t o thank Mr. J . W a l t e r s , D i r e c - of the U n i v e r s i t y of B r i t i s h Columbia Research F o r e s t (UBCRF) f o r the use of f a c i l i t i e s on the F o r e s t . I am a l s o t h a n k f u l to the f o l l o w i n g persons f o r their help: the members of FERIC and the members of the UBCRF who a s s i s t e d me i n the f i e l d work. Mr. D. Myhrman, mechanical engineer at FERIC, f o r his guidance and c o n s t r u c t i v e Mr. D. Anderson criticism. f o r h i s a s s i s t a n c e i n the field. Mr. K. Vatsag f o r h i s e x c e l l e n t machining work. Messrs. H. J o l l i f f e and J . Walters f o r r e v i e w i n g my and thesis. Mrs. C. van Beusekom f o r her f a s t and a c c u r a t e typing. E X P E R I M E N T A L STUDY OF CABLE LOGGING CHAPTER SYSTEMS 1 INTRODUCTION Cable systems f o r h a n d l i n g and t r a n s p o r t i n g logs are widely u t i l i z e d by the f o r e s t i n d u s t r y i n the P a c i f i c Northwest. E a r l y i n the h i s t o r y of l o g g i n g , c a b l e s were used to h a r v e s t timber to and c a b l e systems have been improved i n c r e a s e the e f f i c i e n c y o f the o p e r a t i o n . development of these The major systems was c e r t a i n l y the i n t r o d u c t i o n of " h i g h - l e a d " at the t u r n o f the century. t e c h n o l o g i c a l advancements Since then new have been i n t r o d u c e d and the e x i s t i n g systems have c o n s t a n t l y evolved towards the new requirements o f the l o g g i n g i n d u s t r y . A look upon the present s i t u a t i o n i n d i c a t e s a need f o r y a r d i n g systems t h a t w i l l : i ) Reduce f o r e s t road d e n s i t y because of high road c o n s t r u c t i o n c o s t and because o f e n v i r onment c o n s t r a i n t s , i i ) Harvest efficiently timber on s i t e s i n ^ a c c e s s i b l e with c o n v e n t i o n a l iii) Meet the needs f o r improved practices. systems, silvicultural - 2 - i v ) P r o t e c t the environment. Cable l o g g i n g systems can meet t h i s challenge. The most e f f i c i e n t use of e x i s t i n g methods and opportunity of developing new ideas r e q u i r e s t h a t the the engin- e e r i n g c h a r a c t e r i s t i c s of c a b l e systems be w e l l known. Those c h a r a c t e r i s t i c s are s t u d i e d i n cable mechanics. S o l u t i o n s to problems i n cable mechanics f o r the p a r t i c u l a r case of l o g g i n g were attempted a long time Although the b a s i c problem i s easy to formulate mathemati- c a l l y , numerical s o l u t i o n s are d i f f i c u l t to o b t a i n . techniques were developed to circumvent the however, only the more recent and here. Several difficulty, important ones are Lysons and Mann(6) p u b l i s h e d ago. reported a graphical-tabular method to determine what payload a l o g g i n g system can carry over a given p r o f i l e . the Carson and Mann(2) reformulate a n a l y s i s , d e s c r i b i n g the l i n e segment as a catenary, present catenary and an i t e r a t i v e technique f o r the s o l u t i o n of s k y l i n e equations. Another p u b l i c a t i o n by Carson and Mann(3) proposes an a l g o r i t h m to determine the load path of a running s k y l i n e u s i n g a s t r a i g h t l i n e approximation f o r the load d i s t r i b u t i o n on the l i n e segments; t h i s assumption y i e l d s a p a r a b o l i c shape f o r the l i n e segment. The ment of the p a r a b o l i c model f o r the study of cable i s presented i n Appendix 1 and of t h i s t h e s i s . represents developsystems a significant part - 3 - Two major t h e o r i e s , the catenary model and the p a r a b o l i c model, are t h e r e f o r e a v a i l a b l e t o d e s c r i b e systems. cable However, even the most e l a b o r a t e f o r m u l a t i o n i s based on c e r t a i n degrees o f assumptions, and a q u e s t i o n r e mains as t o know how w e l l the t h e o r i e s r e p r e s e n t the a c t u a l systems. F i e l d measurements a r e r e p o r t e d t o have been made on r e a l l o g g i n g systems(8)(9)(10) and p r a c t i c a l t a b l e s were proposed f o r some s p e c i f i c To the author's cases. knowledge no other experimentation has been c a r r i e d out t o i n v e s t i g a t e thoroughly of l o g g i n g c a b l e systems. The need f o r an the mechanics experimental study t o c o n f i r m the t h e o r e t i c a l approaches would t h e r e f o r e seem necessary. The l i m i t a t i o n s o f the mathematical i s another p o i n t t o be c o n s i d e r e d . formulations Most o f the models assume t h a t the c a b l e s and load are f r e e from the ground; only Carson(4) makes an attempt t o model the dragging l o g , but t h a t model should be t e s t e d i n the f i e l d . of a Another l i m i t a t i o n , and c e r t a i n l y the most r e s t r i c t i v e one i s t h a t all s t u d i e s a r e based on the f o r m u l a t i o n o f the s t a t i c e q u i l i b r i u m o f the system when i t i s obvious t h a t the s k i d ding of logs i s a h i g h l y dynamic o p e r a t i o n . No simple a n a l y t i c a l study can model a c c u r a t e l y the behaviour o f a c a b l e system i n s i t u a t i o n s such as l o g hangups, dragging - 4 - logs, from yarding other plement the sources. the theory This mechanics. carry out logs Again and thesis The to validate secondly to extend iments the and study second more phase The gress from towards be the enumerated as i) achieved for hanging system flyer of these small as The portion to the and limits f i r s t to experphase of of the thesis. the the simplest This cable Gravity tion system study was to cable progression prosystem can follows: Free system Standing Running and the for sophisticated. i i i ) Binkley for this problems. formulation, those com- primarily situations. a of cable limitations beyond selected load V.W. were Only Clamped v) study loading can practical the results ii) iv) the same shock approach of dynamic the or study the investigation most of mathematical complex of ground a research approach the of completed. was some within the analysis were answer the the empirical describes tests theories from an objectives f i e l d investigate free D.D. basic on a - (Figure single Live (Figure skyline la). line skyline; (Figure lb). shotgun or l c ) . with haulback line, skyline. Studier(l) systems and present their a complete numerous descrip- variations. - 5 - The running skyline experiment hanging studied regarded suspended step on cable as the of the u t i l i z i n g a the should component i t of of a load be of gravity yarding carried of any on a and on f i r s t can be free cable single system the system line with and can a be f u l l y primarily as a system. high system out the The considered problems and assumptions highlead the in system. should the cable be modelling clamped line considered basic study real haulback gravity The but not the simulation Because tests were with simpler the f i r s t . load, towards most is a skyline systems since investigated was standing i t an cost was involved realized experimental in that physical model. After parabolic a brief models, f i e l d model f i e l d measurements i s t i c s line bolic with of and free the models this and the thesis presents and hanging gravity are are the to describes the the comparative catenary and experiment on analysis theoretical values for cable, load system. also analysis. Recommendations introduction compared The the clamped The catenary to dynamic stated in the each tests are the on model other of in the charactera single and para- p a r a l l e l presented. conclusion. the - 6 - Figure 1 - The three cable systems experimented and analysed: a) b) c) free hanging cable clamped load on a single l i n e gravity system - ,7 Sky!ine S k y l i ne mped load Skyline Ma in 1 i ne Carr i a CHAPTER 2 INTRODUCTION TO CABLE MECHANICS T h i s chapter presents t e r i s t i c s o f a c a b l e system. c a b l e mechanics are then 2.1 General and d e f i n e s the b a s i c The t h e o r e t i c a l approaches t o introduced. D e s c r i p t i o n of the System. F i g u r e 2 i l l u s t r a t e s the important c a b l e y a r d i n g system. classified charac- f e a t u r e s of a The c h a r a c t e r i s t i c s shown can be i n two groups; the f i r s t group d e f i n e s the dimen- s i o n s and geometry o f the system; the second group d e s c r i b e s the f o r c e s a c t i n g on i t . The nomenclature i n t r o d u c e d i n the p r e s e n t a t i o n o f these c h a r a c t e r i s t i c s w i l l be used throughout the remainder of t h i s Geometrical thesis. characteristics: ;C: Carriage A: Lower support o f the c a b l e / B: Upper support L: Span; h o r i z o n t a l d i s t a n c e between the supports E: D i f f e r e n c e i n e l e v a t i o n between the supports AB: o f the c a b l e _ J Chord 0: Angle of the chord with the h o r i z o n t a l D: D e f l e c t i o n ; v e r t i c a l d i s t a n c e between the - 9 - Figure 2 - Characteristics of a cable yarding system c a b l e and t h e c h o r d a t any point along the cable X: H o r i z o n t a l p o s i t i o n o f a p o i n t on t h e i n the coordinate Y: system cable (X,Y) V e r t i c a l p o s i t i o n o f a p o i n t on t h e c a b l e i n the coordinate system (X,Y) S: Cable a: Angle of the cable w i t h the h o r i z o n t a l a t point. length any - 11-- Force characteristics: co: Weight o f the c a b l e per u n i t l e n g t h R: Weight o f the c a r r i a g e and l o a d T: Tension H: H o r i z o n t a l t e n s i o n i n the c a b l e i n the c a b l e R e l a t i o n s h i p s between the above c h a r a c t e r i s t i c s can be d e r i v e d u s i n g b a s i c mechanics p r i n c i p l e s . V a r i o u s mathema- t i c a l models have been proposed depending on the u n d e r l y i n g assumptions made. 2.2 Modelling Assumptions. The g e n e r a l d e r i v a t i o n s o f the e x i s t i n g formula- t i o n s , the catenary model and the p a r a b o l i c model, a r e c l a s s i c a l a p p l i e d mechanics problems and have been by I n g l i s ( 5 ) . Both these models a r e based on the assumption t h a t the c a b l e i s an i n f i n i t e l y f l e x i b l e body which t h a t no bending r e s i s t a n c e i s considered of the f o r c e s . described implies i n the accounting Another assumption i s made as t o how the u n i f o r m l y d i s t r i b u t e d weight, w, a c t s on the system. catenary model c o n s i d e r s co as u n i f o r m l y d i s t r i b u t e d The along the c a b l e l e n g t h whereas the p a r a b o l i c model s i m p l i f i e s the problem and assumes w d i s t r i b u t e d on the chord o f the system (Figure 3 and F i g u r e 4 ) . distinct T h i s b a s i c d i f f e r e n c e leads t o formulations. I t i s one of the o b j e c t i v e s of t h i s study pare the r e s u l t s o f both t h e o r i e s a p p l i e d t o c a b l e t o comlogging - 12 - Figure 3 - Loading assumption and reference frames f o r the derivation of the catenary model Figure 4 - Loading assumption and reference frames f o r the derivation of the parabolic model - 13 - - 14 - systems and mainder of parabolic to compare both with the experiment. the chapter describes the catenary Catenary This (x', the y') below to basic the point cable of so system, from the expressions ment OP A. The given catenary = The is the sion, is is in coordinate that origin sag the (Figure of the static 0' is 3). cable at In Both system a distance this shape equilibrium co- derived of the seg- catenary: x' — m-- with equation with m cosh i t s determined of same - m distance, the i t s the Mann(2). m = can H/u) be origin (a) translated at the to lower a co- support becomes: y-intercept H, the presented equation m cosh (x,y) is and the The y and theory Carson of by catenary equation = equation (x',y') the y' system the maximum ordinate is of Inglis(5) positioned ordinate and Model. summary reference define model re- model. 2.3 with The a, by at from E, the m cosh m L (x,y) and m, catenary. any — point on (b) to the original where m, Since horizontal the equal cable, m is frame to H/w ten- a m - 15 - constant and becomes a convenient formulation. expressed The parameter f o r the catenary other system c h a r a c t e r i s t i c s can e a s i l y i n terms of the parameter m. The t e n s i o n T^ and and the angles of the l i n e with the h o r i z o n t a l a n given a t the lower and upper supports and by: (c) (d) T„ = oom cosh B (e) (f) 13 and Tg, rv , are T, = com cosh — and tga,. = s i n h — A m A m m be tga„ = s i n h — — ^ B m I t can e a s i l y be shown t h a t the d i f f e r e n c e between the ten- s i o n s at the supports' i s d e f i n e d by the simple r e l a t i o n s h i p : T B - T A = ooE (g) T h i s i s a very u s e f u l e x p r e s s i o n i n the catenary f o r m u l a t i o n of c a b l e mechanics. The Dm And d e f l e c t i o n a t mid-span can be c a l c u l a t e d from: E = 2 , L/2-a - m cosh — m (h) i t i s easy to express the c a b l e l e n g t h as: S = m (sinh — m + s i n h -J) m (i) T h e r e f o r e , most of the system c h a r a c t e r i s t i c s are expressed tions. simply u s i n g the h y p e r b o l i c s i n e and c o s i n e func- However, the t r a n s c e n d e n t a l p r o p e r t y of the hyper- b o l i c f u n c t i o n s render t h e i r use i m p r a c t i c a b l e f o r the of c a b l e systems without has been devoted the a i d of a computer. t o develop study Much work i t e r a t i v e techniques and com- puter programs t o p r o v i d e numerical s o l u t i o n s t o catenary models of s k y l i n e problems. The l a t e s t and most e l a b o r a t e i s t h a t by Carson and Mann(2) who i t e r a t i v e technique adopted this developed the " r i g i d link" f o r the catenary a n a l y s i s i n paper. 2.4 P a r a b o l i c Model. The p a r a b o l i c theory as i t a p p l i e s to c a b l e l o g - ging systems i s developed i n Appendix 1. The development presented progresses from the b a s i c f r e e hanging cable to the more complex f i v e - l i n e system and emphasize each major r e s u l t w i t h numerical examples. Only the b a s i c f e a t u r e s and r e s u l t s of t h i s theory are summarized i n t h i s The equation of the f r e e hanging developed i n the c o o r d i n a t e system (x', y ) 1 section. c a b l e shape, u s i n g the equa- t i o n s of s t a t i c e q u i l i b r i u m of the segment of c a b l e OP (Figure 4) i s - This (x,y) defines equation, the co where H, meter. system the The can V The L of the expressed tensions " the port line are with defined T A = A T the — "and A _ J _ cosa ^ and the the coL P > ,, .. considered as (b ) 1 J be and T_, characteristics and = ^ A tga B the f- £ of the a_ a para- of the H. angles ti lower and upper sup- relationships: - _ 2 L = and terms A following tgcv in a,, H at system as: ) x cbs9H Q coordinate cable conveniently horizontal, by cosa = B the of other A of o 2 to tension, can expressions be shape 2 . ,E + (7- x horizontal also translated parabolic 2 cosGH J 17 a ) L n „ (c') cosGH (d') v <e'> + ' <f > D X5 The deflection The length L/ Oy of at the mid-span is cable derived is simply expressed from: as: - The - s o l u t i o n of t h i s i s g i v e n i n A p p e n d i x 2. approximate formula s = — ^cose The 18 I f the c a b l e i s t i g h t the c a n be (1 and following used: + 3 fL ^ ) ) 2 p a r a b o l i c equations simple to manipulate. are r e l a t i v e l y easy However, t h e e x p r e s s i o n s o f a n g l e s w i t h t h e h o r i z o n t a l and cumbersome and i n t e g r a l i s not t r i v i a l and line the cable length formula c o n s t i t u t e a drawback t o the theory. are - 19 - CHAPTER 3 DESCRIPTION OF T H E F I E L D The the major f i e l d desired and model components instrumentation was designed of a gravity was i n t e g r a t e d variables. the various MODEL system. In t o t h e system The g e n e r a l pieces t o incorporate a l l layout o f equipment addition, t o measure of the f i e l d are described the model i n this chapter. 3.1 Site A at location the University (UBCRF). forest done the Dimensions and had been mid-span could recently the choice Research logged. end of the The f i n a l was i n s t a l l e d found Forest i n the northern surveying, and adjusted, Span L = 131.95 i n elevation, E = 23.05 deflection be obtained Other in Columbia was gave dimensions: Difference span) f o r the experiment was l o c a t e d t h e equipment following A maximum suitable of B r i t i s h The s i t e after and C h a r a c t e r i s t i c s . elements of this Dm = f o r a free were site. also metres metres. 1 0 . 5 m ( i . e . 8% o f t h e hanging taken Anchorings cable. into were consideration available a t - each end for rocky base lower and lation path to the for to of under walk. the The conditions of upper cable, ground very the supports. that and the cleared p r o f i l e similar to cables, There end, equipment the - rigging the the 20 as was well easy of of branches allowed those of the a firm to the the i n s t a l - the tests. and snags tests real a access, f a c i l i t a t e d execution as to was be yarding The easy run in oper- ation. Figure 5 A plan view and p r o f i l e and 6. Each element of of the the site model are w i l l shown now be in des- cribed. 3.2 Cables. The widely used cables in the were 6x19 logging different diametres were 7/16-inch diameters cables eye at one end. The marked precisely ment. The 7/16 the carriage are summarized 2-metre from the long cable along in cable supplier's 135 Both came with a cable, used as was Table type industry. metre the a obtained. 5/8 every IWRC with skyline. I. samples The The are not information. the per of two made length mainline and flemish skyline, for cables rope 5/8-inch factory the weight wire metres the paint, u t i l i z e d as of was measure- to move characteristics metre significantly measured different on - 21 - Figure 5 - Sketch of plan view of the f i e l d model Figure 6 - Sketch of s i d e view of the f i e l d model - 22 - Upper support - 23 - Table I . Cable c h a r a c t e r i s t i c s . w (kg/m) #(inch) Measured Catalogue Difference % 5/8 1.071 1.027 4 7/16 0.521 0.506 3 JZ5 B.S. (N) (NxlOOO) E N/m 2 io 6 50 3.1 188 76 10 7.6 188 J.C'a'b'le.-.diameter Weight per metre BS Breaking s t r e n g t h Tm Maximum s t a t i c t e n s i o n expected during SF Safety i SF 157 w E Tm (N) (NxlOOO) i n newtons the experiment BS f a c t o r f o r the experiment c o n d i t i o n s = ^ E l a s t i c modulus of the cable 3.3 Carriage The (Plate 2) c a r r i a g e had a s i n g l e b a l l bearing-mounted sheave t h a t f i t s the s k y l i n e . mainline w i t h a shackle means o f a clamp. I t c o u l d be connected t o the o r immobilized on the s k y l i n e by A basket was attached t o the c a r r i a g e t o r e c e i v e up t o 40 l e a d weights t o c o n s t i t u t e the l o a d . The maximum l o a d weight i n c l u d i n g the c a r r i a g e and basket was 535 kilograms. 