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UBC Theses and Dissertations

Optimum Michell frames Johnson, Eric William 1970

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O P T I M U M  M I C H E L L  F R A M E S  BY  ERIC WILLIAM JOHNSON B.Sc. ( E n g ) , L o n d o n , 1950 M.Sc. U n i v e r s i t y o f A l b e r t a , 1965  A THESIS THE  SUBMITTED IN PARTIAL FULFILLMENT  OF  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in  t h e Department of  MECHANICAL ENGINEERING  We a c c e p t required  this  thesis  as c o n f o r m i n g  to the  standard  THE UNIVERSITY  OF BRITISH  J u n e , 1970  COLUMBIA  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 l m e n t of the  requirements f o r an advanced degree a t the U n i v e r s i t y 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 i t f r e e l y a v a i l a b l e for reference agree t h a t p e r m i s s i o n  and study.  f o r extensive  I further  copying o f t h i s  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood  that  p u b l i c a t i o n , i n p a r t o r i n whole, o r the copying o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be allowed w i t h o u t my written  permission.  ERIC WILLIAM JOHNSON Department o f M e c h a n i c a l  Engineering  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada Date  ii  ABSTRACT  The space  theory u n d e r l y i n g the d e s i g n of minimum volume  frameworks i s reviewed.  r e l a t i o n s proposed  Fundamental theorems and  and developed  by Maxwell, M i c h e l l ,  Chan, Hemp, Johnson and B a r n e t t are d i s c u s s e d and from a p r a c t i c a l Two  examined  viewpoint.  d i m e n s i o n a l frames, which are c l o s e geometric  approximations d e f i n e d and  Cox,  t o the t h e o r e t i c a l concepts of M i c h e l l ,  formulas e s t a b l i s h e d f o r t h e i r d e s i g n  complete s o l u t i o n .  are  and  Computer programmes are e s t a b l i s h e d f o r  the a n a l y s i s of p i n j o i n t e d t r u s s e s having a wide range of parameters.  The  e f f e c t of changing  parameters on  the  structural properties i s discussed. R i g i d frames are analyzed by the use of STRUDL, [STRUctural Design Language], a m u l t i - p u r p o s e programme f o r the c a l c u l a t i o n of f o r c e s and in rigid  structures.  The  computer  displacements  e f f e c t s of b i a x i a l s t r e s s i n the  j o i n t s and o t h e r d e v i a t i o n s from the t h e o r e t i c a l are examined.  Comparison i s made w i t h o t h e r  concept  structural  d e s i g n s t o e s t a b l i s h the s u p e r i o r i t y and economy of the M i c h e l l design. B i r e f r i n g e n t models are made, u s i n g a n u m e r i c a l l y c o n t r o l l e d m i l l i n g machine, and t e s t e d under l o a d i n a  p o l a r i s c o p e , t o c o n f i r m the p r e d i c t e d s t r e s s l e v e l s i n the members and s t r e s s c o n c e n t r a t i o n s i n the j o i n t s . Examples a r e g i v e n o f p r a c t i c a l a p p l i c a t i o n o f M i c h e l l space frames.  The d e s i g n o f a h i g h t e n s i o n t r a n s -  m i s s i o n tower and o f a l i g h t w e i g h t a s t r o n o m i c a l m i r r o r support a r e c o n s i d e r e d .  A l t e r n a t i v e s o l u t i o n s t o both  problems a r e suggested t o p r o v i d e a b a s i s f o r more d e t a i l e d design.  iv  TABLE  OF  CONTENTS  Chapter  1  Page  SOME THEOREMS OF  OPTIMUM  STRUCTURAL  DESIGN  1  Maxwell's  Theorem  1  Michell's  Theorem  4  Criteria  f o r Design  Structure Volume  of  Optimum  . . .  o f Symmetrical  8 Structures  .Chan's C a l c u l a t i o n o f O p t i m u m Volume Optimum Volume f o r C a n t i l e v e r P o i n t Load Analogy Plastic  with Flow  Stiffness  2  GEOMETRY OF Support Main  Slipline  o f an Optimum  APPROXIMATE fans  Structure  Notation Geometry Lengths Joint  Fields :  . . . .  9  Structure with  End  12 19  i n Plane  Structure  STRUCTURES  22 . . . .  24  26 30 31 32  of Structure o f Members  Coordinates  Deflection  33 35 35 36  v Chapter  3  Page  FORCES IN A STRUCTURE  STATICALLY  DETERMINATE  MICHELL 38  Forces  Due  t o Weight  Forces  i n t h e Members  Convention  f o r Load  o f Members  . . . .  40  Angles  at a  Typical  Joint Size  41 o f Members  Reactions 4  A N A L Y S I S OF CANTILEVERS  47  a t Supports  47  SOME P I N J O I N T E D M I C H E L L .  Selection  of Design  Symmetrical Fan  Fibre  50  Parameters  Cantilevers,  Single  Angle  50 Load  . .  .  57  Index  57  Effect  of Variation  o f Span  Symmetrical Load  Cantilevers,  Symmetrical  Cantilevers,  Single  66 Tilted 68  Multiple  Loads  73  Tilted  5  THE  Symmetrical  Cantilevers,  Cantilevers  DESIGN  OF  J O I N T S AND  MODIFICATIONS Pinned Rigid  54 56  Angle  Volume  Skew  39  Joints Joints  74 , .  76  OTHER 80 81 89  vi Chapter  Page  Biaxial  Stress  Systems  i n Joint  Areas  98  Deflection Biaxial  of Joints  Stress  Semi-Rigid Elastic  Subjected  to  .'  102  Joints  Buckling  . . . . o f Members a n d  Frames  107  Selection 6  COMPARABLE  of Joint  Type  . . .  Array  F i v e and Three levers . . .  Fibre  115 Michell  Canti116  Truss  116  Two B a r C a n t i l e v e r  117  :  Cantilever  of Parabolic  Triangular  Plate  Cantilevers  Section  I l l  . . . .  of Uniform  118  Cross  Section  MANUFACTURE Five Solid  119  AND  Fibre  TESTING  121  Cantilever  . . . .  Cantilever With  Five  Rigid  Fibre Fibre  Calibration Level  OF MODELS  Pin Jointed  Cantilever  Three  109 113  Fibre  Warren  and Design  STRUCTURES  Infinite  7  105  122 126  Lightening  Rigid  Holes  . . . .  Cantilever  129  Cantilevers  of Fringes  i n Terms  127  133 of  Stress 142  vii Chapter 8  Page TOWER FOR HIGH TENSION  TRANSMISSION  LINE  145  Specification  f o r Tower D e s i g n  Three F i b r e C a n t i l e v e r Four F i b r e C a n t i l e v e r 9  146  Design  148  Design  151  DESIGN OF A MIRROR SUBSTRATE Geometric  Design of C a n t i l e v e r  Symmetrical  Cantilever  Skew C a n t i l e v e r  15 3 Ribs  . . .  Rib  157  Rib  157  L o a d s on R i b s and Member S i z e s Multi-Point  Appendix  156  159  S u s p e n s i o n on S i x R i b s  . . .  165  (Second Volume)  A  NOTATION  B  EQUATIONS  FOR STRUCTURES  1  GOVERNING GEOMETRY OF  STRUCTURE  6  C  COMPUTER PROGRAMMES FOR STRUCTURE  DESIGN  D  FORCE  E  DATA FOR SELECTED MICHELL CANTILEVERS  F  BIAXIAL STRESS IN JOINTS  75  G  DETAILS OF COMPARABLE STRUCTURES  83  H  MANUFACTURE OF PHOTOELASTIC MODELS  97  J  DEFLECTION OF UNIFORMLY LOADED PLATES  SYSTEM IN MICHELL CANTILEVERS  . .  17  . . . .  25  . . .  42  . . .  118  v i i i  LIST  OF  FIGURES  Figure  Page  1.1  Forces  acting  on a t y p i c a l s t r u c t u r e  1.2  Curvilinear  1.3  Layout  2.1  P r a c t i c a l approximation  coordinate  . . . .  system  9 12  o f f i b r e network  16 to a  theoretical  optimum  structure  27  2.2  Typical  Michell  29  3.1  Details  o f a t y p i c a l member  3.2  Convention  cantilever  f o r load  angles  39 at a typical  joint  41  3.3  Forces  acting  at joint  J„  . . .  3.4  Forces  acting  at typical  joint,  3.5  at typical  'A'  fan  joint,  at typical  'B'  fan  joint,  3.7  Forces acting a , l . . Forces acting J, l,b Forces acting  3.8  External  A T  41  J ^  . . . .  44  J  3.6  45  at joint  equilibrium  46  of a  Michell  cantilever  48  4.1  Symmetrical  4.2 4.3  4.4  cantilever  with  point  load  Variation fibres Variation fibres  of fan angle  with  number  Variation  o f volume index  . . .  Volume  5 "  55  of 58  of f i b r e angle  with  number  of 59  with  number  of  fibres 4.4a  43  60 index  f o r symmetrical 1  cantilevers, 6  1  XX  Figure 4.4b  Page Volume i n d e x f o r s y m m e t r i c a l c a n t i l e v e r s , § = 2  4.4c  Volume i n d e x f o r s y m m e t r i c a l c a n t i l e v e r s ,  £ = 4.4d  62  4  6 3  Volume i n d e x f o r s y m m e t r i c a l c a n t i l e v e r s , 5  "  •  5  •  •  •  6  4  4.4e  Volume i n d e x f o r s y m m e t r i c a l c a n t i l e v e r s ,  4.5  V a r i a t i o n o f s t r u c t u r e volume w i t h span . .  67  4.6  Symmetrical M i c h e l l c a n t i l e v e r with load  tilted 68  V a r i a t i o n of s u p p o r t r e a c t i o n s of t i l t  angle  4.7 4.8  with  70  V a r i a t i o n o f s t r u c t u r e volume w i t h  angle  of t i l t  71  4.9  Cantilever  4.10  T i l t e d symmetrical M i c h e l l c a n t i l e v e r  4.11  T y p i c a l skew M i c h e l l c a n t i l e v e r  4.12  V a r i a t i o n of support r e a c t i o n s with of skew ' V a r i a t i o n o f f i b r e a n g l e and r a d i u s w i t h degree o f skew . . .  4.13 5.1 5.2a  Possible member  loaded j o i n t s  end d e s i g n f o r t y p i c a l  73 ...  75 77  degree 77 ratio 79  pin-jointed 81  Volumes and j o i n t a l l o w a n c e f o r c a n t i l e v e r s ]j = 1,2  5.2b  with  .  85  Volumes and j o i n t a l l o w a n c e f o r c a n t i l e v e r s 5 -  4  8  6  x Figure  Page  5.2c  Volumes and j o i n t allowance f o r c a n t i l e v e r s  5.2d  Volumes  and  5.3  Typical  joint  5.4  Mohr's stress  5.5  joint  circle  allowance  i n a rigid  for cantilevers  Michell  for uniaxial  and  framework.  biaxial  Deflection  of Michell  and b i a x i a l  due  104  6.1  Comparative  7.1  Five  7.2A  Stress patterns in pin-jointed cantilever subjected t o 110 l b I n n e r end  Michell load.  Stress patterns in pin-jointed cantilever subjected t o 110 l b O u t e r end  Michell load.  7.4  'Solid'  pin-jointed  with  7.5  Five  rigid  7.6a  Five fibre 27 l b l o a d Stresses  framework  . .  structures  Cantilever 20 l b  7.6b  in a Michell  cantilever  fibre  to  stress  Semi-rigid  fibre  joints  cantilevers  5.6  7.3  99  100  uniaxial  7.2B  .  -  106 114  Michell. cantilever  load  25  lightening  .  .  123  123  125  lb  126  holes  -  load 127  Michell  Michell  cantilever  cantilever  130  - J-_ with 7  131  at joint  132  2 7.7 7.8  •7.9  Three f i b r e M i c h e l l c a n t i l e v e r f i l l e t s - 25 l b l o a d Joint details cantilever Three load  fibre  i n three  Michell  fibre  with  radius 134  Michell 135  cantilevers  -  20 l b .  136  xi Figure  Page  7.10  Buckled  7.11  Three ^V"  three  fibre Michell  fibre Michell  radius  cantilever  cantilever  . . .  138  with  fillets  140  16  7.12  Detail  of stress  J  3  7.13  Tensile  test  7.14  Fringe  8.1  Clearance  8.2  Possible I  design  Possible II  design  8.3  9.1  9.2  9.3  Figures  pattern 2  in a  f o r high  joint  1  specimen  orders  4  1  for calibration  loaded  . .  specimen  tension  for a  .  143 144  wires  147  transmission  tower  149  for a  transmission  tower  151  Tentative substrate  design  f o r large  mirror 155  Three f i b r e rib .'  symmetrical  Three  skew  fibre  (Second  around  Michell  cantilever 158  Michell  cantilever  r i b  . .  160  Volume)  Al  Notation  for Michell  structures  A2  Members  i n a t y p i c a l panel  3  Bl  Members  i n fan  6  B2  Typical  quadrilateral  B3  Joints  B4  Five  B5  Deflection  Dl  Forces  panel  in typical Michell  fibre  symmetrical of a Michell  acting  2  7  cantilever  . . . .  cantilever  12  frame  on a M i c h e l l  cantilever  9  15 . . .  28  xii  Figure  D2  Page  Forces  at  joint  J.,„  29  NN  J  D3  Forces  at  a  typical  inner  D4  Forces J _ a,l  at  a  typical  'A'  Forces J, , l,b  at  D6  Forces  at  D7  Freebody  D5  joint  fan  J  ^  . . .  joint, 34  a  typical  'B'  fan  joint, 36  joint  37  diagrams  for  forces  at  supports  39  Fl  Biaxial stresses  at  F2  Approximation  a  to  a  typical  biaxially  joint  . . .  77  F3  Extension  Gl  Warren  G2  W i l i o t .diagram  bar  76  stressed  joint  Warren  32  of  a  typical  member  80  truss  84 for  d e f l e c t i o n of  a  truss  86  G3  Two  cantilever  G4  Cantilever  of  G'5  Triangular  plate  G6  I-beam  HI H2 H3  The e f f e c t o f r o t a t i o n Geometry of c u t t e r o f f s e t s O r d e r and d i r e c t i o n o f c u t s  Jl  machining a Loaded d i s c  parabolic  87 section  cantilever  89 91  cantilever  94  for  typical Michell cantilever. . and s u p p o r t s . . . . . . . . . .  101 102 105  xiii  ACKNOWLEDGEMENTS  The a u t h o r w i s h e s t o e x p r e s s h i s g r a t i t u d e t o Dr. J . P . Duncan f o r h i s i n v a l u a b l e a d v i c e , guidance and encouragement t h r o u g h o u t a l l phases o f t h i s  investigation.  