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(1)S I M P L I F I E D T E N S I O N. C A L C U L A T I O N I N. O F. S U S P E N S I O N. C A B L E. B R I D G E S. by. KENNETH. MARVIN. B.A.Sc. The. A. University. THESIS THE. of. SUBMITTED. in. accept. the. B r i t i s h. Columbia,. this. 1959. PARTIAL. FULFILLMENT. FOR. DEGREE. THE. OF A P P L I E D. the. required. THE. Eng.). IN. OF. SCIENCE. Department. . CIVIL. We. (Civil. REQUIREMENTS MASTER. RICHMOND. of. ENGINEERING. thesis. as. conforming. standard. UNIVERSITY. OF. BRITISH. September,. 1963. COLUMBIA. to. OF.

(2) In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t. of. the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t. freely. a v a i l a b l e for reference. per-. and study.. I f u r t h e r agree. mission for extensive copying of t h i s. t h e s i s for. that. scholarly. purposes may be granted by the Head of my Department or by h i s representativeso. It. i s understood that copying, or p u b l i -. c a t i o n of t h i s t h e s i s for 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 p e r m i s s i o n .. Department The U n i v e r s i t y of B r i t i s h Columbia-, Vancouver 8 , Canada..

(3) ii. ABSTRACT. This. thesis. d e t e r m i n a t i o n of the set of. p r e s e n t s a method w h i c h f a c i l i t a t e s. rapid. cable tension i n suspension bridges.. A. of t a b l e s and c u r v e s the. method.. or t r u s s e s. either' hinged at. bridges the. supports. s u p e r p o s i t i o n method i s d i s c u s s e d a n d. of i n f l u e n c e l i n e s f o r c a b l e. sion bridges. is. tension i n non-linear. the. suspen-. demonstrated.. A d e r i v a t i o n of the. suspension bridge equations. i n c l u d e d and v a r i o u s r e f i n e m e n t s. i n the. A computer program to a n a l y s e w r i t t e n as a n a i d i n t h e the. i n the a p p l i c a t i o n. continuous. A modified. use. i n c l u d e d f o r use. The method i s v a l i d f o r s u s p e n s i o n. with stiffening girders or. is. research. m a n u a l method p r o p o s e d .. included along with. Its. theory are. discussed.. suspension bridges. and f o r. is. the purpose. of. was testing. A d e s c r i p t i o n of the program i s. Fortran. listing..

(4) vii. . ACKNOWLEDGEMENTS. The a u t h o r i s i n d e b t e d. t o D r . R. P. H o o l e y f o r t h e. a s s i s t a n c e , guidance and encouragement g i v e n and i n t h e p r e p a r a t i o n. of this. thesis.. g r a t e f u l t o the N a t i o n a l Research Council money a v a i l a b l e f o r a r e s e a r c h Columbia E l e c t r i c. during. Also,. the research. the author i s. o f Canada f o r m a k i n g. a s s i s t a n t s h i p , and t o t h e B r i t i s h. Company f o r t h e d o n a t i o n o f $500 i n t h e f o r m. of a s c h o l a r s h i p .. K. M. R.. September. 1 6 ,. ±9&3. Vancouver, B r i t i s h. Columbia.

(5) iii. TABLE OF CONTENTS Page CHAPTER 1.. INTRODUCTION. 1. CHAPTER 2.. THEORY AND REFINEMENTS. 5. General Cable Equation Girder Equation S o l u t i o n of E q u a t i o n s E f f e c t of Refinements CHAPTER 3.. COMPUTER PROGRAM. S o l u t i o n of the I n t e g r a t i o n of Program Linkage Input Data f o r F i n a l N o t e s on CHAPTER 4.. ( l l ) and ( 3 5 ) i n T h e o r y on A c c u r a c y .. Girder Equation the Cable E q u a t i o n the the. DETERMINATION. Program Computer P r o g r a m. OF H. General S u p e r p o s i t i o n of P a r t i a l L o a d i n g Cases S i n g l e Span Three-Span Bridge w i t h Hinged Supports Three-Span Bridge w i t h Continuous G i r d e r Variable EI CHAPTER 5. APPENDIX 1 .. CONCLUSIONS. 5 8 12 19 21 25 27 32 33 3^ 36 37 37 38 39 44 45 50 52. BLOCK DIAGRAM AND FORTRAN L I S T I N G FOR COMPUTER PROGRAM. 55. APPENDIX 2 .. TABLES OF CONSTANTS. 60. APPENDIX 3.. NUMERICAL EXAMPLES OF CALCULATION OF H. 68. BIBLIOGRAPHY. 8l.

(6) iv. TABLE OF SYMBOLS. Geometry L. =. Length of. span. B. =. Difference. x. =. A b s c i s s a of u n d e f l e c t e d. cable. y. =. Ordinate. c a b l e measured. i n e l e v a t i o n of c a b l e. of u n d e f l e c t e d. ing undeflected dx. =. Increment. in x. dy. =. Increment. in y. ds. = .Incremental. L. T. =. I f 1^-1 Jo. L. e. =. IJ. d. ^|j. v efle= D c t i oVnes r t i c a l. 3. d. x. x. corresponding. join-. t o dx a n d dy. f o r a l l spans. f o r a l l spans. d e f l e c t i o n o f c a b l e and. h. =. H o r i z o n t a l d e f l e c t i o n of. h&. -. H o r i z o n t a l d e f l e c t i o n of l e f t. .hg. ='. H o r i z o n t a l d e f l e c t i o n of r i g h t. A. =. E q u i v a l e n t support (includes effect.of cable). from chord. supports. l e n g t h of cable. Wj. L. cable. supports. girder. cable cable. displacement. cable for. support support inextensible. t e m p e r a t u r e and s t r e s s. cable. e l o n g a t i o n of.

(7) V. Forces w. =. U n i f o r m l y d i s t r i b u t e d dead l o a d of. p. =. Distributed live. q. =. Distributed load equivalent. =. G i r d e r support. r e a c t i o n at. left. Rj3. =. G i r d e r support. r e a c t i o n at. r i g h t end o f. H. =. T o t a l h o r i z o n t a l component. Hp. =. H o r i z o n t a l component. o f dead l o a d c a b l e. H-^. =. H o r i z o n t a l component. of cable. l o a d on b r i d g e. t e m p e r a t u r e change H ' L. =. bridge. to suspender. of cable. of cable. span span. tension tension. t e n s i o n due t o l i v e. and s u p p o r t. H o r i z o n t a l component. end o f. forces. load,. displacement t e n s i o n due t o. on e q u i v a l e n t b r i d g e w i t h i n e x t e n s i b l e. liwe. load. c a b l e and i m m o v a b l e. supports 8H. =. C o r r e c t i o n t o H-^' t o a c c o u n t support. for. e x t e n s i o n of cable. movement. B e n d i n g Moments =. B e n d i n g moment i n g i r d e r. =. B e n d i n g moment i n g i r d e r a t. left. Mg. =. B e n d i n g moment i n g i r d e r a t. right. M'. =. B e n d i n g moment i n e q u i v a l e n t g i r d e r w i t h no. M'i. E l a s t i c and T h e r m a l. support support cable. Properties. 6. =. C o e f f i c i e n t of t h e r m a l expansion f o r. t. =. Temperature. A.. =. C r o s s - s e c t i o n a l area. E. =. Y o u n g ' s Modulus. I. =. Moment o f i n e r t i a o f. m. =. C o e f f i c i e n t of shear d i s t o r t i o n f o r g i r d e r or. cable. rise of. cable. girder truss. and.

(8) vi A. =. C r o s s - s e c t i o n a l area. G. =. Shear. A^. =. C r o s s - s e c t i o n a l area. 0. =. A n g l e measured. w. of g i r d e r. web. modulus of t r u s s. from t r u s s. diagonal(s). v e r t i c a l to. diagonal(s). equations. approximating. Computer Program a-A. =. C o e f f i c i e n t s of d i f f e r e n c e. girder. equation D. F. P. f o r D e f l e c t i o n Theory s o l u t i o n. =1. h. s. =. 0 f o r E l a s t i c Theory s o l u t i o n. =. 1 to i n c l u d e e f f e c t. =. 0 to d e l e t e. effect. = " 1 t o i n c l u d e change 0. t o d e l e t e. =. effect. of h o r i z o n t a l d e f l e c t i o n of h o r i z o n t a l d e f l e c t i o n i n cable. slope. of c a b l e. slope. i n cable. equation. change. Miscellaneous. a. =. E_ E II E R a t i o of side. b. =. f sf. V. L. 7. EI EI. span l e n g t h ' L. g. to main span l e n g t h L.

(9) S I M P L I F I E D CALCULATION OF CABLE TENSION I N SUSPENSION BRIDGES. . CHAPTER 1 INTRODUCTION. This new t h o u g h t s subject. thesis. adds a f e w new w o r d s ,. to an a r e a. o f s t u d y w h i c h has. of a c o n s i d e r a b l e. analysis. amount. of s u s p e n s i o n b r i d g e s. from the u s u a l ' p r o b l e m s a n d somewhat differences. and t h e. i s a p r o b l e m somewhat by the. to s o l v e .. d i f f i c u l t i e s that. It. the. structural. engineer. i s because of. the. been. i n v o l v e d and t o. d i f f i c u l t i e s i n a n a l y s i n g and d e s i g n i n g. The. different. so much w o r k has. b o t h t o e x p l o r e e x t e n s i v e l y the problems come t h e. a l r e a d y been. o f s t u d y and l i t e r a t u r e .. encountered. more d i f f i c u l t. and p e r h a p s a f e w. done. over-. suspension. bridges. The p r o b l e m i n a n a l y s i s result. of t h e i r r e l a t i v e f l e x i b i l i t y. to d e f l e c t in. the. of s u s p e n s i o n b r i d g e s. i n s u c h a manner as. stiffening girder.. and t h e i r. to m i n i m i z e the. D o u b l i n g the. tures.. suspension bridges. are. desirable bending. the b e n d i n g. s a i d t o be n o n - l i n e a r. stresses sus-. moments. struc-. That i s , , t h e r e i s a n o n - l i n e a r r e l a t i o n s h i p between 1. a. ability. load a p p l i e d to a. p e n s i o n b r i d g e does n o t n e c e s s a r i l y d o u b l e Therefore,. Is. load.