3.4 Winches ( P l a t e 3) The requirements f o r the winches, shown i n t a b l e I I , were d i c t a t e d by the cables c h a r a c t e r i s t i c s . - 24 - Plate 1 — F i e l d model seen from the spar at the lower support. Plate 2 — View of the carriage and load, looking toward the upper support. - 26 - Table Winch II. Cable diameter (inch) Requirements for the winches. Yes 1 5/8 5 50,000 Slow 2 7/6 130 10,000 0 to 2 After conditions of The were proportion f i n a l between having is not a A everytime Its to were of slow resulted and a Gearmatic accept i t s i t 19 the load The 6-ton speed was on 6-ton winch had t o variable skidder. which be Comelong manual to the speed and the However on a line pull was:just mainline. created was than lack winch. capacity of lowered appreciated laborious drum 120 m e t r e s a more the tensions. mounted hand these of and expected Comelong i t s that because 19 w i n c h supplied Desired obvious exactly diameter but i n f i n i t e l y reversible in the mounted became Gearmatic the experiment sufficient assets a i t t o meet cable were skidder The for research d i f f i c u l t choices rubber-tired needed some Reversible Pull Speed Tensions (m/s) (Newtons) Pull length (metre) Its main convenience the Gearmatic problems and a hazard slowly. connected set handling. to precise the skyline. tensions 19 but - 27 - Plate 3 - Mainline directed to the Gearmatic 19 winch with a block at the upper support. Plate 4 - Skyline passing the top of the spar at the lower support and connected to the load-cell. - 29 = Dynamometers 3.5 The line and meters. tensions at the No force Rugged portable in the f i e l d . er for the those on d i g i t a l recorder. described the layout of the only load-cell off the introduce no to to carriage The to reference support then as the gave load section i t to was a taken of to the at can the way the top of read tape are point the simple of the to The mounted load-cell allowed relieved. the of so the The that lower the in straight- bearing cable to used cable hook or be or key a l l being tension. the swivel record- of was the b a l l a satisfy weight and applied height a l l the shape connect run was keep problems chart a support a enough A lower of was i n s t a l l a t i o n to with c e l l a dynamo- avoid advantage on sky- carriage. load-cells on use to load run tension could point sheave. The a the cable cable to the the with the dynamometers the of load-cells recorded The end compatible from of was in at required present hanging the stump. spar and l e t and a spin 1.2-metre at problem free done be or each mainline Electronic the disturbance f i r s t , to output skyline the was equipment. The line the was in were indicator rigging forward. anchored The 3. of at had also Appendix on sheave and 4 characterisitcs load-cell solution also used. The measured support tests. gauge in 5) dynamometers They The were and measurement requirements commonly a upper dynamic very (Plates the end. lower - 30 - Plate 5 Plate 6 — — Skyline and mainline tensiometers at the upper support. Reading of the tensions i n the two l i n e s at the upper support on the strain-gage indicators connected to the tensiometer. -31- - 32 - For meter, referred built. of The on a the with to mainline steel the the were with tension (Plate The locating and the A a surveyed in most the instrument mark knoll surveying T'. T The of top on to f i r s t They given middle be simply and bolted After their upper Shape and Carriage experiment required to leaving respective sheave the a the type foundations. be capable skyline this to are is can the of had 3, cable lever Both the dynamo- lines. action directed of as the of any point where the circumstances. layout stood at are the horizontal so as to read to plane of the of theodolite from justed the the a line. were Cable Salmoragy on Appendix running the of of reference the sky- point B. Position 7) position easily. and concrete taken of in or type tensiometer", tensiometers The was 11 principle: lines object purpose, of to blocks. Surveying 3.6 fixed the through the as l o a d - c e l l by in secured tensiometer in sheaves a special thesis mechanical to a described tension run frame winches support this three tensiometers line in transmitted related a to simple deflection sheave upper tensiometers, measuring work the zero cable. for the Since plan in a the view to serve could and 7 and theodolite TM 1 of accurately this be p r o f i l e The 8. pre-surveyed direction this up, cable Figures of of cable set entire v e r t i c a l vernier was The shown the a method bench- was ad- perpendicular d i r e c t i o n was not Figure 7 - Plan view and dimensions of the surveying layout Figure £ - Side view and dimensions of the surveying layout - 35 - l o c a t a b l e on the t e r r a i n or on the c a b l e the a c t u a l a d j u s t ment of t h e h o r i z o n t a l v e r n i e r was done a t 41° 18.3' w i t h the t e l e s c o p e of the t h e o d o l i t e p o i n t i n g the lower ence A. Simple refer- geometric d e r i v a t i o n y i e l d s the f o l l o w i n g equation f o r the h o r i z o n t a l d i s t a n c e X from a p o i n t of the c a b l e t o the lower support A. X = A'M' - TM 1 tg(Alpha) where tg(Alpha) represents the tangent of the angle read on the h o r i z o n t a l v e r n i e r of the instrument. With the dimensions i n metres the p r e v i o u s equation becomes: X = 48.411 - 55.094 tg(Alpha)(metres) The o r i g i n of the v e r t i c a l v e r n i e r was a d j u s t e d a t zero w i t h the a x i s o f the t e l e s c o p e of the t h e o d o l i t e i n the h o r i z o n t a l direction. The f o l l o w i n g r e l a t i o n s h i p g i v e s the v e r t i c a l d i s t a n c e Y from a p o i n t of the c a b l e t o the lower support A. Y = AA' - (TM'/cos(Alpha)) x tg(Beta) where cos(Alpha) i s the c o s i n e of the h o r i z o n t a l v e r n i e r r e a d i n g and tg(Beta) i s the tangent of the angle read on the v e r t i c a l With the dimensions Y = vernier. i n metres the p r e v i o u s equation becomes: TT*+14.975-(55.094/cos(Alpha))xtg(Beta)(metres) TT' the h e i g h t of the instrument a x i s t o the bench-mark T' was remeasured a f t e r every s e t t i n g of the instrument. The shape of the c a b l e s and the c a r r i a g e p o s i t i o n - were were using this For large deflections taken by surveyed v i s i b l e direct bench 3.7 from anics marks on taking at and simply line, the clinometer gave The upper lower measurements noted number that a the the angles of few their height from line with the horizontal. they have in cable the with supports and at carriage the This good was pre-, cables This the points position the placing was body level rather the performed of the tube to unusual horizontal level with instrument the mech- a in c l i n o - d i r e c t l y horizontal u t i l i z a t i o n of and a results. painted supports, the and a of unstretched of cable cable. importance mainline. were the their the Unstretched skyline. the the reading. Marks to the of adjusting 3.7.2 and under Angle skyline the of 3.7.1 of of theodolite measurements. recorded by the Other theories, meter technique. measurement Because the - surveyed not were 36 two whole negligible line every length BA, was fractions metres error length metre of the 8) on untensioned of metres A the cable obtained between is (Plate and introduced by between addition at the B. It since the of the extremes should the dis- be - Plate 7 — 37 - S u r v e y i n g of the cable and c a r r i a g e posit i o n s w i t h the t h e o d o l i t e . Plate S Measurement of the f r a c t i o n o f metre between the lower support r e f e r e n c e p o i n t and the f i r s t p a i n t mark on the c a b l e . - tances 39 - at the extremes were measured on the c a b l e under t e n - sion. 3.8 A c c u r a c i e s of instruments and expected e r r o r s i n the measurements. The magnitudes of the e r r o r s a f f e c t i n g the various measurements have to be known i n order to make any conclu- s i o n i n the a n a l y s i s of the r e s u l t s . errors have v a r i o u s o r i g i n s , they may p e r s o n a l or n a t u r a l . Experimental be i n s t r u m e n t a l , Instrumental procedural, e r r o r s r e s u l t from i n s t r u - ment i m p e r f e c t i o n s and non-adjustments. The magnitude o f the p r o c e d u r a l e r r o r i n c r e a s e s with the number of steps performed and number of pre-measurements needed f o r the determination of a given v a r i a b l e . from human l i m i t a t i o n s and were the most d i f f i c u l t Personal errors r e s u l t accidents. to apprehend. The natural errors The weather c o n d i - t i o n s i n p a r t i c u l a r a f f e c t e d the r e s u l t s as v a r i a t i o n s of the t e n s i o n and sag r e s u l t e d from the expansion or c o n t r a c t - i o n of the c a b l e from changes i n temperature. wind had rain and the e f f e c t of i n c r e a s i n g the c a b l e weight per metre r e s u l t i n g i n an i n c r e a s e i n t e n s i o n . Other n a t u r a l e r r o r s r e s u l t e d from the y i e l d i n g of the anchorings ment of the t h e o d o l i t e t r i p o d . observed and to them. The recorded The and the settle- n a t u r a l e r r o r s were but no numerical values were attached - 40 - The remainder of t h i s s e c t i o n e v a l u a t e s the ex- pected e r r o r f o r each types of measurement. 3.8.1 E r r o r s i n the t e n s i o n s measurements. A complete t e s t c a r r i e d out on the tensiometers used a t the upper support i s r e p o r t e d i n Appendix 3. With the recommendations formulated i n Appendix 3, l e s s than one percent e r r o r can be obtained f o r t e n s i o n s g r e a t e r than 5 000 newtons i n the s k y l i n e and f o r t e n s i o n s g r e a t e r than newtons i n the m a i n l i n e . 1000 Although no s p e c i f i c t e s t was done on the r i g g i n g a t the lower support the l o a d c e l l a t t h a t p o i n t i s a l s o expected to be accurate to p l u s or minus one percent. 3.8.2 E r r o r i n the c a b l e p o s i t i o n . Instrumental and p e r s o n a l e r r o r s are c l o s e l y r e l a t e d i n t h i s case. The angular accuracy of the Salmoragy t h e o d o l i t e i s g i v e n i n the s u p p l i e r ' s catalogue as 1/10 a minute. However the experience has proved t h a t 1/5 minute was p r o b a b l y a more r e a l i s t i c of of a l i m i t f o r the angular d e f i n i t i o n of the instrument, because the v e r n i e r s were difficult t o read and the l e v e l l i n g r e q u i r e d d e x t e r i t y . The e r r o r i n the c a b l e p o s i t i o n c r e a t e d by the angular e r r o r depends on the d i s t a n c e L from the t h e o d o l i t e to the cable and i s d e f i n e d by: error = * ;li = 3 ^ n 5x60x180 .000058L - For the longest and equal to 6 mm. simplification cable value errors error of of each the computation. The the telescope than cedural the points is but error is analysis is 6+12 of = 3.8.3 the 18 The be positions of the the line more error the a The the instrumental error; depends on the dimensions AA', on the reasonable. mm. of The cable the be of less of position the mea- total error the in v e r t i c a l to The of used height effect total from improper the r e a l i s t i c .2 measurement clinometer positioning is the of pro- adopted cable of the line angle horizontal. instrument. in a of averaging d i f f i c u l t y measurement the error the and in accuracy e l i m i n a t e d by on point estimated appraise be 12 is sake theodolite. dimension v e r t i c a l position Error instrumental this maximum mm. with The to inaccuracies therefore the is the any reproduced error to to positions the d i r e c t l y 6 mm s e e m s to for pre-measured in This to error is analysis cable the the error added d i f f i c u l t dimensional surements in It be error is model distance of measurement. 6 mm. other axis, the this measured affecting position in to - maximum i t s has the on This applied independent procedural the distance 41 zero reading However the is of a adjustment from two instrument for degree. could inverted experience expectation degree. 1/10 has shown correctly the angle on - 3.8.4 Error Because of points error the on the of as high the the as r e s u l t s from i n a c c u r a c y reasonable length be cable to think that 27 stretched mm fraction 5 mm. .02 percent this reference cable the of metre a t each Another the of the cable. total III. e r r o r , y i e l d i n g an The o r m i n u s 37 expected experimental unmm. errors i s Experimental errors a f f e c t i n g the measured v a r i a b l e s . Nomenclature Variable T Tension Vertical Angle of w i t h the Cable position the l i n e .horizontal length It cable shown i n t a b l e I I I . Table of procedural metres l e n g t h . i s t h e n known t o p l u s A summary o f t h e the i n marking of f o r a maximum 135 length of the marks on i s t h e maximum m a g n i t u d e o f error of length. poor d e f i n i t i o n s h e a v e s and extremes can - i n the i n t h e measurement o f error, is the 42 Magnitude of error the 1% Y 18 a .2 S 37 mm degree mm - Dimensional 3.9 a real The any two-line length of gravity For the line slackline economical model had yarding that to be representative done and for map r^ and metres elevation quently of were the this ratio. unit of for were a real scaled lationship and f u l l the system. by r^ one by and i f were i f or operated. than one scaling metre of The span to the same the represented and the same ratio true. the i f the be is for the lengths of the system. scale for in conse- scaled r a t i o of a represents and units weight w i l l difference ratio force shows scaling model and real model the d e f l e c t i o n were by a the analysis the of the shot-gun scale similar load be small the the scaled model to a smaller ratios, The assumed or system is system. are and simulation system dimensional scale down a winches was determine length for with that scaled model p r a c t i c a l considerations down ratio real Forces is two independent means down the the cable force i f and forces the constant; obtained the the as skyline a length simply considered Nevertheless, Two the The kept system of properly. one was scaled results be standing purposes system. the a between system. could system; - similitude yarding model 43 down One r^. of the following by force cable re- 4.4 r P • R W 0) X payload R weight of W weight per real i r P in 2 r kilogram carriage unit for and the load length real in system. kilogram. of the cable in the length, of the cable in the system. weight co - per unit model, r^ length force Angles with a one example, model two are systems to r e a l almost slopes ratio in any and without systems described of ratio. and one ratio. which table size by unitless values are any distortion. As can be IV. The merely represented model can changing related an by the simulate the two scale ratios. The also part of ground the u t i l i z e d inside could of study. be configuration general the prime and similitude. l i m i t s of importance this in terrain Although thesis, future aspect not those extensions were very two of much points the -45 Table IV. - Example o f two systems t h a t the model c a n s i m u l a t e . . System , Model characteristics 132 Span (metres) Slackline system Long r e a c h standing skyline 600 1500 4.55 11.36 Length r a t i o ( r ^ ) - D i f f e r e n c e s i n e l e v a t i o n (metres) 23 105 261 10.5 48 119 Maximum d e f l e c t i o n (metres) Maximum d e f l e c t i o n (%) Average ground s l o p e (%) S k y l i n e l e n g t h of maximum d e f l e c t i o n Load r a t i o ( r ^ S k y l i n e diameter (inch) S k y l i n e weight p e r metre M a i n l i n e diameter (kilogram) 8 8 17 17 17 136 619 1545 - 14:19. 65.64 5/8 1 1/8 1 1/2 1 3.4 6 3/4 1 1.5 2.6 7/16 (inch) M a i n l i n e weight per metre (m) 8 (kilogram) .5 CHAPTER FREE HANGING 4 CABLE The purpose of t h i s experiment was to investigate the c h a r a c t e r i s t i c s of a f r e e hanging l i n e segment. The s i n g l e l i n e segment i s the b a s i c element of a c a b l e system s i n c e the most complex system can always be c o n s i d e r e d as a more or l e s s i n t r i c a t e arrangement of c a b l e segments hanging f r e e l y between the d i f f e r e n t p o i n t s of attachment. This chapter d e s c r i b e s the f r e e hanging c a b l e experiment, compares the f i e l d and t h e o r e t i c a l r e s u l t s and a l s o compares the c a t e n a r y and p a r a b o l i c 4.1 models. D e s c r i p t i o n of the Experiment. For t h i s t e s t the s k y l i n e was r i g g e d so as t o hang f r e e l y , under i t s own weight, between the lower and the upper supports and was t e n s i o n e d w i t h the Gearmatic 19 winch. 4.1.1 Procedure and Data Collection. Nine f r e e hanging c a b l e t e s t s were executed f o r a range of t e n s i o n s a t the upper support between 2700 t o 11000 newtons, and r e s u l t i n g d e f l e c t i o n s at mid-span between 7.1 and 1.6 p e r c e n t of the span l e n g t h . p o r t e d i n Appendix 4. The r e s u l t s are r e - Each t e s t l a s t e d one hour on average and planned t o be executed by t h r e e o p e r a t o r s : d o l i t e , one a t the upper port. produced sup- recorded i n i n d i v i d u a l Sample pages of these note books are r e - i n Appendix Readings one a t the theo- support and one a t the lower The r e q u i r e d i n f o r m a t i o n was f i e l d note books. was 4. of the v e r t i c a l p o s i t i o n of the c a b l e were taken a t s t a t i o n s f i v e metres apart along the h o r i z o n t a l span. The p o s i t i o n a t mid-span was :.. measured a t the b e g i n - n i n g and a t the end of the t e s t t o check the v a r i a t i o n w i t h time. The t e n s i o n s , the angles of the l i n e w i t h the h o r i z o n t a l and the c a b l e l e n g t h were recorded every 15 a t the upper minutes and lower support, as they proved t o change s l i g h t l y w i t h time. Those v a r i a t i o n s were a t t r i b u t e d t o n a t u r a l phenomena l i k e the y i e l d i n g of the anchorings and the changes i n atmospheric c o n d i t i o n s as sun, r a i n or wind. For most of the t e s t s the upper and lower support measure- ments were done by the same o p e r a t o r which r e s u l t e d i n the i m p o s s i b i l i t y of o b t a i n i n g t r u l y simultaneous r e a d i n g s ; the time l a g between the readings was 4.2 about 5 minutes. A n a l y s i s o f the R e s u l t s . The a n a l y s i s of the f r e e hanging c a b l e i s based on - 48 - the comparison of the f i e l d measurements and p r e d i c t e d by the catenary the r e s u l t s as and p a r a b o l i c models f o r the f o l l o w i n g s i x c h a r a c t e r i s t i c s of the system: Cable shape d e f i n e d by v e r t i c a l Y - p o s i t i o n s of p o i n t s of the Dm: cable d e f l e c t i o n at mid-span T.. and T„: A t e n s i o n s at the CL. and A OL,:- angles of the l i n e with is h o r i z o n t a l a t the S: supports 15 the supports skyline length. Only one of these c h a r a c t e r i s t i c s has t o be g i v e n as a parameter to determine the system and the other v a r i a b l e s com- pletely. T as the Parameter. The measured t e n s i o n at the upper support taken as a parameter. The mined u s i n g the catenary by the f o l l o w i n g c h a r t : other v a r i a b l e s c o u l d be T , was deter- and p a r a b o l i c models as d e s c r i b e d - 49 - Figure 9 - D i f f e r e n c e s i n the Y - p o s i t i o n o f the c a b l e between experiment and c a t e n a r y model, and between experiment and p a r a b o l i c model v e r s u s X - p o s i t i o n s on span, f o r f r e e hangi n g t e s t number 4. - 50 - Computation u s i n g Catenary and P a r a b o l a theories Measured Parameter T B Theoretical values f o r the V a r i a b l e s Y, Dm, T , a , a , S Measured v a l u e s f o r the Variables Y, Dm, T , a , a , S A A A B A B Conclusion OL r E E T3 O E >I ^ 20 O • LA] a. x I a> >o o o LA! Yexp.