The a s s i s t a n c e r e n d e r e d by P r o f e s s o r W.O. Richmond, D r . C R . H a z e l l and Dr. H. Vaughan i n t h e s o l u t i o n o f problems a r i s i n g from t h e t h e o r e t i c a l a n a l y s i s i s much a p p r e c i a t e d , as a r e t h e many s u g g e s t i o n s toward t h e s i m p l i f i c a t i o n o f t h e computer  programmes made by Mr. S. S i k k a .  Mr. P. Hurren and Mr. J . Hoar, C h i e f T e c h n i c i a n s , and t h e i r s t a f f , were o f g r e a t p r a c t i c a l h e l p w i t h t h e manufacture and t e s t i n g o f models. The r e s e a r c h was s u p p o r t e d by g r a n t s from t h e N a t i o n a l Research C o u n c i l , f o r which thanks a r e due. Computing  f a c i l i t i e s were p r o v i d e d by t h e U n i v e r s i t y Comput-  ing Centre. S p e c i a l a p p r e c i a t i o n i s e x p r e s s e d t o Mrs. M. E l l i s who c o n v e r t e d my rough n o t e s i n t o t h e f i n a l typed copy.  xiv SYMBOLS  A  member, a n d l e n g t h  a  suffix  B  member, a n d l e n g t h  b  suffix  C  constant  c  constant  to identify  to identify  abbreviation D  support  E  strain  e FA,  o f member,  1  fibre  joint o f m e m b e r , i n 'B'  fibre  joint  f o r cos 6 i n Appendix  B  spacing energy,  Young's  modulus  dilatation FB  forces  i n members A a n d B f o r ( 2 r s i n 6 / 2 ) i n Appendix  f  abbreviation  h  position  I  second  J K  joint i n Michell structure a p p r e v i a t i o n f o r l a r g e term equations i n Appendix B  L  span o f  1  length  M  bending  N  number o f f i b r e s  P  moment o f a r e a  of a  cross-section  repeated  i n several  structure  o f members  (Maxwell's  theorem)  moment i n a Michell  structure  force pin.diameter  r  fan- r a d i u s  s  abbreviation sB  B  vector  p  sA,  i n 'A  i n non-rigid  joints  f o r s i n 0 i n Appendix  c r o s s - s e c t i o n a l area  B  o f members A a n d B  XV t  thickness  o f members  U  direction  vector,  curvilinear  V  volume  V  direction  vector,  curvilinear  W  work  WA,  WB  OJ x,  o f members  WA,  coordinate  WB  deflection y, z  V a,  width  coordinate  Cartesian rise  3  coordinates  o f skew  curvilinear  structure coordinates  6  f a n angle  e  linear  n  arbitary  9  angle  v  Poisson's  ratio  p  radius  curvature  a  normal  T  shear  ty  an  ty  angle  strain fraction  between  of  ( M i c h e l l ' s theorem)  fibres  stress stress  angle between  member  and C a r t e s i a n  axes  xyi.  GLOSSARY  C e r t a i n words a r e used i n t h i s work i n a s p e c i f i c sense. terms.  These are l i s t e d below t o g e t h e r w i t h a few s p e c i a l i z e d A reference  i s g i v e n t o t h e i r more formal  introduction  i n the body o f the paper. Fan  Members r a d i a t i n g from a support p o i n t and not forming p a r t o f the q u a d r i l a t e r a l network o f the main s t r u c t u r e ,  Fibre  (page 30)  A continuous sequence o f members j o i n e d end to end.  These n o r m a l l y o r i g i n a t e a t a support  p o i n t and are continuous t o the o u t e r boundary. In F i g u r e  2.2, AF, FK and KM c o n s t i t u t e one o f  t h r e e f i b r e s r a d i a t i n g from A. Frame  (page 27)  Any  s t r u c t u r e , n o r m a l l y b u t not n e c e s s a r i l y  two  d i m e n s i o n a l , made from s t r a i g h t members  j o i n i n g only rigid  a t t h e i r ends.  The j o i n t s are  and can t r a n s f e r bending moments between  members. STRUDL  (STRuctural Design Language) - A computer routine  developed a t Massachusetts I n s t i t u t e  o f Technology f o r the s o l u t i o n o f indeterminate  structures.  (page 89 )  Truss  Any  s t r u c t u r e , normally b u t n o t n e c e s s a r i l y  two  d i m e n s i o n a l , made from s t r a i g h t members  j o i n i n g o n l y a t t h e i r ends. rotate  freely.  transformed Volume Index Va  (  Bending moments can not be  between members.  T h i s i s the d i m e n s i o n l e s s  )  The j o i n t s can  q u a n t i t y which  appears as t h e l e f t hand s i d e o f e q u a t i o n 1.35.  I t i s used i n the p r e s e n t a t i o n o f  r e s u l t s t o g i v e more g e n e r a l  applicability,  (page 57) , The  remaining  terms, when used as a d j e c t i v e s , have the f o l l o w i n g  meaning:Michell  The  s t r u c t u r e (frame o r t r u s s ) so d e s i g n a t e d  has a g e o m e t r i c a l l a y o u t approximating t o . the t h e o r e t i c a l concepts  e n u n c i a t e d by  M i c h e l l , as d e f i n e d i n Chapter Optimum  2.  (page 2 8)  The volume o f m a t e r i a l i n the s t r u c t u r e so i d e n t i f i e d i s t o be a minimum.  (In o t h e r  works optimum may be used t o mean minimum c o s t , assembly time o r o t h e r Symmetric  criteria).  A c a n t i l e v e r i s s a i d t o be symmetric i f i t s o u t e r end, L ( J  N N  ) l i e s on the x a x i s , which  i s p e r p e n d i c u l a r t o the plane o f the supports  xviii  and e q u a l l y structures  spaced between them.  Such  a r e symmetric about the x a x i s ,  (page 3 0 ) Skew  A skew c a n t i l e v e r i s not symmetric, F i g u r e 2 . 2 i l l u s t r a t e s a skew c a n t i l e v e r , which may be regarded as the g e n e r a l case.  The symmetric  c a n t i l e v e r i s a s p e c i a l type o f s t r u c t u r e w i t h zero skew.  xix  INTRODUCTION OPTIMISATION OF  From e a r l i e s t times, man all  types,  f o r s h e l t e r and  and  industry.  STRUCTURES  has  f o r use  b u i l t structures  in agriculture,  More r e c e n t l y , economic and  of  transport  technical  p r e s s u r e s have demanded a more r a t i o n a l a n a l y s i s ,  rather  than the e m p i r i c a l or s e m i - r a t i o n a l methods which  long  governed s t r u c t u r a l d e s i g n . In g e n e r a l loads  terms, s t r u c t u r e s are d e s i g n e d t o support  a c t i n g at d e s i g n a t e d p o i n t s  i n space and  t h e i r e f f e c t to s e l e c t e d f o u n d a t i o n p o i n t s .  FOUNDATION fc.nfc may precise  Figure  1  The  General  be  to t r a n s f e r  In a d d i t i o n  specif^  or QS a boondon  Structure  to  XX these s p e c i f i e d e x t e r n a l l o a d s , the s t r u c t u r e has a l s o t o support i t s own weight and f o r c e s such as wind or snow loads  applied to i t s surface. Walls,  abutments, dams, f i l l s  are i n t r i n s i c a l l y  and s i m i l a r  structures  s o l i d and t h e i r shape i s u s u a l l y w e l l  d e f i n e d by t h e i r purpose.  The m a j o r i t y  o f s t r u c t u r e s , how-  e v e r , a r e n o t s o l i d and may be c o n s i d e r e d  as an assembly o f  i n d i v i d u a l members so arranged as t o r e s i s t t h e a p p l i e d forces without  failure.  In most cases,  the geometric d e s i g n  i s not s p e c i f i e d uniquely.  of the s t r u c t u r e  Normally t h e requirement, t h a t  c e r t a i n p o i n t s i n space be connected t o support the a p p l i e d loads  can be achieved  member c o n f i g u r a t i o n s .  by any one o f an i n f i n i t e range o f The f i r s t  stage o f d e s i g n  of t h e s e l e c t i o n o f a s p e c i f i c geometric l a y o u t .  thus c o n s i s t s Each con-  s t i t u e n t member may then be designed t o conform w i t h such w e l l e s t a b l i s h e d bases as r e s i s t a n c e t o f a i l u r e o r t o b u c k l i n g , as gauged by e l a s t i c or p l a s t i c t h e o r i e s o f m a t e r i a l b e h a v i o u r . It  i s a t t h i s stage t h a t any o f many such d e s i g n  criteria,  p a r t i c u l a r l y upper bounds on s t r e s s or d e f l e c t i o n or lower bounds on n a t u r a l frequency, may dominate t h e d e t a i l The  optimum d e s i g n  design.  i s u s u a l l y taken t o be t h a t which  minimises the long term c o s t o f c o n s t r u c t i o n and i n s t a l l a t i o n and, and  f r e q u e n t l y , the o v e r a l l l i f e c o s t , i n c l u d i n g maintenance repair.  These c o s t s a r e complex f u n c t i o n s o f the amount  of m a t e r i a l , labour  and overhead expenses i n v e s t e d  manufacture and maintenance. s t r u c t u r a l design  Most p u b l i s h e d  i n design,  work on optimum  i s concerned w i t h the c r i t e r i o n o f  minimum volume o r weight, which are o f t e n i n d i c e s o f c o s t . In a e r o n a u t i c a l , and p a r t i c u l a r l y i n space a p p l i c a t i o n s , the requirement o f minimum weight o f t e n dominates d e s i g n f o r t e c h n i c a l reasons.  Here, the c o s t o f f u e l r e q u i r e d t o  a c c e l e r a t e unnecessary weight, and t h e simultaneous o f pay l o a d may overshadow a l l other cost  reduction  considerations,  even  o f manufacture. In the above d i s c u s s i o n i t was t a c i t l y assumed t h a t  each s t r u c t u r e was s u s t a i n i n g a s i n g l e , constant  l o a d system.  I t i s more common i n p r a c t i c e f o r a s t r u c t u r e t o be at various  times t o d i f f e r e n t s p e c i f i e d l o a d i n g s .  subjected In such  a case, t h e r e may be a t l e a s t two approaches t o the p r e f e r r e d design:a)  Maximum S t r e s s L e v e l Design A chosen geometric l a y o u t i s s o l v e d f o r i n d i v i d u a l  member p r o p e r t i e s , w h i l e s a t i s f y i n g t h e requirements o f f o r c e e q u i l i b r i u m , displacement and  c o n t i n u i t y , boundary  p e r m i s s i b l e working s t r e s s .  conditions  T h i s i s done f o r each l o a d  c o n d i t i o n t o which the s t r u c t u r e i s t o be s u b j e c t e d , i n g more e q u i l i b r i u m e q u a t i o n s than unknowns.  yield-  These s o l u t i o n s  may then be examined member by member t o s e l e c t the f i n a l design.  xxii T h i s p r o c e s s i s repeated f o r d i f f e r e n t l a y o u t s t o determine all  geometric  the optimum minimum volume d e s i g n of  those c o n s i d e r e d .  T h i s procedure i s perhaps  of most  v a l u e f o r cases where no s i n g l e l o a d i n g i s c l e a r l y dominant. b)  U n i f o r m l y S t r e s s e d Dominant Load  Design  T h i s method i s of p a r t i c u l a r v a l u e when a g i v e n l o a d ing  c o n d i t i o n predominates.  A geometric l a y o u t i s s e l e c t e d ,  e i t h e r f o l l o w i n g p r e v i o u s d e s i g n s or as a new  conception.  The member f o r c e s and s t r u c t u r a l d i s p l a c e m e n t s are then determined by a s t a t i c a l l y ant  loading.  determinate a n a l y s i s f o r the domin-  Member p r o p e r t i e s may  then be s e l e c t e d t o  conform w i t h such d e s i g n c r i t e r i a as u n i f o r m s t r e s s ,  resistance  to b u c k l i n g , e t c .  a  statically will  Once member s i z e s are determined,  indeterminate analysis  ( i f the j o i n t s are r i g i d )  i n d i c a t e the magnitude of l o c a l  stress concentrations  caused by bending and shear f o r c e s i n the frame. The s t r u c t u r e i s then a n a l y s e d f o r each of the other loads which i t i s r e q u i r e d t o c a r r y .  C e r t a i n members  may  be s u b j e c t e d t o g r e a t e r loads under these c o n d i t i o n s than by the dominant l o a d i n g , and t h e i r c r o s s s e c t i o n s require modification.  will  A f t e r member l o a d i n g s are rechecked,  the procedure i s i t e r a t e d t o a r r i v e a t a f i n a l d e s i g n . Other geometric l a y o u t s may same way  until  then be examined i n the  a minimum volume s o l u t i o n i s o b t a i n e d .  XXI  A large astronomical dominant l o a d i n g c o n d i t i o n .  mirror  i s an example o f a  O p t i c a l designers  are f a m i l i a r  w i t h t h e f a c t t h a t such a m i r r o r , when h o r i z o n t a l , i s subjected  t o a more c r i t i c a l  l o a d i n g than a t any other  angle. A 1967 t h e s i s by Soosar  [21] surveys approaches o f  t h i s k i n d t o t h e problem o f o p t i m i s a t i o n w i t h p a r t i c u l a r reference  t o s t r u c t u r e s o f predetermined form, and e x e m p l i f i e s  the method.  HISTORICAL BACKGROUND TO OPTIMISATION The  p r i n c i p l e s which govern t h e d e s i g n  o f optimum  minimum volume s t r u c t u r e s were e n u n c i a t e d i n a theorem by J.C.  Maxwell i n 1890 [ 1 ] , f o r simple cases i n which u n i a x i a l l y  s t r e s s e d s t r u c t u r a l members a r e e i t h e r a l l i n t e n s i o n o r a l l i n compression.  T h i s theorem was then extended by A.G.M.  