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Analysis of flexible hingeless arch by an influence line method. Lee, Richard Way Mah 1958

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ANALYSIS OP F L E X I B L E HINGELESS ARCH BY AN INFLUENCE L I N E METHOD  by RICHARD WAY B.Sc.  MAH  LEE  ( C i v i l Engineering), University  A THESIS SUBMITTED  of Manitoba, 1 9 ^ 6  IN PARTIAL FULFILMENT  OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e Department o f CIVIL  We  accept  required  this  ENGINEERING  thesis  as c o n f o r m i n g t o t h e  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1 9 ^ 8  ABSTRACT  An i n f l u e n c e l i n e method f o r t h e a n a l y s i s o f f l e x i b l hingeless  arch by t h e d e f l e c t i o n t h e o r y  thesis.  To f a c i l i t a t e  fication  factors are provided.  lines  o f t h e a r c h may b e The  measured by a dimen-  | TTTf' \  n  rise  magni  tables, influence  much i n t h e o r d i n a r y way.  o f t h e a r c h was c o n v e n i e n t l y  sionless ratio, the  Prom t h e s e  the f l e x i b i l i t y  drawn and u s e d v e r y  flexibility  i n this  t h e work, t a b l e s o f d i m e n s i o n l e s s  t a k i n g i n t o account  readily  i s presented  ~ /-—• , and c a l l e d t h e s t i f f n e s s f a c t o r o f i EI The t a b l e s a r e f o r p a r a b o l i c h i n g e l e s s a r c h e s h a v i n  arch. ratios  p  ~,  of o  variable EI. f o r fe> = 7«  f k ~,  constant  EI or a prescribed  6 4 3 a r e g i v e n f o r fi = 3 a n d 5 w i t h  Values Also  with  the tables  some  contain magnification f a c t o r s  f o r maximum moments a t e l e v e n  points  i n t h e a r c h , when t h e  arch  load.  Although the given  are  Is loaded good o n l y  with  a uniform  f o r parabolic hingeless  or a p r e s c r i b e d v a r i a t i o n  from a parabola  i n moment  of i n e r t i a ,  with  constant  i n E I , t h e t a b l e s may b e  extended t o other h i n g e l e s s different  arches  arches  tables EI  reasonably  whose s h a p e s a r e n o t t o o  and t o a w i d e v s L r i e t y o f v a r i a t i o n  provided  these  v a r i a t i o n s are not  unrealist ic. The p o s s i b i l i t y d e f l e c t i o n theory  of using  i s based on the f a c t  showed t h e h o r i z o n t a l t h r u s t imately  superposition i n the that c a l c u l a t i o n s  a c t i n g on t h e arch  t h e same e i t h e r b y t h e d e f l e c t i o n t h e o r y  was  approx-  or the  iii elastic  theory.  independent bending  of  of an  Because of t h i s , d e f l e c t i o n and  the  the  a r c h is. l i n e a r .  horizontal  differential  thrust  becomes  equation  Thus s u p e r p o s i t i o n  may  for  be  used. The culation. by the  differential  Instead, the  e q u a t i o n was  solutions  a numerical procedure of c o n j u g a t e beam c o n c e p t .  hot  i n the  successive This  tables  procedure  e l e c t r o n i c computer, t h e  the  British  of  approximation, the vertical theory the  Columbia.  were z e r o .  a n a l y s i s . In. s u b s e q u e n t  arch from previous  a p p r o x i m a t i o n as  This  such l e d to  was  using  conveniently  ALWAC I I I E , first  of  and  deflected  assumed.  at  cycle  horizontal  the  elastic shape  the  deflection  theory.  indicated deflection may  be  that  the  error  t h e o r y was  small,  used f o r a n a l y s i s  shown i n t h i s  introduced and  the  of f l e x i b l e  by  the  thesis  linearized  influence arches.  of  Successive  a s o l u t i o n b a s e d on  T h r e e n u m e r i c a l examples  cal-  calculated  r e p r e s e n t e d the  cycles,  analysis  was  In the  programme assumed t h e  deflections  were  approximations,  programmed f o r an University  convenient f o r  line  method  In the  presenting  this thesis  r e q u i r e m e n t s f o r an  of  British  it  freely available  agree that for  Columbia,  that  copying  gain  shall  Department  by or  not  of  advanced degree at  his  s h a l l make  for reference  and  study.  I  for  Sept. 3 0 ,  extensive be  by  a l l o w e d w i t h o u t my  Columbia,  further  of t h i s  the  Head o f  thesis my  It i s understood  of t h i s thesis  C i v i l Engineering  1958  copying  granted  representative.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a . Date  University  Library  publication be  the  of  the  p u r p o s e s may  Department o r  fulfilment  I agree t h a t  permission  scholarly  in partial  for financial  written  permission.  iv  TABLE. OP  CONTENTS, Page  CHAPTER CHAPTER  I.  INTRODUCTION  1  I I . DIMENSIONAL ANALYSIS OF A HINGELESS ARCH  CHAPTER I I I .  THEORY.  CHAPTER  IV.  METHOD OF SOLUTION  CHAPTER  V.  CHAPTER  VI.  6 lij. 23 .  USE OF APPENDIX  38  NUMERICAL EXAMPLE  BIBLIOGRAPHY APPENDIX I .  £l PROGRAMME FOR ANALYSIS OF  SYMMETRICAL.  HINGELESS ARCH UNDER VERTICAL LOAD APPENDIX  3 1  . . .  £3  I I . INFLUENCE TABLES OF MAGNIFICATION . FACTORS  APPENDIX I I I .  TYPICAL INFLUENCE L I N E DIAGRAMS  7£ 119  APPENDIX  I V . TABLES OF MAXIMUM MOMENT  12£  APPENDIX  V. CURVES. OF MAXIMUM MOMENT  129  V  ACKNOWLEDGEMENT  The  author  wishes t o express  h i s thanks t o h i s  adviser,  D r . R.F. H o o l e y , f o r h i s v a l u a b l e  constant  guidance.  to  work u n d e r  indebtedness University  months o f  I t was a g r e a t  ;his s u p e r v i s i o n . to the staff  experience  The a u t h o r  and p l e a s u r e  also expresses h i s  o f t h e Computing C e n t r e  of B r i t i s h Columbia f o r t h e i r  help.  of this  195>8  and was s p o n s o r e d f o r t h r e e months b y t h e  a s s i s t a n c e was g r e a t l y Also this  r e s e a r c h was e x t e n d e d i n t o t h e summer  Their  financial  appreciated.  r e s e a r c h was j o i n t l y done b y Mr. T . E .  and t h e a u t h o r .  As such,  the materials  covered i n  t h i s t h e s i s a r e a l s o i n c l u d e d i n Mr. T . E . P e l t o n ' s thesis It  submitted  was a g r e a t  September,  at t h e  Part  N a t i o n a l R e s e a r c h C o u n c i l o f Canada.  Pelton  s u g g e s t i o n s and  to the University of B r i t i s h  pleasure  Columbia.  t o work w i t h Mr. T . E . P e l t o n .  1 9 ^ 8 . .  Vancouver, B r i t i s h  MasterTs  Columbia.  VI  NOTATIONS  °^  Distribution  P  S t i f f n e s s f a c t o r of the arch. Vertical  %  o f average E I along  component o f d e f l e c t i o n .  H o r i z o n t a l component o f d e f l e c t i o n .  $  Magnification factor.  0, ©j_,©|>  Angle which the tangent t o arch horizontal  A©  Change i n 0 due t o moment. Modulus o f e l a s t i c i t y .  H  Horizontal thrust.  I  Moment o f i n e r t i a .  I  A v e r a g e moment o f i n e r t i a .  K  Constant  Mp  D e f l e c t i o n theory m^  (  E l a s t i c theory  b e n d i n g moment.  b e n d i n g moment.  P  Concentrated  w  i n t e n s i t y o f l o a d on t h e a r c h .  w  a x i s makes w i t h  line.  E  Mg,  the arch r i b .  d  Dead l o a d p e r u n i t  V  Vertical  f  Rise  L.  Span  x, . y  load.  length.  shear.  o f arch.  Rectangular  co-ordinates.  a  CHAPTER I . INTRODUCTION  When a d e s i g n e r h a s t o a n a l y z e  an a r c h , he h a s  at h i s d i s p o s a l two t h e o r i e s ,  namely t h e " e l a s t i c "  theory  and  Fundamentally,  i s no  the " d e f l e c t i o n " theory.  difference  between t h e two, except  both theories  being based  elasticity.  is  due t o t h e i r a s s u m p t i o n s  effect rigid it  theory  arches.  takes  used  The d i f f e r e n c e  assumes t h a t  on i n t e r n a l f o r c e s  into  t h e degree o f accuracy,  on t h e a s s u m p t i o n s o f t h e t h e o r y  of  elastic  i n t h e i r degree o f accuracy  on t h e e f f e c t o f d e f l e c t i o n .  and i s u s e d  the deflections  i n the design of f l e x i b l e  even t h o u g h s m a l l ,  combine w i t h  1  The  d e f l e c t i o n s have n e g l i g i b l e  The d e f l e c t i o n t h e o r y account  there  i n the analysis of i s more a c c u r a t e  because  o f t h e a r c h , and i s  a r c h e s , where  deflections,  a large h o r i z o n t a l thrust to  2  c a u s e l a r g e and i m p o r t a n t An  stability.  similar to a straight t h r u s t . I n both  in  the  loads  and an  axial  d e f l e c t i o n o f t h e s t r u c t u r e s produces  moments w h i c h i n t u r n p r o d u c e f u r t h e r  The r a t e o f i n c r e a s e may be r e p r e s e n t e d b y an  s e r i e s w h i c h may o r may n o t c o n v e r g e .  s t r u c t u r e w o u l d be u n s t a b l e .  value  lateral  and moments i n c r e a s e u n t i l t h e s t r u c t u r e s come  equilibrium.  infinite  a  I n many r e s p e c t s , i t s b e h a v i o r i s  beam u n d e r b o t h  cases,  i n bending  deflections,  moments.  arch under a h o r i z o n t a l t h r u s t i s e s s e n t i a l l y  problem o f e l a s t i c  increases  secondary  Just  of axial thrust f o r a straight  I f i t diverges,  as t h e r e  is a  beam, t h e r e  critical  i s a l s o f o r an  arch a .'critical h o r i z o n t a l t h r u s t , under which the arch becomes u n s t a b l e ,  and moments and d e f l e c t i o n s t e n d t o i n f i n i t y . >  While the a n a l y s i s of a s t r a i g h t involves, only a simple  beam i s s i m p l e ,  differential  equation,  o f a h i n g e l e s s a r c h by t h e d e f l e c t i o n t h e o r y because  i t involves five  unknowns.  The r e s u l t i n g  complicated the  i n form.  i s very  equations  differential  equation  theory  difficult,  i n five i s extremely  does n o t c o n s i d e r  change i n g e o m e t r y o f t h e a r c h , and a s s u c h t h e r e  stability  criterion.  rigid  arch.  times  imperative.  For this  To on b e n d i n g less  the analysis  simultaneous  The e l a s t i c  since i t  This theory reason,  illustrate  moment  only f o r a very  the d e f l e c t i o n theory  numerically the effect  i n an a r c h ,  arch i n f i g u r e ( l . l ) ,  is valid  i s no  i s some-  of d e f l e c t i o n  consider the parabolic hinge-  with dimension  a n d l o a d as shown.  3  F i g u r e (1.1)  According to the e l a s t i c V  EL  the  7 -  1  =  8  moment  MDL=  2k  l  P>  =  2  ^  P-  k l  0.192  feet  quarter  MD  §  t  h  deflection  e  V  deflection  ^  respectively.  D  L  =  theory,  and h o r i z o n t a l  are -0.396  The b e n d i n g  kip, H =  18.88  feet  moment  and  at the  point i s :  = jT(V x  - wx2J-  (-MLE  = |^17.82 ( $ 0 ) -. 18£2 ($of}-  +  y  at t h e q u a r t e r p o i n t  = (MgL + A M J + ^ V  E L  B  = -22.18 f t . - k i p ,  ft.-kip,  -30.82  2j? k i p , a n d t h e v e r t i c a l deflection  theory,  .06(^0+.192)  E  L  + A v)(x + I  +  Hy^+jv £ E I  ) - wx -i2±Jj +AV(x+g)  ( 2 2 . l8+2£ ( 2 ^ ) ) J +j±7.82  -.18^2 ($0)  -  ^MJ  ~ H(y+ ^ ) wj£ -  (. 192)  ( .192)-2^(-.396)-8.61^/  H  =  [659.30  =  1 8 . 9 2  The  61^7.18]  -  two  terms r e p r e s e n t the  quarter point. difference due  as  While  i s of the  were v e r y  illustrated. percent  n i z e d the the  over the  series  M e l a n and E n g e s s e r b e t w e e n 1 9 0 0 and 1 9 3 1  In  193^,  two  elastic  variation  are  the  large, last  the  geometry  of  w became a p p r e c i a b l e  i n c r e a s e was  of the  of d e f l e c t i o n of papers  1 9 0 6 and  and  their  approximately  theory. theory of  on t h e  on t h e  a p p e a r and  i n 193^^  or h i n g e l e s s arches  arches  recog-  internal forces  d e f l e c t i o n theory  by  were p u b l i s h e d  in 1 9 2 £ . ^ 1 » 2 » 3 » 4 ) "  Freudenthal presented  hinged  at  Even though  V,  case, the  began t o  and F r i t z  arch.  w i t h H,  investigators  importance  arch. A  in  numerical values  s m a l l compared w i t h t h e  In this  Early  e l a s t i c moment  same o r d e r o f m a g n i t u d e as t h e  arch, t h e i r products  fifty  in  their  to d e f l e c t i o n o f the  deflections the  [ 6 . 8 0 ]  ft.-kip.  first  term  -  Kasarnowsky  a d v a n c e d M e l a n ' s work f u r t h e r . a method o f a n a l y s i s ^ ^ with  for  a p a r a b o l i c a x i s and  o f moment o f i n e r t i a I = I g S e c S .  In a  a  later  (8) paper,  he  extended  this  work t o i n c l u d e t h e  effect  of  plasticity  i n concrete arches.  U n l i k e M e l a n and  Dischinger  i n 1 9 3 7 advanced the  d e f l e c t i o n theory to i n c l u d e  both  elastic  equation  and  plastic  i n t e r m s o f an  arches, infinite  others,  (9)  and  s o l v e d the  convergent  resultant  series.  ~' Numbers i n b r a c k e t r e f e r t o r e f e r e n c e s i n Bibliography. c  Using  approximately developed  t h e same a p p r o a c h as D i s c h i n g e r , S t e r n  a method u t i l i z i n g  the F o u r i e r series  later  f o r the  (10)  solution extended  •k wa, H i s work was (11) i n 19^1 b y L u t o i n c l u d e h i n g e l e s s a r c h e s .  o f a two h i n g e d  elastic  arch.  The s e r i e s methods h a v e t h e a d v a n t a g e o f p l a c i n g no tion  on v a r i a t i o n In  zontal tion  is valid  simplify the  or loadings.  a l l o f t h e methods m e n t i o n e d above, t h e h o r i -  component  i n Chapter  i n moment o f i n e r t i a  restric-  o f d e f l e c t i o n was n e g l e c t e d .  only f o r very f l a t  III.  But t h i s  arches,  assumption  assump-  as w i l l be shown  was made I n o r d e r t o  and i n t e g r a t e t h e d i f f e r e n t i a l  complexity  This  equation. N a t u r a l l y  of the d i f f e r e n t i a l equation  l e d other  investigators  to consider the p o s s i b i l i t y  integration.  I n 19if8., C h a t t e r j e e i n h i s d o c t o r a l t h e s i s  developed  of numerical  a m e t h o d f o r a n a l y s i s o f two h i n g e d  a r c h by t h e  (12) use  o f Newmark's n u m e r i c a l  American Concrete  Institute  procedure.  ;  I n 195>0, t h e  Committee on R e i n f o r c e d Con(13)  crete  proposed  of these  are based  approximation. finite  a method f o r h i n g e l e s s a r c h e s . on a n u m e r i c a l p r o c e d u r e  A direct  differences  was  and b e n d i n g equation. with  suggested  b y H i r s c h and Popov i n a  The d e f l e c t i o n s  a set o f simultaneous  moments a r e f o u n d  from  the second  The n u m e r i c a l methods may  any v a r i a t i o n  of successive  method u s i n g t h e e q u a t i o n s o f  p a p e r p u b l i s h e d i n 19$$^^^ are o b t a i n e d by s o l v i n g  Both  i n the arch equations, difference  be a p p l i e d t o an a r c h  i n r i b p r o p e r t i e s and l o a d i n g , and a l s o  t h e y may a c c o u n t f o r h o r i z o n t a l d e f l e c t i o n the  deflection  high rise  i s deemed n e c e s s a r y ,  ratio.  i n t h e amount  such  The d i s a d v a n t a g e s  of the arch i f  as i n a r c h e s  of numerical  with  methods l i e  o f work r e q u i r e d .  A valuable  d i s c u s s i o n o f t h e many a p p r o x i m a t e  methods f o r d e f l e c t i o n  t h e o r y moment i s c o n t a i n e d  presented  G - a r r e l t s , and H e d r i c k  by Hardesty,  nection with the design  o f t h e Rainbow A r c h  i n a paper  etc. i n conBridge  over t h e  (15") Niagara Chart  River.  In  195>3  . Rowe d e v e l o p e d  f o rstress i n flexible  method f o r s t e e l  arches  steel  arches.  an A m p l i f i c a t i o n ( 1 6 )  using the interaction  Another d i a g r a m s was  ( 1 7 )  given  i n a paper by M i k l o f s k y Written  theory  ini . d i f f e r e n t i a l form, t h e d e f l e c t i o n  i s u n i q u e , b u t methods o f i n t e g r a t i n g t h e d i f f e r e n t i a l  equation  may b e d i f f e r e n t  order to simplify the  and S o t i l l o .  exception  because terms a r e ne.glected i n  the integration.  But u n f o r t u n a t e l y ,  with  o f t h e a p p r o x i m a t e m e t h o d s , a l l t h e methods  developed  so f a r s t i l l  require the designer  elaborate  analysis before  a solution  t o do a l o n g a n d  i s available.  Also  t h e t i m e a n d l a b o r r e q u i r e d t o d e t e r m i n e t h e maximum a n d minimum moments at any p o i n t are  i n the arch by t r i a l  and e r r o r  tremendous. An i n f l u e n c e l i n e m e t h o d o f a r c h  d e f l e c t i o n theory to f a c i l i t a t e  i s presented  i n this  a n a l y s i s by the  thesis.  