-Ycat. Yexp.-Ypar. kO ± 18 mm 60 80 100 1 2 ^X» m - T a b l e V. .51- F i e l d and computed r e s u l t s . f o r f r e e h a n g i n g c a b l e T e s t #4. FREE HANGING CABLE TEST # 04 DATE: WEATHER: 31/08/76 SUNNY HOT CREW D. Guimier D. Anderson D. Anderson - THEODOLITE - WINCH - SPAR TEST STARTED AT: COMPLETED AT: 13:00 Y position o f t h e c a b l e on the span. X POSITION IN METRES Y POSITION IN METRES 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 -60.0 65.0 65.97 70.0 75.0 80.0 85.0 90.0 95.0 100.0 105.0 110.0 115.0 120.0 125.0 130.0 131.95 EXPERIM. CATENARY PARABOLA -0.013 -0.332 -0.558 -0.696 -0.741 -0.686 -0.548 -0.305 0.026 0.466 0.978 1.579 2.282 3.079 3.251 3.975 4.964 6.049 7.231 8.499 9.873 11.342 12.903 14.567 16.323 18.192 20.145 22.200 23.024 -0.000 -0.318 -0.543 -0.675 -0.714 -0.661 -0.514 -0.275 0.057 0.483 1.001 1.613 2.319 3.118 3.285 4.012 5.000 6.083 7.261 8.534 9.904 11.370 12.933 14.593 16.352 18.209 20.166 22.223 23.052 0.000 -0.320 -0.546 -0.677 -0.715 -0.659 -0.509 -0.265 0.073 0.506 1.032 1.652 2.366 3.174 3.343 4.076 5.073 6.163 7.347 8.625 9.997 11.464 13.024 14.678 16.426 18.268 20.205 22.235 23.052 14:00 DIFFERENCES IN METRES EXP-CAT EXP-PAR CAT-PAR -0.013 •^0.014 -0.015 -0.021 -0.027 -0.025 -0.034 -0.030 -0.031 -0.017 -0.023 -0.034 -0.037 -0.039 -0.034 -0.037 -0.036 -0.034 -0.030 -0.035 -0.031 -0.028 -0.030 -0.026 -0.029 -0.017 -0.021 -0.023 -0.028 -0.013 -0.012 -0.012 -0.019 -0.026 -0.027 -0.039 -0.040 -0.047 -0.040 -0.054 -0.073 -0.084 -0.095 -0.092 -0.101 -0.109 -0.114 -0.116 -0.126 -0.124 -0.122 -0.121 -0.111 -0.103 -0.076 -0.060 -0.035 -0.028 -0.000 0.002 0.003 0.003 0.001 -0.002 -0.005 -0.010 -0.016 -0.023 -0.031 -0.039 -0.047 -0.056 -0.058 -0.065 -0.073 -0.030 -0.086 -0.091 -0.094 -0.094 -0.091 -0.085 -0.074 -0.059 -0.039 -0.012 0.000 - Table 52 - V. (continued) DEFLECTION AT MIDSPAN AS A PERCENT OF THE SPAN DEFLECTION IN PERCENT EXPERIM. C CATENARY PARABOLA 6.202 6.246 6..2 7.1 DIFFERENCES IN PERCENT EXP-CAT EXP-PAR CAT-PAR 0.026 0.070 0.044 TENSIONS AT THE SUPPORTS TENSIONS IN NEWTONS EXPERIM. DIFFERENCES IN NEWTONS CATENARY. PARABOLA i EXP-CAT iEXP-PAR , CAT-PAR 0 0 UPPER SUP. 3080 3080 3080 0 LOWER SUP. 2815 2838 2844 -22 -29 -6 HORIZONTAL * 2830 2837 * * -6 * HORIZONTAL TENSION WAS NOT MEASURED ANGLES OF THE LINES WITH THE HORIZONTAL ANGLES IN DEGREES DIFFERENCES IN DEGREES EXPERIM. CATENARY PARABOLA EXP-CAT EXP-PAR CAT-PAR UPPER END -23.000 -23.232 -22.917 0.232 0.083 0.314 LOWER END —4.000 -4.169 -4.196 0.169 0.196 0.027 SKYLINE LENGTH LENGTH IN METRES DIFFERENCES IN METRES .CATENARY' • PARABOLA' EXP-CAT' EXP-PAR:. !'GAT-PAR ! EXPERIM'. ' 135.140 135.252 135.232 -0.017 -0.003 0.020 - The 5 3. - comparison of the r e s u l t s c o n s i s t s i n the eva- l u a t i o n o f the d i f f e r e n c e s between the measured and c a l c u l ated v a l u e s f o r each o f the v a r i a b l e s . As an example, the r e s u l t s of the computation f o r Test'#4 i s shown i n Table V. The comparison of the measured and t h e o r e t i c a l Y- p o s i t i o n s of the cable Figure cable i s a l s o presented g r a p h i c a l l y i n 9 where the d i f f e r e n c e s i n the Y - p o s i t i o n s a r e p l o t t e d f o r the e n t i r e span l e n g t h . none o f the t h e o r i e s of the Apparently seem t o agree c l o s e l y w i t h the f i e l d r e s u l t s s i n c e the maximum d i f f e r e n c e f o r each of the model i s much g r e a t e r than the maximum expected e r r o r the c a b l e p o s i t i o n s . However no c o n c l u s i o n can be drawn b e f o r e the i n f l u e n c e o f t h e e r r o r i n t e n s i o n 4.3 Error The (18 mm) i n i s examined. Analysis. expected experimental e r r o r s a f f e c t i n g t h e measured v a r i a b l e s and parameters a r e summarized i n Table V I . Table VI. Designation Experimental e r r o r s a f f e c t i n g the measured v a r i a b l e s . Nomenclature Parameter T ± 1% B ± 18 mm Y T Variables a A ± 1% A , a S Experimental e r r o r B ± 0.2 degree ± 37 mm - 54 - The e r r o r s on the parameter T are a f f e c t i n g B the c a l c u l a t e d v a l u e s o f the v a r i a b l e s . F i g u r e 10 and F i g u r e 11 show the same curves as F i g u r e 9 f o r the catenary model and the p a r a b o l i c model u s i n g v a l u e s of T one percent g r e a t e r and one percent' s m a l l e r than the measured v a l u e . It can be seen t h a t one percent e r r o r i n the t e n s i o n measurement has t o be c o n s i d e r e d i n the a n a l y s i s . The Y - p o s i t i o n of the c a b l e at 100 metres from lower support i s used as an example t o e x p l a i n the of the experimental e r r o r s used, the treatment i n the a n a l y s i s , f o r Y and a l l the o t h e r v a r i a b l e s . The e r r o r i n the parameter T_, allows the c a l c u l a t e d and Y~ 2 p o s i t i o n of the c a b l e Y to vary between Y+e^, shown i n F i g u r e s 10 and e 11 f o r the two models. a I f the s seg- ment AB o v e r l a p s the e r r o r zone the theory agrees w i t h the experiment w i t h i n the margin of experimental e r r o r s . examples shown i n F i g u r e s 10 and w i t h the experiment In the 11 the catenary model agrees f o r the Y - p o s i t i o n of the c a b l e a t 100 metres from the lower support and the p a r a b o l a does not agree w i t h the experiment. A more convenient.way t o r e p r e - sent the same c o n d i t i o n i s to e n l a r g e the e r r o r zone as shown i n F i g u r e 12 where the new d e f i n e d by the a d d i t i o n o f e^ and limits. The boundaries t o t n e D and E are previous error theory agrees w i t h the experiment i f point C - 55 - F i g u r e 10 — D i f f e r e n c e s i n the Y - p o s i t i p n o f the cable between experiment and catenary model versus X - p o s i t i o n on span, f o r f r e e hanging t e s t number 4. The t h r e e graphs a r e drawn: u s i n g the measured t e n s i o n Tg a t the upper support as a parameter; u s i n g Tg one percent g r e a t e r than the measured v a l u e ; and u s i n g T F i g u r e 11 - R one percent smaller. D i f f e r e n c e s i n the Y - p o s i t i o n o f the cable between experiment and p a r a b o l i c model versus X - p o s i t i o n on the span, f o r f r e e hanging t e s t number 4. The t h r e e graphs are drawn: u s i n g the measured t e n s i o n Tg a t the upper support as a parameter; u s i n g Tg one percent g r e a t e r than the measured v a l u e ; and u s i n g T R one percent smaller. - 56 - - 57 - (or C ) f a l l s i n s i d e the new e r r o r zone. T h i s procedure was used f o r the a n a l y s i s of a l l the v a r i a b l e s f o r a l l the t e s t s . The e r r o r - z o n e i s d e f i n e d by d a s h - l i n e s i n the d i f f e r e n t f i g u r e s throughout The the t h e s i s . a n a l y s i s has shown t h a t the values e^ and e.^ are almost identical, showing t h a t the system i s l i n e a r f o r s m a l l v a r - i a t i o n s o f the parameter. found The values e^ and e'^ were a l s o i d e n t i c a l i n the a n a l y s i s which allows t o d e f i n e a common e r r o r - z o n e f o r the catenary and p a r a b o l a models. 4.4 R e s u l t s and C o n c l u s i o n s . 4.4.1 Y - p o s i t i o n o f P o i n t s o f the Cable Experiment versus Models. As shown i n F i g u r e 12 f o r T e s t #4, the average v a l u e s of the a b s o l u t e d i f f e r e n c e s between Y measured along the e n t i r e span and Y c a l c u l a t e d w i t h the catenary and p a r a b o l i c models a r e r e s p e c t i v e l y 27 mm and 70 mm. The maximum o f those d i f f e r e n c e s i s 39 mm f o r the catenary model and 126 mm f o r the p a r a b o l i c model. i s 10 8 mm. The maximum h a l f e r r o r - z o n e width Those c h a r a c t e r i s t i c values c a l c u l a t e d f o r t h e 9 t e s t s a r e shown i n F i g u r e 13. The e r r o r s w i t h t h e catenary model are g e n e r a l l y s m a l l e r than t h a t of the p a r a b o l i c model. Except f o r T e s t #1 the average a b s o l u t e d i f f e r e n c e s a r e s m a l l e r than the maximum e r r o r . The maximum d i f f e r e n c e s - 58 - F i g u r e 12 — D i f f e r e n c e s i n t h e Y - p o s i t i o n o f the cable between experiment and catenary model, and between experiment and p a r a b o l i c model versus X - p o s i t i o n s on span, f o r f r e e hangi n g t e s t number 4. The graph shows t h e e r r o r zone taken i n account t h e e r r o r i n the parameter T . R F i g u r e 13 — Average and maximum d i f f e r e n c e s i n t h e Y - p o s i t i o n s o f p o i n t s o f the cable between experiment and catenary model, and between experiment and p a r a b o l i c model f o r the nine f r e e hanging t e s t s . width f o r each t e s t . Maximum e r r o r zone - points f a l l 60 - i n s i d e the e r r o r boundaries f o r 50% of the f o r both the catenary and the p a r a b o l i c model. The as to whether the c a b l e hangs c l o s e r to a catenary test question or a p a r a b o l i c shape cannot be answered c l e a r l y at t h i s p o i n t because of the dependence of the a n a l y s i s on the e r r o r i n the t e n s i o n T^. A d i f f e r e n t approach w i l l now be used t o i n v e s t i g a t e the shape of the f r e e hanging c a b l e . 4.4.2 The catenary Cable Shape: Catenary or f o l l o w i n g approach c o n s i s t s of f i n d i n g curve and experimental Parabola. the the p a r a b o l i c curve t h a t best f i t the p o s i t i o n measurements of the p o i n t s of the f r e e hanging c a b l e . For example, the catenary f i t s the experiment i s the curve curve t h a t b e s t f o r which the sum of the squares of the d i s c r e p a n c i e s between the measured Y - p o s i t i o n s of the c a b l e and i s minimum. support was t h a t d e f i n e d by the equation of the curve T_.c r e p r e s e n t i n g the t e n s i o n at the upper used as the parameter and the problem was to f i n d what t e n s i o n s at the upper support would g i v e best agreement between the experimental and a catenary shape, and between the experimental a p a r a b o l i c shape. cases was shape of the The the cable shape and search f o r the optimum T_,c f o r a l l implemented using a b i n a r y chop technique F i b o n a c c i golden s e c t i o n s . The imental shape and then based on agreement between the exper- the b e s t - f i t curve i s c h a r a c t e r i z e d by average a b s o l u t e value of the d i s c r e p a n c i e s between the the two. - 61 - F i g u r e 14 — Average d i s c r e p a n c i e s between the measured Y - p o s i t i o n s of p o i n t s of the c a b l e and t h a t p r e d i c t e d by the b e s t - f i t catenary curve and by the b e s t - f i t p a r a b o l i c c u r v e , f o r the nine f r e e hanging t e s t s . F i g u r e 15 — Percent d i f f e r e n c e between Tg, measured t e n s i o n a t the upper support, and parameter c a l c u l a t e d f o r the Tg^ best-fit catenary curve and the b e s t - f i t p a r a b o l i c curve, f o r the nine f r e e hanging t e s t s . -" 63 The nary for is and for average the b e s t - f i t a l l the tests. less for the b e s t - f i t parabola the b e s t - f i t For a l l nine where This the is bola, interesting computed the actual between measured cent the a l l of the curve. both 1% and are for the expected the As curve Except computed the measured tests. parabolic and models for the for tests (see compare both the results computed tensions a 18 and The 3, as p l o t t e d on expected, the tensions computed that the Test #1, error in the the differences tension the are tension experiment. the para- differences expressed for 5 4). the are than is 4, of tension smaller mm. parabolic values Tg. the results Appendix the for cable number catenary tension the than by 14 discrepancy hanging than cate- Figure zone, free largest to the smaller curve so in average error always b e s t - f i t shown the measured measured and the catenary the are the Whereas of p a r t i c u l a r l y is T_,c, and a are shape for tests from d e f l e c t i o n was It optimum by 9 shape. catenary the parabola a l l diverge tests represented curve. For catenary for best discrepancies Figure a 15 perfor for the catenary between the s u f f i c i e n t l y close to give confidence to in - 64 - F i g u r e 16 - D i f f e r e n c e s i n the Y - P o s i t i o n s of the c a b l e between catenary and p a r a b o l i c models versus X - p o s i t i o n s on span, f o r f r e e hanging Test #4. F i g u r e 17 - Sketches of the f r e e hanging c a b l e d e r i v e d from the catenary models. shapes and p a r a b o l i c - 66 - F i g u r e IS — D i f f e r e n c e s i n Dm, d e f l e c t i o n a t mid- span between the c a t e n a r y model and the p a r a b o l i c model versus Tg, t e n s i o n a t the upper support, and versus Dm mid-span d e f l e c t i o n i n the f r e e hanging cable. 4.4.3 Y - p o s i t i o n o f P o i n t s o f the Cable. Catenary Model versus P a r a b o l i c Model. A t y p i c a l graph of the d i f f e r e n c e s between the two models, o b t a i n e d with.the F i g u r e 16. c o n d i t i o n s o f . T e s t #4, i s shown i n The p a r a b o l a i s s l i g h t l y under the catenary a t the lower end but i s p l a i n l y above towards.the upper end. T h i s p o i n t i s i l l u s t r a t e d on the s k e t c h o f the c a b l e shapes shown i n F i g u r e 17. In the example, the p a r a b o l a i s 60 mm above the catenary a t mid^span. The d i f f e r e n c e a t mid-span between the two t h e o r i e s i s p l o t t e d i n F i g u r e 18 versus values o f the t e n s i o n a t the upper support T .. The d i f f e r - ' D ences decrease (i.e. r a p i d l y when the t e n s i o n i n -the l i n e i n c r e a s e s when the percent d e f l e c t i o n a t mid-span d e c r e a s e s ) . o o o E Cvl E ro Ci E o •M O ro o u <— oE 5 ' I -i 10 8 h TR, 4 H X 6 1000 3 D e f l e c t i o n at mid-span, % 10 - I f t h e d e f l e c t i o n c a t e n a r y b y i s o n l y 2 4.4.4 t a k e n a s d i s c r e p a n c y t e n s i o n m o d e l s T ^ a n d s h o w n i n i m e n t a n d w h i c h a n e r r o r z o n e h o w e v e r e n c e m i g h t i n a t t h e c o m p a r e d b o t h 19 b e t o t h e s h o w a m o r e t h e i s a was l was a b o v e t h e c a n t e s t s b e t h a t b o t h b e t w e e n #1 e x p e r - f o r T h e m a x i m u m t h a n 2%, p e r c e n t t h e T h e r e s u l t s e x c e p t o n e , s h o w n . T h e t a k e n . n n o w i t h l a r g e r w i t h i n T t h e r e f o r e a g r e e m e n t t h e s h o w i n g t h a n s u p p o r t , v a l u e . s l i g h t l y f a l l M o d e l s . e v a l u a t e d p r o b a b l y v e r y p o i n t s u p p e r p o i n t g o o d l v e r s u s a n a l y s i s , t h a t v e r y m e a s u r e m e n t s 4.4.5 p a r a b o l a t h e m e a s u r e d f o r i s a c c u r a t e t h e a t r e a d i n g h a l f - w i d t h o f i n a t s u p p o r t t h e o r i e s e r r o n e o u s t h e t e n s i o n l o w e r t h e E x p e r i m e n t t e n s i o n t h e 3% m i d - s p a n . p a r a m e t e r F i g u r e f r o m a t m e a s u r e d a m o s t t h a n T e n s i o n s . T h e was l e s s mm - 68 d i f f e r - d y n a m o m e t e r s e x p e c t e d . T e n s i o n s : C a t e n a r y M o d e l v e r s u s P a r a b o l i c M o d e l . T h e s u p p o r t c o m p u t e d m o d e l a r e u p p e r s u p p o r t . i s a l w a y s d e f l e c t i o n t h e t w o d i f f e r e n c e s w i t h p l o t t e d T h e l a r g e r o f o n t h e t h e o r i e s c a t e n a r y F i g u r e 20 t e n s i o n t h a n t h a t c a b l e i s b e t w e e n a t a b o u t t h e m o d e l w i t h t h e m i d - s p a n 1% a n d v e r s u s o b t a i n e d a n d t e n s i o n s w i t h t h e w i t h t h e t e n s i o n t h e c a t e n a r y t h e a t t h e p a r a b o l i c a t t h e p a r a b o l i c m o d e l . d i s c r e p a n c y d e c r e a s e s l o w e r r a p i d l y A t m o d e l 1 0 % b e t w e e n t o . 0 3 % - 69 - Figure 19 - Differences i n T , tension at the lower A support, between experiment and catenarymodel, and between experiment and parabolic model, f o r the nine free hanging t e s t s . Figure 20 — Differences i n T^, tension at the lower support, between the catenary model and parabolic model versus Tg, tension at the upper support, and versus Dm, mid-span d e f l e c t i o n i n the free hanging cable. -70. - ^-Catenary •fParabol i c Mid-span d e f l e c t i o n , % - 71 - f o r a d e f l e c t i o n a t mid-span o f 3%. The same a n a l y s i s c a r r i e d f o r the h o r i z o n t a l t e n ^ s i o n i n the l i n e y i e l d s the same c o n c l u s i o n s as f o r the t e n s i o n a t the lower 4.4.6 support. Angles o f the Gable w i t h the H o r i z o n t a l . Experiment versus Models. The h i s t o r i g r a m s o f the d i f f e r e n c e s between the measured angles and the c a l c u l a t e d angles are shown i n F i g u r e s 21 and 2 2 f o r both model and both end o f the c a b l e . 85% of the p o i n t s are w i t h i n the e r r o r zone a t the upper support and o n l y 45% a t the lower support. of t h i s disagreement sheaves An e x p l a n a t i o n i s found c o n s i d e r i n g the r a d i u s of the a t the upper and lower support not taken i n t o con- s i d e r a t i o n i n the t h e o r e t i c a l models. 4.4.7 Angles o f the Cable with the H o r i z o n t a l . Catenary Model versus P a r a b o l i c Model. The d i s c r e p a n c y between the angles o f the l i n e a t the upper support c a l c u l a t e d with both model i s p l o t t e d i n F i g u r e 23 versus t e n s i o n a t the upper support. The p a r a b o l a i s always above the catenary a t the upper support but the d i f f e r e n c e gets very s m a l l as the t e n s i o n i n c r e a s e s . The e q u i v a l e n t curve i s shown i n F i g u r e 2 4 f o r the angle a t the lower support. For l a r g e d e f l e c t i o n at mid-span ( i . e . i f the - Figure 21 — 72 - Differences i n « , angle of the l i n e with B the horizontal at the upper support, between experiment and catenary model, and between experiment and parabolic model, f o r the nine free hanging t e s t s . Figure 22 — Differences i n o^, angle of the l i n e with the horizontal at the lower support, between experiment and catenary model, and between experiment and parabolic model, f o r the nine free hanging t e s t s . - 74 - F i g u r e 23 — D i f f e r e n c e s i n dg, angle of the l i n e w i t h the h o r i z o n t a l a t the upper support, between catenary model and p a r a b o l i c model, versus Tg, t e n s i o n a t the upper support, and v e r s u s Dm, mid-span d e f l e c t i o n i n the f r e e cable. hanging Sketch of the r e l a t i v e p o s i t i o n o f the c a t e n a r y and p a r a b o l a . Figure 24 — Differences i n a , A angle of the l i n e w i t h the h o r i z o n t a l a t the lower support, between c a t e n a r y model and p a r a b o l i c model, versus Tg, t e n s i o n a t the upper support, and versus Dm, mid-span d e f l e c t i o n i n the f r e e cable. Sketches of the r e l a t i v e of the c a t e n a r y and p a r a b o l a . hanging positions -,7 5 - 76 t e n s i o n i s l e s s than catenary 2750 n e w t o n s ) t h e p a r a b o l a a l o n g the e n t i r e span. the p a r a b o l a t h e two curves i n t e r c e p t a l o n g the span. i s horizontal. span d e f l e c t i o n i s The i s above the For d e f l e c t i o n l e s s than i s under the c a t e n a r y a t the lower f l e c t i o n at which the tangent support - 7% 7% support and i s not the de- t o the cable at the lower T h i s s i t u a t i o n o c c u r s when t h e mid- 4.3%. d i s c r e p a n c i e s between the angles o f the lines a s c o m p u t e d w i t h t h e c a t e n a r y m o d e l and w i t h t h e p a r a b o l i c m o d e l c a n be c o n s i d e r e d as n e g l e c t a b l e f o r m i d - s p a n t i o n s l e s s than 4.4.8 The 6%. Cable Length: Experiment versus l e n g t h measurement gave t h e t o t a l length of the cable. Models. unstretched To o b t a i n a v a l u e c o m p a r a b l e w i t h t h e o r e t i c a l r e s u l t s a c o r r e c t i o n f o r e l o n g a t i o n has a p p l i e d t o the f i e l d the approximate S = deflec- results. The to be correction i s defined relationship: Sm the (1 + g | ) S elongated length Sm measured u n s t r e t c h e d T_, t e n s i o n i n the l i n e at the upper E e l a s t i c modulus o f t h e A c r o s s - s e c t i o n area length cable support by - Figure 25 — 77 - Differences i n S, cable length, between experiment and catenary model, and between experiment and parabolic model, f o r the nine free hanging t e s t s . Figure 26 — Difference i n S, cable length, between catenary model and parabolic model, versus Tg, tension at the upper support, and versus Dm, mid-span d e f l e c t i o n i n the free hanging cable. - 79 - t h e s k y l i n e , S i s given by: For S = Sm (1 + T^/37000) w i t h T D e x p r e s s e d i n newtons. The d i f f e r e n c e s b e t w e e n t h e m e a s u r e d elongated l e n g t h a n d t h e c o m p u t e d t h e o r e t i c a l l e n g t h a r e shown i n Figure 25 f o r t h e n i n e f r e e hanging t e s t s . I f T e s t #1. i s ignored, the r e s u l t s demonstrate the v a l i d i t y of both the catenary and p a r a b o l i c models l e n g t h f o r m u l a t i o n s confirm the value and a l s o chosen f o r t h e e l a s t i c modulus o f t h e - cable. 