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 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. 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. 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 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 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 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 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 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 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 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 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 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 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 ) 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 TO Figure 2. FOLLOW PAGE 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) 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) 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 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) 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, TO Figure 4. FOLLOW PAGE 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 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- 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) 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 ) 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 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 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 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 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 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 ) TO Figure 5; FOLLOW PAGE 22 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 ) 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. 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. 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 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 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 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 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 . 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 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 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: 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 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 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 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 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 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 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) 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 TO CASE I Figure 8. FOLLOW PAGE 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 ) 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 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 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 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 44 Three-Span Bridge w i t h Hinged It i s a simple matter to extend method d e s c r i b e d a b o v e hinges at the g i r d e r solved to f i n d a total a = ratio 2 s £2 values of spans which with (60) must be is ... ( 7 8 ) span l e n g t h L to main span l e n g t h L g EI b C L ) of b ("^ l) s against (CL) for selected 2 b. line f o r H£ i n t h e main span i s then X v- L' (58) a n d A shows c u r v e s The I n f l u e n c e H Equations bridge the EX - 14(a) case of a t h r e e - s p a n A ^ f o r a l l the of s i d e ( A l l s = "S ! ( Figure a p p l i c a t i o n of 2a b ( A . ) 1 s + L the 3 EI b to the supports. 1 where: Supports (79) f where X 1 + 2a3b(A ... ( 8 0 ) ) L - a The i n f l u e n c e L f line f o r H£ i n t h e s i d e span Is v- (81) 1/s The s u b s c r i p t s i n the term - v l indicates that the influence A 1/s curve f o r main span, X f o r the g the side span i s g e n e r a l l y d i f f e r e n t due t o t h e d i f f e r e n t v a l u e o f C L . s i d e span Is g i v e n by from that f o r The multiplier the 45 2 a b ( A ]_) x s ..: 1 A~l g (79) a n d the ( A 1^s b Figures 1 " > x ... ( 8 2 ) '• and sag f o r against t 2a3b(Ai). + In equations X s (8l), L and f a r e the main span. F i g u r e 14(b) for values selected and X , the multipliers for the of span shows X and X of a. and 3.4(b) be u s e d t o g e t h e r 14(a) values It is plotted intended i n order influence ^line g to length that determine ordinates from F i g u r e 10. I n the shown t h a t the . / IL SH case of the m u l t i p l e span b r i d g e , it c a n be e q u a t i o n f o r 8H i s g i v e n b y \ 1 ... ( 8 3 ) A l I X As i n t h e case equations ( 7 l ) and f o r 8H t o give 8H 2 f L EI Values A s i n g l e span, s u b s t i t u t i o n c a n be made 1 a b from (73) a n d t h e r e s u l t i n g e q u a t i o n c a n be s o l v e d (84) A ,1 , L e — + X of of the AE ( ^ 1^s are tabulated i n T a b l e 2, A p p e n d i x 2, and A~l Appendix 3 contains a n u m e r i c a l e x a m p l e s h o w i n g how t h e o r F i g u r e s 10 a n d 14 c a n be u s e d t o d e t e r m i n e span b r i d g e w i t h h i n g e d s u p p o r t s . procedure is the same as is that for a single Three-Span Bridge with Continuous Up t o t h i s It point, H for a clear that tables three- the general span. Girder c o n s i d e r a t i o n has been restricted 46 to s i n g l e and m u l t i p l e span b r i d g e s ' w i t h h i n g e s The p r o b l e m becomes continuous at the supports, manner. Equation equations (85) to Eld^v-L Hv somewhat at all more c o m p l i c a t e d i f t h e but it supports. girder may be s o l v e d i n a s i m i l a r (43) c a n be r e p l a c e d by e q u a t i o n (57) and by (87) below: Hjy 1 (85) dx' Eld v Hv 2 MgX 2 dx 2 . (86) L E I d,2^ v HV3 3 M (L-x) 3 dx^ (85) to l o a d moments action but of the (87) r e p r e s e n t a s i n g l e w i t h end moments partial The t h r e e e q u a t i o n s equation It i s necessary exists. single 1 Equation span and i t lijfL on a c o n t i n u o u s equivalent span. to the In a n a l y s i s to e q u a l i z e ( 8 5 ) has been c a n be shown t h a t from of a continuous where solved for the slopes girder. single end s l o p e s the the contin- case of a are / dv > 1 EI dx together are s p a n w i t h no a p p l i e d Mg and M^ r e s u l t i n g t e n s i o n H]_ a c t i n g (58) f o r a s i n g l e structure, uity ... ( 8 7 ) L Equations dv is ... ( 8 8 ) \dx where 4 dv 1 (CL)3 dx / o Equation 4 .