M i c h e l l i n 1904 [2] t o frameworks c o n t a i n i n g both and  compressive members, and s u b j e c t e d  condition.  t o a s i n g l e load  He showed t h a t such frameworks must c o n s i s t  of an i n f i n i t e number o f members a l i g n e d along mutually perpendicular foundation  tensile  a network o f  curved l i n e s s p r i n g i n g from t h e  and spanning t h e domain t o the l o a d s .  Such a  t h e o r e t i c a l s t r u c t u r e cannot be manufactured e x a c t l y b u t can be approximated by a s e r i e s o f chords o f t h e c u r v e s . The  t h e o r e t i c a l model p r o v i d e s  a standard  t o which these  xxiv r e a l s t r u c t u r e s may Implicit s t a t e d by him, is the  be  i n M i c h e l l ' s proof,  i s the  a l s o more s t i f f same s p a c i a l  compared. although not  directly  f a c t t h a t t h i s minimum volume  than any  structure  other s t r u c t u r e contained  domain and  subjected  in  to the same l o a d s .  M i c h e l l * s work seems t o have been o v e r l o o k e d f o r nearly its  f i f t y y e a r s u n t i l H.L.  Cox  [ 3 ] , [ 4 ] , i n 1958  a p p l i c a t i o n to some simple d e s i g n problems.  p o r t i o n o f Cox's l a t e r book [16] was of M i c h e l l ' s theorem and n a u t i c a l and  A  devoted t o an  has  space d e s i g n .  In 1960,  M i c h e l l s t r u c t u r e s , and loadings.  In 1966,  structural  design,  R.L.  analysis  A.S.L. Chan [6]  f o r t h e i r minimum volume.  examined c r i t i c a l l y  considerable  i t s p r a c t i c a l a p p l i c a t i o n i n aero-  c a n t i l e v e r s conforming t o the M i c h e l l c r i t e r i a and an e x p r e s s i o n  illustrated  the  Ghista,  Barnett  t h e i r use  derived  [19],  l i m i t s of a p p l i c a b i l i t y  described  discussed  [20],  of  f o r some simple  [9] surveyed optimum  d e s c r i b i n g the works mentioned above, as  a sequence o f i n v e s t i g a t i o n s u s i n g  e l a s t i c d e s i g n to a c h i e v e  minimum weight.  a l t e r n a t e l i n e s of d e v e l o p -  ment u s i n g  He  also discussed  as c r i t e r i a p l a s t i c s t r e n g t h  simultaneous b u c k l i n g deflection.  He  different  failure  under a l l p o s s i b l e modes and  by  minimum  demonstrated t h a t the M i c h e l l s t r u c t u r e i s  a l s o optimum f o r t h i s In 1969  design,  last  condition.  Hegemier and  Prager  methods Maxwell's and  [17]  demonstrated  M i c h e l l ' s theorems and  by proved  XXV  t h a t the M i c h e l l s t r u c t u r e i s a l s o optimum  (has minimum  volume) f o r l o a d i n g w i t h s t a t i o n a r y creep and n a t u r a l frequency.  In 1969,  Sheu and Prager  f o r a given [18]  developments i n optimum s t r u c t u r a l d e s i g n between and  reviewed 1962  1968. The  above paragraphs  have reviewed  the development  of Maxwell's and M i c h e l l ' s theorems and t h e i r a p p l i c a t i o n t o the s e l e c t i o n of optimum minimum volume s o l u t i o n s t o v a r i o u s s t r u c t u r a l l o a d i n g problems.  In r e c e n t y e a r s a p a r a l l e l  development of the t h e o r y has been used t o s o l v e problems in  plasticity. In 1960,  and,  W.S.  Hemp [5] reexamined M i c h e l l ' s work  f o r the f i r s t time, demonstrated i t s analogy w i t h  l i n e f i e l d s i n plane p l a s t i c  flow.  In m a t e r i a l loaded  slipso  t h a t i t deforms p l a s t i c a l l y , shear occurs along o r t h o g o n a l , curved s l i p  lines.  These form the boundaries  of s m a l l  b l o c k s o f the m a t e r i a l whose s i z e tends t o zero as a continuum i s approached.  Along these l i n e s t h e r e i s r e l a t i v e  movement of the b l o c k s and no  linear  strain.  Thus, f o r a  g i v e n c o n f i g u r a t i o n of boundary t r a c t i o n s , t h e r e i s an i n f i n i t e network of s l i p l i n e s .  This i s exactly  analogous  to M i c h e l l ' s i n f i n i t e s t r u c t u r a l network, whose o r t h o g o n a l c u r v i l i n e a r l i n e s of constant f o u n d a t i o n s and  linear  loads i n space.  connect  In both cases  t h e o r e t i c a l metworks are approximated of s t r a i g h t l i n e s which are chords coordinates.  strain  rigid  these  by a l i m i t e d number  t o the  curvilinear  xxv i  Figure 2  Elementary S l i p L i n e F i e l d f o r E x t r u s i o n Through a D i e ( a f t e r Johnson [23])  C o n s i d e r f o r example, p l a s t i c flow o f m a t e r i a l i n a d i e w i t h p e r f e c t l y rough, plane w a l l s .  There w i l l be no  r e l a t i v e motion between the w a l l s and the a d j a c e n t m a t e r i a l , or  along l i n e s a t r i g h t angles t o the w a l l s .  These are the  p r i n c i p a l d i r e c t i o n s i n the m a t e r i a l b e i n g deformed.  The  m u t u a l l y p e r p e n d i c u l a r s l i p l i n e s — d i r e c t i o n s of r e l a t i v e movement between elements o f the m a t e r i a l — w i l l be i n c l i n e d at  45° t o these p r i n c i p a l d i r e c t i o n s and thus t o the d i e  s u r f a c e and w i l l spread out i n an i n f i n i t e  network  compatible w i t h o t h e r boundary and l o a d i n g c o n d i t i o n s . T h i s corresponds  e x a c t l y w i t h the l a y o u t of the  members forming a M i c h e l l i n f i n i t e  fibre structure.  are i n c l i n e d a t 45° t o the s u r f a c e o f any r i g i d  These  foundation  xxvii subjected  to a d i s t r i b u t e d load.  They form an o r t h o g o n a l  c u r v i l i n e a r network which spans the domain and extends o u t ward t o support the a p p l i e d l o a d s .  The o u t e r f i b r e s d e f i n e  the boundaries of the s t r u c t u r a l domain. In s t u d i e s o f p l a s t i c i t y the t h e o r e t i c a l s l i p  line  f i e l d s a r e approximated by g r a p h i c a l s o l u t i o n s , u s i n g a coarse g r i d o f s t r a i g h t l i n e s l a i d out a c c o r d i n g a r e g u l a r manner.  These l a y o u t s  to rule i n  a r e e x a c t l y analogous t o  those o f r e a l s t r u c t u r e s approximating the t h e o r e t i c a l , minimum volume, i n f i n i t e  f i b r e arrays  domain- f o u n d a t i o n arrangements. further explained  f o r associated  load-  T h i s . a p p r o x i m a t i o n w i l l be  i n this" t h e s i s .  In 1961 W. Johnson  [8] extended t h i s analogy.  He  demonstrated t h a t i n a d d i t i o n t o the correspondence between s t r u c t u r a l l a y o u t and s l i p l i n e s i n the t h e o r e t i c a l case, the v e l o c i t y diagram o f the s l i p  line field  i s identical  i n shape w i t h the f o r c e diagram o f t h e s t r u c t u r e i n the approximate s o l u t i o n s .  Bow's n o t a t i o n may be used i n both  cases when u s i n g g r a p h i c a l s o l u t i o n s .  T h i s correspondence  may be used t o o b t a i n upper bound e s t i m a t e s o f the volume o f optimum frames from known s l i p l i n e f i e l d s o r v i c e  versa.  In a book by Johnson, Sowerby and Haddow [15] t h i s analogy i s r e s t a t e d and i l l u s t r a t e d w i t h examples  of s l i p  l i n e f i e l d s generated by e x t r u s i o n through rough d i e s o f various  shapes.  In p a r t i c u l a r , e x t r u s i o n through a rough  45° wedge shaped d i e produces a s l i p l i n e f i e l d  identical  xxviii w i t h the base fans of the c a n t i l e v e r s d e s c r i b e d l a t e r i n the p r e v i o u s work.  The main s t r u c t u r e o f these c a n t i l e v e r s  i s suggested by the s l i p l i n e f i e l d s  f o r compression o f a  s l a b between p a r a l l e l p l a n e rough p l a t e s . In 1969  a paper by W.  Johnson  [23] d e s c r i b e d the use  of these a n a l o g i e s f o r the s o l u t i o n of p l a t e - b e n d i n g and o t h e r problems.  V e l o c i t y diagrams from s l i p l i n e f i e l d s are  used to d e s c r i b e k i n e m a t i c modes o f d e f o r m a t i o n t o o b t a i n upper bounds f o r the c o l l a p s e of t r a n s v e r s e l y loaded plates.  O u t l i n e P l a n of Study The p r e s e n t work c o n c e n t r a t e s on c a n t i l e v e r s as a b a s i c element i n simple and compounded s t r u c t u r a l forms. example, as Chan demonstrated combined  [6] f o u r c a n t i l e v e r s may  For  be  t o form a beam capable o f r e s i s t i n g pure bending.  The major p a r a m e t e r s — s p a n , number of members, p o s i t i o n and type o f l o a d i n g — a r e v a r i e d to determine t h e i r effects.  relative  Formula are d e r i v e d from elementary t h e o r y and  computer programmes d e s c r i b e d which may s o l u t i o n of c a n t i l e v e r s o f any d e s i r e d  be used f o r the size.  The r e a l s t r u c t u r e s d i s c u s s e d are approximations to the c o r r e s p o n d i n g t h e o r e t i c a l optimum frames, and  their  d e s i g n i n t r o d u c e s secondary problems not c o n s i d e r e d i n the i d e a l case.  The study d e s c r i b e d i n t h i s t h e s i s c o u l d be  c o n s i d e r e d as an e x p l o r a t i o n o f the d e g r a d a t i o n s o f the i d e a l s t r u c t u r e f o r a g i v e n domain and l o a d i n g system.  These  are i n e v i t a b l y i n c u r r e d as t h i s i d e a l i s m o d i f i e d t o s u i t f e a s i b l e methods o f d e s i g n , c o n s t r u c t i o n and use. In p a r t i c u l a r ,  j o i n t d e s i g n r a i s e s s e v e r a l problems.  I f u n i f o r m u n i a x i a l s t r e s s d i s t r i b u t i o n i s t o be s a t i s f i e d , the j o i n t s should be pinned, r e q u i r i n g e x t r a m a t e r i a l and c a r e f u l p r o f i l i n g t o achieve a smooth flow o f s t r e s s .  Rigid  j o i n t s i n t r o d u c e areas o f b i a x i a l s t r e s s which m o d i f i e s the s t r u c t u r a l d e f l e c t i o n . if  Additional material i s required  the j o i n t a r e a i s i n c r e a s e d t o m a i n t a i n a u n i f o r m  level.  In e i t h e r case, the r i g i d  bending  and shear  j o i n t induces  stress  secondary  stresses.  S t r u c t u r a l members a r e o f t e n assumed t o be w e i g h t l e s s f o r elementary  analysis.  I f c o n s i d e r e d , the s e l f weight  f o r c e s are i n t r o d u c e d as c o n c e n t r a t e d loads a t the end joints.  T h i s s i m p l i f i c a t i o n i g n o r e s the bending  s t r e s s e s generated secondary  by the s e l f weight f o r c e s .  and shear  These  e f f e c t s were examined u s i n g computer programmes  (STRUDL) developed  a t Massachusetts  I n s t i t u t e o f Technology  [22]. I n d i v i d u a l members s u b j e c t e d t o compressive may f a i l by b u c k l i n g .  loading  The s t r u c t u r e s c o n s i d e r e d i n t h i s  work were examined f o r t h i s p o s s i b i l i t y .  The E u l e r c r i t i c a l  l o a d f o r each member, assumed t o have f r e e ends, was taken as a lower bound c r i t e r i o n .  In most cases where  failure  by b u c k l i n g was  possible, redesign  a s l i g h t increase  of the c r o s s s e c t i o n or  i n the amount of m a t e r i a l used  a satisfactory solution.  Buckling  i f made as a t h i n p l a t e , may  provided  of the whole s t r u c t u r e ,  be prevented by  the use  of  p a r a l l e l systems, s u i t a b l y c r o s s  connected, i n e x a c t l y  same manner as more c o n v e n t i o n a l  t r u s s e s and  frames  two the  are  designed. A s e r i e s of models of t y p i c a l M i c h e l l were t e s t e d p h o t o e l a s t i c a l l y .  The  structures  fringe patterns  observed  corresponded c l o s e l y w i t h the s t r e s s d i s t r i b u t i o n p r e d i c t e d by  theory.  along  The  s t r e s s l e v e l was  the g r e a t e r p a r t of the  between members.  found to be  l e n g t h of each member  to b i a x i a l s t r e s s exhibited s t r e s s  of the same type as p r e d i c t e d by The of r i g i d  and  L o c a l e l a s t i c b u c k l i n g of the t h i n models  c o u l d account f o r the observed d i s c r e p a n c i e s . subjected  s e n s i b l y uniform  J o i n t areas concentrations  the t h e o r e t i c a l a n a l y s i s .  