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Analysis of flexible arches Sled, John James 1959

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ANALYSIS OF FLEXIBLE ARCHES  by JOHN JAMES SLED B.Sc.  ( C i v i l Engineering), University of Saskatchewan, 1955  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in the Department of CIVIL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1959  ii ABSTRACT  A method of analysis of f l e x i b l e arches under the action of a x i a l deformation, support movements, and f a b r i c a t i o n errors by the d e f l e c t i o n theory method i s presented i n this thesis.  The e l a s t i c moments  and dimensionless magnification factors f o r parabolic hingeless arches with rise-span r a t i o s of 1/8, 1/6, 1/4 and 1/3 are given. Although the data i s given f o r parabolic hingeless arches with a constant EI and one prescribed v a r i a t i o n of EI i t i s shown, by B  numerical examples, that the tables may be used f o r other arches whose shapes do not d i f f e r greatly from a parabola and, by i n t e r p o l a t i o n , to other variations of EI.  I t i s also shown that these solutions f o r hingeless  arches may be used to obtain the solution of one and two hinged archeso I t i s shown by theory and by numerical tests that the d e f l e c t i o n theory moments are d i r e c t l y proportional to the magnitude of the a x i a l deformation, support movement, or f a b r i c a t i o n e r r o r .  I t i s also  shown that these moments, when determined separately, may be added to each other and to moments due to load to obtain the correct t o t a l moment. The solutions i n the tables were calculated by a numerical procedure of successive approximations. ALWAC I I I E  (  The e l e c t r o n i c computer, the  at the University of B r i t i s h Columbia was used to perform  the large amount of numerical work required.  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  the r e q u i r e m e n t s f o r an advanced degree at the  University  o f B r i t i s h Columbia, I agree t h a t the  L i b r a r y s h a l l make  it  study.  freely  a v a i l a b l e f o r r e f e r e n c e and  agree t h a t p e r m i s s i o n f o r e x t e n s i v e f o r s c h o l a r l y purposes may  I further  c o p y i n g of t h i s  be g r a n t e d by t h e Head of  Department o r by h i s r e p r e s e n t a t i v e s .  g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n  A  ^Uf^  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 3, Canada.  my  I t i s understood  t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r  Department of 6-.^WfS  thesis  financial  permission.  iii  TABLE OF CONTENTS PAGE INTRODUCTION  1  CHAPTER I  THE GOVERNING DIFFERENTIAL EQUATION  4  CHAPTER II  PROGRAM FOR AUTOMATIC COMPUTATION  17  CHAPTER III  NUMERICAL VERIFICATION OF THEORY  24  Part 1  V a r i a t i o n of Moment With A x i a l Deformation  25  Part 2  V a r i a t i o n of Moment With V e r t i c a l Support Movement  27  Part 3  V a r i a t i o n of Moment With Rotation of a Support  Part 4  Comparison of Uniform A x i a l Deformation,  30  Fabrication Error Producing a Horizontal Gap at the Crown, and Horizontal Movements of Supports Part 5  32  Comparison of a Fabrication Error Which Produces a V e r t i c a l Gap at the Crown With a V e r t i c a l Movement of a Support  Part 6  Superposition of A x i a l Deformation and Support Movements With Load  CHAPTER IV  34  39  NUMERICAL DATA  Part 1  Tables of Magnification Factors  46  Part 2  Deflection Theory Moment Curves  59  Part 3  Influence Lines f o r E l a s t i c Thrust  76  CHAPTER V  NUMERICAL EXAMPLES  Part 1  Analysis of a Hingeless Arch  78  Part 2  Analysis of a Two Hinged Arch  85  iv  ACKNOWLEDGEMENT  The author wishes to express his sincere thanks to h i s adviser, Dr. R. F. Hooley, for h i s expert guidance and help.  The  contribution of h i s great knowledge and valuable time was greatly appreciated. The author also wishes to thank the s t a f f of the Computing Centre at the University of B r i t i s h Columbia for their assistance i n the use of the computer. This research was carried on during the summer of 1959 and was sponsored by the National Research Council of Canada. f i n a n c i a l assistance was g r a t e f u l l y received.  September, 1959 Vancouver, B r i t i s h Columbia  Their generous  V  NOTATIONS  (i  F l e x i b i l i t y factor of the arch  A  Fabrication error of the arch  (SH  Horizontal d e f l e c t i o n of a support  6y  V e r t i c a l d e f l e c t i o n of a support  £  Horizontal component of d e f l e c t i o n  *l  V e r t i c a l component of d e f l e c t i o n  9  Angle which the tangent to arch axis makes with a horizontal line  d©  Change of 8 due to moment  A  Coefficient of a x i a l deformation  0  Magnification factor  U)  Rotation of a support  E  Modulus of e l a s t i c i t y  f  Rise of arch  H  Horizontal thrust  1  Moment of i n e r t i a  I  a v  £( )  Average moment of i n e r t i a Linear  function  L  Span of arch  MQ  Deflection theory bending moment  Mjr  E l a s t i c theory bending moment  P  Concentrated load  w  Intensity of load on arch  Uniform loading on arch V e r t i c a l shear Horizontal coordinate of arch axis V e r t i c a l coordinate of arch axis  1  INTRODUCTION  The elementary and most commonly used theory of arch analysis Is the e l a s t i c theory.  This theory neglects the e f f e c t of  changes i n geometry on the stresses i n the arch.  The d e f l e c t i o n  theory takes these changes of geometry into account.  In f l e x i b l e  arches the geometry changes, when combined with the a x i a l thrust, produce moments of s i g n i f i c a n t magnitude. The f l e x i b i l i t y of an arch i s measured by the dimensionless r a t i o (i =J £j  .  The secondary moments due to d e f l e c t i o n may  usually be neglected i n fixed arches with a value of ft less than 1. A load which produces a /3 of 7 i s approaching the c r i t i c a l buckling load. The behaviour of an arch under load i s similar to that of a beam under a l a t e r a l load and an a x i a l thrust.  In both cases  the deflections combined with the a x i a l forces produce additional moments which i n turn produce additional deflections and moments u n t i l the structure comes to equilibrium. In the case of a beam under load i t i s possible to solve the d i f f e r e n t i a l equation and obtain an exact expression f o r the stresses i n the beam*.  In the case of the arch, however, due to the  complexity of the d i f f e r e n t i a l equation a general solution i s not available i n tabulated functions and numerical methods must be used. Lee and Pelton  presented tables and graphs to  f a c i l i t a t e the analysis of hingeless f l e x i b l e arches under v e r t i c a l  2  load.  This work i s to be extended to include the e f f e c t s of a x i a l de-  formation and support movements. The r e s u l t s are presented as dimensionless magnification factors.  A magnification f a c t o r i s the r a t i o of the d e f l e c t i o n theory  moment to the e l a s t i c theory moment.  Tables for parabolic hingeless  arches having rise-span r a t i o s of 1/8, 1/6, 1/4, and 1/3 are given i n Chapter IV. Values of the magnification factors 9 are given f o r /9 13 •= 5, and i n some cases f or ft *» 7. moments are also included.  s3  3,  Graphs of the d e f l e c t i o n theory  The tables and graphs are given f o r a constant  EI and one p a r t i c u l a r v a r i a t i o n of EI.  For other v a r i a t i o n s of EI  i n t e r p o l a t i o n i s necessary. Theory and numerical examples are presented  to show that  the magnification factors are p r a c t i c a l l y independent of the magnitude of the a x i a l deformation  or support movements.  I t w i l l be shown that  these moments may be superimposed with the moments due to other loadings. The object of this thesis i s to present magnification f a c t o r s , not d e f l e c t i o n theory moments, since the moments are s e n s i t i v e to v a r i a t i o n s i n EI while the magnification f a c t o r s are not nearly as sensitive.  Also, for arches whose axes d i f f e r l i t t l e from a true  parabola i t may be assumed that their magnification factors are approximately presented  the same as those f o r the parabolic arches which are  i n this thesis, although their e l a s t i c moments may d i f f e r .  I t i s assumed that the designer can f i n d the e l a s t i c moments of the arch and choose the proper magnification f a c t o r s , by i n t e r p o l a t i o n , f o r r i s e span r a t i o s and v a r i a t i o n s of EI other than those given. The program f o r the e l e c t r o n i c computer, the ALWAC III E,  3  used by Lee and Pelton was altered to enable the c a l c u l a t i o n of the e f f e c t s of a x i a l deformation and support movements. modifications are given i n Chapter I I .  Details of the  4  CHAPTER I THE GOVERNING DIFFERENTIAL EQUATION  The action of an arch under load, a x i a l deformation, and movement of the supports i s best studied by i t s d i f f e r e n t i a l equation. From the d i f f e r e n t i a l equation certain conclusions w i l l be drawn which w i l l be v e r i f i e d by numerical examples i n Chapter I I I . W-C/X  M  1-  H  C/M  Figure(1-1) In Figure ( l - l ) an element of an arch, i n i t i a l l y i n position  A B  T  deflects under the action of a load w to a position  AJBJ,  5 The horizontal d e f l e c t i o n of point A i s •? and the v e r t i c a l d e f l e c t i o n i s £.  The forces are shown acting on the element i n i t s deflected p o s i t i o n .  The moment, v e r t i c a l shear and horizontal thrust acting on the element are M, V, and H respectively.  Equations of S t a t i c s 5>  - 0  x  Since the load i s assumed to act v e r t i c a l l y the horizontal thrust H i s constant. ZFy  - 0 w dx + dV - 0 or  V' ™ -w  where ' represents _d_ dx  ^M H(dy  -  B I  0  • d?) - V(dx + d?) + wdx ~  + dM - 0  A f t e r d i v i d i n g by dx we obtain H( « y  + ? ') - V ( l + 4») • M« - 0 To combine the preceeding equations the l a t t e r must be  divided by 1 y'  + ? » 1 • «•  J -  V +  M«  o  0  1 D i f f e r e n t i a t i n g with respect to x we obtain  i+S«  ( i • s «) ^  1+s*  z  (i +  Multiplying by ( l + £ ' ) and substituting f o r V M"  -  M  '  1 +£ •  +  H(y" ••?") . H V '  M  '  g"  1 +£»  -  -w(l + ^ » )  gives (la)  6 Equations of A x i a l Deformation The i n i t i a l length of the element AB i s ds  -  /(dx)  • (dy)  2  2  The length of the element a f t e r deformation i s  J (dx + d?) • (dy • d ? )  dsi -  2  2  Assuming that the change i n length of the element due to temperature change and/or a x i a l thrust i s A d s : d s ( l • A) or  J(dx)  «= d s  t  + ( d y ) ( l + A)  2  2  - J (dx)  2  + 2dxd£ + ( d £ )  2  + ( d y ) • 2dyd? + ( d ? ) 2  Squaring both sides gives (dx)  2  + (dy)  2  + (dy)  2  1 + 2A + A | -  ( d x ) • 2dsd£+ ( d £ ) + ( d y ) + 2dyd? + ( d ? )  2  2  2  2  or (dx)  2  2X+  V  - 2dxd£  Dividing by (dx) 1  +  (y') ] [2 A 2  A J  +  -  2  +  (d£)  • 2dyd?  2  +  (d'?)  2  we obtain  2£»  +  (E»)  2  •  2y»f«  •  ( »?')  