I n order  t h e work, d l m e r i s i o n l e s s t a b l e s o f m a g n i f i -  c a t i o n f a c t o r s which i n c l u d e t h e e f f e c t s  of horizontal  7 deflection  and  shifting  o f l o a d are p r o v i d e d .  The  deflection  theory bending  moments a r e o b t a i n e d by  theory bending  moments by t h e m a g n i f i c a t i o n f a c t o r s .  such,  the  of the  designer  elastic  i s r e q u i r e d t o do  theory.  w i t h a wide v a r i e t y w i t h any  loading.  m u l t i p l y i n g the  no .ana l y s i s  T h i s method may  be  Although  the  t o be  true  different  also f o r h i n g e l e s s arches from a true parabola.  ations r e q u i r e d f o r these  The  great  t a b l e s , was  computer were t h e n  so t h a t t h e y may  be  put  reduced  into  be  these  ratios  Chapter  I I I by  the  development  the  dimensionless  s h o u l d be  used.  was  valid  s m a l l and  I n Chapters  for practical  IV  too  obtained r a t i o s i".  dimensional  chapter  will  show  of the i n f l u e n c e shown i n  deflection  and V,  t a b l e s were s e t up  that the  assumed  calcul-  Data  The  validity  it will  and how  examples i n C h a p t e r  i n d i c a t e d t h a t e r r o r i n t r o d u c e d by t h e t h e o r y was  good  electronic  to dimensionless  of a l i n e a r i z e d  Three numerical  only  amount o f  s u p e r p o s i t i o n i s u s e d w i l l be  theory f o r arch analysis. shown how  The  and  reasonably  Columbia.  general use.  were a r r a n g e d .  l i n e m e t h o d and how  inertia  done on t h e  a n a l y s i s o f a h i n g e l e s s a r c h i n the next how  arches  whose s h a p e s a r e not  computer a t t h e . U n i v e r s i t y o f B r i t i s h from the  beyond t h a t  given t a b l e s are  f o r h i n g e l e s s p a r a b o l i c a r c h e s , t h e y may  As  a p p l i e d to  i n v a r i a t i o n o f moment o f  elastic  linearized  be  they VI  deflection  i n f l u e n c e method o f a n a l y s i s  application.  CHAPTER I I DIMENSIONAL ANALYSIS  The dimensionless the  purpose o f t h i s ratios  chapter  i n the analysis  i s to introduce the of f l e x i b l e  a r c h e s by  i n f l u e n c e l i n e method and t o a s c e r t a i n t h e n a t u r e  o f the  magnification factor,  even though i t s a l g e b r a i c r e l a t i o n s h i p  may n o t b e o b t a i n e d .  These d i m e n s i o n l e s s  arguments i n t h e g i v e n The  analysis  As  o f arches by t h e i n f l u e n c e l i n e  that the s t i f f n e s s  w i l l be shown, t h i s  i s t h e same as i n t h e e l a s t i c  thrust  stiffness,  acting there  method theory,  o f t h e a r c h needs t o be c o n s i d e r e d .  stiffness factor  i s measured by t h e  m a g n i t u d e and d i s t r i b u t i o n o f E I , t h e s p a n , tal  formed the  tables.  in the d e f l e c t i o n theory except  ratios  on t h e a r c h . corresponds  and t h e h o r i z o n -  F o r a p a r t i c u l a r value of  a definite  8  influence line.  In  9  the  e l a s t i c theory,  a unit  load across  influence  line  the arch;  corresponding  may be s i m i l a r l y arch,  an i n f l u e n c e l i n e  obtained  i s obtained  by m o v i n g  i n the d e f l e c t i o n theory, to a c e r t a i n s t i f f n e s s  b y moving  a unit  load  an  factor  across t h e  b u t at t h e same t i m e p u t t i n g a d e a d l o a d on t h e a r c h so  as t o m a i n t a i n  a constant  H.  This  w i l l be f u l l y  explained i n  Chapter I I I . But,  f o r the present,  case as shown i n f i g u r e  consider t h i s  (2.1).  P  L F i g u r e (2.1)  fundamental  10 The  parabolic hingeless  and  by i t s own dead w e i g h t . U n d e r t h e s e  in  the arch  considers  will  arch  deflect to point  the deflected  of P from l e f t  x  = Distance  of point  f  = rise  L  = Span o f a r c h .  cK  The moment M ,  at p o i n t  A'  springing,  A from l e f t  springing,  of arch.  = A v e r a g e o f p r o d u c t s o f Y o u n g ' s M o d u l u s and moment o f i n e r t i a . = D i s t r i b u t i o n o f average E I along  T h e s e v a r i a b l e s may be combined i n t o s e v e n ratios  which  D  load  = Distance  Av.  any p o i n t A  load.  Xp  EI  loads,  load P  quantities:  = Concentrated = Dead  A'.  by c o n c e n t r a t e d  shape o f t h e s t r u c t u r e ,  depends on t h e f o l l o w i n g P  i s loaded  arch  axis.  dimensionless  and t h e i r r e l a t i o n s h i p e x p r e s s e d a s :  P L  V EI  A v  .L  L  L  w d  L  .  or  In the e l a s t i c is A  theory,  the d e f l e c t i o n o f the arch  assumed t o be n e g l i g i b l e , and t h e moment Mg a t any p o i n t i s shown b y d i m e n s i o n a l a n a l y s i s t o be  J  J L = PL g / — , ^p_ , f , _ P _ 1 3  \  L  L  L  w  dL  )  (2.2)  11 The  h o r i z o n t a l t h r u s t H i n d u c e d by t h e l o a d s  sum o f H , h o r i z o n t a l t h r u s t  caused by dead l o a d ,  d  horizontal thrust ations  c a u s e d by l i v e  showed t h a t  differs  the values  l i t t l e from that  load  P.  H i s independent  arch,  b a s e d on t h e e l a s t i c  In the e l a s t i c  Preliminary  theory.  may be assumed t h a t and i t s v a l u e  and H , 1  o f H by t h e d e f l e c t i o n  of the e l a s t i c  i s the  calcultheory  As such, i t  of the d e f l e c t i o n of the theory  may b e u s e d .  theory,  H = P $(^> , X , V L  L  Prom e q u a t i o n s  ,«*) d  L  (2.3)  '  (2.2) a n d ( 2 . 3 ) ,  equation  (2.1a)  may be w r i t t e n a s :  The  square root  of  — — Av.  i s called the stiffness  factor of  E I  the  arch.  The a r i t h m e t i c  because p r e l i m i n a r y ness o f t h e arch rib  axis  given  ness o f an a r c h variation  investigations indicated that  w i t h a v a r i a b l e moment o f i n e r t i a  by t h i s  the s t i f f along i t s  E I , provided the  i n E I was n o t u n r e a s o n a b l e . (2.1) shows t h a t  t h e b e n d i n g moment  e q u a l t o Mg m u l t i p l i e d by a f u n c t i o n  called the magnification seven d i m e n s i o n l e s s ations  term,  a v e r a g e E I was c l o s e s t t o t h e s t i f f -  w i t h t h e same c o n s t a n t  Equation is  a v e r a g e E I was c h o s e n i n t h i s  showed t h a t  factor.  ratios.  This  w h i c h i s commonly ^ i s a function of  However, p r e l i m i n a r y  may be c o n s i d e r e d  calcul-  t o be i n d e p e n d e n t o f  the or  dimensionless  ratio  i f the d e f l e c t i o n  , i f this r a t i o i s very small, dL i s small. F i g u r e (2.2) i s a p l o t o f t h e  m a g n i f i c a t i o n f a c t o r <f> a t a number o f p o i n t s i n t h e a r c h versus  the dimensionless  r a t i o — £ — . I t c a n be s e e n t h a t d constant without t o o great e r r o r i f L  $ may be c o n s i d e r e d w  P  i s s m a l l near the o r i g i n .  Thus, f o r s m a l l  deflection,  T d  equation  (2.1) becomes  %  = ^ * (— *  The  \EI  function  > f A  v  L  ' ^  <*1  ( 2  i n equation  (2.if)  i n t h e Appendix.  these  t a b u l a t e d f u n c t i o n s may be u s e d f o r s o l u t i o n loading  t i o n theory  as d e v e l o p e d  b a s i s ' o f u s i n g these  i n t h e next  simplified  )  i s tabu-  be shown i n C h a p t e r V  c o n d i t i o n on t h e a r c h .  '^  )  lated  general  It will  >  L  L  as g i v e n  • f  how  o f more  The l i n e a r i z e d d e f l e c -  chapter  functions.  will  form the  13  CHAPTER I I I THEORY  Iritroduction The  b e h a v i o r o f an a r c h u n d e r l o a d  by  i t sdifferential  of  an a r c h b y t h e d e f l e c t i o n t h e o r y i s i n g e n e r a l  However,  equation.  i s best  i t w i l l be shown, t h a t  The e q u a t i o n f o r t h e a n a l y s i s  by simple  such t h e p r i n c i p l e o f s u p e r p o s i t i o n C o n s i d e r an element  figure  (3.1).  Under l o a d , on and  t h e element  t h e element  a r e shown  respectively  horizontal  may b e  valid  l i n e a r , and applied.  o f t h e a r c h r i b as shown i n  t h e element  i s at p o s i t i o n AB.  d e f l e c t s t o A'B'. i n i t s deflected  H a r e t h e moment, v e r t i c a l  acting and  Initially  non-linear.  assumptions,  i n most p r a c t i c a l c a s e s , t h i s , e q u a t i o n becomes as  studied  shear,  The f o r c e s  p o s i t i o n . M, V,  and h o r i z o n t a l  a t any s e c t i o n ; ^ and ^  components o f d e f l e c t i o n .  acting  thrust  are t h e v e r t i c a l  15  Figure  Equation  of E q u i l i b r i u m There are three  w is vertical,  equations of e q u i l i b r i u m .  t h e h o r i z o n t a l t h r u s t s at A»  Equating to  (3.1)  t h e sum  of vertical  are  f o r c e s on t h e  equal.  element  zero, wdx  or  + dv = 0 v i =  where  » represents e  -w  (3.1)  -r— dx  Another r e l a t i o n s h i p sum  and. B»  Since  o f moments about  p o i n t B?  i s obtained to  zero.  by  equating  the  1 6  dr\) - V ( d x + d$ ) + dM =  H.(dy  +  H(y»  + n j ) - V ( l + 3 ») + M«  0  or  There the  equations  a r e now two e q u a t i o n s  are not s u f f i c i e n t  t h e p r o b l e m becomes  o f t h e a r c h and H o o k e r s  doing  s o , t h r e e more e q u a t i o n s  with f i v e  unknowns;  f o r a solution.  Therefore,  and t h e d e f o r m -  law must be c o n s i d e r e d . involving  the f i v e  In  unknowns  obtained.  Equation  of  Deformation  Initially  t h e l e n g t h o f t h e e l e m e n t AB i s 2  (ds) After  ( 3 . 2 )  0  s t a t i c a l l y indeterminate  ation  may be  =  L  = (dx>  2  + (dy)  deformation, 2.  2  (dsi) =  (dx + d$ )  Prom t h e s e  +  (dy + dnj  two r e l a t i o n s h i p s  and n e g l e c t i n g r i b  shortening,  Since  deflections  are s m a l l , square  t e r m s may be  neglected,  and  J ' = -y*iv»  ( 3 . 3 )  17  In  (3.2),  figure  t a n (9 +  A  e)- = 1  +  S '  + ...  = 1 + y • nj  1  1+5'  tan(69 + 46) = y» + nj + yt r^i  and  tan  = y»  T h e s e two e q u a t i o n s  give the f a m i l i a r  expression  A© = ri» P r o m Hooke's  l a w and n e g l e c t i n g t h e e f f e c t  of  M a x i a l and s h e a r  deformation, ^  3-  s  e q u a l t o t h e change i n  curvature of t h e element.  -  <a.)'  cose  Hence  -ii EI  ^  A relationship from the condition that deflection supports five is  i n v o l v i n g H may the h o r i z o n t a l  a t one s u p p o r t  do n o t y i e l d .  unknowns  (3.1]-)  n" cos 9 be o b t a i n e d  component o f  i s known o r e q u a l t o z e r o i f t h e  This w i l l  provide f i v e  equations i n  and a r i g o r o u s s o l u t i o n f o r M, V, H,  p o s s i b l e b u t w i l l be e x t r e m e l y  Basic D i f f e r e n t i a l  ^,  and £,  complicated.  Equation  Differentiate  equation  (3.2) and e l i m i n a t e V  1  18 from the  equation,  M" - M' £  11  1+f  Numerical  + H ( y " + rt") - H(xL+_£lil i 1 +i'  claculations  showed t h a t  M  1 + J ¥hen may  neglected  side. Thus,  On  the  M"  Substitute  back into  compared t o  right  hand  (3.5a)  equation  equation  a similar  side, ^ '  )1"  (3-k)  deflection arches. the  an  (3.5)  arch.  of the  The  t e r m Hy"  i s small  =  The  as n o t e d  v a l u e s o b t a i n e d by  be  as  used without  great  the value error.  1.  = -w -Hy" (3-5)  o n l y be  equation f o r to  horizontal  neglected f o r  flat  a p p e a r t o be unknown i n  i n Chapter  elastic  same  (3.5b),  II,  calculations  o n l y a s m a l l d i f f e r e n c e between  the  such,  the  (3.5b)  differential  can  on  compared t o  t e r m Hy» '*\ " i s due  a r c h , and  showed t h a t t h e r e was  and  i s the  Hy'y"/^!  -w  equation  horizontal thrust H  e q u a t i o n , , but  theory,  >  to  + Hy"  into  (3.5a),  equation  simplifies  + H ( l + y»  Equation of  in ^  •  [EI cos eit"]" + H(l + y' bending  and  and h i g h e r t e r m s  + Hy«y"  n  small  1  substituting this be  #7_  was  +i*  A l s o , n e g l e c t i n g square  Hfy-t + ri») i " = Hy»  (3.5a)  ^  >  1  negligible.  = - w ( l + *«)  n  theory o f H by  and the  the  the  deflection  e l a s t i c theory  Because o f t h i s f a c t ,  H  is  may  19  assumed t o be i n d e p e n d e n t becomes  a linear  variable becomes  k  and  show how  >  a  n  equation ( 3 . 5 )  d  differential  equation  with  and a method o f s u p e r p o s i t i o n  s u p e r p o s i t i o n i s used,  consider the three  ( 3 . 2 ) i n the next  of loading i n figure  page.  s u c h /that t h e h o r i z o n t a l t h r u s t  I I a r e e q u a l t o H. L e t ^ , » ^ 2 >  to the d i f f e r e n t i a l In  £  3 1 1 ( 1  applicable.  are constants  2  fourth order  coefficients,  To cases  of ^  equation  k]_ and  i n Cases I  ^ 3 be t h e s o l u t i o n  corresponding  t o each  case.  case I , [EI c o s  6»t  1  + H ( l + yt  .]  1$^ n = - k ^  - Hy"  (3.6)  in  case I I , It  [EI c o s 6 4  n  1 1  + H ( l + y ? ) \f\  ..]  2  = -k w  z  2  -Hy"  2  (3-7)  and  i n case I I I , [EI  cos  e»t  + H ( l + y ' )»|  ]  3  =- -(w +w )--Hy 1  tt  2  (3.8)  Divide  ( 3 . 6 ) by k  equations  ( 3 - 7 ) by k  and e q u a t i o n  2  and  add t h e two t o g e t h e r ,  EI cos e  (M ia). +  \ %  k  2  +  H  (  i  +  y.)  V  / = - (  W l  + w) 2  k  - Hy"  l  k  2  j ( 3 . 9 )  20  CASE  II  Figure (3.2)  21  By  comparing  equations  (3.8) and ( 3 . 9 ) ,  l  k  k  2  MD3 = M D I+  S i n c e moment a n d *\ a r e p r o p o r t i o n a l t o e a c h  l  k  k  other,  2  or M  3D = *  l  M  Let  l  E  l  k  + ^2 1 E '2 M  (.3.10)  k  jq-]_-p; and n^-g be e l a s t i c  t h e o r y moment due t o W]_ and w  2  respectively. Thus, M  1E  =  k  lLE  = k  m  m^  2  S u b s t i t u t e t h e two r e l a t i o n s h i p s  % ^  and ^  value  a  = ' l r  e  M  in  ( 3  practical  case.  corresponding  -  1 : L )  t o the  They a r e t a b u l a t e d I n t h e  analysis. a p p l i c a t i o n of equation  w i t h h i g h t h r u s t may meet w i t h  such cases  (3.10),  m  magnification factors  appendix t o f a c i l i t a t e  arches  equation  1 E * >2 2 E  of H f o r the t o t a l  The  into  k ^ a n d k^ a r e v e r y  (3.11) t o  difficulty,  large.  because  When k]_ and k  2  a r e t o o l a r g e , t h e d e f l e c t i o n o f t h e a r c h may become s o b i g that t h e s o l u t i o n s are not v a l i d , true only f o rsmall d e f l e c t i o n .  f o r the theory  holds  Instead of increasing the  22 thrust  b y k-j_ and k , t h e t h r u s t may be i n c r e a s e d b y p l a c i n g 2  a dead l o a d on t h e a r c h , The  value  influence to  and e q u a t i o n  o f d e a d l o a d r e q u i r e d may be o b t a i n e d line  f o r H.  The above c o n c l u s i o n s  c a s e s o f more t h a n two l o a d i n g s ,  influence  (3.11) s t i l l  holds.  from t h e  may b e e x t e n d e d  a n d t h u s t h e method o f  l i n e may be u s e d f o r a n a l y s i s o f f l e x i b l e  a r c h e s. An  exact  and a c l o s e d f o r m s o l u t i o n o f  (3-5)  i s difficult  given  t o shape o f a r c h  the  and i m p r a c t i c a b l e ,  d e f l e c t i o n theory  moment,  magnification appendix.  Also  section of  To o b t a i n t h e  a numerical  a p p r o x i m a t i o n s was u s e d i n t h i s the n e x t daapter.  when c o n s i d e r a t i o n i s  axis, v a r i a t i o n i n cross  r i b , and d i s c o n t i n u i t y i n l o a d .  equation  procedure of successive  t h e s i s , as d e s c r i b e d i n  t h e next c h a p t e r  f a c t o r s were o b t a i n e d  will  show how t h e  and t a b u l a t e d  i n the .  CHAPTER I V . METHOD OF SOLUTION  Numerical  Procedure  The  d i f f e r e n t i a l equation f o r bending  arch derived i n the last  chapter  o f an  i s valuable to  how t h e p r i n c i p l e o f s u p e r p o s i t i o n may be u s e d , convenient procedure  In this thesis,  of successive approximation  electronic tions  f o r calculation.  computer was U s e d .  were c a l c u l a t e d  twenty e q u a l  4x  moment  the  beam method.  (ij..l). F o r t h e purpose  t h e a r c h was s u b d i v i d e d  intervals.  interval.  into  I n each i n t e r v a l , t h e  and E I were assumed t o b e c o n s t a n t .  