i n v e s t i g a t i o n demonstrates t h a t p r a c t i c a l  forms  j o i n t e d M i c h e l l s t r u c t u r e s have volumes which  approach those of the c o r r e s p o n d i n g t h e o r e t i c a l optima s o l u t i o n s f o r a s p e c i f i e d load-domain-foundation c o n d i t i o n . I t i s probable t h a t they w i l l p r o v i d e  the c l o s e s t p r a c t i c a l  s o l u t i o n , s i n c e t h e i r b a s i c geometric l a y o u t i s a l r e a d y known t o be optimum.  Since  the types of d e g r a d a t i o n r e f e r r e d  t o above are a s s o c i a t e d w i t h j o i n t s which occur i n a l l designs, design  the c o n v e r s i o n  of any  i n v o l v e s some i n c r e a s e  geometric l a y o u t to a p r a c t i c a l i n volume from t h a t c a l c u l a t e d  xxx i by an elementary  analysis.  At the l e a s t , the i d e a l M i c h e l l  s o l u t i o n i s l i k e l y to be an e x c e l l e n t f i r s t  approximation  and p r o v i d e s a s t a r t i n g p o i n t f o r f u r t h e r computer as d e s c r i b e d f o r example i n the l a s t s e c t i o n of the  analysis, review  by B a r n e t t [ 9 ] . Minimum volume s t r u c t u r e s have a wide a p p l i c a t i o n to many f i e l d s of d e s i g n .  They show to b e s t advantage where  minimum weight i s of paramount importance. c o n s i d e r e d f o r use  i n a i r c r a f t and  They should  be  space v e h i c l e s , where  t h e i r shape p e r m i t s , the M i c h e l l l a y o u t b e i n g i n h e r e n t l y deeper i n p r o p o r t i o n t o span than most other t r u s s d e s i g n s . In t h i s c o n n e c t i o n l i g h t e n i n g h o l e s have been used i n a i r f r a m e and  s h i p r i b s f o r many years i n an e m p i r i c a l manner.  M i c h e l l l a y o u t s c o u l d p r o v i d e g u i d e l i n e s f o r t h e i r more e f f e c t i v e d e s i g n by " c h a n n e l i n g " the s t r u c t u r a l s t r a i n  into  mutually perpendicular d i r e c t i o n s . In more s t a t i c a p p l i c a t i o n s , economy i n m a t e r i a l can show s a v i n g s i n i n s t a l l a t i o n i n remote a r e a s , where t r a n s p o r t c o s t s can be  significant.  A p o s s i b l e design f o r a high tension transmission tower i s i n c l u d e d as an i l l u s t r a t i o n of t h i s These f a m i l i a r s t r u c t u r e s are designed  application.  to support known  l o a d i n g s at f i x e d p o i n t s i n space w h i l e p r o v i d i n g s u i t a b l e clearance for e l e c t r i c a l i n s u l a t i o n .  It i s increasingly  common f o r such towers t o be s i t e d i n remote a r e a s , o f t e n b e i n g p l a c e d on s i t e by h e l i c o p t e r . o b v i o u s l y of advantage.  Minimum weight i s  xxxii A more  specialised  framework  concerns  substrate  serves  and  to  maintain  attitudes. the part  of  is  deflection  loadings of  the  imply has  adequate  in  the  the  self  load  in  mirror  in  is  for  the  deformation.  low  level to  of  resist  manufacture  and  by  assumes  forces  it  is  various  form  The  The  the  stress  in  the  the  self  tolerable  optical  of larger  Michell  optimum  reduces  the  The  surface  distributing  since turn  mirrors.  minimum d e f l e c t i o n  weight  this  Michell  reflecting  normal use.  suitable  substrate,  reserve  of  present the  frameworks. familiar maximum  primary  for  a  weight deflections  requirements— the  structure,  different  transport  These  stress.  to will  work  have  which  loadings  to  site—terrestrial  While  of  yield  applications,  theoretical  of is  subjected  savings  in  material  economics  the  in  volume  the  and  Their and,  in  same  shape,  forward  competitors.  and  stress less  to  unfamiliar straight  their  continued  of  significantly  forms  structures that  provides  uniform d i s t r i b u t i o n  structural  Michell  fields  the  for  the  space.  evidence  similar  as  surface—limited  during  The  of  large  and  generally  incurred or  causing  mirror a  and  the  systems  criterion  structural  over  of  support  shape  particularly  reactions given  to  design  surface  the  layout  only its  The  mirror  support  application  experimental Michell than  loading  more and  fabrication generally  use for  operation.  in  many  dynamic  1  CHAPTER 1 SOME THEOREMS OF OPTIMUM STRUCTURAL DESIGN  MAXWELL'S THEOREM I n 1890 which s p e c i f i e d  C l e r k Maxwell p u b l i s h e d a theorem t h e t o t a l volume o f a framework  t o f o r c e s i m p o s e d by an e x t e r n a l f o r c e s y s t e m . was  [1]  subjected This  an e a r l y a t t e m p t t o d e f i n e optimum s t r u c t u r e s .  proof  i s s h o r t and i s r e p r o d u c e d h e r e w i t h m i n o r  for consistency  The  changes  i n symbols.  A framework  i s composed o f members c a r r y i n g e i t h e r  compressive or t e n s i l e  forces.  I t i s i n equilibrum  the a c t i o n of a system of f o r c e s P  under  , acting at points i n  s p a c e , whose p o s i t i o n v e c t o r s a r e h" , r e f e r r e d t o a N  ient origin,  paper  conven-  0.  L e t t h e framework v i r t u a l displacement, space of magnitude ordinate origin,  3e.  be s u b j e c t e d t o an  infinitesimal  c o n s i s t i n g of a uniform d i l a t i o n This  of  i s so a p p l i e d t h a t t h e c o -  0, i s u n d i s t u r b e d .  s p a c e i s t h u s e x t e n d e d by a s t r a i n  Every l i n e a r element of e.  Then -  work done by a p p l i e d f o r c e s  =  . . . . (1.1)  ,t Strain  energy  where length.  F  The  stored  by  frame  =  Q  represents  the force  s u f f i c e s  and c  t  i n a  indicate  member  and  Ii t s  respectively  the  * tensile  and By  by  compressive  members.  the p r i n c i p l e of V i r t u a l  t h e above  equations  are equal.  Work,  the energies  expressed  Thus  (1.2) t = l  C s >  Suppose so  proportioned  everywhere  a ^  t  everywhere  a  .  M a i  that  the cross-section  that  the stress  and  that  o f each  i n a l l tensile  i n the compressive  member  i s  members  members  i s  i s  Then  • C v  where and  V , t  sive V,  V  c  s  are the  members  i s  represents  respectively.  area  volumes  of the tensile  The  volume  t o t a l  •  (1.3)  c  the cross-sectional  total  %  o f a and  of the  member compres-  structure  thus  F^_ a n d F the d i f f e r e n c e i n t of the compressive tensile forces are c  represent t h e magnitudes of these forces, h e i r e f f e c t being allowed f o r by the sign term, f o l l o w i n g t h e u s u a l convention that p o s i t i v e .  3  V=V V t+  If  equations  .  c  1.3 a r e s u b s t i t u t e d  into  . . . (1.4)  1.2  (1.5)  The on  constant C i s a function  the s t r u c t u r e  framework i s  imposed  and i s i n d e p e n d e n t o f t h e way i n w h i c h t h e  constructed.  Equations to  of the loading  1.3 and 1.5 may be s u b s t i t u t e d  into  1.4  yield  (1.6)  (1.7)  where  It  s h o u l d be n o t e d  1.6 and 1.7 i s independent C i s a function  that  t h e second  term i n e q u a t i o n s  o f t h e d e s i g n o f t h e framework.  of the external  loading  while  3  arbitrary  uniform  stresses.  A minimum volume s t r u c t u r e deduced  o*. and a a r e t c  from e q u a t i o n  conditions  1.6 by c o n s i d e r i n g  may  t h u s be  t h e minimum  value  of  the f i r s t  solution  term  to this  alone.  o b t a i n e d by p l a c i n g  to  zero.  or  o n l y compression  a)  Such s t r u c t u r e s w o u l d  ties and  b)  Maxwell c o n s i d e r e d the simple  members.  or struts  triangular applied  or V  equal  c o n t a i n o n l y t e n s i o n members  Examples o f t h e s e a r e  (and r o p e s )  opposite a x i a l  either V  subjected t o equal  forces,  o r t e t r a h e d a l frames w i t h  at their vertices  forces  and a c t i n g  along  l i n e s which i n t e r s e c t w i t h i n t h e frame, c)  c a t e n a r i e s and a r c h e s . Such s t r u c t u r e s may be c o n s i d e r e d as t r i v i a l  present  i n v e s t i g a t i o n which i s concerned  volume s t r u c t u r e c o n t a i n i n g b o t h  tensile  'mixed' f r a m e s ,  f o rthe  with  a minimum  and  compressive  members.  F o r such  Maxwell!s theorem had  no d i r e c t  u s e , and i t r e c e i v e d l i t t l e a t t e n t i o n  f o r some  year  MICHELL'S THEOREM M a x w e l l ' s t h e o r e m , as e x p r e s s e d generalized cover  by A.G.M. M i c h e l l  'mixed' f r a m e w o r k s .  repeated  below i n m o d i f i e d Consider  in  i n 1904  Again  by e q u a t i o n  [2]  the proof  1.5, was  and e x t e n d e d  to  i s s h o r t and i s  form  a g a i n t h e same g e n e r a l i z e d f o r c e s y s t e m a s  t h e Maxwell theorem, b u t a p p l i e d i n t u r n t o a s e r i e s o f  different  frames occupying  bounded o r i n f i n i t e ) .  a certain  space  (which may be  5  L e t t h i s space, and the t r u s s e n c l o s e d , arbitrary  deformation such t h a t no l i n e a r element o f the  space s u f f e r s an e x t e n s i o n than  T){SZ)  ,  an a r b i t r a r y arbitrary  undergo an  where  or contraction numerically  ( 6 £ ) i s the length  small f r a c t i o n .  greater  o f the element and n i s  [Note t h a t t h i s i s an  d e f o r m a t i o n and need n o t be u n i f o r m as r e q u i r e d  by Maxwell's theorem.] In a t y p i c a l of l e n g t h the  truss, A, c o n t a i n i n g N members, a member  w i l l undergo a s m a l l change o f l e n g t h  f o r c e a c t i n g on t h i s member i s  increase  If  ( a x i a l l y ) , then the  i n s t r a i n energy s t o r e d i n the framework d u r i n g the  deformation i s c l e a r l y  The  work done by the e x t e r n a l f o r c e s d u r i n g  d e f o r m a t i o n i s S^.  this  I t i s independent o f the shape o f the  framework and equals the change i n s t r a i n energy Remembering t h a t by d e f i n i t i o n a l l  | e, | <_ n ,  stored. then  .  •  ( 1 . 9 )  6 Suppose, however, t h e r e e x i s t s a truss, M, i n which a l l members a r e s u b j e c t e d t o a uniform s t r a i n e q u a l t o n . Then, i n e x a c t l y the same way as i n 1 . 9  Equation 1 . 7  .  .  .  -.  a  9  e  a  ( 1  « 1 1  )  demonstrates t h a t i f the volume o f a  s t r u c t u r e i s t o be a minimum, the term  n [ Z (Fl n=l n  )J  n  must be a minimum, s i n c e a l l o t h e r v a r i a b l e s i n t h i s are independent  ( 1 . 1 0 )  o f the framework.  Thus E q u a t i o n  equation  1-10  i n d i c a t e s t h a t o f a l l the p o s s i b l e frames t h a t may be used, t h a t having u n i f o r m s t r a i n i n a l l members w i l l have the minimum volume. T h i s i s e q u i v a l e n t t o s p e c i f y i n g uniform s t r e s s i n a l l members f o r frameworks manufactured from  isotropic  m a t e r i a l s and s u b j e c t e d t o s m a l l d e f o r m a t i o n s .  Unless  s p e c i f i c a l l y mentioned, i t w i l l be assumed t h a t the s t r e s s e s i n t e n s i l e and compressive  members are n u m e r i c a l l y equal  .  .  .  .  ( 1 . 1 2 )  The p r e c e d i n g proof s p e c i f i e d no l i m i t on the s i z e of the space t o be occupied by the frameworks under  comparison.  7 If  t h e space  the  condition  than the  any  specified  less  volume  direct  contrary  they  at  an  the  are of  angle  0  a  Thus  9  e  lying  of  of  be  I f  strain  w i l l  w i t h i n the.  a  frame  of  t h e optimum strain  i n the v i r t u a l  I f this i n a  larger  structure,  were  n,  de-  not so,  member  than  M,  f o r which which  i s  conditions.  p r i n c i p a l  the strain  opposite  less  bounds.  point  magnitude  volume  frames  the  a t a  a  satisfying  i n i t smembers.  however,  beyond  found  have  uniform  i n the proof.  had  t o one  €  may,  frames  with  of p r i n c i p a l  (equal)  point,  frames  a l l other  to the specified  a t a  If  be  w i l l  strains  t h e members  lines  strain  If sign  then  extending  could  i n f i n i t e ,  strain  There  considered  direction  as  unequal  than  general  l i e along  formation  the  having  boundary.  In must  uniform  i s bounded,  volumes  smaller  considered  of  frame  space  have  a  be  strains  i s t h e same  signs, them,  = £c (29) OS  i s not exceeded  then Eg,  are of i n a l l  the strain  i s given  by  < £  i n either  case.  the  same  directions. i n a  direction  8 CRITERIA  FOR  The must  DESIGN  members  l i e along  follow  two  divided a)  systems  b)  optimum  of  of  of  curve  and  volume)  value  framework  stress. Such  STRUCTURES  They  sets  must  may  be  involutes  derived  (evolute),  systems  of  concentric  networks  minimum  curves.  