+ e ( 8 6 ) c a n be s o l v e d t o C L (CL-2) e CL -CL - e e" C L (CL+2) (89) give o — v 2 M 2 Ir EI where V 2 . (90) 47 x Vr (CL) Then the 2 dx C x - e L " e C L - e~ end s l o p e s Mg L dVg e _ C x ... ( 9 1 ) C L c a n be f o u n d from dVg . (92) E I dx where /dV \ 1 1 CL CL 1 1 CL CL 2 •o /dv^ \dx L j e e q = x (v ) _ 2 L CL . (93) -CL - e + e -CL ... ( 9 4 ) ~CL ^CL e - e F r o m symmetry o f t h e (v ) OL girder, it is clear that . (95) x (96) ... ( 9 7 ) I n the case of a s y m m e t r i c a l t h r e e - s p a n l o a d moment, t h e b e n d i n g moments a t the bridge, towers w i t h no a p p l i e d are equal. unknown moment M c a n be f o u n d b y e q u a l i z i n g end s l o p e s towers at The the and i s g i v e n b y ab / dv- M H l ,dx / L s f dv^ dx It to + a Vdx j Ls c a n be shown t h a t where the elastic ... ( 9 8 ) b / dv 2 i n the theory case of an e x t r e m e l y s t i f f is valid, equation (98) c a n be girder reduced 48 H l 1 + ab: f ... 3 b 2 a In the (99) — + — will be n e c e s s a r y shown t h a t of i t e r a t i v e p r o c e d u r e r e q u i r e d t o compute H , it t o compute M a number o f t i m e s , a n d i t w i l l i s advantageous H a n d c o r r e c t b y means m i n e d f o r e a c h new t r i a l t o compute M w h i c h i s e value of H . Values of K are these parameters p u t e d and K i s found from the b and (CL) . i n F i g u r e 16. tables or c u r v e s , A s o l u t i o n s to equations (86) a n d also When M i s com- M i s - f o u n d from (87), i t M .. f L X 3 i s found that ... ( 1 0 1 ) EI A3 K is ... ( 1 0 0 ) Mg f IL X 2 2 deter- tabulated g M = K JVL From the be independent o f a m u l t i p l i e r K w h i c h must be i n Appendix 2 f o r s e l e c t e d values of a, plotted against it 3 ... ( 1 0 2 ) EI where X 3 (CL) The to the 4 + e (CL-2) - 4 ~A~ 2 C L "3 C L (103) -CL ,CL e t o t a l v a l u e of the s o l u t i o n of equations e" (CL+2) s u p p o r t movement c o r r e s p o n d i n g (85) to (87) f o r a l l t h r e e s p a n s g i v e n by A A 2. H-^f L T t ... EI (10-4) where T 1 3 1 A t M 2a b(A ), + 2A 2ab(A ). 2 2 + A 1 : ... ( 1 0 5 ) is 49 where M 1 + ab K ... ( 1 0 6 ) 3 b — + — a 2 Values of b ( ^ l ^ s are three-span dix the same as bridge with hinged supports 2 and F i g u r e 1 4 ( a ) . also and a r e A 2 and V a l u e s , of b (CL) i n F i g u r e 15 f o r the case of a found i n Appen- ^ A 2^s a r e A ,2 against those used f o r plotted A l selected values of b and are i n c l u d e d i n A p p e n d i x 2. Influence l i n e s f o r H' i n a continuous suspension li b r i d g e must be f o u n d b y s u p e r i m p o s i n g two i n f l u e n c e first is the same i n f l u e n c e second i s a c o r r e c t i o n f o r m a i n s pL a n X t hve I n f l u e n c e 1 2 3 Y v A i The l i n e used f o r hinged g i r d e r s . continuity. I n the case of The the l i n e , i s g i v e n by + v + f lines. A 2 + (107) A3 where X 1 ... ( 1 0 8 ) T Y 2X2 M (109) T Curves of 2 v A~ values are + v 3 are + A~ 2 shown p l o t t e d i n F i g u r e 12 a n d tabulated 3 f o u n d i n A p p e n d i x 2. For the left side span, the influence l i n e f o r H-^ i s g i v e n by L X s / V l A Y + ii sf ^2 \ A 2/ ... ( 1 1 0 ) 50 where ... (ill) T Y S = Mab Figure for ( Ao) ^ s 11 shows c u r v e s selected values of 2 of course but hand. a n d A p p e n d i x 2 has v of s i d e span i s opposite (CL) . i t e r a t i o n procedure of the tables (112) tabulated values The i n f l u e n c e l i n e f o r t h e 2 similar to t h a t f o r the left side The n u m e r i c a l e x a m p l e i n A p p e n d i x 3 shows t h a t use , ... f o r d e t e r m i n a t i o n of H converges o r c u r v e s makes t h e right span the r a p i d l y and calculations simple. V a r i a b l e EI... It (86) and c a n be s e e n t h a t (87) a r e g i v e n f o r t h e r i g i d i t y E I w i t h i n the constants constant the tabulated span i s is possible numerically, for and t a b u l a t e similar to equations s p e c i a l case constant. (58), i n which the Therefore, i n A p p e n d i x 2 depends E I w i t h i n each It solutions on t h e use t o s o l v e the equations, other p a r t i c u l a r v a r i a t i o n s d a t a f o r use at least in girder in analysis. rigidity A suitable a t y p i c a l mode o f v a r i a t i o n o f s u c h t h a t most o r a l l s u s p e n s i o n b r i d g e h a v e ,a s t i f f n e s s v a r i a t i o n which l i e s w i t h i n a range data. Analysis d e t e r m i n e d by i n t e r p o l a t i o n between However, it Is the assumption of girder stiffness of t a b u l a t e d of span. a p p r o a c h . m i g h t be t o d e t e r m i n e two s e t s girder suggested the that constants tabulated girders d e f i n e d by might then be values. the a s s u m p t i o n of constant 51 girder rigidity culations. i s a reasonable and d e s i r a b l e Some n u m e r i c a l e x a m p l e s were s o l v e d u s i n g computer program d e s c r i b e d i n the p r e c e d i n g s p a n b r i d g e was a n a l y s e d as a c o n t i n u o u s at the supports. minimum o f one f o r hand The m a i n e p a n g i r d e r .5 t i m e s the maximum o f 1.5 t i m e s A three- g i r d e r and w i t h midspan s t i f f n e s s the.mid-span the chapter. stiffness at stiffness the towers at. the e q u a l to the average study that a reasonably accurate assuming an average H-^ e n c o u n t e r e d recommended value f o r were l e s s order the of H girder stiffness. than 2 per cent. Therefore, to determine the average indicated by Errors it in is girder v a l u e be assumed f o r e a c h value f o r H. points. rigidity c a n be d e t e r m i n e d t h a t f o r a l l hand c a l c u l a t i o n s , a c o n s t a n t r i g i d i t y E I e q u a l to the in value of the to a quarter girder The r e s u l t s hinges v a r i e d from a The same b r i d g e was a n a l y s e d a s