equal t o t h e average  t o an  The moments and d e f l e c -  Consider t h e arch In f i g u r e integration,  but not  a numerical  adopted  by the conjugate  of numerical  indicate  Their values  o f moments o r E I at t h e two ends o f  The p l o a d s were assumed t o a c t a t t h e  23  Figure  (4.1)  middle  o f each i n t e r v a l ,  r e p l a c e d by the  same  equivalent  and  P  f o r uniform  loads  loads, they  a c t i n g at t h e m i d d l e  of  interval. In the f i r s t  cycle,  the v e r t i c a l  and h o r i z o n t a l  components o f d e f l e c t i o n were assumed t o be Deflection the  were  and  elastic  bending  theory  moment t h u s  analysis.  zero.  calculated  I n subsequent  represented  cycles,  the  deflected  shape o f t h e  s t r u c t u r e from previous  analysis  assumed.  This process  continued u n t i l  agreement  was  o b t a i n e d between the  calculated ential  shape.  equation  assumed s h a p e o f t h e  In Chapter  f o r bending  but of  since they any  are  solution  iterations bility was  procedure,  of the  factor  concluded  arch.  twelve,  the  simplify  the  not  affect  i n Chapter  into  the  III.  arches,  account,  The  number o f  this  flexinumber  and f o r o r d i n a r y a r c h whose m a g n i f i c a t i o n  i s less than  1.5,  five  or l e s s  Iterations  were  sufficient. Except  f o r one  f e a t u r e , the  f o r m e d d u r i n g e a c h c y c l e were t h e theory. arose  the  In  computations  same as  c a l c u l a t i n g the d e f l e c t i o n  q u e s t i o n o f whether the  e l e m e n t be f r o m t h e  initial  i n the  of the  r o t a t i o n arm  shape o r t h e f i n a l  In  linearity  depended o n t h e  For very f l e x i b l e  small  equation.  t e r m s were t a k e n do  the  differ-  a r c h , a number o f  r e q u i r e d f o r convergence  of the  over  as  these  small they  a r c h and  I I I , i n deriving  n o n - l i n e a r t e r m s were n e g l e c t e d t o the numerical  a close  was  perelastic  arch, of  there  an  shape  of  26  the  arch.  I t was d e c i d e d  t h a n e i t h e r o f them; t h i s ation of figure  (i|..2).  d e c i s i o n was s u p p o r t e d b y  Initially  element  i s x ^ and f i n a l l y  of t h i s  small  element  t o use t h e average geometry  x^.  area  i s better + x±)  consider-  t h e r o t a t i o n arm o f an  D e f l e c t i o n due t o r o t a t i o n  i s equal t o t h e a r e a under t h e curve AA».  Figure  This  rather  2)  a p p r o x i m a t e d b y a t r a p z o i d whose a r e a i s  equal to i  (x  electronic  computer t o u s e t h e a v e r a g e g e o m e t r y i n s t e a d o f  the  initial  f  ( A 9).  or the f i n a l  one was  included  insignificant,  i n t h e programme.  and t h e  operation  was  the  g e o m e t r y was u s e d i n c a l c u l a t i n g b e n d i n g  final  easily  The t i m e r e q u i r e d by t h e  Of  course, moments.  27  A f t e r the less  arch i s s t i l l  To f i n d be  d e f l e c t i o n s have b e e n assumed, t h e  cut  the three  a s t r u c t u r e w i t h t h r e e redundant f o r c e s . redundants,  at t h e m i d d l e  each h a l f of the  t h r u s t , and formed the  shear  internal  and  The  s t r u c t u r e was  vertical  a r c h due  f o r c e s at the  crown.  deflections  Electronic  equations  With these  e a s i l y o b t a i n e d by  c e d u r e was  stated before, p e r f o r m e d on  available  an  the  t w e n t y one  As  channel  and  gave t h e  of  three  three forces calculated  conjugate  done so t h a t  aforementioned  beam method.  on f o u r  the  stored independently any  ing the o t h e r s .  of the  t h r e e may  A f t e r this, data  computer p r o c e e d s t o  are  Input  c a l c u l a t e the  bending  This  80 t o 9i+.  into  changed without  i s s t o r e d , on  The  working  of each o t h e r . T h i s be  Alwac  page.  arch, the v a r i a t i o n of P loads  pro-  Columbia,.  i s shown o n n e x t  i s s t o r e d i n m a i n memory c h a n n e l s  and  numerical  computer, t h e  programme o p e r a t e s  arch axis,  m a c h i n e and  the  cases  equations  at the U n i v e r s i t y o f B r i t i s h  shown, t h e g e o m e t r y o f t h e  along the  the  crown  crown moment,  a r c h were  electronic  f l o w d i a g r a m f o r t h e programme  channels  at t h e  Computer  As  I I I E,  and  simultaneous  s o l u t i o n o f these  Rotation,,  These i n d i v i d u a l  known, moments a t o t h e r p o i n t s i n t h e and  (I4..I).  to P loads  of three  assumed t o  deflection  were c a l c u l a t e d .  coefficients  continuity.  the  as shown i n f i g u r e  horizontal deflection, for  hinge-  EI the was affect-  a coded moment,  command,  28  FLOW CHART  Form A  Copy x, y t o x y and x , y  S  IT  Input l o a d i n g 20 v a l u e s  Input E I 10 v a l u e s  I n p u t x, y , s 10 v a l u e s o f each  a >  a  F i n d crown r o t a t i o n a n d d e f l e c t i o n f o r H r 10  f  Make c h a n n e l of zeros  f  Make c y c l e number z e r o  Add one t o c y c l e number  <  F i n d crown r o t a t i o n and = 1 deflection for V c  F i n d crown r o t a t i o n a n d deflection for M - 100  Type o u t 20 v a l u e s of 20 v a l u e s o f  $  j\  c  Normal  Jump F i n d crown r o t a t i o n a n d deflection for loading <<  Type o u t 2 0 moments  Switch 2  Jump  Normal  _____  Find coefficients f o r3 equations of c o n t i n u i t y  Switch 1 Type o u t c y c l e number  I t e r a t e once I n s o l v i n g f o r M , H, a n d V_  _  Type o u t M , c  H, a n d V  =31  Stop  c  Form new v a l u e s o f x  Wait f o r i n p u t o f one number, Q  Q = 1  a»  a  n  d  x  f ' f y  Find t o t a l values f o r M, $, Yj  Q » 1  Figure (4.3)  29  h o r i z o n t a l t h r u s t , and v e r t i c a l required f o r continuity.  s h e a r f o r c e a t t h e crown  A f t e r t y p i n g these  computer c o n t i n u e s , t o c a l c u l a t e t h e b e n d i n g  three out, the moments a t  t w e n t y one p o i n t s i n .the a r c h a n d t h e n h o r i z o n t a l and v e r t i cal  deflections  (4.1).  figure  at twenty points, i n t h e a r c h  On c o m p l e t i o n  of this  o p e r a t i o n , t h e programme  command i n C h a n n e l 9 0 .  stops  on a . l b  panel  s e t t i n g s , as shown i n t h e f l o w  Start  switch to Start  appropriate  a n d t h e n N o r m a l , t h e computer  t h e moment o r d e f l e c t i o n o r b o t h ,  the  calculated  types  and p r o c e e d s t o u s e  shape o f t h e a r c h as a s u b s e q u e n t  m a t i o n f o r d e f l e c t i o n and c o n t i n u e s  to other  approxi-  cycle o f  cal-  I f o n l y t h e d e f l e c t i o n t h e o r y moment I s d e s i r a b l e ,  the Normal-Start computer  With  d i a g r a m and N o r m a l -  out  culation.  as shown i n  s w i t c h may be i n i t i a l l y  continues  s e t at S t a r t  i t s sequence o f o p e r a t i o n without  At  t h e end o f e a c h c y c l e , t h e c o m p u t e r t y p e s  of  c y c l e s t h a t t h e i t e r a t i o n has been  and t h e stopping.  o u t t h e number  performed.  A f t e r t h e programme h a d b e e n d e b u g g e d , t h e f i r s t results  were c h e c k e d on a d e s k c a l c u l a t o r t o i n s u r e t h e  correctness its  o f t h e programme.  specification  Influence  The c o m p l e t e programme  i s p r o v i d e d i n Appendix I f o r r e f e r e n c e .  Tables I n Chapter  I I I , I t was shown how  a modified influence  l i n e method m i g h t be u s e d f o r d e f l e c t i o n t h e o r y To  with  facilitate  this  analysis.  work, i n f l u e n c e t a b l e s o f m a g n i f i c a t i o n  f a c t o r s have been c a l c u l a t e d  and t a b u l a t e d i n A p p e n d i x I I  30  and  III.  T h e s e t a b l e s were s e t up by moving  the  arch, but v a r y i n g  t h e dead l o a d so as t o k e e p H  The. m a g n i f i c a t i o n f a c t o r s were o b t a i n e d d e f l e c t i o n theory theory that  moments.  a P load  moments, by t h e i r The t a b l e s  constant.  by d i v i d i n g t h e  corresponding  are i n dimensionless  r e s u l t s may be u s e d f o r s i m i l a r  across  arches.  elastic f o r m , so  The a d v a n t a g e  of t a b u l a t i n g m a g n i f i c a t i o n f a c t o r s , i n s t e a d of d e f l e c t i o n theory  moments, a r e t w o f o l d .  axes d i f f e r ably  little  assumed t h a t  Firstly,  from a t r u e  f o r arches  parabola,  i t may be  whose reason-  t h e i r magnification f a c t o r s are approxi-  m a t e l y t h e same, e v e n t h o u g h t h e i r M-g a r e d i f f e r e n t . Secondly,  f o r p a r a b o l i c arches  f r o m t h e two v a r i a t i o n s g i v e n to obtain The  whose v a r i a t i o n  i n the.tables, i t i s possible  a good i n t e r p o l a t i o n o f m a g n i f i c a t i o n f a c t o r s .  Use o f Appendix i n t h e next  fully.  In EI differs  chapter  will  explain  this  CHAPTER  V.  U S E OF APPENDIX  A p p e n d i x I - Programme f o r A r c h A n a l y s i s The  c o m p l e t e programme  used i n the i n v e s t i g a t i o n used t o analyse the be  programme  other  and i t s s p e c i f i c a t i o n  are provided f o r reference.  cases  not covered  i s a l s o good f o r e l a s t i c  economically  used to c a l c u l a t e  in this theory  elastic  I t may be  thesis.  analysis.,  theory  Since i t may  moments  and  deflections.  A d e t a i l e d d e s c r i p t i o n o f t h e programme and  how  i t s h o u l d be u s e d a r e g i v e n i n t h e s p e c i f i c a t i o n .  Appendix I I - Tables  of Magnification Factors  These a r e i n f l u e n c e t a b l e s f o r m a g n i f i c a t i o n f a c t o r s of bending  moments.  As m e n t i o n e d , t h e y were o b t a i n e d by  m o v i n g a P l o a d o n t h e a r c h and v a r y i n g t h e d e a d l o a d so  31  32  t h a t H may the  law  be  constant.  For the  case  of v a r i a b l e EI  of d i s t r i b u t i o n f o l l o w s t h a t of f i g u r e  arranging  the  dimensionless  ratios,  i t was  given,  (5.1).  found  In  convenient  30+  Zc-  ±1_ £1 1-2 2.0OS  c  H  • lo  1  -2o  X  6  1  1  -3o  .4c  r— .sro  /-  Figure  for  i n t e r p o l a t i o n to l e t  The  rest  o f t h e t a b l e s s h o u l d be  illustrative loads  and  example,  (5.1)  self-explanatory.  c o n s i d e r the  d i m e n s i o n s as  shown.  arch  in figure  As (5«2)  an with  33  _ _  2oo'  •———  (5-2)  Figure  Assume t h a t u n d e r t h e g i v e n able  »f—  l o a d s , H = 25 k i p s .  to obtain the d e f l e c t i o n theory  point.  With-'J•=? i  ,  0.25,  J : =  X P l  moment  the designer  t o o b t a i n tf, and  fa. M  where m^_, a n d m ^ point  enters  D  n  L  the appropriate  d e f l e c t i o n theory  e  = h  m  l E  +  table  and column  moment  ^2 2E m  a r e e l a s t i c b e n d i n g moments a t t h e q u a r t e r  due t o P-j_ and P  conditions,  T  at t h e q u a r t e r  = .10, ;fP2 = O.I4.O, and  ~f ^=5,  I t Is d e s i r -  2  r e s p e c t i v e l y . F o r more g e n e r a l  i n f l u e n c e l i n e may b.e e a s i l y  drawn f r o m  loading  these  tables. Variation  i n Rise The  other  rise  Ratio  tables  ratio,  contain only four r i s e  r a t i o s . F o r any  i n t e r p o l a t i o n i s necessary.  T h i s may be  3k  done g r a p h i c a l l y by  plotting  is tedious. Instead, recommended.  The  $ versus'—  a straight  line  e r r o r w o u l d be  .  However,  interpolation is  s m a l l , as  i n d i c a t e d by  s m a l l d i f f e r e n c e s i n $ between c o n s e c u t i v e — L Variation  i s impossible  variations of EI  to  o f y6 and  shape,  considerable v a r i a t i o n  a small effect  where t h e  on t h e  For high values ten percent.  b e t w e e n t h e two  with  interpolate  ratios.  tabulated,  an  a l l the p o s s i b l e t a b l e s show t h a t f o r  influence lines  are of the  of EI  a r c h r i b has  along the  o f <j/>. They a r e  o f yB , say f$ = 5 , However any  almost  other v a r i a t i o n  withjudgment the  magnidesigner  i n E I between t h e  two  accuracy.  any  Factor  other values  accurate value secant  of  different  from  those  o f m a g n i f i c a t i o n f a c t o r s may curves  i n figure  (5.3)  next  P r e l i m i n a r y i n v e s t i g a t i o n s showed t h a t t h e J3  only  i n EI  u s i n g the  of $ v e r s u s  same  the d i f f e r e n c e  o b t a i n e d by page.  low  identi-  s m a l l e r d i f f e r e n c e i n the  f o r other v a r i a t i o n  in Stiffness For  the  Because o f t h i s ,  reasonable  Variation  But  two  values  would g i v e  fication factor. may  consider  i n designs.  values  c o u l d be  the  i n EI It  cal.  this  followed approximately ^ - Sec.f  a secant  v a r i a b l e but  each p a r t i c u l a r  v a l u e o f k s h o u l d be  The  the curve  law,  ky6j  where k i s a d i m e n s i o n l e s s case.  on  be  n e a r l y a constant chosen  in  from  35  36  the  nearest  should  the  designer's  the  two.  ner  obtains the  locate  For  the  k  to p = 3 . 5 of 3 . 5 , =$. may  fe  tabulated  value  example, value  curve  of V  p o i n t e d out  lies  b e n d s up much f a s t e r t h a n t h e  If  k  curve  = i| i n s t e a d  almost  as t h e  the  f a r beyond  and  the  true magnification factor secant  curve  beyond  7.  Appendix I I I  In arch  - T y p i c a l Influence  design,  the  a r e u s u a l l y at t h e  Typical  influence lines  in this  appendix.  lines for his influence gross  is  this  and  same. I t  are  curve  and  = 3,  desig-  k"»s  e x t r a p o l a t i o n too  valid  The  = 3  range i s not  that  o f $/.  ?s.  between  a v e r a g e o f k fovp  given  A=  Following  desired value  that  approximately  tables f o r p  (5.3).  t h e two  tabulated  a c a s e f o r f2 = 3 . 5 -  from the  Uses t h e  I n many c a s e s ,  be  of  in figure  designer  a v e r a g e o f two  Consider  locates the  the  or the  The  e r r o r and a l l the  on  f o r these designer  points  designer  the on  Diagram  sections i n a  hingeless  s p r i n g i n g , the. h a u n c h , and  an  draw s i m i l a r  i n Appendix I I . a partial  net  Crown.  influence By  check  o f i n t e r p o l a t i o n by  influence line the  the  s e c t i o n s have b e e n p l o t t e d  may  has  results  f o r p a r a b o l i c arches zero.  governing  arch from t a b l e s  l i n e s , the  Line  l i e on  a r e a u n d e r an  drawing against the  fact  a smooth influence  curve line  37 A p p e n d i x I V - T a b l e s o f Maximum Moment These t a b l e s g i v e t h e m a g n i f i c a t i o n f a c t o r s f o r maximum b e n d i n g  moments o f v a r i o u s p o i n t s i n t h e a r c h ,  when t h e a r c h i s l o a d e d w i t h a u n i f o r m l o a d . They were o b t a i n e d by c a l c u l a t i n g p l o t t e d from ratio,  data  stiffness  than those  t h e areas  under t h e i n f l u e n c e l i n e s  i n Appendix I I . factor  F o r other values  , and d i f f e r e n t  tabulated, interpolation  variation  i s used  of rise of EI  as i n A p p e n d i x  II. Appendix V - Curves These from  o f Maximum Moment  curves  o f maximum b e n d i n g  data In Appendix I I .  governing  They  moment  show t h e d e s i g n e r where t h e  s e c t i o n s o f t h e a r c h r i b a r e and where  ment o r s t e e l  a r e drawn  c o v e r p l a t e s may b e c u t .  reinforce-  CHAPTER V I . NUMERICAL EXAMPLES  Example  1. The  solutions This  object of this  g i v e n by t h e c o m p u t e r  i s t o show, t h a t t h e  are s u f f i c i e n t l y  accurate.  i s . done b y s h o w i n g t h a t t h e s o l u t i o n s , s a t i s f y t h e  differential  equation  a certain-arch with solutions stituted  f o r b e n d i n g o f an a r c h .  a specific  back i n t o  To do t h i s . ,  load i s considered.  g i v e n b y t h e Computer f o r t h i s , c a s e  difference  the d i f f e r e n t i a l  equation,  The  a r e subusing  finite  equations. Consider  as  example  shown i n f i g u r e  the arch with (6.1).  This  3 8  i t s l o a d i n g and d i m e n s i o n l o a d i n g p r o d u c e s maximum  3 9  moment a t t h e l e f t ratio  springing.  The d e a d l o a d t o l i v e  load  i s one.  ./263 Mf>perfect  Figure ( 6 . 1 )  Under t h i s  loading,  t h e Computer s o l v e s  b e n d i n g moments f o r t w e n t y one e q u a l l y in  the arch,  s t a r t i n g from the l e f t  i n t h e second stitute for  c o l u m n of. t h e t a b l e  f o r H = 25 k i p s . The spaced h o r i z o n t a l  springing,  i n figure  are tabulated  (6.2).  