classes  tangents  (minimum  p r i n c i p a l  orthogonal  orthogonal  rectangular  an  general  any  Sets  class  of  two  from  OPTIMUM  directions  sets  into  of  OF  of  equiangular  c i r c l e s  straight  and  s p i r a l s .  their  lines,  are  r a d i i ,  and  special  cases  of  b. This  i n f i n i t e  implies  number  of  that  the  members,  optimum  framework  i f orthogonal  contains  curves  are  to  an be  followed. A ent the  framework  systems,  may  provided  be  constructed  that  from  displacements  parts  are  of  d i f f e r -  compatible  along  interfaces. Point  which that  forces  d i s t r i b u t e their  i n f i n i t e  frames  may  conditions. would level,  points  numbers By  have a.  the  using be  of of  may load  fibres  d i f f e r e n t  the  such same  supported  into  action  specified In  be  to  cases, minimum  the  are  by  special  network,  singular  or  members,  by  points,  arranging where  converge. sets  of  s a t i s f y from  orthogonal given  boundary  equations  volume  for  a  curves,  1.7  given  and  and  several loading  1.10,  uniform  a l l stress  9  S p e c i a l Case f o r Volume of S t r u c t u r e Consider any two d i m e n s i o n a l s t r u c t u r e as i n d i c a t e d in Figure  1.1  The s t r u c t u r e i s assumed to be supported a t  two f i x e d p o i n t s A and B, which are symmetrically  placed  about the x a x i s .  Figure  1.1  Forces A c t i n g on a T y p i c a l  A t y p i c a l load, P , N  Structure  i s shown, which i s a c t i n g a t L , any N  p o i n t on the x a x i s Then  RA a a  Total  load  A L X  Bjc L  + Ay J +  By j  ( 1  1 3 )  10 where t h e s u f f i c e s x and y i n d i c a t e components these  forces p a r a l l e l  By e l e m e n t a r y s t a t i c s  of  t o t h e x and y a x e s . -  . . . . (1.14)  . . . . (1.15)  (1.16)  These e q u a t i o n s  yield,  by  substitution  D  ™ L  No A  and B y  . . (1.17)  c o n c l u s i o n s c a n be drawn a b o u t t h e m a g n i t u d e s without  knowledge  o f the s t r u c t u r a l  of  arrangement.  y However, c o n s i d e r  are  2 J  applied perpendicular  everywhere  zero.  Then  the s p e c i a l  case  i n which a l l loads  t o t h e x a x i s , so t h a t  applying equation  1.5  -  (P^)  is  11  . . . . (1.18)  For vertical side the  it  of  any  structure  reactions this  constant  follows  at  the  equation C  is  i n w h i c h A =B ;(that y y supports  becomes  zero  that  and  zero.  since  as  works  of  the  tensile  the  total  volume  of  the  compressive  who  some  reviewed  showed  Hencky's  forty  years  Cox  [3],  by  that  structural  special  these  design  theorem  for  Many w o r k s extensive Barnett can  list  [9].  provide  of The  the  have  lines  been  references  Hemp  slip-line  inspiration for  with in  . . . .  (1.19)  is  the  thus  members  in  plane in  there  design  of  lay  frame-  for  others  optimum  expressed  plastic this in  dormant  and  [5]  those  included  layouts the  1.12,  theory  conditions  published is  1.5,  members  Michell  [4],  analagous  slip  equation  conditions.  geometrical  were  hand  t  volume  these  right  equation  total  For until  from  The  satisfying  the  equal,)the  Thus  from  v =v c  same  are  is,  the  by  flow.  field  and  paper  discussed optimum  in  by  an R.L.  turn  structures.  12 CHAN'S CALCULATION  OF THE VOLUME OF AN OPTIMUM STRUCTURE  A p a p e r by Chan cation of Michell's  [6]  i n 1960 d i s c u s s e d  the a p p l i -  t h e o r e m t o two d i m e n s i o n a l  structures  and  i n p a r t i c u l a r to c a n t i l e v e r s .  for  t h e volume o f an optimum minimum volume s t r u c t u r e .  treatment since  i s lengthy  i t forms a t h e o r e t i c a l b a s i s It  an  and r i g o r o u s ;  follows  lines.  handed c u r v i l i n e a r  Figure  1.2  T h e s e may  to the present that  arrays  be u s e d  s y s t e m as shown  an  expression His  a summary o f h i s p a p e r  from M i c h e l l ' s proof  optimum s t r u c t u r e must l i e a l o n g  perpendicular  He d e r i v e d  C u r v i l i n e a r Coordinate  work.  t h e members o f  of mutually  to define  i n Figure  follows,  1.2.  System  a right  13 P o i n t D has c u r v i l i n e a r c o o r d i n a t e s a, 3. tangent d i r e c t i o n s  to the a and 3 curves a t D are U and V  U making a p o s i t i v e angle cf) w i t h the x a x i s .  respectively,  The i n s e t i n d i c a t e s may  The  be r e p r e s e n t e d  a s m a l l l i n e element DD  which  1  i n terms of the c u r v i l i n e a r c o o r d i n a t e s  da and d3 [Chan 1]  The d i r e c t s t r a i n a t D i n the a d i r e c t i o n w i l l assumed to be +e and t h a t i n the 3 d i r e c t i o n to be -e. shear s t r a i n i s zero s i n c e these  (1.21)  be The  are p r i n c i p a l d i r e c t i o n s .  The r o t a t i o n a t D due to these s t r a i n s w i l l be denoted by c o . The r e l a t i o n s between the displacement and s t r a i n s and r o t a t i o n s , s t a t e d i n Love  1  [ 7 ] . These y i e l d  [Chan 2]  1 ay . JJ aB B d p AB53  [Chan 3]  Set  +  V A B  i n c u r v i l i n e a r c o o r d i n a t e s are  3A 3p>  A  cHJ  components  (1.22)  o  [Chan 4]  [Chan 5]  A l s o , by c o n s i d e r i n g the angle cj) i n F i g u r e  1.2  14  «  M _  M  ,  ^ J doc  [Chan 6]  . .  (1.23)  1 IB 1.22  o f u and v  A £ + V ¥  a  2A  ±  «  Equations derivatives  -  ?  and.1.23 may  be u s e d t o f i n d  the  -  dU ^ bp  - B O O + V ^  [Chan 7]  5J3 (1.24)  Aco-u^; /av , - B e - u ^ v  u and v may  =  jL^GO-2£^ and, f i n a l l y ,  be e l i m i n a t e d  between  JL (cO+2£^==  O  [chan  equations  8]  1.24,  [Chan 9]  . . (1.25)  o f OJ g i v e s  elimination  [Chan 10]  This curvilinear of  t h e optimum  „ v )  f o r the system o f  structure.  solution  be c o n s i d e r e d  a  equation  c o o r d i n a t e s and f o r t h e l a y o u t o f t h e members  For may  i s the c o m p a t i b i l i t y  . (1.26)  3(f)  of t h i s  9<f>  =  W  =  z  e  r  o  fundamental equation three  cases  15 Here a and  3 are both  s y s t e m becomes C a r t e s i a n . in  two  perpendicular  b)  Either  or  will  an  e n v e l o p e t o an  •—-  either be  will  c)  form  is  a or  straight.  The  other  ~-  nor  considered values).  Both s e t s of  s cp e c i f y i n g  s e t of  i n the  cases  one  i n general  form  d e p e n d on  lines  will  be  set  the  curved  evolute.  are  lines  case  arising are  region being  that there  and  zero.  dp  i s t h e more g e n e r a l  (the o t h e r  inflections  They w i l l  —  da  This  3 i s a constant  'involutes' to that  Neither  straight  zero.  ' e v o l u t e ' whose shape w i l l  boundary c o n d i t i o n s . and  coordinate  dp  case  coordinates  the  A l l s t r u c t u r a l members a r e  da  of  and  arrays.  ~  In t h i s  constant  a r e no  and  the  one  from t h i s  as  continuously considered  zeros  special  curved  may  be  3d)  3d)  9a  93  o f TA- o r  that w i l l  and  avoided  inside  the  boundaries. For  s o l u t i o n of the  d i t i o n s must be shown i n F i g u r e  equations,  assumed, and  some b o u n d a r y  Chan c o n s i d e r e d  the  con-  layout  1.3.  From f i x e d  points, A  and  drawn, m e e t i n g a t C s u c h t h a t ACB  B,  arcs of radius r  = 90°.  C  i s the  are  origin  by  be  16  P  F i g u r e 1.3  Layout of F i b r e Network  of the c u r v i l i n e a r c o o r d i n a t e system a , 3 , the two a r c s b e i n g the r e s p e c t i v e base l i n e s .  circular  A typical point,  P i s d e f i n e d by the i n t e r s e c t i o n of the c o o r d i n a t e l i n e s FP and GP,  whose angular v a l u e s are d e f i n e d by the  between the tangent a t F or G and a t C.  The  angle  the a p p r o p r i a t e tangent  network i s bounded by the outer f i b r e s ADL  and  BEL. I f the a r c s c e n t r e d on A and B are of e q u a l r a d i u s , as drawn, the network i s symmetrical  about CL.  I f they  17  differ,  symmetry i s l o s t b u t t h e same g e n e r a l t h e o r y i s  applicable. Many d i f f e r e n t boundary c o n d i t i o n s c o u l d be s p e c i f i e d , each l e a d i n g t o somewhat d i f f e r e n t end e q u a t i o n s .  This  arrangement i s p a r t i c u l a r l y c o n v e n i e n t s i n c e i t r e p r e s e n t s a simple c a n t i l e v e r .  'A' and 'B' a r e s i n g u l a r p o i n t s , through  which pass an i n f i n i t e number o f f i b r e s .  They can thus  r e s i s t c o n c e n t r a t e d l o a d s and r e p r e s e n t t h e s u p p o r t p o i n t s in a real  structure.  A s o l u t i o n o f t h e c o m p a t i b i l i t y e q u a t i o n 1.11 i s  0  0  SB jt oC i ^  ss, — oC +  , and, i n p a r t i c u l a r  p  . . . .  (1.27)  which conforms w i t h t h e c o n v e n t i o n s e s t a b l i s h e d by F i g u r e 1.3. Chan then shows t h a t t h e r a d i i  of curvature, p  and p , o f t h e c o o r d i n a t e l i n e s a t a p o i n t P ( a , B) a r e g i v e n Q  P  by  (1.28)  where Ig and T^ a r e B e s s e l f u n c t i o n s o f t h e zero and f i r s t o r d e r r e s p e c t i v e l y .  18 If an  now  i s o t r o p i c  the  structure  material,  stresses  everywhere  being  opposite  w i l l  of  s i m i l a r l y  points  i n  tion  of  are  sign  be  space,  Then  i s  P  loaded  i n  moving 1.28  the  i n  those  numerically not  i n  TJ  i n  Maxwell  volume  i s  V  within  the  c r i t e r i a . to  c r i t e r i a  same The  evaluate  ments  P^, co.  actual  more  acting as  a  any  boundaries,  Consider forces  than  the at  r e s u l t  but  of  by  the  external  and  not  the B  the  f i b r e s  strains  be  fixed  the  load.  derive  the  d e f l e c -  Thus  -  i s  which can  the to  could now  be  subjected  suffer Then  conforms  optimum.  conforming  J\ , which  i s  63)  carrying  structure,  deformation.  forces  '3'  i n  considered  thus  volume,  l done  being  structures,  optimum points  to  that  of  equation  structure  optimum  p r a c t i c a l  and  from  P  and  less  A  d i r e c t i o n s .  structure  thus  Let  used  and  such  fibres,  may  t h e o r e t i c a l Michell  'a'  application  be  made  those  during  (Chan  the  1.3  manner  equal,  equal.  P -  The  Figure  some  numerically from  equations  point  shown  small  the  to  Its  same  loads  these then  be  used  determined. to  external  displace-  t o t a l  work  19 71. _  §W  =  Y  ft -^A'  . . . .  A t y p i c a l member o f t h e s t r u c t u r e , h a v i n g £  A  and a c r o s s - s e c t i o n a l  strain The  a r e a S^, w i l l  undergo a  a length linear  ± E and b e s u b j e c t e d t o a u n i f o r m n o r m a l s t r e s s ± a .  strain  energy s t o r e d i s thus — AaT  By t h e p r i n c i p l e  r  . . . .  M  (1.31)  ^  where V^. i s t h e volume  1.31  (1.30)  o f t h e optimum  structure.  o f v i r t u a l work e q u a t i o n s  1.30 and  are equal -  V  m  ss JL.  y  P.CO-  . . . .  (1.32)  OPTIMUM VOLUME FOR CANTILEVER WITH END POINT LOAD  Finally,  consider i n particular  a s t r u c t u r e a s shown  i n F i g u r e 1.3, s u b j e c t e d t o a s i n g l e p o i n t l o a d directed  normal t o the l i n e  of P are equal by e q u a t i o n s  s o a = 8 = <$.  of  Thus  The c u r v i l i n e a r  intersecting  a t L, t h a t  the v i r t u a l displacement,  the load i s  T  coordinates  The d i s p l a c e m e n t s d e s c r i b e d  1.29 a r e i n d i r e c t i o n s p a r a l l e l  t o t h e a and 3 l i n e s CL.  CL.  P , a t L,  to the tangents i s a t ±45° t o  co , o f L , i n t h e d i r e c t i o n T  20  J l ^ S ) ! ^ * 2SI,(2gj|  = ]ZEf  (1.33)  S u b s t i t u t i n g i n t o 1.32  V  MW  =^rj|| 2S)l (2S) 2Sl(2X) +  s  +  1  . . . . (1.34) [Chan 66]  In dimensionless terms  = JI [(l 2S)i;(2S)i-2a^)] +  •  •  •  (1.35)  I t should be emphasized t h a t the optimum s t r u c t u r e c o n s i d e r e d above i s i n f a c t p u r e l y t h e o r e t i c a l . i d e a l i z e d concept f o r mathematical  I t i s an  m a n i p u l a t i o n , but i t s  p r a c t i c a l d e s i g n r a i s e s many problems, n o t l e a s t being the j o i n t s between the two f a m i l i e s o f f i b r e s . I t should be noted t h a t Chan's s o l u t i o n the volume i n terms o f f a n angle, 6.  