2  Since the change i n length of the element w i l l be small,  X.  2  may be neglected compared to 2 A and we obtain x[i  +. ( y » ) ] 2  c  »  +  (g')  2  •  •  +  (?  ')  2  (2a)  Equations of Bending Deformation  From Hooke's law the r o t a t i o n of Bj with respect to A^ due to moment i s d(d©)  -  M EI  ds  + dsi 2  2  2  7 or M EI  d(dO)  dx cos 6  1  •  *  7  A temperature change of 80 degrees or an a x i a l stress of 15,000 p . s . i . i n a s t e e l arch gives a value of A of .0005; hence * may be 2 neglected when compared to 1. Dividing by dx gives  (de)'  -  M_ EI cos § Before  tan §  ™  dy_ dx  the load i s applied :  •» y»  After the load i s applied ' tan (8  +  d9) dx  or  •  tan (9  de)  •  d£  • *l + e.'  <° v' l  '  The change i n slope i s d©  •» arctan  / v' 1  * +  C  -  arctan (y')  A f t e r s u b s t i t u t i o n we obtain _, »  arctan / y* 1  + •  *\ C  arctan ( y ) f  M EI cos 8  (3a)  In summary then, the three basic equations of an arch are' M" l  M» S" +  +  H(y"  + ? ")  -  H/y' \ 1  ( g') + y» 2  ( .)2 y  2  arctan  \ I  • s »/  -  arctan (y')  • I'V" + e. *)  *= -w(l  + illi.  2  + £•)  <la) (2a)  2  M EI cos ©  (3a)  8 These equations contain 4 unknowns £ , ®2 , M and H.  The  thrust H i s determined from boundary conditions and the other three unknowns may be obtained by the solution of the three simultaneous equations. equations are highly non l i n e a r .  These  Their solution f o r an arch would correspond  to the " e l a s t i c a " problem of an i n i t i a l l y straight column. In applying these equations to an arch of normal proportions certain approximations may be made. d e f l e c t i o n may be approximated  If i t i s assumed that the horizontal  by the expression £  «* £  0  s i n ^jp- and the  maximum horizontal d e f l e c t i o n i s —h— the maximum value of C' i s -JL- . 2T5£F 200 This i s small compared to 1 and may be neglected i n equations ( l a ) and (3a).  Assuming the maximum deflections are 0  ^ i^*^  200  , the terms ,{  2  and  are very small compared to the f i r s t power of these terms and  be neglected i n equation (2a).  may  In equation (3a), since d8 i s small, the  tangent of d8 may be assumed equal to d8.  Applying these approximations  the three equations become M"  +  A.[l *l"  M»<£"  +  H (y"  ( y ' ) - £*  +  2  -  +  9( ")  -  H(y«  + *?•)£"  «  -w  • y**l*  (lb)  (2b) (3b)  M  EI cos 8  To obtain a better understanding of the action of a f l e x i b l e arch these three equations w i l l be combined into one equation. Rearranging equation (2b) we obtain  and  C"  -  2Ay'y ' -  -  y»<?"  -  y"  Rearranging equation (3b) we obtain M  -  EI cos 8*?" r  and  M" .  ~I "  EI cos 8<? "  9 Substitution into equation (lb) gives [EI COS 8 ? "] " .  H( » y  +  M' C,"  + Jp«)  +  H(y" + ^ ")  (2Ay'y"  y' V"  -  - y"^')  Multiplying and c o l l e c t i n g terras [EI cos e ? " ]  "  +  -  2A<y') ]y"  +  H [l  +  H[y»^»^"  M«£" +  H [l +  2  +  +  H [l  y"(7«) ] 2  w  gives  (y*) ]*?" 2  -  2A] y ' y " f  -w  (4a)  I t i s d i f f i c u l t to show that the term M» £ " i s small compared to the other terms, however, tests made with the computer show that the e f f e c t of this term i s small and may be neglected when the deflections are small.  The e f f e c t of this term w i l l be discussed further i n Chapter I I I .  I f the rise-span r a t i o — i s 1/3 and the c o e f f i c i e n t of a x i a l deformation i s .0005 (a temperature  change of 80°), the value of 2 A i s .001 and the  maximum value of 2A-(y')  2  i s .0018.  pared to 1 and may be neglected.  Both these terms are very small com-  The terms due to a x i a l deformation  thus  disappear from the d i f f e r e n t i a l equation and must enter the solution of the equation only as boundary conditions. The terms containing products of d i f f e r e n t i a l s of *} are small compared to the other terms and may be neglected. The d i f f e r e n t i a l equation now becomes +  +  -  W  - Hy  To f a c i l i t a t e the study of this equation the variables w i l l  +  W  -  j-> y  +  _ y  =  y  L* * ANC  »  10 T3  yt-l  o  and l e t ... - ft *• where av  If we multiply the equation by E I  2  av  E I  ft i s a dimensionless r a t i o defining the f l e x i b i l i t y of the arch, we cos e ^ " J -  -  "  wlP  EI  ft [ 1 1  + -  a  •  (y»)  *)" * ft  2  2  y'y"  obtain  ^»  2»  (4c)  y  a y  I f fi and the d e f l e c t i o n s are both small, the terms which are products of  ft  and the derivatives of ^ are small.  Neglecting these  two  terms gives the equation  wL  I cos 8 fj lav  n  3  - /3y''  <4d)  2  EIav  Equation (4d) i s a fourth order l i n e a r d i f f e r e n t i a l equation with variable c o e f f i c i e n t s .  I t i s the d i f f e r e n t i a l equation of the e l a s t i c  theory of arch analysis and moments obtained by the solution of t h i s equation are the e l a s t i c theory moments. If V and and the moments obtained  the d e f l e c t i o n s are large we have a f l e x i b l e arch from the s o l u t i o n of equation (4c) are the  d e f l e c t i o n theory moments.  Equation (4c) i s a l i n e a r d i f f e r e n t i a l  equation but the structure i t governs i s non-linear.  This Is due  to the  fact that an increase i n the load also changes fi and thus the c o e f f i c i e n t s of the equation as well as the d e f l e c t i o n s .  Because of t h i s change of  c o e f f i c i e n t s , superposition i n the ordinary sense i s not possible. have shown that the thrust due  Tests  to load determined by the e l a s t i c and  d e f l e c t i o n theories are almost the same.  Thus i t i s possible to use  r e a d i l y obtained e l a s t i c thrust In equation (4c). modified method of superposition, to be described  the  Using t h i s thrust a l a t e r , i s possible.  I t w i l l be seen that the larger the value of ft , the larger w i l l be the difference In the moments determined from equation  11 (4c) as compared to those determined by equation (4d).  The r a t i o of the  d e f l e c t i o n theory moments, determined by equation (4c), compared to the e l a s t i c theory moments, determined  by equation (4d), are the magnification  factors. The term/3 ?" i n equation (4d) represents the e f f e c t of 2  horizontal thrust i n the arch.  I f this term i s neglected, the equation  of an arch whose ends are unrestrained horizontally i s obtained : i cos e lav  wL3 Flav  The fact that an a x i a l deformation enters into the solution of the d i f f e r e n t i a l only as a boundary condition indicates that a uniform a x i a l deformation A i s equivalent to a f a b r i c a t i o n error which produces a gap at the crown of A c SH  = AL.  «• A.L or a horizontal movement of a support of  This was found to be true by tests on the computer.  Results  of these tests are given i n Chapter I I I . To i l l u s t r a t e the p r i n c i p l e of superposition consider an arch, shown i n Figure (1-2), which i s loaded by a variable load w^ (load l ) . I t i s desired to combine t h i s loading with a rotation Ct) of the l e f t support (load 2) to obtain load 3.  To superimpose these two cases the horizontal  thrust of each individual case must be made equal to the thrust of the combination.  I t w i l l be assumed that the e l a s t i c theory and the d e f l e c t i o n  theory thrusts are equal i n a l l cases.  A uniform load qj which produces  a thrust equal to the thrust produced by the rotation must be added to the variable load w^. For convenience, equation (4c) w i l l be designated as  L^)  +  tf L (H) 2  2  -  -  _WL3  EIav  -/? y' 2  (5)  12  4  Load 3 Figure (1-2)  13 Letting the d e f l e c t i o n due to the load (wj  +  q^) be ^ a ,  the d i f f e r e n t i a l equation f o r case A becomes  L W ) + /3 L (? ) - - <„. + qi) J s L - /3 r 2  x  C5a)  2  a  fl  FI v 3  where Q  o  2 "3 Elav  =>  L  A uniform load q2 which produces a thrust equal to the thrust produced by the variable load must be combined with the r o t a t i o n to give the correct horizontal thrust. q2 alone be ^  Letting the d e f l e c t i o n due to r o t a t i o n and  the d i f f e r e n t i a l equation f o r case B i s  LltfyJ * /3 L (*? ) - - q _ L L - / y " 2  <5b)  ?2  2  h  2  Elav where  ft  -  2  3 Elav  h  l 2  To combine case A and case B (qj  •  q ) must be considered.  Letting  2  the case of a uniform load  9  f} be the d e f l e c t i o n due to this c  loading the d i f f e r e n t i a l equation i s £l<?C>  # £  +  2  ( 2  ?C>  -  -  (qi  - BT  q )  +  2  2  <5c)  Elav where (3  -  1  H  3 Elav L  Letting tfd be the d e f l e c t i o n due to load 3, the combination of the load w^ and the rotation CaJ, the d i f f e r e n t i a l equation becomes Li<?d) where / ?  2  •/? £ <£ >  -  2  2  -  d  - «i _ t L Elav  - /? y" 2  ( 5 d  >  3 Elav  H  l 2  Adding equations (5a) and (5b) and subtracting equation (5c) we obtain  Ll<fm  H  +  -  -  w  ' l  ^c>  l 3  Elav  -  /?V?a  +  2  ?"  +  H  -  7c> (6  >  14 The addition and subtraction of these equations i s only possible because care was taken to make the value offti d e n t i c a l i n each equation.  Comparing equations (5d) and (6) we see that  - U  ^"b - ^ c  +  In equation ( 5 c ) assuming a parabolic arch t  H3  (qj + q )L2  m  2  M and  (i 2  (  + <*2>  q i  "av also  y  •  and  y  -  Therefore  -  (Lx L  x  * 8fEI L  a v  2)  2  4f (x L  x2)  y"  8f  -  -  r  In arches which have a shape other than parabolic, the d i s t r i b u t i o n of load which produces the correct thrust and zero bending moment must be used. Substituting these quantities into equation (5c) gives  M?">  +  /3 L ^c>  +  2  (q! •  q ) 2  - <qi q> J L +  -  2  c  2  8fEI  8f L  a v  EI  a v  (5c)  or  L < YV  •  X  tf L (? > 2  2  c  -  o  For loads less than the c r i t i c a l buckling load the only solution to t h i s equation i s ?d  -  9  -  ?a  a  •  9b  +  ?b  c  -  0, hence  and  From the equation (3b) the moments are proportional to the  15 deflections and Mud  -  MDa  -  M  -  M  F  Ea  MDb  +  x  A  x  ^ a MEa 0a  +  +  M  MES  Eb  MOb MEb  * 0b  Thus the d e f l e c t i o n theory moments may be determined by multiplying the e l a s t i c theory moments by magnification f a c t o r s . The t o t a l e l a s t i c thrust of the combination must be used i n determining the magnification f a c t o r s . tabulated i n Chapter V.  The magnification factors 0b due to r o t a t i o n are The magnification factors 0a due to load are  tabulated i n the Appendix of Ref. 2. A more d i f f i c u l t problem of superposition i s the analysis of a loaded arch hinged at one support.  The moments i n t h i s arch may be  determined by superimposing the solution determined f o r the r o t a t i o n of a support of a hingeless arch with the solution found f o r the loaded hingeless arch.  To obtain the hinged condition the moments produced i n the  loaded hingeless arch must be superimposed with a r o t a t i o n of a support which produces a moment at the support moment produced by- the load.  equal and of opposite sign to the  The r o t a t i o n i s unknown and since the  rotation produces a thrust, the magnification f a c t o r s which must be applied to the e l a s t i c moments are unknown. To find the moments i n this arch, i t i s necessary to assume a rotation and calculate the t o t a l e l a s t i c thrust in the arch due to the load plus the assumed r o t a t i o n .  