t h e s e moments b a c k i n t o t h e d i f f e r e n t i a l  the arch,  a second order  the d i f f e r e n t i a l equation  M" + — El  Sec  3  i n M.  is first  From e q u a t i o n  6. M + Hy" = -w  M" i s a p p r o x i m a t e d b y a t h r e e equat i o n ,  equation  point  points  To s u b equation  w r i t t e n as  (3-5b), (6.1)  central difference  MJ  =  M  i - 1  "  +  quarter  i  1  +  (4x) where  M  2  i s . . t h e b e n d i n g moment  a t any p o i n t  i .  At the  point, M  t.  _  £ 3 . 8 . 6 1 3  2 ( . 6 9 . 8 0 0 2 )  -  ' d o ) =  - 1 3 . 6 7 2 6  7 2 . 0 6 6 5  +  2  ( 1 0 ) " ^  also, sec  3  and  e y"  Substituting equation  1 . 1 7 0 9  =  - . 0 0 6 6 , 6 7  =  these  values  into the l e f t  hand s i d e o f  ( 6 . 1 ),  - 1 3 . 6 7 2 6 ( 1 0 ) "  2  JJQ^-Q  ' +  +  2 5  -  . 2 5 2 3  ( - . 0 0 6 6 6 7 )  =  A c t u a l w on t h e a r c h i s e q u a l t o percentage,  ( 1 . 1 7 0 9 )  difference =  =  ( 6 9 . 8 . 0 0 2 )  -w  -w.  Therefore,.  . 2 5 2 7 .  . -  2  ^  2  3  "  -^^(lOO)  . 2 5 2 7  = By t h e same p r o c e d u r e ,  t h e s o l u t i o n s at o t h e r ' p o i n t s  a r c h may be shown t o s a t i s f y results As  of these  - 0 . 2 percent.  the d i f f e r e n t i a l  i nthe  equation.  c a l c u l a t i o n s are tabulated i n figure  The ( 6 . 2 ) .  shown> t h e d i f f e r e n c e s b e t w e e n t h e a c t u a l and c a l c u l a t e d  w a r e s m a l l , and s o l u t i o n s g i v e n b y t h e computer may be considered  sufficiently  accurate.  41  Results  .X  M (ft.-kip)  T  Calculated w (kip.-ft.)  D  L  0 .05 .10 • 15 .20 .25 .30 .35 .40 •45 ; .50 .55 , .60 .65 • 70 .75 .80 .85 .90 .95 1.00 :  1  - l l i | . . 1665  - 63.5938 - 16.47&7 24.8234 53.8613 69.8002 72.0665 61.0219 37.7085 9.8677 -14.6355 -34.3700 -48.1634 -55.1226  -54.6753  -46.6339 -31.2812 - 9.4648 17.3141 46.8312 76.1313  o f Example  '  .2543 .2589 .2585 ,.2556 .2523 .2499 .2490 .1877 .1271 .1281 . 1289 .1293 .1291 . 1287 .1277 .1217 . 1250 .1235 . 1225  Actual w(kip.-ft.)  .2527 .2527 .2527 .2527 .2527 • 25.27 .2527 .1895 . 1263 .1263 .1263 .1263 .1263 .1263 .1263 . 1263 . 1263 .1263 . 1263  i Figure  1.  (6.2)  Percentage difference  .6 2.5 2.3 1.1 -.2 -1.1 -1.5 -.9 .6 1.4 2.1 2.4' 2.2 1.9 1.1 -3.6 -1.0-2.2 -3.0  k-22.  Example  Consider the previous  analysis., the bending  the a i d o f the to  same a r c h as  Computer.  show t h a t t h e  i n example 1.  I n the  moments were o b t a i n e d  I t i s the purpose o f t h i s  same s o l u t i o n s may  be  o b t a i n e d by  with example  the  i n f l u e n c e l i n e method u s i n g t h e t a b l e s i n A p p e n d i x I I . As Ft.  2  i n example 1,  noted  H = 25  k i p s , EI = i|0,000  -kip.  y  V EI  40,000  = 5 At  the  left  springing,  following table. table  The  a r e o b t a i n e d as shown i n . t h e  magnification factors  were o b t a i n e d f r o m  c o r r e s p o n d i n g t o ^-'r-  a table in  , ^ = 0,  •Appendix I I  and  8=  5-  C a l c u l a t i o n f o r Moment at ~ = L i v e Load I n f l u e n c e l i n e Ord. M ( F t . - k i p ) (kip) (Ft.)  Space  E  1 2 3  1.263 1.263 1.263 1.263 1.263 1.263 1.263,1.263  4 5  6 7 8 X_M  E  =  -3.94  -9.92 -12.60 -12.80 -11.26. - 8.56 - 5-26  -  1.7k  - 8 3 . 5 0  Figure  - 4-97 -12.54 -15.92 -16.18 -14.22  -10.82 - 6.64 -2.21 £M ( 6 . 3 )  D  i n this,  =  4 1.08. 1.13 1.21 1.29 1.39  1-49 1.66  .2.34 -111.9  o  Mp  =  ^  (Ft.-kip )  -5.4 -14.2 -19 .-3 -20.9 -19.8 -16.1 -11.0 - 5.2  The  s o l u t i o n g i v e n b y t h e Computer f o r t h e l e f t  is  -114.16.  to  114-16 - 111.9  Therefore  the percentage  springing  d i f f e r e n c e i s equal  (100) •= 1.9&% S i n c e t h i s l o a d i n g p r o d u c e s  114.16 maximum moment  at t h e l e f t  u s e d t o o b t a i n t h e moment  s p r i n g i n g , A p p e n d i x I V may b e at t h i s  p o i n t . Prom A p p e n d i x I V ,  ^ = 1.34 M  D  = -83.50 = 112.0  (1.34) ft.-kip  w h i c h I s t h e same, as b e f o r e . For other points a t e d as i n d i c a t e d  i n t h e a r c h , moments a r e c a l c u l -  i n t h e above t a b l e .  shown i n t h e b e n d i n g  moment  curves  The r e s u l t s a r e  on t h e n e x t  page. A s  shown, s o l u t i o n s o b t a i n e d b y t h e i n f l u e n c e l i n e followed  closely  analysis f o rthis  t h e ones g i v e n b y t h e Computer. case  e r r o r o f approximately it for  may b e c o n c l u d e d practical  method The  i n d i c a t e d t h a t t h e r e was an a v e r a g e one and a h a l f p e r c e n t .  As  such,  t h a t t h e I n f l u e n c e l i n e method i s v a l i d  application.  Example 3 . I n t h e p r e v i o u s example, t h e v a l u e o f H was s u c h yB was e x a c t l y e q u a l t o 5 , and m a g n i f i c a t i o n f a c t o r s  that  were t h u s  directly  obtained f r o m t h e t a b l e s without  inter-  polation.  To i n d i c a t e t h e d e g r e e o f a c c u r a c y b y u s i n g  the  law. f o r i n t e r p o l a t i o n  Vj  secant  as e x p l a i n e d i n C h a p t e r 1 , but with a d i f f -  c o n s i d e r t h e same a r c h a s i n example  erent  d e a d l o a d and l i v e  foot.  load both  equal t o  k i p per  c a s e , H = 16 k i p s and g - F^-{ 2.0-0 = 4 . T h e  Inthis  '  deflection  0 . 0 8 . 0 8 5  t h e o r y moment M  D  V i j . 0 , 0 0 . 0  at t h e l e f t  by t h e Computer i s -6J4..O9 f t . - k i p .  springing given  The f o l l o w i n g c a l c u l -  ations  a r e f o r v a l u e o f M^ a t t h e same p o i n t , u s i n g t h e  secant  law f o r  interpolation. C a l c u l a t i o n f o r Moment a t ^ = 0 .  $2f=Sec.  Space M ( f t . - k i p ) E  1 • 2  -  3 4  k for  = 3  k for  = 5  k  Av.  k  Av#  3 . 1 8  . 0 7 3  . 0 7 7  . 0 7 5  1 . 0 5  -  8 . 0 2  . 0 9 3  . 0 9 8 -  . 0 9 6  1 . 0 8  -  -10.18  . 1 1 3  . 1 1 9  . 1 1 6  1.12  - 1 1 . 4 0  - 1 0 . 3 5  -  3 - 3 4 8 , 6 5  . 1 3 3  . 1 3 8  . 1 3 6  1 . 1 7  -12.11  9 . 1 0  . 1 4 7  . 1 5 4  . 1 5 0  1.22  -11.05  6  r -  6 . 9 3  . 1 6 3  . 1 6 6  .166  1 . 2 7  -  8 . 8 . 0  7  -  i | . 2 i f  . 1 8 3  . 1 8 5  . 1 8 5  1 . 3 5  -  5 . 7 3  8  -  l.lil  .244  .226  . 2 3 5  1.7.0  -  2.U.0  - 6 3 . I f 8.  F t .-kip  5  Z  M"_ = E  5 3. 4 4  F t . -kip Figure  Percentage  difference *  = (6.5) 6 4 . 0 9  -  6 3 . 4 8  6 4 . 0 9  =  1 %  46  I n t h e above secant  curves  illustrate $ values the  table,  t h e k v a l u e s were o b t a i n e d f r o m t h e  (5.3).  i n figure  the procedures  i n column  They  i n interpolation.  s i x c o u l d be d i r e c t l y  Actually, the  obtained from  same f i g u r e . As n o t e d  i n the previous  be u s e d f o r f i n d i n g  at l e f t  example,  t o A p p e n d i x I V , when fi= .137.  Similarly,  Average k =  . U+2,  3y  A p p e n d i x I V may  springing, since the loading  on t h e a r c h p r o d u c e s maximum moment  k =  are t a b u l a t e d t o  at t h i s  point. According  $ = 1.0.9, f r o m f i g u r e  when p = 5 ,  4 = 1.34  and k =  (5.3), . L+6.  and  V  = S e c . k'^5 = S e c . L+2 (4) = 1.19  Thus,  Mp = - 5 3 - 4 4 (1.19) = -63.51 As  illustrated,  ate f o r i n t e r p o l a t i o n .  ft.-kips.  t h e secant  law i s s u f f i c i e n t l y  accur-  4-  Example  In the  s e c o n d example., t h e  were s u c h t h a t usable. For and  the  the  tables  cas e o f  tables,  are  are  a p p e n d i x were d i r e c t l y  r a t i o , s t i f f n e s s f a c t o r )5  little  a general  whose v a r i a t i o n i n E I  (6.7).  arch  more d i f f i c u l t  in  , the  because  necessary.  C o n s i d e r as  figure  of the  d i f f e r e n t from those t a b u l a t e d  calculations  interpolations  rise  ;  v a r i a t i o n in EI  i n the  properties  The  case the  along the  dead l o a d  arch i n f i g u r e  arch r i b i s given  to  live  load  (6.6)  in  ratio is  one.  L  = .1 . L 7 Under t h i s  loading,  H = 16  kips.  The  value  of  a v e r a g e E I as c a l c u l a t e d f r o m f i g u r e (6.7) i s lj.0,000 f t . - kips. P> = (200) = i i . C a l c u l a t i o n s , f o r moment 2  16  V ij.0,000  r  at t h e  left  (6.8).  The  the  springing M_,!s  for £=y  3  5,  and  given  and  between ^ then f o r  cases. EI,  jB=  of 3,  $  constant instead  As  ^=IL*  Similarly  using  -  — =''-y, — = 0, orily a s m a l l  p=  i t was  In  and  i  arch.  figure  3,  and  hy  secant  ,  straight curves  =  line  4> and  difference  load  at  two  space  6,  when f*> =  interpolation.  (5.3),  i n the  s p a c e 6,  between t h e  at  and  in figure  c a s e o f v a r i a b l e EI  p, = L L b e t w e e n  load  1.12,  values  interpolate  between the  = 0,  from  f i n d the  , second f o r  a v a r i a t i o n i n EI  £ = y  To  necessary to  example, f o r ' 7  the  f o r the  o f the  an  1.47  table  s e c o n d column were o b t a i n e d  t h i r d column,  first  is  i n the  shown i n t h e  e l a s t i c theory analysis  o f c/ i n t h e  at  are  two,  5 Then  = 1.2.$.'  tables,  ^ = 1.26. and  / =  There 1.25  48  f \  -  •  (£r)ef^~  32.000  I l l -  p.***.  2oo'  3.G -2-7  2.0 _____  /•0-  __, /. ss r — V - 7  \ I  . t'2o \ /•'<?  f  <^oijft7sf/ _=_"  •  •/O  •2o  -3o  • 4o  _ /^,ycsr^  (67)  .3d  0.80  4 was  s e l e c t e d f o r t h e case  When t h e d i f f e r e n c e was a v e r a g e o f t h e two EI  tfls  of variation  larger was  i n EI i n this  example.  as i n o t h e r l o a d s p a c e s , t h e  s e l e c t e d because t h e v a l u e s of  a t e a c h p o i n t i n t h e a r c h l a y b e t w e e n t h e ones f o r  constant  E I arid v a r i a b l e E I as shown i n f i g u r e . (6.7) . C a l c u l a t i o n f o r Moment a t f = 0 Interpolated  Space  Mg  1 2 • 3 4 5 6 7 8  (ft.-kip)  -2.97  =  9  E  -3.09 -8.39 -11.63 -13.00 -12.36 -10.02 - 6.66 - 2.84  1.04  1.07 1.10 1.15 1.20 1.25 1.34 1.72  -10.58 -11.30 -10.30 - 8.02 - 4-98 - 1.65  ^M (ft.-kip)  IMD=  -67.99 f t . - k i p s JThe s o l u t i o n g i v e n by t h e Computer f o r moment a t t h e l e f t Figure (6.8) s p r i n g i n g I s -67.88 f t . - k i p s . . The d i f f e r e n c e i s v e r y s m a l l . The and  bending  moment c u r v e s  a c c o r d i n g t o t h e Computer  t h e i n f l u e n c e l i n e method u s i n g  shown i n t h e n e x t  page.  i n t e r p o l a t i o n are  As shown, i t may be  concluded  t h a t by i n t e r p o l a t i o n bending  moments f o r a r c h e s  v a r i a t i o n s i n E I are d i f f e r e n t  those  obtained with  reasonable  accuracy.  whose  t a b u l a t e d c o u l d be  9  .51 BIBLIOGRAPHY 1.  E n g e s s e r , F r . "Uber den E i n f l u s s d e r F o r m a n d e r u n g e n a u f den k r a f t e p l a n s t a t i s c h b.estimmter S y s t e m e , i n besondere der Dreigelenkhogen." Z e i t s c h r i f t f u r A r c h i t e k t u r und I n g e n i e u r w e s e n , ' 1903, s. 178..  2.  M e l a n , J . " D i e E r m i t t l u n g d e r Spannungen im D r e i g d e n k b o g e n und i n dem d u r c h ein.en g.eraden B a l k e n m i t M i t t e l gelenk v e r s t e i f t e n Hangetragen m i t . B e r u k s i c h t i g u n g s e i n e r F o r m a h d e r u n g . " O s t . Wschr. O f f e n t l . B a u d i e n s t , 1903, a. 438.  3.  M e l a n , J . "G-enavere T h e o r i e des Z w e i g e l e n k b o g e n s m i t B e r u k s i c h t i g u n g der durch die Belastung erzeugten Formahderung." Hanb. I n g e n i e u r w i s s e n a c h . , I I Bd. 5 A b t . , Kap. X I I , 1906.  4.  M e l a n , J . "Der Heft  5.  4,.  biegsam eingespannte  1925,  s.  Bogen."  Bauingenieur,  143.  K a s o r n o w s k y , S. " B e i t r a g z u r The.orie • w e i t g e s p a n n t e r Brukenbogen mit kampfergelenken." S t a r i l b a u , H e f t 1931, a. 61.  .6,  6. F t i t z , B. " T h e o r i e u n d B e r e c h n u n g y o l l w a n d i g e r Bogentrager b e i B e r u . c k s i c h t i g u n g des E i n f l u s s e - s d e r s y s t e m y e r f o r m u h g " J u l i u s S p r i n g e r , B e r l i n , . 1934• 7.  F r e u d e n t a l , A . " D e f l e c t i o n Theory f o r Arches." P u b l i c a t i o n o f the. I n t e r n a t i o n a l A s s o c i a t i o n f o r B r i d g e and S t r u c t u r a l E n g i n e e r i n g , v o l . 3, 1935, p. 117.  8. F r e u d e n t a l , A. " T h e o r i e des G r a n d e s V o u t r e s on B e t o n . " P u b l i c a t i o n of the I n t e r n a t i o n a l A s s o c i a t i o n f o r B r i d g e and S t r u c t u r a l E n g i n e e r i n g , v o l . 4> 1936, p . 249. 9.  10.  D i s c h i n g e r , F r . "Uritersuchungen u b e r d i e k n i c k s i c h e r h e i t , d i e e l a s t i s c h e V e r f o r m u h g und das k r i e s c h e n des Betons b e i Bogenbrucken." Bauingenieur, Heft 18, 1937, s. 487. S t e r n , G. "A d e f l e c t i o n T h e o r y o f Two-Hinged A r c h R i b s . " T h e s i s No. 516, D e p a r t m e n t o f C i v i l E n g i n e e r i n g , C o l u m b i a U n i v e r s i t y , New Y o r k , N. Y.  52 11.  L u , W.F. " D e f l e c t i o n M e t h o d f o r A r c h A n a l y s i s . " Thesis submitted i n p a r t i a l f u l f i l l m e n t of the requirements f o r degree o f D o c t o r o f S c i e n c e i n the  University of Michigan.  (195l).  12.  G h a t t e r j e e , P.N.. "On t h e D e f l e c t i o n T h e o r y o f R i b b e d Two-Hinged E l a s t i c A r c h e s . " Thesis submitted f o r t h e degree o f D o c t o r o f P h i l o s o p h y , U n i v e r s i t y o f I l l i n o i s , 1949.  13.  W h i t n e y , C.S. " P l a i n a n d R e i n f o r c e d C o n c r e t e A r c h e s . " J o u r n a l , American Concrete I n s t i t u t e , t i t l e no.  47-46,  14.  1950-1951.  H i r s c h , E.G., Popov, E . P . " A n a l y s i s o f A r c h e s b y F i n i t e Differences." Proceedings, American S o c i e t y of  C i v i l E n g i n e e r s , p a p e r n o . 829  15.  (1955).  H a r d e s t y , S., G a r r e l t s . , J.M., and H e d r i c k , I.G. "Rainbow A r c h B r i d g e o v e r N i a g a r a G o r g e . " T r a n s a c t i o n s , American S o c i e t y o f C i v i l E n g i n e e r s ,  v o l . 110 ( 1 9 4 5 ) .  16. Rowe, R.S. "The A m p l i f i c a t i o n o f S t r e s s i n F l e x i b l e S t e e l A r c h e s . " T r a n s a c t i o n s , American S o c i e t y o f  C i v i l E n g i n e e r s , v o l . 119  17.  (1954).  M i k l o f s k y , H.A., S o t i l l o , O . J . " D e s i g n o f F l e x i b l e S t e e l Arches. By I n t e r a c t i o n D i a g r a m s . " P r o c e e d i n g s , A m e r i c a n S o c i e t y o f C i v i l E n g i n e e r s , p a p e r n o . 1190  (1957).  APPENDIX I PROGRAMME FOR ANALYSIS OF SYMMETRICAL HINGELESS ARCH UNDER VERTICAL LOAD  SPECIFICATION  Working  Channel:  •  Channels I , I I , I I I , I V , and M  Subroutine Required:  Start  Drum s t o r a g e :  C h a n n e l s 80 t o 93  Machine  Two m i n u t e s p e r c y c l e  Time:  routine,  O l j D T I , l e ; DTO  ARCH DIVIDED INTO TWENTY SEGMENTS 53  Summary: The  program s o l v e s f o r d e f l e c t i o n t h e o r y moments  by a n u m e r i c a l procedure o f s u c c e s s i v e the f i r s t  approximation. I n  c y c l e t h e v e r t i c a l and h o r i z o n t a l components o f  d e f l e c t i o n are assumed t o be zero.  Bending moments and  d e f l e c t i o n s thus c a l c u l a t e d represent the e l a s t i c analysis.  theory  I n subsequent approximations, the d e f l e c t e d  shape o f the arch from t h e p r e v i o u s c y c l e Is assumed. Successive  assumptions as such l e a d t o a s o l u t i o n based on t h e  d e f l e c t i o n theory.  The extent t o which the i t e r a t i o n i s  c a r r i e d depends on the accuracy r e q u i r e d .  Calculations  can be stopped at any c y c l e . F o r t h e purpose o f n u m e r i c a l i n t e g r a t i o n , t h i s programme assumes that t h e arch Is d i v i d e d i n t o twenty d i v i s i o n s , not n e c e s s a r i l y o f equal l e n g t h . d i v i s i o n o f length be  constant.  I n each  ds t h e f l e x u r a l r i g i d i t y i s assumed t o  The load P i s assumed t o act at t h e c e n t e r  of t h e d i v i s i o n . The  programme accepts and s t o r e s  Independently  the geometry, the f l e x u r a l r i g i d i t y and the l o a d s along t h e arch a x i s . A f t e r t h i s data i s s t o r e d , t h e computer begins it8 It  c a l c u l a t i o n s with t h e command, " 9 3 1 6 c a r r i a g e first  c a l c u l a t e s the bending moment M , t h e h o r i z o n t a l  t h r u s t H, and the v e r t i c a l shear f o r c e V for  c  c  that  are, r e q u i r e d  c o n t i n u i t y at the crown. With these t h r e e v a l u e s the  computer c a l c u l a t e s t h e bending moments at p o i n t s and  return."  