(10) 2. and r e s u l t a n t is. that. A direct. s u p e r p o s i t i o n of r e s u l t s. methods able. stresses.. of a n a l y s i s. i n the. result. of t h i s. of p a r t i a l. non-linearity. loadings,. d e p e n d e n t on s u p e r p o s i t i o n a r e. analysis. of suspension b r i d g e s .. It. and. not. applic-. w i l l be. shown. h e r e t h a t a m o d i f i e d s u p e r p o s i t i o n method c a n be a d a p t e d solution. of s u s p e n s i o n b r i d g e Investigation. by t w o . o b j e c t i v e s developed.. least. theory. lems,. so i n the. exact. theory. of a n a l y s i s .. analysis. been the. solutions. Theory, which takes account. vior. of s u s p e n s i o n b r i d g e s .. been. the. simplification. been. development. of the. use. of suspension b r i d g e. theory. l a b o r r e q u i r e d f o r a n a l y s i s and d e s i g n . the. a l i n e a r r e l a t i o n s h i p between. the. depending. on t h e. give different flexibility. i n order. . Chapter 2 i s devoted t i o n Theory or forms accepted in. the. standard. field. of i t .. in. geo-'. under. and t h e. As m i g h t be. results. usual. expected,. which can v a r y. widely. bridge.. to a development. of the. Deflec-.. T h e r e seems t o be no u n i v e r s a l l y. D e f l e c t i o n Theory.. favors. to. A result. changes. l o a d and s t r e s s. of the. has. Thus', t h e E l a s t i c T h e o r y. o f s u p e r p o s i t i o n c a n be u s e d .. two t h e o r i e s. the. beha-. Another g o a l of i n v e s t i g a t o r s. l o a d and t e m p e r a t u r e c h a n g e s .. methods. of. non-linear. metry r e s u l t i n g from d e f l e c t i o n , of a suspension b r i d g e. is. be. However,. c a n be o b t a i n e d b y t h e. been the E l a s t i c T h e o r y , w h i c h i g n o r e s. live. prob-. a completely. t o use f o r d e s i g n p u r p o s e s .. Deflection. has. have. inspired. i m p o s s i b l e t o d e v e l o p and w o u l d. accurate. the. been. As i n most e n g i n e e r i n g. reasonably. reduce. has. of s u s p e n s i o n b r i d g e s ,. is virtually. e x t r e m e l y cumbersome. has. two m a i n t h e o r i e s. One g o a l o f i n v e s t i g a t o r s. of an exact. the. problems.. of s u s p e n s i o n b r i d g e s. and a t. to. Each of the. a slightly different. version.. many. experts. Various.

(11) 3 refinements. i n the. r a c y of the. calculated results. equations. may t a k e. desired. effect. different. and thus the. on t h e. equations. i s shown.. v e r s i o n o f the. is a quantitative. the. tion. is. D e f l e c t i o n Theory...,. f o r the. s h o u l d be n o t e d. is. shown in-. on a c c u r a c y. or n e g l e c t. t o be f o u n d. following. that. in provide. throughout. this. work c o n s i d e r a loadings.. i s g i v e n h e r e t o t h e more c o m p l e x c o n s i d e r a t i o n s. It. will. be s e e n. of. chapters.. c o n d i t i o n s and s t a t i c. d y n a m i c l o a d i n g s on s u s p e n s i o n. solution. Also. effects. of i n c l u s i o n. . No new t h e o r y. confined to s t a t i c. attention. the. development has been i n c l u d e d h e r e t o. a framework of r e f e r e n c e It. Theory. accuracy. d i s c u s s e d and. i n d i c a t i o n of the. refinements.. Chapter 2 but. are. accu-. The E l a s t i c T h e o r y i s. w h i c h m i g h t be e x p e c t e d as a r e s u l t some o f t h e. Deflection. f o r m s d e p e n d i n g on t h e. Some o f t h e s e r e f i n e m e n t s. as a s i m p l i f i e d cluded. t h e o r y may be i n c l u d e d t o i m p r o v e t h e. No of. bridges.. i n development. of the. theory. of a s u s p e n s i o n b r i d g e p r o b l e m i n v o l v e s the. that. simultaneous. s o l u t i o n o f a d i f f e r e n t i a l e q u a t i o n and a n i n t e g r a l. equation.. In the. necessary. more g e n e r a l and more e x a c t. to r e s o r t. t o n u m e r i c a l methods. equations.. The s i m u l t a n e o u s. t r y method.. solutions,. f o r the. s o l u t i o n of each of. these. and. H e n c e , s o l u t i o n o f a n u m e r i c a l e x a m p l e c a n become. Fortunately, i s no l o n g e r n e c e s s a r y. procedure. because of the. on a c o m p u t e r .. was w r i t t e n f o r t h e. by hand c a l c u l a t i o n s .. existence. of computers. t o p e r f o r m a l l c a l c u l a t i o n s by hand.. The p r o b l e m o f s u s p e n s i o n b r i d g e a n a l y s i s. investigate. is. s o l u t i o n i s found by a c u t. a n e x t r e m e l y l e n g t h y and t e d i o u s. solution. it. is well. Chapter 3 describes. I B M 1620. digital. suited. a program which. computer I n o r d e r. suspension bridge a n a l y s i s .. It. for. to. is believed. that. it.

(12) 4 the. methods e m p l o y e d I n t h e p r o g r a m a r e. analysis.. For that reason,. a listing. has. been i n c l u d e d i n the hopes t h a t. the. preparation. of s u s p e n s i o n b r i d g e. The k e y t o a s i m p l i f i e d problems. is a rapid determination. tension.. it. I n the. more e x a c t. a c u t and t r y m e t h o d .. well. suited for. of the. Fortran. may s e r v e. the. of the. v a l u e o f the. cable. dimensionless examples. the. ratios.. use. These are. i l l u s t r a t i n g the. the. total. value. t o be d e t e r m i n e d a n d i s quired. to i n i t i a t e. therefore. calculations.. or curves. bridges. with continuous. either. hinged at. the. supports.. of c a b l e. tension the. Use o f. relating. certain. included, along with numerical. which i s. method.. The method. shown t o be. valid,. Since H is. the. unknown, an e s t i m a t e The i n i t i a l. improved by a r a p i d l y c o n v e r g i n g i t e r a t i v e of H .. believed. method i s d e v e l o p e d .. of H i s known.. accurate value. cable. a method,. a p p l i c a t i o n of the. employs a form of s u p e r p o s i t i o n , providing. bridge. The p r i n c i p l e s u p o n w h i c h. of t a b l e s. in. t e n s i o n i s found by. Chapter 4 describes. shown a n d t h e. method r e q u i r e s. as a g u i d e. programs.. t o be new, w h e r e b y H , t h e h o r i z o n t a l component. method depends a r e. program. s o l u t i o n to suspension. methods,. c a n be f o u n d e x t r e m e l y q u i c k l y .. computer. value. is.re-. e s t i m a t e of H i s. procedure. The method may be a p p l i e d t o stiffening girders. to g i v e an suspension. or w i t h. girders.

(13) 5. CHAPTER 2 THEORY AND REFINEMENTS. General The f o l l o w i n g of. a loaded g i r d e r ,. suspended is. concerned with. or e q u i v a l e n t plane. truss,. the. tower tops or anchorages.. simplifying. assumptions. are. I n the. The s u s p e n d e r s a r e. Inextensible.. 2.. The s u s p e n d e r s a r e. so c l o s e. together. may be r e p l a c e d b y a c o n t i n u o u s The d e a d the 4.. is. dead l o a d a l o n e ,. cable. is. Less exact. u s e u s u a l l y make t h e. they. along. are. forms. following. usual for. be. neglected.. no b e n d i n g. action moment.. and. hence. the. so-called. of suspension. bridge. D e f l e c t i o n T h e o r y i n common. additional. s m a l l compared w i t h. the. f o r each span,. theory. of the. under. parabolic.. The h o r i z o n t a l d e f l e c t i o n s. can. the. fastening.. straight. constant. " D e f l e c t i o n T h e o r y " o r more e x a c t. 6.. that. is distributed. and c a r r i e s. initially. The above a s s u m p t i o n s. analysis.. bridge. initially. The d e a d l o a d i s the. analysis,. girders.. The g i r d e r of. 5.. l o a d of the. which. made:. 1.. 3.. case. o f known r i g i d i t y ,. by v e r t i c a l suspenders from a p e r f e c t l y c a b l e ,. anchored at. following. derivation is. the. assumptions: of the. cable are. very. vertical deflections,. and.

(14) 6. 7.. Deflections of the cable are very small compared with cable ordinates, and their effect on cable slope can be neglected in calculation of cable extension.. 8.. Shear deflections in the girders are very small compared with bending deflection and can be neglected .. Assumptions 6, 1 and 8 may be excluded with l i t t l e difficulty in the derivation, and may even be excluded in an analysis by digital computer.. Therefore, the effects of hori-. zontal deflections, cable slope change, and shear deflection are included here and discussed briefly.. It is'not to be thought. that their inclusion results in a complete theory, but perhaps these are some of the more important refinements which can be made.. Others* have discussed the effect of the above refine-. ments,- and in addition have introduced, or at least mentioned, other refinements such as tower horizontal force, tower shortening cable lock at midspan, effect of loads between hangers, temperature differentials between girder flanges, finite hanger spacing, weight of cable and hangers, variation of horizontal component of cable tension with hanger inclination, and so forth. The Deflection Theory of suspension bridge analysis results in a non-linear relationship between forces and deflections and hence the principle of superposition and methods dependent on superposition are not applicable in the usual manner.. In order to simplify the force-deflection relationship. "Into a linear one, i t is necessary:to make a further simplifying * Reference ( 1 2 ).