s u m i n g a c o n s t a n t value. cal- span TO FOLLOW PAGE 51 Figure 13 52 CHAPTER 5 CONCLUSIONS It i s d o u b t f u l t h a t D e f l e c t i o n Theory a n a l y s i s of suspension bridges procedure b y hand c a l c u l a t i o n s w i l l o r one t h a t enthusiastically. solutions does structural engineer w i l l approach However, i t has been f o u n d , t h a t Theory s o l u t i o n s are i n many c a s e s . the e v e r be a s i m p l e too i n a c c u r a t e Therefore, exist. It Elastic even f o r p r e l i m i n a r y d e s i g n the need f o r D e f l e c t i o n w o u l d be e x p e d i e n t Theory to t u r n the o v e r t o a computer and a v o i d a l l hand c a l c u l a t i o n s , but not always a p r a c t i c a l procedure. a design, tions, there w i l l and i t w i t h the ments that (43) a n d that the i s hoped t h a t t h e s e w i l l was a p p a r e n t the i n the simpler Deflection chapter Theory represented Reference (l) It Refine- by is equations question- by f u r t h e r is of the D e f l e c t i o n computer a n a l y s i s . be f o u n d t a b u l a t e d calculaeasier on T h e o r y a n d the a d d i t i o n a l a c c u r a c y a t t a i n e d f o r use some h a n d of h i g h a c c u r a c y . more r e f i n e d v e r s i o n s attractive is e a r l y stages of be made somewhat j u s t i f i e d i n hand c a l c u l a t i o n s , and i t reserved for that here. ( l i b ) gives results a b l e whether ment i s a l w a y s be a n e c e s s i t y f o r methods p r e s e n t e d It D u r i n g the task refine- recommended T h e o r y be E q u a t i o n (43) i s r e l a t i v e l y i n hand c a l c u l a t i o n s s i n c e s o l u t i o n s a r e i n Steinman's text* on s u s p e n s i o n to bridges. 53 Once t h e cable tension for a total possible to superimpose solutions for p a r t i a l as g i v e n i n S t e i n m a n ' s Equations method p r e s e n t e d the of d e t e r m i n i n g inherent c a n be s e e n Appendix 3 that the ( l i b ) f o r m the the i n the the total the more t h a n girder presents p a r t l y p a i d f o r by e f f o r t theory for of cable hinged at the but A no C o n t i n u i t y i n any s t r u c t u r e is in.analysis. s t e p t o w a r d an a c c u r a t e , simplified tension. the cable It i n A p p e n d i x 3 meets t h e the method simple is believed method d e v e l o p e d i n C h a p t e r 3 a n d i l l u s t r a t e d i n Therefore, especially supports. some a d d i t i o n a l d i f f i c u l t y method o f d e t e r m i n i n g simplicity. for calculated to a p p l y , must be a n a c c u r a t e , calculations tension. i s made tension of a n a l y s i n g s u s p e n s i o n b r i d g e s the the calculations given i n e x t r e m e l y easy s h o u l d be e x p e c t e d . The f i r s t conditions accuracy. sample case of a b r i d g e w i t h g i r d e r s continuous value v a l u e of c a b l e i n the method i s basic above e q u a t i o n s method has a r e l a t i v e l y h i g h It in and method g i v e n a n d h e n c e by t h i s loading is text. (43) No a p p r o x i m a t i o n n o t l o a d i n g c a s e i s known i t objectives that sample o f a c c u r a c y and method s h o u l d be u s e f u l as p a r t of • a t o t a l method o f a n a l y s i s . One a p p r o a c h to a s i m p l i f i e d be a d e t e r m i n a t i o n a n d t a b u l a t i o n factors his i n a manner thesis similar on n o n - l i n e a r whatever might o f i n f o r m a t i o n on a m p l i f i c a t i o n to t h a t d e s c r i b e d by A . F r a n k l i n in arches. methods to recognize method o f a n a l y s i s are is important in suspension bridge a n a l y s i s , of suspension bridges. used to complete t h a t methods despite So l o n g as the the a n a l y s i s , of s u p e r p o s i t i o n are the non-linear total.value it valid behavior of the cable 54 tension is known and applied in the equations for the partial loadings, the bending moments and deflections for the partial loading cases may be superimposed to give the total values. The key, then, is the determination of the total cable tension, and a simple, accurate method of determining the cable tension has been presented In this work. 55 APPENDIX 1 COMPUTER PROGRAM C C C C C C KEN RICHMOND CIVIL ENGINEERING THESIS PROGRAM TO ANALYSE SUSPENSION BRIDGES AND COMPUTE MAGNIFICATION FACTORS. MAIN LINE PROGRAM 43 1 2 3 4 5 6 7 8 10 9 11 12 14 13 15 DIMENSION E l ( 5 3 ) , P(53) , A B ( 1 3 , 5 3 ) , A ( 5 3 , 5 ) DIMENSION E(53) , B ( 5 3 ) , BM(2,53) READ 1 , KODE , C , R , S FORMAT ( 12 / ( E l 4 . 7 ) ) READ 2 , F , SIDE , RISE , T , SLIP FORMAT ( F 6.4 / F 6 . 4 / F 6 . 4 / ( E 1 4 . 7 ) ) PRINT 3 , KODE , R , S FORMAT(//6H KODE= 14 ,3H R= E14.7 ,3H S= E14.7 / ) PRINT 4 , F , SIDE , RISE FORMAT ( 3H F= F 7 . 4 , 6H SIDE= F11.4 , 6H RISE= F l 1 . 4 / PRINT 5 , T , SLIP FORMAT ( 3H T= El 7.7 , 6H SLIP= El 7.7 ) EIO =1.0 / (8.0 * F * C ) IA= 17.0 - SIDE * 20.0 ID= 37 + 17 - IA IF ( KODE - 10 ) 8 , 8 , 6 KODE = KODE - 10 DO 7 I = IA , ID , 1 E l ( I ) = EIO GO TO 11 DO 9 I = IA, ID, 1 READ 10 , E l ( I ) FORMAT ( F 7 . 4 ) E l ( I ) = EIO * E l ( I ) PRINT 12 FORMAT(/23H I P El / ) DO 13 I = I A , I D , 1 READ 14 , P ( I ) FORMAT ( F 8 . 4 ) PRINT 1 5 , 1 , P ( D , E l ( l ) FORMAT ( 13 , F 9 . 4 , E17.7 ) DF = 0.0 SF = 0.0 GO TO ( 1 6 , 1 7 , 1 8 , 1 9 , 1 6 , 1 7 , 1 8 , 1 9 ), KODE ) 56 16 DF = 1.0 17 SF = 1.0 GO TO 19 18 DF =1.0 19 N2 = 1 GO TO 200 20 D = 0.0 HD = 0.125 / F HL = 0.1 DO 36 N = 1,2 K = 0 21 HT = HL + HD IF (KODE - k ) 22,22,23 22 Hk = 2 I I = IA IE = ID GO TO 400 23 Hk = 1 N2 = 3 GO TO 200 2k ERROR = 0.0 N2 = 2 GO TO 200 25 ERROR = ERROR + SLIP PRINT 26 , HL , ERROR 26 FORMAT ( A H HL= E17.7 , 7H ERROR= El 7.7 ) IF(K-I) 27,28,27 27 HL1 = HL ERR = ERROR HL = HL + .1 * HD K = 1 GO TO 21 28 I F ( A B S ( E R R 0 R ) - 1 . 0 E - 5 ) 3 0 , 3 0 , 2 9 29 DELH = ERROR *(HL - HL1)/(ERROR - ERR) ERR = ERROR HL1 = HL HL = HL - DELH GO TO 21 30 E ( I A - 1 ) = -E(IA+1) DO 31 l = I A , I D , 1 BM(N,I) = ( 400.0*(2.0*E(I)-E(I-1)-E(I+1)))*(1.0+D*HT*SM) 31 BM(N,I) = (BM(N,I)-SM*(P(I)+1.0+HT*D2Y))*EI(I) BM(N,IA) = 0.0 BM(N,ID) = 0.0 IF ( KODE -k ) 3 3 , 3 3 , 3 2 32 BM(N,17) = 0.0 BM(N,37) = 0.0 33 SUM = 0 Q = ID - IA + 1 DO 3k I = IA,ID,1 3k SUM = SUM + E l ( I ) 57 35 36 41 39 38 37 42 C C c AVG = SUM / Q CDL = HD / AVG CLL = HL / AVG CTOT = HT / A V G PRINT 35 , CDL , CLL , CTOT FORMAT (/ 5H CDL= E l 7.7 , 5H CLL= E l 7 . 