0 t o 20  t h e h o r i z o n t a l and v e r t i c a l d e f l e c t i o n s at p o i n t s  1 to  55 19.  On completion o f t h i s o p e r a t i o n the programme s t o p s .  Then, w i t h a p p r o p r i a t e p a n e l s e t t i n g s and Normal-Start switch t o S t a r t  and Normal, t h e computer types out the  moments o r d e f l e c t i o n s o r both, and proceeds t o another cycle of i t e r a t i o n .  At the end o f each c y c l e , t h e computer  types out the c y c l e number. Inputs:The t h r e e i n p u t s are independent Geometry - 8200 c a r r i a g e r e t u r n .  starts.  The computer accepts  the geometry o f a r c h as f o l l o w s :  XT^  *^1  x  3  * * * 10 x  *^3 "' * ^10  ds, ds ds ... ds 1 2 3 10 _L. x, y and ds During i n p u t , the computer forms t h e sum o f 30 i n p u t s and types out t h e d i f f e r e n c e between the sum and X  •  II* t h e d i f f e r e n c e i s zero o r  s m a l l due t o round o f f e r r o r , i t i n d i c a t e s that the computer accepted the data EI  8300 c a r r i a g e r e t u r n .  correctly.  The computer accepts  the average EI In each ds d i v i s i o n as f o l l o w s : EI, •L The  EI_  d  EI, 3  ... ET  10  £EI /  computer w i l l type out the d i f f e r e n c e i n  sums as b e f o r e .  56  8305 c a r r i a g e r e t u r n .  Load  The computer accepts  the loads P as f o l l o w s : P  l  The In  P  2  P  3 *•*  P  20  f  Pn  computer w i l l type out t h e d i f f e r e n c e  sum as b e f o r e .  Details: 1. The computer s o l v e s t h e t h r e e simultaneous of  c o n t i n u i t y at the crown by I t e r a t i o n , so that  a f t e r t y p i n g out t h e f i r s t  approximation, the  computer stops and waits f o r a coded to  equations  be typed i n by t h e o p e r a t o r .  Instruction  I f the number 1  i s t y p e d I n , t h e computer w i l l I t e r a t e a g a i n and type out another approximation.  I f a number g r e a t e r  than 1 i s typed i n , the computer continues i n i t s sequence o f o p e r a t i o n s . T h e r e f o r e , t h e o p e r a t o r should s a t i s f y h i m s e l f f i r s t have converged  that t h e s o l u t i o n s  before i n s t r u c t i n g t h e computer t o  proceed f u r t h e r .  The computer types out t h e s o l u -  t i o n s as f o l l o w s : c 100  M  .  JL10  2. With Jump s w i t c h 1 at Jump and Normal-Start to S t a r t  switch  and Normal, t h e computer types out bending  moments at p o i n t s 0 t o 20 as f o l l o w s . moment produces  compression  Positive  i n top f i b r e .  M  M  M  MX  Q  5  M  6  20  M  l5  M  M  M  M  M3  2  7  19  lk  M  M  M  8  18  13  M  9  M  17  M  12  M  M  16  l l  With Jump switch 2 at Jump and Normal-Start swit to Start and Normal, the computer types out the horizontal deflection  and v e r t i c a l deflection  at points 1 to 19 with t h e i r sign conventions as  follows:  *1  h  *3  7  *8  S  positive *10  to  left  AS  positive to right  V  *3  V **>  positive upward  /J12  *11  «_>  58  ]+. With both Jump Switches 1 and 2 at Jump and NormalS t a r t Switch t o S t a r t first  and then normal the computer  types out the bending moments and then the  horizontal  and v e r t i c a l  deflections.  5. The s c a l i n g and maximum v a l u e s o f t h e v a r i a b l e s s r e as  follows: x, y, dx,  Scaling  dy, ds  Maximum Value  128  EI  2*  <_, EI  &  262,000 .000488  2  23  512  V  2  23  512  H  2^  256  M  2  32,000  Mds EI  2 7  P  1 7  2  32  Note: Care s h o u l d be taken that t h e u n i t s used i n any problem s h o u l d produce v a l u e s l e s s than o r equal t o those above. 6. An example o f c a l c u l a t i o n Is shown i n the f o l l o w i n g pages.  These c a l c u l a t i o n s  of Chapter V I .  were done f o r example 1  59  k  r>f*  Example  200, 10~^0 30 L>0 50 60(2PJ80 90 100 6.33333 12 17 21.33333 25 28" 30.33331I3_V33 33.33333 . .33333_( 10.4478 10.2702 11.8389 11.4953 11.1815 10.9001 (To. 6^6 __U-_39^10_;_0iiLj^.0074 — -<$95T3T1^ ^_0092g  >ooo 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 rv^OOOgp - . 0 0 0  40000  Pn 2 3 2 6 6 5 2.52665 2.52665 2.52665 2.52665 2.52665 2.52665 2.52665 1.263325 1.263325 1.263325 1.263325 1.263325 1.263325 1.263325 1.263225 1.263325 1.263325 1.263325(3g373> .000008 Start C°rt t^o r\cJ __ /9 c'/femj'/ai1 e<r*ss-*ry 132629  «.  <JT49999^  H  t  /Q  —  ^32349g|r^  -fat- f'rS*-  -83.5067 46.4799  -40.3096 46.6777  -5.7125 38.2757  20.2848 21.2738  56,0582 -29.0201  30.9758 -33.9355  9.9266 -34.8175  -7.0890 -31.6660  -.0580 -.4099  -.1806 . -.3777  Oyc/e. 37.6823]* 1.98873 He t^iti —. -20.0712"?* -24.4812 )  -.3002 -.3393  -.3823 -.3110  -.4168V -.3013)  ,Z2$Z  .3002 .3100  .3hk9V .3013)  .0408 .3593  .130k  -.0916 -.8078  -.3079 -.6696  -.5471 -.4391  -.7366 -.1563  -.8306\> .1340}  .0644 .7551  .2226 .7180  .4121 .5946  .5852 .3937  .7070V .1340 5  .3506  r^res^nf  .3301  <_V?_j//c  /-A<±esy  tfia/ys't.  6^  •L- r C  •.1468055 .148766 •.148766 •.148766  2.485044 2.486736 2.486738 2.486738  •1.3550786 •1.2038230 •1.2037208 •1.2037208  He  lb  •.I452IO •.I48689 -.148697 •.I48697  2.4850k9 2.488060 2.488064 2.488064  -1.3660300 -1.1644958 -1.1642537 -1.1642535  •.14336k .147476 .147476 -.147476 .147476  2.485875 2.489429 2.489434 2.489434 2.489434  -1.3698306 -I.1508725 -1.1505620 -1.1505617 -1.1505617  14241+8 .146790 .146797 .146797 .146797  2.486277 2.490034  •1.37H623  2.49004o 2.490o4o 2.490o4o  -1.1461086 •1.1457715 •1.1457710 •1.1457710  .142105 .146530 .146538 146538 .146538  2.486432 2.490261 2.490267 2.490267 2.490267  •1.3716302 •1.1444387 •I.1440918 •I.1440914 •1.1440914  1 1 1 1 11  1 1 1 1 4  -.141968 •.146416 •.146423 •.146423 •.146423  2.486507 2.490361 2.490367 2.490367 2.490367  •1.3717948 •1.1438520 •1.1435016 •1.1435012 •1.1435012  •.141922 •.146378 •.146385 -.146385 -.146385  2.486523 2.490386 2.490392 2.490392 2.490392  •1.3718525 •1.1436464 •1.1432947 •I.1432941 •I.1432941  -.141884 -.146355 -.146355 -.146355 -.146355  2.486540 2.490407 2.490413 2.490413 2.490413  -I.3718729 -I.1435735 -I.1432216 -I.1432210 -1.1432210  -.141891 -.146362 -.146362 -.146362 -.146362  2.486541 2.490408 2.49o4i4 2.490414 2.490414  -1.3718803 -1.1435479 -1.1431959 -1.1431953 -1.1431953  •.141884 .146347 •.146355 .146355 .146355 >146355  2.486534 2.490403 2.49o4o9 2.490409 2.490409 2.490409  -1.3718827 -1.1435397 -1.1431874 -1.1431869 -1.1431869 -1.1431869  8,  l l l l  4 9.  10.  11.  e s s e  1  Con  ve  62  H e c t i c  ^eor1  £ 0  1-1; M  •114.1665 69.8002  -63.5938  72.0665  -15.4787 61.0219  24.8234 37.7085  76.1315 -46.6339  46.8312 -54.6753  17.3141 -55.1226  -9.4648 -48.1634  53.86130 9.8677/ •31.2812 •34.3700 5  -.0824 -.6283  -.2585 -.5850  -.4367 -.5252  -.5662 -.4745  .0580 .5332  .1898 .5156  .3335 .4824  .448k .4546  -.1321 -I.3141  -.4548 -1.1356  .0907 1.0876  .3173 1.0107  -.8282 .8097  .5949 .7975  -I.1402 -.3915 .8508 .4669  .leff  -.6285 -.4496 • s  .5151 .4496  1  kff  -1.3149^ .0542 / 1.0279 .0542  Storage Scheme The f o l l o w i n g c h a r t s show the storage scheme used In the programme.  Chart  1 shows the main memory  whioh were used f o r s t o r i n g v a r i o u s d a t a .  The  words  occupied i n each channel are shown i n Chart 2. Channel IV was  Working  a l s o used as temporary storage f o r c o e f f i -  c i e n t s i n the c o n t i n u i t y equations. t i o n s are used:  channels  The f o l l o w i n g n o t a -  ( S u b s c r i p t c r e f e r s t o crown, L and R  r e f e r t o l e f t h a l f and r i g h t h a l f of t h e arch r e s p e c t i v e l y )  x, y  Rectangular  x, y  P i n a l x,  ds  Length measured along the a r c h r i b .  EI  Produot of modulus o f e l a s t i c i t y and moment of inertia.  P. n  Concentrated  H  Horizontal thrust.  c  c o - o r d i n a t e s ( i n i t i a l geometry),  y.  loads.  Shear f o r c e . M,M ,M ,M C  L  R  ^»*c'^L'^R  Moment. Horizontal deflection. Vertical  deflection.  R o t a t i o n at crown.  64  ^VL*  "^VR  Horizontal  d e f l e c t i o n at crown o f the l e f t  and the r i g h t h a l f of t h e arch due t o V  ^ML  c  respectively  = 1.  "^HR  Horizontal  d e f l e c t i o n at crown due t o R* = 10.  ^MR  Horizontal  d e f l e c t i o n at crown due t o M  /p  Horizontal  d e f l e c t i o n at crown due t o P .  R  c  ^HR*j V e r t i c a l d e f l e c t i o n at crown due t o V  n •ML  ^  M  R  j)  H  Q  = 10, M  c  c  = 100 and P  c  * 1,  respectively.  n  -PL ^VL  V^HL  ^HR  R o t a t i o n at orown due t o V  ^ML  ^  H  M  R  ,j  Q  = 10, M  Q  = 100, and P  n  c  = 1,  respectively  f PR j _5y,f ,f ,$p  Relative  M  h o r i z o n t a l d e f l e c t i o n at  crown due t o V „ , H = 10, M = 100, and C c * c P ^V'^H'^M'^P  n  respectively.  Relative  v e r t i c a l d e f l e c t i o n at crown.  f V » ^ H " » ^ M » R e l a t i v e r o t a t i o n at crown.  v=  ft  U  * M  5H  » 100.  n  IHL  H  half  65  Chart I - Drum Storage o f Data Channel  da KE n  P  H =10 c  a3  a6  X  a8  a9  y  aa  ab  ac  ad  t[  ae  af  M  bo  bl  4  b2  63  1  bk  b£  M  b6  b7  4  b8  b9  *?  ba  bb  M  be  bd  %  be  bf  do  dl  M  d2  d3  4  dk  d$  n  d6  dl  v. M  d8  d9  V and P  c  BZ  y  M =100 c  M  al  ak  c  c»  aO  X  v =i  H  n  da  Channel o f zero 4c» *?c> %  f  o  r  e  a  c  h  h  a  l  f  a r c h due t o H , V , M , c  Relative  ^,  c  iq , <p c  c  c  f  P  db  n  dc  EI  dd  due t o H , V , M , c  V  c  c  c  P  n  de eo  storage  x + _|.  c  o  ds  Temporary  Run Number, H ,  Right h a l f of f a r c h  Left half of a r c h  Items  ei  e2  e3  ek 00  66  Chart 2 - Channel Storage a. Numbers i n the words r e f e r t o moment, d e f l e c t i o n , x,  and y at each point i n t h e arch o r t o load P, ds ds, EI and at each space. 0  1  2  3  i+  $  6  7  8  9  10  •  b. Channel db  ^VR  IVR  ^VR  *HL  ^HL  <pHL .  ^HR  ^HR  <jpHR  ^ML  ^ML  ^ML  -*MR  ^MR  ^MR  ^PL  tfpL  <?PL  ^PR  ^R  fpR  •  c. Channel de  in  IK  iM  1M  /p  tp  f?  d. Channel 00  Cycle Number  H  V c  c  68  d. Working C h a n n e l l IV  u 1-U 1 1-U  JL.  U-l  -___p  %  Equations o f ^Continuity at Crown  Coded Programme  A copy o f the coded programme l a p r o v i d e d h e r e . The o p e r a t i o n s c a r r i e d out i n each channel are shown i n the f o l l o w i n g b l o c k diagrams.. The programme operates In working Channel I w i t h t h e e x c e p t i o n o f Channel 8e which operates i n working Channel I I I .  71  Stored  Stored  i n 80  S u b r o u t i n e t o f i n d M, H. o f h a l f  an a r c h  $ ,  canti-  i n 85  Calculate ^HR  a n c  *  ii  H  R  ,  ^HR*  store  leved from the springing  Stored  i n 81  Stored  Finish  of subroutine i n  Calculate  Channel  80.  ^YL  a  I n 86  n  d  ^yL* s t o r e  '  C a l c u l a t e -?yRf  Stored  i n 82  Decimal input  x. y . ds.  Stored  i n 87  Finish  calculation  ^VR'  fvR  Calculate and Stored  i n 83  Calculate  Decimal input  and  Stored Form Add  and  1 to cycle  Calculate and  store  g^,  and  number. q^,  ft  jf y^,  j_f MR *  ?MR»  ^MR  /?  ^  i n 89  Calculate  store.  store  d  store  Stored  i n 81+  n  for/yp,  i n 88  D e c i m a l Input E I n  a  ^VR  store  Stored  P  '?VR>  store  i  p  L  ,  p L  ,  72  S t o r e d i n 8a  jf ,  Calculate and  S t o r e d i n 8f  /? ,  p R  k  ppJ  PR  ^  PR  store  Calculate to t o t a l  M, R  Calculate/ ,  (f , ..  v  x +/  , y +  store.  Decimal output c y c l e  number,  (12 c o e f f i c i e n t s )  C a l c u l a t e U, 1-U,  i  jyjj,  n  1-U  Stored i n 9 1  S t o r e d i n 8c Calculate!"-  /  1  )  Decimal output j f ^ ,  <fn U - u /  [  etc.(7  coefficients) Stored i n 9 2  S t o r e d i n 8d Solve M , c  H , c  V  Complete d e c i m a l o u t p u t ^ .  y  R  Decimal output M , c  Hg,  V  c  Stored i n 8e(operates i n I I I ' D e c i m a l i n p u t a number Q. S t o r e E Q , V" i n C h a n n e l 0 0 . c  Calculate total  due  R  Stored i n 9 0  S t o r e d I n 8b  etc.  ^  R  l o a d i n g and s t o r e .  Calculate and  $ ,  M^  f  i^, ^  due t o  l o a d i n g and s t o r e .  Decimal output  ty^,  ^2-,  Stored i n 93 Calculate  x + -L  , y +  J^.  73 Booh 11237921 U9W800 571f6o6b 17886121 U9k03flt  8504  6o4b3aOO U86bl71d 571*79^ 786b666a U27571f 66^_3000 3000e500 786b6o6a e7226o6e 48W6llb 3000e64b U86al783 l*91bl79c _0.02U84b 11250000 666b484b 791b6520 170a8deO 00000000 mbU2k ^960571f 280o494o 00000000 571-78 *- 78ce6o»*a 81811100 OOOaOOOO  571a7820 571a78la 17892800 SbdbllOl 1«8301780 48281781 49204921 87db7928 81801100 571a78l5 790c4922 496c7929 oaoooooo 1*8301102 81801100 81861100 87e21110 8fa__12d 87e4ll95 8f*_112f 81851107 87<Sa_l8o 8 7 a 9 U l c 8 f b l l l 2 6 8 5 a l l l 8 6 85-»bll03 00050000 8 7 e l U 1 0 8 f a e l l 2 d 87e3H95 8 f a c l l 2 f 8185HO3  8104  86o4  1  496d792a 496e2800 l*9228fdb 79144921 57151101 00800000 8fb6ll26 8fb4ll2d  78lbl»828 17015715 7820U830 I709OOOO 81801100 00050000 85a01l86 87e3U95  87db7928 ^607929 4961792a 49628fdb 28001103 87dall80 85aall03 8fb2112f  49204922 791^921 00000000 81871100 5b6db6da 87a8lllc 87ellll0 81861102  28004940 871e571f 5blbll60 l*84ba504 6117^917 17088da4 8da5571f 5bl31l60  8704 5715781b 484ba504 5bObll60 00000000 61174917 6717871f 0115070© U8281780 57157820 171c8da6 5b071l60 OOI50006 1*8301788 8da7571-* 87a48fel 00190015 81801100 5bOfll60 8fe287a6 00190001 4923U02 H84ba504 8fe38fe4 00000000 8fb71126 61X1^911 lbOOOOOO 0019000c 8fb5U2d 171l8ddc 00000000 OOOaOOOO  87db7928 49647929 4965792a 49668fdb 79031U4 00050000 85alll86 87e4U95  7907^24 790b4925 790f4926 7913^927 8188IIOO 87«3all80 85abll03 8fb3112f  87<iall80 87a8lllc 8fbcll26 85a01l86 85aall03 87a9lllc 87e21U0 81871101  8304 871e571f 5blbll60 l*84ba503 6II74917 170U8ddd 5bl3U60 6717871f 5b0_1160  lbOlOOOO a50U6l07 0017009d 871e571f 49071799 00000000 5b0bll60 8da35ble 0017000d H84ba504 II606707 010b0301 6107^907 5bla871f OOObOOlB 17098da2 Il601b96 00000000 571f5b03 OII30696 OOOeOOOB ll6oU84b 0013008e OOOaOOOO  8804 1*7820 301780 28004921 49221101 00000000 OOcSOOOO 85alll86 8fball2d  791^20 8180U00 87db7928 49701102 00000000 87dall80 85abll03 87e31195  7929^71 7 9 1 ^ 2 0 792a4972 28004921 8fdb571a 49220000 781*1*828 81891100 170ell03 00000000 87a91Uc 8fbdU26 00050000 8 7 e l U 1 0 8 f b 8 U 2 f 81881109  84o4 85-c87dd 571f2800 3000784b ea6bc44b 178WdaO 8dal87a4 8fa88fa9 87a68f_a  8fab571b 28004860 17058dda 8dOOOOO0 83007920 6117^920 8b002800 49204921  8904 571a7820 1*8301780 81801100 87-bll01 87a2U80 85aall03 8fbell2f 8fbbll2d  7928497 * 7929^975 792a4976 8fdbll02 87a8lllc 87elU10 8l8all00 87e4U95  571*7815 178a2800 48281782 49204921 571a78la 49220000 1*8301103 81801100 8fd21126 85a01l86 8fd0112d 87e31195 00050000 87e21U0 8fb9U2f 8189110c  571f2800 a304607a 6o4bU84b U87bl790 170479% 571f787b 1117112b l»86bl789 1103784b 796al*929 60%3&00 112cllOf 786b666a 571-784b 3000e500 60Aa3a00 8204  786b666a 28004970 3000e5O0 571*1-90 a304607a 00000000 **87bl799 28004970 571-787b 1U90000 U86bl792 492a2800 796a4928 4970118c 112e0000 OOOaOOOO  7913^22 85a01l86 790-4923 8fb01126 790b4924 8 7 a 8 l U c 7907^925 87a_JLL80 7903^26 OaOOOOOO 791e^927 00000001 81851100 00200000 85aall03 OOOaOOOO  S  1  74  8f04 79212e00 81801100 49215715 87a485d4 78lb4828 5713784b 17085715 646b486b 78204830 1789U02 17101101 00050000 8fd9U26 85alll86 8fd7112d 87e4ll95  8fa887a5 87a685d6 85d55713 5713784b 784b646b 606b486b 486bl70a I7878190 8fa91103 OOOaOOOO 87a3U80 87a9111c 85abll03 87e21110 8fd5112f 8l8fll05  7969676d 6176496a 2930791* 49657971 797a6l7e al05eb67 67754969 496e8fde e56be763 7979677d 4l68e766 a3052e00 496d7962 eb642800 496f4l6e 67664962 a31beb6e e76beb6a 796a£l6e c5633000 8l8cll00 49667972 651*4967 08000000  9004 8fablb04 81931100 871fb500 5b0bll60 7907*781 13l8ll8f 85d85703 5ble7845  1160791a f78ll71c 7907*781 57035bl6 784all60 791»f78l 178d7907 f78285d9  57035bl2 00050000 78451160 5b6db6da 791a*78l 02000010 17828191 000a8l91 0511040a 8faa87a7 05110415 85d7570f 59OOOO00 784b6o6b 05U0481 486bl71b  8c04 312ea50a 4l60e76f 312ea503 497f2800 4973416c eb68312o 6l7b497b 4l6da317 e76feb68 a50a6l77 4l60e76b eb6l312e 312ea50a 4977416c eb643l2e 497c2800 61734973 «76beb64 e50a497* 4l65»317 4l62e76b 312ea503 4l62e76f ob6l312e eb6a312e 497b4l6e eb66312e 497<S0000 »50a4977 e76feb66 a50a6l7* 8l8dUOO  9104 85d4570b 5b0a7846 ll60790e f78ll704 790ff78l 570b5b06 784bll60 790ef78l  1794790* 01190311 17961113 f78285d5 0119031c 051104le 570b5b02 OII90388 00050000 78461160 59000000 5b6db6da 