(15) 7 assumption. for. the. It. is. "Elastic. errors. which. it. for. not. the. small. as. to. metry. of. the. ness. the. is. Much. end. to. here. cable. as. relating This. on. the. geo-. moment. arm. of. long. passed. out. has. that. by. been. cable. yield. of. the. of. is. two. at. and. loaded. end. reactions. and. end. moments. applied. support.. In q. a. Rg, to. of. design.. attempts. of. high. results. of. to. i n Were. useful-. simplify. accuracy. with. Theory;. and. i t. is. to. Theory. consists. of. the. a. and. is. to. end. hinged. addition,. equivalent. Is. an. that. d i f f e r e n t i a l and. as. cable. the. tension.. deflections. girder. are. i n i t i a l l y. dead. moments. the. or. girder. the. the. load M^ a n d the is. a. Mg,. with. are. posi-. supported. span w,. equation.. bridge. and. to. to. equation. girder. suspension. equal. referred. deflections. forces. girder. to. f i r s t. cable. loads here. constant. and. the. span. distance a. The. relates. girder. single. cable. with. is. to. distances,. by. girder. and. equation. a. separated. The. in. equations.. referred. 1 shows. Both. load. Deflection. deflection. shown.. economy. since. E l a s t i c. Theory. devoted.. second. as. E l a s t i c. results. equation,. equation. the. expended. to. the. solution. The. have. is. tive. distributed. hence. the. force.. would. A l l. the. and. on. so. calculations,. loads.. at. cable. effect. are. Theory. applied. B. negligible. girder. Deflection. Figure. A. and. of. girder. second. a. cable. follows:. lengthiness. thesis. loads.. the. as. basis. satisfy. Theory. the. have. stated. the. to. Solution simultaneous. of. be. is. large. too. energy. this. may. which. that. approaching that. assumption. known. Theory. Deflection. ease. cable. well. are. the. E l a s t i c. It. deflections. the It. further. Theory".. The. 9.. this. length live which. L.. load. p,. may. be. result. of. continuity. subject. to. the. suspender. forces.. The.

(16) TO. Figure. 2.. FOLLOW. PAGE.

(17) cable is connected to the girder by vertical suspenders and carries the distributed load q.. The cable is in tension, the. horizontal component of which is constant and is equal to H. At the supports, the vertical components of the cable tension are y ' and Vg'. A. Under the action of live load and temperature. changes, the cable and girder deflect from the positions shown in solid lines to the positions indicated by dashed lines. The cable supports deflect horizontally the distances h^ and hg. The original cable position is given by co-ordinates x and y measured horizontally from A and vertically from the chord joining the undeflected cable supports at A and B.. A point P on. the cable deflects from its initial position to a point P' horizontally a distance h and vertically a distance v. A point Q, on the girder deflects from Its initial position vertically below P to a position Q,' vertically a distance v. Cable Equation Figure 2 shows an elemental length of the cable at point P.. Its undeflected position is shown as a solid line,. while its deflected position is shown as a dashed line.. The. length of the element in the undeflected position is given by (ds). (dx). 2. •~. (dy. 2. +. \. Bdx\. -I. 2. J. -•. (. 1. >. Under the action of live loads the cable deflects as shown and the length of the same element of cable in the deflected position is given by (ds + Sds). 2. (dx + dh). /dy. 2. +. Bdx. dv\. 2. +. .... (2).

(18) Subtracting (l) from (2) and rearranging terms, i t is found that ds S'ds -. 1 6ds. fl. —. +. -. dh. —. 1. 1. dx. dh\. dv /dy. 2 dx/. B. 1. -. dx Vdx. L. +. dv ]. 2 dx.. . . .. (3). dx dx \ 2 dx Since ^ds and ^h both extremely small compared with unity, dx dx they may be dropped from the terms 1 + i ^ and 1 + 1 ^ 1 . 2 dx 2 dx The term i is generally small compared with - ^ over 2 dx dx li most of the span, but may be significant, especially in very a r e. s. flat cables. ds 6ds dh. Expression ( 3 ) then reduces to dv /dy B 1 dv'. dx. dx \dx. dx. dx. L. 2 dx>. The extension of the cable 8ds as caused by temperature expansion and stress is given by 1 8ds. 6tds. dx where:. 2. ds. dx. (5). AE dx. 6. = coefficient of thermal expansion. t. = temperature rise. . H-^ = change in horizontal component of cable tension due to application of live load, temperature changes, support movement, etc. A = cross-sectional area of cable E. = Young's Modulus for cable material. Again, since fUl is extremely small compared with unity, i t may /. d X. \2. be deleted from- the term ( 1 + ^±\ . Then, the binomial theorem V bracketed dx/ All but the first can be applied to expand the term, <. two terms can be neglected, giving 8 d.s dx. £ tds dx. H. L. ds. AE dx. 1. 1. /dy. 2 Vdx. B' L,. +. dv /dy. B. 1. dv\. dx Vdx. L. 2 dx/. (6).

(19) 10. or, since' ds. 1. dx. 2. 1. /dy B>. 2. \dx L,. (7). then 6ds. etds. dx. dx. H ds. ds. L. +. AE dx dx. dv / dy. B. dx. L. V dx. 1 2. dv>. . (8). dxy. A combination of equation (4) representing cable geometry and equation (8) representing Hooke's Law gives the cable equation as dh dx. et. /ds\. H fas\. 2. 3. *~. T. +. ^dxy. —. AE VdxJ. HL. ds). AE Vdxi. 2. dv / dy. B. dx I dx. L. dv. A. 1 2. dx, (9). The above cable equation may be simplified significantly i f i t is observed that the term [OS- - — + In expression (5) is \dx L dx/ normally less than .2, and £LY_ is generally small compared with dx dy - I L Hence, dv_--j_- ^ y significant in the total expresdx L dx sion and can reasonably be neglected. Since — has already dx s n. o. v e r. been neglected compared with unity in the same expression, this amounts, to neglect of the effect of deflections on cable slope 6ds etds 'dsVbecomes H (5) and expression dx. -. -H T. +. (5a). dx AE \dx/ When (5a) is combined with ( 4 ) , the simplified cable equation is dh. et /ds\. dx. idxy. 2. H +. L. AE. /ds\ \dx>. 3. dv (dy dx \dx. B. I d v' L 2 dx,. (9a). It can be seen that neglect of the change of cable slope is reflected in expression (9) by neglect of the term ^ (4J AE \dxI.

(20) 11 _J± is usually of the order .001 and — AE Is normally not much larger than 1, so a term of order .001 has. compared with unity.. d x. been neglected compared with 1. argues that i t is negligible.. On this basis, Timoshenko However, i t is not difficult to. see that a given percentage error in one term of expression (9) could be magnified by subtracting that term from another of similar magnitude to give a larger percentage error in ^Jl. Expression (9) can be further simplified i f — — is 2.Mx. neglected compared with. - 5. in expression ( 4 ) . dx L. Then the. cable equation becomes dh et /ds\ H Ids\ 3 dv /dy B\ — = — + dx \dx/ AE \dx/ dx \dx L/ This final expression gives a linear relationship between hori2. T. (. 9. b. ). zontal and vertical deflections. It should be noted that the above linear relationship between horizontal and vertical deflections does not imply that the structure is linear.. The cable equation has been reduced. to a linear equation, but a non-linear relationship can and does s t i l l exist between stresses and applied loads. If the cable equation is integrated over the span length and the horizontal displacements of the supports are inserted as constants of integration, the following expression results: h. B. -h. r. L. A. J. r. L. o. Gt /ds\ dx 2. o HL. ds. AE Vdx,. dx. (10).

(21) 12 |ds_^ dx. L. or, i f. denoted by L, and. 2. i s. L. f _^.^ d. 3. d x. is denoted. _^ o \ y. by L. , then Q. h. B "A h. €. t. L. =. C. t. +. L. %. e AE L. H /ds. v. 2. L. AE \dx',. dv /dy. B. 1. dv. dx \dx. L. 2 dx. dx. (li)}. If the change in cable slope can be neglected, then - h. G. n. tL<_. H. T. L_. f. dv / dy. L. B. 1. dv \ dx. (•lla). +. AE. dx \ dx 2 dx o If the term ^ can be neglected compared with Q*L - ^ then 2 dx, dx L h - h„ G tL H L • dv / dy B \ dx B A t L e _ . .. (lib) AE LJ o dx \dx r. =. L. +. The cable equation (ll) or the simplified forms (lla) and (lib) relate cable displacements to cable loads.. It will. be seen that the refinement represented in equation (ll) is difficult to justify, considering the additional accuracy attained and the effort required to solve- the equation.. Equation. (lla) requires considerably less effort to solve, but is also difficult to justify.. Equation (lib) will be found to be. sufficiently accurate for most purposes. The cable equation is one of the two equations to be solved in analysis of suspension bridges.. Cable deflections. must be consistent with support movements and with girder deflections as given by the girder equation. Girder Equation In order to derive the girder equation, i t is necessary to consider separately the two main components of the bridge,.

(22) TO. Figure. 4.. FOLLOW. PAGE.

(23) 13. the cable and the girder.. Figure 3 shows a free-body diagram. of the girder under the action of applied loads.. The reaction. R can be found from A. R. M - M. A. B. A. (p + w - q) (L - a) da. +. ... (12). Then, the bending moment in the girder at x, denoted by M Is given by M. M. A. +. (M -M )x x B. A. (p+w-q)(L-a)da. L. — +. f (p+w-q)(x-a)da... (13) x. L. o For simplicity define a quantity M' equal to the bending moment produced by a l l loads except those applied by the cable.. This. is given by M' M (M -M )x = + — L A. B. A. +. x. r. L (p+w)(L-a)da. r. x. (p+w)(x-a)da. ... (14). o. Then M. M. 1. x^. L. q(L-a) da. T. x. q(x-a) da ... (15). -Jo. Jo. L. Now, i t is necessary to consider the static equilibrium of the cable under the applied loads.. Figure 4 shows the. forces acting on the deflected cable, indicated by a dashed line. The cable tensions at the supports are resolved into components in the direction of the chord joining the deflected points of support of the cable, and vertical components V^ and Vg. vertical component V^ is given by r L q ( L + h - (a+h)) da VA. The. B. L + h -h B. . (16). A. It will be noted that the forces in the direction of the closing chord have horizontal components equal to H, the horizontal component of cable tension.'. Then, for the equilibrium of the.