Using  t h i s thrust,  the magnification factors to be applied to the e l a s t i c moments at the hinged support are chosen.  If the e l a s t i c moments due to load and  rotation m u l t i p l i e d by the proper magnification factors are not of opposite sign and equal, a new value of the r o t a t i o n must be assumed  16 and the procedure repeated.  When the r o t a t i o n at the hinged support and  the thrust i s determined, the proper magnification factors may be applied to the e l a s t i c moments at the other points i n the arch to determine the governing moments. I t was found that i t i s permissible to neglect the thrust due to moment for small r o t a t i o n s .  17  CHAPTER II PROGRAM FOR AUTOMATIC COMPUTATION  To check the superposition  p r i n c i p l e s and to produce the  tables and graphs presented i n Chapter IV, the program for the ALKAC III E e l e c t r o n i c computer, which was used by Lee and Pelton, was altered to enable the e f f e c t to a x i a l deformation and support movements to be determined.  This program finds the d e f l e c t i o n s , moments and thrust i n the  arch by the conjugate beam method.  The program begins by finding the  e l a s t i c theory deflections and forces with the deflections assumed to be zero.  The deflections theory moments and thrust are found by successive  application of the conjugate beam formulae using the geometry altered by the deflections determined i n the previous cycle. The program i s set up to analyse an arch divided into 20 d i v i s i o n s not necessarily of equal length. shape.  The arch may have any symmetrical  The location of the end of each d i v i s i o n must be s p e c i f i e d .  In  each d i v i s i o n the f l e x u r a l r i g i d i t y i s constant; however, any r i g i d i t y may be assigned to each d i v i s i o n .  The loads are assumed to act at the centre  of each d i v i s i o n and they may have any desired d i s t r i b u t i o n .  The moment  and d e f l e c t i o n at the end of each d i v i s i o n , the shear at the crown, and the horizontal thrust i s determined by the computer.  The complete d e t a i l s of  the program are given i n Ref. 2. A diagram of the arch i s given i n Figure (2-1). The  steps which the program follows i n computing the moment,  thrust, and shear at the crown and then the moment and deflections at the  Forces  ,  ft  »,  i  and Geometry of  V JJ - J  IIOR  fe ts  the In  Arch i)n  In  1l  Deflections  of  the  Arch 03  Figure (2-1)  19 end of each d i v i s i o n , using the geometry altered by the deflections determined i n the previous cycle, are as follows: (1)  Assuming the arch cut at the crown, the horizontal and v e r t i c a l deflections and the r o t a t i o n at the crown due to a unit thrust applied to each half of the arch at the crown are found.  (2)  Horizontal and v e r t i c a l deflections and the r o t a t i o n at the crown due to a unit shear applied to each half of the arch at the crown are found.  (3)  The horizontal and v e r t i c a l deflections and the rotation at the crown due to a unit moment applied to each half of the arch at the crown are determined.  (4)  The horizontal and v e r t i c a l deflections and the r o t a t i o n at the crown due to the applied loads on each h a l f of the arch are calculated.  (5)  The r e l a t i v e horizontal and v e r t i c a l deflections and rotations at the crown due to the unit loads and the actual loads are determined.  (6)  The c o e f f i c i e n t s required f o r the three simultaneous equations necessary to f i n d the moment, thrust, and shear at the crown are determined.  (7)  The three simultaneous equations are solved by i t e r a t i o n to f i n d the moment, thrust, and shear at the crown.  The computer types out the  values of M , H,and V" . By typing the proper code, the computer may c  c  be made to perform another cycle of i t e r a t i o n of the three simultaneous equations and type out new values of M  c >  H,and V" or to continue on c  in the program. (8)  Having the moment, thrust,and shear a t the crown, the moment, horizontal d e f l e c t i o n , and v e r t i c a l d e f l e c t i o n at the end of each d i v i s i o n of the  20  arch i s now (9)  computed.  By the proper setting of the jump switches, e i t h e r the moments or  the  deflections or both the moments and deflections at the end of each d i v i s i o n may  be typed  out.  (10) The deflections which have been determined are added to the  initial  geometry for use i n computing the moments i n the next cycle.  The  new  deflections divided by 2 are added to the i n i t i a l geometry f o r obtaining (11) The  the deflections due  to the moments.  cycle i s repeated u n t i l the arch comes to equilibrium i n the  deflected  position. The a x i a l deformation and support movements being boundary  conditions, i t i s permissible occur.  to neglect  Since i t i s the r a t i o ~  the small changes i n ds which  which i s used i n the conjugate beam method  EI  of a n a l y s i s , a temperature change of 80 degrees i n a steel or concrete arch i s equivalent  to a change i n the moment of i n e r t i a l of .05 percent which i s  negligible. The free deflections of the half arches due  to a x i a l  deformation and support movements must be added to the deflections r e s u l t i n g from the applied  loads.  To include these deflections the program must be  altered i n three places. To enter these deflections into the main memory of computer i t was  necessary to add a d e f l e c t i o n input routine.  i s stored i n channel 95 with the l a s t few  routine  instructions in channel 96.  routine i s begun by the command 9500 carriage return.  The  The routine accepts  the a x i a l deformation and support movement deflections and the following sequence:  This  the  rotations i n  21  *1.  719»  *2*  »  »  9  ^18»  »  i  »  ?2»  »  >  *  »  *  »  fyl8»  >  »  >  •  * lOL*  »  »  .  *10R.  »  »  »  »  ^10L»  >  >  »  »  ?10R»  (Check Sum) Horizontal deflections  t  o  Z\c\i  a  r  e  P  o s i t i v e  t  o  Horizontal deflections £jg to £JQ are positive to the r i g h t . r  deflections are p o s i t i v e upward. &>L positive clockwise.  i  t n  e left.  Vertical  p o s i t i v e counterclockwise. (t)^ i s  s  During input the computer forms the sum of the forty  deflections plus 4 times the sum of the two rotations and then outputs the difference between this sum and the check sum.  The same storage scheme as  that used i n the o r i g i n a l program i s used i n storing the deflections i n the channels.  The deflections £j to C^gL  Deflections  £^9  to  CioR  a  r  e  s t o r e <  *  i  a  r  e  stored i n the channel CO.  channel C l .  n  Deflections  are stored i n channel C2 and deflections ^19 to ^JQR C3. 6^L and  to  *?JQL  are stored i n channel  are also stored in words If and l b respectively of channel 25  C3.  As i n the o r i g i n a l program the deflections are scaled 2 27  and the  rotations are scaled 2 No provision was made f o r consideration of deflections at the supports i n the o r i g i n a l program.  The deflections at the support are  assumed equal to zero and^when the geometry i s input, the coordinates of the support points are automatically  set equal to zero.  Rather than make  extensive program changes i t was found to be easier to introduce  a deflection  at the supports by setting the coordinates of the supports equal to the desired d e f l e c t i o n by making "one word changes".  The coordinates of the  supports are stored i n word 00 of their respective channels.  The horizontal  22  and v e r t i c a l coordinates of the l e f t support are stored i n channels a4 and a6 respectively.  The horizontal and v e r t i c a l coordinates of the r i g h t  support are stored i n channels a5 and al respectively. The coordinates, supports must be introduced  representing  the d e f l e c t i o n s , of the  a f t e r the e l a s t i c moment, thrust, and shear at  the crown i s computed but before the moment and d e f l e c t i o n of every point i s calculated. I t was found most convenient to do this by stopping the program a f t e r i t has typed out the moment, thrust, and shear at the crown of the f i r s t cycle. 2  2 5  i.e.  The deflections are entered i n hexadecimal form scaled  .10  ^  00333333  .20  ~  00666666  1.00  - ~ 02000000  Care must be taken i n choosing the correct signs. i s re-entered by typing the code 8b00 carriage return.  The program  The moment, thrust,  and shear are typed out again and the program continues i n i t s usual manner. The second instance where the program must be altered i s i n the computation of the r e l a t i v e d e f l e c t i o n s and rotations at the crown due to the unit loads and the applied loads (Step 5).  The r e l a t i v e deflections  and rotations at the crown due to a x i a l deformation and support movements must be included i n computing the moment, thrust, and shear at the crown. Since no provision was made i n the o r i g i n a l program for additional f a c t o r s , i t i s necessary to add the deflections and rotations at the crown due to a x i a l deformation and support movements to those which have been determined for the applied loads.  The actual a l t e r a t i o n i s accomplished by changing  word 12 i n channel 8a from 818bll00 to 81961107 and the addition of one  23 channel of program operating i n channel 96. The third a l t e r a t i o n of the program i s i n the computation of the deflections at the end of each d i v i s i o n (Step 8 ) . The deflections due to a x i a l deformation and support movements must be added to the deflections due to load.  This i s accomplished by changing word 05 i n channel 8f from  87a485d4 to 81971103 and the addition of one channel of program operating in channel 97. The three additional channels of program with entry codes are as follows: 9504 280a4940 871e5500 5blfll60 484b6103 49031708 8dc05500 5blbll60 484b6l03  49031718 8dcl5500 5bl7H60 484b6l03 49031709 8dc25500 5bl31l60 484b6l03  49031719 5b0fll60 495f6l03 49035b0b 1160495b 8dc36l03 81961102 00000000  00000000 00000000 001b0092 OOlbOOOa 001900Id 0019000d 0019001c 0019000c  9604 87db85c0 7978614a 497885cl 797c6l4a 497c85c2 7979614a 497985c3 797d614a  497d797a 615f497a 797e6l5b 497e8fdb 83408541 87420000 818bll00 00000000  49035blf 11606703 871f5blb H601b00 00000000 00000000 00000000 00000000  00000000 8b408d41 8f421100 00000000 00000000 00000000 0219058e 00190086  9704 85d487c0 55 If784b 606b484b 17848dd4 85d587cl 55 If784b 606b484b 17948dd5  85d687c2 55 If784b 606b484b 17858dd6 85d787c3 551f784b 606b484b 17958dd7  83408541 87420000 87a485d4 818fU09 00000000 00000000 00000000 00000000  8b408d41 8f421100 00000000 00000000 00000000 00000000 00000000 OOOaOOOO  24 CHAPTER I I I  NUMERICAL VERIFICATION OF THEORY  With the program altered and checked, a number of runs were made to check the ideas set forth i n Chapter I.  