790ef78l 790f*78l 790ff783 1789790* 570b5b07 15000000 f78l0000 784all6o 81841111 81921100 790ef78l 00000000  8d04 28004169 a317eb6l 312e497e 797e4917 4llbe77e all13000 61174917 28004Uf  9204 571f5blb 784bll60 7917*781 17807913 f78385d6 571f5bOf 7846U60 7917*781  17947913 f78l571f 5bOb784b U607917 f78ll709 7913*782 85d7571* 5b077846  8a04 87db7928 17905715 49787929 78204830 4979792a 17050000 497a8fdb 81801100 5715781b 00000000 1*8281101 00050000 8fd31126 85alll86 8*dU12d 87e4ll95 8b04 87db7960 67644960 7968616c 49647970 61744968 7978617c 496c796l 61654961  87db7928 00000000 497c7929 00000000 497d792a 00000000 497e8fdb 00000000 8l8bll00 00000000 87a31l80 87a9111c 85abU03 87e21110 8fbfU2f 8l8all02  e77da308 6llf491f ll60791e 61174917 8fe0871f 858ell4o 7973491b 5b0f791b 02170783 4ll7e777 II602800 0311068e a3096llb f7835bl3 0318061a 491b797b 791*1160 00000000 491f4U7 2800f783 00000000 e77fa310 5b0b7917 00000000  8e04 f78l871e 8b005755 5b531l60 785b4828 674fld4b 17455755 83007917 78604830 4921791* 174dll42 49221141 00050000 8*d8U26 85a01l86 8fd6ll2d 87e3U95  11607917 01190316 f78l!71d OII90382 7913*781 0119038d 571*5b03 0119031c 784bU6o 5b6db6da 7917*781 59000000 178eOOOO 01190308 81841111 00050000  9304 87a485d4 17l88fe2 6o6b486b 81841100 4llbe747 818OUOO e747c520 0000000a 571*784b 87a685d6 171d8fe4 5b6db6da 79174921 87e0110c a50l646b 571*784b 7907*782 00000000 4llfe747 00020000 486bl784 a50l606b 81901108 00000000 c5221l43 00100048 8fel87a5 486bl789 00000000 00000000 85d5571* 8fe387a7 87a48fel 00000000 87a21l80 87a8lllc 85aall03 87ellll0 784ba501 85d7571* 8*e287a6 00000000 8fd4ll2* 818*1100 646b486b 784ba501 8fe38fe4 OOOaOOOO  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  M-ri  _P L  ,025 .075 .125 .175 .225 .275 .325 .375 .425 .475 .525 .575 .625 .675 .725 • 775 .825 .875 .925 .975  -a PL  rf f o r p = 3  x 100  Const. EI  -1.976 -If. 996 -6.367 -6.1|93 -5.727 -4.372 -2.690 - .898 .826  2.341 3.542 4.358 4.754 4.727 4.311 3.573 2.623 1.604 .704 .155  f - 1 7 ~ A  x — = 0  Var. EI  -2.221 -6.036 -8.510 -9.498 -8.966 -7.132 -4-468 -1.479 1.429 3.946 5.849 7.004 7.36k 6.966 5.943 4.508 2.961 1.597 .619 .123  rf f o r  Const. EI  Var. EI  1.02 1.03 1.05 1.08 1.10 1.12  1.02 1.03 1.05 1.07  1.16  1.31 .89 1.04 1.07 1.09 1.10 1.10 1.10 1.10 1.09 1.09 1.08 1.08  1.09  1.13 1.17 '1.33 .90 1.05 1.09 1.10 1.08 1.10 1.10 1.09  1.10 1.07 1.06 1.05  Const. EI  1.07 1.11 1.18 1.26  1.34 1.44 1.59 2.19 •k9  1.11 1.24 1.31 1.34 1.36 1.36 1.36 1.35 1.33 1.31 1.29  = 5 Var. EI  1.05 1.10 1.16  1.24 1.34 1.46 1.64 2.31 .51 1.15 1.30 1.36 1.39 1.39 1.37 1.34 1.31 1.28  1.24 1.21  rf f o r yB =  7  Const. EI 1.20  1.34 1.57 1.83 2.14 2.54 3.22  5.83  -1.61 1.12  1.73 1.88  2.19 2.29  2.34 2.35 2.32 2.27 2.22 2.16  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  x X  P  L .029  .075 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .625 .675 .725 • 775 .825 .875 .925 .975  £ = 0.05"  -  L  L  rf f o r p =  ioo =  .L26  - .456 -2.467 -3.413 -3.561 -3.144 -2.360  -1.380 - .344 6.33 1.464 2.084 2.456 2.565 2.421 2.058 1.541 .957 .425 .094  Var. EI .226  -1.311 -4.245 -5.938 -6.329 -5.560 -3.997 -2.041 - .017 1.820  3.283 4.251 4.668 4.545 3.957 3.048 2.025 1.101 .429 .085  Const. EI  .95 1.23 1.09 1.10 1.12 1.13 1.15 1.18 1.33 1.05 1.11 1.13 1.14 1.14 1.14 1.14 1.14 1.13 1.13 1.13  i  8  rf f o r p>= 5  3  PT  Const. E I  =  Var. EI  .90 1.09  1.07  1.09  1.11 1.13 1.16 1.22  11.29  1.06 1.11 1.13 1.14 1.13 1.13 1.12 1.11 1.10 1.10 1.09  Const. E I  .82 1.77 1.32 1.35 1.41 1.47 1.55 1.68 2.33 1.11 1.38 1.46 1.50 1.5-1 1.52 1.51 1.50 1.48 1.46 1.44  Var. EI  .63 1.32 1.25 1.31 1.40 1.50 1.62  1.88 47.46 1.17 1.40 1.48 1.51 1.51 1.50 1.47 1.44 1.40 1.37 1.33  rf f o r p>- 7 Const. EI  .42 3.50 2.03 2.17 2.41  2.69 3.06  3.67 6.90 .80 2.17 2.58 2.77 2.87 2.91 2.91 2.86 2.81 2.76 2.68  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P r = 0.10 .  t = I L 8  ^ f o r p> = 3  f o r p- 5  Moment at  MT? T,  -1  x 100 =  Const. E I  .025 .075 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  .337 1.624 1.525 - .176 -1.167 -1.617 -1.671 -1.453 -1.071 - .613 - .150 .255 .568 .763 .829  .772 .617 .402 .184 .042  Var*. E I  .178 .939 .085 -2.253 -3.490 -3.704 -3.161 -2.168  - .979 .204 1.227 1.981 2.407 2.488 2.256 1.789 1.214 .670 .264 .053  Const. E I  1.01 1.00 .98 1.56 1.14 1.12 1.12 1.12 1.11 1.08 .85 1.37 1.25 1.23 1.21 1.20 1.20 1.19 1.18 1.19  Var. E I  .95 .94 -.84 1.14 1.13 1.14 1.16  1.17 1.20 1.05 1.17 1.18 1.18 1.18 1.17 1.17 1.16  1.15 l.l4  1.14  ^ f o r p= 7  Const. E I  Var. EI  Const. E I  1.00 .98 .89  .79 .78 -5.80 1.50 1.48 1.53 1.60 1.68 1.82 . .90  .88 .84 .50 9.56  3.16  1.50 1.1+4 1.1+4 1.43 1.40 1.28  •34 2.37 1.95 1.84 1.79 1.75 1.72 1.69 1.66 1.64  1.60  I.67 1.69 1.69 I.67 1.63 1.60  1.57 1.53 1.48  2.92  2.64 • 2.59 2.56 2.43 1.95 -1.74 6.20 4.54 4-13 3.91 3.75 3.62  3.50 3.41 3.31  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P f = °*15  £ =  # f o r /3=3  # f o r ,8=5  Moment at  L  .025 .075 .125 .175 .225 .275 .325 .375 .1+25 .475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  x 100 = PL Var. EI Const. EI  .256 1.21+3 3.109 3.220 1.456 .207  - .622  -1.116 -1.353 -1.397 -1.306 -1.130 - .910 - .680 - .465 - .285 - .149  - .061  - .017 - .003  .135 .714 1.979 1.557 - .451 -1.562 -1.958 -1.860 -1.456  - .901  -  .319 .196 .580 .797 .840 .732 .528 .303 .123 .025  Const. EI  Var. E I  Const. EI  1.06  1.01 1.01 1.00 .96 1.32 1.15 l.l4 1.14 1.09 1.10 .93 1.63 1.33 1.28  1.18  1.05 1.0k 1.05 1.08 1.28 1.01 1.05 1.07 1.08 1.08 1.08 1.07 1.06 1.03 .99 .93  .82 .60  .08  1.26  1.24 1.23 1.22 1.21 1.21  1.16  1.12 1.14  1.26  1.95 1.00 1.15 1.21 1.24 1.26 1.26  1.23 1.19 1.11 .99 .79 .40 -.30 -1.48  1  for p -  Var. EI  1.00 .78 .96 .82 2.32 1.57 1.53 1.52 1.49 1.37 .68 3.53 2.33 2.11 2.00 1.92 1.88 1.83 1.79 1.71  7  Const. EI  1.43 1.38 1.30 1.35 1.66 3.71 .85 1.30 1.49 1.61  1.68 1.71 1.70 1.63 1.52 1.28  .73 -.19  -1.82  -4.87  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P r = ° -  Moment at  L  % x 100 = PL Const. E I  .025 .075 .125 .175 .225 .275 .325 .375 .if25 .475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  .183  .900  Var. E I .096  • 513 1.438  2.992 2.790  .787 - .370  - .390 -1.118  -1.718  -1.497  -1.190  -1.997  -2.071  -1.979  -1.763 -1.4.61 -1.113 - .757 - 432 - .180 - .038  £ = i  0  rf f o r p, = 3  2.285 4-274 I4..308 2.329  2  .864  -1.450  -1.355 -1.105  - .812  - .528  - .291  - .125 - .033 .002 .006  .002  Const. E I  1.12 1.10  1.09  1.07 1.08 1.13  1.26  .76 1.02 1.07 1.10 1.11 1.12 1.13 1.13 1.13 1.13 1.12 1.12 1.10  . Var. E I 1.09  1.08 1.07 1.05 1.06 1.11 1.00 1.08  1.09  1.10 1.10  1.09  1.07 1.03  .95 l.lk  .05  12.79  2.55 2.34  rf f o r J3> = 7  rf f o r  Const. E I  1.38 1.35 1.30 1.24 1.27 1.45  1.96  -.01 1.02 1.22 1.32  1.39  1-44 1.47 1.48  1.49 1.49  1.47 1.45 1.1+3  Var. E I  1.25 1.24 1.21  1.16  1.17 1.34 1.00 1.2k 1.31 1.35 1.35 1.31 1.23 1.08 .78 .10 -2.48 43.27 6.23 3.61  Const. E I  2.13 2.05  1.91  1.74 1.86 2.53  k.62  -3.72  .60  l.k5 1.92  2.26  2.52 2.73 2.88 2.98  3.00  2.98 2.96 2.89  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  Us. T,  PL  x ioo =  Const. E I  .025 .075 .125 .175 .225 .275 .325 .375 •k25 .475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  .119 .597 1.553 2.986 4.888 4-750 2.555 .785 -.583 -1.576 -2.226 -2.567 -2.639 -2.486 -2.159 -1.712 -1.206 - .712' - .303 - .065  Var. EI  .062 .336 .962 2.052 3.733 3.576 1.545 .059 -.959 -1.582 -1.881 -1.922 -1.769 -1.487 -1.137 - .780 - .469 - .235 - .086 - .016  f- = 0.25  T= 1  ft f o r £ = 3  ^ f o r p> = 5  Const. E I  1.18 1.16 1.14 1.12 1.09 1.10 1.15 1.30 .88 1.06 1.10 1.12 1.14 i.i5 i.i5 i.i5 i.i5 i.i5 1.15 1.14  Var. E I  1.17 1.16 1.14 1.12 1.09 1.10 1.18 3.14 1.03 1.10 1.13 l.Ui  1.15 1.16 1.15 1.14 1.13 1.12 1.10 1.07  Const. EI  1.62 1.57 1.51 1.43 1.33 1.35 1.54 2.15 .45 1.16 1.34 1.43 1.50 1.54 1.57 1.58 1.58 1.57 1.55 1.54  # f o r p> - 7  Var. EI  Const. E I  1.57 1.55 1.51 1.43 1.33 1.36 1.66 10.22 1.00 1.31 1.44 1.53 1.58 1.60 1.60 1.58 1.53 1.48 1.43 1.45  3.07 2.92 2.68 2.40 2.10 2.20 2.98 5.59 -1.93 1.22 2.00 2.46 2.76 3.00 3.17 3.26 3.28 3.28 3.27 3.19  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES' UNDER THE ACTION OF A CONCENTRATED LOAD P  & x 100 = PL L .025 .075 .125 .175 .225 .275 .325 .375 425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  Const. E I  Var. E I  .063 .332 .913 1.855 3.198 4.968" 4.683 2.350 .k69 -.972 -1.992 -2.618 -2.889 -2.850 -2.559 -2.082 -l.k98 - .899 - .388 - .08k  .033 .18k .551 1.237 2.376 2.072 3.8i|5 1.670 .016 -1.158 -I.896 -2.25k -2.292 -2.081 -1.698 -1.23k - .780 " 407 - .15k - .030  f L  7- = 0.30  Moment at  rf f o r ys = 3  Const. E I 1.28 1.25 1.21 1.18 1.1k 1.10 1.10 1.15 1.38 1.02 1.10 1.13 1.1k 1.15 1.16 1.16 1.16 1.16 1.16 1.15  Var. EI 1.29 1.27 1.2k 1.21 1.17 1.12 1.12 1.20 9.03 1.07 1.13 1.16 1.17 1.18 1.19 1.18 1.18 1.17 1.16 1.1k  1 8  rf f o r p>= $  rf f o r  /3  = 7  Const. E I  Var. EI  Const. EI  2.01 1.91 1.78 1.65 1.51 1.37 1.37 1.56 2,44 1.02 1.33 1.45 1.51 1.56 1.59 1.60 1.61 1.60 1.59 1.57  2.0k 2.01 1.90 1.77 1.61 144 145 1.76 36.28 1.18 146 1.59 1.67 1.71 1.73 1.72 1.70 1.67 1.6'k 1.65  k.73 k.36 3.64 3.31 2.81 2.32 2.33 3.08 6.94 .66 2.02 2.54 2.85 3.08 3.23 3.32 3.34 3.33 3.32 3.25  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  i X  P L  .025 .075 .125 .175 .225 .275 .325 .375 .425 •475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  x 100 =  for  PL  Const. E I  .016 .106  .365 .883 1.737 2.984 4.670 4.325 1.965 .094 -1.296  -2.226  -2.730 -2.855  -2.661  -2.223 -1.632 - .994 - .434 - .095  Var. EI  .008 .056 .204 .547 1.221 2.353 4-011 3.717 1.475 -.223 -1.402 -2.103 -2.381 -2.308 -1.975 -1.487 - .965 - .514 - .197 - .039  £ = 0.35  Const. EI  1.76 1.53 1.38 1.28  1.21 1.15 1.10 1.10 1.14 1.83 1.09 1.12 1.14 1.15  1.16 1.16  1.16  1.16 1.16  1.15  ,6=3 Var. EI  fl  for  Const. EI  1.83 1.61 1-47 1.36  3.89 3.05 2.46  1.19 1.13 1.12 1.19 .78 1.12 1.16 1.18 1.19 1.19 1.19 1.19 1.18 1.18 1.16  1.54  1.26  2.06  1.76  1.36 1.34 1.49 4.10 1.30 1.44 1.50 1.55 1.58 1.59 1.60  1.59 1.58 1.57  ft= 5 Var. EI  3.94 3.48 2.89 2.42 2.04 1.73 1.47 1.45 1.72 -.01 1.43 1.60  1.68 1.73 1.76 1.76 1.75 1.72 1.70 1.70  j&  for  p>=  1  Const. E I  12.89 9.43 6.78 5.07 3.91 3.00 2.31 2.22 2.76 -12.75 2.02 2.56 2.84 3.02 3.15 3.23 3.23 3.23 3.22 3.15  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment a t "  f- = O.kO L  Mg — x 100 = PL Const. EI  .025 .075 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  -.02k -.081 -.092 .069 .505 1.299 2.517 4.209 3.905 1.623 -.137 -1.388 -2.161 -2.499 -2.465 -2.135  -1.608  - .997 - .441 - .097  rf f o r p = 3  Var. EI  -.012 -.047 -.077 -.018  .266 .920  2.043 3.697 3.418 1.222 - .397 -1.467 -2.034 -2.170 -1.966 -1.539  -1.026  - .556  - .216  - .043  Const. E I .78 .69 .26 2.93  1.43 1.23 1.14  1.09 1.08  1.11  .96  1.11 1.13 1.14 1.15 1.15 1.15 1.15 1.15 1.14  ~ = « L 8  ^ f o r >s = 5  Var. E I  Const. EI  Var. EI  .77 .69 •43 -4.75 1.70 1.32 1.19 1.12 1.10 1.15 1.07 1.15 1.17 1.18 1.19 1.19 1.19 1.18 1.18 1.17  .08 -.36 -2.11 8.86 2.64 1.85 1.52 1.31 1.27 1.36 1.05 1.41 1.47 1.50 1.53 1.54 1.55 1.55 1.54 1.53  -.04 -.49 -1.63 -24.38 3.98  2.29  1.74 1.43 1.38 1.55 1.29  1.60 1.64 1.69 1.72 1.72 1.72 1.70 1.68 1.69  ^ f o r /3 = 7 Const. E I  -3.69 -5.73 -13.46 -35.11 7.86 4-34 2.89 2.07 1.85 2.08 2.97 2.59 2.72 2.84 2.92 2.97 2.98 2.97 2.96 2.91  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  f  =  °-45  f -  L  M  5t L  .025 .075 .125 .175 .225 .275 .325 .375 .1|25  .475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  R  x  PL Const. E I  -.055 -.229 -.456 -.587 -.499 -.089 .724 2.002 3-790 3.614 1.485 -.107 -1.183 -1.785 -1.971 -1.819 -1.426 - .909 - .409 - .091  >  100 = Var. E I -.026 -.126  -.294 -.457 -.487 -.229 .441  1.612  3.346 3.177 1.117 -.347 -1.253 -1.665 -1.672 -1.389  - .961  - -534 - .210 - .042  Const. EI  1.03 1.02 1.01 .98 .89 -.25 1.25 1.12 1.07 1.06 1.08 1.03 1.10 1.11 1.12 1.13 1.13 1.13 1.13 1.13  for £ = 3 Var. EI  1.04 l.Oli , 1.03 1.00 .94 .59 1.42 1.17 1.09 1.08 1.12 1.09 1.13 1.15 1.16  1.17 1.17  1.16  1.16 1.15  I  =  T  for Const. E I  1.05 1.02 .95  .82  .51 -3.99 1.92 1.42 1.24 1.20 1.26  1.47 1.38  1.4l  1.44 1.45 1.46 1.46 1.46 1.45  I  8  y B  =  5  $ f o r p> = 7  Var. EI  1.13 1.06 1.01 .89 .59 -.93 2.70 1.64 1.34 1.29 1.39 1.45 1.52 1.57 1.60 1.61 1.62 1.61  1.60  1.62  Const. EI  .59 .42 .17 -.36 -1.81 -21.55 4.52  2.42 1.71 1.55 1.56  5.89 2.46 2.4k 2.48 2.51 2.53 2.83 2.53 2.50  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  1=1  7~ = 0.$0  L  Ik x T  II  .025 .075 .125 .175 .225 .275 ..325 .375  .1+25 .1+75  .525 .575 .625 .675 .725 .775 .825 .875 .925 .972  100 =  rf f o r p = 3  PL  Const. E I  - .077 - .339 - .729 -1.086 -1.273 -1.182 - .710 .205 1.620  3.568 3.568 1.620  .205 - .710 -1.182 -1.273 -1.086 - .729 - .339 - .077  Var. E I  - .037 - .180 - .kk6 - .773 -1.039 -1.093  - .795 - .038 1.257 3.H+2 1.2^7 - .038 - .795 -1.093 -1.039 - .773 -  .W>  - .180 - .037  Const. E I  1.10 1.10 1.10 1.09 1.08 1.06 1.03 1.31 1.09  1.06 1.06 1.09  1.31 1.03 1.06  1.08 1.09 1.10 1.10  1.10  Var. EI  1.13 1.12 1.12 1.12 1.11 1.10 1.06  -.1+2  1.13 1.08 1.08 1.13 -.1+2  1.06 1.10 1.11 1.12 1.12 1.12 1.13  8 rf f o r p> = 7  rf f o r p = 5 Const. E I  1.32 1.32 1.31 1.28  1.26  1.21 1.08 2.03 1.30 1.19 1.19 1.30 2.03 1.08  1.21  1.26  1.28 1.31 1.32 1.32  Var. EI  1.1+5 1.1+3 1.1+3 1.1+1  1.37 1.31 1.18  -1+.20 1.1+6  1.27 1.27  1.1+6 -1+.20  1.18 1.31 1.37  1.1+1 1.1+3 1.1+3  145  Const. E I 1.81+ 1.82  1.79 1.73 1.61  1.1+6  1.08 3.91 1.77 1.1+7 1.1+7  1.77 3.91 1.08  1.1+6  1.61 1.73 1.79  1.82 1.81+  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  M  _P L .025 .075 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .525 .675 .725 .775 .825 .875 .925 .975  JUL  x 100 =  L  x L  = 1 6  L  fi  f o r p>= 3  f o r p=  PL Const. E I  Var. EI  Const. EI  V a r . EI  Const. EI  -1.969 -4.964 -6.3OI -6.1+02 -5.628 -4.284 -2.628 - .874 .817 2.298 3.473 4.276 4.670 4.654 4.255 3.539 2.606 1.600 .704 .155  -2.215 -6.008 -8.445 -9.397 -8.847 -7.020 -4.390 -1.451 l.k03 3.872 5.744 6.890 7.260 6.893 5.899 ' 4.494 2.965 1.606 .625 .124  1.02 1.04 1.06 1.08 1.11 1.13 1.18 1.34 .87 l.Ok 1.08 1.09 1.11 1.11 1.11 1.11 1.10 1.10 1.09 1.08  1.02 1.03 1.05 1.08 1.10 1.13 1.18 1.35 .89 1.06 1.09 1.11 1.11 1.11 1.11 1.10 1.09 1.08 1.07 1.06  1.08 1.13 1.21 1.29 1.39 1.49 1.66 2.34 • 43 1.11 1.27 1.34 1.38 1.40 1.41 1.39 1.40 1.3-8 1.36 1.34  $ Var. E I 1.06 l.ll 1.18 1.27 1.38 1.51 1.71 2.46 .44 1.16 1.33 l.4o  1.43 1.43 1.42 1.39 1.36 1.33 1.29 1.25  for  p= 7  Const. EI 1.24 1.41 I.67 2.02 2.42 2.92 3.72 7.10 -2.27 1.10 1.83 2.21 2.46 2.57 2.68 2.71 2.67 2.53 2.56 2.47  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment a t  5> .025 .079 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  x 100 = PL Const. E I Var. EI  .432 • .429 -2.414 -3.339 -3.479 -3.068 -2.303 -1.349 - .344 .605 1.413 2.020 2.387 2.500 2.368 .2.021 1.519 .947 .422 .094  .231 -1.288 -4.192 -5.856 -6.230 -5.465 -3.927 -2.010 - .030 1.767 3.202 4-158 4.581 4.475 3.912 3.027 2.021 1.103 .432 .086  J=o.o  § = £  5  rf f o r p - 3 Const. EI  Var. EI  .95 1.25 1.10 1.11 1.13 1.14  .89 1.10 1.08 1.10 1.12 1.15  1.19 1.34  1.24 7.22 1.07 1.12 1.14 1.15 1.15 1.14 1.14 1.13 1.12 1.11 1.10  1.16  1.06  1.13 1.15  1.16 1.16  1.16 1.16 1.15 1.15 1.15 1.15  1.18  rf f o r B= /  Const. E I  .80 1.92 1.36 1.39 1.48 1.53  1.61  1.75 2.44 1.12 1.47 1.52 1.56 1.58 1.59 1.55 1.59 1.57 1.54 1.52  5  ^ f o r f> = 7  Var. EI  .60 1.37 1.28  1.35 1.45 1.55 1.70 1.97 30.08 1.19 1.57 1.58 1.56 1.54 i.5i 1.47 1.43 1.39  Const. E I  .32  4.19  2.23 2.44 2.76 3.11 3.5i .32  '.60"  2.37 2.90 3.18 3.29 3.1+0 3.42 3.37 3.0k 3.2k 3.16  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  T  J-J  .025 .075 .125 .175 .225 .275 .325 .375 .1+25 •475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  & x 100 = PL Const. E I Var. E I  .31+2  1.61+1+  1.566 - .117 -1.100 -1.553 -1.619 -1.1+20  -1.061  -  .627 .187 .206 .512 .706 .779 .733 .590 .387 .179 .01+1  .182 .957  .126  -2.188 -3.1+10 -3.62k  -3.098 -2.135 - .980 .170 1.168  1.910 2.335 2.426  2.211  1.763 1.203 .667 .264 .053  f- = 0.10  ? = r  JJ  L  4 for Const. EI  1.01 1.00 .98 1.8k 1.15 1.13 1.13 1.13 1.13 1.10  .92  i.4o 1.28  1.24 1.23 1.22 1.21 1.20 1.17 1.17  jrf f cr Var. E I  .95 .94 -.31 i.i5 1.14 i.i5 1.17 1.13 1.21 1.04 1.18 1,19 - 1.20 1.20 1.19 1.18 1.17 1.16  1.16 1.15  6  Const. E I  1.00 .98 .90  4-30 1.60 1.49 1-47 1.47 1.44 1.32 2.68 2.07 1.93 1.88 1.78 1.80 1.77 1.73 1.70  f o r ft= 7  p = 5 Var. E I  Const. E I  .78 • 77 -3.99 1.56 1.53 1.59 . 