7 , 6H CTOT= E l 7.7) D = 1.0 PRINT 41 F0RMAT(/48H I DEFLECTION BM ELASTIC BM PHI DO 37 I = I A , I D , 1 IF ( B M ( 1 , U ) 3 8 , 3 9 , 3 8 PHI = 1.0 GO TO 37 PHI = B M ( 2 , I ) / BM(1,1) PRINT 42 , I , B M ( 2 , I ) , B M ( 1 , I ) , P H I , E ( I ) FORMAT(I3#F14.8,F17.8,F17.8,E20.7 ) GO TO 43 /) SUBROUTINE 1 100 101 C C C E I ( I A - I ) = EI(IA+1) EI(ID+1) = E l ( I D - 1 ) SM = S / EIO D2Y = - 8 . 0 * F RAE = R * R / EIO DO 101 I =1 I , I E , 1 DEI = 10.0 * ( E l O + 1 ) EK - 1 ) ) D2EI = 4 0 0 . 0 * ( E l ( 1 - 1 ) - 2, 0 * El (I) + E l ( 1 + 1 ) ) X = I - II X = .05 * X DY = 4 . 0 * F * (AL - 2 . 0 * X ) - BL / AL DS = S Q R ( 1 . 0 + DY * DY ) AB(1, = 1.6E5 * E l ( I ) - 8 . 0 E 3 * DEI AB(3, = - 6 . 4 E 5 * E l ( l ) + 1.6E4 * DEI + 4 0 0 . 0 * D2EI AB(4, = SM*AB(3,D + 400.0*(-1.0-DF*DY*DY)+20.0*DF*DY*D2Y AB(5, = 9 . 6 E 5 * E l ( I ) - 8 0 0 . 0 * D2EI AB(6, = S M * A B ( 5 , D + 8 0 0 . 0 * ( 1 . 0 + DF*DY*DY ) AB(7, = - 6 . 4 E 5 * E l ( I ) - 1.6E4 * DEI + 4 0 0 . 0 * D2EI AB(8, = SM*AB(7,I) + 400.0 * (-1,0-DF*DY*DY)-20.0*DF*DY*D2Y AB(9, = 1.6E5 * E l ( I ) + 8 . 0 E 3 * DEI AB(10,I = ( P ( l ) + 1 . 0 ) * ( 1 . 0 - SM * D2EI) AB(11,1 = D2Y * ( 1 . 0 - SM * D2EI) AB(12,1 = - D F * T * D 2 Y * ( 3 . 0 * D Y * D Y + 1.0 ) = - D F * R A E * D 2 Y * D S * ( 4 . 0 * D Y * D Y + 1.0 ) AB(13,I GO TO 201,202,203),N1 SUBROUTINE 2 200 II = IA IE = 17 AL = SIDE 201 202 C C C 204 203 BL N1 GO II IE AL BL N1 GO II IE AL BL N1 GO GO = RISE = 1 TO 204 = 17 = 37 = 1.0 = 0.0 = 2 TO 204 = 37 = ID = SIDE = -RISE = 3 TO ( 1 0 0 , 3 0 0 , 4 0 0 ) , N 2 TO ( 2 0 , 2 5 , 2 4 ) , N 2 SUBROUTINE 3 300 DO 301 I = I I , IE , 1 X = I - II X = .05 * X DY = 4 . 0 * F * (AL - 2 . 0 * X ) - BL / AL DS = S Q R ( 1 . 0 + DY * DY) RAE = R * R / EIO IF(I - I I ) 302 , 302 , 303 302 DE = 2 0 . 0 * E ( l + 1 ) GO TO 307 303 IF (I - IE) 305 , 304 , 304 304 DE = - 2 0 . 0 * E ( l - 1 ) GO TO 307 305 DE = 10.0 * ( E ( l + 1 ) - E ( l - 1 ) ) 307 B ( I ) = H L * R A E * S F * D S * D S * ( D Y * D E + . 5 * D E * D E ) - . 5 * D E * D E 301 B ( l ) = D*B(I)+RAE*HL*DS**3+T*DS*DS-DY*DE IS = IE - 2 DO 306 I = I I ,IS , 2 306 ERROR = ERROR +(.05/3.0) * ( B ( l ) + 4 . 0 * B ( l + 1 ) + B ( l + 2 ) ) GO TO ( 2 0 1 , 202 , 2 0 3 ) , N1 C C C SUBROUTINE 4 400 DO 401 I = I I , IE , 1 A ( l , 1 ) = A B ( 1 , I ) * ( 1.0 + SM*D*HT ) A ( l , 2 ) = AB(3,D + D * HT * A B ( 4 , 1 ) A ( l , 3 ) = AB(5,D + D * HT * A B ( 6 , I ) A ( l , 4 ) = AB(7,D + D * HT * A B ( 8 , 1 ) A ( l , 5 ) = AB(9,D * ( 1 . 0 + SM*D*HT ) 401 B ( l ) = A B ( 1 0 , I ) + H T * A B ( 1 1 , 1 ) + D * H T * A B ( 1 2 , 1 ) + D * H T * H L * A B ( 1 3 , 1 ) IM = II -1 IN = IE +1 B(IM) = - ( P ( l l + 1 ) + 1.0 + HT *D2Y) * ( SM / ( 1 . 0 + HT * SM oo # O A _ ,—^ s—s in + o < U J • < S ' — 1— - — . '—* —• >CM O * »—• + —> — — X + CM L U * - * 4 < * N * — CM C O • *— + — ' — s C A O A «» a: a: I I t—- . 1 ->v. L—U .—, < U J — * —_ * MD < •—•> C Q ' — - o — CM ctL 5 —+ + < * — 1— U J — CM —> —^ < 1 CO 0 N L A K U J -5 — — ' ' « — L U o -4-4 — »— r— O I «• LU L A — Cu 0 • O 0 O O ^ 0 0 - •O O •O 0 • • O • •»< < * + O A + I S Z - ^ ^ - J - — >— » — C M ^ . II • • • I I 4 4 I 4 N II CM OA < I — »•» r- CM II II II II II II II II II II II II II II II I — + — II CO + ^-.^-s^^^-,^^^^^^->^->^^Z II OQ " J *———* ^—v <—O n * * * * * * * * * * * * * - N M i J Z 4 - N M i l - N M i l I S 2 Z Z Z < -4" M ' — . SI •» ' — * CM -4" O ^ — CM CM »•> —CO Z O Q ^ ^ OO I - 4 CM C Q - 3 w || || »'—* L A -4 I » w CO OA O •CM — LU —» l l ^*>_ - , CM - 3 - > " J —) 0 A 0 A O A 0 A O A - 4 L A * — CM O A — O A < C » Z l ^ - 0 ^ '—- L U < C — ^ - - Z — — ' II —5 ' — II 3 || || || || || » w + CM"—' + — — • O OO O O 0 0 O O O O O ' - ' - ^ ^ J - + » C M <C I —*—'—' «— » 0O O O O O O O O O O O O O — ' 1 — C O » « 1 r- » I — Q ; ^ O ^ > <t || — O O '— *— O CM CM — I OO 1 1 - 4 II II - 4 + + - 4 + + II Z 1 1 1 — ' — - — ' I — H —~—'—-——-——^—'iz: O C M O ' O—"—*—— ' — ' II L L O O '— ^—'O — - — 0 D C D C D C D C O Q < < < < < < < < < < < < < < CM O -4 — Q 0 t D i Q < C O Q < C D c C U J L l J -4" 0 -4 0 L A OA 0 -4-4 — UJ 0 vD 0 Z O CJ LU r-~-co 0 -4 - 4 - 4 TO MAIN LINE FOLLOW PROGRAM \ 1 READ: KODE, C , R , S , F, SIDE,RISE,T, SLIP PRINT: K(j)DE, R,S F, SIDE, RISE T , SLIP COMPUTE EIO, I A , ID K<!)DE> 10 KODE < 10 1= IA ,ID READ EI (I) EIO COMPUTE E I ( I) I = IA.ID READ P(I) PRINT I,P(I),EI(I) SHEET "I OF 6 PAGE 59 TO FOLLOW PAGE TO FOLLOW PAGE INITIALIZE ) I ERROR=0 \ , N2 = 2 ,200) 25 ERROR = ERROR + SLIP PRINT: HL,ERROR S W I T C H X y K K S ° / S \ W I T C H K=I V NEW ERRORl < I O L 5 H H TO (N,IA) BM L = + .1 H D INTERPOLATE COMPUTE BM L ESTIMATE NEW ESTIMATE H, (N,ID) COMPUTE AVERAGE EI CDL.CLL.CTOT PRINT N = 1,2 CDL, C L L , CT(|)T SHEET .3 OF 6 -<3 TO 1= IA , ID BM(I,I ) * 0 PRINT 1, BM(I,I), BM(2,I), PHI, E(I) SUBROUTINE I. Compute and store constants AB(I,I) to AB(I3,I) COMPUTE S M , D2Y, • RAE I = II, COMPUTE AB ( 1,1) TO AB ( 13,1) SHEET 4 OF 6 IE FOLLOW PAGE 59 TO SUBROUTINE © SPECIFY II, I E , L , B SIDE Integrate by SPECIFY SPECIFY II,IE, L, B II,IE, L , B RIGHT SPAN SIDE SPAN 3 cable Simpson's © (202) MAIN SPAN SUBROUTINE PAGE 2 (200) LEFT FOLLOW equation Rule 1= I I , < ERROR 3 I B(I) = ERROR I E - 2 ,2 + -h 4 B ( I + I) + B (I +2)J @ SHEET 5 OF 6 TO SUBROUTINE 4 I Compute at all deflections points to I E + I in E(I) and II-l store = II, IE COMPUTE A (1,1) TO A(T,5) a B( I ) COMPUTE BOUNDARY CONDITIONS I = I I - I, I E - I R E D U C :E A ( I + 1 ,2) aA( 1 + 2,1 ) TO Z ERO ? <!s REDUCE A ( I E + 1,2) TO ZERO COMPUTE E( I E +1), E (IE) SHEET 6 OF 6 FOLLOW PAGE 59 6o APPENDIX 2 TABLES TABLE 1, F- INPLUENCE CURVES A- ll r L - dx V]_ A1 A P. rx 'o r x v m 2 T HL EI 2 1. X L ."4843 + 2 v 3 + ~A~ 2 X Q ^ Vo A Pl .0311 .0613 .0899 ..1160 .1391 .1588 .1744 .1859 .40 .45 ' .1928 '.1952 • 50 . 1928 .55 .60 .1859 . 1744 .65 .1588 .70 .1391 .75 .80 .1160 .85 .0899 .90 .0613 .0311 .95 .05 .10 .15 .20 .25 .30 . .35 A m L v dx dx + V-: 2 Ai .0360 .0245 .0681 .0487 .0722 .0961 .0946 .1203 .1156 .1407 .1348 .1573 .1702 .1519 . 