4.4.9 Cable Length:, Parabolic Catenary Model Model. The d i f f e r e n c e s i n t h e c a b l e both models:are p l o t t e d i n Figure upper support. the catenary versus lengths computed with 26.versus t e n s i o n a t the A r a p i d d e c r e a s e o f t h e d i f f e r e n c e between model and t h e p a r a b o l i c model i s n o t e d as t h e tension increases. The p a r a b o l i c l e n g t h i s s h o r t e r b y 180 a t 10% m i d - s p a n d e f l e c t i o n . : a h d o n l y by l e s s t h a n 1 mm d e f l e c t i o n a t mid-span i s l e s s than 3%. mm i f the - 80 - CHAPTER 5 CLAMPED LOAD ON A S I N G L E The LINE f o l l o w i n g experiment was designed t o study the e f f e c t o f a c o n c e n t r a t e d v e r t i c a l l o a d clamped a t a known distance along the span o f a s i n g l e l i n e . t i o n occurs i n a c t u a l y a r d i n g This configura- systems when the c a r r i a g e i s equipped with a s k y l i n e stop or when a c a r r i a g e bumper i s clamped on the s k y l i n e . The same s i t u a t i o n i s a l s o found when the chokers a r e attached d i r e c t l y on the l i n e as i n highlead. 5.1 D e s c r i p t i o n o f the Experiment. The 5/8-inch s k y l i n e was used f o r t h i s t e s t . c a r r i a g e and l e a d weights c o n s t i t u t e d the v e r t i c a l The load and a s m a l l clamp t h a t f i t the 5/8-inch s k y l i n e was manufactured to stop t h e c a r r i a g e from r o l l i n g . A convenient way t o execute the experiment was t o proceed as f o l l o w s : i ) choose a c a r r i a g e p o s i t i o n a t about 1/8 of the cable span s t a r t i n g from the upper support; ii) choose a load from 100 Kg t o 500 Kg by steps of 100 Kg; iii) increase the t e n s i o n i n the s k y l i n e a t the upper support from the minimum p o s s i b l e t o about 30,000 N i n 4 t o 6 steps; - 81 - i v ) lower the c a r r i a g e and change the l o a d . the e n t i r e range of load values has If been i n v e s t i g a t e d change the c a r r i a g e p o s i t i o n . While t h i s procedure i m p l i e s 175 only 29 were a c t u a l l y done and u l a t e d i n Appendix 5.2 different tests, the r e s u l t s of these are tab- 4. A n a l y s i s of the R e s u l t s . 5.2.1 Pre-considerations. A s k y l i n e with a s i n g l e concentrated load i s shown i n F i g u r e 27 f o r three d i f f e r e n t p o s i t i o n s of the l o a d . Such a system can be c o n s i d e r e d to be comprised of two segments f r e e l y hanging between the load and.the two For the three p o s i t i o n s presented cable supports. the t o t a l d e f l e c t i o n a t the load i s about 7 percent but the d e f l e c t i o n at the midp o i n t i n each of the f r e e hanging segments i s l e s s than percent. Therefore u s i n g the r e s u l t s of the a n a l y s i s of f r e e hanging c a b l e presented 2.5 the i n Chapter 4, the f o l l o w i n g c o n c l u s i o n s can be drawn f o r the expected d i f f e r e n c e s b e t ween experiment catenary characteristics. and parabola f o r the two segments - 82' " Table V I I . Expected d i s c r e p a n c i e s i n the c h a r a c t e r i s t i c s of the f r e e hanging segments. Differences between experiment and theories Differences between catenary and parabola less than .02% Tension Within the margin Deflection at mid-point 33 mm of Angles of the lines with the horizontal .04° experimental errors 1 mm Cable length Table VII shows t h a t no d e t e c t a b l e introduced theories; on the i n d i v i d u a l l i n e segments by the two thus the study can be r e s t r i c t e d only a n a l y s i s of the load p o s i t i o n s tensions d i f f e r e n c e s are t o the f o r d i f f e r e n t loads and and a d e t a i l e d a n a l y s i s o f each l i n e segment i s unnecessary. 5.2.2 Method of A n a l y s i s . F i v e elements, shown i n F i g u r e consideration 27, are taken i n t o i n the a n a l y s i s : - R load weight - T_, tension support i n the s k y l i n e a t the upper - - T 83 tension A - in the skyline at the lower support - X horizontal - Y vertical The these parameters. T_. a n d This as system X as is The parameters procedure was position position completely usual and adopted way solve and the the defined of load load. by any proceeding for the of of the is variables analysis can three to take Y be of and R, T . summarized follows: Measured Parameters R, T_ and X B Computation u s i n g nary and p a r a b o l a catetheories Measured V a r i a b l e s Y and A T h e o r e t i c a l v a l u e s f o r the v a r i a b l e s Yc and T.c A Conclusion 5.2.3 The iables and Error Analysis. experimental parameters are errors recalled affecting in Table the measured VIII. var- - 84 - Table VIII. Expected e r r o r s i n the measured v a r i a b l e s and parameters of the clamped l o a d on a s i n g l e l i n e system. Experimental Nomenclature Designation 1% R Parameters Variables the same way for the iations treatment as for measured in the in the Y 18 mm the variables experimental hanging T^ are calculated values of Y the calculated variables Y parameters upper is support R and T T_) The T^ The with is done error enlarged and . obtained (1.01 errors cable. and the the 1% A free errors at Neglected of the sion X T The 1% B T error by zones the resulting upper the combined with from limit largest the var- for ten- small- 13 est load opposite extreme had The extremes. The lower error limit is obtained zone is defined with by the those cases. It load R). (.99 a was found considerable that small influence errors on the in the tension results. and - 85 - Figure 27 — Skyline with a single concentrated load f o r three d i f f e r e n t positions of the clamped load. Figure 28 - Differences i n Y-position of the load, between experiment and catenary and parabolic models, f o r the 29 clamped load on a single l i n e t e s t s . - 87 - 5.3 Results 5.3.1 and Conclusions. Y - p o s i t i o n o f the Load: Experiment versus Models. The d i f f e r e n c e s between the measured Y and the Y c a l c u l a t e d w i t h the catenary on F i g u r e and p a r a b o l i c models are p l o t t e d 28 f o r the 29 t e s t s . The d i f f e r e n c e between the models i s so small t h a t i t does not appear on t h i s graph. Figure and 28 shows complete agreement between the experiment the models f o r a l l the t e s t s but one. 5.3.2 Y - p o s i t i o n o f the Load - Catenary Model versus P a r a b o l i c Model. The discrepancies between the two models w i l l be maximum when the d e f l e c t i o n a t mid-point i n the l i n e segments i s maximum. T h i s s i t u a t i o n occurs f o r the s m a l l e s t (R = 100 Kg) and the minimum t e n s i o n clearance f o r the complete load path. load (T_, = 5886N) t h a t gives The d i f f e r e n c e s , i n those c o n d i t i o n s , between the two t h e o r i e s are p l o t t e d on Figure 29 versus h o r i z o n t a l p o s i t i o n s o f the load along the e n t i r e span. The p a r a b o l i c model over-estimates the d e f l e c t - ion of the load r e l a t i v e t o the catenary p a r t of the load-path upper end. model i n the lower and under-estimates i t towards the T h i s p o i n t i s i l l u s t r a t e d on a sketch paths shown i n F i g u r e 30. of the load The maximum d i f f e r e n c e i s 15 mm. - 88 - Figure 29 — Differences i n the Y-positi;bns of the load, between catenary and parabolic models versus X-positions on the span Tg R tension at the upper support = 5885 N load = 100 Kg. Figure 30 — Sketches of the catenary and parabolic model load paths f o r the clamped load on a single l i n e . - 89 - - 90 - Figure 31 - a) Force balance a t the l e v e l i n a clamped load on a s i n g l e l i n e system, u s i n g catenary and p a r a b o l i c models. b) Unbalanced f o r c e s at the load l e v e l i n a clamped load on a s i n g l e l i n e , using the p a r a b o l i c model and a Y - p o s i t i o n of the load higher than t h a t a t e q u i l i b r i u m . - 91 - I t w o u l d be l o g i c a l t o t h i n k t h a t because t h e l i n e . weights are under-estimated with should yield cases. a d e f l e c t i o n smaller this than t h e catenary theory in a l l H o w e v e r , t h e r e s u l t s show t h a t t h i s a f f i r m a t i o n i s false for a large proportion o f t h e span. t i o n i s investigated i n Figure the the parabola, 31. Figure forces a t the load f o r the catenary equilibrium. i s 107 mm a b o v e t h e p a r a b o l a . parabola has been a r t i f i c i a l l y catenary; contradic- 31a r e p r e s e n t s and t h e p a r a b o l a a t F o r t h e load p o s i t i o n chosen catenary This (X = 65 m) t h e In Figure 31b t h e s e t a t t h e same l e v e l a s t h e t h e f o r c e s c a l c u l a t i o n shows t h a t t h e l i f t i n g c a p a c i t y o f t h e system i n t h i s p o s i t i o n i s s m a l l e r than t h e l o a d a n d t h e s y s t e m i s f o r c e d t o s a g more b e c a u s e o f t h e unbalanced force. a) b) X - p o s i t i o n of R weight of T R the load on the span : 6 5 m the load t e n s i o n at the upper support : 100 Kg : 5886 N - 92 - 5.3.3 t e n s i o n a t the lower Experiment support: versus Models. The d i f f e r e n c e between the measured T and the t e n s i o n c a l c u l a t e d w i t h the catenary and p a r a b o l i c models are p l o t t e d on F i g u r e 32 f o r the 29 t e s t s . The d i f f e r e n c e between catenary and parabola i s s m a l l and does not appear on the graph. The r e s u l t s are between the upper and lower e r r o r boundaries f o r a l l the t e s t s but one. T h e r e f o r e the t h e o r i e s a r e c o n s i d e r e d t o agree w i t h the a c t u a l measurements . 5.3.4 Tension a t the lower support - Catenary Model versus P a r a b o l i c Model. The d i s c r e p a n c i e s between the two t h e o r i e s are p l o t t e d on F i g u r e 33 f o r p o s i t i o n s o f the l o a d along the e n t i r e span. The t e n s i o n at the lower support computed with the p a r a b o l i c model are l a r g e r than t h a t with the catenary model f o r any p o s i t i o n o f the l o a d . The d i f f e r e n c e i s however always very s m a l l and l e s s than 3N (.05%) f o r the study case. The agreement between the two t h e o r i e s i s best when the load i s i n the mid-span area. - 93 - F i g u r e 32 - Differences i n T , A support, t e n s i o n at the lower between experiment and p a r a b o l i c models, f o r the 29 and l o a d on a s i n g l e l i n e F i g u r e 33 - catenary Differences i n T , A support, clamped tests. t e n s i o n at the between catenary and lower parabolic model versus X - p o s i t i o n s on the span. Tg t e n s i o n at the upper support = 586*5 N R load = 100 Kg. - 94 - 6 CHAPTER GRAVITY SYSTEM The the experiment described v e r y s i m p l e and i n t h i s Chapter commonly u s e d g r a v i t y g r a v i t y system a c a r r i a g e r u n n i n g on to the the point upper support w i t h o f maximum r e a c h by 6.1 Description The and the of stretched length the skyline could system. The be tensioned with on the Set at the ii) upper Choose a l o a d of iii) length 100 Let go the The r e s u l t s of the be the The un- kept constant the in a live tension skyline c a r r i e d out as to in gravity follows: the skyline support, f r o m 100 Kg t o 500 Kg by steps i t can Kg. carriage run down as f r e e l y under i t s w e i g h t . iv) P u l l 6-ton Comelong (or t e n s i o n ) of p o s i t i o n i s thus of the system or m o n i t o r e d as the pulled to i t s G e a r m a t i c 19. skyline could t e s t s were t h e r e f o r e i) returns the Experiment. similate a fixed skyline gravity the In skyline i s m a i n l i n e and the stored of system. gravity. s k y l i n e was m a i n l i n e was the tested t e s t s are of The starting determined, carriage a b o u t 1/8 f a r as along the the span reported skyline in length. i n Appendix 4. steps - 96 - 6.2 A n a l y s i s o f the R e s u l t s . 6.2.1 The Pre-considerations. system can be (very s i m i l a r l y t o the clamped load on a s i n g l e l i n e s t u d i e d three i n Chapter 5) considered as s i n g l e l i n e segments f r e e l y hanging between the c a r - r i a g e and each o f the supports. The s k y l i n e segments are very t i g h t , w i t h d e f l e c t i o n a t mid-span l e s s than 2.5%, i n most of the cases p e r m i t t e d by the l i m i t s o f the study. As concluded i n the a n a l y s i s o f the clamped load on a s i n g l e l i n e , u n d e t e c t a b l e d i f f e r e n c e s w i l l be i n t r o d u c e d by the p a r a b o l i c and catenary models on the c h a r a c t e r i s t i c s of the s k y l i n e f r e e hanging segments. The t e n s i o n i n the m a i n l i n e i s u s u a l l y l a r g e enough t o keep the m a i n l i n e segment t i g h t when the c a r r i a g e approaches the upper support; however as the c a r r i a g e runs out the t e n s i o n and i n the m a i n l i n e decreases i s very small when the c a r r i a g e e v e n t u a l l y stops. The d e f l e c t i o n i n the m a i n l i n e can then be l a r g e enough t o result i n noticeable discrepancies The e f f e c t o f those d i s c r e p a n c i e s and on the t e n s i o n s 6.2.2 between the two t h e o r i e s . on the c a r r i a g e p o s i t i o n i s i n v e s t i g a t e d i n the a n a l y s i s . Method of A n a l y s i s . S i x c h a r a c t e r i s t i c s o f the system are taken i n t o consideration i n the a n a l y s i s : - R - T c a r r i a g e and load weight D tension i n the s k y l i n e a t the upper support " tension A T i n the s k y l i n e a t the lower support " T t e n s i o n i n the m a i n l i n e B3 a t the upper support - X h o r i z o n t a l p o s i t i o n o f the c a r r i a g e - Y v e r t i c a l p o s i t i o n o f the c a r r i a g e Any combination o f three o f these c h a r a c t e r i s t i c s t h a t does not i n c l u d e the p a i r completely. R,X,T D ( T , T ) d e f i n e s the system g were taken as parameters, i n the a n a l - y s i s , t o determine the v a r i a b l e s T_, T catenary and p a r a b o l i c f o r m u l a t i o n s . y s i s procedure i s given D O and Y u s i n g the A c h a r t o f the a n a l - as f o l l o w s : Measured Parameters R,X,T Computation u s i n g catenary and p a r a b o l i c theories Measured V a r i a b l e s T T , Y T h e o r e t i c a l values f o r the v a r i a b l e s Y 0 r 3 Conclusion - 98 - 6.2.3 The Analysis. computation of the experimental determination of the error-zone as d e s c r i b e d line. Error e r r o r s and t h e b o u n d a r i e s w e r e done e x a c t l y i n C h a p t e r 5 f o r t h e clamped l o a d on a s i n g l e The same i n f l u e n c e o f t h e e r r o r s i n t h e t e n s i o n a n d l o a d on t h e t h e o r e t i c a l v a l u e s 6.3 Results 6.3.1 and o f t h e v a r i a b l e s was f o u n d . Conclusions. Y-position of the Carriage - Experiment versus Models. The d i f f e r e n c e s between t h e measured Y and t h e Y calculated with Figure 34. the catenary and p a r a b o l a F o r 13 c a r r i a g e p o s i t i o n s f r o m t h e l o w e r t o t h e upper support. T e s t #1 was a t t h e p o i n t o f maximum r e a c h o f the system f o r t h e given tension the ground f o r t h e f i r s t 6 tests. The and t h e m a i n l i n e t h e o r i e s agree w i t h margin o f experimental The are plotted i n touched the experiment w i t h i n the e r r o r f o r t h e l a s t 2/3 o f t h e s e r i e s . f a c t that the theories diverge boundaries f o rthe f i r s t g r e a t l y from t h e e r r o r t e s t s c o u l d be e x p e c t e d s i n c e no p r o v i s i o n i s made i n t h e a s s u m p t i o n s o f t h e t h e o r i e s f o r t h e n a t u r a l l i m i t a t i o n s o f t h e ground p r o f i l e . mainline as c a l c u l a t e d by t h e p a r a b o l a The s h a p e o f t h e and c a t e n a r y i s purely - 99 - Figure 34 - D i f f e r e n c e s i n Y - p o s i t i o n o f t h e c a r r i a g e , between e x p e r i m e n t and c a t e n a r y and p a r a - b o l i c m o d e l s , f o r t h e 13 g r a v i t y s y s t e m tests. Figure 35 - M a i n l i n e s h a p e s a s p r e d i c t e d by t h e t h e o r e t i c a l models f o r t h r e e o f t h e g r a v i t y s y s t e m tests„showing i n t e r s e c t i o n w i t h ground p r o f i l e . - 100 - - 101 - t h e o r e t i c a l f o r t h e f i r s t 6 t e s t s a s shown i n F i g u r e 3 5 . The m a i n l i n e i s c a l c u l a t e d t o be f r e e f r o m t h e g r o u n d f o r t e s t 7 compatibly with the a c t u a l s i t u a t i o n i n the f i e l d during the test. Another deduction the parabola catenary does n o t d i v e r g e does. 34 i s t h a t from F i g u r e f r o m t h e t e s t s a s much a s t h e T h i s c a n be e x p l a i n e d c o n s i d e r i n g t h e f o r c e s a c t i n g on t h e c a r r i a g e i n t h e a r e a o f maximum r e a c h system. As t h e c a r r i a g e a p p r o a c h e s t h e l o w e r h o r i z o n t a l t e n s i o n i n the mainline decreases component o f t h e t e n s i o n i n t h i s support l i n e at the c a r r i a g e This v e r t i c a l the v e r t i c a l component a d d i n g formulation the v e r t i c a l in the mainline stops With the component o f t h e t e n s i o n i n c r e a s e s r a p i d l y towards i n f i n i t y r e s u l t of the combination to the In the component o f t h e m a i n l i n e t e n s i o n i n c r e a s i n g when t h e m a i n l i n e t o u c h e s t h e g r o u n d . catenary level of the l i n e w e i g h t o f t h e l o a d c a u s e s t h e s y s t e m t o s a g more. field the but the v e r t i c a l i n c r e a s e s because o f the i n c r e a s e o f the angle with the h o r i z o n t a l . of the of the l i n e angle as a and l i n e weight. For the p a r a b o l i c model, the t o t a l weight o f the m a i n l i n e i s assumed t o be d i s t r i b u t e d on t h e c h o r d constant as t h e l i n e and t h e r e f o r e r e m a i n s s a g s more a n d c o n s e q u e n t l y the v e r t i c a l component o f t h e t e n s i o n i n t h e m a i n l i n e c o n v e r g e s t o w a r d s a f i n i t e value. reality better. As a r e s u l t t h e p a r a b o l i c model represents - 102 6.3.2 - Y - p o s i t i o n of the C a r r i a g e Model v e r s u s P a r a b o l i c S i m i l a r l y t o the - Catenary- Model. a n a l y s i s of the clamped l o a d a s i n g l e l i n e i n C h a p t e r 5, the c o m p a r i s o n was conditions Kg) and support of load (T-£ (R = 100 = 6867N) t h a t r e s u l t s i n t h e ment b e t w e e n t h e two shows d i s c r e p a n c i e s the theories. f o r the The as l a s t s t a t i o n s of the s e c t i o n of the load path. l e v e l was two load as the upper disagreeanalysis s e v e r a l metres i n 36 up The to parabola carriage for a short T h i s phenomenon r e s u l t i n g f r o m lines with explained largest load path. the d e f l e c t i o n at the angles of the at the shown i n F i g u r e over-estimates the done w i t h r e s u l t s of t h i s t h a t r a n g e as h i g h a r e a o f maximum r e a c h , and 10 mm tension on the h o r i z o n t a l at the i n C h a p t e r 5. Figure 37 carriage sketches the paths. 