1.66 1.75 1.90  .86 .82 .47 15.73 3.38 2.98 2.87 2.85 2.66 2.10 -1.19 8.06 5.45 4.80 4.59 4.39 4.23 3.91 3.94 3.76  .82  1.67 1.75 1.77 1.77 1.75 1.71 1.69 1.65 1.61 1.57  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD Moment at  PL x L  .025 .075 .125 .175 .225 .275 .325 .375 •425 475 .525 .575 .625 .675 .725 .775 .82$ .875 .925 .975  .259 1.257 3.139  3.26k  1.508 .261 - .575 -1.083 -1.336  -1.399 -1.326 -1.165 - .955 -.729 - .512 - .325 - .179 - .080 - .025 - .004  Li  Var. EI  .138 .727 2.010 1.607 - .387 -1-497 -1.90k -1.826  -1.448 -  .919  - .358 .143 .522 .743 .796 .702 .511 .296 .120 .025  Const. EI  £ - I  0.15"  rf f o r p  100 =  Const. E I  f =  L " 6  rf f o r p= 5  = 3 Var. EI  1.09  1.02 1.01 1.00 .97 1.36 1.16 1.15 1.15 1.14 1.17  1.09  1.83 1.37 1.31  1.06 1.06  1.05 1.05 1.09 1.29  .99 1.05 1.08 1.10 1.10 1.10  1.07 1.05 1.01 -95 1.08 -95  .96  1.28  1.27 1.25 1.24 1.23 1.23  Const. EI  1.21 1.19 1.15 1.17 1.30 2.07 .91 1.13 1.22 1.27 1.30 1.31 1.31 1.29  1.25 1.2k 1.06  .85 .56 5.86  rf f o r  Var. E I 1.01  1.00 .97 .84 2.57 1.62 1.57 1.56 1.53  1.40  .76 4.46 2.51 2.24 2.12 2.03 1.98 1.93 1.88 1.82  p=  7  Const. EI  1.54 1.48 1.38 1.46 1.90  4-58 .35 1.15 1.46 1.66 1.81 1.94 2.0k 2.09  2.15 2.15 1.81 1.31 .87 .34  APPENDIX *EI MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P ' Moment at  V  L  .025 .075 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  M™ - a x 100 = PL Const. E I Var. E I  .167 .910 2.306 lf.305 4.347 2.373 .829 - .336 -1.167 -1.709 -2.004 -2.092 -2.012 -1.804 -1.519 -1.153 - .789 - .454 - .190 - .040  .098  .522 1.460  3.028  2.838 .916 - .343 -1.084 -1.433 -1.500 -1.377 -I.141 - .857 - .575 - .333 - .157 - .054 - .010 .002 .001  7" = 0.20  ~ =y  rf f o r p = 3 Const. EI  1.26 1.12 1.10 1.08 1.09 1.15 1.29 .66 1.01 1.07 1.10 1.12 1.14 1.14 1.14 1.15 1.15 1.14 1.11 1.10  Var. E I  1.10 1.09  1.08 1.06 1.06 1.27 .96 1.07 1.10 1.11 1.11 1.11 1.09  1.06  .99 .83 .52 -.97 5.57 3.25  rf f o r p = 5 Const. E I  1.60 1.41 1.35  1.28  1.31 1.52 2.07 -.38 .99 1.23 1.35 1.44 1.50 1.53 1.56 1.59 1.57 1.56 1.54 1.52  rf f o r p= 7  Var. E I  1.30 1.28 1.25 1.19 1.21 1.1+2 .82  1.22 1.32 1.37 1.37 1.37 1.32 1.20 .98 .53 -.75 -6.63 19.15 5.78  Const. E I  2.67 2.30 2.10  1.92 2.09  2.93 5.44 -6.17 • 31 1.45 2.05 2.52 2.90 3.17 3.41 3.63 3.64 349 3.61  3-47 o  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  - a x 100 = PL L .02$ .01$ .12$  .175 .225 .275 .325 .375 .1+25 .475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  Const. EI  .120 .602 i.56k 3.005 k.915 k-783 2.591 .819 - .556 -1.559 -2.221 -2.576  -2.662  -2.520 -2.193 -1.751 -1.21+0 - .735 - .315 - .068  Var. EI  .063 .343 .976 2.076 3.766 3.614 1.584 .092 - .937 -1.574 -1.888 -1.943 -1.802 -1.526 -I.176 - .814 - .493 - .249 - .092  - .018  f _ 1 L 6  f- = 0.25  fi f o rft= 3 Const. EI  1.19 1.17 1.16  1.13 1.10 1.11  1.16  1.32 .87 1.06 1.11 1.14 1.15  1.16  1.17 1.17 1.18 1.17 1.18 1.18  Var. EI  1.19 1.17  1.16  1.13 1.10 1.11 1.19  2.62  1.12 1.10 1.13 1.16 1.17 1.17 1.17  1.16  1.15 1.14 1.12 1.10  ^ f o r P= Const. EI  1.70 1.64 1.57 1.47 1.37 1.39 1.59 2.25 .32 1.17 1.37  l.kg  1.55 1.60 1.64  1.65 1.67 1.66 1.65 1.64  5  Var. EI  1.65  1.62  1.57 1.49 1.37 1.41 1.74 8.02 .95 1.32 1.48 1.58 1.65 1.68 1.69 1.68 1.64 1.59 1.53 1.52  fi f o rft= 7 Const. EI  3.51 3.35 2.98 2.72 2.36 2.48 3.39 6.58 -3.00 1.18 2.14 2.72  3.16  3.42 3.71 3.88 3.89 3.73 3.88 3.82  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  1%  x 100 =  PL Const. E I  .025 .075 .125 .175 .225 .275 .325 .375 .425  •m  .575 .625 .675 .725 .775 .825 .875 .925 .975  .063 .333 .916 1.86k 3.214 4-992 4-712 2.383 .499 - .948 -1.978 -2.618  ,-2.902  -2.876 -2.593 -2.119 -1.532 - .923 - .400 - .087  Var. EI  .033 .186 .557 1.250 2.396 4.099 3.876 1.702 .042 -1.140 -1.891 -2.263 -2.314 -2.112 -1.734  -1.268  - .805 - .423 - .161 - .031  t _ 1  ~ = 0.30  L " 5  ^ f o r p> = 3 Const. EI 1.29  1.25 1.23 1.19 1.15 1.11 1.11 1.16 1.37 1.02 1.11 1.14 1.16 1.17 1.17 1.18 1.18 1.18 1.19 1.18  V a r . EI  1.32 1.29  1.26  1.22 1.18 1.13 1.13 1.21 4.39 1.07 1.14 1.17 1.19 1.20 1.20 1.20 1.19 1.19 1.17 1.16  4 f o r yB= 5 Const. EI  2.14 2.01 1.87 1.71 1.57 1.41 1.40 1.63 2.52 .99 1.35 1.48 1.56 1.61  1.65 1.66 1.69 1.66 I.67 1.65  Var. E I  2.19 2.12 2.00 1.86 1.68 1.48 I.50 1.84 16.19 1.17 1.49 1.64 1.73 I.78 1.81 1.81 1.79 1.76 1.72 1.71  4 for  = 7  Const. EI  5.10 5.10 4.31 3.81 3.21 2.59 2.58 3.50 7.81  .48 2.14 2.79 3.22 3.47 3.72 3-87 3.88 3.73 3.86 3.79  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  MR  - a x 100 =  PL Const. E I  .025 .075 .125 .175 .225 .275 .325 .Jf25 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  .015 .103 .361 .883 1.742 2.999 4.694 4.355 1.997 .124 -1.274  -2.216  -2.734 -2.873 -2.690 -2.257 -1.665 -1.019 - .446 - .098  Var. EI  .008 .056 .204 .550 1.229 2.369 4.034 3.745 1.504 - .199 -1.387 -2.100 -2.392 -2.332 -2.005 -1.519 - .991 - .530 - .204 - .040  £ = 0.35 L  f L  rf f o r je> = 3 Const. E I  1.95 1.65 1.42 1.30 1.22 1.16  1.11 1.10 1.15 1.78 1.08 1.12 1.15 1.16  1.17 1.17 1.17 1.17  1.16  1.15  =  Var. E I  1.91 1.66 1.50 1.38  1.28  1.20 1.13 1.13 1.20 .71 1.12 1.16 1.19 1.20 1.21 1.21 1.20 1.20 1.17 1.17  1 6  rf f o r ft  Const. EI  445 3.38 2.65 2.17 1.86 1.59 1.39 1.37 1.53 3.78 1.31 1.46 1.54 1.59 1.63 1.63 1.65 1.65 1.64 1.62  = 5  for  £ = 7  Var. E I  Const. EI  449 3.80 3.10 2.56 2.13 1.79 1.51 149 1.77 -.33 145 1.64 1.74 1.80  16.61  1.82  1.83 1.83 1.80 1.77 1.76  11.66 7.81  5.96  4.54 3.41 2.53 2.43 3.0k 11.75 2.12 2.80 3.17 3.36 3.57 3.69 3.70 3.55 3.65 3.55  <  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  Mp»  IPLx L  .025 .075 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  Const. E I  .  - .015 - .087 - .101 .060 .501 1.305 2.534 4.235 3.937 1.656 - .110 -1.371 -2.157 -2.510 -2.488 -2.166 -1.638 -1.021 - .453 - .100  100 = Var. EI  -.012 - .050 - .083 - .023 .265 .925 2.059 3.721 3.I47 1.251 - .375 -1455 -2.037 -2.185 -1.991 -1.567 -1.051 - .572 - .223 - .044  7- = O.I4.O  ft f o r p = 3 Const. EI  1.37 .76 .30 3.32 1.45 1.24 1.15 1.09 1.08 1.11 .96 1.11 1.14 1.15 1.16 1.16 1.16 1.16  1.17  1.16  </ f o r /3= 5  Var. EI  .76 .69 •44  -3.57 1.74 1.13 1.20 1.12 1.11  1.16 1.06  1.15 1.17 1.19 1.20 1.20 1.20 1.19 1.19 1.17  Const. EI  .07 -.39 -2.07  10.61  2.90  1.92 1.55  1.33 1.29 1.38 .98 1.44  1.50 1.54 1.57 1.57  1.61  1.60  1.60  1.58  Var. EI  -.11 -.56 -1.75 -20.32 4.30  2.40 1.79 1.46 1.4l 1.57 1.29 1.60 1.69 1.74 1.77 1.78 I.78 1.77 1.74 1.73  ^ f o r /3 = 7 Const. EI  -7.35 -6.71  -14.I8  48.72 9.50 5.00 3.19 2.25 2.01 2.19 3.67 2.85 3.02 3.12  3.26  3.34 3.35 3.23 3.32  3.26  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  PL L .025 .075 .125 .175 .225 .275 .325 .375 .1+25 .575 .625 .675 .725 .775 .825 .875 .925 .975  x  Const. E I  • .238 - .1+71 - .603 - .511 -  .092 .731+  2.021+ 3.821 3.61+9 1.516  - .081+  -1.172 -1.788 -1.988 -1.8L1+ -l.i+52 - .931 - .1+21 - .078  100 Var. E I 7.028 - .130 - .301+  - .1+70 - .1+97 - .233 .1+1+9 1.631 3.373 3.208 l.H+5 - .328 -1.21+7 -1.673 -1.691 -1.1+13 - .983 - .51+9 - ;:217 - .01+1+  f =I L 6  ~ = O.I+5  4 for p= 3 Const. E I 1.05 1.05 1.02 .98 .90 -.23 1.25 1.13 1.07 1.06 1.08 1.07 1.11 1.12 1.13 1.11+ 1.11+  i:S 1.39  Var. EI 1.01+ 1.05 1.01+ 1.01 .91+ .58 1.1+3 1.18 1.10 1.09 1.13 1.08 l.lif 1.16 1.17 1.17 1.18 1.17 1.17 1.16  ^ f o r /3= 7  ^ f o r /5= 5 Const. E I 1.05 1.02 .96  :S -1+.29 1.96 1.1+5 1.25 1.21 1.27 1.53 1.1+0 1.1+3 1.1+7 145 i.5i 1.52 1.50 1.81  Var. E I 1.12 1.06 1.01 .88 .55  -1.08 2.80 1.68 1.36 1.31 1.1+1 1.1+7 1.55 1.60 1.61+ 1.65 1.67 1.66 1.61+ 1.65  Const. ET .1+6 .05 -.68 -2.50 -26.18 5.01+ 2.62 1.79 1.60 1.60 8.1+1+ 2.73 2.65 2.71 2.75 2.79 2.71 2.78 3.32  sO VJ1  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  L  .025 .075 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .625 .675 .725 • 775 .825 .875 .925 .975  ~ x 100 = PL Const. E I  - .080 - .349 - .747 -1.108  -1.292  -1.189 - .707 .222 1.633 3.602  3.602  1.633 .222 - .707 -1.189 -1.292 -1.108 - .747 - .349 - .080  Var. E I  - .038 - .186 - .459 - .790 -1.057 -1.105 - .795 - .025  1.281  3.173 3.173 1.281 - .025 - .795 -1.105 -1.057 - .790 - .459 - .186 - .038  & = 0.50 L rf f o r  Const. E I  1.07 1.08 1.10 1.09 1.08 1.07 1.02 1.34 1.11 1.06 1.06 1.11 1.34 1.02 1.07 1.08 1.09 1.10 1.08 1.07  - = L 6  3  Var.EI  1.12 1.13 1.13 1.13 1.12 1.10  1.06  -1.36 I.14 1.08  1.08  1.14 -1.36  1.06  1.10 1.12 1.13 1.10  1.08  1.07  rf f o r  Const. E I  1.33 1.33 1.32 1.30 1.24 1.21 1.06 2.08 1.33 1.20 1.20 1.33 2.08 1.06 1.21 1.24 1.30 1.32 1.33 1.33  ^=5  rf f o r p~ 7  Var. E I  Const. EI  1.46 1.46 1.46 1.44 1.39 1.33 1.18 -7.76 1.48 1.28 1.28 1.48 -7.76 1.18 1.33 1.39 1.44 1.46 1.46 1.1+6  I.87 1.89 1.85 1.80 1.64 147 1.03 4.11 1.86 1.51 1.51 1.86 k.ll  1.03  l.kj  1.64 1.80 1.86 1.89 I.87  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  •»  p L  .025 .075  .125 .175 .225 .275 .325 .375 .1+25 .475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  - S x  PL  100 =  f  - = 0 L  L  d for 3 = 3  =  1 4 ^ for P = 5  Const. EI  Var. EI  Const. EI  Var. EI  Const. E I  -1.953 -1+.897 -6.167 -6.216 -5.424 -4.100 -2.495 - .810 .796 2.205 3.326 4.101 4-495 4.501 4.141 3.468 2.574 1.592 .706 .156  -2.201 -5.947 -8.306 -9.180 -8.586 -6.773 -4.214 -1.385 1.348 3.712 5.515 6.638 7.032 6.720 5.802 4-463 2.973 1.625 .638 .128  1.03 1.05 1.08 1.11 1.14 1.17 1.22 l  1.02 1.04 1.07  1.11 1.19 1.30  1.13 1.17 1.22 1.43 .87  1.53 1.68 1.94 2.94 .17 1.13 1.35 1.46 1.52 1.56 1.59  %  1.05 1.10 1.12 1.13 1.14 1.14 1.14 1.14 1.14 1.13 1.13  1.16  1.07  1.11 1.13 1.14 1.14 1.14 1.13 1.12 1.11 1.10 1.09  l.kl  1.60 1.60  1.58 1.56 1.53  Var. EI  1.08 1.15 1.25 1.37 1.51 1.68 1.96 2.98 .20 1.19 I.41 1.51 1.57 1.59 1.59 1.57 1.54 1.49 1.45 1.41  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  x  _P_ L  .025 .075 .125 .175 .225 .275 .325 • 375 .425 475 .529 .575 .625 .675 .725 .775 .825 .875 .925 .975  2& x  100 =  PL Const. EI  -.377 -2.307 -3.189 -3.310 -2.909 -2.181 -1.283 - .340 .548 1.308 1.884 2.242 2.365 2.257 1.943 1.472 .926  .415 .093  — = L  ^ forp = 3  Var. EI  Const. EI  .242 -1.239  .93 1.39 1.14 1.15 1.17  - 4 . O 8 O  -5.679 -6.014 -5.255 -3.769 -1.937 - .054 1.655 3.028 3.957 4.388 4.322 3.813 2.980 2.010 1.108 •437 .088  £ L  0.05  1.19 1.21  1.25 1-43  1.06  1.15 1.18  1.19. 1.20 1.20 1.21 1.20 1.20 1.19 1.19  =  l 1+  ^ for  V a r . EI .86  1.14  1.10 1.12  1.15  1.18 1.22 1.29 5.02 1.08  1.15 1.17  1.18 1.19 1.19 1.18  1.17  1.16  1.15  1.13  Const. EI  • 73 2.47 1.52 1.57 1.65 1.74 I.87 2.06 3.01 1.09 1.57 1.70 1.77 1.81 1.84 1.86 1.85 1.83 1.80 1.77  p  = 5 Var. EI .47  1.52  1.39  1.48  1.61 1.75 1.94  2.31  22.63  1.21  1.41 1.70  1.76 1.79 1.80 1.78 1.74  1.70  1.65 1.61  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  -S. x 100 = _p L  .025 .075 .125 .175 .225 .275 .325 .375 •425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 i975  Const. E I  Var. E I  .351 1.684 1.648 .000 - .965  .190 .995 .2l4 -2.048 -3.235 -3.449 -2.959 -2.057 - .978 .101 1.043 1.754 2.176 2.289 2.111 1.704 1.175 .658 .263 .053  -1.508 -1.349 -1.037 - .652 - .255 .106 .395 .588 .673 .649 .533 .355 . .166 .038  f = 1 L 4  0.10  f o r ft = 3  PL  -1.420  ^ -  Const. E l 1.02 1.01 .99  1.17  1.16 1.15 1.15 1.14 1.11 .97 1.80 1.37  1.31  1.29 1.28  1.27  1.26 1.26 1.26  ^ f o rft= 5  Var. EI  .94 .93 .09 1.19 1.17 1.19 1.20 1.22 1.25 1.00 1.22 1.24 1.24 1.24 1.24 1.24  1.23 1.22 1.20 1.18  Const. EI  Var. EI  1.03 1.00  .73 .71 -2.83 1.77 1.72 1.78 1.88 1.99 2.17 .05 1.88 1.99 2.03  .92  1.72  1.61 1.61 1.60  1.55 1.41  .78  4.40  2.53  2.29 2.22  2.17  2.12 2.08 2.04 2.00  2.o4  2.04 2.02 1.98 1.93 1.88 1.83  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at M T  Li  .025 .075 .125 .175 .225 .275 .325 .375 .lf25 475 .525 .575 .625 .675 .725 • 775 .825 .875 .925 .975  v  PL  K  Const. EI .266  1.285 3.198 3.351 1.613 .370 - 476 -1.008 -1.294 -1.395 -1.360 -1.231 -1.014 - .830  - .612  - 412 - .2k6 - .122 - .014 - .008  100 = Var. EI  • 43 .755 2.075 1.713 - .251 -1.355 -1.782 -1.747 -1424 - .952 - .440 .029 .395 .623 .698 .633 471 .277 .114 .024  7- = 0.15"  rf f o r  r = r  p= 3  rf f o r  j5= 5  Const. EI  Var. EI  Const. EI  Var. EI  1.10 1.09 1.07 1.08 1.4 1.4 -.89 1.03 1.08 1.10 1.12 1.13  1.03 1.02 1.01 .99 1.51 1.17 1.17 1.17  1.33  1.05 1.03 .99 .86 3.56 1.76 1.71 1.69 1.65 1.51 .95  i:S 1.15 1.14 1.13 1.11 1.08 1.01  1.16  1.13 1.01 4.73 1.49 1.39 1.35 1.33 1.31 1.30 1.28 1.25  1.29  1.23 1.27 1.52 2.58 45 1.04 1.22 1.33 141 1.48 1.54 1.59 1.61 1.62 1.61  1.57 1.53 1.14  18.16  3.16 2.67 2.51 2.40 2.33  2.26  2.19 2.13  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at f = 0.20  f = I L  L  __P L  .025 .075 .125 .175 .225 .275 .325 .375 .1+25 .475 .525 .575 .625 .675 • 725 .775 .825 .875 .925 .975  |f  PL-Const. E I  100 =  4 ^ for £ = 5  for £ = 3  , !  .189 .926  2.31+3 4.361+  1+.1+23  2.1+59 .915 - .260 -1.112 -1.682 -2.009 -2.129 -2.078 -1.888 -1.597 -1.21+1 - .862 - .503 - .212 -  .01+6  Var. E I  .102  .51+1 1.505 3.101+  2.939 1.027 - .239  -1.005  -1.392 -1.502  -1.1+20 -1.218 - .955  - .678 - .1+28  - .232  - .101+  - .037 - .009 - .001  Const. E I  1.17 1.16 1.U+ 1.11 1.12 1.19 1.36 .38 1.00 1.09 1.13 1.16 1.18 1.19 1.21 1.21 1.22 1.22 1.21 1.20  Var. E I  1.12 1.12 1.11 1.08 1.09 1.19 .73 1.06 1.10 1.13 1.12+ 1.15 1.11+ 1.13 1.09 1.03 .92 .71 .31 .10  Const. E I  Var. E I  1.63 1.58 1.50 1.1+0 1.1+5 1.71+ 2.50 -1.98 .87 1.27  1.1+4 1.1+1  1.60 1.71 1.80 1.86  1.56 1.57  1.1+7  1.91  1.91+ 1.91+ 1.91+  1.91  1.36 1.28 1.32 1.70 -.22 1.13  1.31+ 1.1+5 1.52  1.55 1.1+4  1.26 .86 .16  -1.14  -3.89  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  - = 0.25  £ = i L 1+  L  M-p - £ x 100 = PL  L  Const. E I  .025 .075 .125 .175 .225 .275 .325 .375 .1+25 .1+75 .525 .575 .625 ;675 .725 .775 .825 .875 .925 .975  .122 .609 1.582 3.039 I+.966 1+.8I+8 2.665 .891+ - .1+90 -1.512 -2.201 -2.588 -2.70k -2.588 -2.282 -1.838 -1.316 - .788 - .31+0 - .071+  rf f o r  Var. E I .065 .353 1.003 2.12k 3.831+ 3.697 1.670 .169 - .881 -1.51+9 -1.898 -1.987 -1.873 -l.$13 -1.265 - .891 - .550 - 2282 - .105 - .020  Const. E I 1.25 1.23 1.20 1.17 1.13 1.13 1.20 1.39 .75 1.06 1.13 1.16 1.19 1.20 1.22 1.22 1.23 1.23 1.22 1.22  y5  =3  rf f o r  Var. E I 1.22 1.22 1.20 1.17 1.13 1.11+ 1.21+ 2.23 .99 1.11 1.16 1.19 1.21 1.22 1.23 1.22 1.22 1.20 1.19 1.21  Const. EI 1.99 1.91 1.80 1.66 1.50 1.53 1.82 2.66 -.29 1.16 1.31+ 1.32 1.31 1.31 1.30 1.29 1.27 1.21+ 1.21 1.18  5 = 5 Var. E I 1.90 1.86 1.78 1.67 1.51 1.56 2.01 6.95 • 77 1.35 1.60 1.76 1.88 1.96 2.01 2.03 2.00 1.96 1.91 I.87  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  i  PL  L  .025 .075 .125 .175  .22$ .275  .325 .375 425 • 475  .$2$ .$7$ .62$ .675 .725 .775 .825 .875 .925 .975  yf  x 100 =  Const. EI  .063 .331 .918 1.875 3.2k0 5.036 4-774 2.454 • 572 - .886 -1.937 -2.606 -2.924 -2.928 -2.668 -2.203 -1.609 - .979 - .428 - .094  f = 0.30  Var. E I  .034 .190 .568 1.273 2.436 4.155 3.945 1.774 .107 -1.095 -1.874 -2.278 -2.360 -2.181 -1.815 -1.345 - .866 - .459 - .176 - .035  7- = rr  for  p=  3  for  Const. EI  Var. E I  Const. EI  1.39 1.34 1.29 1.24 1.19 1.13 1.13 1.19 1-44 1.00 1.12  1.36 1.35 1.32 1.27 1.21 1.15 1.15 1.25 2.78  2.60 2.43 2.20 1.97 1.74 1.53 1.53 1.79 2.87 .89 1.42 1.61 1.72  1.16  1.18 1.20 1.21 1.22 1.22 1.22 1.22 1.21  1.06  1.15 1.19 1.22 1.24 1.2? 