S p e c i f i c a l l y we wish to  check the v a r i a t i o n of the d e f l e c t i o n theory moments with a x i a l deformation A » v e r t i c a l movement of a support 6y and rotation of a support (SL> .  Tests  were made to compare the moments due to a x i a l deformation A , horizontal movements of the supports SJJ and a horizontal gap at the crown £ J I . c  The  moments due to a v e r t i c a l support movement were compared to the moments produced by a f a b r i c a t i o n error which produces a v e r t i c a l gap at the crown. F i n a l l y , the superposition of a x i a l deformation and support movements with various load conditions are checked.  25 Part 1 Variation of Moment With A x i a l  Deformation  I t was found that A enters the solution of the d i f f e r e n t i a l equation of the arch only as a boundary condition. expected to X.  Hence i t would be  that f o r a constant /3 the moments s h a l l be d i r e c t l y proportional Several runs were made f o r various values of X f o r d i f f e r e n t  values of /3 .  The r e s u l t s are given i n Table  (3-1)  The r e s u l t s show that, for constant (i ,  i s for a l l XEI  p r a c t i c a l purposes a constant.  The magnification factors 0 are thus  independent of X and need only be computed f o r various values of  B .  These runs were made with a uniform load applied producing thrust to give values of (i equal to 3, 5 and 7 with the thrust due to the a x i a l deformation neglected.  This was found to be more correct than  including the e l a s t i c thrust due to a x i a l deformation i n the determination of 3 .  This i s due to the f a c t that an increase i n the length of the arch  produces a positive thrust when computed by the e l a s t i c theory; however, on subsequent cycles the increase In the rise-span r a t i o reduces  this  thrust or, i n the case of the more f l e x i b l e arches, the thrust due to an increase i n length becomes negative. to a x i a l deformation may  Therefore, the change i n thrust due  be neglected.  The v e r t i c a l d e f l e c t i o n at the crown i s : 1.56  XL  f o r /3 - 0  1.58  XL  for  fS " 3  1.62  XL  for  0  - 5  1.70  AL  for 3  - 7  -~-  DUE TO UNIFORM AXIAL DEFORMATION f - 1, CONSTANT EI L 6  ftX  L  A - -.001  ft-  3 A - .001  A - -.001  ft -  5 A - .001  A • -.001  A - -.0005  7 A - .0005  A - .001  .00  37.57  37.00  31.66  31.23  20.15  20.13  20.12  20.08  .05  28.19  27.86  27.15  26.83  24.24  24.17  24.06  24.00  .10  18.80  18.65  20.48  20.28  23.43  23.35  23.18  23.11  .15  9.84  9.82  12.56  12.47  18.41  18.33  18.15  18.09  .20  1.61  1.68  4.21  4.21  10.36  10.29  10.17  10.12  .25  - 5.64  - 5.52  - 3.87  - 3.79  .70  .68  .60  .60  .30  -11.74  -11.59  -11.12  -10.98  - 9.20  - 9.19  - 9.16  - 9.13  .35  -16.58  -16.42  -17.14  -16.96  -18.16  -18.10  -17.99  -17.92  .40  -20.09  -19.92  -21.63  -21.41  -25.23  -25.13  -24.93  -24.85  .45  -24.21  -22.04  -24.39  -24.15  -29.73  -29.61  -29.36  -29.26  .50  =22.92  -22.75  -25.32  -25.08  -31.27  -31.14  -30.87  -30.77  Table (3-1)  27 Part 2 Variation of Moment With V e r t i c a l Support Movement It i s impossible f o r v e r t i c a l support movements 5y  to enter  the solution of the d i f f e r e n t i a l equation of the arch other than as a boundary condition; hence i t would be expected that, f o r constant /3 , the moments w i l l be d i r e c t l y proportional to <5y.  ^  n e  r e s u l t s of several runs,  which were made to compare the moments produced by various values of 6 \j for a constant /3 , are given i n Table  (3-2)  The r e s u l t s of these runs show a small degree of nonlinearity.  The d e f l e c t i o n theory moment due to 6yj at a point i n the arch,  for constant /3 , may  MD - (  Cl  •  be expressed  as  c - v \ 6v 6  2  L / L  \  or i n terms of magnification factors as  The factor 0i i s a function of the f l e x i b i l i t y /3 of the arch as well as the geometry and d i s t r i b u t i o n of EI and thus i s s i m i l a r to the usual magnification f a c t o r .  The factor 0  2  i s a constant which i s  p r a c t i c a l l y independent of /3 but depends mainly on the rise-span r a t i o and d i s t r i b u t i o n of EI.  The term 0j i s much larger than the term 0  deflections of usual magnitude. half of the arch are equal.  -jY. f o r  The 0\B for corresponding points on each  The 0 s f o r corresponding points on each half 2  of the arch are of equal magnitude but of opposite sign. l i n e a r i t y appears to be mainly due to the term M ' £ Chapter I .  2  M  This non-  of equation ( l b ) of  This non-linearity appears only i n cases where there i s non-  28  symmetry and shear i n the arch.  At the points i n the arch where 0 2 - 0 ,  i . e . no non-linearity, both the second derivative of the horizontal deflections and the f i r s t derivative of the moments are very small.  It  w i l l be seen that this term i s independent of the f l e x i b i l i t y of the arch. Bearing i n mind that 6y  =» -.005L represents a support settlement of 1 foot  i n an arch of 200 foot span, this non-linearity may be neglected i n p r a c t i c a l cases. A v e r t i c a l movement i s p o s i t i v e upward.  The e l a s t i c thrust  due to the v e r t i c a l movement of one support i s zero and the d e f l e c t i o n theory thrust i s n e g l i g i b l e .  M  P  L  6 EI  DUE TO A VERTICAL MOVEMENT OF THE LEFT SUPPORT  V  f - 1, CONSTANT E I , /? . 5 L 6 X  L .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00  6vi  - -.005L  -2.45 -3.40 -4.04 -4.35 -4.35 -4.07 -3.56 -2.85 -2.01 -1.07 - .08 .91 1.86 2.74 3.48 4.04 4.37 4.43 4.18 3.58 2.67  6  V L  - -.002L  -2.51 -3.45 -4.08 -4.38 -4.36 -4.06 -3.54 -2.82 -1.97 -1.02 - .03 .96 1.91 2.77 3.50 4.05 4.36 4.40 4.13 3.53 2.60  <5  VL  *" -.OOIL  -2.53 -3.47 -4.09 -4.38 -4.36 -4.06 -3.53 -2.81 -1.95 -1.00 - .01 .98 1.92 2.78 3.51 4.05 4.36 4.40 4.12 3.51 2.58 Table (3-2)  6  V L  - .001L  -2.58 -3.51 -4.12 -4.40 -4.36 -4.05 -3.51 -2.78 -1.92 - .98 .01 1.00 1.95 2.81 3.53 4.06 4.36 4.38 4.09 3.47 2.53  6  V L  - .002L  -2.60 -3.53 -4.13 -4.40 -4.36 -4.05 -3.50 -2.77 -1.91 - .96 .03 1.02 1.97 2.82 3.54 4.06 4.36 4.38 4.08 3.45 2.51  6  V L  - .005L  -2.67 -3.58 -4.18 -4.43 -4.37 -4.04 -3.48 -2.74 -1.86 - .91 .08 1.07 2.01 2.85 3.56 4.07 4.35 4.35 4.04 3.40 2.45  30 Part 3 Variation of Moment With Rotation of a Support A r o t a t i o n of a support also enters the d i f f e r e n t i a l equation of the arch only as a boundary condition; hence i t may be assumed that, f o r constant /3 , the moment due to the rotation of a support w i l l be d i r e c t l y proportional  to the magnitude of the rotation (s). A number of runs were  made to determine the moments produced by various values of  The runs  were made f o r a constant /? , the thrust due to the rotation being taken into account i n the determination of fi .  The r e s u l t s are given i n Table  (3-3). The moments due to a rotation also show a small degree of non-linearity. MD  -  M  I f the d e f l e c t i o n theory moment i s expressed as E  (0x  neither 0^ nor 0 the arch. arch.  2  +  0 &J) 2  are equal for corresponding points on the two halves of-  The factor 0  2  also varies somewhat with the f l e x i b i l i t y of the  At points i n the arch where the second derivative of the horizontal  deflections i s zero the non-linearity i s n e g l i g i b l e .  There i s , however,  non-linearity at the points where the f i r s t derivative of the moments i s zero.  This suggests that the two terms involving the products of the  derivatives of the v e r t i c a l deflections i n equation (4a) also contribute to the  non-linearity.  ^ Ct>EI  DUE TO A ROTATION OF THE LEFT SUPPORT f - 1, CONSTANT EI, /3 «=• 5 L 6  X  L .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00  (t) m .03 L  -4.95 -5.34 -5.26 -4.72 -3.91 -2.90 -1.80 - .69 .35 1.24 1.95 2.44 2.67 2.63 2.31 1.73 .90 - .12 -1.26 -2.40 -2.43  U) - .01 L  -4.95 -5.32 -5.21 -4.69 -3.88 -2.88 -1.80 - .70 .32 1.20 1.90 2.38 2.62 2.58 2.28 1.71 .90 - .10 -1.21 -2.34 -2.36  cu  L  - -.003  -4.94 -5.30 -5.18 -4.67 -3.87 -2.88 -1.80 - .71 .29 1.17 1.87 2.35 2.59 2.56 2.26 1.70 .90 - .08 -1.19 -2.31 -2.32  Table (3-3)  & i  L  = -.01 -4.94 -5.29 -5.17 -4.66 -3.86 -2.87 -1.80 - .72 .29 1.16 1.86 2.33 2.57 2.54 2.25 1.69 .90 - .08 -1.17 -2.29 -2.30  dJl " -.03 -4.94 -5.26 -5.14 -4.63 -3.83 -2.86 1.80 - .74 .26 1.12 1.81 2.29 2.53 2.50 2.22 1.68 .90 - .06 -1.13 -2.23 -2.24  32 Part 4 Comparison of Uniform Axial Deformation  X, Fabrication  Error Producing a -Horizontal Gap at the Crown A H , C  and Horizontal Movement of Supports 6 H It was found that, for equal changes in horizontal span length, the moments produced by these three effects are practically equal. This MQL  confirms the findings of the theoretical analysis. A comparison of « T » 2 A ti. ^^ET' * O*~E"T * l °ffi given in Table (3-4). In a l l H 3N(  v a r  o u s  v  a  u  e  s  a  r  e  cases the thrust due to A , 8R or A ^ v a s neglected i n the determination of (3. The larger discrepancy of the moment at the crown due to a gap at the crown i s due to the fact that there is a discontinuity in the arch axis at this point because the two sections of the arch meeting at this point are too short.  In the case of a uniform axial deformation or  a support movement the two halves of the arch retain their parabolic shape. From these results i t may be concluded that any effects which produce or tend to produce changes in the span length of the arch may be represented by a uniform axial deformation which produces a free horizontal deflection of the same amount. A variable symmetrical axial deformation may be considered as a uniform axial deformation which produces the same horizontal free deflection.  COMPARISON OF J V ; , ^ AND D X E I AU^I • SttVI M  £ «• l L  L  X"  -.001  A H  CONSTANT EI  6  P -5  • 3 X  s  L  C  •»  .001L A - -.001  A H  /?-  C  - .001L  A -  -.001  6H "  7  -.0011 A H  C  » .001L  .00  37.57  37.56  31.66  31.68  20.15  20.13  20.23  .05  28.19  28.18  27.15  27.16  24.24  24.22  24.28  .10  18.80  18.79  20.48  20.48  23.43  23.42  23.43  .15  9.84  9.83  12.56  12.54  18.41  18.40  18.36  .20  1.61  1.60  4.21  4.18  10.36  10.36  10.28  .70  .70  .60  .25  - 5.64  - 5.64  - 3.87  - 3.90  .30  -11.74  -11.75  -11.12  -11.16  - 9.20  - 9.20  -  .35  -16.58  -16.59  -17.14  -17.18  -18.16  -18.16  -18.25  .40  -20.09  -20.09  -21.63  -21.66  -25.23  -25.22  -25.28  .45  -22.21  -22.21  -24.39  -24.41  -29.73  -29.72  -29.74  .50  -22.92  -22.77  -25.32  -24.91  -31.27  -31.26  -30.44  Table ( 3 - 4 )  9.31  34 Part 5 Comparison of a Fabrication Error Which Produces a V e r t i c a l Gap at the Crown A y  c  With a  V e r t i c a l Movement of a Support 6y The d e f l e c t i o n theory moments due to a f a b r i c a t i o n error which produces a total v e r t i c a l gap at the crown A.y were also determined. c  I t was  assumed that the free v e r t i c a l deflections of each half of the arch vary uniformly from zero at the supports to ± ^ V  c  at the crown, i . e . both  halves of the arch r e t a i n their parabolic shape although t h e i r rise-span r a t i o s are d i f f e r e n t .  