6.3.3 Tension i n the Support: The S k y l i n e at the Lower Experiment versus Models. d i f f e r e n c e s b e t w e e n t h e m e a s u r e d T, and the the parabola are A tension calculated with shown i n F i g u r e catenary on the and f o r the 13 and tests. p a r a b o l i c models i s s m a l l graph. w i t h i n the 38 the catenary The two and t h e o r i e s agree w i t h l i m i t s of experimental errors. noted f o r the v e r t i c a l p o s i t i o n of the present f o r the The tension, T . & This d i f f e r e n c e between does not the experiment The divergence carriage i s should appear be not expected con- - F i g u r e 36 - 103 - D i f f e r e n c e s i n the Y - p o s i t i o n s of the c a r r i a g e , between catenary and parabolic models versus X - p o s i t i o n s on the Tg R Figure 37 — t e n s i o n at the upper span. support 6867 i n the s k y l i n e = Carriage plus load = 100 Sketches of the catenary and parabolic models l o a d paths f o r the g r a v i t y system. N Kg. Or 104 - - 105 - F i g u r e 38 - D i f f e r e n c e s i n T^, support t e n s i o n at the lower between experiment and catenary and p a r a b o l i c models, f o r the 13 gravity system t e s t s . F i g u r e 39 - D i f f e r e n c e s i n T^, support, t e n s i o n a t the lower between catenary and parabolic models versus X - p o s i t i o n s of the c a r r i a g e on the Tg R span. t e n s i o n i n the s k y l i n e at the 6867 upper support = Carriage plus load = 100 N Kg. - 106 - X on the span, m - 107 - s i d e r i n g the catenary theory f o r which the d i f f e r e n c e between and depends on to and E o n l y . 6.3.4 Tension i n the S k y l i n e a t the Lower Support: Catenary Model versus P a r a b o l i c Model. The graph of the d i f f e r e n c e s between the v a l u e s of T A c a l c u l a t e d w i t h the two models f o r p o s i t i o n s of the c a r - r i a g e along the e n t i r e l o a d path i s shown i n F i g u r e 39. The catenary y i e l d s the l a r g e s t v a l u e s f o r the t e n s i o n s f o r a l l the p o s i t i o n s of the c a r r i a g e . The maximum d i f f e r e n c e i s obtained a t each end of the l o a d path and i s 7 newtons or (.1%). The two t h e o r i e s agree almost p e r f e c t l y when the c a r r i a g e i s a t mid-span. 6.3.5 Tg^ Tension i n the M a i n l i n e a t the Upper Support: Experiment Since the t e n s i o n was when the l i n e was versus Models. not recorded i n the m a i n l i n e touching the ground, F i g u r e 40 shows the h i s t o r i g r a m of the d i f f e r e n c e s between experiment i e s f o r the l a s t 7 t e s t s . the e r r o r boundaries; and H a l f of the t e s t s d i v e r g e however, the tensiometer was theorfrom utilized to measure t e n s i o n s l e s s than 1000N f o r those t e s t s and f o r t h i s range of t e n s i o n a one percent expected optimistic. e r r o r i s too - Figure 40 - Differences upper gravity 41 - and the upper the carriage T , 3 the between 0 0 , mainline at experiment models, tension support, model, on tension upper R B in for the and the 13 tests. in parabolic 0 T parabolic and T - tension system Differences at in support, catenary Figure 108 the in between versus the mainline catenary X-positions model of span. the skyline support Carriage in plus load at the = 6867 = 100 N Kg. 109 GO - - 6.3.6 n o - Tension i n the M a i n l i n e Support: a t the Upper Catenary Model versus P a r a b o l i c Model. The mainline p l o t of the d i f f e r e n c e s f o r the t e n s i o n i n the c a l c u l a t e d w i t h both t h e o r i e s and shown i n F i g u r e 41 demonstrates, as expected, t h a t the disagreement between the two t h e o r i e s i s small f o r p o s i t i o n o f the c a r r i a g e towards the upper support. The maximum value crepancy i s 8N a t the p o i n t o f maximum reach. of the d i s - 7 CHAPTER D Y N A M I C T E S T S The tems few of was n o t one o f dynamic potential idea and study of tests of 7.1 the prime the field model and equipment of results are presented f i r s t sys- the thesis. approach to The A show t h e and o b t a i n t o be expected. i n this cable an equipment chapter. Equipment. input and output were changed. volts DC. The output of variable speed devices f o r the tensiometer, The l o a d - c e l l s were excited was c o n n e c t e d to a strip-chart and s e n s i t i v i t y (Plate by a 10 record- 9). Tests. Various presented a of of as only, 7.2 objectives tried The er behaviour were the type tests t h e dynamic types of tests were carried o u t ; two are here: i) V e r t i c a l ii) Instantaneous by 7.2.1 The o s c i l l a t i o n stop of of the load the carriage running gravity. V e r t i c a l position of Oscillation the load of being the Load. surveyed by the - 112 - Plate 9 — Strip-chart recorder, generator and transformer-regulator used f o r the recording of the tensions at the upper support. Plate 10 — Manual i n i t i a t i o n of the v e r t i c a l tory motion of the clamped load. oscilla- -113- - 114 usual technique manually the system i s brought i n t o v e r t i c a l ( P l a t e 10). r e a c h e s a b o u t 150% - motion When t h e t e n s i o n i n t h e s k y l i n e of i t s s t a t i c value the e x c i t a t i o n i s s t o p p e d and the t e n s i o n a t the upper support i s recorded u n t i l damping b r i n g s i t back t o i t s o r i g i n a l s t a t i c value. A t y p i c a l c h a r t i s shown i n F i g u r e Conclusion The e x a m p l e has ii) of the 42. Test. v i b r a t i o n recorded the i) i n the s k y l i n e f o r t h e t e s t g i v e n , as an following characteristics: frequency .475 damping r a t i o Hertz ( p e r i o d 2.10 seconds) .05. No m a j o r d i f f e r e n c e s w e r e n o t i c e d w h e t h e r t h e c a r r i a g e clamped or hooked t o the 7.2.2 mainline. Instantaneous R u n n i n g by For was Stop of the Carriage Gravity. t h i s t e s t the c a r r i a g e i s f r e e d t o run from a known p o s i t i o n , t h e n s t o p p e d s u d d e n l y w i t h t h e m a i n l i n e w i t h a c l a m p on t h e recorded skyline. skyline. a t the upper support The The 43. The the t e n s i o n s i s on b o t h t h e m a i n l i n e f i r s t e x a m p l e was d e s c r i b e d on F i g u r e e f f e c t on performed f o r the r e s u l t s are or and the conditions shown i n F i g u r e 45. - 115 Figure 42 - - Chart-recording of the tension i n the skyline at the upper support during v e r t i c a l o s c i l l a t i o n of the load of the gravity system. Recorder: - input - output - chartspeed = 10 V = 20 mV = 10 sec/inch Carriage and load: - 495 Kg. - located at mid-span - deflection 6.7$ - 117 - F i g u r e 43 - Sketch of dynamic t e s t . C a r r i a g e stopped by the main l i n e . S t a r t i n g p o s i t i o n o f the c a r r i a g e . X = 106.09 m Y = 11.OS m Figure 4 4 - S k e t c h of dynamic t e s t . C a r r i a g e stopped w i t h a clamp on the s k y l i n e . S t a r t i n g p o s i t i o n o f the c a r r i a g e . X = 72.02 Y = 2.96 - 119 - C o n c l u s i o n of the T e s t . Tension i n the s k y l i n e . J u s t a f t e r the shock, a drop t o 60% of the o r i g i n a l s t a t i c tension rapidly f o l l o w s a s m a l l peak a t 135%. The maximum t e n s i o n reached i s 150% of the s t a t i c t e n s i o n and a complex v i b r a t i o n phenomenon takes p l a c e i n the l i n e with a dominant low frequency of .67 Hertz (period 1.5 seconds). Tension i n the m a i n l i n e . An i n c r e a s e o f more than 700% from the o r i g i n a l s t a t i c t e n s i o n i s recorded a f t e r the shock. e f f e c t i s repeated p e r i o d i c a l l y This snapping every 1.5 seconds w i t h a r a p i d l y d e c r e a s i n g magnitude (about 300% f o r the t h i r d peak). Other v i b r a t i o n s of h i g h e r f r e q u e n c i e s interfere with t h i s b a s i c pattern. The second example was performed d e s c r i b e d i n F i g u r e 44. The r e s u l t s i n the c o n d i t i o n s are shown i n F i g u r e 46. C o n c l u s i o n of the t e s t . The example i s presented i n p a r a l l e l w i t h the r e s u l t o b t a i n e d f o r a t e s t o f the f i r s t type. The b a s i c -.120 Figure 45 - - Chart-recordings of tensions i n the l i n e s of a gravity system during dynamic t e s t s . a) i n the skyline b) i n the mainline Carriage stopped with the mainline. Time, seconds - Figure 4 6 — 122 - Chart-recordings of tensions i n the skyline at the upper support. a) carriage stopped with a clamp b) carriage stopped with the mainline - 123 - - 124 - difference following i s t h e s h a r p p e a k o f 155% o f t h e s t a t i c t e n s i o n t h e s h o c k on t h e c l a m p . d i s t u r b e d i n t h i s second example The s k y l i n e and complex i s much more vibration phenomena o c c u r i n t h e s y s t e m l o n g a f t e r t h e s h o c k . 7.3 Conclusion. D y n a m i c t e s t s w e r e e a s i l y p e r f o r m e d on t h e f i e l d model. The f i r s t results show t h a t c o n s i d e r a b l e dynamic t e n s i o n s c a n be d e v e l o p e d i n t h e l i n e s a n d d e m o n s t r a t e c l e a r l y t h e need f o r f u r t h e r research i n this area. - 125 - CHAPTER D I S C U S S I O N A N D 8 C O N C L U S I O N Both the r e s u l t s o f the a n a l y s i s and the p r a c t i c a l aspect o f c a b l e l o g g i n g problems should be taken i n t o cons i d e r a t i o n to propose an o p e r a t i o n a l t h e o r e t i c a l model f o r c a b l e l o g g i n g systems. The catenary w i t h an a c t u a l f i e l d configurations. and the p a r a b o l i c model were compared t e s t f o r three t y p i c a l c a b l e system The f a c t , shown i n Chapter 4, t h a t the shape of a f r e e hanging c a b l e segments i s c l o s e r t o a catenary a parabola ing i s probably not a s u r p r i s e t o an engineer than trust- the b a s i c laws o f mechanics; the catenary model des- c r i b e s a c a b l e y a r d i n g system with b e t t e r p r e c i s i o n than the p a r a b o l i c model does, but i s t h i s p r e c i s i o n needed f o r pract i c a l applications? Cable y a r d i n g systems do not operate i n i d e a l experim- e n t a l c o n d i t i o n s and much u n c e r t a i n t y i s attached i n the field to t h e i r various c h a r a c t e r i s t i c s . cases the ground p r o f i l e i s known with a p r e c i s i o n of about 300 mm (1 foot) . In the best o f the The u n c e r t a i n t y on the v a l u e s of the a t i n g t e n s i o n s i s as l a r g e as 10% on the e x i s t i n g oper-.j. yarders even equipped with the most s o p h i s t i c a t e d t e n s i o n c o n t r o l - 126 - systems and the i n a c c u r a c y i n the s c a l i n g of the l o g s can r e s u l t i n an i n p r e c i s i o n of 10% i n the payload v a l u e . Table IX compares these a c t u a l p r e c i s i o n s w i t h the p r e c i s i o n s i n the c h a r a c t e r i s t i c s of the f i e l d model, and w i t h the differences between catenary and p a r a b o l i c Table IX. System Characteristics models. P r e c i s i o n i n the knowledge of the c h a r a c t e r i s t i c s o f : the f i e l d model a r e a l y a r d i n g system D i s c r e p a n c i e s between catenary model and p a r a b o l i c model f o r the same characteristics. Nomenclature F i e l d model precision R e a l system precision Discrepancies Cat. - Par. Ground p r o f i l e Vertical position Y Operating tension T 1% 10% l e s s than Payload R 1% 10% l e s s than 1% 18 mm 300 15 mm mm 1% The f i e l d model c h a r a c t e r i s t i c s are ten times more a c c u r a t e than a r e a l y a r d i n g system; t h i s i s j u s t i f i e d by the s c i e n t i f i c aspect o f the experimental approach. the catenary and p a r a b o l i c Both model compute the system charac- t e r i s t i c s w i t h a p r e c i s i o n f a r beyond what i s known and - 127 needed i n t h e f i e l d , of - t h e r e f o r e any m o d e l o r any combination t h e m a i n a s s e t s o f b o t h m o d e l c a n be u s e d t o investigate the s t a t i c c h a r a c t e r i s t i c s of a cable logging B r i e f D e s c r i p t i o n of the Model The model proposed system. Proposed. uses equation g (Chapter 2) from the c a t e n a r y f o r m u l a t i o n , t o t r a n s f e r a l l the f o r c e s known i n t h e s y s t e m , bolic to the c a r r i a g e l e v e l . Then, t h e model i s used t o f o r m u l a t e the e q u a t i o n s of brium of the c a r r i a g e . i m p l e m e n t e d by C a r s o n This procedure was para- equili- successfully and Mann(3) f o r a r u n n i n g skyline system. A s i m p l e and sufficiently a c c u r a t e model i s a v a i l - a b l e to d e s c r i b e a c a b l e y a r d i n g system tions. T h i s model would s a t i s f y d e s i g n e r s and i m p l i e d by t h e m o d e l . i s not e x a c t l y the case, c a b l e system u s e r s have t r a d i t i o n a l l y static The r e s u l t s to account system i n the Conscious designers that and a p p l i e d a f a c t o r of s a f e t y to the f o r the v a r i o u s u n c e r t a i n t i e s . f a c t o r 5 recommended by t h e Workmen's C o m p e n s a t i o n B o a r d ( 1 1 ) i s more " i g n o r a n c e " t h a n " s a f e t y " and more r e s e a r c h s h o u l d be done i n t h e s t u d y o f t h e of condi- the needs of c a b l e u s e r s i f c a b l e systems were o p e r a t e d conditions of s t a t i c this in static cable logging systems. proves that behaviour - 128 - The c o n c l u s i o n s from the dynamic t e s t s d e s c r i b e d i n Chapter limited. 7 are t e n t a t i v e s i n c e the experiment was I t was noted t h a t the t e n s i o n s were very much d i s t u r b e d by the c a r r i a g e motion and 700% of the s t a t i c value was a t e n s i o n as h i g h as recorded i n the m a i n l i n e . should be noted, however, t h a t the m a i n l i n e was initially. very It understressed The dynamic t e s t s y i e l d more q u e s t i o n s than answers to the problem at t h a t p o i n t of the study but clearly show the need f o r f u r t h e r i n v e s t i g a t i o n s i n t h i s f i e l d and a l s o demonstrate the p o t e n t i a l u s e f u l n e s s of the f i e l d model to c a r r y those investigations. Conclusion. The study i s s u c c e s s f u l i n s e l e c t i n g a simple t h e o r e t i c a l model f o r the d e t e r m i n a t i o n of c a b l e y a r d i n g system s t a t i c c h a r a c t e r i s t i c s . i s based on the comparative field The s e l e c t i o n of the model a n a l y s i s of the r e s u l t s of the t e s t s and the t h e o r i e s , showing t h a t although the shape of a f r e e hanging c a b l e i s b e t t e r d e s c r i b e d as a c a t e nary than a p a r a b o l a both t h e o r e t i c a l models are accurate enough to s o l v e p r a c t i c a l c a b l e system problems. The t h e s i s a l s o shows the importance of the dynamic f o r c e s a c t i n g on the system. t h i s area i s r e q u i r e d . Further research i n - 129 - L I T E R A T U R E C I T E D B i n k l e y , V.W. and S t u d i e r , D.D. 1974. Cable Logging Systems. USDA F o r e s t S e r v i c e , P o r t l a n d , Oregon, 190 p. Carson, W.W. and Mann, C.N. 1970. A technique f o r t h e s o l u t i o n of s k y l i n e catenary equations. PNW-110, 18 p., i l l u s . P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n , P o r t l a n d , Oregon. Carson, W.W. and .'.Mann, C.N. 1971. An a n a l y s i s of runn i n g s k y l i n e l o a d path. PNW-120, 9 p., i l l u s . P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n , P o r t l a n d , Oregon. Carson, W.W. 1975. A n a l y s i s o f running s k y l i n e with . :: drag. PNW-193, 8 p., i l l u s . P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n , P o r t l a n d , Oregon. I n g l i s , S i r Charles Edward. 1963. A p p l i e d Mechanics f o r Engineers. 404 p., i l l u s . New York Dover P u b l i c a t i o n s , Inc. 6. Lysons, H.H. and Mann, C.N. 1967. S k y l i n e Tension and D e f l e c t i o n Handbook. U.S. F o r e s t Serv. Res. Pap. PNW-39, 44 p.,. i l l u s . P a c i f i c Northwest F o r e s t and Range Experiment S t a t i o n , P o r t l a n d , Oregon. 7. Selby, S.M. 1964. Standard Mathematical T a b l e s . The Chemical Rubber Co., C l e v e l a n d , Ohio. 632 p. Veda, M., S a i t o , T., Tominaga, M.,and S h i b a t a , J . 1962. S t u d i e s on the Main Cable i n S k y l i n e Logging. F i r s t Report. B u l l e t i n of the Government F o r e s t Experimental S t a t i o n no.188. Tokyo, Japan. Veda, M. and S a i t o , T. 1965. S t u d i e s on the Main Cable i n S k y l i n e Logging. Second Report. B u l l e t i n o f the Government F o r e s t Experiment S t a t i o n no.188. Tokyo, Japan. 10 Veda, M. 1966. Studies on the Main Cable i n S k y l i n e Logging. T h i r d Report. B u l l e t i n o f the Government F o r e s t Experiment S t a t i o n no.18 8. Tokyo, Japan. - 130 - 11. Workmen's C o m p e n s a t i o n B o a r d . 1972. A c c i d e n t P r e v e n tion Regulations. 298 p. Workmen's C o m p e n s a t i o n Board o f B r i t i s h Columbia. 12. W i r e r o p e Handbook. 1959. U n i t e d S t a t e s S t e e l C o r p o r a t i o n , San F r a n c i s c o , C a l i f o r n i a . 