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Simplified calculation of cable tension in suspension bridges Richmond, Kenneth Marvin 1963

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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).  

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