(24) 14. cable. x q(x. H(y-8_+5) = V (x-h ) ]. 2. A. A. where y - S]_ + S. - (a+h))da. .. ( 1 7 ). is the vertical distance from the deflected. 2. position of the closing chord to the deflected cable.. It is clear. from geometry that 8. B|h. 1. (h -h )x. A. B. V. L. >. A. . (18) L. V. Reference to Figure 2 will show that 82. v. h /dy. B'. Idx. L. .. ( 1 9 ). The term V (x-h )in equation (17). now becomes A. V (x-hA') A. A. x. fl. r. q(L-a)da. L. L. q(hg-h) da. V.h AA A. +. (20). o ^ B" A o A When i t is observed that the horizontal deflections are very h. h. h. small compared with the dimensions L, a and x, i t can be seen that the terms. x. f. ^(hB~h) da ^ v h an(. are very small compared. a. L+ho-hfl da. and can be approximated with negligible j o L+hB^A error in the total term V (x-h ) as follows:. with. x. I. ^ QA-k-a). Jo. fl. R. A. x ^ q(h -h) L. B. da. x. L. R. A. q(h -h) da B. ... ( 2 1 ). o ^ B ^ A. Vh A. A =. h H dy> A. ... ( 2 2 ). VdXy. A Then, the term . f q( ~a) of equation (20) can be expanded, Jo L+hB'hA neglecting a l l but the first two terms to give x ^ ^ q(L-a) da x q(L-a) da x(h -h ) f q(L-a) da x. L. L. d a. L. L. A. o. L+h. B" A h. L. +. fi. L. ... (23). Note that the second term on the right hand side of expression (23) is much smaller than the first term.. It is therefore per-.

(25) 15. mlssible to approximate i t as follows x(h -h ) r L q(L-a) da x(h -h ) H / dyi A. B. A. o. B. ... ( 2 4 ). \ dx ~ vi A. ~. If substitutions from expressions ( 2 1 ) , ( 2 2 ) , (23) and (24) are made in expression ( 2 0 ) , i t is found that V (x~h ) A. xf q(L-a)da L. A. x(h -h ) H / dy^ ffi. B. L +. x. r. Lq(h -h) da. \ dx, A. h H /dy^. B. A. L. \dxJ. (25). .. A. If substitutions from expressions ( l 8 ) , (19) and (25) are made in expression (17) and the result is combined with expression (15)j the following expression results: M. H. W. y. r x. v. h H /dy) A. .dxj A. +. x. H /dy\. h rh A. B. \dx/AL. +. r +. q(h -h) da. L. B. Jo. L. hqda. ... ( 2 6 ). Since terms involving the horizontal deflections are small, i t is permissible to make some approximations in these terms. Specifically, i t is permissible to approximate the suspender forces q by dy -Hdx'. ... ( 2 7 ). If the above is substituted in (26) the expression for bending moment in the girder becomes M. M-< H. hA. +. y. +. v. B. hA. L. (h -h )x B. +. x. 'dy\. L. \dx/ A. A. ^dx (h -h ) A. B. B \ r hd^y da. h /dy. x. L. d^y(h -h)da B. 2~. o dx. o. dx. 2. . (28).

(26) 16. It can be shown that, If a l l horizontal displacements are increased by a constant amount h* , the bending moment in the girder at a l l points will be unchanged.. Hence, i f h is set 0. equal to -h^, h^ may be replaced by zero in expression ( 2 8 ) , hg may be replaced by hg - h , and h may be replaced by h - h . A. A. Expression (28) may be differentiated twice to give dM. -p -w - H d y. dv. d h /dy. B\. dh d y. dx'. dx. dx. dx \dx. Lj. dx dx. 2. 2. 2. 2. 2. 2. 2. 2. 2. ... ( 2 9 ). If horizontal deflections are neglected in the girder equation, expressions (28) and (29) reduce to M - M-' - H(y + v) dM. -p -w -H /d y. 2. T. \~?. +. 2. ~?~\. \dx. 2. (28a). d v\. 2. =. dx. .... 2. •". (. 2. 9. a. ). dx / 2. If vertical deflections are neglected in the girder equation, expressions (28) and (29) reduce to the Elastic Theory expressions M = M"' - Hy 1. dM dx — ~ 2. -p -w -Hd y dx'-. .... (28b). .... (29b). 2. =. Expression (13) can be differentiated twice to give dM 2. q -p - w ... ( 3 0 ). dx'. Then equations (29b) and (30) can be combined to give expression ( 2 7 ) , an approximate relationship which was used earlier. From elementary strength of materials, the basic differential equation relating deflection to bending moment and shear in a girder is EId v. -M + EImd M. dx'. dx<. 2. 2. ... ( 3 1 ).

(27) 17. where:. E. Young's Modulus for the girder material. I. moment of inertia of the girder. m. coefficient of shear distortion m = ,. 1. for a girder. m = A E sin 0 cos 0 for a plane truss w = cross-sectional area of girder web G = shear modulus D. A. cross-sectional area of the diagonal member (s). AT,D =. angle measured from vertical to diagonal member (s) The deflections due to shear are small compared with the deflections due to bending.. Therefore, negligible error. 2. results i f %—^ is represented by the approximate expression dx^. (29a),. If expressions (28) and (29a) are substituted in expression ( 3 0 ) , 'and the resulting equation is differentiated twice, the following fourth-order differential equation is found: 2d(El) d 3 d (El) d v (1 + Hm) Eld^v p +w 2. 2. v. dx 4. dy. +. dx^. +. dx3. dx. dv 2. 2. H. +. H d^h. dx^. dx. 2. +. dx^. dx<. /dy. , B. dh d y. Idx. L. dx d x. 2. 2. d (El) m p + w + Hd y 2. dx'. .. ( 3 2 ). dx'. The above differential equation reduces in the Elastic Theory to EId v. 2d(El) d 3. 4. +. dx 4. md (El) 2. dx'. dx. dx-. p. +. w. d (El) d v 2. v. +. dx'. 2. dx. 2. p. +. w. Hd y 2. +. dx'. Hd y 2. +. dx'. S u b s t i t u t i o n from the cable equation. ... ( 3 3 ) c a n be made, i n e x p r e s s i o n.

(28) 18. (32) to give the horizontal deflections in terms of the vertical deflections.. Cable equation (9b) is sufficiently accurate here.. Remember that horizontal deflections, have a very small effect on bending moments.. It is a simple matter to prove that. 0. d h / dy dx \dx 2. +. H d^y ds L. AE dx dx 2. p. d v / dy. ... ( 3 4 ). p. dx \ dx Expression (34) can be substituted in equation (32) and terms may be collected to give the following fourth-order linear differential equation h. d^v 4 dx. E l ( l + Hm). 2d(El)(l + Hm) dx -2Hd y /dy. H. dx^ HGtd y. [dx. HH ds. 2. L. dv 2. +. dx' B^ L/. 2. p. dx. +. w. 2 Hd y + dx 2. 2. dx 2 p. w. Hd y dx^. 'dry. AE dx dx md (El). d (El)(l + Hm). 2. (35). Equation (35) is the girder equation.in a general form. It includes the effect of variable girder stiffness, horizontal ^ cable deflection and shear deflection. equation are H and v. unknowns H and v.. Unknowns present•in the. The cable equation (ll) also relates. Therefore, simultaneous solution of equations.

(29) 19. (ll) and (35) will yield the values of H and v. The ordinates of the cable in the undeflected position can be found by considering the equilibrium of the cable and girder under the action of dead load alone.. No bending moment. is present anywhere in the girder and the applied moment M' is given by M'. wx(L - x) ... ( 3 6 ). 2. From equation (28) Hrjy". wx(L - x) (37). If the cable ordinate at mid-span is denoted by f, then H. wL. 2. D. (39). 8f. and y. 4f. x(L - x) L. =. dy = 4f(L dx. L. dy 2. dx. 2. •. (40). 2. 2x). (41). 2. 8f. L. (42). 2. Solution of Equations (ll) and (35) Solutions to equations (ll) and (35) individually cannot be evaluated in terms of simple functions except in a few very special cases.. Therefore, s.ome method of numerical. analysis must be resorted to in most cases. whether the evaluation is by algebraic or numerical methods, the general approach is the same.. First, some estimate.

(30) 20. Then, equation (35) must be solved to give. must be made of H.. values of v corresponding to the estimated value of H.. Then,. equation (ll) can be solved to give a value of hg - h^ which must be equal to a value consistent with external conditions (for example, zero for fixed cable anchorages).. In general, the. first estimate of H will result in an error in the computed value of h-Q - h^;. and so a second estimate of H must be made,. and the procedure repeated to give a second error in the computed value of hg - h^.. Subsequent estimates of H can be made. by interpolation between previous estimates to give finally a tolerable error in h - h , and a sufficiently accurate value of ri. H.. A. The last solution of equation (35) thus obtained can be used. to evaluate bending .moments at a l l points, and shearing forces at a l l points. In multiple span bridges, the procedure for analysis Equation (35) must be. is similar to that for a single span.. solved to give values of v for a l l spans, corresponding to an estimated value of H.. Then equation (ll) must be evaluated for. a l l spans to give the total value of h - h where h and h. 13. A. D. A. here represent the horizontal displacements of the cable anchorages at the ends of the bridge. ' In doing this, L and L-jrepresent the sums of values of e. respectively, for a l l spans. If numerical analysis is resorted to in the solution of equations (ll) and (35)* no mathematical difficulties are encountered due to inclusion of the effects; of cable slope change, horizontal deflection of the cable, and shear deflection in the girder.. Except that the calculations are a l i t t l e longer, i t. is as convenient to include these refinements as to leave them.

(31) 21. out.. Equations (ll) and (35) are suitable for solution by. numerical analysis on a digital computer, and In the program outlined in the next chapter, the refinements mentioned above may be included or left out. at will. It is not to be thought that the reduction in labor due to simplification of equations (ll) and (35) is insignificant. In-fact, If the error introduced is tolerable, considerable saving in labor is possible by the use of equation (lib) in place of (ll) and by the omission of horizontal deflection and shear deflection from the girder equation.. The girder equation may. then be used in the form EId v 2. —. dx^. Hv --. Hy =. M' -. ... ( 4 3 ). Equation (43) is usually considered to be the basic differential equation for suspension bridges.. Solutions to equation (43) are. to be found tabulated in Steinman's text*and elsewhere for loading conditions commonly encountered in design.. Numerical. analysis is unnecessary except for programming a digital computer. Effect of Refinements in Theory on Accuracy It may be of some value to indicate here briefly the order of magnitude of error introduced by neglect of the effects of change in slope of the cable, horizontal deflection of the cable and shear deflection in the girder. It has been shown that the error resulting from neglect of the change of cable slope in the calculation of cable extension is largely dependent on the ratio Mi. Test calculations AE were performed with the aid of the computer program outlined in TT. the next chapter. Reference (l). In the test case, the value of _L. w a s. #. Q23l.