An error which leaves the l e f t half of the arch  higher than the right i s considered p o s i t i v e .  The moments produced by  various values of ^ v f ° a constant ft are given i n Table (3-5). c  r  A com-  parison of the moments produced by a v e r t i c a l support movement and a v e r t i c a l gap at the crown f o r /3 » 7 are given i n Table (3-6). A comparison of Tables (3-2) and (3-5) and a study of Table (3-6) shows that the d e f l e c t i o n theory moments produced by these two conditions are p r a c t i c a l l y i d e n t i c a l .  The non-linearity of the moments  due to a gap at the crown i s l e s s . The moments due to a non-uniform temperature change i n an arch may thus be determined support movements.  by the superposition of v e r t i c a l and horizontal  The moments due to the horizontal change i n length are  equal to the moments produced by a horizontal movement of a support of the same amount. The moments produced by a difference i n the v e r t i c a l deflections of the two halves of the arch are equal to the moments produced by a v e r t i c a l support movement of the same amount. An example of this  method i s given i n Table (3-7). to those i n Part 6.  The table i s set up i n a s i m i l a r manner  —^wr ^V E I  DUE TO A VERTICAL GAP AT THE CROWN  C  f - 1 „ CONSTANT EI, L 6  X  L  A v - -.005L Ay *c  c  - -.001L  A\J  C  =. 5  - .001L  Aw  " .005L  .00  -2.54  -2.55  -2.56  -2.58  .05  -3.47  -3.49  -3.50  -3.51  .10  -4.09  -4.10  -4.11  -4.12  .15  -4.38  -4.39  -4.39  -4.40  .20  -4.36  -4.36  -4.36  -4.37  .25  -4.06  -4.06  -4,06  -4.05  .30  -3.52  -3.52  -3.52  -3.51  .35  -2.81  -2.80  -2.79  -2.78  .40  -1.95  -1,94  1.93  -1.92  .45  -1.01  - .99  - .99  - .97  .50  - .02  .00  .00  .02  .55  .97  .99  .99  1.01  .60  1.92  1.93  1.94  1.95  .65  2.78  2.79  2.80  2.81  .70  3.51  3.52  3.52  3.52  .75  4.05  4.06  4.06  4.06  .80  4.37  4.36  4.36  4.36  .85  4.40  4.39  4.39  4.38  .90  4.12  4.11  4.10  4.09  .95  3.51  3.50  3.49  3.47  1.00  2.58  2.56  2.55  2.54  Table (3-5)  COMPARISON OF  2 ft^ AND  2  f..- 1, CONSTANT EI, ft ~ 7 L 6 X  L  6v  L  • -.001L  Av  c  - -.001L Uniform  .00  3.62  3.61  .05  .51  .50  .10  -2.69  -2.70  . 15  -5.43  -5.44  .20  -7.34  -7.34  .25  -8.24  -8.24  .30  -8.09  -8.08  .35  =6.99  -6.98  .40  -5.12  -5.11  .45  -2.71  -2.69  .50  - .02  .00  .55  2.66  2.68  .60  5.07  5.09  .65  6.95  6.96  .70  8.05  8.06  .75  8.22  8.22  .80  7.33  7.33  .85  5.44  5.44  .90  2.72  2.71  .95  - .47  - .48  1.00  -3.58  -3.59  Table (3-6)  38 SUPERPOSITION OF HORIZONTAL AND VERTICAL MOVEMENTS OF THE LEFT SUPPORT TO GIVE THE MOMENTS DUE TO A UNIFORM AXIAL DEFORMATION OF THE LEFT HALF OF THE ARCH L ~ 200 f t . , f - 1, CONSTANT EI - 40,000 kip f t . , 0 - 7 L 6 Uniform Load 2  Col. 1 X  L .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00  Col. 2 5  H l  *" -.0005L -2.013 -2.422 -2.342 -1.840 -1.036 - .070 .920 1.816 2.522 2.972 3.126 2.972 2.522 1.816 .920 - .070 -1.036 -1.840 -2.342 -2.422 -2.013  Col. 3  <5v L  -.0005f  - .121 - .017 .090 .181 .245 .275 .270 .233 .171 .090 .001 - .089 - .169 - .231 - .268 - .274 - .244 - .181 - .091 .016 .119  Table (3-7)  Col. 4  2(2)  • (3)  -2.134 -2.439 -2.252 -1.659 - .791 .205 1.190 2.049 2.693 3.062 3.127 2.883 2.353 1.585 .652 - .344 -1.280 -2.021 -2.433 -2.406 -1.894  Col. 5 A  L  - -.001 -2.135 -2.434 -2.244 -1.650 - .784 .208 1.189 2.043 2.683 3.050 3.113 2.871 2.343 1.579 .651 - .340 -1.273 -2.013 -2.424 -2.401 -1.893  39  Part 6 Superposition of A x i a l Deformation and Support Movements With Load A number of runs were made to check the superposition of a x i a l deformation and support movements with various load conditions. The moments determined separately by the computer due to two e f f e c t s are added together to be compared to the moments which are obtained by the computer when the two e f f e c t s are combined.  In each of the following tables of  comparison two conditions are compared. location of the point on the arch.  The f i r s t column gives the  The next two columns give the d e f l e c t i o n  theory moments due to each condition when determined separately with the computer.  The fourth column gives the sum of the moments of these two  conditions.determined  independently.  The f i f t h column gives the moments  which were determined by the computer when the two cases were combined. In a l l cases the thrust due to A o r ^ was neglected i n the determination of /3 . The units of the moments are f t . kips. In a l l cases the discrepancy due to determining of the two effects separately i s n e g l i g i b l e .  the moments  I t i s thus shown that the  conclusions of Chapter I are correct and the sum of the e l a s t i c moments of each e f f e c t , m u l t i p l i e d by the correct magnification f a c t o r , gives the correct t o t a l moment.  40  SUPERPOSITION OF A LOAD PRODUCING A MAXIMUM POSITIVE MOMENT AT THE CROWN WITH A NEGATIVE UNIFORM AXIAL DEFORMATION L - 200 f t . , f - _L, CONSTANT EI - 40,000 k i p f t - , LL » 1,0- 7 L 6 DL 2  Live Load Placed on 6 Central Divisions of Arch Col. 1 X  L  Col. 2  Col. 3  Load Alone  A - -.001 Alone  Col. 4  2 (2) • (3)  Col. 5 Load and A - — .001  .00  94.08  -4.03  90.05  90.19  .05  36.70  -4.85  31.85  31.78  .10  -15.47  -4.69  -20.16  -20.43  .15  -52.95  -3.68  -56.63  -57.03  .20  -70.11  -2.07  -72.18  -72.57  .25  -65.14  - .14  -65.28  -65.54  .30  -39.56  1.84  -37.72  -37.77  .35  2.60  3.63  6.23  6.43  .40  45.27  5.04  50.31  50.73  .45  72.90  5.94  78.84  79.40  .50  82.44  6.25  88.69  89.31  E l a s t i c ^ at crown - -.860 f t . Deflection theory 1 at crown - -1.374 f t .  Table (3-8)  41  SUPERPOSITION OP A LOAD PRODUCING A MAXIMUM POSITIVE MOMENT AT THE CROWN WITH A NEGATIVE UNIFORM AXIAL DEFORMATION L « 200 f t . , f - 1, VARIABLE E I , EL^=> 80,000 k i p f t - , LL - l,/3= 5 L 3 DL 2  Live Load Placed on 6 Central Divisions of Arch  Col. 1 X  L  Col. 2 Load Alone  Col. 3 A - -.001 Alone  Col. 5  Col. 4 £ ( 2 ) • (3)  Load and X - -.001  .00  260.28  -5.69  254.59  254.41  .05  138.44  -5.74  132.70  132.56  .10  29.95  -5.20  24.75  24.61  .15  - 54.65  -4.09  - 58.74  - 58.88  .20  -103.76  -2.51  -106.27  -106.39  .25  -109.20  - .70  -109.90  -109.96  .30  - 72.41  1.04  - 71.37  - 71.33  .35  - 2.48  2.49  .01  .11  .40  69.05  3.53  72.58  72.72  .45  113.89  4.15  118.04  118.19  .50  129.08  4.36  133.44  133.59  Maximum e l a s t i c *j «• -.949 f t .  ) ) Maximum d e f l e c t i o n theory V "* -1-450 f t . )  Table (3-9)  at crown  42  SUPERPOSITION OF A LOAD PRODUCING A MAXIMUM NEGATIVE MOMENT AT THE LEFT SUPPORT WITH A NEGATIVE UNIFORM AXIAL DEFORMATION L - 200 f t . , f - 1, CONSTANT EI = 40,000 kip f t . , LL - 1, Q m 5 L 6 DL 2  Live Load Placed on 8 L e f t Divisions of Arch Col. 1 X  L  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 Notes  Col. 2  Col. 3  Load Alone  X - -.001 Alone  -114.17 - 63.59 - 15.48 24.82 53.86 69.80 72.07 61.02 37.71 9.87 - 14.64 - 34.37 - 48.16 - 55.12 - 54.68 - 46.63 - 31.28 - 9.46 17.31 46.83 76.13  Col. 4 £  Col. 5  (2) + (3)  -120.30 - 68.89 - 19.50 22.31 52.96 70.46 74.15 64.31 41.92 14.69 - 9.57 - 29.43 - 43.74 - 51.58 - 52.34 - 45.78 - 32.08 - 11.98 13.17 41.31 69.66  -120.50 - 69.02 - 19.58 22.31 53.02 70.57 74.29 64.45 42.03 14.75 - 9.58 - 29.49 - 43.84 - 51.69 - 52.46 - 45.86 - 32.12 - 11.97 13.21 41.40 69.80  -6.33 -5.43 -4.10 -2.51 - .84 .77 2.22 3.43 4.32 4.88 5.06 4.88 4.32 3.43 2.22 .77 - .84 -2.51 -4.10 -5.43 -6.33  Load and A " -.001  This load also produces a maximum p o s i t i v e moment at the 1/4 point. This i s the load studied i n Example 1 of Ref. 2.  Maximum e l a s t i c  »  -1.035 f t .  Maximum d e f l e c t i o n theory  «  ) ) -1.539 f t . )  Table (3-10)  at f  -  .30  SUPERPOSITION OF A LOAD PRODUCING A MAXIMUM NEGATIVE MOMENT AT THE LEFT SUPPORT WITH A SETTLEMENT OF THE LEFT SUPPORT L . 100 f t . , f - i , VARIABLE EI, E I L J  «. 40,000 k i p . f t . , LL ~ 1,0 DL 4 2  a v  - 5  Live Load Placed on 9 Left Divisions of Arch Col. 1 X  L .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00  Col. 2 Load Alone -243.26 -175.62 - 99.96 - 24.36 40.63 84.93 103.78 98.62 73.42 32.58 - 12.34 - 50.20 - 77.73 - 91.73 - 89.08 - 67.47 - 26.95 27.60 88.03 146.64 198.01  Col, 3  6v  L  - -.20 Alone  f t .  - .59 .67 1.84 2.78 3.34 3.40 3.04 2.41 1.66 .84 .01 - .82 -1.64 -2.40 -3.03 -3.40 -3.35 -2.81 -1.88 - .71 .53  Notet  Col. 4  Col. 5  £ < 2 ) + <3>  Load and SVL - -.20  -243.85 -174.95 - 98.12 - 21.58 43.97 88.33 106.82 101.03 75.08 33.42 - 12.33 - 51.02 - 79.37 - 94.13 - 92.11 - 70.87 - 30.30 24.79 86.15 145.93 198.54  f t .  -244.14 -175.20 - 98.32 - 21.70 43.92 88.35 106.86 101.11 75.17 33.53 - 12.21 - 50.91 - 79.26 - 94.05 - 92.06 - 70.87 - 30.37 24.65 85.94 145.67 198.26  This load also produces a maximum p o s i t i v e moment at x «= .30 L Maximum e l a s t i c - -.381 f t . ) ) at 2, . .30 Maximum d e f l e c t i o n theory *} - -.821 f t . )  Table (3-11)  44  SUPERPOSITION OF A NEGATIVE ROTATION OF ONE WITH AN EQUAL POSITIVE ROTATION OF THE OTHER SUPPORT  L » 200 f t . , f - 1, CONSTANT EI - 40,000 kip. f t . , L 6 Uniform Load  0-7  2  Col. 1 X  L .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00  Col. 2 OJ  L  - -.003  - .455 1.290 2.733 3.674 4.032 3.825 3.139 2.104 .871 - .409 -1.589 -2.539 -3.153 -3.352 -3.095 -2.384 -1.282 .087 1.529 2.796 3.612  Col. 3 (&)R  -  .003  -3.612 -2.796 -1.529 - .087 1.282 2.384 3.095 3.352 3.153 2.539 1.589 .409 - .871 -2.104 -3.139 -3.825 -4.032 -3.674 -2.733 -1.290 .455  Table (3-12)  Col. 4 2(2) • (3) -4.067 -1.506 1.204 3.587 5.314 6.209 6.234 5.456 4.024 2.130 .000 -2.130 -4.024 -5.456 -6.234 -6.209 -5.314 -3.587 -1.204 1.506 4.067  Col. 5  iaJ - -.063 L  U)R  »  .003  -4.076 -1.521 1.185 3.568 5.298 6.200 6.232 5.464 4.039 2.153 .028 -2.098 -3.990 -5.426 -6.211 -6.197 -5.315 -3.602 -1.232 1.469 4.029  45 SUPERPOSITION OF A LOAD PRODUCING A MAXIMUM NEGATIVE MOMENT AT THE LEFT SUPPORT WITH A ROTATION OF THE LEFT SUPPORT L m 200 f t . , f « 1, CONSTANT EI - 40,000 kip. f t . , LL - 1, /3 - 5 L 6 DL Live Load Placed on 8 Left Divisions of Arch 2  Col. 1 X  L .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00 Note:  Col. 2  Col. 3  Col. 5  Col. 4  Load Alone  cVL - -.003 Alone  -114.17 - 63.59 - 15.48 24.82 53.86 69.80 72.07 61.02 37.71 9.87 - 14.64 - 34.37 - 48.16 - 55.12 - 54.68 - 46.63 - 31.28 - 9.46 17.31 46.83 76.13  2.95 3.18 3.11 2.80 2.32 1.73 1.08 .43 - .18 - .70 -1.12 -1.41 -1.55 -1.54 -1.36 -1.02 - .54 .05 .71 1.38 1.99  X<2)  • (3)  Load and au °> -.003 L  - 111.22 - 60.41 - 12.37 27.62 56.18 71.53 73.15 61.45 37.53 9.17 - 15.76 - 35.78 - 49.71 - 56.66 - 56.04 - 47.65 - 31.82 - 9.41 18.02 48.21 78.12  -111.56 - 60.65 - 12.49 27.61 56.26 71.67 73.32 61.64 37.71 9.31 - 15.66 - 35.73 - 49.72 - 56.70 - 56.10 - 47.73 - 31.90 - 9.48 17.98 48.18 78.10  This load also produces a maximum p o s i t i v e moment at the 1/4 point. This i s the load studied i n Example 1 of Ref. 2.  Maximum e l a s t i c  f  »  -.861  Maximum d e f l e c t i o n theory V "  ) ) -1.364 f t . )  Table (3-13)  at £ - .25  46 CHAPTER IV NUMERICAL DATA Part 1 Tables of Magnification Factors Curves of magnification factors due to axial and support movements are given.  