1665 .1794 . 1848 .1782 .1867 .1867 .1848 .1915 .1922 .1794 .1884 • ..1702 .1798 .1573 . .1658 .1407 . 1461 .1203 -.1201 .0961 . .0681 .0875 .0360 .0476 L .0007 .0030 .0068 .0120 .0184 .0259 .0342 .0432 .0527 .0625 .0722 .0817 .0907 .0990 .1065 .1129 .1181 .1219 .1242 + A 3 L As .0006 .0024 .0054 .0096 .0149 .0211 .0283 .0363 .0449 -.0541 .0635 .0732 .0827 .0919 .1006 .1084 .1151 .1203 .1237 A m .0009 .0035 .0076 .0130 .0196 .0271 .0353 .0440 .0531 .0625 .0718 .0809 .0896 .0978 .1053 ..1119 .1173 .1214 . 1240 61 TABLE 1 HL EI 2 Al 2. .4435 3. .4091 x L .05 .10 .15 ••.20 .25 .30 .35 .40 .45 .50 .55 .60 .65 • .70 .75 .80 .85 .90 95 .05 .10 . .15 .20 .25 .30 .35 .40 .45 .50 .55 . 60 .65 .70; .75 .80 .85 .90 .95 F l (CONTINUED) p s P A m A l .0241 .0479 .0711 .0933 .1141 .1333 .1504 .1652 .1771 .1859 .1912 . 1924 .1892 .1811 ..1676 .1482 .1223 .0894 .0488 .0365 .0687 .0967 .1207 . 1409 .1572 .1698 .1788 .1841 .1859 .1841 .1788 .1698 .1572 .1409 . 1207 .0967 .0687 .0365 .000? .0031 .0068 .0120 .0184 .0259 .0342 .0432 .0527 .0624 .0722 .0817 .0907 .0990 .1065 .1129 .1181 .1218 . 1242 .0006 .0024 .0053 .0095 .0147 .0209 .0280 • .0359 .0444 .0535 .0630 .0726 .0821 .0914 . 1002 .1081 .1149 .1202 .1237 .0009 .0035 .0077 .0131 '.0197 .0272 .0354 .0441 .0532 .0625 .0717 .0808 .0895 .0977 .1052 .1118 .1172 . 1214 .1240 .0238 .0311 .0614^ .0473 .0900 .0701 .1161 .0921 .1128 .1392 .1588 .1318 .1744 • .1490 .1858 .1639 .1761 .1927 .1852 .1950 . 1908 .1927 .1858 .1926 .1744 .1899 .1588 .1823 .1692 .1392 .1161 . 1501 .0900 .1244 .0614 .0912 .0500 .0311 .0369 • .0692 .0972 .1211 .l4io .1571 .1694 .1782 .1835 .1852 .1835 . 1782 .1694 .1571 . 1410 . 1211 .0972 .0692 .0369 .0007 .0031 .0069 .0120 .0184 .0259 .0342 .0433 .0527 .0625 . 0722 .0816 .0907 .0990 .1065 .1129 .1180 .1218 . 1242 .0005' .0023 .0053 .0093 .0145 .0206 .0276 .0354 .0440 .0530 .0624 .0720 .0816 .0909 .0998 .1078 .1147 .1201 .1237 .0009 .0036 .0078 .0132 .0198 .0273 .0354 .0442 .0532 .0625 .0717 .0807 .0895 .0976 .1051 .1117 . 1171 .1213 .1240 .0311 .06l4 .0899 .1161 .1392 .1588 • .1744 .1858 .1928 .1951 . .1928 .1858 .1744 .1588 . .1392 .1161 .0899 .06l4 .0311 62 TABLE HL 2 EI 5. 7. AT X . 1 L .35^2 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 .3122 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 • .65 .70 .75 .80 • • .85 .90 .95 1 FT (CONTINUED) F AT 1 s .0312 .0615 .0901 .1162 .' .1392 .1587 • .1743 .1856 .1925 .1948 .1925 .1856 .1743 .1587 .1392 .1162 .0901 .0615 .0312 .0231 .0460 .0683 .0898 .1102 .1291 .1463 .1614 .1740 .1838 . 1902 .1928 .1911 .1845 .1723 .1538 .1283 .0948 .0524 .0377 .0704 .0983 .1218 .1412 .1568 .1687 .1771 .1821 .1838 .1821 .1771 .1687 . 1568 .1412 .1218 .0983 .0704 .0377 .0007 .0031 .0069 .0120 .0184 .0259 .0343 .0433 .0527 .0625 .0722 .0816 ' .0906 .0990 .1065 .1129 . 1180 .1218 .0313 - . 0 2 2 5 .0616 .0448 .0902 .0667 .0878 .1163 .1079 .1393 .1267 .1587 .1742 .1439 .1592 .1855 .1924 .1722 .1824 .1947 .1924 .1895 .1930 . 1855 .1742 .1921 .1864 .1587 .1750 .1393 .1572 .1163 .0902 .1319 .0616 .0981 .0546 .0313 .0386 .0715 .0993 .1225 .1415 .1566 .1680 .1761 .1809 .1824 .1809 . 1761 .1680 . 1566 .1415 .1225 .0993 .0715 ..0386 .0007 .0031 .0069 .0121 .0185 .0259 .0343 .0433 .0528 .0625 .0721 .0816 .0906 .0990 .1064 .1128 .1180 .1218 . 1242 m 1 .1242 A„ ..0005 - .0023 .0051 .0091 .0141 .0201 .0270 .0347 .0431 .0521 .0614 .0710 .0806 .0900 .0990 .1072 A m .1199 .1236 .0009 .0036 .0079 .0134 .0200 .0275 .0356 .0443 .0533 .0625 .0716 .0806 .0893 .0974 . 1049 . 1115 .1170 .1213 .1240 .0005 .0022 .0050 .0089 .0138 .0196 .0264 ' .0340 .0423 .0512 .0605 .0701 .0797 .0892 .0983 .1066 .1139 .1197 .1235 .0009 .0037 .0080 .0136 .0202 .0277 .0358 .0444 .0534 .0625 .0715 .0805 .0891 .0972 .1047 .1113 .1169 .1212 . 1240 .1143 63 ' TABLE 1 HL EI 2 10. 20. Al ..2652 .1766 f X L p l x (CONTINUED) s .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 • .55 .60 .65 • .70 .75 .80 .85 .90 .95 .0314 .0618 .0904 . 1164 .1394 .1587 .1741 .1854 .1922 .1944 .1922 -.1854 .1741 .1587 .1394 .1164 .0904 .0618 .0314 .0218 ..0434 • .0645 .0851 .1048 .1234 .1406 .1562 .1696 .1805 .1886 .05 .10 .15 .20 .25 .30 .35 .40 .45 ' .50 • .55 .60 .65 .70 .75 .80 .85 .90 .95 .0316 .0622 .0908 .1168 .1396 .1587 .1739 .1849 .1915 .1938 • .1915 .1849 .1739 .1587 .1396 .1168 .0908 .0622 .0316 F A m l A m .1931 .1934 . 1889 .1787 .1618 .1369 .1028 .0577 .0397 .0731 .1007 .1234 .1418 .1562 .1670 .1746 .1791 .1805 .1791 .1746 .1670 .1562 .1418 .1234 .1007 .0731 .0397 .0007 .0031 .0069 .0121 .0185 .0260 .0343 .0433 .0528 .0625 .0721 .0816 .0906 .0989 . 1064 .1128 .1180 .1218 .1242 .0005 .0021 .0048 .0086 .0133 .0190 .0257 .0331 .0412 .0500 .0593 .0688 .0785 ' .0881 .0973 .1058 .1133 .1194 .1234 .0010 .0038 .0082 .0138 .0205 .0279 .0360 .0446 .0534 .0625 .0715 .0803 .0889 .0970 .1044 .1111 .1167 .1211 .1239 .0199 .0397 .0592 .0784 .0971 .1151 .1322 .1481 . 1625 .1751 .1853 .1925 .1961 .1949 .1880 .1738 .1505 .1157 .0668 .0433 .0777 .1049 .1261 .1426 .1550 .1641 .1703 .1739 .1751 .1739 .1703 .1641 .1550 .1426 .1261 .1049 .0777 .0433 .0007 .0031 .0069 .0121 .0186 .0260 .0344 .0434 .0528 .0625 .0721 .0815 .0905 .0989 . 1063' .1128 .1180 .1218 .1242 ,ooo4 .0019 .0044 .0079 .0123 .0176 .0238 .0308 .0385 .0470 .0560 .0655 .0752 .0850 .0946 .1037 .1119 .1186 .1232 .0011 .oo4i .0087 .0145 .0213 .0287 .0367 .0451 .0537 .0625 .0712 .0798 .0882 .0962 .1036 .1104 .1162 .1208 .1238 64 TABLE 1 HL EI . X 2 V;L F l (CONTINUED) p s P A m 30. .1324 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 -.70 • 75 .80 .85 .90 .95 - .0319 .0626 .0912 .1171 .1397 .1586 .1736 .1845 .1910 .1932 . 1910 .1845 .1736 .1586 .1397 .1171 .0912 .0626 .0319 .0186 .0373 .0557 .0740 .0919 .1094 .1262 • .1422 .1572 .1707 .1823 .1913 .1971 .1985 . 1943 .1824 .1608 . 1261 .0742 .0464 .0817 .1082 .1282 .1431 .1540 .1617 ' .1668 .1697 .1707 .1697 .1668 .1617 .1540 . 1431 .1282 .1082 .0817 . .0464 50. .0882 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 • .65 .70 .75 .80 .85 • .90 .95 .0322 .0632 .0919 .1176 . i4oo .1586 .1732 .1838 .1902• .1923 .1902 .1838 .1732 .1586 .i4oo .1176 .0919 .0632 .0322 .0172 .0343 .0515 .0685 .0854 .1020 .1184 .1343 .1496 .l64o .1771 .1884 .1971 . 2020 .2018 .1939 .1754 .1417 .0863 .0517 .0880 .1135 .1312 ..1436 .1520 .1577 .1613 .1633 . 