193 p. - 131 - APPENDIX 1 Appendix 1 p r e s e n t s the p a r a b o l i c model developed by G.G. Young"*" and the author, f o r the study o f c a b l e l o g ging system problems. Appendix 1 was p r i m a r i l y produced as a t e a c h i n g a i d f o r a u n i v e r s i t y course i n f o r e s t h a r v e s t i n g . I t s t u d i e s c a b l e mechanics from the very b a s i c f r e e c a b l e t o the more complex f i v e l i n e system. hanging Because the course can be attended by students w i t h a l i m i t e d knowledge i n mechanics the development i s very d e t a i l e d i n the f i r s t c h a p t e r s , t o become reasonably s u c c i n c t towards the end as the student p r o g r e s s e s . Numerical examples a r e presented a f t e r every major development. Although intended p r i m a r i l y f o r f o r e s t r y students, we b e l i e v e t h a t t h i s paper can be o f use t o anyone d e a l i n g w i t h problems i n c a b l e mechanics. Assistant Professor H a r v e s t i n g and Operations Research F a c u l t y of F o r e s t r y U n i v e r s i t y of B r i t i s h Columbia - 132 - CONTENTS Introduction A. B. Free hanging cable 1. General d e s c r i p t i o n of 2. Models of 3. Equation of 4. T e n s i o n s and d e f l e c t i o n system 2 the c a b l e shape 3 7 concentrated load Governing equations for The c a b l e w e i g h t assumed t o on the 1 the system Cable with a s i n g l e 1. the is the system act subchords 16 2. Sag and d e f l e c t i o n 3. Problem 1 - 19 Given the sag • ( o r d e f l e c t i o n D^) a t x = x^ and t h e clamped l o a d R what the 4. t e n s i o n s T ^ and T ^ a t Problem 2 - Given the are the supports? 22 sag ( o r d e f l e c t i o n D^) a t x = x^ a n d the tension T at the upper support B what i s the l i f t i n g c a p a c i t y R o f t h e 5. Problem 3 - Given the u p p e r s u p p o r t and the what i s at tension T B at system? 27 the l o a d R a t x = x^ t h e s a g S ( o r d e f l e c t i o n D) the p o s i t i v e of the load? 31 133 - C. Standing s k y l i n e . Gravity 1. Description of 2. Governing equations 3. Problem 1 (or the Given the d e f l e c t i o n what i s the - system system 34 f o r the system the • 35 sag S D) and t h e t e n s i o n T^ at l o a d R- B in the skyline? 4. 37 Problem 2 (or in Given the d e f l e c t i o n the s k y l i n e a t c a p a c i t y R of 5. the what i s at sag S• D) a n d t h e B what i s tension the lift the system? Problem 3 - G i v e n the in D. the 40 tension T f i 2 s k y l i n e a t B and the l o a d R the sag S (or the deflection) the c a r r i a g e ? Five line 1. D e s c r i p t i o n of 2. Governing 3. Mathematical 43 system skyline the system equations for the 44 system s i m i l i t u d e between and t h e f i v e l i n e system the 45 standing ; 47 CABLE MECHANICS Free hanging 1. cable General d e s c r i p t i o n of ; the system V •y ••• : B c. ^ \ E B ' VA' y' A H ^ \ C = cable . '." * o : x' - L ^ • .. . 1.1 9 : - -—' -•- ' '-' ----- • •:• -- - - Geometry C = cable B = upper support of the cable A = lower the cable L = span; support of horizontal distance E = difference in- e l e v a t i o n between the between t h e supports supports AB = c h o r d 6 = angle between t h e P = any p o i n t on t h e c h o r d and t h e 1.2 x', y' = coordinate [6 = a r c t g (E/L)] cable 0 = p o i n t of h o r i z o n t a l tangent 0, horizontal of the cable system Forces The t e n s i o n of the at cable. any p o i n t o n a c a b l e ' a c t s a l o n g the tangent at that point - 135 - T P H P V P T A T B tension = horizontal = vertical Models of the cable at component o f component o f P the tha at the lower tension at the upper support of the cable tension tension tension = weight 2. i n the at at P P support per u n i t length system D i f f e r e n t e q u a t i o n s c a n b e d e v e l o p e d d e p e n d i n g o n t h e a s s u m p t i o n s made a s how t h e u n i f o r m l y d i s t r i b u t e d c a b l e w e i g h t a c t s o n t h e c a b l e . Three d i f f e r e n t models c o u l d be to used:- model 1 : cable weight i s of the system u n i f o r m l y d i s t r i b u t e d over the h o r i z o n t a l span model 2 : cable weight system u n i f o r m l y d i s t r i b u t e d over model 3 : cable weight i s cable i t s e l f model 1 is the u n i f o r m l y d i s t r i b u t e d over the model 2 chord of length of model the the 3 T h e b e s t a c c u r a c y i s o b t a i n e d w i t h m o d e l 3 , b u t on t h e o t h e r h a n d m o d e l s and 2 p r o v i d e a s i m p l e r m a t h e m a t i c a l f o r m u l a t i o n f o r the s y s t e m . 1 The d e v e l o p m e n t o f t h e e q u a t i o n s f o r t h e s y s t e m i n t h i s p a p e r i s b a s e d on m o d e l 2. I t g i v e s a v e r y good d e s c r i p t i o n o f l o g g i n g c a b l e s i n most c a s e s . T h e e q u a t i o n s a r e most a c c u r a t e f o r t i g h t c a b l e s . - 136 - 3. Equation of 3.1 the cable Equation of shape the cable The o r i g i n 0 i s shape • a c t i n g on t h e * H tension * **P * W = x'u/cosG + ^P c o o r d i n a t e system 0, x', y' t h e p o i n t o f maximum s a g . The s y s t e m c o n s i d e r e d i s Forces i n the = ^P the c a b l e b e t w e e n 0 and P . system: at 0, horizontal tension at P assumed w e i g h t of the c a b l e between 0 and P - 137 - Equations of e q u i l i b r i u m * Sum o f v e r t i c a l forces V * Sum o f - p x'w/cose = 0 = 0 horizontal forces = 0 - H + H p = 0 of * Sum o f the cable moments a b o u t Hy' cos6 The l a s t y' tension 3_ component constant throughout the cable 0 to This x' 2 horizontal 2 c a n be a r r a n g e d t o 2cos6H is i.e. P = 0 equation = .Hp=H is the yield equation of the cable in c o o r d i n a t e system O x ' y ' . Conclusion: c a b l e hangs l i k e a p a r a b o l a . the the The above e q u a t i o n i s r e l a t i v e t o a c o o r d i n a t e s y s t e m w i t h o r i g i n a t t h e p o i n t o f maximum s a g i n t h e c a b l e . S i n c e t h i s p o i n t moves w i t h v a r y i n g t e n s i o n , i t i s not a very u s e f u l e x p r e s s i o n . N o r m a l l y one r e q u i r e s a c o o r d i n a t e s y s t e m w i t h o r i g i n a t one o f t h e s u p p o r t s . Equation of the cable shape i n the coordinate system A , x, y. - 138 - The t r a n s l a t i o n of rx = x ' the coordinate + a o Substituting system i s defined by: rx' = x - a iy' = y + b r f o r x' and y' i n y' x' yields, 2 the new equation 2cos9H of the cable i n the system A , x, y + b = (x - oi a) 2 2cos6H ^ Now t o determine Limit the values conditions, of a.and b: x= 0 when y = 0 and x = L when y = E o) give: and a' 2cos9H E + b = <" ( 2cos6H L " a) S u b t r a c t i n g t h o s e two 2 from which b = to 2cos6H y + to • ( L 2cos9H L 2 and y + to (L 2cos6H 2 + a - 2La) 2 - 2La) cos6HE uL L - COSSHE! aiL 2 to 2cosGH and b i n t h e equation of fL - COS9HE! 2 2 2 equations E = Substituting"a = the x - toL cable: L 2 2cos6H cos9HE uL x + fL - cs*efl|T - 2x 2 2 to 2cos8H toL L - 2cos6H y = T h i s i s the p a r a b o l i c e q u a t i o n at the l e f t h a n d s u p p o r t . u x 2cos6H of the cos6H 2 + E L cable cosBHE toL cos9HE toL -2 c o s 6toL H with L 2 the X coordinate syste - 139 - T h e above e q u a t i o n can be r e a r r a n g e d s o t h a t i t s components m e a n i n g f u l from a p h y s i c a l s t a n d p o i n t y i e l d i n g : - u x ( L - x) 2cos8H From t h e above Ex L -MX(L - x) 2cos6H figure it c a n be seen that: NP = y position of the cable at x NM = y position of the chord at x NP - NM = NP + MN = MP referred of the at to point as cable at the x. x = L/2 Tangents the to cable and vertical This x = D - distance quantity DEFLECTION o f x At midspan more + E_x L and c a b l e Deflection are is the cable = MX(L - x) 2cos6H Dm = t o L / 2 ( L - L / 2 ) 2cos6H = coL 8cos6H 2 between chord classically at x. - 140 - Since the necessary t e n s i o n a c t s on t h e t a n g e n t o f t h e c a b l e i t i s v e r y o f t e n t o know t h e s l o p e o f t h i s t a n g e n t a t a g i v e n p o i n t P . The t a n g e n t a i s g i v e n b y t h e v a l u e c a b l e e q u a t i o n w i t h r e s p e c t t o x: the 2cosGH (ii L x + - cos8H Example: the the slope at uiL ] 2cos6H A is: = tga = Slope = E - tga slope at B is: tga = D Tensions and derivative of the 2cosGHj L 4. first 01 y = dx of of the cable at point x. oiL 2cos9H oiL + E cos6H — L E 2cos9H uL + L 2cos6H deflection G i v e n the geometry o f the system ( E and L ) and the type of c a b l e ( u ) e q u a t i o n s o f t h e c a b l e s 1 and 2 o n l y d i f f e r b e c a u s e o f t h e v a l u e s o f horizontal tensions and H ^ . the the The h o r i z o n t a l t e n s i o n i s a v e r y h a n d y p a r a m e t e r f o r t h e d e r i v a t i o n o f t h e e q u a t i o n s b u t i t i s n o t o f r e a l i n t e r e s t when s o l v i n g a p r a c t i c a l p r o b l e m . In p r a c t i c e the important f a c t o r s are: - deflection at - tension the at a given point supports - 141 - 4.1 P r o b l e m 1: Given tensions and 4.1.1 the deflection at Horizontal the tension The e q u a t i o n for D = at point is D x = in the the X are the cable deflection = 1 • Then H = " what supports? 1_ 2cos6H 1 l> 2D cos9 ( L x 1 If in 4.1.2 the the e q u a t i o n of the c a b l e i s needed equation d e r i v e d at 3.2. Tensions T f i at = _ H _ cosa o D the and plug this value supports T A = H cosa. A and f r o m 3 . 3 , t with H = U X 1 " l 2D,cos9 ± & r> ~ j± + L • IOL 2cos6H tga. = F, L oiL 2cos6H 0L ( L X ) of H - 142 - The d e f l e c t i o n D = Dm i s In H = this case OJL known a t m i d s p a n . x^ = L / 2 D = Dm 2 8Dmcos8 and tga tga. The t e n s i o n s a t T f i = = _E + ooL L 2cos6 (j, = E L h)L 2cos9 8Dmcos6 2 the 8Dmcos9 = E + 4Dm supports wL . 8 c o s a „ Dmcos9 and 2 = w L c a n be T^ = K NOTE: L 2 E - 4Dm L obtained from o)L 8 c o s a . Dmcos9 2 A A n o t h e r way o f d e r i v i n g t h e s l o p e o f t h e c a b l e a t i s to use the f o l l o w i n g p r o p e r t y of p a r a b o l a s : The t a n g e n t s a t A and B c r o s s t h r o u g h m i d s p a n and PQ = MP at Q on the Then: tga. = JEL AN tga = -2Dm + E / 2 = E - 4Dm L/2 L = J £ _ = -(2Dm + E / 2 ) T5R -(V2) = 4Dm + E L N A and B vertical - 143 - o If the cable i s very different. Assuming the chord With t h i s Then T = B B 1.5 the the angle and 8 a r e t e n s i o n at B acts not along assumption H = a cose D = Dm i s T = tight = 8 i s assuming that of the system. uix, ( L - x , ) 1 1_ 2DjCose H = if sufficiently ) , and X l ( L " X ' c o s a ^ = cos8 recall tg8 = E_ L l> aD^cose) 2 g i v e n at inidspan coL 8Dm(cos8) 2 2 Example Problem A f r e e h a n g i n g c a b l e has a 500-meter span w i t h a d i f f e r e n c e of 200 m e t e r s b e t w e e n t h e s u p p o r t s . The w e i g h t p e r m e t e r o f t h e 1" c a b l e i s a b o u t 6 l b s . The m i d s p a n d e f l e c t i o n i s 2%. a) Assuming t h a t the b) tensions Check t h a t is close to the at the the tensions both act a l o n g the cable, what are ends? difference difference i n t e n s i o n between in elevation the t i m e s oi. two supports - 144 - c) Assume now t h a t tension at the the cable is tight enough and c a l c u l a t e the upper s u p p o r t . Solution: a) from 4.1.3 Horizontal tension <oL 8Dmcose u) = 6 L lbs/meter Dm = .02 x 500 = 500 m 9 = Arctg = 10 (200/500) .2 6 x (500) 8 x 10 x c o s 2 1 . 8 = 21.8° 20194 lbs Tens i o n s Lower T. support = tga H cosaA = 200 - 4-x 500 10 = .32 10 .48 a . = 17.74' A 20194 cosl7.74 Upper = 21203 lbs support tga_ H cosa. B = 200 + 4 x 500 25.64 20194 cos25.64 b) 22400 E x (o 200 x 6 22400 v lbs 21203 = 1197 lbs 1200 lbs NOTE: One o f t h e c o n c l u s i o n s o f t h e c a t e n a r y f o r m u l a t i o n f o r t h e c a b l e ( b a s e d on m o d e l 3 , w e i g h t d i s t r i b u t e d o v e r t h e l e n g t h o f the c a b l e i t s e l f ) i s t h a t t h e d i f f e r e n c e i n t e n s i o n b e t w e e n two p o i n t s of a f r e e h a n g i n g c a b l e i s e q u a l t o the w e i g h t p e r u n i t l e n g t h o f the c a b l e t i m e s the d i f f e r e n c e i n e l e v a t i o n between the two p o i n t s . Because in this any m o d e l . characteristic is s i m p l e and exact it c a n be used - 145 - c) from 4.1.A 8Dm(cosO) B T' = B This 6 x . Given the point cable? the 2 (cos21.8) 2 = 21749 assumption underestimates P r o b l e m 2: of (500) 8 x 10 x s u p p o r t by 4.2 2 S i n c e the e q u a t i o n of (22400 tension 21749) T„ at lbs the = 650 B what tension lbs is at the upper o r 3%. the deflection of each " the deflection is D = tox(L - x ) 2cos6H for a given ' i g e o m e t r y o f t h e s y s t e m and t y p e H only. The p r o b l e m i s t h e n t o 4.2.1 Horizontal H = T cosa B and t g a Recall: R B tension i n the cable B = E_ + uiL L 2cos6H (from 3.3) cosa. B V1 Then: of cable the d e f l e c t i o n find H given T . H = T V + tg a 2 B D 1 + E + coL L 2cos6H at x depends on - 146 - H yi + T_ = E + - uL L 2cos6H H (1 + ]E_ + L Squaring both s i d e s : H 2 1 oil 2cos6H 'E' 2 + + • u>L 2cosG L H 2 'l E L + 2i + H J This quadratic equation aH c a n be + bH + c = 0 with is given H + Eto cosf3 Ed) cos6H oiL 2 - T B = 0 2cos6^ a = 1 + Eoi cosO oiL 2cos6 solution + 2 written b = The 1 2 -< 2 H = -b V7" by + 4ac 2a 4.2.2 Deflection H is t h e n known and t h e deflection at point x can be D = oix(L - x ) 2cos6H 4.2.3 To c a l c u l a t e H = T coso B tga i deflection at midspan given T ^ B = E + _ u L _ L 2cos9H H c a n be and the d e r i v e d as .2 Dm = <oL 8cos0H (from in 3.3) 4.2.1 (from 3.2) calculated from - 147 - 2.4 Tight T D cable c a n be assumed t o a c t a l o n g AB Tgcosctg = T c o s e Then: B The d e f l e c t i o n D = at any. p o i n t x i s uix(L - recall t£ = E L given by x) 2(cos6) T 2 2.5 B Example Problem A f r e e h a n g i n g c a b l e h a s a 500 m e t e r s p a n w i t h a d i f f e r e n c e o f 200 meters between the s u p p o r t s . The w e i g h t p e r m e t e r o f t h e 1" c a b l e i s a b o u t 6 l b s and t h e b r e a k i n g s t r e n g t h 4 4 . 8 t o n s . Use a s a f e t y f a c t o r o f 4. a) Assuming t h a t the t e n s i o n s act midspan d e f l e c t i o n i f the l i n e b) Assume now t h a t t h e c a b l e i s t i g h t enough a n d c a l c u l a t e m i d s p a n d e f l e c t i o n f o r t h e same c o n d i t i o n s a s b e f o r e . Solution a) from 4.2 Horizontal tension a l o n g t h e c a b l e what i s t h e i s tensioned to c a p a c i t y . the - 148 - H is a solution 1 + + H E = 200 T of: L = 500 That „ 2 B 8 = Arctg (200/500) = 21.8 lbs : yields H fl 2 + f200l 500 1.16H 2 + 1292H - 5 x 10 2 from (22400) x 5 x 10 8 = 1.16 4.2.3 6 x (500) = 10 8 c o s 2 1 . 8 x 20212 Deflection a t midspan = 10 500 meters = 2% 4.2.4 Horizontal = 0 = 0 lbs Dm = a) x L 8cos9H or 8 + 4 x 1.16 2 x H = 20212 6 x 500 2cos21.8 + H 200 x 6 + cos21.8 H = -1292 + Vl292 from ~ to = 6 = 4 4 . 8 x 2000 = 22400 a 2 uL 2cos6 Eh) + cos8 tension H = T^cosE H = 22400 x c o s 2 1 . 8 = 20800 l b s =9.7 6 x (500)' 8 x c o s 2 1 . 8 x 20800 Dm = oiL 8cos6H or Deflection a t m i d s p a n = 9 . 7 = 1.9% 500 This assumption underestimates the meters deflection. - 1 2 9 2 + 48184 2.32 - 149 - B. Cable with a s i n g l e 1. concentrated Governing equations the 1.1 f o r the load system. The c a b l e w e i g h t is assumed t o a c t subchords Additional notation C = point of R = weight attachment of the of S = sag of the l o a d on the cable load x = h o r i z o n t a l p o s i t i o n of cable 6^ = a n g l e b e t w e e n the at the load point C (Distance ' from x axis) s u b c h o r d CA and t h e h o r i z o n t a l 9^^ = A r c t g [ ( S / x ) ] &2 = a n g l e b e t w e e n e OL„ a r 0 2 the s u b c h o r d CB and t h e h o r i z o n t a l = A r c t g [(S + E)/(L - x)] = a n g l e between cable 1 and t h e h o r i z o n t a l a t C = a n g l e between cable 2 and t h e h o r i z o n t a l a t C on - 150 - The p o s i t i o n o f w i t h two o f t h e Two e q u a t i o n s 1.2 Equation - deflection - load R - tension static at one o f the supports relate these v a r i a b l e s : e q u i l i b r i u m of C. Sum o f h o r i z o n t a l forces The e q u i l i b r i u m o f H (tga the x components o f component o f Sum o f v e r t i c a l R - or sag at C can be w r i t t e n t o horizontal * t y p e o f c a b l e (oi) a l o n g the system c o m p l e t e l y . (1) : Express the * the- s u p p o r t s ( E , L ) and t h e following variables define r 1 the forces + tga ) =0 cable the tension forces is shows t h a t a constant H. the - 151 - The sections of cable 1 and 2 are two f r e e hanging cables. Then t g a can be derived from the formula i n Part A, Section 3.3. r E and L have to be s u b s t i t u t e d by t h e i r corresponding each s e c t i o n . x t g a r values f o r 2cos6 H 9 = S + E - oi(L'- x) L - x 2cos9 H 2 This y i e l d s R = H (S x 1 x R = Equation (2): 2COS6JH + S + E L - x + S + E - oi(L - x) L - x 2cosG H 2 2cos9, HL S + HE L - x x(L - x) oi (L - x) 2cos8„ oix 2cos6, oi(L - x) 2cos8„ (equation 1) - 152 - Express the t e n s i o n at B i n terms of H. Tg = H/cosa B T. = H/cosa ='H V l + (tgctg) 2 = H \]l + (tga ) 2 From Part A, Section 3.3, we can c a l c u l a t e t g a ^ f o r the f r e e hanging s e c t i o n • of cable 2, and t g a ^ f o r s e c t i o n 1. ^ x, t = a 'S * 01 = E + S •+ L - x .§ x A 2 aix 2cos9 H + 1 E + S L - x Tg = H | / l + T - x) 2cos8 H OJ(L _S x = H|/l + io(L - x) 2cos9 H (equation 2) 2 (equation 2 ) MX 2cos9 H Squaring both s i d e s of equation 2 2 2 Tg - H 2 2 1 E + S + \2 L - x 0 = H' (L - x) 2cos6 H + f + ) + (E + S)m [ L - xj J cos9 1 E s E + S L - x 2 j(L - x) cos9 H 2 [ai(L - x)] _ T [ 2cos9 ) 2 2 H 2 2 2 H can be e a s i l y obtained i f needed by s o l v i n g t h i s quadratic equation. Sag and d e f l e c t i o n Further development of equation 1 y i e l d s : S = Rx(L - x) - xE + HL i n which cos9^ and c o s f l 2 2 (L - x) + o)x(L - x) 2HLcos9, 2HL-COS0, MX are dependent on S. d e r i v e d without f u r t h e r assumptions. 2 No simple expression f o r S can be - 153 - C a b l e weight assumed A s s e e n on t h e figure cose^ = c o s 6 The p r e v i o u s S to 2 act above on t h e this c h o r d AB s i m p l i f i c a t i o n assumes = cos9 expression for S becomes: = R x ( L - x) - x E + a ) x a - x ) + iox(L - x ) 2 2 HL = foC - = Kx(L HL 1 L ~ *) HL x) S = R x ( L - x) HL that: 2HLcos9 2 - x E + cox L L - 3 aix x E + mxL(L - "3 o + aixL + aix 2HLcos6 o - 2mx I x) 2HLcos6 + ti)x(L - x) 2Hcos0 - xE L Sag a t at concentrated a distance x from load R lower support - 154 - This e q u a t i o n c a n be b r o k e n i n t o 1) O J X ( L - x) 2cos0H ii) iii) Rx(L HL x) three = MP = d e f l e c t i o n components: due t o c a b l e w e i g h t = PC = a d d i t i o n a l d e f l e c t i o n due t o - x E = NM (Part A, Section 3.2) load - L s a g = 1TC = P 1 : + M P + ' N M The d e f l e c t i o n MC p r o d u c e d a t t o be Deflection = Rx(L HL At midspan x = L / 2 2 Dm = RL + mL 4H 8cos9H x) the concentrated + oix(L - x ) 2cos6H = MC l o a d can thus easily be seen - 155 - 3. Problem 1 - Given load 3.1 the R what a r e t h e Horizontal Solving ( o r d e f l e c t i o n D^) a t x = x ^ and t h e sag tensions T and A t e n s i o n i n the at cable f o r H i n e q u a t i o n (1) yields LS, R + x (L - 1 x (L + x E l x co(L - + 2cos6, = R ux + x ) x M < ~ l> 2cos6, L x x ) ± + x 1 + cox. 1 ] =R - x )J E X x (L - + 2cos0, + Ex /L)-| and S ^ l x^ 1 L S the s u p p o r t s ? 2cos9, x + co(L - 2cos6, x Deflection X ; L 2cos9 ) r x = x^ at L H = x l ^ L ~ LD, x p R + u x ] ^ ~ L x i ^ + ^ ( L 2LD cos6 1 1 2LD - x 1 ) ' cos8 2 clamped - 156 - Recall from 1.1 tg6 = S /x 1 then 1 1 V 1 cosO, tg6 = (S 2 + E)/(L - 1 X;L then ) ^i'*!* + __1 1 + cos9. H is 3.2 then completely T e n s i o n s at the defined. supports t From 1 . 3 (equation 2 and 2 ) T and T A T A = H 1+ s * T B = Hl/1 + where H h a s 2003 1 E + S 3.3 If R is g i v e n and the 1 + U x^ the by: D ^ 1 + L - are given sag 6^ (L - x) ± 2cos9 H 2 value defined Sm i s in 3.1. known a t midspan, i . e . x^ = L / 2 S = Sm 3.3.1 Horizontal tension (from 3.1) Dm = Sm + E / 2 = D e f l e c t i o n a t m i d s p a n or Sm = Dm - H = LR + 4Dm E/2 toL + 2 16Dmcos9 and coL 2 16Dmcos9 1 + 2Sm 1 + 2(Sm + E) = LR + tuL 1 + 2 f 1 16Dm I c o s 9 4Dm + 2Dm - E /l + f2Dm + 1 cos9. 