1.25 1.24 1.24 1.23 1.23  1.81  1.87  1.92  1.94 1.94 1.94 1.92  p=  5 Var. EI 2.62  2.51 2.34 2.13 1.89 1.63 1.65 2.08 9.71 1.13 1.59 1.80 1.94 2.03 2.09 2.11 2.10 2.07 2.03 2.00  8  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  5L  PL  L  Const. E I  .025 .075 .125 .175 .225 .275 .325 .375  .012 .094 .343 .873 1.747 3.02k  .i+25  475 .525 .575 .625 .675 .725 • 775 .825 .875 .925 .975  k.742  4.421 2.073 .197  -1.216  -2.185 -2.737 -2.910 -2.754 -2.335 -1.740 -1.075 - .475 - .105  100 = Var. E I  .007 .054 .202 .552 1.243 2.401 4.086 3.810  1.573 - .138 -1.347 -2.090 -2.415 -2.38k -2.076  -1.592 -1.052 - .569 - .221 - .044  f = °-35  f = r  rf f o r p> = 3  rf f o r p>= 5  Const. EI  Var. E I  Const. E I  Var. E I  2.24 1.78 1.52 1.37  2.04 1.82  6.57 4.52 3.28 2.56  6.31 4.92 3.84 3.05 2.45 1.99 1.64 1.61 1.95  1.26  1.19 1.13 1.12 1.17  1.61  1.09 1.14 1.17 1.19 1.20 1.21 1.21 1.21 1.21 1.20  1.61  1.45 1.33 1.23 1.15 1.15 1.22 .39 1.13 1.19 1.21 1.23 1.2k 1.25 1.25 1.24 1.23 1.23  2.09  1.75 1.50 1.46 1.66 3.43 1.36 1.57 1.68 1.75  1.81  1.85 1.87 1.87 1.87 1.85  -1.92  1.52 1.77 1.91 2.00  2.06  2.09 2.08  2.06  2.02 1.99  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  X  P  L  .025 .075 .125 .175 .22$  — x 100 = PL Var. E I Const. E I  - .029 - .102 - .127 .033 .1+86  .325 .375  1.312 2.569 k.295  .475 .525 .575 ..625 .675 .725 .775 .825 .875 .925 .975  1.737 - .039 -1.325 -2.H4 -2.532 -2.540 -2.235 -1.709 -1.076 - 481 - .107  .21$  .1+25  - .014 - .057  - .097  - .039 .256 .935 2.092 3-777 3.517 1.322 - .318 -1.425 -2.039 -2.221 r2.o5o -1.634 -1.109  - .610  - .240 - .048  7- = O.ILO  -  L  L  =  f o r JB = 5  $ f o r /6 = 3 Const. E I  Var. E I  Const. E I  Var. E I  .79 .69 .34 6.02 1.55  .81 .69 •44 -2.09 1.89 1.38 1.22 1.13 1.12 1.18 1.02 1.16 1.19 1.21 1.22 1.25 1.23 1.23 1.22 1.22  -.14 -.64 -2.31 24.37 3.41 2.16 1.69 1.41 1.35 1.46 -.21 1.52  -.36 -.83 -2.10  1.28  1.17 1.11  1.09  1.13 .30 1.12 1.15  1.17  1.18 1.19 1.19 1.19 1.19 1.19  1.60  1.66 1.72 1.75 1.77 1.77 1.77  .82  -15.16  5.34 2.73 1.97 1.55 1.48 1.68 1.27 1.71  1.82  1.90 1.96 1.99 1.99 1.97 1.94 1.91  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P f _ 1 L  Moment at  L  .025  .075  .125  .175  .225  .275 .325 .375 .1+25  .475  .525  .575 .625 .675 • 725 .775 .825 .875 .925 .975  ikxlOO = PL Const. E I  - .062 - .258 - .507 - .61+5  -  .51+3  - .101 .755 2.075 3.895 3.733 1.595 - .025  -1.11+1+  -1.796  -2.026  -1.903 -1.516 - .981 - .1+1+7  - .101  rf f o r  Var. EI  -.030 -  .11+2  - .327 - .501 - .525 - .21+3  1.676  3.1+4o  3.283 1.21I+ - .281 -1.232 -1.691 -1.736 -1.1+70 -1.036 - .581+ - .233 - .01+7  Const. E I  1.01+  1.01+  1.02 .98 .89 -.37 1.30 l.lii  1.08 1.07 1.10 .1+1+  1.11 1.13  1.15 1.16  1.16 1.16 1.16 1.16  rf  =3 Var. E I  1.08 1.06 1.05  1.02 .91+  .55  1.1+7 1.19 1.11 1.10 1.14 1.06 1.15  1.17 1.19 1.20 1.20 1.20 1.19 1.20  for  Const. E I 1.05  1.00 .93 .76 .35  -5.13  2.22 1.51+  1.30 1.25  1.32 1.91+ 1.1+7  p=5 Var. EI  1.09 1.05  .99 .83 .1+6 -1.1+9 3.18 1.80 1.1+2 1.35 147 1.51+ 1.61+  1.57  1.71 1.77 1.81  1.63 1.63  1.81 1.79 1.77  1.52  1.60 1.62 1.62  1.82  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  & x PL L  .025 .075 .125 .175 .225 .275 .325 .375 425 .1+75 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  Const. E I  - .086 - .373 - .792 -1.161 -1.339 -1.213 - .700 .262 1.715 3.686 3.686 1.715 .262  - .700 -1.213 -1.339  . -1.161  - .792 - .373 - .086  100 =  r = 0.50  ~ = r L ii  ^ for p = 3  Var. E I - .01+1  - .200 - 48<? - .833 -1.101 -1.133 - .796 .006  1.31+0 3.21+7 3.21+7 1.31+0 .006  - .796 -1.133 -1.101 - .833 - .1+89 - .200  - .01+1  Const. EI  1.12 1.12 1.12  1.11  1.10 1.08 1.02  1.33 1.11  1.07 1.07  1.11 1.33  1.02 1.08 1.10  1.11  1.12 1.12 1.12  of f o r B> = 5  Var. E I  1.15 1.15 1.15 1.15  1.11+  1.12 1.07  12.1+3  1.15 1.09 1.09 1.15  12.1+3  1.07 1.12  i:3 1.15 1.15 1.15  Const. EI 1.1+2 1.1+1 1.1+0  1.38  1.33 1.26  1.03 2.18 1.37 1.23 1.23 1.37 2.18 1.03 1.26 1.33  1.38  1.1+0 1.1+1 1.1+2  Var. E I  1.53 1.51+ 1.51+  1.52 1.1+7 1.39 1.17  1+6.1+1+ 1.55  1.32 1.32  1.55 1+6.1+1+  1.17 1.39 1.1+7 1.52  1.51+ 1.51+  1.53  H  O ^3  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  5s PL  L  f  .025 .075 .125 .175 .225 .275 .325 .375 425 475 .525 .575 .625 .675 .725 • 775 .825. .875 .925 .975  -1.938 4.837 -6.0k5 -6.047 -5.237 -3.930 -2.373 - .756 .776 2.117 3.189 3.938 4.333 4-362 4.039 3407 2.548 1.587 .708 .158  ~ = T  =  rf f o r p = 5  rf f o r p = 3  x 100  Const. EI  r- .0  Var. EI  -2.189 -5.891 -8.177 -8.977 -8.339 -6.536 -4.044 -I.318 1.299 3.560 5.295 6.396 6.84 6.559 5.7H  4434 2.981 1.643 .649 .131  Var. E I  Const. E I  Var. EI  1.05 1.07  1.03  1.15  1.13  1.18 1.30 1.48 1.65 1.85 2.09 2.49 4.23 -42 1.4 1.51 1.69 1.81 1.89 1.95 1.98 2.00 2.0b 1.96 1.96  1.13 1.23 1.38 1.57 1.78 2.03 2.45 4.10 -.35 1.21 1.57 1.74 1.85 1.91 1.94 1.94 1.91 1.85 1.79 1.72  Const. E I  1.11  1.19 1.23  1.30  1.61 .79 1.06  1.13  1.16 1.18 1.19 1.20 1.20 1.20 1.20 1.19 1.19  1.06  1.09 1.17  1.22 1.29  1.56  • 83 1.08  1.4  1.16 1.18 1.18 1.19 1.18  1.17  1.16  1.4  1.13  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  MT?  _P L  .025 .075 .125 .175 .225 .275 .325 .375 425 .475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  - f i x 100 = PL  Const. E I  456 - .330 -2.212 -3.054 -3.158 -2.765 -2.069 -1.220 - .336 • 497 1.212 1.760 2.110 2.242 2.157 1.872 1.431 .907 . 410 .092  ft  Var. E I  .251 -1.194 -3.977 -5.514 -5.810 -5.055 -3.617 -1.863 - .073 1.551  2.862  3.765 4.205 4.176 3.718 2.935 1.998 1.111 .442 .089  t = 1  2. _ o . 0 5  for  L  p  Const. E I  .90 1.60 1.19 1.21  1.23  1.25  1.28 1.33  1.56 1.08 1.21 1.24  1.26  1.27  1.28 1.29  1.29 1.29 1.28 1.27  =3  J  jrf f o rft-5 Var. E I  .81 1.19  1.14  1.17  1.20 1.24 1.29  1.37 4.78 1.10 1.19 1.22 1.24 1.25 1.25 1.25 1.24  1.37  1.21 1.20  Const. EI  Var. E I  .58 3.59 1.85 1.91 2.03 2.19 2.37 2.69 4.19 1.00 1.85 2.09 2.22 2.30 2.36 2.39 240 2.41 2.36 2.36  .22 1.83  1.62  1.75 1.94 2.15 2.43 3.02  25.96  1.20 1.81 1.74  2.16  2.23 2.27 •2.27 2.24 2.18 2.11 2.04  H O  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  %  x  100  f = 0.10 ^  jrf f o r  —  t = I L ->  fi=3  jrf f o r  3=5  PL L  Const. E I  .025 .075 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  .359 1.717 1.718 .103 -  .81+4  -1.300 -1.407 -1.280 -1.011 - .670 - .312 .019 .291  .481 .575 .571 .478 .324 .153 .035  Var. EI  Const. E I  .197 1.029 .293  I.04  -3.073 -3.283  1.21 1.19 1.19 1.19 1.18 1.15 1.03 5.01 1.51 1.41 1.38 1.37 1.36 1.35 1.34 1.33  -1.919  -2.821+  -1.980 - .970 .039 .927 1.608 2.026  2.159 2.015 1.645 1.147 .649 .261 .053  1.02. 1.01 .28  Var. EI  .93  .92  .17 1.25 1.23 1.24 1.26 1.28  1.31 .65  1.28  1.31 1.32 1.32 1.32 1.32 1.31  1.29 1.28 1.26  Const. E I  1.07 1.03 .93 -3.64 2.03 1.86 1.83 1.82  1.77  1.60  .98 22.11 3.50 2.99 2.84 2.73 2.68 2.64 2.57 2.56  Var. EI .62  .59 -3.13 2.21 2.09  2.17 2.30  2.1+6  2.72  -1+.88  2.27 1.74 2.57  2.61  2.63  2.62  2.58 2.52  2.1+3  2.35  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  M ^ x 100 PL Const. EI Var. E I v  X  P L  .025 .075 .125 .175 .225 .275 • 325 .375 .1+25 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  .271 1.307 3.21+5 3.424 1.704 .467 - .386 - .935 -1.250 -1.384 -1.384 -1.286  -1.123 - .921 - .705 - .496 - .310  - .162 - .062  -  .012  .148 • 779 2.133 1.809 - .127 -1.221 -1.664 -1.666 -1.394 - .976 - .512 - .076 .276 .508  .602  .564 .429 .257' .107 .022  r" = 0. If?  rf f o r Const. EI  1.15 1.13 1.10 1.12 1.22 1.55 .68 1.00 1.08 1.13 1.17 1.19 1.22 1.24  1.26 1.28 1.29  1.29 1.30  1.29  rf f o r js, = 5  p= =3 Var. EI 1.06  1.05 1.03 1.01 1.88 1.20 1.20 1.20 1.19 1.17 1.07 -.15 1.68 1.49 1.44 1.1+2 1.40 1.37 1.28 1.34  Const. EI  Var. EI  1.56 1.49 1.39 1.48  1.11 1.08 1.02 .88 6.82 2.02 1.93 1.91 1.86 1.70 1.14 -5.91 4.65 3.54 3.25  1.92 3.60  -.91 .75 1.18  1.43 1.63  1.81 1.98 2.15 2.32 2.47 2.61 2.76  2.82  2.93  3.09  2.97 2.87 2.76 2.67  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  M -1 x 100 PL Const. EI Var. EI s  L  .025 .075 .125 .175 .225 .275 .325 .375 425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  .191 .938 2.370 if. 4 1 0 4486 2.534 .994 - .186 -1.053 -1.647 -2.003 -2.155 -2.132 -1.965 -1.685 -1.327 - -934 - .551 - .235 - .051  .105 .557 1.543 3.170  3.029  1.131 - .139 - .924 -1.314 -1.494 -1.453 -1.286  -1.045 - .777 - .521 - .307 - .156 - .065 1 .020 - .003  f L  2- = 0.20 L  rf f o r  p= 3  =  1 3  rf f o r p = 5  Const. E I  Var. EI  Const. E I  1.25 1.22 1.19 1.15 1.17  1.18 1.17 1.15 1.12 1.13 1.27 -.03 1.02 1.11  2.05 1.94  1.26  148 -.37 .97 1.11 1.17 1.27 1.25 1.27 1.30 1.31 1.32 1.33 1.33 1.32  1.16  1.19 1.21 1.22 1.23 1.23 1.21 1.17 1.12 1.03 .97  1.82  1.64 1.72 2.20 3.38 -6.91 .55 1.30 1.67 1.93 2.4 2.31 2.46 2.57 2.66 2.73 2.72 2.75  Var. EI 1.69  1.65 1.58 1.45 1.54 2.21 -4.71 .83 1.31 1.56 1.75 1.90  2.04 2.15 2.24 2.30 2.28  2.82  2.67 1.81  H  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  5a x PL  L  .025 .075 .125 .175 .225 .275 .325 .375 .1+25 475 • 525 .575 .625 .675 .725 .775 .825 .875 .925 .975  Const. EI  .122 .611  1.592 3.060 5.003 4-902 2.733 .968 - 419 -1456 -2.170 -2.588 -2.737 -2.650 -2.363 -1.925 -1.393  - .81+3  - .367 - .080  f _ 1 L 3  - = 0.25 L  ft f o r p = 3  100 = Var. EI  .067 .361 1.024 2.163 . 3.893 3.773 1.753 .247  - .820  -1.515 -1.898 -2.021 -1.937 -1.696 -1.354 - .970 - .608 - .317 - .120 - .023  <& f o r ,9=5  Const. EI  Var. E I  1.34 1.32 1.28 1.23 1.17 1.18  1.31 1.29 1.27 1.23 1.17 1.18 1.31 2.24 .95 1.13 1.20 1.24 1.27 1.30 1.31 1.32 1.32 1.31 1.30  1.26 1  42  .55 3.70 1.16 1.21 1.24 1.27 1.29 1.31 1.32 1.32 1.32 1.31  1.29  Const. EI 2.61  2.44  2.26  2.02 1.77 1.81 2.25 3.48 -1.74 1.11 1.63 1.91 ' 2.12 2.28 2.42 2.51 2.58 2.64 2.62  2.64  Var. EI  2.39 2.32 2.21 2.02 1.78 1.86 2.55 8.09  .29  1.38  1.82  2.11 2.34 2.51 2.65 2.75 2.78 2.76 2.70 2.63  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  L  .025 .075 .125 .175 .225 .275 .325  .375 425  475  .525  .575 .625 .675 .725  • 775 .825 .875 .925  .975  HE. x 100 PL Var. E I  .061 .326  .034  .192 • 574  .915  1.289  1.875  2.467  3.255  5.072 . ...  4.205 4.010  1.848 .178  -I.040 -1.846  -1.885 -2.584  -2.938 -2.976 -2.741 -2.287 -1.688 -1.037 - .467 - .101  = 0.30  f _ L  # for p  Const. E I  4.830 2.527 .651 - .814  3£  -2.283  -2.400 -2.249  :  . -1.896 -1.424  - .928 - .498 - .193 - .038  Const. EI  =3  jrf f o r p  Var. EI  1.52 1.46  1.48  1.31 1.24 1.17 1.17 1.24 1.52  1.27  1.31 2.48  1.14  1.18  1.38  .96  1.19 1.23 1.25 1.27  1.291.30 1.30 1.30  1.29  1 3  1.45 1.40  1.34  1.19  1.19  1.05  1.24 1.27 1.30 1.32 1.32  1.33 1.32 1.31 1.30  Const. E I  3.64 3.25 2.86 2.47 2.10 1.79  1.77  2.15  3.57 .63 1.56 1.85  2.04 2.19 2.31 2.38 2.45  2.49 2.47  2.49  =5 Var. E I  3.48  3.30 3.04 2.70 2.31  1.91 1.93 2.55 9.53 1.00 1.77.. 2.11 2.34 2.51  2.64 2.73 2.75 2.73 2.67 2.61  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at 7- =  !k x  p L  .025 .075 .125 .175 .225 .275 .325 .375 425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  PL Const. EI  .009 .082 .327 .854 1.741 3.042 4.786 4.490 2.157 .282  -1.147 -2.144 -2.733 -2.943 -2.818 -2.416 -1.818 -1.134 - .505 - .112  0.35  4 f o r p> = 3  100 = Var. EI .006  .049 .195 .547 1.249 2.427 4.134 3.877 1.649 - .067  -1.296  -2.072 -2.433 -2.435 -2.149 -1.669  -1.116 - .610  - .239 - .048  rf f o r  /B = 5  Const. E I  Var. EI  Const . EI  Var. EI  3.H  2.58 2.11 1.74 1.56 1.40  13.10 7.22 4-73 3.38  10.54 7.46 5-49 4.08  2.08 1.70 1.64 1.90 3.67 1.45 1.78 1.95 2.06 2.16 2.23 2.28 2.32 2.30 2.31  2.40  2.14 1.70 1.47 1.33 1.23 1.15 1.14 1.20  1.60  1.10 1.17 1.20 1.23 1.25  1.26  1.27 1.27 1.27 1.27  1.28  1.18 1.17 1.27 -.89 1.14 1.22 1.25 1.28 1.30 1.31 1.31 1.30 1.29  1.29  2.60  3.H  1.89  1.83 2.29 -8.97 1.65 2.04 2.25 2.40 2.51 2.59  2.61  2.59 2.54 2.48  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  V  L  .025 .075 .125 .175 .225 .275  ?*E PL  .625 .675 .725  • 775  .825 .875 .925 .975  100 =  Const. E I  - .034 - .120 - .160 - .003 .k62  1.313 2.601 4.359 4.099  .325 .375  425 475 .525 .575  x  1.829  .Ohk  -1.268  -2.124 -2.552  -2.59L: -2.309 -1.784 -1.133 -  .511 .115  f _ 1 L "* 3  f r 0.40,  L  rf f o r p= 3 Var. EI -  .016 .066 . 115 .062  .240 .939 2.124 3.836 3.595 1.403 - .250 -1.386 -2.038 -2.256 -2.112 -1.705 -1.170 - .65! - .258 - .052  Const. E I  .80 .69  .36 -77.38 1.71 1.35 1.21  1.13 1.11  1.15  2.22 1.14  1.17 1.20 1.22 1.23 1.24 1.24 1.24 1.24  rf f o r 3=5  Var. EI  .79 .69 •45 -1.35 2.14 1.45 1.26 1.16 1.14  1.21 .95 1.18 1.22  1.25  1.27  1.28 1.28 1.28  1.27  1.26  Const. E I  Var. EI  -.61 -I.14  -.83 -1.47  -3.10  --  4.70  2.67 1.94  1.54 1.46 1.60 3.28  1.70  1.82  1.91 1.98  2.04  2.08 2.11 2.10 2.10  -3.04 -14.62  7.72  3.42 2.29  1.73 1.63 1.87 1.24  1.93 2.10 2.22 2.31 2.37 2.39 2.38  2.34 2.29  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P f . 1 L " 3  Moment at  & P L  .025 .075 .125 .175 .225 .275 .325 .375 425 .475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  x  100 =  Const. E I  -  .068 .281 .549 .695 .583 .115 .775 2.131 3.978 3.829 1.686 .044 -1.110 -1.802 -2.068 -1.968 -1.585 -1.036 - .476 - .108  4 forB = 5  $ f o r p = .3  PL Var. E I  -  .033 .154 .354 .538 .560 .259 .480 1.724 3.514 3.369 1.293  - .226  -1.213 -1.710 -1.784 -1.532 -1.092  - .622  - .250 - .051  Const. E I  1.06 1.05 1.03 .99 .88 -•43 1.35 1.17 1.10 1.09 1.12 1.73 1.12 1.15 1.17 1.19 1.20 1.20 1.20 1.20  Var. E I 1.09  1.08 1.06 1.03 .94 1.55 1.22 1.13 1.11 1.17 1.01 1.17 1.20 1.22 1.23 1.24 1.24 1.23 1.23  Const. E I  1.00 .96 .84  .62  .04 -6.99 2.62  1.70 1.38 1.32  1.4o .56  1.62  1.69 1.75 1.80 1.84 1.87 1.86 1.86  Var. E I  1.07 1.01 .90 • 70 .20 -2.37 3.85 2.01 1.52 1.44 1.56 1.82  1.84 1.93 2.02 2.08 2.10 2.10 2.07 2.04  -~3  APPENDIX I I MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A CONCENTRATED LOAD P Moment at  PTT  rf f o r j& = 3  x 100  p L  Const. E I  .025 .075 .125 .175 .225 .275 .325 .375 .425 475 .525 .575 .625 .675 .725 .775 .825 .875 .925 .975  - .092 - .399 - .81+1 -1.222 -1.393 -1.21+2 - .693 .308 1.793 3.782 3.782 1.793 .308 - .693 -1.21+2 -1.393 -1.222 - .81+1 - .399 - .092  x  f- = 0.50  Var. E I - .01+6 - .216 - .523 - .881 -1.150 -1.167 - .798 .OiJ.1 1.1+07 3.332 3.332 1-407 .041 - .798 -1.167 -1.150 - .881 - .523 - .216 - -046  Const. E I 1.15 1.15 1.15 1.14 1.12 1.09 1.01 1.37 1.13 1.08 1.08 1.13 1.37 1.01 1.09 1.12 1.14 1.15 1.15 1.15  4 f or /3 = 5 : Var. E I 1.14 1.18 1.18 1.18 1..16 1.13 1.07 3.30 1.18 1.11 1.11 1.18 3.30 1.07 1.13 1.16 1.18 1.18 1.18 1.14  Const. E I 1.55 1.54 1.52 1.48 l.io 1.29 1.01 2.38 1.46 1.28 1.28 1.46 2.38 1.01 1.29 1.40 1.48 1.52 1.54 1.55  Var. EI 1.62 1.69 1.69 1.66 1.59 1.47 1.18 10.33 1.66 1.39 1.39 1.66 10.33 1.18 1.47 1.59 1.66 1.69 1.69 1.62  APPENDIX IV MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A UNIFORM LOAD WHICH PRODUCES MAXIMUM BENDING MOMENT f _ 1 L " 8  X  L  0 .05 .10  .15 .20 .25 .30 .35  4o 45  • 5o  X  3  wL Const. E I  16.76  8.57 3.96  k-p  7.5k 9.12 9 42 8.56 7.07 5.81 540  K  1000 = Var. EI  2k. 16  4-72  7.88  k.25 k.35 6.16 6.99  6.80 5.78  k.85  k.ko  4 for Const. EI  1.08 1.13 1.13 1.06 1.10 1.12  1.4 1.4 1.13 1.09 1.08  J*>  =3  4 for  Var. EI  1.09 1.12 1.15 1.13 1.07 1.13 1.16  1.17 1.16  1.12  1.10  Const. E I  1.30 146 l.kk  1.18  1.3k  l.kk  1.50 1.52 149 1.33 1.25  p=5 Var. EI  1.31 146 1.59 i.5o  1.23 147  1.61  1.65 1.65 145 1.3k  4 for  p-1  Const. E I  2.07 2.6k 2.63 147 2.20 2.60  2.82 2.95 2.85 2.17 1.61  APPENDIX IV MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A UNIFORM LOAD WHICH PRODUCES MAXIMUM BENDING MOMENT  X  L  0 .05 .10 .15 .20 .25 .30 .35  .1+0  45  .So  X* wL  Const. EI  16.53 8.36  3.81+  1+.85 7.61 9.20  9.1+9 8.61+  1000 =  rf f o r yQ =  3  rf f o r  p= 5  rf f o r  p-  7  e  7.11 5.87 5.1+6  Var. EI  Const. EI  23.89 H+.50 7.72 1+.17 1+43  1.09 1.11+ 1.13 1.07 1.11  7.07 6.85  1.15 1.15 1.11+ 1.10 1.08  6.26  5.81+  1+.90  1+.1+5  1.U+  Var. E I  1.09 1.13 1.16 1.11+ 1.09 1.11+ 1.17 1.18 1.17 1.13 1.10  Const. E I 1.31+  1.52 1.1+8 1.22 1.1+0  1.1+9 1.55 1.57 1.52 1.35 1.26  Var. EI  1.35 i.5i 1.65 1.55 1.26  1.52 1.67 1.72 1.70 1.1+8 1.36  Const. E I  2.31 3.03  2.91+  1.63 2.52 2.97 3.20 3.32 3.17  2.31+  1.67  APPENDIX I V MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A UNIFORM LOAD WHICH PRODUCES MAXIMUM BENDING MOMENT f L  =  1 IT  X  L  0 .05 .10 .15 .20 .25 .30 .35  4o  45 .50  2B  wL*  x  Const. E I  16.03  7.95 3.59 5.0k 7.81 9.36 9.63 8.76 7.22 6.02 5.66  1000 = Var. EI  33.30 14.01  7.36 3.97  k.61  6.k6 7.24 6.97 5.95 5.04 4.59  rf  for  Const. EI  1.12 1.18 1.15 1.10 1.15 1.17 1.18 1.18  1.16  1.11 1.09  p= 3  rf  Var. EI  1.12 1.17 1.20 1.16 1.10 1.18 1.21 1.21 1.19 1.14 1.12  for  Const. EI  1.47 1.74  1.62  1.36 1.59 1.68 1.74 1.74 1.66 1.42 1.31  p=5 Var. EI  1.47 1.70 1.87 1.68 1.38 1.72 I.87  1.92 1.86 1.57 143  APPENDIX IV MAGNIFICATION FACTORS FOR HINGELESS ARCHES UNDER THE ACTION OF A UNIFORM LOAD WHICH PRODUCES MAXIMUM BENDING MOMENT f  1 3  =  L  x L  0 .05 ,10 .15 .20 .25 .30 .35 .40  ¥  .5o  x 1000 = wL Const. E I  15.58 7-57  3-i+l  5.21 7.96 9.50 9.35 8.89 7-35 6.22 5.88  Var. EI  22.74  13.55  7.02 3.82  4-77 6.64 7-40 7-06 6.07 5.19 4.78  $ for Const. E I  1.16 1.25 1.18 1.15 1.21 1.23 1.23 1.23 1.19 1.13 1.11  £=3 Var. EI  1.15 1.22 1.25  1.18  1.15  1.23  1.26 1.26  1.23  1.16  1.14  jzf f o r Const. EI  1.76 2.18 1.83 1.66 1.99 2.08 2.08 2.06 1.92 1.55 1.39  yB= 5 Var. E I  1.73 2.08 2.33 1.89 1.68 2.11 2.29 2.31 2.17 1.74 1.53  

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