1640 .1633 . 1613 .1577 .1520 .1436 .1312 .1135 .0880 .0517 l .0008 .0031 .0070 .0122 .0186 .0261 .0344 .0434 • .0528 •' . 0 6 2 5 .0721• .0815 .0905 .0988 .1063 .1127 .1179 .1218 .1241 .0008 .0032 .0070 .0123 .0188 ..0262 .0345 .0435 .0529 .0625 .0720 .0814 .0904 .0987 .1061 .1126 .1179 .1217 .1241 A s A .ooo4 .0018 .0041 .0074 .0115 .0166 . .0225 .0292 .0367 .0449 .0537 .0631 .0728 .0827 .0926 .1020 .1107 .1179 .1230 .0012 .0044 .0092 .0151 .0219 .0294 .0373 .0455 .0539 .0625 .0710 .0794 .0876 .0955 .1030 .1098 .1157 .1205 .1237 .0004 .0017 .0038 .0068 .0107 .0154 .0209 .0272 .0343 .0421 .0507 .0598 .0695 .0795 .0896 .0995 .1088 .1168 . 1226 .0013 .0049 .0099 .0161 .0230 .0304 .0381 .0461 .0543 .0625 ' .0706 .0788 .0868 .0945 .1019 .1088 .1150 . 1200 .1236 65 TABLE 1 HL EI 2 A -, a l X • L 1 P (CONTINUED) s- P • m A n 1 A s A m 70. .0662 .05 .10 .15' .20 .25 .30 .35 .40 .45 .50 • .55 .60 • .65 .70 .75 .80 . .85 .90 .95 .0325 .0636 . .0923 .1180 • .l4oi .1585 . .1730 .1833 .1896 .1917 .1896 .1833 .1730 .1585 .1401 .1180 .0923 .0636 .0325 .0163 .0327 .0490 .0653 .0815 ..0976 .1135 .1292 .1445 .1592 .1730 .1855 .1959 .2032 .2058 .2011 .1855 .1533 .0958 .0561 . .0008 .0032 .0930 . 1172 .0071 .0124 .1332 .1436 .0188 .1504 .0263 .0346 .1547 . 0436 .1573 .1587 .0529 .1592 .0625 .0720 .1587 .0813 .1573 .1547 .0903 .1504 .0986 . 1061 ..1436 .1125 .1332 .1172 .1178 • .0930 .1217 .0561 : .1241 .ooo4 .0016 .. . 0 0 3 6 .0065 .0102 .0146 .0199 .0260 .0328 .0404 .0487 .0577 .0673 .0773 .0875 , .0977 •.1074 .1160 .1224 .0015 .0052 .0105 .0168 .0238 .0311 .0388 .0466 .0545 -.0625 .0704 .0783 .0861 .0938 '.1011 . 1081 .1144 .1197 .1234 100. .0482 .05 .10 .15 20 .25 .30 .35 .40 .45 .50 .55 . 60 .65 .70 .75 . .80 .85 .90 .95 .0328 .0156 .0312 .0468 .0623 .0779 .0934 .1089 .1242 .1393 .1541 .1683 .1817 .1936 .2031 .2087 .2077 .1958 .1662 .1073 .06l4 .0008 .0032 .0987 .0072 .1213 .0124 .1350 .0189 .1433 .0264 .1483 .1512 .0347 .0436 .1529 .1538 .0529 .1541 . .0625 • .0720 .1538 .0813 .1529 .0902 .1512 .1483 .0985 . 1060 .1433 .1125 .1350 .1213 .1177 .1217 .0987 .06l4 . 1241 .0003 .0015 .0035 .0062 .0097 .0140 .0190 .0249 .0315 .0388 .0469 .0556 .0650 .0750 .0853 .0957 .1059 .1150 . 1220 • .0016 .0057 • .0112 .0177 .0247 .0320 .0395 .0471 .0547 .0625 .0702 .0778 .0854 .0929 . 1002 • .1072 .1137 .1192 .1233 .o64l .0928 .1184 .1403 .1584 .1726 .1828 .1889 .1910 .1889 .1828 .1726 .1584 .1403 .1184 .0928 .0641 .0328 66 b( ' TABLE 2 . 2 a .25 .30 .35 .40 • .45 . 163 . 2 1 6 .269 .321 .372 .423 .474 .524 .175 .231 .286 .340 .393 .445 .496 .546 . .064 .127 .187 .246 .303 .358 .412 .465 .516 .566 .05 .10 .15 1 .056 .109 2 .060 .118 3 1 b HL - EI L ) .20 .50 5 .073 .143 .210 .273 .334 .392 .448 .501 .552 .601 7 .082 .159 .232 .299 .363 .423 .479 .532 .583 .631 10 .096 .183 .262 .334 .401 .463 .520 .573 .622 .668 20 .137 .251 .348 .430 .501 .564 .619 .668 .712 .751 30 .175 .309 .415 .501 .573 .633 .684 .728 .767' .801 50 .241 .4oi .515 .601 .667- .720 .764. .800 .831 .857 70 .297 .471 .586. .667 .727 .774 .811 .842 .867' .888 100 .367 .550 .660 .785 .824 .854 .879 .899 .916 .733 TABLE 3 . HL EI _ b( A ) 2 s b 2 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 1.00 1 .034 .068 .102 .135 .168 .200 .233 .265 .296 .327 .626 2 .037 .074 .109 .144 .179 .213 .246 .278 .310 .342 .626 3 .040 .079 .117 .154 .189 .224 .258 .291 .323 .354 .627 • .046 .090 .131 .171 .209 .246 .280 .314 .346 .377 .629 5 7 .052 . 100 . 1 4 5 .187 '.227 .265 .300 .334 .366 .396 .630 10 .060 .114, .164 .210 .251 .290 .326 .360 .391 .420 .632 20 .086 .157 .218 .270 .315 .355 .390 .421 .449 .474 .637 30 .109 .194 .261 .315 .361 .399 .432 .461 .486 .508 .642 50 .151 .252 .325 .380 .423 .457 .486 .510 .531 .548 .650 70 .187 .297 .370 .423 .463 .494 .519 .540 .557 .572 .656 100 .231 .348 .419 .467 .502 .529 .551 .568 .583 .595 .664 TABLE 4. K = I}- TOWER MOMENTS HL EI a .0 .1 .2 3 .5 any .2 .3 .4 .5 .6 .2 .3 .4 .5 .6 .2 .3 .4 .5 .6 .2 .3 .4 .5 .6 .984 .967 .937 .973 .948 .977 .955 .979 .960 .981 .964 .961 .925 .968 .939 .973- .948 .955 .977 .961 .980 .921 .958 .935 .966 .945 .971 .953 .975 .960 .979 .924 .960 .935 .966 .970 ..944 .974 .951 .957 .978 3 5 7 10 20 30 50 70 .953 .910 .925 .934 .941 .947 .893 .912 .925 .935 .944 .888 .907 .921 . 932 .926 .860 .882 .897 .908 .916 .837 .900 .817 ..865 .762 .796 .820 .838 .852 .732 .772 .802 .825 .773 .632 .678 .712 .737 .759 .602 .652 .691 .723 .750 .600 .705 .550 .601 .638 .668 .693 .524 .576 .618 .653 .546 .716 .743 .617 .653 .683 .709 .731 .646 .610 .449 .502 .543 .576 .604 .430 .482 .525 .562 .594 .432 .479 .519 .553 .584 .451 .941 .892 .907 .920 .930 .938 .864 .884 .899 .911 .832 .858 .878 .894 .908 .838 .860 .877 .891 .904 .845 .864 .878 .790 .824 .848 .867 .883 .784 .816 .841 .862 .879 .793 .819 .840' .858 .873 .845 .727 .765 .794 .819 .840 .739 .769 .794 .815 .833 .646 .684 .684 .524 ..572 .611 .676 .542 .579 .610 .637 .660 .486 .516 .542 .565 .389 .442 .482 .516 .546 .374 .424 .466 .502 .534 .377 .421 .459 .493 .522 .395 .427 .456 .480 .503 100 .480 .332 .382 .422 .456 .485 .321 .367 .406 .441 .472 .324 .365 .400 .431 .459 .340 .370 .396 .419 .439 APPENDIX 3 NUMERICAL EXAMPLES Example 1. Single Span D I * r .4 25 > k/ft. f \M r > < „ 2 5 0 . 'Je 2 5 Q t\ f\ k/ft ' .1 1000' Given: Length of span L = 1,000 ft.. Sag of cable f = 100 ft.. EI = 1..5 ( 1 0 ) K ft. /girder 8 2 AE of cable = 7 - 0 (lO ) K 5 *L = e 1,082 ft. *L .= 1,054 f t . t Dead load w = 1.0 K/ft. Live load p = .4 K/ft. distributed where shown + 25 K at quarter point as shown et = 3.25 ( 1 0 ~ ) 4 Support displacement hg - h^ = -.5 f t . Formulae given in References (l) and (6) 69 Step 1. H L Compute 1.0 (IOOO) 8f 8 (100) 1250 K 2 (1000) 2 EI L wL 2 D and design constants. .OO667 2 1.5 ( 1 0 ) 8 1082 e .0015 AE 7 (10 ) f L (100) 5 2 EI 2 1.5 (IOOO) .0667 (IO ) 8 Step 2. Compute H-^ . 