deformation  The e l a s t i c moments due to each e f f e c t  are also included. The magnification factors due to a x i a l deformation were obtained by subjecting the arch to a small negative uniform a x i a l  deformation  and applying a uniform v e r t i c a l load to obtain the desired value of /3 „ the thrust due to the a x i a l deformation being neglected i n the determination of /3 .  I t was  shown i n Chapter H I  that these moments and magnification  factors are applicable to any effect which changes or tends to change the span length of the arch.  The e l a s t i c moments are exact f o r any condition  which produces a change i n horizontal length of the arch of S H •=» XL. The e l a s t i c moments and magnification factors are symmetrical  thus the  values f o r one-half of the arch need only be given. The moments and magnification factors due to a v e r t i c a l movement of a support were obtained by subjecting the arch to small v e r t i c a l movement of one support and applying a uniform load to give the desired value of 13 .  The non-linearity discussed i n Chapter III was neglected.  The  average of the magnification factors f o r corresponding points on each half of the arch are given f o r one half of the arch.  The e l a s t i c moments f o r  corresponding points on each half of the arch are equal and of opposite  47  sign.  It was shown i n Chapter III that these moments and magnification  factors are applicable to any e f f e c t which produces or tends to produce unequal but uniform v e r t i c a l deflections of the two halves of the arch. The moments and magnification factors due to a r o t a t i o n of the l e f t support were obtained by subjecting the arch to a small negative rotation of the l e f t support and applying a uniform load to give the desired value of /3 , the thrust due to the rotation being neglected i n the determination of ft .  Variation i n Rise-Span Ratio Magnification factors are given for four rise-span r a t i o s . For other rise-span r a t i o s interpolation i s necessary. p l o t t i n g 0 versus ^.  This may be done by  However, since the difference i n 0 i s small between  consecutive ^ r a t i o s , a straight l i n e interpolation w i l l r e s u l t i n a very small error. Variation i n D i s t r i b u t i o n of EI The magnification factors have been determined for two d i s t r i b u t i o n s of E I Figure ( 4 - l ) .  B  a constant EI and a v a r i a t i o n of EI as given i n  In arches with a v a r i a b l e EI the average EI considered t  along the horizontal axis of the arch, has been used i n the determination of Q .  For low values of ft the tables show that the change i n the  d i s t r i b u t i o n of EI has only a small e f f e c t on the value of 0. values of ft the difference i n 0 i s larger.  For larger  With judgment the designer  interpolate between the two f o r other d i s t r i b u t i o n s of EI.  may  48  3.02.7  2.0-  I. 7 £7  Variable  El  1.2 In-  i.o  constant  El  0.8 0.6  OO  .00  20  .10  .30  .50  .40  L D i s t r i b u t i o n of EI i n Tables Figure (4-1) V a r i a t i o n of F l e x i b i l i t y Factor Magnification factors are given for/3 some cases, f o r 13 «• 7.  a  3,13  a  5, and, i n  The magnification factor f o r 13 •» 0 i s , of course, 1,  For other values of $ interpolation i s necessary. p l o t t i n g 0 versus (3 or 13 . 2  This may be done by  I t i s found, i n most cases, f o r 0>1 that the  curve of 0 versus Q may be approximated by the curve  49  0  Where 0 <1  -  sec k/3  i t i s found that the magnification factors may usually be  approximated  by the curve 0  -  2 - sec k/3  In both cases k i s a dimensionless variable which i s nearly constant at each point i n the arch.  These two curves are symmetrical about the  axis for equal values of k. side of the axis. in Figure (4-2). of 0 f o r 0 ( 1 .  0 - 1  Hence these curves need only be plotted on one  Plots of these curves f o r various values of k are given Values of 0 f o r 0)1  are given to the l e f t of the values  MAGNIFICATION FACTORS FOR PARABOLIC HINGELESS ARCHES UNDER THE ACTION OF A UNIFORM AXIAL DEFORMATION MEL XEI  X  9  for 0 - 3  0  X  for 0 - 5  0 for/3 - 7  L Const. EI  — i | OO 0  M->  B  Var. EI  Const. EI  Var. EI  Const, EI  Var. EI  Const. EI  Var. EI  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50  56.29 39.96 25.35 12,46 1.29 - 8.17 -15.90 -21.92 -26.22 -28.80 -29.66  84.70 63.68 44.86 28.26 13.87 1.70 - 8.26 -16.01 -21.54 -24.86 -25.97  .94 1.00 1.05 1.14 2.19 .88 .99 1.02 1.04 1.05 1.06  .92 .96 1.01 1.07 1.17 2.11 .89 1.02 1.06 1.08 1.09  .80 .96 1.13 1.43 5.04 .56 .92 1.05 1.12 1.16 1.17  .71 .83 .98 1.19 1.56 5.05 .55 1.02 1.18 1.26 1.29  .53 .85 1.28 2.05 11.49 - .22 .74 l.ll 1.31 1.41 1.45  .25 .50 .86 1.43 2.54 13.26 - .55 .95 1.51 1.79 1.88  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50  40.18 28.40 17.87 8.57 .51 - 6.31 -11.89 -16.23 -19.33 -21.19 -21.81  60.76 45.60 32.02 20.05 9.67 .89 - 6.30 -11.88 -15.88 -18.27 -19.07  .93 .99 1.03 1.15 3.17 .89 .99 1.02 1.04 1.05 1.05  .91 .95 1.00 1.07 1.18 2.57 .90 1.02 1.06 1.08 1.08  .79 .96 1.15 1.47 8.27 .61 .94 1.06 1.12 1.15 1.16  .69 .82 .98 1.20 1.60 6.66 .58 1.03 1.19 1.26 1.29  .50 .85 1.31 2.15 20.26 - .11 .78 1.12 1.30 1.40 1.43  .21 .47 .86 1.47 2.68 18.53 - .47 .98 1.54 1.80 1.89  Table <4-l)  MAGNIFICATION FACTORS FOR PARABOLIC HINGELESS ARCHES UNDER ACTION OF A UNIFORM AXIAL DEFORMATION A  0  MRL  for B - 3  0  0  for 0 - 5  ?CET  for 0 - 7  L Const. E I  i  I  Var.  E I  Const. E I Var.  E I  Const. E I Var.  E I  Const. .41 .86 1.44 2.48 -21.36 .12 .87 1.17 1.32 1.39 1.42  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50  23.89 16.71 10.29 4.63 - .28 - 4.43 - 7.83 -10.47 -12.36 -13.49 -13.87  36.46 27.24 18.98 12.705.39 .05 - 4.32 - 7.72 -10.15 -11.60 -12.09  .92 1.00 1.07 1.17 -1.19 .93 1.00 1.02 1.03 1.04 1.04  .89 .95 1.01 1.08 1.21 17.60 .92 1.02 1.06 1.07 1.08  .76 .97 1.19 1.56 -6.62 .73 .98 1.07 1.11 1.13 1.14  .65 .80 1.00 1.26 1.73 63.05 .68 1.06 1.20 1.26 1.28  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50  15.89 10.99 6.61 2.75 - .60 - 3.43 - 5.75 - 7.55 - 8.84 - 9.62 - 9.87  24.43 18.16 12.55 7.59 3.30 - .33 - 3.30 - 5.61 - 7.26 - 8.25 - 8.58  .91 1.00 1.08 1.19 .49 .98 1.02 1.03 1.03 1.03 1.03  .87 .94 1.02 1.11 1.24 - .54 .96 1.04 1.06 1.07 1.07  .71 1.00 1.26 1.67 --.98 .88 1.04 1.09 1.10 1.10 1.10  .58 .79 1.04 1.35 1.90 - 5.26 .79 1.11 1.22 1.26 1.27  Table (4-2)  EI  Var.  E I  .05 .38 .88 1.66 3.25 219.27 - .35 1.07 1.64 1.90 1.98  MAGNIFICATION FACTORS FOR PARABOLIC HINGELESS ARCHES UNDER THE ACTION OF A VERTICAL MOVEMENT OF A SUPPORT cSy MEL <5y EI 2  X  L  a  a  0 f o rft- 3  L  Const. EI  Var. EI  Const. EI  Var. EI  0 for  Const. EI  (3 = 5 Var. EI  0 for /3 «, 7  Const. EI  Var. EI  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50  -5.613 -5.052 -4.490 -3.929 -3.368 -2.806 -2.245 -1.684 -1.122 - .562 .000  -6.429 -5.786 -5.144 -4.500 -3.858 -3.215 -2.572 -1.929 -1.286 - .643 .000  .84 .92 .99 1.05 1.10 1.14 1.18 1.20 1.22 •1.23 .00  .72 .80 .88 .97 1,06 1.14 1.20 1.25 1.29 1.31 .00  .48 .71 .94 1.15 1.35 1.52 1.67 1.78 1.87 1.91 .00  .07 .29 .55 .85 1.18 1.50 1.78 2.02 2.19 2.29 .00  - .55 - .03 .63 1,37 2.14 2.89 3.56 4.13 4.56 4.81 .00  -2.33 -1.85 -1.10 - .02 1.39 3.02 4.67 6.15 7.34 8.09 .00  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50  -5.352 -4.817 -4.282 -3.747 -3.212 -2.676 -2.141 -1.606 -1.070 - .535 .000  -6.190 -5.572 -4.952 -4.334 -3.714 -3.095 -2.476 -1.857 -1.238 - .619 .000  .84 .93 1.00 1.06 1.10 1.14 1.17 1.20 1.20 1.21 .00  .72 .80 .89 .98 1.07 1.14 1.20 1.25 1.28 1.30 .00  .48 .72 .96 1.17 1.36 1.52 1.64 1.74 1.81 1.85 .00  .04 .28 .57 .87 1.20 1.53 1.80 2.03 2.19 2.29 .00  - .67 - .10 .63 1.45 2.28 3.08 3.77 4.35 4.75 5.02 .00  -2.82 -2.29 -1.42 - .14 1.53 3.45 5.33 7.04 8.35 9.19 .00  Table <4-3)  MAGNIFICATION FACTORS FOR PARABOLIC HINGELESS ARCHES UNDER THE ACTION OF A VERTICAL MOVEMENT OF A SUPPORT  8  V  L2  0 f o r Ii - 3  5v Ei  X  L  0 f o r (3 = 5  T"  Li  Const. EI  .1  df>  •  V  11.1.  B H  lull  1  Var. EI  Const. EI  Var. EI  Const. EI  Var. EI  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50  -4.776 -4.298 -3.821 -3.344 -2.866 -2.388 -1.910 -1.433 - .955 - .478 .000  -5.639 -5.075 -4.511 -3.948 -3.384 -2.820 -2.256 -1.692 -1.128 - .564 .000  .85 .94 1.02 1.06 1.09 1.11 1.12 1.12 1.12 1.12 .00  .71 .81 .90 1.00 1.08 1.15 1.19 1.22 1.24 1.24 .00  .48 .77 1.02 1.21 1.36 1.45 1.51 1.55 1.57 1.58 .00  - .03 .24 .56 .91 1.27 1.59 1.85 2.03 2.16 2.22 ..00  .00 .05 .10 . .15 " .20 .25 .30 .35 .40 .45 .50  -4.218 -3.796 -3.374 -2.952 -2.530 -2.109 -1.687 -1.266 - .844 - .422 .000  -5.075 -4.568 -4.060 -3.552 -3.045 -2.538 -2.030 -1.522 -1.015 - .508 .000  .86 .98 1.04 1.07 1.06 1.04 1.02 .99 .96 .95 .00  .71 .82 .93 1.02 1.10 1.14 1.16 1.16 1.16 1.15 .00  .53 .88 1.12 1.24 1.27 1.24 1.17 1.09 1.02 .97 .00  - .14 .19 .57 .98 1.37 1.67 1.93 1.98 2.03 2.05 .00  Table (4-4)  MAGNIFICATION FACTORS FOR PARABOLIC HINGELESS ARCHES UNDER THE ACTION OF A ROTATION OF THE LEFT SUPPORT <0  L  ML E  X  L  r-<|CC  P  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00  0 for  fc) EI L  Const. FI -8.377 -6.759 -5.282 -3.946 -2.751 -1.696 - .782 - .009 .623 1.115 1.466 1.676 1.746 1.675 1.463 1.110 .617 - .017 - .792 -1.708 -2.764  Var. EI  Const. EI  -12.064 -9.731 -7.610 -5.700 -4.002 -2.516 -1.242 - .179 .672 1.311 1.738 1.954 1.958 1.750 1.330 .698 - .145 -1.200 -2.467 -3.945 -5.635  .89 .95 1.01 1.07 1.13 1.20 1.36 21.22 .89 1.04 1.09 1.12 1.13 1.14 1.14 1.13 1.09 3.15 1.15 1.10 1.06  fi -  3  Var. EI .85 .89 .95 1.01 1.09 1.18 1.33 2.53 .84 1.05 1.11 1.15 1.16 1.16 1.14 1.07 1.68 1.18 1.11 1.07 1.03  0 f or Const. EI .66 .84 1.02 1.21 1.43 1.72 2.33 79.82 .49 1.07 1.30 1.35 1.49 1.52 1.52 1.48 1.36 8.38 1.56 1.38 1.23  IS  = 5  Var. EI .52 .64 .80 1.01 1.28 1.62 2.22 7.11 .25 1.11 1.40 1.55 1.62 1.62 1.55 1.31 3.38 1.68 1.44 1.29 1.16  0 for 3 - 7 Const. EI . .09 .45 .92 1.51 2.27 3.41 5.94 333.57 - 1.75 .76 1.79 2.38 2.76 2.98 3.08 3.02 2.73 18.66 3.03 2.47 1.99  Var. EI -  .50 .27 .11 .73 1.70 3.21 6.22 32.65 - 4.31 .41 2.17 3.20 3.80 4.07 4.02 3.38 8.45 3.92 3.12 2.53 2.06  Table (4-5) Ul  MAGNIFICATION FACTORS FOR PARABOLIC HINGELESS ARCHES UNDER THE ACTION OF A ROTATION OF THE LEFT SUPPORT CtJi MEL <tf EI  X  L  8  twM  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00  0  L  Const. EI  Var. EI  -7,952 -6.410 -5.006 -3.734 -2.595 -1.591 - .720 .016 .618 1.087 1.422 1.622 1.689 1.622 1.420 1.085 .616 .013 - .724 -1.595 -2.600  -11.536 --9.302 - 7.270 - 5.442 - 3.816 - 2.392 - 1.171 - .153 ,663 1.276 1.687 1.896 1.901 1.704 1.305 .703 - .101 - 1.108 - 2.318 - 3.730 - 5.345  for p - 3  Const. EI .88 .95 1.02 1.08 1.14 1.23 1.40 -10.35 .89 1.04 1.09 1.13 1.14 1.15 1.15 1.14 1.11 - 1.38 1.18 1.12 1.07  0  f o r fi - 5  Var. EI  Const. EI  .83 .89 .95 1.02 1.10 1.20 1.35 2.77 .85 1.05 1.12 1.16 1.17 1.17 1.15 1.09 1.87 1.20 1.13 1.08 1.04  .62 .82 1.04 1.25 1.49 1.81 2.50 -54.54 .48 1.08 1.32 1.46 1.54 1.58 1.59 1.57 1.47 - 6.31 1.64 1.45 1.28  0  f o r /3 - 7  Var. EI  Cons t. EI  .46 .61 .79 1.03 1.32 1.70 2.36 8.42 .22 1.11 1.43 1.59 1.67 1.69 1.63 1.41 4.07 1.76 1.51 1.34 1.20  .10 .33 .91 1.64 2.59 4.02 7.26 -219.21 - 2.35 .63 1.86 2.61 3.11 3.45 3.63 3.63 3.48 - 11.01 3.53 2.92 2.31  Var. EI -  .80 .54 .08 .66 1.86 3.76 7.60 45.58 - 5.62 .10 2.30 3.63 4.46 4.93 4.98 4.50 9.94 4.75 3.80 3.09 2.50  Table (4-6) ON  MAGNIFICATION FACTORS FOR PARABOLIC HINGELESS ARCHES UNDER THE ACTION OF A ROTATION OF THE LEFT SUPPORT (t)-. ML E  0  4>7ET L  -<l<t  .00 .05 .10 .15 ,20 .25 .30 .35 .40 .45 .50 .55 .60 .65 ,70 ,75 .80 .85 .90 .95 1.00  Const. E I -7.038 -5.664 -4.410 -3.275 -2.260 -1.364 - .588 .068 .606 1.024 1.322 1.502 1.561 1.501 1.322 1.024 ..605 .068 - .589 -1.366 -2.261  Var.  