The span o f the  s t r u c t u r e does n o t enter h i s equations. on page 2 3 t h a t the span as quoted  expresses  Indeed he notes  i nhis results i s  measured d i r e c t l y from l a y o u t drawings,  which a r e o f the  corresponding M i c h e l l c a n t i l e v e r s having s m a l l -values o f N. T h i s i n t r o d u c e s a s m a l l e r r o r i n e v a l u a t i n g the span o f the infinite fibre  structure.  21 Conversely, first  specified  designer. angle of  as  i n the being  Methods a r e the  o f more p r a c t i c a l  described  appropriate  o f N.  as  l o w e r bound e s t i m a t e  infinite is  fibre  designs  small,  I t i s o f no  structural  the v a l u e o f the  structure.  s i m i l a r and  value.  the  Arbitrarily  The  volume i s u s e d o n l y  may  be  To  quote  to  for a  actual value  20  of  i s used  6 in  an  approaches  0.1%  of  consequence s i n c e a standard  fan  finite  6 for N =  'error' i n both  as  the  by  the  the  true  optimum  which  other  evaluated.  volume o f an  Chan  If  " . . . . the  knowledge  optimum s t r u c t u r e , when i t c a n  i s most v a l u a b l e ' . efficiency.  of  amounting t o perhaps  practical  use  is  f o r c a l c u l a t i o n o f the  Michell structure  value an  f o l l o w i n g work, t h e span  I t represents  . . . another  stituted,  the  penalty  comparing  the weight of the  optimum s t r u c t u r e . "  involved  the  form of  ultimate  be  of found,  structural;  construction  i s sub-  is readily calculated  proposed  by  construction with  the  22  ANALOGY WITH SLIP  Equations  + 2£jzS)  dp  express the c u r v i l i n e a r  o  c o o r d i n a t e s , a and 8.  i n plane p l a s t i c  m a t i c a l Theory  =  a geometrical r e s t r i c t i o n  form w i t h those e x p r e s s e d  lines  IN PLANE PLASTIC FLOW  1.2 5  JL. v  in  LINE FIELDS  on t h e f o r m o f  They a r e i d e n t i c a l  i n Hencky's Theorem  flow.  for slip  F o r e x a m p l e , i n "The Mathe-  of P l a s t i c i t y , "  C l a r e n d o n P r e s s , 1950) on page  by R. H i l l  (Oxford,  135, t h e s e a r e q u o t e d  Hill  as .  e q n . 12  '3 x and y a r e C a r t e s i a n and  k i s the y i e l d  considered. equations.  axes,  stress  p i s t h e mean c o m p r e s s i v e  i n shear o f the m a t e r i a l  cj) h a s t h e same m e a n i n g  i n both  sets of  stress  being  23 M a t e r i a l which i s loaded  so  greater  than the  longer  and  s a i d t o be  are  mutually An  plastic.  perpendicular  expansion or In  lines)  this  material  material  lines  — w i l l  be  enclosed  contraction  lines  of  The  mutually  of r e l a t i v e inclined  In  at  45°  i n developing  in  2.  p a p e r by  W.  The  Johnson  shears  called  along  those  slip  a rigid,  lines.  lines  under-  lines. (slip Consider  p e r f e c t l y rough  perpendicular  rough s u r f a c e  line  the  fields  Michell structures  Hill  [8], contain in this  mentioned  above,  many r e f e r e n c e s field.  will  slip-lines—  s o l u t i o n s which correspond the  surface.  material  surface.  theoretical slip  book by  v o l u m e o f work r e c o r d e d  along  slip  velocity.  with the  rough  are  elastically  c u r v i l i n e a r coordinates  t o the  graphical  those used Chapter  behave  m o t i o n between e l e m e n t s o f  p r a c t i c e the  a p p r o x i m a t e d by  are  stresses  between s l i p  constant  moving p a s t  the  material  which  a c t u a l l y i n contact  stationary.  directions  The  analogy, the  represent  plastic  be  s t r e s s no  element of m a t e r i a l  goes no  The  yield  that  are to  described and  a  t o the  large  24 Since  the analogy between s l i p l i n e f i e l d s and  optimum s t r u c t u r e l a y o u t i s e x a c t , possible layouts  these s o l u t i o n s  provide  f o r M i c h e l l s t r u c t u r e s which c o u l d be used  to support some system o f l o a d i n g , as determined by boundary conditions. STIFFNESS OF AN OPTIMUM STRUCTURE  One f i n a l i n v e s t i g a t i o n should [9]  be noted.  Barnett  i n 1966 reviewed the Maxwell and M i c h e l l t h e o r i e s a l -  ready d e s c r i b e d .  He  demonstrated t h a t the optimum  s t r u c t u r e o f minimum volume i s a l s o the  stiffest  occupying the same r e g i o n o f space and s u b j e c t e d  structure t o the  same system o f l o a d s . T h i s may be shown by m o d i f i c a t i o n o f e q u a t i o n 1.32. C o n s i d e r an optimum s t r u c t u r e s u b j e c t e d t r a t e d l o a d P.  t o a s i n g l e concen-  Then from e q u a t i o n 1.32  .  . . . (1.36)  where co i s the displacement of the p o i n t of a p p l i c a t i o n o f P.  Since  i t has a l r e a d y  the volume o f any other  been shown t h a t  i s l e s s than  s t r u c t u r e , i t f o l l o w s t h a t the d e f l e c -  t i o n o> i s l e s s than t h a t f o r any other  s t r u c t u r e having the  same maximum s t r e s s and s t r a i n and s u b j e c t e d  t o the same  l o a d , P. I f the s t r u c t u r e i s s u b j e c t e d each may be c o n s i d e r e d  separately  t o s e v e r a l loads a t once,  and the volume o f the optimum  s t r u c t u r e r e q u i r e d to support i t alone may from equation 1.34. may  be c a l c u l a t e d  For each o f these the same c o n c l u s i o n  be drawn t h a t the d e f l e c t i o n i s l e s s than f o r a l l other  comparable s t r u c t u r e s .  These loads may  then be combined  the volume o f the optimum s t r u c t u r e then r e q u i r e d w i l l equal to the sum for  be  of the volumes of the optimum s t r u c t u r e s  each i n d i v i d u a l l o a d .  of s u p e r - p o s i t i o n  T h i s f o l l o w s from the p r i n c i p l e  s i n c e i t has  d e f l e c t i o n s are small and I t now  and  been s p e c i f i e d t h a t  the e l a s t i c l i m i t  remains to c o n s i d e r  not  the  exceeded.  some more p r a c t i c a l  s t r u c t u r e s which are approximations to the optimum frames considered  above.  e t i c a l design w i l l  The  r e s u l t i n g compromise with the  i n v o l v e an i n c r e a s e  s t r u c t u r e s w i l l no longer be optimum.  theor-  i n volume s i n c e  the  26  CHAPTER 2  GEOMETRY OF  The the p r e v i o u s nature.  STRUCTURES  optimum o r minimum v o l u m e s t r u c t u r e d i s c u s s e d chapter  is theoretical  I t s d e s i g n may  whose v o l u m e w i l l  be  Modern m a n u f a c t u r i n g sive  APPROXIMATE  m a t e r i a l s and  weights a l l help  be  r a t h e r than  approximated  practical  than  techniques,  i n c r e a s e d use  the  i n the  continuing pressure  t o make t h e u s e  in  in real structures  somewhat l a r g e r the  in  ideal of  t o reduce  case expen-  component  a f e y e r more c o m p l e x  shapes  economical. The did  not  Michell  consider  sets of  fibres  transfer  joints  b e t w e e n them. criteria  further  discrepancy The  The  trarily curves that an  Any  ideal  successive tangents 8 to each  a series  of  introduce  two  no the  a  case.  as  spacing,  along  the  i s thus  i l l u s t r a t e d i n Figure  a r e drawn a t t h e p o i n t s o f  chapter  s t r u c t u r e b a s e d on  1.3,  readily  i l l u s t r a t e d in Figure  curvilinear  chosen equiangular  angle  previous  there  which w i l l  approximation  indicate  and  practical  from the  coordinates  8.  The  a t an  tangents  intersection.  a coordinate are  to  2.1.  arbithese  It will  be  inclined  at  seen  other.  Each c u r v i l i n e a r by  angles  general concept,  lines  i n the  a t a l l , s i n c e t h e members o f  must h a v e j o i n t s  a practical  dashed  discussed  cross at r i g h t  Michell  suggests  concept  straight  coordinate  l i n e can  be  l i n e s drawn b e t w e e n t h e s e  approximated p o i n t s of  27 intersection.  For example PQ,  QS,  3.  approximate the curve  Each of these chords r e p r e s e n t s , i n the approximate s t r u c t u r e , one of a l i m i t e d number of concentrated members, which  replace  the bundle of f i b r e s l y i n g between the c o o r d i n a t e s 3 ± j • MNRSQP r e p r e s e n t s a s m a l l s e c t i o n of such an approximate s t r u c t u r e .  The members form two  four-sided panels, i n  each o f which two o p p o s i t e i n t e r n a l angles are 9 0 ° w h i l e others are obvious  (90 + 8 ) and  (90 - 8 ) .  I t i s thus  the  immediately  t h a t such a s t r u c t u r e does not s a t i s f y the M i c h e l l  criteria,  s i n c e the members are no longer o r t h o g o n a l .  Its  v olume must thus be l a r g e r than t h a t of the optimum s t r u c t u r e . I t c o u l d reasonably be expected  t h a t the d i f f e r e n c e i n volume  w i l l be small s i n c e the framework f o l l o w s the g e n e r a l l a y o u t  F i g u r e 2.1  P r a c t i c a l Approximation Structure  to a T h e o r e t i c a l Optimum  28 o f t h e optimum s t r u c t u r e , d i f f e r i n g tribution  only  i n the l o c a l  of m a t e r i a l .  It  i s proposed  to refer  t o such  approximate s t r u c t u r e s  as MICHELL  s t r u c t u r e s , t o mark h i s a s s o c i a t i o n  underlying  theory.  The  exact  arrangement o f a M i c h e l l  depend on t h e s p a n , l o a d i n g , number work and on t h e g e o m e t r i c a l ments o f c o m p a t i b i l i t y . important ditions  This l a s t  since i t determines  f o r the system.  distributed  load—and  structure.  In t h i s  be e x c l u d e d .  a fixed material  thus  elastic  con-  support—or  a  be u s e d as a b o u n d a r y t o t h e  t h e members r a d i a t i n g  from B would  continuous  of s l i p  i s being  support  lines,.ACFE  than  the p o i n t s t  i s p e r h a p s more  for structural  use-  analysis.  i n F i g u r e 1.3 w o u l d  represent  as a d i e f a c e , p a s t w h i c h t h e p l a s t i c  extruded.  This w i l l  obtained,  fibres  forces.  boundary concept  flow analogy  numbers o f  They r e p r e s e n t  I n t h e p r e s e n t work, c a n t i l e v e r s  results  i s particularly  F o r example i n F i g u r e 1.3, t h e l i n e  case  s u r f a c e such  be s t u d i e d .  o f frame-  t h e b o u n d a r y and s u p p o r t  of concentrated  i n the p l a s t i c  In t h e s t u d y  structure w i l l  A s has a l r e a d y b e e n s t a t e d A and B i n t h i s ,  of a p p l i c a t i o n  ful  factor  i n t h e optimum s t r u c t u r e .  The  their  o f members, t y p e  a r e s i n g u l a r p o i n t s where i n f i n i t e  converge,  with  c o n d i t i o n s imposed by t h e r e q u i r e -  ACFE c o u l d r e p r e s e n t a c o n t i n u o u s  drawing  dis-  not g r e a t l y l i m i t  in particular the value  s i n c e many o f t h e e q u a t i o n s  will  of the  derived are  applicable  to other  arrangement  types o f M i c h e l l s t r u c t u r e .  o f the s t r u c t u r e s  t o be c o n s i d e r e d  The  general  i s shown i n  Figure 2.2.  Figure  2.2  Typical Michell  The c a n t i l e v e r i s s u p p o r t e d and B, and may be s u b j e c t e d more  joints  b e t w e e n members.  assumed t o be p i n n e d axial  forces  t o any  only.  so t h a t  Cantilever  a t two f i x e d p o i n t s , loadings  These  joints  a l l members  a t one o r are i n i t i a l l y  are subjected  to  A  30 The segments a)  cantilever  may  be  divided  into  three  d i s t i n c t  a  fan of equal  length  members,  r  , radiating  from  A, b)  a  similar  f a n o f member  network  forming  length,  r  , radiating  from  B, c)  a  from  the fans  Support  quadrilateral  t o complete  arrays  into  the structure.  that  they  serve  to d i s t r i b u t e  The members  are perpendicular  symmetric  the cantilever  i s  as  i n Figure  the cantilever  on  the x  axis.  members,  9,  The  of  concentric no  loads  the  structure.  The  length  remaining a s AC  reactions  a r e so  proportioned  I f they  are o f  equal  i s  the x  The  axis;  that  w i l l  unequal,  and L  w i l l  not l i e  radiating  from  A  angle  between  members  t o be  CFD,  i n tension a r e needed  resisted  and  B  successive -.  the lines  be  length,  i f  skew  members  and CB.  the support  a n d CB  about  CGE,  the curvilinear coordinates  members  of  i s constant.  Note lines  2.2,  AC  a t C.  r,  t h e same  t h e span  extending  Fans  These  are  panels  or  approximate  shown  i n Figure  compression;  i n these  the  zones  1.3.-  c l e a r l y since  base  no  there  are  circumferentially.  * I t may b e n o t e d t h a t a l t e r n a t i v e s u p p o r t s y s t e m s may be u s e d t o r e p l a c e these segments o f t h e s t r u c t u r e . In particular a fixed continuous support c o u l d be p r o v i d e d along t h e a r c s CFD a n d CGE. In such cases these f i v e p o i n t s would be f i x e d a n d s u f f e r no d e f l e c t i o n u n d e r l o a d . This would modify the deformation o f the structure but would not a f f e c t its geometry.  31  Main S t r u c t u r e As i l l u s t r a t e d  i n F i g u r e s 2.1 and 2.2, the remainder  of the s t r u c t u r e i s composed o f q u a d r i l a t e r a l frames, approximating the c u r v i l i n e a r network.  A l l these frames a r e s i m i l a r ,  having the same arrangement of a n g l e s . The  angle between members i n the f a n s , 9,  determines  the angular r e l a t i o n between a l l other members i n the framework.  I t w i l l be noted  t h a t each c h a i n o f members, t y p i c a l l y  AF, FK, KM or BG, GK, KH, t u r n s t e a d i l y i n one d i r e c t i o n , always toward the centre l i n e . s t a t e d i n Chapter a or 3 l i n e s .  T h i s s a t i s f i e s the requirement  1 t h a t there should be no i n f l e c t i o n s i n the  The angle between s u c c e s s i v e members i n a  c h a i n i s always 6, except a t the f a n boundaries 9/2.  where i t i s  T h i s a r i s e s from the g e o m e t r i c a l requirements  specified  i n F i g u r e 2.1. Loads may be a p p l i e d t o the M i c h e l l c a n t i l e v e r a t any or a l l o f the j o i n t s . members  As drawn, w i t h equal spacing o f the  (9 c o n s t a n t ) , the c o o r d i n a t e s o f the j o i n t p o i n t s , D  to M, depend on r , r are s p e c i f i e d .  and 0 and a r e f i x e d once these v a r i a b l e s  I f a M i c h e l l s t r u c t u r e i s r e q u i r e d t o support  f o r c e s a t random p o i n t s , a s t r u c t u r e would have t o be s u i t a b l y designed with unequal spacing between members.  In such a  case the panels would not a l l be s i m i l a r as i n the p r e s e n t work. T h i s case has not been pursued loads would r e q u i r e a unique s o l u t i o n .  as each arrangement o f The g e n e r a l  argu-  ments d e v e l o p e d b e l o w w o u l d , however, a p p l y procedure could  Notation For labelling An  be f o l l o w e d  to solve  as i n F i g u r e  system f o r l a b e l l i n g  ponent p r o p e r t y  Briefly,  two  a and b , t h e f i r s t  suffices  the chain  2.2, i s i m p r a c t i c a l .  each j o i n t ,  was t h e r e f o r e d e v i s e d  A p p e n d i x A.  and i s d e t a i l e d i n  i tconsists of identifying relating  with  t h e 'B' c h a i n .  These a r e used i n  t h e f o l l o w i n g symbols -  l e n g t h o f member i n 'A' c h a i n ,  B ^  l e n g t h o f member i n 'B' c h a i n ,  a  , ,FB , ab' ab J  , ab  f o r c e i n member A , , B , , ab' ab' J  joint,  sA,sB  c r o s s - s e c t i o n a l a r e a o f member,  WA,WB  w i d t h o f member.  Members, o r t h e i r coordinates support  of the j o i n t  points  j o i n t s by  to i t s position  k~  A  member and com-  o f members r a d i a t i n g f r o m A, a n d t h e s e c o n d  to i t s p o s i t i o n along  FA  such s t r u c t u r e s .  f o r Structures  of the j o i n t s ,  connection  similar  a l l but the smallest of structures, a r b i t r a r y  organized  along  and a  properties, are i d e n t i f i e d  by t h e  a t i t s i n n e r end, t h a t n e a r e s t t h e  as measured a l o n g  the chain  o f members.  32a Graphical  Construction  Graphical support  spacing,  lengths, of  of  Structure  solution of a structure to determine the f i b r e  i s n o t recommended.  It will  values  giving  o f 8.  This  i s continued  angle  require  s e v e r a l diagrams each c h a r a c t e r i s e d  trial  o f known  low s i n c e  •repetitive  construction.  For imate values  illustrative  be f o l l o w e d .  Figure  2,2.  until  of 8  m a g n i f i e d by t h e  purposes o r f o r o b t a i n i n g  The p r o c e s s  approxbelow  i s illustrated in  ( l a r g e r than  actual  i f possible)  Construct  t h e r i g h t a n g l e ACB w i t h AC = CB f o r a  symmetric  cantilever.  cantilever, 3.  the value  Accuracy i s  errors are  L o c a t e A, B and L , t o s c a l e size,  2.  the construction  of parameters, the procedure described  should  1.  graphical  6 and member  by one o f s e v e r a l  t h e s p e c i f i e d span i s d e t e r m i n e d .  relatively  s p a n and  With centres  these  F o r an  lengths  unsymmetric  are unequal.  A and B draw c i r c u l a r  arcs  of  radius  AC and BC r e s p e c t i v e l y . 4.  Lay out the angles  8 around these a r c s (N-l)  times t o produce t h e base  fans,  ADFC a n d  BCGE. 5.  Extend at  each r a d i a l  line  so c o n s t r u c t e d  an a n g l e t o i t s p r e v i o u s  outward  d i r e c t i o n , turning  32b always i n the same d i r e c t i o n .  The f i r s t  extension  g  i s at  t o the r a d i a l l i n e s , w h i l e a l l f u r t h e r  e x t e n s i o n s t u r n through 8. The d i r e c t i o n o f t u r n i n g i s always toward the c e n t r e l i n e so t h a t l i n e s r a d i a t i n g from A t u r n 'downwards' and those from B t u r n upwards, 8°  For  example AC i s produced a t ^  to  form the l i n e CG,  at  G.  'clockwise'  intersecting radial line  BG  GE i s then drawn t o 8° t o CG t o i n t e r s e c t  BE a t E.  (The p o i n t s C,F,D, and C,G,E  are  a l r e a d y known s i n c e they l i e on the c i r c u m f e r e n c e of  the c i r c l e s drawn i n s t e p  3.)  S i m i l a r l y AF i s extended outward  'clockwise'  8°  at TJ-  t o form FK w h i l e BG i s extended  'counterclockwise'  to form GK.  outward  T h i s completes the  f i r s t q u a d r i l a t e r a l p a n e l CFKG.  This routine i s  f o l l o w e d s y s t e m a t i c a l l y t o complete the s t r u c t u r e , the  l a s t two e x t e n s i o n s i n t e r s e c t i n g a t L. I t w i l l be seen t h a t t h e r e are sequences o f  p e r p e n d i c u l a r l i n e s , such as CG, GK, Another s e t i s CF, FK, KH and HL.  KM and  T h i s forms an  a l t e r n a t i v e method of c o n s t r u c t i o n , once the e x t e n s i o n s have been made.  ML.  first  C a l c u l a t i o n of Geometry of  Structures  A r e p r e s e n t a t i v e M i c h e l l s t r u c t u r e i s shown i n F i g u r e 2.2.  As can be  seen the c o n s t r u c t i o n i s r e p e t i t i v e ,  major p a r t of the framework c o n s i s t i n g of f o u r s i d e d a l l of the same angular  panels  shape.  From t h i s diagram the nine v a r i a b l e s may These are l i s t e d below, together represent  the  with  be  specified.  the symbols which  them i n the computer programmes used f o r s o l u t i o n of  the many s t r u c t u r e s examined. L  (XSPAN)  span of s t r u c t u r e ; d i s t a n c e along  x-axis,  from plane o f supports t o plane of  outer-  most p o i n t of s t r u c t u r e , Y  (YSPAN)  r i s e of s t r u c t u r e ; d i s t a n c e from  x-axis  to outermost p o i n t of s t r u c t u r e , D r  (D) A  (RADA)  support  spacing;  length  "upper fan r a d i u s ' ; l e n g t h of members A r a d i a t i n g from support  r  B  (RADB)  AB,  A,  'lower fan r a d i u s ' ; l e n g t h of members 3  r a d i a t i n g from support 0  (THETA)  'fibre angle ; 1  r a d i a l members i n f a n s .  'fan a n g l e ; members of N  (N)  one  consecutive  9 i s taken  constant  structure,  angle i n c l u d e d between  1  B, (o,b)  B,  angle between  i n t h i s work, f o r any  (a,o)  outer  fans,  number of chains each support  of members r a d i a t i n g from  point,  34 6  (BETAA)  angle  In a symmetrical  ABC. s t r u c t u r e these reduce  to s i x , since  then  Y  The  -  45°  =.  ZERO  ....  (2.1)  s i x remaining v a r i a b l e s are not a l l independent  and t h r e e more may  S =* tanfi)  =  be e l i m i n a t e d by the f o l l o w i n g  (N-I)9  equations  . . . . (2.2)  A  There are thus s i x independent parameters t h a t must be specified  f o r a skew M i c h e l l c a n t i l e v e r , or t h r e e f o r the  symmetrical  structure.  In p r a c t i c e , i t i s u s u a l t o d e s i g n a s t r u c t u r e w i t h a known span and support s p a c i n g , and w i t h a known number of components, so t h a t normally N, L, Y and D would be The remaining two  equations t o complete  the  specified. solution  c o u l d t h e o r e t i c a l l y be w r i t t e n by c o n s i d e r a t i o n of the geometry of the s t r u c t u r e , by d e r i v i n g e x p r e s s i o n s f o r L and Y i n terms of r,. , r  D  and  N increases.  0.  In p r a c t i c e t h i s becomes h i g h l y complex as  T h i s i s d i s c u s s e d i n Appendix B and  are g i v e n f o r N as l a r g e as 5 .  The  examples  equation  f o r the span of an N f i b r e d s t r u c t u r e i n v o l v e s both s i n  0  35 and cos 8 t o the power  (N + 2) .  Chan s t a t e s t h a t a  general  equation f o r the span c o u l d not be found and the present work s u b s t a n t i a t e s h i s remark. For a s p e c i f i c v a l u e of N, as d i s t i n c t from the g e n e r a l case, an i t e r a t i v e method was  found to be p r a c t i c a l and  itself  Values of 0 and  to computer s o l u t i o n .  (—) B  lent  were assumed  r  and used t o c a l c u l a t e v a l u e s of L and Y. compared w i t h the r e q u i r e d v a l u e s and changed i n a s y s t e m a t i c way T h i s procedure was  These  8 and  v a l u e s were  ( r / r ) then,  so t h a t L and Y may  a  be  recalculated.  repeated u n t i l the e r r o r became l e s s  than  a specified value. The computer programme t o c a r r y out t h i s search i s d e t a i l e d i n Appendix C. c a l c u l a t e s parameters few c y c l e s .  The  I t i s q u i t e r a p i d l y convergent, and w i t h an e r r o r of l e s s than 0.01%  same programme, w i t h v a r i a n t s i s used  in a to  c a l c u l a t e many other items of i n f o r m a t i o n about the s t r u c t u r e being  investigated. Lengths of Members The  lengths of a l l the members of the framework may  c a l c u l a t e d from f i r s t p r i n c i p l e s of geometry from F i g u r e  2.2.  The a c t u a l equations are s t a t e d i n Appendix B.  Joint  Coordinates  As f o r the member  l e n g t h s , j o i n t c o o r d i n a t e s may  d e r i v e d d i r e c t l y from F i g u r e 2.2.  The  a c t u a l equations  be  be  are  36  s t a t e d i n Appendix B.  D e f l e c t i o n of Structures Somewhat p a r a d o x i c a l l y , the d e f l e c t i o n of the s t r u c t u r e depends e n t i r e l y upon i t s geometry and i s independent o f the l o a d i n g . the  T h i s f o l l o w s from the d e s i g n  condition  that  s t r e s s and s t r a i n a r e uniform throughout the s t r u c t u r e .  Each member i s p r o p o r t i o n e d  to s a t i s f y t h i s condition.  Thus  once the s t r a i n i s s p e c i f i e d , the change o f l e n g t h o f each member i s known and thus the d e f l e c t i o n can be c a l c u l a t e d . A W i l i o t diagram was used t o determine the d e f l e c t i o n s , the diagram being  solved  t r i g n o m e t r i c a l l y to maintain accuracy.  I t has been assumed t h a t the support p o i n t s A and B a r e f i x e d i n space and t h a t the s t r u c t u r a l d e f l e c t i o n s a r e e n t i r e l y due  t o d e f o r m a t i o n o f members.  The e q u a t i o n s a r e s t a t e d i n  Appendix B. I t should  be noted t h a t the above statement  d e f l e c t i o n i s independent of loading) members a r e under s t r e s s . inner  (that  i s true only i f a l l  I f a s t r u c t u r e were loaded  j o i n t , some members would then be u n s t r e s s e d  formed, i f s e l f - w e i g h t were n e g l e c t e d .  a t an  and unde-  In such case  the  s t r a i n i s not u n i f o r m i n a l l members and the W i l i o t diagram i s modified. Chapter 4.  This point i s discussed  i n a s e c t i o n of  37  R i g i d i t y of The  Structures  s t r u c t u r e shown i n F i g u r e  p i n - j o i n t e d , a c t s l i k e a mechanism. replaced the  by a r o l l e r ,  2.2,  i f assumed to  I f j o i n t B,  f r e e to move i n any  say,  t h a t both support p o i n t s of the  s t r u c t u r e and I f the  specified direction,  It i s therefore  essential  are f i x e d to m a i n t a i n the i n t e g r i t y  j o i n t s were r i g i d ,  the  s t r u c t u r e can  independent of the  support  maintain  points.  