(32) 22. considerably larger than the usual range of values of ^L. The AE. test bridge was a three-span bridge, loaded over the entire main span.. The resultant error in H was 0.1%, and the maximum L. error in bending moment was 0.11%.. Therefore, i t is reasonable. to expect that neglect of the effect of cable slope change will never result in errors larger than 1% in cable tension and bending moments. Shear deflections have been investigated by C. D. Crosthwaite.*. It was found that.the largest errors in bending. moments and shears occur near the supports and that the effect of shear deformation diminishes toward mid-span.. Neglect of. shear deformation was shown to account for errors as large as 9.5% In shears and 6.5% in bending moments.. The writer has. compared the results of calculations for the case of a threespan bridge loaded over one half of the main span.. The magni-. tude of the percentage error depends on the ratio of shear flexibility to bending flexibility, in the girder, and in addition depends to some extent on the cable tension and cable area. Figure 5 indicates the relationship between percentage error and given dimension ratios.. The computed quantity under con-. sideration is the- bending moment at the quarter point on the loaded portion of the main span. Within the normal range of values of errors in computed values of the bending moment L can be as high as 15%. For a l l values of the dimensionless 2. ratios within the range of normal values shown in Figure 5, i t was found that the errors in the value of H determined by ij. neglecting shear deflection were less than 1%. Reference ( l 8 ).

(33) TO. Figure 5;. FOLLOW. PAGE. 22.

(34) 23. The. effect of longitudinal deflection has been investigated by H. H. Rode * and he has reported that the reduction in bending moment will usually be between 5 $ and 2$.. The writer. has examined a single case of a three-span bridge loaded over half the main span and the adjacent side span.. The bridge TJ. under consideration was very flexible, having a ratio 100.. 2. T. D of EI. The errors' in bending moments due to neglect of the. horizontal deflection were found to be as high as 8.5$ of the maximum bending moment in the girder.. The error in. was. 0.32$.. The writer has compared also the effect of using cable equation,(lib) for a particular example calculation. examined was a three-span bridge having ratio. The bridge. of 100 for • EI. the main span.. It carried a live load of .4 times the dead. load over•one half of the main span.. It was found that. approximation of the cable equation by equation (lib) Instead of equation (lla) resulted in. an error in. H-^. of 3.7$.'. This was. reflected in the girder moments by an error of 1.1$ of the maximum moment. It was observed that any approximation of the cable equation resulted in an error in H-^ which was reflected by an error of reduced significance in the girder bending moments. On the other hand, approximation of the girder equation by neglecting shear deflection or horizontal cable deflection results in small errors in moments.. but larger errors in bending. These larger errors in bending moment are due not to. the inaccuracy in H but to the omission of terms in the girder equation. Hence, i t would seem reasonable to determine H-^ by Reference ( l 6 ).

(35) 24. using approximate equations (43) and either (lla) or (lib). Then, the value of the H found by these approximate equations can be used with the more exact girder equation (35) or a simplified version of i t to yield bending moments of required accuracy. Usually/ the value of H found by the approximate equations will L. be in error by less than 1% due to the neglect of cable slope change, shear deformation and horizontal deflection.. Choice. of equation (lla) or (lib) will depend on the flexibility of the girder and the ratio of live load to dead load; accuracy desired.. and on the. It is suggested that equation (lib) will be. satisfactory for a l l but very flexible girders and heavy loads..

(36) 25. CHAPTER 3 COMPUTER PROGRAM The analysis of suspension bridges is a problem especially suitable for solution on a digital computer.. Each. of the equations (ll) and (35) may be readily programmed for ' numerical solution, and their simultaneous solution by a cut and try method reduces simply to an Iteration procedure.. It is a. simple matter to delete terms in either equation to test the effect of neglecting refinements In the theory.. Also the elastic. theory solution can be readily obtained from the deflection theory equations merely by deleting terms which apply only to the deflection theory.. Therefore, the program has been written. to give, for each loading case, both a deflection theory solution and an elastic theory solution. The computer used is an IBM 1620 desk-size digital computer, at the University of British Columbia Computing Centre. The core-storage memory holds up to 40,000 digits of data and instructions.. Available input devices are the console type-,. writer and a card-reader.. Output is by console typewriter,. card punch or on-line printer. It was found to be convenient to write the program ' using Fortran, the simplified scientific language available for IBM computers.. The particular version of Fortran used is desig-. nated by the U.B.C. Computing Centre as Fortran lA..

(37) 26 All input data required for the program, and a l l output are in dimensionless form.. It is convenient, to think of. the unit of length as being L, the length of the main span, and the unit of load or force is wL, the total uniformly distributed dead load on the main span. The program was written originally for the purpose of investigating the relationship between elastic theory bending moments and deflection theory bending moments.. It analyses a. three-span suspension bridge using first the elastic theory, and then the deflection theory, and compares the computed bending moments.. In both cases, a cut and try method is used to deter-. mine H, the horizontal component of cable tension.. In general,. each trial value of H produced an error In the computed value of the relative support movement.. Each trial value of H and its L. accompanying value of error Is printed out.. The actual values. printed out are Eli and A error^ dimensionless form. When wL L 2 2 2 the error is practically zero, the values of D , L and EI EI EI are printed and the bending moments at several points on the i n. H. girder are computed.. L. H. L. H L. When the bending moment has been computed. by both methods, the program prints the deflection theory bending moment, the elastic theory bending moment, the ratio between the two, and the deflection.at points along the girder. The program analyses a three-span suspension bridge with constant or variable girder stiffness. either continuous at the towers or hinged.. The girder may be In the numerical. procedures used, the main span is divided into twenty equal partial lengths equal in length to the main span sections.. The bending. moments, deflections and so forth are computed for the points separating these partial lengths of girder.. The points are.

(38) 27 numbered points. c o n s e c u t i v e l y from l e f t. at. ively.. the. left. In i t s. and r i g h t. to r i g h t. the. program r e q u i r e s. s p a n s be d i v i d e d i n t o a n e v e n number. span. length. w o u l d be. limit. block diagram f o r that. the. differential for as. the exact. respect-. that. and. for. it. the. to main. acceptable. ratios. theory. solution.. of c a b l e .. s o l u t i o n of. the. and t h e. are. integral. i n the. equation. i n c l u d i n g such. found. elastic. i n the. theory. by the. same. i n the. deflec-. an. theory. deflection. technique.. Girder Equation. Two a p p r o a c h e s t o n u m e r i c a l s o l u t i o n o f t h e e q u a t i o n were. and. equations. or u n i t y to g i v e a d e f l e c t i o n refinements. in. refine-. of c a b l e. D, w h i c h i s made z e r o t o g i v e. i n c l u d e d or d e l e t e d. be. i n c l u d e d i n the program. Terms w h i c h a r e. but not. solution,. S o l u t i o n of the. the. general can. horizontal deflection. The a b o v e - m e n t i o n e d. theory are. i n the It. i s deemed p r a c t i c a l ,. shear d e f l e c t i o n ,. equations. girder. equations. m u l t i p l i e d by a f a c t o r. initial. side. storage. p r o g r a m shown i n F i g u r e 6.. These. a f o r m as. t i o n theory. "initial. the. span l e n g t h. illustrated. of the program i s. equation. change i n s l o p e. elastic. of s i d e. Therefore,. .7.. of s o l u t i o n i s. the. heart. cable.. ments as. are. ratio. of p a r t s ,. with. .7, .6,. .5,. . 3 , .2 o r . 1 . The method. seen. the. t o a. maximum o f. 17 a n d 37}. t o w e r numbered. present form,. limitations further. on t h e g i r d e r. considered.. The e q u a t i o n. c o u l d be t r e a t e d as. v a l u e " p r o b l e m , a n d some a s s u m p t i o n as conditions. c o u l d be made i n o r d e r Then, f o r. dition,. a different. o f the. assumed. t o g i v e an a d d i t i o n a l s o l u t i o n to the. a s o l u t i o n by. e a c h unknown i n i t i a l. initial. an. t o unknown. to f i n d. the R u n g e - K u t t a method. value. girder. con-. c o n d i t i o n c o u l d be equation.. The.

(39) TO. READ INPUT DATA. s. PAGE. 27. INITIAL. PRINT V. FOLLOW. v. INPUT. D = 0. ESTIMATE H = O.lwL L. Note •• D = 0 for Elastic. Theory.. D= I for Deflection Theory.. /. "K = 0. M SWITCH O F F ) ,. S O L V E GIRDER EQUATION ( S E E FIG. 7 ). I. SOLVE C A B L E EQUATION FOR ERROR IN h. B. ". h. A NEW ESTIMATE. PRINT H. ERROR. H. L. = H +0.lwL L. AV. K=0. NO. K =l (Switch on).. FIND N E W TRIAL V A L U E H|_ BY INTERPOLATION. COMPUTE BENDING MOMENTS. D= 0 D=l COMPUTE MAGNIFICATION FACTORS. Figure. 6.. PRINT BM'S MAG. FACT.. END. DEFLECTION. JOB. OF.

(40) 28. solutions thus found could then be superimposed so as to satisfy all boundary equations.. Another approach is to deal with the. equation directly as a "boundary condition" problem, by '.-substituting for the differential equation a system of difference equations, which may be solved to give the deflections at points along the girder.. The latter method was thought to be most. suitable for this problem and was adopted. At a point i on the girder, the derivatives in the girder equation may be approximated by finite differences as given below: (vi_ -. 1. 2. d. / 1Y _ ±. d. J. 3. 1. dx. 2. _ 'where v. ,. ±. V. ±. +. \2. /_ d. i - 2. - 4v. ±. 1. +. 1. lV. 1. +v. 1 + 2. ,. 1 + 1. -. +. +. 2. 2,. (44). - 2v. ±. +. v. i+l). 2. 1. i. 1. V_. 2. fv^. d. /I. ±. 4. 1. dx / i. 4 v _ + 6v. l v. i-l. l v. V 2. i+l 2. Vj__]_, v-j_, v i + 1 ,. v^. +2. are deflections at points. separated by a small distance d. If substitution from equations (44) is made in equation (35)* given in the preceding chapter, i t is possible at each point on the girder to write a difference equation of the form: a. il i-2 + i2 i-l + i3 i v. a. v. a. v. +. a. i4 i+l + i5 i+2 = v. a. v. b. i. ••• ( 5 ) 4.