EI  -10.359 - 8.346 - 6.514 - 4.865 - 3.398 - 2.113 - 1.011 - .091 .647 1.202 1.576 1.766 1.775 1.601 1.245 .706 - .014 - .917 - 2.003 - 3.270 - 4.720  f o r ft " 3  Const. E I Var. .84 .94 1.03 1.11 1.19 1.29 1.51 -1.65 .90 1.05 1.11 1.15 1.17 1.19 1.20 1.20 1.18 .92 1.22 1.16 1.10  E I  .79 .86 .95 1.04 1.14 1.24 1.42 4.00 .86 1.06 1.14 1.17 1.20 1.21 1.20 1.15 5.36 1.25 1.17 1.12 1.07  Table (4-7)  0  0 for/3  for (3 - 5  Const. E I .46 .77 1.08 1.39 1.71 2.16 3.18 -11.52 .38 1.09 1.39 1.57 1.69 1-.78 1.84 1.87 1.84 1.11 1.89 1.68 1.44  Var.  E I  .29 .49 .75 1.07 1.46 1.96 2.83 15.18 .09 1.12 1.51 1.72 1.84 1,91 1.91 1.77 14.22 2.02 1.74 1.52 1.33  Const. E I -  1.06 .38 .73 2.28 4.38 7.62 15.78 -102.25 - 6.41 .48 2.15 3.84 5.14 6.19 7.04 7.72 8.28 13.62 6.31 5.76 4.44  =7  MAGNIFICATION FACTORS FOR PARABOLIC HINGELESS ARCHES UNDER THE ACTION OF A ROTATION OF THE LEFT SUPPORT (eJ  L  ML E  0 f or Ii - 3  x L Const. EI  •Ho 8  .00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00  -6.179 -4.962 -3.851 -2.846 -1.946 -1.153 - .465 .116 .592 .962 1.226 1.384 1.436 1.382 1.222 .956 .585 .107 - ,476 -1.166 -1.961  Var. EI -9.217 -7.416 -5.778 -4.303 -2.990 -1.841 - .854 - .030 .630 1.128 1.464 1.636 1.645 1.492 1.176 .697 .055 - .750 -1.718 -2.848 -4.142  Const. EI .77 .93 1.06 1.17 1.27 1.40 1.71 . .64 .91 1.07 1.14 1.18 1.22 1.25 1.27 1.29 1.30 1.28 1.29 1.24 1.15  Var. EI .72 .83 .95 1.07 1.19 1.32 1.55 10.19 .87 1.08 1.16 1.21 1.24 1.26 1.27 1.25 .60 1.32 1.24 1.17 1.10  0  f o r /9 - 5  Const. EI .17 .65 1.15 1.65 2.18 2.89 4.76 .8.81 .11 1.08 1.50 1.78 2.00 2.19 2.37 2.52 2.64 3.05 2.36 2.19 1.81  Var. EI .  .04 .26 . .66 1.14 1.73 2.48 3.84 57.64 . .26 1.10 1.64 1.95 2.19 2.36 2.48 2.50 1.25 2.51 2.21 1.91 1.61  Table (4-8) CO  59  Part 2 Deflection Theory Moment Curves For convenience, curves of the d e f l e c t i o n theory moments are also included.  In a l l cases they are the moments which were found i n the  determination of the magnification f a c t o r s .  In the case of uniform a x i a l  deformation and v e r t i c a l movement of a support the moments determined with a constant EI are plotted to the l e f t of the centreline and the moments with the variable EI are plotted to the r i g h t .  In the case of the r o t a t i o n  of the l e f t support, the moments f o r constant and variable EI are plotted on separate sheets.  In a l l cases the horizontal scale i s 1"  =>  .10 L.  •  urn :|±ix ti.cr ttit  TP  t+PF  •nit -pp-t  ±t+ itit  r -l-!-r'  1  tit  SHE  -FFH  TXT:  - i ;  tin  St  H - h -  i-i-H  S t ©  ill  ii:!3t IRISES.  xrix  .11TT-t  xnxt  tttfi  P±F±  Part 3 Influence The  Lines f o r E l a s t i c Thrust  influence lines of the e l a s t i c thrust of the eight  arches studied i n this thesis are given in Table (4-9). The e l a s t i c 2 thrust due to rotation "^ i s given i n the l a s t l i n e of the table.  INFLUENCE LINES FOR ELASTIC THRUST f - I L 8  f - 1 L 3  f - 1 4 L  f - 1 6 L  Const. EI  Var. EI  Const. EI  Var. EI  Const. EI  Var. EI  Const. EI  Var. EI  .025  ..033303  ..019014  .025377  .014574  ..017502  .010170  .013558  .007955  .075  .155391  .097737  .118131  .074700  .081089  .051832  .062549  .040351  .125  .367761  .259174  .278746  .197370  .190166  .135959  .145871  .105182  .175  .632067  .500280  .477554  .379426  .323609  .259180  .246677  .199009  .225  .915796  .804759  .689856  .607817  .464451  .411549  .351837  .313434  .275  1.191869  1.139876  .895451  .857639  .599355  .575904  .451410  .435120  .325  1.438289  1.463575  1.078212  1.097717  .718115  .731960  .538132  .549182  .375  1.637834  1.738404  1.225720  1.300752  .813194  .862685  .606919  .643708  .425  1.777814  1.936749  1.328945  1.44687 5  .879326  .956108  .654418  .710711  .475  1.849876  2.04043 2  1.382008  1.523130  .913193  1.004653  .678629  .745348  I.  10.000000  10.000000  7.500000  7.500000  5.000000  5.000000  3.750000  3.750000  -56.29  -84.70  -40.15  -60.77  -23.89  -36.46  -15.89  -24.43  L  HL  2  Table (4-9)  78 CHAPTER V NUMERICAL EXAMPLES Part 1  Analysis of a Hingeless Arch To i l l u s t r a t e the use of the tables presented  i n Chapter IV  and the tables given i n the appendix of Ref. 2, the moments and magnification factors at a support of the Rainbow Arch Bridge  3  w i l l be determined.  The  moments and magnification factors due to l i v e load, a temperature drop of 60 degrees, a horizontal support movement of .20 f e e t , and a support settlement  of 1.0 feet w i l l be found. This arch has a span of 950 feet and a r i s e of 150 feet.  Each  r i b i s subject to a uniform l i v e load of 1.38 kips per foot and a concentrated l i v e load of 26.7 kips.  The dead load of the arch i s not uniform; hence  the arch i s not a true parabola.  The greatest deviation of the arch axis  from a parabola which passes through the crown and the supports i s approximately 1.75 feet. which i n turn support  The arch supports 24 equally spaced columns  the deck of the bridge.  For the purposes of analysis Hardesty et a l considered the arch divided into 24 d i v i s i o n s of equal horizontal length, the mid-point of each d i v i s i o n coinciding with a column.  Since our tables are set up  for the analysis of an arch divided into 20 d i v i s i o n s of equal horizontal length, we s h a l l consider the arch as having 20 equally spaced columns and subject to the same l i v e load. 1.38  x  2^9. 20  The l i v e load per panel now becomes ~  65.5 kips  79 Since the shape of the arch i s quite close to that of a parabola the moments and magnification factors which have been determined hingeless arches w i l l be used i n the analysis.  f o r parabolic  The table on Page 86 and  the graph on Page 119, both of Ref. 2, show that l i v e load should be placed on the 8 d i v i s i o n s of the arch closest to the support to produce a maximum negative moment at that support. placed at  The concentrated l i v e load should be  = ,175. The moment of i n e r t i a at the mid-point of each d i v i s i o n i s  given.  The value of E i s , of course, constant and w i l l be taken as 30 x 10^  psi => 4.32 x 10° k i p / f t , . axis i s 94.5 f t . \ (5-1).  The average moment of i n e r t i a along the horizontal  The d i s t r i b u t i o n of EI of the arch i s given i n Figure  The d i s t r i b u t i o n of EI which was used i n the tables i s also shown. 2..7Q  2.0 — /.70  t.50  Constant  1.0  /f/-^  .325  .85  .80 .60  0.0 .00  JO  20  .30  D i s t r i b u t i o n of EI Figure (5-1)  .40  .50  80 This figure shows that the v a r i a t i o n from constant HI of the EI of the Rainbow Arch Bridge i s approximately 1/4 as much as that f o r the variable EI used i n the tables.  For i n t e r p o l a t i o n i n the tables i t w i l l be assumed  that the moments and magnification factors are composed of 75 percent of the value determined for a constant EI and 25 percent of the corresponding value found for the variable E I . The dead load thrust, as determined by Hardesty et a l , i s 6,062 kips.  This corresponds to a uniform dead load of 8.06 k i p s / f t . on a  parabolic arch.  Since the ^  r a t i o i s small (1/5,8) the thrust due to l i v e  load may be determined with s u f f i c i e n t accuracy by using Table (4-9) which i s for parabolic arches. ratio £ = ^ 0  By a s t r a i g h t line interpolation f o r a rise-span  and the. assumed d i s t r i b u t i o n of E I an influence l i n e for (  e l a s t i c thrust as given i n Table (5-1)  .025 .075 .125 .175 .225 .275 .325 .375 .425 .475  i s obtained.  .024 .115 .276 .484 .716 .949 1.140 1.347 1.459 1.520  .375 =  5.051  .025  Influence Line of E l a s t i c Thrust Table (5-1) A l i v e load of 65.5 kips/panel  and a l i v e load of 26.7 kips at  produces a thrust of 5.051 x 65.5 • .484 x 26.7 thrust becomes 6,062 + 344  =  6,406 kips.  »  344 kips.  » .175 The t o t a l  The f l e x i b i l i t y factor  81  f o r the arch under this load i s thus  Q °  M  /jfi£ . / 6406 ^ 95Q2 ~ _ J EIav J 4.32 x 10° x 94.5 The influence l i n e of e l a s t i c moments at the support due to  load may be determined by a two way i n t e r p o l a t i o n f o r a rise-span r a t i o — «• 150 and the assumed d i s t r i b u t i o n of EI using tables on Pages 75 and 86 L 950 of Ref. 2.  The ordinates of the influence l i n e f o r the d i v i s i o n s which are  loaded are given i n Column 2 of Table (5-2). and 26.7 kips at  L  Live loads of 65.5 kips/panel  • .175 produce an e l a s t i c moment at the support of  -.3679 x 65.5 x 950  -  22,900  -.0716x26.7x950  -  1,800 24,700 f t . kips  Col. 1  Col. 3  Col. 2 M  Col. 4 MD  E  0  L  PL  .025 .075 .125 .175 .225 .275 .325 .375  -.0203 -.0522 -.0685 -.0716 -.0645 -.0498 -.0308 -.0102  1.03 1.06 1.09 1.14 1.16 1.23 1.30 1.54  PL -.0209 -.0554 -.0746 -.0816 -.0748 -.0614 -.0400 -.0157 -.4244  -.3679 Moments a t x •=» .00 L Table (5-2)  The magnification factors f o r a value of /3 ° 3.77 span r a t i o j - m A?0  flnd  t  h  e  a s s u t n e f  9  a rise-  j d i s t r i b u t i o n of EI are obtained by  82  interpolation and the use of Figure (4-2). of Ref. 2 are again used. of Table (5-2).  The tables on Pages 75 and 86  The magnification factors are given i n Column 3  Multiplying the ordinates of the e l a s t i c influence l i n e  by these magnification factors gives the ordinates of the d e f l e c t i o n theory influence l i n e which are shown i n Column 4 of Table (5-2).  The d e f l e c t i o n  theory moment at the support due to the same load thus becomes .4244 x 65.5 x 950  -  26,400  .0816 x 26.7 x 950  -  2,100 28,500  The o v e r a l l magnification factor of the e l a s t i c moment at a support due to this load i s thus  28.500  «  1.15  24,700 By straight l i n e i n t e r p o l a t i o n i n Table (4-l) f o r the r i s e span r a t i o and the assumed d i s t r i b u t i o n of EI a value of  M  g  L  » 49.4 i s  AEI obtained f o r the e l a s t i c moment at a support due to a uniform a x i a l deformation.  Assuming a temperature drop of 60 degrees and a c o e f f i c i e n t  of expansion of .0000065 the value of A i s -60 x .0000065 - -.00039.  Hence  the e l a s t i c moment at the support due to a temperature drop of 60 degrees is M  E  - 49.4 <= 49.4 -.00039 x 4.32 x 10 x 94.5 950 •=> 8,300 f t . kips 6  The magnification factor f o r Q « 3.77, ~ «  and the  assumed d i s t r i b u t i o n of EI i s obtained by i n t e r p o l a t i o n from Table (5-l) and with the a i d of Figure (4-2).  