1 Here, i t is necessary to estimate H in order to select an influence line. H = H. Then D HL 2 It is sufficiently accurate to use = 1250 (.00667) = 8.3' EI H^' is found from the influence lines plotted in Figure 10. The influence line ordinate at the location of the point load is .139 L . is The area under the curve from .25 to .50 (.0625 - .Ol85)L . Therefore f H' 2 5 (.139) (1000) L = .4 (.0625 - .0185) H 100 (1000) 100 = 35 + 176 = 2 1 1 K Step 3. Compute A '•• A ' = B *A h h " f t L H ^e 'L t " "L". AE ='- .5 - 3.25 ( 1 0 = -1.17 ft. T - 4 ) € (1054) - 2 1 1 (.0015) 2 70 Step 4. Compute SH. Here, another estimate of H must be made in order to determine . A - ^ . It is sufficiently accurate to estimate H =H +H . Then 1 D HL 2 L = ( 1 2 5 0 + 2 1 1 ) .OO667 = 9.75 EI Figure 13 is used to determine A-, for • EI H T 1 found that ^ SH A f L 2 = .277. " EI Step 5. -1.17 L E -59K . 2 7 7 ' ( . 0 6 6 7 ) + .0015 AE Compute H. H = H D + H ' + 8H L = 1250 + 2 1 1 - 59 = 1402 Step 6. = 9.75 and i t is Then 8H can be found from < A.} 2 K Compare the value of H' computed in step 5 with the value estimated in step 4, and repeat steps 4 and 5 until they are the same. HL 2 = 1402 (.00667) =9.37 EI From'Figure 13, i ^ = .284 8H -1.17 .284 -58 K (.0667) + .0015 H = 1403 K In the case of a single span, one repetition of steps 4 and 5 should give a sufficiently accurate value of H. Example 2. Three-Span Bridge with Hinged Supports Q 25 k fA k/ft i 1 f\ 750' Given: Main span length L = 1000 f t . Main span sag f = 100 f t . Side span length L„ = 500 f t . EI main span = 5.0 (lO?) K ft. /girder 2 EI side span = 2 . 5 ( 1 0 ? ) K ft. /girder 2 AE of cable = 7 . 5 (lO ) K 5 Side span rise = 112.1 f t . L e = 2 l 8 l ft. L = t 2116 ft. Dead load 1 K/ft. Live load .4 K/ft. on main span as shown + 25 K on side span where shown et = 3.25 (IO ) -4 Support displacement H B -h A =0 k/ft 72 Step 1. Compute H and design constants D Prom equation ( 3 9 ) H 1.0 D (1000) 8 L EI 5.0 a 500 K (100) (1000) 2 . 1250 2 .020 2 -(10?) .5 1000 b 500 2 5.0 (io?) 2.5 (10 ) 1000 f L 2 (100) EI 7 (1000) . . 2 0 (lO ) 7 5 Step 2 . 2 .5 Compute H-^' Estimate H = H = 1250 K D HL = 2 1250 (.020) =25.0 EI Figure 14(a) shows values of ( A i ) plotted against Mil— EI Al for selected values of b. For HL> = 2 5 . 0 and b = . 5 , the EI ordinate '-^^ is . 7 8 0 . Figure 14(b) shows values of "Al the multipliers X and X plotted as abscissae against the ordinate ( A j) for selected values of a. For a =• . 5 - — . • "A~l and ( A i ) = . 7 8 0 , the multipliers are: b s , — N s s 10 s b s x x "Al .836 = a = .082 The main span influence line area from . 0 0 to . 7 5 is found 2 from Figure 10 to be .IO63 — X. Therefore the contribuf is. tio'n to H-j^ from the main span 1 H< .1063 ( 1 0 0 0 ) L (.836) (.4) 2 356 K 100 The influence line ordinate for the point load on the side span is . 1 9 4 1 X . Therefore, the contribution to H ' G L from the side span is H» L .194 (1000) (.082) ( 2 5 K) 4 K 100 The total H ' = 356 + 4 = 360 K. L Step 3. Compute A '. From equation (71) A' = 0 - 3.25 (IO ) ( 2 1 1 6 ) - 360 (.0029) -4 = -1.72 Step 4. ft. Compute 8H. Estimate H = H + H < = D HL 2 L 1250 + 360 = l 6 l 0 K = 1610 (.020) = 3 2 . 2 EI 2 Figure 13 shows "A" plotted against h= ± . EI 1 2 For H L = 32.2, EI .126 From equation (75) '8H -52 K -1.72 .126 (.20) .0029 .832 Step 5. Compute H. H = 1250 + 360 - 52 = 1558 K Step 6. Compare the value of H computed in step 5 with the value estimated in step 4. convergence. HL EI 2 = 1558 (.020) = 31.2 Repeat steps 4 and 5 until 74 From Figure 13, .A ^ = .130 6H -1.72 .130 (.20) + -50 K .0029 .833 H = 1250 + 360 - 50 = 1560 K Compare the value of H computed at the end of step 6 Step 7. with the value estimated in step 2. Repeat steps 2 and 5 to convergence. From Figure 14 X = x s H .833 = .083 < L = .1063 . (1000) 2 (.833) (.4) .194 ( 1 0 0 0 ) (.083) (25) + 100 = 354 + 4 = 358 100 K ' H = 1250 + 358 - 50 = 1 5 5 8 K Step 7 will.seldom produce any significant improvement in the accuracy of H. For most suspension bridges a is usually less than .5 and b is usually less than .5. There- fore, X Since 0 is small and not sensitive to changes in H. X is equal to 1-2 X , i t also is not sensitive to changes • in X. Example Continuous 3. 400' 100' Suspension Bridge 250' • 750' < > Given: Main span length Main span sag Side span length Main span EI = 1.5 (lO ) K f t . 2 Side span EI = 7.5 (io?) K f t . 2 AE of Side L e L t cable span = 2l8l f - L 100 ft. = 500 a ft. 1000 8 (105) ft. K ft. 112.1 ft. 2116 ft. Dead load =1.0 Live load = = = =7.5 rise L = K/ft. K/ft. .4 + 25 K at distributed main span -4 displacement hg - h^ = shown quarter et = 3.25 ( i o ) Support as 0 point 76 Step 1. Compute H (1000) D and d e s i g n (1.0) 2 constants. 1250 K 8 (100) L (1000) 2 EI L .00667 2 (IO ) 1.5 8 2l8l e _ AE (IO ) 7.5 5 (100) f L 2 EI ' (1000) 2 500 .0667 (IO ) 1.5 a .0029 _ 8 .5 1000 b 500 2 1000 M e Step 1.5 (lO ) 7.5 (io ) .1 + ( . 5 ) 7 (.5) f 1.5 + . 5 / . 5 2. Compute H ^ ' Estimate H L 2 .5 8 .500 = 1 2 5 0 (.OO667) = 8 . 3 3 EI Figure of H L 1 6 shows K p l o t t e d a and b . against TJT 2 f o r selected values EI . F o r a = . 5 and b = . 5 ( F i g u r e 1 6 ( d ) ) a n d = 8 . 3 3 , K i s f o u n d t o be .836. EI (106) M = .836 (.500) = .418 From F i g u r e 1 4 , ( A i ) b From F i g u r e " 1 5 , s = .623 A2 = -.630 "Al b(~~A ) 2 "Al s = -.405 Then f r o m equation 77 By equation (105) T = 1 + 2 (.5 ) (.623) + .418 2 (-.630) + 2 (.5) (-.405) 3 = 1 + 2 T= (.078) - .526 - 2 (.085) .460 The 'multipliers X and Y for the main span influence line ordinates are found from equations (108) and (109) .X 1 2.17 .460 Y .526 -1.14 .460 ~ The multipliers X and Y for the side span influence line g g ordinates are found from equations ( i l l ) and (112) X s .078 .170 .460 ~ Y s .085 -.185 .460 ~ The main span influence line is made up by superimposing curves from Figure 10 and Figure 12 in accordance with equation (107). The contribution to H-^' from the main span is H< L 2.17 (1000) .4 (1000) (.106) + .139 ( 2 5 ) .4 (1000) (.106) + .142 ( 2 5 ) 100 1.14 (1000) 100 H< = L 997 - .524 = 4 7 3 K The side span influence line is made up by superimposing curves from Figure 10 and Figure 11 in accordance with equation (llO). span is The contribution to H-^' from the side H' L .170 ( 1 0 0 0 ) .4 (.125 - .019) 2 100 • .185 ( 1 0 0 0 ) .4 (.125 - .014) 2 100 H ' = 77 - 82 = -5 K L The total value of H^' from main and side spans is H < = 473 - 5 = 468 K L Step 3. Compute A ' . Prom equation ( 7 l ) A' = 0 - 3.25 = -2.03 Step 4. (IO ) -4 (2116) - 468 (.0029) ft. Compute 8H. Estimate H = 1250 + 468 = 1718 K HL 2 = 1718 (.00667) = 11.