(41) 29 where ai l. a. EI. 1 d(El). d4. d3 dx. 4EI. 12. (1 + DHm). 2. d(El). d^. d. dx. 3. 1 d (EI). ^. +. d. DH. B\. +. a. ,2 6EI. 13 =. d. 4. d (EI). d. dx. 2. DH. +. d. d EI. c. dx B. /dy. dx. 2. d. 2. /dy. B. Vdx. L. (46). 2. d y /dy 2. h. d. B. d x Vdx 2. dx. 3. H. 2. G t d y (1 +. dx'. ds d y 2. In equations d e f l e c t i o n theory. d y / 1 2. +. 3 /dy. 2. L. d'. h. ( l + DHm). AE dx dx. for. P. 2. md (El). H. 2P. ( l + DHm). 2. dx 2. +. \dx. 2. DH. 1 d (El) d. 1 d(El). (p + w). HPh. dx. 2. 2. d. i. d. d Idx. 2. B'. 2. +. _\. 1. d y /dy. h. +. ^ J. P. 2. (1 + DHm). 2 d(El). H. 2. Ll. Vdx. 2. 2. F d. b. d. 4EI. a .14. a.. dx. 2. ( l + DHm). x. 2. dx. dx' B. md (El)\ 2. J. 2. ,dx. I1 \. +. 4 /dy. B. \dx. L. (46) a b o v e , or e l a s t i c. N. 2>. D may be e i t h e r. theory,. I n a d d i t i o n P ^ may be one o r z e r o t o. one o r. zero,. as p r e v i o u s l y d e s c r i b e d .. Include or d e l e t e. the. effect.

(42) . 30 of h o r i z o n t a l d e f l e c t i o n. of the. cable.. The e f f e c t. of. shear. d e f o r m a t i o n may be d e l e t e d - s i m p l y b y a s s u m i n g a z e r o v a l u e m, t h e. shear f l e x i b i l i t y. flexural. rigidity. differences,. as. of the. of the. girder,. In equations. constant E I , derivatives In order equation,. it. is. (44).. I n the. For. case of a g i r d e r. to express (45).. c a s e o f z e r o moment a t. following M. equation,. differential. boundary. in. The c a s e o f z e r o d e f l e c t i o n. at. i n the. form. If. the. *. 1 v.. 2 v.. "2. .,. -"2. t\. +. (47). If. the. equal p a r t i a l girder,. involving. it. the. girder. lengths,. Note t h a t the exterior. supports. the. difference. 2. 1. -2. each of the. "...(49). dx / i 2. .. the. p o i n t under. on e a c h s i d e for. N - 1 i n t e r i o r points. t o w r i t e an e q u a t i o n. the. i n v o l v e the. f i c t i t i o u s points. Is p o s s i b l e. at. d e f l e c t i o n at. equation. (48). 2. i s d i v i d e d i n t o a f i n i t e number N o f. is. possible. two a d j a c e n t p o i n t s. (31).. V. i , then. 1 + DHm \. d. the. ( p + w + Hd y \. :. L\. d. a point. -m =. ~2. (29a) and. dx. zero at. 1 v . ,-,. &. d. the. \. b e n d i n g moment i s becomes. i s made o f. 2. /. equation. use. Elm /p + w + Hd y. 2. 2. a point,. d e r i v e d from equations. d v ( E l ( l + DHm)\ dx. of. conditions. obviously expressed. the. the. .... the. the. finite. = o. Vi. at. may be r e p l a c e d by. to o b t a i n a s o l u t i o n to. i s necessary. supports. D e r i v a t i v e s of E I ,. of E I v a n i s h .. s i m i l a r form to equation the. girder.. for. o f the. p o i n t under. deflections the. (45). at. the. supports.. The f o u r a d d i t i o n a l. at. consideration.. nearest. the. supports Therefore,. to write N - 1 t y p i c a l i n t e r i o r equations. N + 3 unknown d e f l e c t i o n s .. form. c o n s i d e r a t i o n and. i n t e r i o r points. exterior, to. of the. on. and it. involving. equations.

(43) 31 required that. a r e p r o v i d e d a c c o r d i n g t o the boundary. I s , z e r o d e f l e c t i o n at. e a c h e x t e r i o r. bending. moment a t e a c h e x t e r i o r. tinuous girder, support. point that be. support.. I n the case. the t y p i c a l i n t e r i o r. point.. An.array f o r a girder. Coefficients. w i t h no i n t e r i o r. there i s a. equation at. d i f f e r e n t from. supports. that. of. the form. equation.. t h i r d and f o u r t h a multiple. The f i r s t. Then, t h e f i r s t equations. determined. determined. t o z e r o by. This process the l a s t. of the. of. subtracting elimination. equation i s. a s i n g l e unknown. deflection. Then, a l l o t h e r d e f l e c t i o n s. by s u b s t i t u t i o n. into preceding. a multiple. non-zero c o e f f i c i e n t s i n the. of the second e q u a t i o n .. which i s r e a d i l y determined.. i s evident. c o e f f i c i e n t of the t h i r d. c a n be r e d u c e d. t o an- e q u a t i o n i n v o l v i n g. quickly. the equations. t o z e r o by s u b t r a c t i n g. o f c o e f f i c i e n t s c a n be c o n t i n u e d u n t i l reduced. b y a n x.. Condensed. of the array.. e q u a t i o n c a n be r e d u c e d. equations.. can. of v a l u e s as they a r e The p r o c e d u r e. described. a b o v e i s known a s t r i a n g u l a r i z a t i o n and b a c k s u b s t i t u t i o n . relatively. simple.and. to. i s indicated. zero are indicated. - A s y s t e m a t i c method o f s o l v i n g. is. con-. of c o e f f i c i e n t s f o r a s e t of equations. Coefficients. be. of a. i s s i m p l y r e p l a c e d by a n e q u a t i o n f o r z e r o d e f l e c t i o n a t. below.. first. s u p p o r t , and z e r o. a t one o r more o f t h e i n t e r i o r p o i n t s. I n that, case,. written. from. conditionsj. f a s t here. since. t h e r e i s a band. It. width.

(44) 32 of. only f i v e. that. the. non-zero c o e f f i c i e n t s .. values. of the. coefficients. a r r a y remain zero throughout ming,. use. allotted storage. i s made o f t h i s I n the. i n the. shown t o t h e. necessary. computer,. the. of the. solution.. it. to. note. of. the a r r a y full. i s condensed. the. In program-. k n o w l e d g e , a n d no memory s p a c e. to the. is For. form. array.. value of H .. times,. The e q u a t i o n s. i s worthwhile to f a c t o r. c o e f f i c i e n t s a^j. compute and s t o r e. values. v a l u e . of H , the. and b ^ .. out terms Then i t. of constants. once. is. for. (46) f o r a_ ... t i m e - c o n s u m i n g c a l c u l a t i o n s , e v e n on a. each of the. as. entire. shaded p a r t. to s o l v e the g i r d e r e q u a t i o n . s e v e r a l. Therefore,. new t r i a l. i n the. important. s o l u t i o n of a s u s p e n s i o n b r i d g e problem, i t. e a c h e a c h new t r i a l represent. the. is. computer f o r these z e r o c o e f f i c i e n t s .. right. In. It. computer. involving H in. is possible. ab. ., such that f o r. c o e f f i c i e n t s a^^ a n d bj_ a r e. to each. computed. follows:. a . ,1 = a b .i . i (1 + DHm) I v. a . „ = ab . „ + DHab . .. i2 l3 14 a. i3. =. a b. i5. +. D H a b. i6. a . . = ab._ + D H a b . i4 17 10 a^ - ab^g ( l + DHm). .... Q. b.. = a b . . , ^ + Hab.-,-, + D H a b . + DHH ab . ^ llO i l l il2 L 113. The. c o m p o s i t i o n of the. 1. of. n o. equations. Integration. (4.6). of the. terms a b . .. T. is. n. obvious from an e x a m i n a t i o n. a n d n e e d n o t be w r i t t e n. here.. Cable Equation. Integration. of the. applying Simpson's Rule,. (50). c a b l e e q u a t i o n i s performed by. w h i c h may be s t a t e d. as. follows:.

(45) 33 B. r. f(x). dx. .. A. E. f. _ + 4 f . + f. , _ i-l i i+i. 1=2,4/6. where, of. N. I n the. case. o f the. i n t e g r a t i o n are. the. ... ( 5 1 ). dh i s — and the l i m i t s dx span under c o n s i d e r a t i o n .. cable equation,. two ends o f t h e. f. dh. I n the program, dh. H. dx. AE V d x /. L. D. +. /ds\. equation for — dx. G t (<ls\. 3. F H /ds \ s li / \. 2. is s. slope. is. dy d v. 2. dx. dx dx. ( d y dv I. AE \ dx. where F of. the. _|_. dx dx. 1 /dv. x. 2 \ dx. cable.. (9) when t h e f a c t o r s. F. the. effect • •.. E q u a t i o n (52) i s e q u i v a l e n t a. and D are. noted that Simpson's Rule requires. made u n i t y .. It. d has b e e n made e q u a l t o o n e - t w e n t i e t h. be. that. the. ratio. of. to. change. equation. s h o u l d be. t h a t e a c h s p a n be, d i v i d e d. i n t o a n e v e n number o f e q u a l p a r t i a l l e n g t h s .. required. ... ( 5 2 ). 2 Vdx,. one o r z e r o t o i n c l u d e . o r d e l e t e •. i n the. 2 ,.. 1 /dv'. 2. of the. S i n c e the main span,. length it. of s i d e span l e n g t h to main span. is. length. . 1 , .2, . 3 , e t c .. Program Linkage• Four main p a r t s t h a n once a r e. of the program which are. w r i t t e n i n the. form of subroutines. l i n k e d up l a r g e l y b y t h e u s e One o f t h e s e s u b r o u t i n e s from the. left. computes. Another subroutine span from I I eqiiation,. to I E .. either. support. more. and t h e y. are. o f " c o m p u t e d GO TO" s t a t e m e n t s . the. array ab.. _L. span,. used. at. integrates. II the. for a single. J. to the r i g h t s u p p o r t. at. IE.. cable equation f o r a s i n g l e. A t h i r d subroutine. solves. f o r a s i n g l e span from I I. the. girder. to I E , or f o r a.