The d e f l e c t i o n theory moment at a support due  to the temperature drop becomes Mjj  -  x 0  -  -8,300 x .87  «  -7,200 f t . kips  83  A temperature drop of 60 degrees produces the same moments as the r i b shortening due to an a x i a l stress of A E  »  .00039 x 30 x 10^  =  11,700 p s i .  It was shown i n Chapter III that the moments due to axial deformation and horizontal support movements are p r a c t i c a l l y  identical.  The e l a s t i c moment due to a horizontal outward support movement of . 2 0 f t . i s -  ME  -49.4 6 n L  E I  2  -  -49.4  FF  20  x 4.32 x 1Q x 94.5 6  950 -  - 4 , 5 0 0 f t . kips  The d e f l e c t i o n MD  -  2  theory moment i s  ME x 0  -  - 4 , 5 0 0 x .87  By straight  -  - 3 , 9 0 0 f t . kips  line i n t e r p o l a t i o n i n Table ( 4 - 3 ) a value of  MnL A" - ' - 5 . 6 1 i s obtained f o r the e l a s t i c moment at a support due to the DyEI v e r t i c a l movement of a support. The e l a s t i c moment due to a v e r t i c a l 2  1  m  settlement of 1.0 f t . of the f a r support i s M  E  »  -5.615yEI L2  "  - * 5  6 1  L O x 4.32 x 10 950  -  6  x 94.5  2  - 2 , 5 0 0 f t . kips  By applying the proper magnification factor the deflection theory moment i s obtained MQ  -  Mg x 0  -  - 2 , 5 0 0 x .78  -  - 2 , 0 0 0 f t . kips  The moments at a support of the Rainbow Arch Bridge are summarized i n Table ( 3 - 3 ) .  84  EFFECT  M  0  E  Uniform LL - 1.38 Cone. LL -  26.7  k  Temp, drop =• 60°  J  -24,700  -28,500  1.15  - 8,300  - 7,200  .87  H  °  -20  - 4,500  - 3,900  .87  v  -  1.0  - 2,500  - 2,000  .78  Moments at a Support of Rainbow Arch Bridge Table (3-3)  85 Part 2 Analysis of a Two Hinged Arch The use of the tables w i l l be further i l l u s t r a t e d  by the  determination of the moments and magnification factors at the 1/4 point of the two hinged arch which i s also discussed  i n Ref. 3. As before the arch  has a span of 950 feet and a r i s e of 150 f e e t . uniform l i v e load of 1.375 k i p s / f t . The coordinates  Each r i b i s subject to a  and a concentrated load of 26.7 kips.  of the arch axis are not given; however, i n a l l calculations  except the determination of the dead load thrust i t w i l l be assumed that the arch axis i s a parabola. In the analysis by Hardesty et a l the uniform l i v e load was placed on the 10 d i v i s i o n s closest to one support of the 24 d i v i s i o n arch. The concentrated l i v e load was placed at  •» .229. For our analysis the  uniform l i v e load w i l l be placed on the 8 d i v i s i o n s closest to one support of a 20 d i v i s i o n arch.  The concentrated l i v e load w i l l be placed at  j» 950 _£ - .225. The load per panel due to uniform l i v e load becomes 1.375 x -rrX*  ra  65.4 kips/panel.  The moment of i n e r t i a of the arch i s given as 116.14 f t . ^  at the crown and 120.24 f t . at the 1/4 point. 4  I t w i l l be assumed that the  arch has a constant moment of i n e r t i a of 116.14 + 120.24 2  -  118.2 f t .  The dead load e l a s t i c 7,126 kips.  4  thrust determined by Hardesty et a l i s  This corresponds to a uniform dead load of 9.50 k i p s / f t .  parabolic arch.  on a  The thrust due to the l i v e load w i l l be determined by the  use of Table (4-9).  Interpolating f o r a rise-span r a t i o £ =>  ordinates of the influence line f o r e l a s t i c  , the  thrust as shown i n Table (5-4)  86 are obtained.  The thrust due to a load of 65.4 kips/panel and a concentrated  L  P  .025 .075 .125 .175 .225 .275 .325 .375 .425 .475  .299  |  .375  2  1  .959  .025  -  5.122  Influence Line of E l a s t i c Thrust Table (5-4) load of 26.7 kips at  - .225 i s  5.122 x 65.4 + .739 x 26.7  -  355 kips  The t o t a l e l a s t i c thrust becomes 7,126 • 355  -  7,481 kips.  Neglecting  thrust due to r o t a t i o n of the supports the f l e x i b i l i t y f a c t o r 3 due to this load i s /3 -  / HL2__ J EI a v  -  / J  7.481 x 950 4.32 x 10 x 118.2 2  -  3.64  6  To f i n d the solution of a two hinged arch, having solutions for load and support r o t a t i o n of a hingeless arch, i t i s necessary to combine rotations of the supports with the load so that the moments at the supports of the arch are equal to zero.  The e l a s t i c s o l u t i o n w i l l be  found f i r s t . By i n t e r p o l a t i o n f o r £ - i|£  of the tables on Pages 75 and 86  of Ref. 2 the ordinates of the e l a s t i c influence l i n e s of a hingeless arch  87 for moments at the two supports are obtained.  These are given i n Column 2  of Tables (5-5) and (5-7) respectively. The ordinates of an e l a s t i c influence l i n e of a hingeless arch f o r moment at the 1/4 point i s obtained by interpolation i n tables on Pages 80 and 91 of Ref. 2.  These are given  i n Column 2 of Table (5-6) The e l a s t i c moment at — - .00 i n a hingeless arch due to the l i v e load i s ME  -  -.3315 x 65.4 x 950  -  -20,600  -.0565 x 26.7 x 950  -  - 1.400 22,000 f t . kips  The e l a s t i c moment at ~ ~ .25 due to the l i v e load i s %  -  .1836 x 65.4 x 950  -  11,400  .0491 x 26.7 x 950  -  1.200 12,600 f t . kips  Col. 1 Xp  Col. 2 M  Col. 3  L  PL  n V  .025 .075 .125 .175 .225 .275 .325 .375  -.0197 -.0497 -.0632 -.0642 -.0565 -.0430 -.0264 -.0088  1.03 1.06 1.10 1.13 1.17 1.20 1.27 1.48  E  -.3315  Col. 4  MD  PL -.0203 -.0526 -.0695 -.0725 -.0661 -.0516 -.0335 -.0130 -.3791  Moments at j- - .00  Table (5-5)  Col.  1  2>  Col.  2  Col.  ME  L  3  0  PL  .025 .075 .125  .0012  .175 .225 .275 .325  .0300 .0491 .0478 .0258  .375  .0081  4  PL .0015  1.28 1.26 1.24 1.21  .0060 .0156  Col.  .0076 .0194 .0363 .0565  1.15 1.17  .0559 .0322 .0122  1.25 1.50  .1836  .2216  Moments at — - . 2 5 Table ( 5 - 6 )  Col. x  p  L  1  Col. M  2  E  .0016  .075  .0070 .0160 .0261  .275 .325 .375  3  0  PL  .025 .125 .175 .225  Col.  1.13 1.14 1.15 1.16  .0355 .0427 .0467 .0469  Col.  4  & PL  .0018 .0080 .0184  1.17  .0303 .0415  1.17 1.17 1.16  .0500 .0546 .0544  .2225  .2590  Moments at - - 1 . 0 0  Table ( 5 - 7 )  89  The e l a s t i c moment at — - 1.00 due to the l i v e load i s L  M  -  E  .2225 x 65.4 x 950 .0355 x 26.7 x 950  -  13,800  900 14,700 f t . kips  By interpolation f o r i . - 150 i n Table (4-5) the e l a s t i c moments 950 due to support rotations of a hingeless arch may be determined. The e l a s t i c moment at a support due to a rotation of that support i s found to be M  E  -  -8.04  <J>EI  The e l a s t i c moment at the 1/4 point due to a rotation of the closest support is -  ME  -1.61 cam  The e l a s t i c moment at a 1/4 -  ME  point due to a rotation of the f a r support i s  1.09 ^ E I L  The e l a s t i c moment at a support due to a rotation of the other support i s  The sum of the e l a s t i c moments at — - 1.00 must also equal zero; hence -2.64  £gl -8.04.2Jf!i  • 14,700  -  0  Solution of these two simultaneous equations gives L  uV EI R  L  -3,760  3,050  90  The e l a s t i c moment at the 1/4 point of the two hinged arch i s thus Mg  -  12,600  -  1.61  -  22,000 f t . kips  (-3,760)  •  1.09  (3,050)  Hardesty et a l found this moment to be 22,594 f t . kips By i n t e r p o l a t i o n i n Table ( 4 - 9 ) the thrust due to support rotation of a hingeless arch i s found to be H  -  -43.58  -  -43.58 (-3,760 + 3,050)  MM  950  -  35 kips  The rotations at the two supports being of almost equal magnitude and of opposite sign results i n a n e g l i g i b l e thrust due to r o t a t i o n . To obtain the d e f l e c t i o n theory solution of the two hinged arch, the d e f l e c t i o n theory moments due to load and support rotations of the hingeless arch must be determined.  They are obtained by multiplying the  e l a s t i c theory moments by a magnification factor corresponding to fi «• 3 . 6 6 and j - - ^-Q>  -The magnification factors are found i n the same  tables as the corresponding e l a s t i c moments.  The magnification factors  are given i n Column 3 of Tables ( 5 - 5 ) , ( 5 - 6 ) and ( 5 - 7 ) .  The ordinates of  the d e f l e c t i o n theory influence lines due to load are shown i n Column 4 of Tables ( 5 - 5 ) ,  (5-6)  and ( 5 - 7 ) .  The d e f l e c t i o n theory moment at j- • .00 i n a hingeless arch due to l i v e load i s MQ  -  - . 3 7 9 1 x 6 5 . 4 x 950  -  23,500  - . 0 6 6 1 x 26.7 x 950  -  1,700 2 5 , 2 0 0 f t . kips  91  The d e f l e c t i o n theory moment at £ - .25 due to the l i v e load i s MD  -  .2216 x 65.4 x 950  -  13.800  .0565 x 26.7 x 950  -  1.400  15,200 f t . kips The d e f l e c t i o n theory moment a t *• - 1.00 due to the l i v e load i s Mp  -  .2590 x 65.4 x 950  -  16,100  .0415 x 26.7 x 950  -  1,000  17,100 f t . kips The d e f l e c t i o n theory moments due to support rotations are found by multiplying the e l a s t i c theory moments by the correct magnification factors.  and ft » 3 . 6 6 .  They must be determined f o r — -  The d e f l e c t i o n  theory moment at a support due to a r o t a t i o n of that support i s  MD - .8.04  L  x .83 - -6.65 4L£i L  The d e f l e c t i o n theory moment a t the 1/4 point due to a rotation of the closest support i s MD  -  -1.61  ^Ei x  1.33  -  -2.14  ^  The d e f l e c t i o n theory moment a t a 1/4 point due to a rotation of the far support i s MD  -  1.09  ^Fix  1.25  -  1.36  £EL  The d e f l e c t i o n theory moment a t a support due to a rotation of the other support i s MD  -  - 2 . 6 4 x ^ x 1 . 1 2  -  -2.96  ^Ei  As before, the sum of the d e f l e c t i o n theory moments a t both supports must be equal to zero; hence -6.65 — ~ —  - 2.96 — £ — - 25,200  -  0  92  and 2.96  — ~  6.65 — |  17,100  -  0  Solution of these two simultaneous equations gives L ^REI  L  -  -6,160  -  5,310  The d e f l e c t i o n theory moment at 1 / 4 point of the two hinged arch thus becomes MQ  -  15,200  -  2.14 (-6,160)  -  35,600  f t . kips  •  1.36  Hardesty et a l found t h i s moment to be  (5,310)  37,123  f t . kips.  The o v e r a l l magnification factor due to t h i s load i s 0  -  2& "E  -  35.600 22,000  -  1.62  The overall magnification factor found by Hardesty et a l i s 0  -  37.123 22,594  -  1.64  By a s i m i l a r analysis of the same arch with a hinge at one support the magnification factor f o r the maximum moment at the 1 / 4 point i s found to be 0  -  MD M E  23.400 17,000  -  1.38  If this were a hingeless arch,the magnification factor of the maximum moment at the 1 / 4 point i s 0 . -  " 0 M E  -  15.200 12,600  -  1.21  93 BIBLIOGRAPHY  Timoshenko, S.:  "Strength of Materials", D. Van Nostrand Company, Inc.,  New York, (1955) Lee, R. W. M.:  "Analysis of F l e x i b l e Hingeless Arch by an Influence  Line Method".  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 of the degree of Master of Applied Science i n the University of B r i t i s h Columbia Pelton, T. E.:  (1958).  "Magnification Factors f o r Hingeless Arches".  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 of the degree of Master of Applied Science i n the University of B r i t i s h Columbia  (1958).  Hardesty, S., G a r r e l t s , J . M., and Hedrick, I. G.: Over Niagara Gorge".  "Rainbow Arch Bridge  Transactions, American Society of C i v i l  Engineers, v o l . 110 (1945).  

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