-4 EI From Figure 13, "/T^ = .252 By equation (104) 8H -2.03 -191 K .252 (.460) (.0667) + .0029' Step 5. Compute H. H = 1250 + 468 - 191.= 1527 K Step 6. Compare the value of H computed in step 5 with the value estimated in step 4 and repeat steps 4 and 5 until convergence. HL 2 = 1527 (.00667) = 1 0 . 1 EI From Figure 13, 8H = .270 -2.03 .270 (.460) (.0667) + .0029 -181 K 79 H = 1250 + 468 - 181 = 1537 K Step 7. Compare the value of H computed at the end of step 6 with the value estimated at the beginning of step 2 and repeat steps 2 to 6 until convergence. HL = 1537 (.OO667) = 1 0 . 2 2 EI From Figure 16(d), K = . 8 l 2 M = .812 (.500) = ,406 From Figure 14, ( ^ l ^ s = .636 b From Figure 1 5 , - A 2 = -.632 b("A" ) . = -.420 2 s T = 1 + 2 ( . 5 ) (.636) +• .403 3 2 (-.632) + 2 (..5) (-.420) = 1 + 2 ,(.080) - .509 - 2 .(.O85) T = .481 X =: 1 2.08 .481 ~ Y .509 -1.06 .481 ~ X •• .080 s .166 .481 ~ Y c o H .085 -.177 .481 ~ 1 L 2.08 (1000) .4 (1000) (.106) + .139 ( 2 5 ) 100 - 1.06 (1000) 100 .4 (1000) (.106) + .142 ( 2 5 ) 8o .166 + (1000) 2 (.4) (.111) 2 (.4) (.111) 100 .177 (1000) 100 = 945 - A' 487 + 74 = 0 - 3.25 = -2.05 (IO ) -4 - 69 = 473 (2116) - K 473 (.0029) ft. From Figure 1 3 "A^ = . 2 7 0 8H -2.05 .270 (.481) (.0667) + .0029 = -178 K H = 1250 + 473 - 178 = 1545 K Here, the improvement in accuracy from 1537-K to 1545 K represents a relatively large improvement compared with what might usually be expected. For shorter or more rigid side spans, the Improvement will be less significant and step 1 might reasonably be omitted In many cases. 81 BIBLIOGRAPHY 1. Steinman, D. B., A Practical Treatise on Suspension Bridges, .2nd Ed., London, Wiley, 1929. 2. Pugsley, Sir Alfred Grenville, The Theory of Suspension Bridges, London, Edward Arnold, 1957. 3. Johnson, Bryan and Tourniere, Theory and Practice of Modern Framed Structures, 1 0 t h Ed., New York, Wiley, 1928. 4. Priester, G. C., "Application of Trigonometric Series to Cable Stress Analysis in Suspension Bridges", Engineering Research Bulletin No. 12, Ann Arbor, Michigan, University of Michigan, 1929. 5. Timoshenko, S., "Theory of Suspension Bridges", Journal of the Franklin Institute, Vol. 6. 235 (March 1943). Timoshenko, S., "The Stiffness of Suspension Bridges", Transactions, A.S.C.E., Vol. 95 (1930). 7. Steinman, B. D., "A Generalized Deflection Theory for Suspension Bridges", Transactions, A.S.C.E., Vol. 8. 100, (1935). Shortridge-Hardesty and Wessman, H. E., "Preliminary design of Suspension Bridges", Transactions, A.S.C.E., Vol. 101 (1936). 9. Westergaard, H. M., "On the Method of Complementary Energy and its Applications to Structures Stressed Beyond the Proportional Limit, "to Buckling and Vibrations, and to Suspension Bridges'.', Transactions, A.S.C.E., Vol. 107 (1942). 82 10. Tsien, Ling-Hi, "A Simplified Method of Analysing Suspension Bridges", Transactions, A.S.C.E., Vol. 11. (1949). Gavarini,.C., "Considerations on Suspension Bridges", AcierStahl-Steel, Vol. 12. 114 26, No. 2 (March 196l), Brussels, Belgium. Szidarovszky, J., "Corrected Deflection Theory of Suspension Bridges", Proceedings, A.S.C.E., Vol. 86, No. st 11 (Nov. I960). 13. Szidarovszky, J., "Practical Solution for Stiffened Suspension Bridges of Variable Inertia Moment and its Application to Influence Line Analysis", Acta Technica, Vol. 19, No. 14. 3-4, Budapest Heilug, R., "Eine Bernerkung zur Haengebruechentheorie", Der Stahlbau, Vol. 15. (1958). 26, No. 2, Berlin (Feb. 1957). Muller-Breslau, "Theorie der durcheinen Balken versteiften Kette", Zeitschrift der Arch un Ing, Vereins zu Hannover, 1881. 16. Rode, H. H.,, "New Deflection Theory", Kgl. Norske Viden skabers Selskabs Skrifter, Oslo, 1930. 17. Atkinson, R. J. and Southwell, R. V., "On the Problem of Stiffened Suspension Bridges and its Treatment by Relaxation Methods", Journal, Inst. Civ. Engrs., Longon, 1939. 18. Crosthwaite, C. D., "The Corrected Theory of the Stiffened Suspension Bridge", Journal, Inst. Civ. Engrs., London, 1947. 19. Crosthwaite, C. D., "Shear Deflections", Publications,. • I.A.B.S., Vol. 20. 12, Zurich (1952). Selberg, A., "Suspension Bridges", Publications, I.A.B.S., Vol. 8, Zurich (1947).
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Simplified calculation of cable tension in suspension...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Simplified calculation of cable tension in suspension bridges Richmond, Kenneth Marvin 1963
pdf
Page Metadata
Item Metadata
Title | Simplified calculation of cable tension in suspension bridges |
Creator |
Richmond, Kenneth Marvin |
Publisher | University of British Columbia |
Date Issued | 1963 |
Description | This thesis presents a method which facilitates rapid determination of the cable tension in suspension bridges. A set of tables and curves is included for use in the application of the method. The method is valid for suspension bridges with stiffening girders or trusses either hinged at the supports or continuous. A modified superposition method is discussed and the use of influence lines for cable tension in non-linear suspension bridges is demonstrated. A derivation of the suspension bridge equations is included and various refinements in the theory are discussed. A computer program to analyse suspension bridges was written as an aid in the research and for the purpose of testing the manual method proposed. A description of the program is included along with Its Fortran listing. |
Subject |
Suspension bridges Bridges -- Design and construction Strains and stresses |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-01-14 |
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 | 10.14288/1.0050635 |
URI | http://hdl.handle.net/2429/40083 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1963_A7 R35 S4.pdf [ 6.91MB ]
- Metadata
- JSON: 831-1.0050635.json
- JSON-LD: 831-1.0050635-ld.json
- RDF/XML (Pretty): 831-1.0050635-rdf.xml
- RDF/JSON: 831-1.0050635-rdf.json
- Turtle: 831-1.0050635-turtle.txt
- N-Triples: 831-1.0050635-rdf-ntriples.txt
- Original Record: 831-1.0050635-source.json
- Full Text
- 831-1.0050635-fulltext.txt
- Citation
- 831-1.0050635.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0050635/manifest