(46) 34 continuous 1=37. and. g i r d e r from I I to IE w i t h i n t e r m e d i a t e supports. 1=17.. Each of the above s u b r o u t i n e s r e q u i r e s the .. l o c a t i o n o f I I , t h e i n i t i a l p o i n t I o f t h e c a l c u l a t i o n and t h e end. at. p o i n t I of the c a l c u l a t i o n .. Two. of the. IE,. subroutines. a l s o r e q u i r e the span l e n g t h L of the span under c o n s i d e r a t i o n and. the d i f f e r e n c e. i n e l e v a t i o n o f t h e a n c h o r a g e s B.. there i s a s u b r o u t i n e which for. specifies. s u c c e s s i v e s p a n s as w e l l as. s p a n f o r use. routine.. the. I t has t h e o r y may F^ and. the. sub-. the l i n k a g e i s e f f e c t e d to s o l v e girder. or f o r hinges. rigidity. Program. been s t a t e d t h a t c e r t a i n r e f i n e m e n t s. be d e l e t e d o r i n c l u d e d a t w i l l F. g. equal to zero or u n i t y .. be e i t h e r h i n g e d. a t the towers. or v a r i a b l e .. two. setting. In. values. F u r t h e r , the g i r d e r. or c o n t i n u o u s , of. data card.. the f o l l o w i n g. the. the. table:. The. may. constant combination. code number i s r e a d i n f r o m. columns of the f i r s t. code number i s shown by. by. I n o r d e r t o s p e c i f y any. the above c h o i c e s , a s i n g l e first. statement. towers.. Input Data f o r the. of. TO". the r o u t e of the program upon e x i t f r o m. the g i r d e r e q u a t i o n f o r e i t h e r a c o n t i n u o u s at. Before. t o s p e c i f y the v a l u e of •. i s t o be u s e d i n a "computed GO. F i g u r e 7 shows how. IE. L f o r each. i n the o t h e r s u b r o u t i n e s d e s c r i b e d above.. a constant which determine. t h e l o c a t i o n o f I I and. t h e v a l u e s o f B and. entering a subroutine, i t i s necessary. to. Therefore,. of. the. meaning of. the.

(47) TO. PORTION OF MAIN LINE. CONTINUOUS. SUBROUTINE. PROGRAM. HINGED. N4 = 2 ^CONTINUOUS GIRDER). N4 = l HINGES. SPECIFY II A N D I E FOR CONTINUOUS GIRDER. N2 = 3 (GIRDER CALC.). SOLVE GIRDER EQUATION II TO I E. CONTINUATION OF MAIN PROGRAM. SUBROUTINE. 2. SPECIFY. SPECIFY. SPECIFY. II,IE, L,B. 11,IE , L , B. II,IE, L , B. LEFT. SIDE. SPAN. FOLLOW. MAIN. RIGHT. SPAN. Figure. SIDE. SPAN. 7.. PAGE. 4. Determine v at all points. 34.

(48) 35 CODE Constant E I. •. £h. F, Continuous. 11. 01. 1. 1. 12. 02. 0. 1. 13. 03. 1. 0. 14. 04. 0. 0. 15. 05. 1. 1. 16. 06. 0. 1. 17. 07. 1. . 0. 18. 08. 0. 0. The n e x t related 2* AEL E I. y. Variable EI. to the a n <. span. data. three data. cards. each c o n t a i n the. elastic properties. v a l u e of a. constant. H L They a r e D. of the b r i d g e .. c a r d and are. i n E14.7. format.. first. 14 c o l u m s o f a. There f o l l o w. s p e c i f y i n g the geometry of the b r i d g e .. three. the main span,. the. main span l e n g t h ,. 'li. l e n g t h of the and t h e. the main span l e n g t h .. slip is. i n E14.7. as a r a t i o. variable,. rise. cards.. format,. g i v e n are. left. anchorage. set. values. cards. is. EI at. of € t If. of data. two d a t a. and the. anchorage. the g i r d e r. rigidity. cards g i v i n g. e a c h p o i n t on t h e. one t o a c a r d , - ' a n d a r e. The the. If. omitted.. the g i r d e r has Finally,. constant. the. cards. the. girder values r a t i o of. t h a t p o i n t to the g i r d e r r i g i d i t y a t. of the main span.. of data. F o l l o w i n g are. t o the r i g h t a n c h o r a g e .. i n F7.4 f o r m a t ,. the. i n F6.4 f o r m a t and o c c u p y. there i s r e q u i r e d a set. the g i r d e r r i g i d i t y a t centre. the. of. s i d e s p a n as a r a t i o - o f. of the main span l e n g t h .. v a l u e of the g i r d e r s t i f f n e s s from the. s i d e s p a n s as a r a t i o. of the. They a r e. s i x columns of the. specifying,. data. They a r e £. f o r. ). first. 2. 3 ^Y?T> where I i s t h e g i r d e r moment o f i n e r t i a a t m i d -. The a b o v e v a l u e s e a c h o c c u p y t h e. cards. Hinged. EI,. a-:-set o f c a r d s. is. the this read.

(49) 36 giving. the. v a l u e of the. live. l o a d on t h e. One c a r d i s r e a d f o r e a c h p o i n t anchorage. to the. F6.4. format.. Final. Notes. g i r d e r as a r a t i o. on t h e g i r d e r f r o m . t h e. r i g h t anchorage.. The r a t i o s. are. P. w. left. given i n. I n A p p e n d i x 1 t h e r e i s a c o m p l e t e b l o c k d i a g r a m and a listing tions. of the. for. this. p r o g r a m as thesis.. it. was w r i t t e n a n d u s e d. No c l a i m i s made t h a t. the b e s t. one t h a t c o u l d h a v e b e e n w r i t t e n f o r. taken.. However, i t. did give satisfactory. a c c u r a c y was g o o d and t h e. amount. it. was n o t. ticated. value. the program the. studies. results.. It. In i t s. may be t h a t. it. some v a l u e as a g u i d e. of. s i m i l a r programs,. and f o r. to others. that reason. i n the. will. therefore. to the. i t has. the. the program. form u s u a l l y r e q u i r e d of a l i b r a r y program.. may have. under-. The. p r e s e n t f o r m , and. considered worthwhile to r e v i s e. it. is. of i n f o r m a t i o n y i e l d e d by. p r o g r a m was e n t i r e l y , a d e q u a t e . be o f l i t t l e f u r t h e r. in investiga-. more. sophis-. However, preparation. been. preserved. here. The m a j o r p o r t i o n o f t h e s o l v e the. set. of equations. Running time f o r each t r i a l minutes.. Since three. Deflection. the. girder. equation.. s o l u t i o n was a p p r o x i m a t e l y two were r e q u i r e d f o r a n E l a s t i c. or f i v e. Theory s o l u t i o n ,. approximately fourteen. representing. trials. T h e o r y s o l u t i o n and f o u r. c o m p u t i n g t i m e was t a k e n. the. trials total. to s i x t e e n. were r e q u i r e d f o r. computing time. minutes.. was. a. to.

(50) 37. CHAPTER 4 DETERMINATION OF H. General It involves Since the. the. the. h a s b e e n shown t h a t a n a l y s i s o f s u s p e n s i o n simultaneous. method o f s o l u t i o n must be a t r i a l. c a l c u l a t i o n s are. determined,, i t deflections, further. of. lengthy.. bending•moments. H.. t o use. ( l i b ) and. are. repeated. procedure,. value of H i s. t o compute a l l. i n the g i r d e r .. f o r most p u r p o s e s ,. equations. The e q u a t i o n s. and s h e a r s. i n H and v .. and e r r o r. H o w e v e r , once t h e. i s a s t r a i g h t f o r w a r d matter. b e e n shown t h a t ,. accurate. s o l u t i o n o f two e q u a t i o n s. bridges. it. (4-3) i n t h e. is. It. has. sufficiently. determination. here f o r convenience of. reference. h. B. - h. e tL. A. EId v. Hv. 2. t. +. Hy. H. l. L. L. r. dv / d y. B \ dx. dx V dx. L. .... e. j o. AE M'. -. —p. dx^. ... (43) It. w i l l be n o t e d. c a b l e due t o t e m p e r a t u r e. i m m e d i a t e l y t h a t e x t e n s i o n of. r i s e and s t r e s s. e l o n g a t i o n has. effect. exactly equivalent. to a s m a l l r e l a t i v e support. If. term A. as. the h-o. B. (lib). hA A. is defined. 6 t L t.. HL T. e. the an. movement.. / s. AE then. A. may be t h o u g h t. o f as. the. equivalent. support. displacements.

(51) 38 of. an i n e x t e n s i b l e c a b l e . r. dv / d y. L. Jo. to. B \ dx. dx \ d x. ... (54). L J. A method b a s e d here,. Equation ( l i b ) reduces. whereby a d e s i g n e r. on e q u a t i o n s can determine. (43) a n d. (54) i s. v e r y q u i c k l y the. of H f o r a s i n g l e span or f o r a m u l t i p l e span b r i d g e hinged at. the. supports. Superposition. or. (43) i s a l i n e a r d i f f e r e n t i a l. v a r i o u s r i g h t hand s i d e. permissible. to replace. then equation. Eld^v. H y. dx. 2. VQ. =. x. +. The s o l u t i o n t o V. H y. Q. a number. expressions... H on t h e. b y H Q + H-pHv. either. continuous.. i s p e r m i s s i b l e to superimpose. for. value. of P a r t i a l Loading C o n d i t i o n s. Since equation it. presented. of s o l u t i o n s. to (43). In p a r t i c u l a r ,. r i g h t hand s i d e. (43) may be. equation,. of the. it. is. equation. written. M' ... (.55). (55) may be g i v e n b y. V]_. +. where EId v 2. Q. Hv. H y. 0. Q. M'. ... ( 5 7 ). dx" Hv-. EId v2. dx. n. ... ( 5 8 ). £. Then A. may be r e p l a c e d b y . A. 0 o A. H y. dv,0. 'dy. B \ dx. dx. \ dx. L. / dy. B \ dx. r L dv o. dx. 1. \ dx. + A ^ i n equation. (54) t o. give. ... ( 5 9 ). ... (60).