1 cos9. c o s 6, S l L - Y) 1 + E x. ] Introducing and = E = s l o p e of the c h o r d L p = Dm = p e r c e n t deflection/100 L s = tg6 1 + H = _K_ + toL 4p 16p 3.3.2 Tensions T A T at the s) 2 + \ / l + |2Sm + oiL L 4cos6 H = H HI + ( 2 ( E + Sm) + L 1 cos 1 + s) 2 and 2 oiL 4cos6„H 1 and i n t r o d u c i n g p and s 2 p yields: cos8^ A= l/ ( - Hl/l (2p - s) H (2p + supports = H Wl + Substituting T (2p - S + : + T B = Hl/1 + where H has '2 P + B + g y i + the v a l u e d e f i n e d Assumption t h a t the t e n s i o n s chords of the system at ( 2 p in + s ) : 3.3.1. the s u p p o r t s A and B a c t a l o n g the sub- - 158.- 3.4.1 Horizontal tension The e x p r e s s i o n • = X l ( L - of x )R 1 + H i s u n c h a n g e d and i s as d e f i n e d 2 2 uxj (L - x ) o) (L - x ) x LD^ 3.4.2 Tensions T at the 1 = H ^1 + (S /x )' Tg = H / c o s 8 2 = H \jl + f(S the the v a l u e sag S is Introducing s H = _R_ + 4p 1 1 + 1 + E)/(L - 1 defined known a t and p oiL 16p 2 supports 1 If 3.1. t 2LD cos6 = H/cos6 A X l 2LD^cos0^ where H h a s 3.4.3 + in in x^' 3.4.1 m i d s p a n (x.^ = L / 2 and (recall (2p - s) s = Sm) 3.3.1) 2 + \jl + (2p + s) 2 and V 3.5 H \jl + (2p - s)' H \ | l + (2p + s)' Example A c a b l e system has 200 m e t e r s b e t w e e n i s about 6 l b s . A d e f l e c t i o n at that a) Assuming t h a t tensions b) a 5 0 0 - m e t e r s p a n w i t h an e l e v a t i o n d i f f e r e n c e the s u p p o r t s . The w e i g h t p e r m e t e r o f t h e 1" 5 , 8 8 5 l b l o a d i s c l a m p e d a t m i d s p a n and the p o i n t i s 10%. the at both Assume now t h a t the tensions act tangent to the c a b l e , what are of cable the ends? the upper s u p p o r t . cable is tight and c a l c u l a t e the tension at - 159 - Solution a) Tensions act along Horizontal the tension cable (3.3.1) H = _R_ + jJL_ (\[ 1 + (2p 4p '16p ' R = 5,885 lbs s) \jl + + 2 (2p +'s) L = 500 m p = .1 2 s = E / L = 200/500 u = 6 lbs/m Then H = 14712 + 4099 = 18811 lbs Tensions (3.3.2) a t the supports Tg = H1 / 1 + = 18811 2p + s + u L j / l + (2p + s ) ' 4H . 2 + . 4 + _ 6 x 500 y 1 + i + ( . 2 + . 4 ) 2 4 x 18811 Tg = 18811 V l . 4 2 = 22400 lbs Similarly T . = 19048 l b s A b) Tensions a c t on t h e s u b c h o r d s Horizontal tension Same as b e f o r e Tg = T A = H\/I H^l (3.4.3) H = 18811 l b s + (2p + s ) + (2p - s) 2 = 2 = 18811 188.11 Vl736 VI704 = 21937 l b s = 19183 lbs 160 - Problem 2 at 4.1 the Given the sag (or x i s the Horizontal tension Recall H equation 1 + + This equation 2 The lifting capacity x = x , and t h e t e n s i o n J. "~ • R o f the system? T •' B cable ( E + S)toH .+ cos9„ x, l/cos9„ the at JL 2i L - aH in D ) (2) E + with deflection S u p p e r s u p p o r t what - oi (L - x) 2cos8„ 2 ~ „, 2 B 1 + is a quadratic of the form: + bH + c = 0 solution for H is Wb H = 2 - 4ac 2a 4.2 Lifting capacity Recall equation HLS, - x^(L where H h a s 4.3 If Tg i s 4.3.1 R f o r l o a d clamped at x ) 1 the value and t h e Equation 2(E oix^ HE L - x^ Horizontal 1 + x^ (1) + given point (2) u(L - 20086^ defined sag at Sm i s Xj) 2cos9, 4.1. R can then e a s i l y known a t midspan be = L/2 found. S = Sm, tension is reduced + Sm)j' H + to: (E + Sm)oiH + cos9„ w i t h Dm = Sm + E / 2 = D e f l e c t i o n at oiL 4cos9, midspan (from cos9. - T B (Sm = Dm - 3.3.1) = 0 E/2) - 161 - Introducing s qH and p , the + / q L a (2p + s ) 2 UJLJ I H + q fwLi where q = 1 + (2p + s) H is A.3.2 the p o s i t i v e Lifting (2) is reduced L 4 by a n a l o g y w i t h P With 4.4.1 the =0 2 the quadratic. R " f 1 cos 9 (Vl+ (2pbe assumption that Horizontal to: + 1 1 ^ cos9 | 2 3.3.1 R can t h e n e a s i l y 4.4 1/cos T„ B coL toL 4cos9^ 4cos02 R = 2H (2Sm + E) - u L 4 = s o l u t i o n of R = 4HSm + 2HE - = 2 - becomes: capacity R Equation R quadratic equation tension the s ) (2P + 2 + s) 2 , found. tension acts on t h e s u b c h o r d CB - 162 - H = T cos0 B with 2 cos6 = 2 1 1 + (S 4.4.2 Lifting R is H L S x^L 4.4.3 If the x = L/2 x from e q u a t i o n 1 + x^ 2 1 sag Sm i s and f r o m - known a t 2 " ^ " V ^1 2cos6 1 2 midspan S = Sm H = T cos9 B _EH 1 ( L - -x.^)' 2 c o s 6 - x ) Introducing s Dm = Sm + E / 2 and p = T B / V l + (2p + s) : 4.3.2 R = 4pH - u L 4 4.5 + E/L - capacity R obtained R - L Vl + - s) (2p 2 + Vl + (2p + s) 2 Example A c a b l e s y s t e m h a s a 5 0 0 - m e t e r s p a n w i t h an e l e v a t i o n d i f f e r e n c e o f 200 meters between the s u p p o r t s . T h e w e i g h t p e r m e t e r o f t h e 1" c a b l e i s a b o u t 6 l b s and t h e b r e a k i n g s t r e n g t h 4 4 . 8 t o n s . The d e f l e c t i o n a t m i d s p a n i m p o s e d b y t h e g r o u n d p r o f i l e i s 10%. What i s t h e l o a d t h e s y s t e m can l i f t a t m i d s p a n i n t h i s c o n d i t i o n ? Use a s a f e t y f a c t o r o f 4. a) Assuming t h a t the tensions at the supports act on t h e cable b) Assuming that the tensions at the supports act on t h e subchord Solution a) Horizontal tension H is qH Z a solution + (4.3) of: co (2p + s ) H + q (wL 2 [4] 2 where with oi = 6 q = l + ( 2 p + s ) L = 500, , s = E / L = 200/500 = .4 T_ — 4 4 . 8 x 2000 = 22400 lbs p = .1 itself 163 - - Then and q = 1 + (.2 the e q u a t i o n + .4) 2 = 1.36 becomes: 1.36H 2 + 1050H + 765000 - 1.36H 2 + 1050H - 501760000 = 0 500995000 = 0 H = - 1 0 5 0 + V ( 1 0 5 0 ) + 4 . x 1.36 2 x 1.36 2 Lifting capacity R = 4pH - wL j ^ | l + (2p - s ) R = 4 x . 1 x 18811 - = 7524 The 'b) lift R is + ( __ 2 ) 2 s) ' 2 + \j ±+ { f > ) 2 5885 l b s . on t h e s u b c h o r d (4.4.3 of: V l + (2p + s) H = 22400/\/l + (-6) Lifting + (2p + tension solution H = T / +Vl 1639 = 5885 l b s capacity Horizontal 2 6 x 500 4 Assuming the t e n s i o n H is x 50099500 = 18811 l b s 2 2 = 19208 lbs capacity R = 4 x . 1 x 19208 - 1639 = 6044 The l i f t i n g is capacity lbs overestimated by t h i s method. - 164 - To s o l v e t h i s p r o b l e m f o r a l l p o s i t i o n s o f t h e d e s c r i b e d by p o i n t C w h i l e the l o a d p r o g r e s s e s r e f e r r e d t o as " l o a d p a t h " a t c o n s t a n t t e n s i o n l o a d i s to f i n d the curve from A to B . This curve i s T . B U n f o r t u n a t e l y p r o b l e m 3 d o e s n o t h a v e any e a s y s o l u t i o n b e i s o l a t e d i n any o f t h e e q u a t i o n s . D e p e n d i n g on t h e a c c u r a c y d e s i r e d and o n t h e means from d i f f e r e n t methods. Two o f 5.1 t h o s e methods Graphical are presented since used,S S cannot can b e obtained here: - graphical solution - iterative technique solution T h i s s o l u t i o n c a n be w o r k e d o u t w i t h a s i m p l e i s as f o l l o w s : calculator. The p r o c e d u r e - 165 - l) make a g u e s s 2) now assume explained plot 3) S 1 at that you are if T , versus t o 2) , 3. and R and d e t e r m i n e I TL BI as T B1 T - is is given for problem 1 i n i A) S: smaller greater to than y o u r given T choose a s m a l l e r t h a n T„ c h o o s e a l a r g e r determine a new p o i n t on the S: S and if and go b a c k curve S versus When y o u e s t i m a t e t h a t y o u h a v e d e f i n e d t h e c u r v e S v e r s u s accuracy f i n d the v a l u e of S f o r your g i v e n T . . B T with T B enough Tension at B Curve T e n s i o n at B versus Sag G i v e n T, (solution for the Sag) T h i s method c a n be v e r y l a b o r i o u s i f the. f i r s t g u e s s e s f o r S d i f f e r g r o s s l y f r o m the f i n a l s o l u t i o n . To a v o i d u n n e c e s s a r y c a l c u l a t i o n t h e u s e r o f t h i s method i s a d v i s e d t o use t h e a s s u m p t i o n 3 . 4 f o r t h e f i r s t g u e s s e s . - 5.2 Iterative 166 - technique Can be w o r k e d w i t h The p r o c e d u r e i s a pocket as c a l c u l a t o r but make a g u e s s 2) assume now t h a t at S: explained determine a computer. S = you are given S 1 and T I 3) more a p p r o p r i a t e f o r follows: .1) as is f o r problem' 2 i n t h e new v a l u e c . new b = p r e v i o u s „ and d e t e r m i n e H and R 1 B 4 f o r S g i v e n by . (R - R.) S-^ + 1 x, 1 (L - x,) 1_ HL where R i s 4) go t o is your given load 1 u n t i l R^ i s n o t s i g n i f i c a n t l y d i f f e r e n t then the v a l u e you are l o o k i n g f o r . T h i s method c o n v e r g e s c h o i c e s o f S. towards the right answer f r o m R. The s a g for S for reasonable first - 167 - Standing s k y l i n e . 1. Gravity D e s c r i p t i o n of 1.1 the Additional system system notation C = carriage a T , a C1' T C2 , to = w e i g h t p e r u n i t length cf u>2 = w e i g h t p e r u n i t length of main a_. T C3 = a n g l e s o f the the c a r r i a g e = t e n s i o n s a t c The s u b s c r i p t s application.of respective 1" t for the h e skyline cables respective line with t h e a n g l e s and t e n s i o n s f o r c e and the l i n e . = angles of subchords 1 = Arctg ((S/x)) 2 = Arctg ((S + E ) / ( L - Q 6 the horizontal at lines Recall 6^ and the x)) define the point of - 1.2 168 - Variables In p r a c t i c a l a p p l i c a t i o n s 4 v a r i a b l e s d e f i n e the system i n a d d i t i o n t o the u s u a l g e o m e t r i c a l p a r a m e t e r s ( E , L ) and t h e w e i g h t s p e r u n i t l e n g t h . S T _ - 2. (or d e f l e c t i o n ) tension upper support - load, x - horizontal position lifting equations c a n be w r i t t e n Nine equations for the that Sum o f h o r i z o n t a l + H 2 3 - U load those v a r i a b l e s 1 c l + H tga C o n t i n u i t y of the whole l and system. c R forces = 0 ± (1) Sum o f v e r t i c a l H tga 2.2 system relate describe --.--i----------- * the line carriage H H the load system are developed E q u i l i b r i u m of * to the i n main capacity of of of system. Governing equations 2.1 at R Various the ( o r D) s a g 2 the forces c 2 + ^ t g a ^ tension in = R the skyline (2) through the carriage describe - 169 - The t e n s i o n through the T T T H 2.3 Cl = T lV 1 = + H /cosa c l H /cosa c 2 2 cl the = H ^ l + (tga ) 2 c l = H \/l + (tga ) 2 c 2 = + (tga 2 (tga ) Angles at ends H \/l 2 of tga t g a c l A l tga + of cable (4) " 21^0036^ (5) x 2 = S + E L - x tga ai(L - x ) 2H cos9 2 (6) 2 = S + E + ai(L - x) L - x 2H cos8 2 (7) 2 Section 3 tga-- tga - S + E L - x M 3 = S + E -r 3 L - x (L 2H cos6 X ' (L X 2 (9) ) 2H cos9 3 (8) ) 3 M p 3 passing (3) 2 1 x Section ) o f t g a h a s b e e n d e r i v e d i n P a r t A , 3.3. by t h e i r c o r r e s p o n d i n g v a l u e s f o r each = S aix x ' 2H^cos9^ = c 2 each s e c t i o n The g e n e r a l e x p r e s s i o n h a v e t o be s u b s t i t u t e d Section i s o n l y d i s t u r b e d i n d i r e c t i o n when carriage. Then: C2 1 Cl C2 i n the s k y l i n e s h e a v e s of t h e 2 .. E and L section. - Problem 1 tension 3.1 Given T„„ at BZ B in Horizontal 3.1.1 the sag the S (or 170 the - deflection D) and t h e l o a d R , what is the skyline? tensions H, is "tga obtained " are (1) H defined + H 2 from the - 3 H by x (4), (6) + H„ x 2H cos6 1 - or and (2) i n which the (8). uix 2cos6 + E L - x 2 S + E L - x + H, co(L - x ) 2H cos6 2 W 3 " 2H cos9 ( L X 3 ) 2 as: - co(L - x ) 2cos8 1 - "3 " 2cos8 ( L 2 S + S + E x L - x H, and fs 1 can be r e a r r a n g e d x| (1) = 0 (2) (2) equations 2cos8 x o,(L - x ) 2cos8 1 + ) (H 2 S + E + H ) 3 L 2 - "3 " 2cos6, ( L 2 X ) - x = R finally l = H R + cox 2cos6, is known. Therefore 3.1.2 H - x) SL + x E + "3 2cos6„ ( L 2 Equation H " lx(L X ) + co(L - x ) 2cos8« n Let is (3): H ^ l + known and t g a us Then H call 2 is . thf> f i r s t the (tga^) = H ^ l + 2 can be calculated term of equation solution of 1 + 11 S q u a r i n g b o t h s i d e s and r e p l a c i n g yields: 1 + S + E L - x io(L - x ) 2H cos8 2 2 (tga )' from equation (3) (4). " a " , which i s (tg<* ) tga^ c 2 = known. a c2 ^ y i t s e x pression equation (6) - 171 - further development H 3.2 2 is the T e n s i o n i n the H (S + E)oo + S + E L - x 1 + gives: cose solution skyline: of this - x) J(L 2 2 a 2 = 0 2cos0, 2 q u a d r a t i c e q u a t i o n as on p a g e 13. T B2 skyline T B2 = H 2^ equation c o s a B2 (7) = H 2 tRa , n ; + ^ t g c l B2^ 2 a 3.3 tension i n the d f r o m = S + E + ui(L - x ) L - x 2H cos9 2 The n skyline at the 2 upper support B i s then completely defined. Example Problem: A g r a v i t y c a b l e s y s t e m h a s a 500 m e t e r s p a n w i t h a n e l e v a t i o n d i f f e r e n c e o f 200 m e t e r s b e t w e e n t h e s u p p o r t s . The w e i g h t p e r p e r m e t e r o f t h e 1" s k y l i n e i s 6 l b s and t h a t o f t h e 3 / 4 " m a i n l i n e 3.4 l b s . T h e 6 2 3 4 . 5 l b l o a d e d c a r r i a g e i s a t m i d s p a n and t h e d e f l e c t i o n a t t h a t p o i n t i s 10%. What i s the tension in the skyline at the upper Solution: .. 3 . 3 . 1 H r R + oix 2cos8, + OJ(L - x' 2cos8„ w.j(L - x) 2cos8„ 2 J x(L - x) SL + xE support? - 172 - R = 6234.5 l b s oj = 6 lbs/tn L = 500 m x = 250 m S = -50 m tg6_ = -50 * 200 = - . 6 500 - 250 l/cos6 2 = 1.1662 tge, = l/cos6 1 = 1.0198 ^ 3.3.2 3 E = 200 m -50 500 - 250 = 20924 l b s H 2 1 + a o> = 3.4 lbs/m 2 = u S + E L - x 2 1 + H (S 2 + E)to cos9 + 2 o>(L - x) 2cos9, a = 0 (tga )' c l tga . = ^50 - 6 x 250 x 1.0198 = -.2365 250 2 x 20924 a = 21501 l b s ( t e n s i o n i n the s k y l i n e a t the c a r r i a g e ) H 3.3.3 2 = 18812 l b s T. B2 T B 2 = H /cosa 2 B 2 tga , = -50 + 200 + 6(250)(1.1662) 250 2 x 18812 T B 2 = 18812/.8397 = 22401 l b s = .6465 - 173 - Problem 2 at 4.1 B what is Given the lift Horizontal T sag S (or capacity the R of deflection the and the tension in the skyline system? 2 B2 = H 2 / c O S O l equation B2 = 2 H (7) t V 1 ( + 8 B2 A = s t g a B 2 + E Solving H 2 and is is for H the is 2 identical solution therefore of the ) 2 a n d f l ">(L - x) 2H cos6 + L - 4.1.2 D) tensions I 4.1.1 the x 2 as what has resulting 2 been quadratic d e r i v e d i n P a r t B, equation, known. «1 Equation H 2 is (3) is us 1 + known and t g everything is Let "xV call 0 1 ^ c a (tga n ^ e c c l a l i + (tga )' c u l a t e < the y from e q u a t i o n (6) where known. "a" the known q u a n t i t y i n the equation. H^ i s i c 2 solution of -xV1 + (tga c l ) = a second term of the 1.3 - 174 - S q u a r i n g b o t h s i d e s and r e p l a c i n g e q u a t i o n (4) yields: tg 0 1 ^ by i t s expression 1 + x further 2H^cos8^ development 1 + _S gives 2 2 4.1.3 H the solution 4.3 Lift 2 - =n0 this quadratic equation. 3 From e q u a t i o n 4.2 of a 2cos8, cos8. is - (1) H+H2 3 capacity R H- n x = 0 Equation (2) The tga are g i v e n by equations (4), the lift capacity defined. Example R = ^ t g a ^ = H, is + H tga completely 2 c 2 + H-jtga^ (6), (8). Therefore, % Problem: A g r a v i t y c a b l e s y s t e m h a s a 500 m e t e r s p a n w i t h a n e l e v a t i o n d i f f e r e n c e o f 200 m e t e r s b e t w e e n t h e s u p p o r t s . The weight p e r m e t e r o f t h e 1" s k y l i n e i s 6 l b s and t h e b r e a k i n g s t r e n g t h 4 4 . 8 tons. The w e i g h t p e r meter o f the 3/4" m a i n l i n e i s 3.4 l b s . T h e d e f l e c t i o n a t m i d s p a n i s 10%. What i s t h e conditions? l o a d the system can l i f t a t midspan i n Use a s a f e t y f a c t o r o f 4. these Solution: T, B2 22400 L 500 m -50 m lbs oo = 6 l b s / m x = 250 m l/cos8 2 = 1.1662 co = 3 . 4 3 lbs/m 200 m 1/cose^^ = 1 . 0 1 9 8 1 7 5 - 4.3.1 H„ Solving in for H I 2 s ^ n a X t h e e x a m p l e page 29 Then H 4.3.2 - 2 1 respects similar to the s o l u t i o n f o r H (4.5). = 18811 l b s H 1 + H So) 2 V X cos8, with a tga = - 5 0 + 200 250 c 2 H \ / 1 + 2 + 2 =n0 2 - a 2cos9, (tgapj)' 6(250) x 2 x 18811 1.1662 .5535 a = 1 8 8 1 l \ / l + ( . 5 5 3 5 ) " = 21500 l b s Then ^ 4.3.3 H H 4.3.4 = 20921 l b s 3 = 20921 - 3 Lift capacity R R = H tga C2 tga l t ga c + H tga l 2 = -50 250 C t g a 18811 = 2110 l b s = " 5 5 3 + ^ t g a ^ 6 x 250 x 1 . 0 1 9 8 2 x 20921 .2365 5 _ = - 5 0 + 200 250 R = 6234.5 c 2 lbs 3.4(250) 2 x x 1.1662 2110 = .3651 - 176 - Problem 3 the sag This Given S (or the tension T^^ i n the d e f l e c t i o n ) problem i s similar to at the the the s k y l i n e at S = f(S) 5.1 can be s o l v e d of one d e s c r i b e d on p a g e (5.) form S = f ( S ) where f is an f(S) expression for S = x(L - S appears i n o n p a g e 37 c a n b e r e a r r a n g e d t o x) R + x) R + LH, 5.2 the 31 f o r S on a c o m p u t e r b y i t e r a t i o n . S = SL + x E = x ( L H. or is function. Derivation The l o a d R , what carriage? S c a n o n l y be o b t a i n e d f r o m a n e x p r e s s i o n o f intricate B and t h e oix 2cos9, cox + io(L - x ) 2cos9, + co(L - x ) 2cos9^ the second + + term i n 3 2cos9„ 3 2cos0„ 2cos82 yield: Ex L the e x p r e s s i o n of H ^ , cos8^, cos92. Iteration No m a t t e r what i t e r a t i v e remains I d e n t i c a l : technique is used the 1) make a g u e s s a t 2) compute t h e v a l u e s of The d e r i v e d as value of 3) Find 4) Check on Go to general procedure S = is cos9^, cos92 and d e s c r i b e d f o r Problem 2 (4.1). t h e new v a l u e f o r S convergence 2 until the sag converges inside given tolerances. - 177 - D. Five line 1. D e s c r i p t i o n of 1.1 system the Additional to system notation weight per u n i t length of haulback u> w e i g h t p e r u n i t length of mainline 3 u^i a weight per u n i t a n g l e of length a cable with of the slackpuller horizontal T h e s u b s c r i p t s f o r t h e a n g l e s and t e n s i o n s of the f o r c e and the l i n e . define the point of application - 178 - 1.2 Variables I n p r a c t i c a l a p p l i c a t i o n s A v a r i a b l e s d e f i n e the system i n a d d i t i o n to t h e u s u a l g e o m e t r i c a l p a r a m e t e r s ( E , L ) and t h e w e i g h t s p e r u n i t l e n g t h . - S ( o r D) - x horizontal position - R load, - T . Governing 2.1 at of capacity of the of load) load the upper support equations the system i n main can be w r i t t e n to line relate those v a r i a b l e s and describe system. equations Tensions that for in lines The t e n s i o n at (or d e f l e c t i o n lifting tension Various the sag in point. the 1 and 1' the haulback r u n n i n g through a b l o c k at A remains T h e r e f o r e the b e t w e e n A and C a r e In p a r t i c u l a r the and a l s o H system =H',. sections of the free hanging lines unchanged 1 and identical. tensions at the carriage T c l and T c l , are the same 1' - 179 - 2.2 Equilibrium Sum o f Sum o f 2.3 the + H 2 l H t g a C l H\Jl H the 2.2 2 2H (1) = 0 X forces C2 t g a + H 3 t g a C3 + H '3 t g a C3 t e n s i o n i n the haulback page = (2) R through the carriage 35) = T C2 H /cosa , 1 to + forces - 3 the v e r t i c a l (similar Cl + H ' 3 Continuity of T carriage the h o r i z o n t a l H 2 of ( 1 = + (tga H /cosa 2 c l ) 2 c 2 = H 2 \ / l+ (tga c 2 )' (3) - 2.4 Angles at the ends of 180 - each s e c t i o n of cable The g e n e r a l e x p r e s s i o n of t g a has b e e n d e r i v e d i n p a r t A ( 3 . 3 ) . E and L and to have t o be s u b s t i t u t e d b y t h e i r c o r r e s p o n d i n g v a l u e s for each Section tga c l section. 1 (and iox. = S g a A l = Section tga tga = x + — 2H cose ] | tga „ = S + E L - x 2 „ = S + E + to(L - x ) L - x 2H cos6 2 section 2 (5) (5') (6) (7) 3 for 2 and to^ f o r to i n the expressions of tga written 2. ^ 3' S u b s t i t u t e 3 ' f o r 2 and to,, f o r to i n t h e o f t g a w r i t t e n f o r s e c t i o n 2. Mathematical (4') 3 Substitute Section Al' (4) i co(L - x ) 2H cos6 Section Cl' 2 2 for tga 2^0080^^ X t 1') s i m i l i t u d e between the standing expressions (8') s k y l i n e and the The g o v e r n i n g e q u a t i o n s d e r i v e d f o r the s t a n d i n g i d e n t i c a l to t h a t o f t h e f i v e l i n e s y s t e m . five line (9') system s k y l i n e a r e i n many ways F o r most o f the c a s e s the s o l v i n g p r o c e d u r e s d e v e l o p e d f o r t h e t h r e e p r o b l e m s s t u d i e d f o r the s t a n d i n g s k y l i n e w i l l be a p p l i c a b l e f o r the f i v e l i n e system w i t h v e r y few c h a n g e s . When t h e i n d i v i d u a l t e n s i o n s i n l i n e s 3 and 3 ' a r e n o t n e e d e d we c a n d e f i n e to^* = to^ + to^,, c o m b i n e d w e i g h t p e r u n i t l e n g t h o f t h e m a i n l i n e a n d s l a c k puller, and co* = 2to. - 1 8 1 With t h e s e new n o t a t i o n s 1 the e x p r e s s i o n + + co(L - x ) 2cosG 2 - of c a n be d e r i v e d f r o m io *(L - x) 3 2cos6 2 w h i c h compares w i t h the e x p r e s s i o n o f 2 on page (1) and (2) x ( L - x) SL + x 37 ( 3 . 1 . 1 ) and „ . • , v co*x + oi(L - x) S = x ( L - x) R + 2H L 2cos9^ 2cos6 T 1 w h i c h compares w i t h This similitude is a l m o s t any k i n d o f , + OJ_*(L 2 the e x p r e s s i o n of used to S on page develop v e r s a t i l e cable yarding system. - 3 2cosG, 4 3 x) xE L ( 5 . 1 ) . computer programs t h a t can handle - 182 - Figure 1+7 - Parabola i n the coordinate system. (x,y) - C A B L E L E N G T H P R O M T H E 18 3 - APPENDIX 2 P A R A B O L I C The p a r a b o l i c e q u a t i o n i n a t e system (x,y) Y = T H E O R Y o f t h e c a b l e i n t h e coord: i s : to 2cos9H 2 . E k L _ o)L , 2cos0H ; (b') The l e n g t h o f a s m a l l e l e m e n t o f c a b l e i s e v a l u a t e d as: ds 2 2 dx- +dy 1 + dx; y n—TI . B - 184 Therefore the t o t a l l e n g t h of the cable i s : S = The gjx l+( cosGH Ids = E L 2cos9H ) dx = square r o o t i n the i n t e g r a l can be expressed P(x) = where The - and ,E _ cuL , L 2cos0H A = B = 2u ,E cosGH L* T C = ( 2Cx + B 4C dx T h i s exact formula cations. as: 1/(1 + A) + Bx + Cx l 2 ; V " 0)L 2cos0H ) ^COS0rT s o l u t i o n to t h i s i n t e g r a l • ( /P(x)dx i s g i v e n by Selby(7) 4AC-B' P(x) o (i log ( 8C as dx ' J o • x^/T l/p(x) ' + -J=.)J i s cumbersome to use f o r p r a c t i c a l appli- An approximate e x p r e s s i o n of the l e n g t h of a t i g h t - 185 - cable i n the case of l e v e l Inglis(5) s u p p o r t s (E = o) i s g i v e n by and t h e W i r e Rope H a n d b o o k ( 1 2 ) a s : s - L ( l + §-'<22> ) 2 I f t h e two s u p p o r t s a r e n o t l e v e l t h e f o l l o w i n g e x p r e s s i o n c a n be used: 2 b cose U + 3 ' - 186 - APPENDIX 3 T E N S I O M E T E R A3.1 I n t r o d u c t i o n . A tensiometer i s d e f i n e d here as an capable of measuring instrument the t e n s i o n i n f i x e d or running lines. The need f o r a tensiometer a r i s e s when a load c e l l or any other s o r t of t e n s i o n measuring d e v i c e cannot be p l a c e d a t the dead end of the rope i n which the t e n s i o n i s t o be measured. Two types of tensiometers have been developed by d i f f e r e n t manufacturers mainly to meet the needs f o r crane i n d i c a t i n g and warning systems. The f i r s t type manufactured by Rucker Company 1 operates on the f i x e d r e l a t i o n s h i p between the t e n s i o n and the n a t u r a l frequency of a wirerope. In t h i s system, an e x c i t o r causes the c a b l e to v i b r a t e . The t e n s i o n i n the line of a g i v e n weight per u n i t l e n g t h i s d e r i v e d from the reading of the frequency w i t h the sensor. 2 D i l l o n and Company produces a tensiometer of the second "'"Rucker C o n t r o l Systems, 47 00 San Pablo Avenue, Oakland, C a l i f o r n i a 94608. 2 D i l l o n and Company, Inc., 14620 Keswick S t r e e t , Van Nuys, C a l i f o r n i a 91407. - 187 - Figure A-8 - Basic P r i n c i p l e of the tensiometer. - 188 - type. T h i s type works on a simpler mechanic p r i n c i p l e . c a b l e i s g i v e n a d e f l e c t i o n w i t h t h r e e sheaves. i n the l i n e i s deduced middle sheave. Two The The tension from the f o r c e thus c r e a t e d on the tensiometers of the second type were built. A3.2 D e s c r i p t i o n o f the Tensiometers. A3.2.1 Principle. The b a s i c mechanical p r i n c i p l e of the tensiometer i s i l l u s t r a t e d i n F i g u r e 48. three sheaves 1, 2 and 3. a l e v e r p i v o t i n g about 0. The c a b l e winds through the The c e n t r e sheave 2 i s mounted on The a c t i o n of the c a b l e on the middle sheave i s thus t r a n s m i t t e d t o the load c e l l to an i n d i c a t o r which measures the t e n s i o n . connected A study of the - 189 f o r c e balance on the cable - l e v e r shows t h a t the t e n s i o n T i n the i s r e l a t e d to the f o r c e F a p p l i e d to the T = load c e l l by: F /(2 x sinG) where 0 i s the angle of the t a l between the l i n e w i t h the horizon- sheaves. I f the geometry of the machine remains the same when the load i s a p p l i e d , sin0 i s a constant and the previous r e l a t i o n s h i p becomes: T = k x F where k i s the constant of the machine. A3 .2.2 Design. A scaled reproduction of the blue p r i n t of the siometer designed f o r t h i s study i s shown i n F i g u r e p a r a l l e l channel bars c o n s t i t u t e the g e n e r a l machine. The outer sheaves and l e v e r i s composed of two b o l t e d on the by f r i c t i o n i n the t r a n s m i s s i o n r o l l e r bearing l e v e r and the frame. The and s h a f t of taken so as to avoid of the f o r c e s . the losses A special i s a l s o used f o r the connection between load c e l l . The the rotate s t e e l p l a t e s welded t o g e t h e r S p e c i a l care was Two s t r u c t u r e of machined to support the b a l l bearings f o r the middle sheave. 49. the t r i a n g u l a r l e v e r on s h a f t s guided by p i l l o w b l o c k s ten- frame i s over designed to the - 190 - Figure 49 - Copy of the blue-print of the tensiometer. - minimize the at the flexion deflection maximum i n 192 and very the load - l i t t l e c e l l error which is does not requirements two tensiometers shown Table X. i n Table were manufactured L i n e diameter Requirements for (inches) Maximum l i n e t e n s i o n A3.2.3 The (newtons) Load heart of electronic the transducers changes in of the load c e l l fed into is calibrated c e l l . Very gauges bonded force d i r e c t l y i n changes the a i n strong steel Tensiometer 2 7/16 50000 5000 an terms a is the that voltage. schematically, to the tensiometers. 5/8 the machine into is for Cells. forces deflection . 3 mm X. Tensiometer 1 are exceed by load. The cells introduced The load the c e l l the resistance of due t o the or load is a output applied to of element. the application strain-gauges i n signal recorder composed steel Load changes electronic but e l a s t i c element c e l l . translate indicator of load which the strainThe of a connected - to form a balanced The in wheatsbone major the tensiometers Table Load c e l l 193 - bridge. characteristics a r e shown XI. Load i n c e l l s Table the load-cells used XI. characteristics. characteristics Brand name of Tensiometer 1 Tensiometer 2 BLH BLH U3G1 U3G1 Capacity ( l b s ) 10,000 11,000 Safe working l o a d ( l b s ) 15,000 1,500 Weight ( l b s ) 10 6 Recommended e x c i t a t i o n ( v o l t s AC o r DC) 12 12 1 Designation Output/input millivolt/volt 3 ( a t maximum c a p a c i t y ) 3 ( a t maximum c a p a c i t y ) 2 P r e c i s i o n f o r the t e n s i o n % BLH This - Electronics, Massachusett figure atory .3 I n c . , 42 F o u r t h 02154. was d e t e r m i n e d conditions. Avenue, experimentally .3 Waltham, i n ideal labor- - A3.3 Tensiometer The relate the indicator relationship and the reading F = and force F on a'R in the b to T = k T = ka'R + T = aR b to tensiometer the force F tension applied indicator is is in to necessary the the to line. load c e l l linear: 1 theory x a the + transmitted of reading between R - Calibration. calibration The 194 the the tension load T c e l l in the are line and related the by: F Therefore: or the and the graph T versus tensiometer sketched hooked and at gave gradually at the on the and between the tension R in shown set-up end ka' R Figure one = reading 1 before The with relationship indicator of + kb' "true" for The the line other end of the indicator and had was to line. consists T. be The of kb T in the example, determined to be the tension the for the is c e l l 1 calibrated was calibrated, with calibration of setting gauge the line linear. load laboratory The 1 calibration and been = an proved indicator tension tensiometer 51 b As tensiometer line the and linear. Figure the to siometer is calibration 50. of a applied a the winch tenfactor - 195 - Figure 50 - Sketch o f the equipment set-up f o r the c a l i b r a t i o n o f the t e n s i o m e t e r . F i g u r e 51 - Graph o f t e n s i o n read by the tensiometer versus t e n s i o n i n the l i n e before bration. cali- - 196 - - and the equal the zero to the value scale so on the indicator tension of the that T = T aR + cuted i f the A proper setting tained when dition is t r i a l s slope the are other R words, of be find the tension relation: = R (i.e. a = can be quickly explained the gauge factor the slope and line gauge of i f x T versus factor the line = for s u f f i c i e n t l y before y far this and hereafter of the 1 b = 0). Procedure. Factor. required reading origin Gauge satisfied taken the the of of and the In Set change the readings T procedure varies that line. factor calibration A3.4.1 so b Calibration The - 2 the previous becomes: A3.4 in gauge the 197 on T on R accurately is followed: the indicator (Figure the R and the apart. value 2 In of 52). indicator versus R exe- is is 1. values gauge ob- This practice, the The of contwo several factor is found. The in Table XII. different t r i a l s for tensiometer 1 are reported - 198 - Figure 52 - Influence of the gage factor of the i n d i cator on the graph of tension read by the tensiometer versus tension i n the l i n e . Figure 53 — Influence of the zero knob adjustment on the graph of tension read by the tensiometer versus tension i n the l i n e . - 199 - - Table X I I . T r i a l number Gauge f a c t o r = 2.02 2 = 2.07 3 2.08 20620 T R ± 5200 T 15420 y GF = 2.09 2 20040 T R ± 5250 T 14790 y 5350 20180 14670 T R ± 5350 T 14620 y 2 ± 20120 5560 14560 R 2 20250 T R ± 5080 T 15170 y R + b 20100 5510 ± 19970 R of equation = 2 2 The r e s u l t of t h i s f i r s t T ± R x 2 14750 R x 4 Tension T (newtons) 2 x GF ••== R e s u l t s o f the gauge f a c t o r adjustment f o r the s k y l i n e 'i tensiometer. R x GF - Tensiometer r e a d i n g R (newtons) 1 GF 200 ± 2 20480 5300 15180 s e t t i n g i s a l i n e T versus - 201 - Figure 54 - a) Graph of tension read by the skyline tensiometer versus tension i n the l i n e after f i n a l calibration. b) Graph of the discrepancies between tension read by the skyline tensiometer and the tension i n the l i n e , f o r increasing tensions and f o r decreasing tensions. - 202 - R tensiometer reading, NX 1000 - 203 - A3.4.2 S e t t i n g the O r i g i n of the Tension to Scale Zero. The a d j u s t m e n t o f t h e o r i g i n k n o b o n t h e i n d i c a t o r translates the l i n e T versus R ( F i g u r e 5 3). The proper s e t t i n g o f t h e o r i g i n knob i s o b t a i n e d when T^ = R 2 / w i t h .. T^ a t e n s i o n c h o s e n i n t h e m i d d l e For tensiometer in of the tensiometer range. 1, T^ was r e a d t o be 2006 on t h e l o a d c e l l 1 t h e l i n e a n d t h e o r i g i n k n o b was s e t t o 5.038 o n t h e t e n - siometer i n d i c a t o r t o r e a d t h e same v a l u e f o r R^. A3.4.3 Checking the C a l i b r a t i o n . To v e r i f y t h e c a l i b r a t i o n t h e c u r v e T v e r s u s R was determined f o r t e n s i o n s f r o m z e r o t o t h e maximum c a p a c i t y o f t h e i n s t r u m e n t and back t o z e r o . for tensiometer The r e s u l t s o f t h i s test 1 i s shown i n F i g u r e 54. A3.5 C o n c l u s i o n . The f i n a l t e s t shows h y s t e r e s i s f o r t h e c u r v e v e r s u s R. the l i n e The t e n s i o m e t e r o v e r - e s t i m a t e s t h e t e n s i o n when i s slackened. To a v o i d t h i s e r r o r t h e t e n s i o n mea- s u r e m e n t s w e r e made o n t h e a s c e n d i n g e s i s curve. then T lifted branch of the hyster- T h i s n e c e s s i t a t e d t h a t t h e l i n e be d r o p p e d and e v e r y t i m e t h e t e n s i o n had t o be a d j u s t e d t o a v a l u e s m a l l e r t h a n t h e p r e v i o u s one. - 204 The - e x p e c t e d e r r o r c a n be d e d u c e d f r o m t h e final t e s t t o be l e s s t h a n 1% f o r t e n s i o n s g r e a t e r t h a n 5000N. - 205 - APPENDIX 4 Note book sample page. Record of the data mea- sured w i t h the t h e o d o l i t e f o r the f r e e hanging t e s t . 1 H o r i z . angle Verti. degree minute degree Station 0 , angle minute 2.57 45 .0 2.S7 _J« •^ 2 41 38 _.1.8_.3._ 14.2 4- 34 53.0 6 31 14.0 8 27 _16.8 . 37-i 10 . ; 2> „°1'3 47-» 12 18 28.7 __i..3__. _ 4 0 . 9 . _ 14 16 •• zsy ^•7 0 -1 18 3 32.6 2?3 20 358 20.9 2£0 eS"'S Ul IV - 0 _3_>JL_. _10_.8_ 348 343 07.3 14.6 28 338 ..1C.3 30 ...334 .. 32 330 34 36 38 40 42 44 322 10.3 ..3_26__ ....24.7 , 319 316 2 66 IS-3 263 0S--I 52.9 2-53 s-*,-«* 46 309 36.2 48 50 307 '305 34.9 43.8 no 27i ^7^ . W e i l lher i-l Tempie.ra... ure. time sltirti: - 400 41 .1 0 4 .6 U IV '.c' ip I ! |l I M ' i 1 Lite. Start ncJrikiJleVel: yd W i>kL Endj- .U?>d_!_!_LLl Ja/C I ! ! i I II!II!I•I 1 _ M i d | a t a t i d n ! horiz. angle i 4 2 19.1 . v e r t i la'ngle :sta_}.t:.L_L. . ZQl: 2 7 . } I ! ! i. i ! i rerti. I ! ; ' ..2.6$ j angle end 1 Heigh't of. i n s j : U.t'. : : , U i l l j i i 111 i j 1 i i Comments, L THeod Statlijoh 52 -3 1 Theodolite! check plummet; / 1 02- Coll. IS-5 46.9 _3_L4._. ._14„._0_ 48.8 311 fl.w..[2!7l]JM J : I! _ie - o l_..<t .57.1. Date. _L-zJrb. s e t t I i '6 -1 14.3. FH ife'rjtlca'll l e v e l V 2. f>L 26i_ Experiment if IjtjUr \H*<\ LS2__ 40.8 22 - ._2.se 8 24 26 Comments 53. u, olite man ' 304 d^ejgrfej [minute 303 Vert degre: 50.4 01.8 Z4-3 angleL plnute 27 4 "02.9 Coaaeiit! jiiT - 206 - N o t e book s a m p l e p a g e . Record o f t h edata at t h e lower support, f o r t h e free hanging Time zero Indie. reading - J X P '.:" i me Tensior i Angle degree • IV. 14. 10 1? -35 11 lit,JO -Si II 0 l -81 1031 10 IS" IS, 00 _IDJ5:„ -S^ 01 test. \l >j if 11 + 6.5" ' -1a t *. i 1" + 6.3, Jae a _L J .116 r n r a t i 11 11 i n a t i i '_t -{e fi i Ini -C a t 3 L! (3 1 ,h .tn..lnJer r e f er e n c e : - L in 16 in t p a i n t ni a r k ...OS oidz&b S JJumbei ..of. ..pai n t marl s_beiwe en t h e Z7 j | ok : i - ! w u. wo A sP u •_. mrt 1 \ 1 0X i - - i i - i ! ovxcj / ,.. ..two„su pp.or.ts.: -- 1 r h 111 - A b i >i'- c 1 !C k losest 1. s 1, J ATI IT f- Vl i R M pc b a t .6 r v I taken J P 1 i i 1 1 i | j i i | i i | i i ; i i •! ! ! i !I i ri i i !I !i 1! 1 i i !i J \ ! 1 j• ij j| 11 1 > - 207 - N o t e book s a m p l e p a g e . Record o f t h e data taken at t h e upper support, f o r t h e f r e e hanging t e s t . i ; Time - i n d i e , Xensioi Comments rsadinf I /, -15 J.13.CL -IU.SL. CUj H i 3 .ttockcahr. J4 1 5-,oo JJJ.5" 1 S\ 15 l.l II 1 ) • ExperiitaeiVtl L DktA "Wei. \r>3\ m i l ! Temperature Ifjinis'h, OMidafcoiJchecM ' ' 'J I I I I I I ' ' ' Gil ! i ! i i i_i I ! I I ! I battery. ! ! ! AbneyLbiieckj: length -from-c l o s e s t p a i n t r to_upp e r ref« rencej. Cpmme.ntS. '/4 Winih 1 i! i ^2.1 '<?r6MC/bta I Ii iM il-f! - Table X I I I . Test # Mid-span deflection % 208 - Summary o f t h e e x p e r i m e n t a l o f t;he f r e e h a n g i n g t e s t . Tension upper support (N) results Tension lower support (N) Angle upper support (deg) Angle lower support (deg) length (m) 3.7 5200 4758 17.8 1.5 134.2 2 1.7 11000 10800 13.2 6.5 133.5 3 7.0 2727 2501 25.0 -5.8 135.5 4 6.3 3080 2815 23.0 -4.0 135.1 5 4.9 3885 3541 20.0 -1.0 134.6 6 2.9 6327 6121 16.1 3.1 134.0 7 2.4 7789 7514 15.0 5.5 133.9 8 2.1 8878 8613 14.2 5.1 133.8 9 1.9 9633 9349 14.0 5.7 133.7 - 209 - Table XIV. Test Summary o f e x p e r i m e n t a l r e s u l t s . Clamped l o a d on a s i n g l e l i n e . Load position (m) Y (m) X Tension upper support (N) Load (Kg) Tension lower support (N) 1 126.57 19.26 3747 100 3099 2 127.13 19.58 5886 200 4738 3 127.34 19.76 8103 300 6651 4 127.85 21.51 19168 300 18089 5 127.98 21.14 19914 495 18050 6 118.45 16.58 18442 495 16706 7 118.42 17.23 21788 495 20081 8 118.58 17.83 25309 495 23671 9 118.55 18.19 28939 495 27399 10 93.24 5.77 7710 200 6965 11 93.05 10.83 14224 200 13577 12 93.15 12.67 21169 200 20512 13 93.06 13.52 27546 200 26957 14 93.84 5.14 15244 495 14185 15 93.48 8.70 21562 495 20581 16 93.51 10.78 29037 495 28184 17 67.92 1.77 19620 495 18472 18 67.96 2.77 21719 495 20512 19 67.96 4.25 25662 495 24515 20 68.00 5.11 28704 495 27624 21 67.81 1.79 12841 300 11997 22 67.78 3.32 14950 300 14185 23 67.84 5.98 21395 300 20620 24 67.83 7.41 27978 300 27203 25 50.03 0.84 23073 500 22376 26 50.07 2.38 28537 500 27781 27 49.90 0.90 15195 300 14548 28 49.92 3.13 21042 300 20394 29 49.92 4.52 27909 300 27124 - 210 - T a b l e XV. Test # Carriage location, , Y (m) "•X (m) Summary o f e x p e r i m e n t a l of t h e g r a v i t y system. results T e n s i o n skyl i n e upper support (N) Load T e n s i o n skyl i n e lower support (N) Tension i n the mainline N 1 11.94 -1.42 11870: 300 11409 * 2 18.47 -1.24 13714 300 * 3 29.87 -0.39 15745 300 13272 * 4 44.64 1.27 17432 300 ft 5 56.50 2.98 18403 300 17118 * 6 65.07 4.38 18540 300 17913 * 7 72.22 5.66 18383 300 ft 313 8 82.92 7.75 17775 300 372 9 96.00 10.64 16510 300 17216 * 10 104.78 12.81 14930 300 529 11 115.32 15.81 12517 300 14616 * 12 125.03 19.26 9545 300 814 13 128.85 21.11 8112 300 9388 * 14 11.28 -1.49 17677 495 17147 ft 15 12.93 -1.51 18266 495 18001 ft 16 35.68 oo:o8 25162 495 24701 * 17 49.94 1.65 26143 495 * ft 18 65.51 3.84 25721 495 627 19 84.97 7.42 24054 495 25378 * 20 106.26 12.49 19982 495 1030 21 123.41 18.08 13380 495 18560 * 1486 22 129.13 20.73 8544 495 * 2020 Not measured. * * 441 637 922 794
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Experimental study of logging cable systems Guimier, Daniel Yves 1977
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Title | Experimental study of logging cable systems |
Creator |
Guimier, Daniel Yves |
Date Issued | 1977 |
Description | Two theoretical formulations of logging cable system problems, the catenary model and the parabolic model are investigated and compared with the results of experiments executed on a gravity system field model. The study shows that although the shape of a free hanging cable is better described as a catenary than a parabola, both theoretical models are accurate enough to solve practical cable system problems. The few dynamic tests tried on the field model show the great importance of the dynamic forces in a logging cable system and the need for further research in this field. |
Subject |
Lumbering Lumbering -- Machinery |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-19 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0094104 |
URI | http://hdl.handle.net/2429/20531 |
Degree |
Master of Applied Science - MASc |
Program |
Forestry |
Affiliation |
Forestry, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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