(52) 39 The p h y s i c a l s i g n i f i c a n c e o f e q u a t i o n s is. difficult. to describe. since. the. a s u p e r p o s i t i o n of mathematical physical. states.. superposition. Then t h e. l o a d moments by H Q .. Q. deflections. v. The d e f l e c t i o n s. exists. It. and t h e the will. e'qual t o z e r o . from the. A. v ^ and. A c a n. two p a r t i a l. loading. be shown t h a t the. it. of cable. H-^ a r e. still. stretch. functions. value. of the. T h e n , H-^ i s. When. compatisum o f. tension. total. A. result-. inextensible. the p o r t i o n of. be r e m e m b e r e d ,. Is. displacement.. Both H. value of H , but. it will. sensitive. cable. to e r r o r. the and. Q. be. i n an. of H .. i f a p p l i e d i n the. tension. the. the. Span. make u s e. of. it will. of H^ i s not. Since superposition valid. to. i s a d v a n t a g e o u s t o make. and s u p p o r t. shown t h a t d e t e r m i n a t i o n. applied. cases.. a p p l i e d l o a d a c t i n g on a b r i d g e w i t h. effect. force. represented. g i v e n by the. p o r t i o n of c a b l e. t e n s i o n r e s u l t i n g from A , which,. Single. tension. be a t t r i b u t e d. s o l u t i o n f o r v Is. Then H Q i s. of the. and A ]_ t o t a l A , t h e n. 0. two. to t h i n k of. by a c o n s t a n t. a result. cable. sum o f. t e n s i o n r e p r e s e n t e d by H ^ .. c a b l e and immovable s u p p o r t s .. estimated. are. Q. M' and a p o r t i o n of the. V Q and V ] _ f o r. ing. and. Q. a n d H ^ t o t a l H and when A. bility. the. H o w e v e r , i t m i g h t be c o n v e n i e n t. a c t i o n o f the p a r t i a l c a b l e H. is r e a l l y only. s o l u t i o n s and n o t. a b r i d g e w i t h movable anchorages r e s t r a i n e d H.. (57) t o (60). manner. of r e s u l t s outlined,. has. b e e n shown t o. be. it. is permissible. to. of the R e c i p r o c a l Theorem i n d e t e r m i n a t i o n F i g u r e 8 shows, t h e. theorem.. attributable. two c a s e s r e q u i r e d f o r. Case 1 i l l u s t r a t e s. to the p a r t i a l c a b l e. the. deflections. tension H. 1 #. This. of. cable. application v-^ and A ]_ corresponds.

(53) TO. CASE. I. Figure. 8.. FOLLOW. PAGE.

(54) 4o t o the live. s o l u t i o n of equations. load. 2 the. tension. ( 5 8 ) and. due t o a u n i t. (60).. load. on t h e. This. (57) and. case of a s i n g l e u n i t. (59) f o r t h e. where H Q i s. the. sum o f H. According. 1. In. L. corresponds. and the. to the. case from. to a s o l u t i o n of. equations. l o a d on t h e. dead l o a d. the. span,. tension H ^ .. r e c i p r o c a l theorem,. the. equation of. i n f l u e n c e l i n e f o r H£ i s g i v e n b y. V v. span. c a b l e i s assumed t o be i n e x t e n s i b l e and s u s p e n d e d. immovable s u p p o r t s .. the. Case 2 shows. v-. A H fL. 2. x. (61) Equation. (58) c a n be r e a d i l y s o l v e d 2. 4. E I (CL). 2. x -. L. L. 2 +. ,. (CL)'. 2((l-e-. C L. )e. (CL)^(e. to give +(e. C x. U i j. -. e. C L. -l)e-. - U i j. C x. ). ) (62). .. ( 6 3 ) Equation v. 1. (62) c a n be w r i t t e n i n a s i m p l e r f o r m. H fL. as. 2. x. . (64). EI where v.. 4. x \. (CL)'. 2. L, When t h e. stitution. +. +. (CL)'. H rE I L. A. 1. (CL) (e 2. C L. -e-. e. C L. C x. ) (65). ). e x p r e s s i o n f o r v ^ i s d i f f e r e n t i a t e d a n d sub-. i s made i n e q u a t i o n 2. v. 2((i-e-CL)eCx ( CL-i)e-. x. (60), i t. Is found. that. ... ( 6 6 ).

(55) 41 where. 64. •1. (CL)'. 12. T. ±. Then t h e H. L. 3. of the. and. is,. t i v e p o s i t i o n on t h e. are. -e". -e-. C L. C L. (CL+2) ... ( 6 7 ). ). i s g i v e n by. A ^ are. d i m e n s i o n l e s s and a r e. func-. d i m e n s i o n l e s s q u a n t i t y C L , where C i s d e f i n e d by. (63).. influence. C L. line for H £. Note t h a t v. equation. (CL-2). \. ±. tions. C L. (CL) (e. influence. L / v. -. 4 + e. line for. tabulated. of course,. span.. a l s o a f u n c t i o n of the. V a l u e s of -. V. l , representing. on a s i n g l e s p a n i n d i m e n s i o n l e s s. i n T a b l e 1 o f A p p e n d i x 2.. shows i n f l u e n c e l i n e s f o r v a l u e s. of. rela-. Also,. the. form. Figure 10 '•. ( C L ) = 1 and 2. :. ( C L ) = 100. 2. To f a c i l i t a t e d e t e r m i n a t i o n o f H f o r d i s t r i b u t e d l o a d s ,. the. area under. been. plotted. i n the. A-^ t o t h e A. f. n 1. = J o. one o f t h e. x. left. v ~. n. i n f l u e n c e curves. same f i g u r e . of p o i n t x ,. The c u r v e shows t h e. area. where ... ( 6 9 ). A 1. By t h e u s e. a l s o i n c l u d e d l i n T a b l e 1. of the. i s p o s s i b l e to determine. curves. or t a b l e s d e s c r i b e d above. H £ and hence H Q ,. i n an i n e x t e n s i b l e c a b l e w i t h. and s u p p o r t. displacement.. effect. 2. Equation. tension that. of c a b l e. it. would Then. stretch. F i g u r e 13 shows a p l o t o f A ^. ( C L ) w h i c h c a n be u s e d t o f i n d. Tabulated values are. the. immovable s u p p o r t s .. a c o r r e c t i o n 8H must be a d d e d f o r t h e. against. partial. dx. Tabulated values are. exist. i n F i g u r e 10 h a s. the. c o r r e c t i o n 8H.. a l s o i n c l u d e d i n T a b l e 1.. (53) shows t h a t A. i s a f u n c t i o n of H. the.

(56) 42 unknown l i v e this. value. load tension.. initially. •. It. is possible. to a v o i d. by r e w r i t i n g (53) i n the. estimating. form. L. e. 6H. A = A. (70). AE. where hB. hA. € tin. H. L'^e. .., ( 7 1 ). AE The c o r r e c t i o n S r l ' I s ,. SH. i n the. case of a s i n g l e. span. AH. (72). where. i t h a s b e e n shown i n e q u a t i o n. (66) t h a t. 1. H-. A. X. ,2 r-L. (73).. —. T A. EI Equations. ( 7 3 ) and. ( 7 l ) c a n be s u b s t i t u t e d. i n equation. (72) t o. give. • 8H L. A -. 8H. C. _. AE_. f. ... ( 7 4 ). A~l. 2 L. EI Equation. (74) c a n t h e n be s o l v e d f o r :8'H t o. 8H. give. A!f L 2. A. 1 1. EI In order. +. . (75). L. -1. AE. to determine. f o r H-^ a n d e s t i m a t e. oE,. the. It. value. i s necessary. t o compute. another. once. o f H t o f i n d TJT^ f r o m F i g u r e 1 3 .. Then 8H and h e n c e H c a n be computed f r o m e q u a t i o n computed v a l u e. A '. o f H does n o t a g r e e w i t h. the. (75).. estimated. c o m p u t a t i o n must be made w i t h a d i f f e r e n t. If. value,. value of. the.

(57) 43 N u m e r i c a l examples •converges. i n A p p e n d i x 3 show t h a t. i. of H and so a n i t e r a t i v e p r o c e d u r e. the. of H£.. l i n e s are. CL.. This. CL.. A t one e x t r e m e. is. +. At. the. is. so f l e x i b l e. extreme as. c a n be shown t h a t. V_. L 3. x. f 4. L. influence line. estimate. the. determina-. as. zero,. x. extreme the. value of. values. elastic. of. theory. i s g i v e n by ... ( 7 6 ). L CL becomes. infinitely. t o o f f e r no r e s i s t a n c e the. of the. because. 4'. 3. 2 / x. L. other. r e l a t i v e l y independent. as CL a p p r o a c h e s. v a l i d and t h e. 8. implied for. i l l u s t r a t e d by a s t u d y of the. x f. Is. on a n i n i t i a l. However, i t e r a t i o n i s u s u a l l y unnecessary. influence. becomes. iteration. rapidly. D e t e r m i n a t i o n o f H£ d e p e n d e d. tion. the. influence line. large,--the. girder. t o d e f l e c t i o n , and. it. equation'is. 2' (77). F i g u r e 9 shows i n f l u e n c e l i n e s f o r H ' f o r. the. extreme. Li. values line. of C L .. F u r t h e r i n v e s t i g a t i o n shows t h a t. ordinates. by the. extreme. intersect.. It. f o r a l l values values except i s apparent. be i n t r o d u c e d b y i n a c c u r a t e. of CL l i e w i t h i n i n the. the. the range. r e g i o n where t h e. t h a t no s i g n i f i c a n t e r r o r. defined. curves i n Hi^ w i l l. value of H . For T-2 ca l l v a l u e s o f CL t h e a r e a u n d e r t h e c u r v e must be — . This 87' p o i n t becomes c l e a r e r when i t i s r e a l i z e d t h a t a u n i f o r m l o a d p covering results pL 8f. the. entire. estimates. influence. of the. s p a n i n t r o d u c e d no g i r d e r b e n d i n g moment and. i n a c a b l e t e n s i o n w i t h a h o r i z o n t a l component e q u a l. to.

(58) TO. FOLLOW. .2 CL = 0 —. «. C L = cx -•. /. .2. .4. .6. .8. x. 17 X L. CL= 0. CL= o o. .05 .10 .1 5 .20 .25 .30 .35 .40 . .45 .50. .0311 .0613 .0899 .1160 .1392 .1588 .1745 .I860 .1926 .1953. .0356 .0675 .0956 .1200 . 1406 .1575 .1706 .1800 .1856 .1875. Figure. 9.. PAGE 43.

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