However, t h i s r e l i e s on the r i g i d i t y of the m a t e r i a l j o i n t s and  any  a p p l i e d l o a d would r a i s e h i g h  i n these a r e a s , i f A and  considered  as an  around  local  stresses  B were not both f i x e d .  I t thus f o l l o w s t h a t the be  an  i t s solution.  a fixed configuration  the  were  s t r u c t u r e would c o l l a p s e by r o t a t i o n about A u n t i l  e q u i l i b r i u m p o s i t i o n i s reached.  be  f i x e d support p o i n t s  e s s e n t i a l f e a t u r e o f the  From the c o n d i t i o n s  structure.  s p e c i f i e d above and  the geometry of any M i c h e l l framework may  be  should  i n Appendix  B,  completely  determined, once the b a s i c parameters such as span, number of f i b r e s , arrangement of supports and  so on are known.  38  CHAPTER 3 FORCES IN A STATICALLY DETERMINATE MICHELL STRUCTURE  We have s e e n i n t h e p r e v i o u s geometry o f a M i c h e l l s t r u c t u r e ments  laid  to define  spacing  a given  i s specified  such a s t r u c t u r e  and number o f f i b r e s  required  f o r a complete  assumed  the other  to satisfy  calculations  solution of a Michell structure  assumed t o be c o n c e n t r a t e d  e a c h end o f t h e member.  or  i s selected  once i t s span,  Thus a l l f o r c e s  and t h e members a r e assumed  c o m p r e s s i o n , w i t h no a p p l i e d It w i l l  i f this  equally at  are applied  t o be i n p u r e  which  determinate.  a p p r o x i m a t i o n , t h e w e i g h t o f e a c h member,  t o be c o n s i d e r e d , i s  joints  uniquely,  t o be p i n - j o i n t e d and t h u s s t a t i c a l l y  As a f i r s t is  These a r e  design. I t now r e m a i n s t o d e s c r i b e  is  by t h e r e q u i r e -  down by M a x w e l l , M i c h e l l and C h a n .  sufficient support  chapter that the  a t the  tension  b e n d i n g moments.  f u r t h e r be assumed  that  the thickness  of  D all is the  members, p e r p e n d i c u l a r constant, axial  tensile  numerically members.  that  to the plane o f the s t r u c t u r e ,  the material  used  o r compressive  constant,  differing  i s i s o t r o p i c and  stress only  i s everywhere  i n sign  between  39 FORCES DUE TO THE WEIGHT OF MEMBERS  For  the purpose o f the f o l l o w i n g  calculations a l l  dimensions w i l l be assumed to be i n i n c h e s , and f o r c e s i n pounds.  The c h o i c e of these u n i t s  serves merely t o d e f i n e  i s somewhat a r b i t r a r y and  p r e c i s e l y c e r t a i n constants i n -  volved i n the c a l c u l a t i o n s . F i g u r e 3.1 shows a t y p i c a l s t r u c t u r a l member made from a material  of s p e c i f i c gravity p  Weight o f member  =  (Aab)fy^A b)"tp Q  . (3.1)  Half of t h i s weight i s assumed t o be c o n c e n t r a t e d at each end j o i n t , J , and J , , . ' ab a, (b+1) J  member i s  F A a  k  s  a n c  The a x i a l f o r c e on the  ^ the uniform s t r e s s i s everywhere a  PAab  F i g u r e 3.1  discussion  D e t a i l s o f a T y p i c a l Member  The e f f e c t s o f t h i s assumption are c o n s i d e r e d i n the o f r i g i d j o i n t s i n Chapter 5.  (WAa^ttf'**  Thus  FAob  . . . . (3.2)  S u b s t i t u t i n g t h i s i n t o 3.1 t o e l i m i n a t e the term it  f o l l o w s t h a t the weight c o n c e n t r a t e d  t^ab^  a t each j o i n t t o  r e p l a c e the s e l f weight o f t h i s member, i s g i v e n by  o  where W i s a c o n s t a n t  e  •  o  (  3 3} o  equal to  F o r c e s i n the Members For the purposes o f the f o l l o w i n g c a l c u l a t i o n s i t w i l l be  assumed t h a t there  i s a l o a d a p p l i e d a t every  LOAD(ab), and t h a t i t a c t s a t an angle T(ab) Further,  t o the y a x i s .  f o r complete g e n e r a l i t y , the whole s t r u c t u r e i s  assumed t o be t i l t e d  so t h a t the s e l f weight f o r c e s a c t a t  an angle + t o the y a x i s . Figure  joint,  3.2.  The equations d e r i v e d below a r e thus com-  p l e t e l y general  and may be a p p l i e d t o any l o a d i n g  For members c o n s i d e r e d to zero as can Load a specific  These d i r e c t i o n s a r e shown i n  t o be w e i g h t l e s s ,  system.  W i s placed  equal  (a,b) i f no e x t e r n a l f o r c e i s a p p l i e d a t  joint.  A n a l y s i s o f the f o r c e s a c t i n g i n the s t r u c t u r a l members commences c o n v e n i e n t l y  a t the outer  end, L, and  Aa,b-.  gure  3.2  C o n v e n t i o n f o r Load A n g l e s a t a Joint  Typi  L O A D  gure  3.3  Forces Acting  a t J o i n t J.  42 proceeds s y s t e m a t i c a l l y inward t o the support p o i n t s . Support r e a c t i o n s w i l l be  considered  i n a succeeding s e c t i o n .  T h i s a n a l y s i s i s the f a m i l i a r method of j o i n t s .  A  force  diagram f o r the complete s t r u c t u r e c o u l d have been drawn a general  formula c o u l d have been w r i t t e n a p p l i c a b l e t o a l l  j o i n t s . For  compilation  of the computer programme i t was  found convenient to c o n s i d e r The  f i v e d i f f e r e n t types of  d e r i v a t i o n of the f o l l o w i n g e q u a t i o n s are g i v e n  Appendix  and  joint. in  D. *  a)  The  Outer J o i n t ,  From F i g u r e this  joint  3 . 3 , which shows the  f o r c e s a c t i n g at  -  L*^W*j-*»D+ (<v* sKw w  9 M / W  J„„ NN  t))]  ' C o s Q - W [ W . S i V i ^ + f - J)+B^Mcosf£-f-f>] (3.4)  * In a l l these cases i t i s assumed t h a t the f o r c e s i n the 'A' members are t e n s i l e (+ve) and t h a t those i n the 'B' members are compressive (-ve). The s i g n of the e q u a t i o n , a f t e r the n u m e r i c a l values are i n s e r t e d , w i l l of course determine the t r u t h of t h i s assumption.  43  b)  Forces a t a Typical  Inner J o i n t , J  ab  .  3 . 4 shows t h e f o r c e s  Figure J a  k«  T  fibres,  n  e  diagram a l s o by p l a c i n g  acting  applies  Figure  3.4  to joints  the appropriate  ^  on a t y p i c a l  (3.5)  joint  on t h e o u t e r  forces  e q u a l t o zero,  L©AD fc» 0  Forces a t t y p i c a l  joint, J  ab  FA^b [ l w(A bcos(i|i+t-0)-B .,. s'm(f+t-^)] +  Q  Q  b  + FBob [sinO* w(B cos(ipt-0) + B .,,t> cos(lp+fj) ob  Q  (cos0 - W J A ^ , c o s ( ^ - 0 ) B , , ^ a ( ^ ^ j ] ) . Q (  +  Q  FA ok [sm9 w(A b Sin(H>H)+Aa,b-»sin(^H-6)] +  a  (3.6)  44 c) F o r c e s a t a J o i n t  o n t h e 'A' F a n ( e x c e p t J , )  a.o  Figure  3.5  Forces a t a Typical  F A a ) l [' "•W(A Ai ,  'A' F a n J o i n t , J  a  c o s ( v f - £ ) - B e - , , , Sinl>*t-f))"  (cosf ^w[A «5 (^t-f)+BQ.,,sia (4>+-f)]) coS  aj  (3.7)  p  A ,  *^(A Sift^-gJ+Aa.,sm(q*t)]  Q(  ap  + FB^, [cos  w(Aa cos(^|)- B , sin (<i>*4)] >0  Qt  (cos | - W[A co5(s> t-|)* B . s / n ao  +  a  u  45 d) F o r c e s  at a Joint  on t h e B ' F a n (except J 1  1  )  J=Ai.b  PBo.b  Figure  3.6  Forces  at a Typical  'B' F a n J o i n t  J,^  FA, |o«|+w|A, bCoa(pi-t-|)- BabSia(V+HB] b  1  FA,, .r kB,. b  b  | i n | + w|B cos(Y+t) + B,.bCos(s^-f ffl ab  (cos f -W[A,, ., cos (^i-|)+8 .b3in(^H)] ) b  0  (3.8)  PA,.b ^iafl+W[Ai.b-.sin(^-e)+Ai,bsinCM>-i-t)]]  (cos | -  W [\  cos(^H-|)+ o b s/n(H»+t3) B  tlH  46  WT.  F i g u r e 3.7  FA FA,  0  -  +  I(I  FB  Forces a t J o i n t  [cosf+WlA,,, sm(B-4)- B c , , c o s ( f - B - f ]j [sin§+ W[B s'in(j3-4) ' c.s'a(p-f-f); +  M  B  M  ((-w[A sin(H  +  l i 0  £>o,.cos(/3-f)] ) . (3.9)  " ^...[sinf+W|A, cos(f &  M)+A cos(H)]] M  + FB,Jos|-.w[A , Siri(B-^ FB..= O.I I  0  (|-W[A,, S1K(BH9 + B cos(£-f)] ) 0  OA  47 Size  o f Members  The determined and  length  of a t y p i c a l  by g e o m e t r i c a l  the forces The  the a x i a l  a  3.4 - 3 . 9 .  The  stress i s uniform.  and o t h e r  from s t a n d a r d  ....  Q  s e c o n d moment  of area  properties  the r e -  Thus  FA,b I F  b  been  as d e t a i l e d i n C h a p t e r . 2 ,  a r e known f r o m e q u a t i o n s  WA k = °member  conditions  w i d t h , WAab, may now be c a l c u l a t e d u s i n g  quirement that  the  member, A ^ , h a s a l r e a d y  (3.10)  o f the s e c t i o n , weight of  d e s i r e d may now be c a l c u l a t e d  equations.  Reactions  a t Supports  A p i n - j o i n t e d M i c h e l l s t r u c t u r e , a s shown i n F i g u r e 2.8  and e l s e w h e r e ,  mechanism. points  right the  I t i s made"rigid  or, i t s equivalent,  at each  2 that  i s not of i t s e l f  rigid  and i s i n f a c t  by t h e u s e o f two f i x e d  two p e r p e n d i c u l a r  support  a  support reactions  support point.  -  This  i s indicated  This  i s implicit  i n Figure 3.8. i n the requirement  the angle ACB—between angle,  This  immediately  and t h u s b o t h  requires  support points  i n Chapter  f a n members — i s  w h i c h was a f u n d a m e n t a l p o i n t  structure.  distance  the inner  stated  a  i n the layout of AB t o b e a f i x e d  must be f i x e d .  ,.  In p r a c t i c e , when r e a l there w i l l For  inevitably  s m a l l relative  infinitesimal,  be some d i s p l a c e m e n t  d i s p l a c e m e n t s , where  3.8  t h e c h a n g e i n ACB i s  i n a l l t h e above  External Equilibrium  A typical perpendicular  of these p o i n t s  t h e geometry o f t h e system i s m a i n t a i n e d ,  assumption which i s i m p l i c i t  Figure  structures are considered,  load  of a Michell  an  calculations.  Cantilever  i s shown i n F i g u r e 3.8, t o g e t h e r w i t h  components  o f the support r e a c t i o n s .  t u d e o f t h e s e components may be d e t e r m i n e d t h e f o r c e s i n t h e members a p p r o a c h i n g  The magni  by r e s o l u t i o n o f  the support  points.  Thus -  A« =|. rFA<, cos[p-(a-l)^ + w(A . FA^)sin(+)j 0  A  v  -1  B H  a 0  [F/Vo sia §-(a-1) 8] + w(A , FA ) os(rj] a 0  J, [FB .  0 b  aj0 6  sin |p+(b-i)0] + w(B FB )sia(tJ| oi  ob  B =£(fB.. cos[^ |»-!)9]+ W(B FBjcos(t)' v  b  +  the p o s i t i v e d i r e c t i o n o f each r e a c t i o n shown i n the above drawing.  ob  component  being  50  CHAPTER  ANALYSIS -  In  SOME P I N - J O I N T E D M I C H E L L  Subjected  to Concentrated  the preceding  structures been  OF  i n general,  discussed  4  chapters  Loadings  the geometry  and o f c a n t i l e v e r s  and e q u a t i o n s  CANTILEVERS  derived  -  of  Michell  in particular,  connecting  has  the various  parameters. The  forces exerted  loading  may  then  be  Chapter  3.  This  i n turn  of  member  each  points  of  number,  loading  analysis data  were  volume  i s tabulated  The limits  and  decisions  enables  calculated  analyzed some  varied.  are discussed  Selection  by t h e a p p l i e d  the equations  laid  out i n  the cross-sectional since a uniform  of cantilevers,  to obtain  structural  member  area  stress  :  ;  at a l l  specified.  A variety fibre  each  calculated  t o be  has been  on  insight The  i n Appendix  design  subjected into  1  and  to different,  t h e way  significant  rise  i n which  results  from  types  the this t h e'  E'.  Parameters  parameters  could  be v a r i e d  therefore necessary  to reduce  span,  i n the following sections while  of Design  i t was  while  of various  t h e volume  of data  t o make that  over some  could  very  wide  arbitrary rapidly  be  accumulated  by u s e o f computer  I t was structures  therefore  decided  solutions. t o base t h e m a j o r i t y  of the  on t h e d e s i g n p a r a m e t e r s o f t h e p h o t o e l a s t i c  m o d e l s t h a t were t o be made f r o m b i r e f r i n g e n t m a t e r i a l tested  to confirm  these c a l c u l a t i o n s .  be  c u t f r o m s h e e t one q u a r t e r  of  a l l members,  the  t , was  The  so t h e  This  t o be c a l c u l a t e d  from e q u a t i o n  birefringent material  u s e d was  rectangular 3.10.  CR39  that  o f CR39 i s 300,000 p s i . and i t s s p e c i f i c  1.31.  The s t r a i n was a r b i t r a r i l y  gravity i s  c h o s e n as 0.001.  v a l u e s were t h u s e s t a b l i s h e d E  300,000  Strain  z  0.001  Uniform a x i a l s t r e s s  a  300  T h i c k n e s s o f a l l members  t  0.25' i n c h e s  Specific  p  1.31  W  = r-~  L  The affect  use o f these  the g e n e r a l i t y  specific  The  f o r a l l models:-  Modulus o f E l a s t i c i t y  S e l f weight constant  Corpor-  t h e modulus o f e l a s -  ticity  Gravity  enabled  p r o d u c e d by t h e H o m a l i t e  The m a n u f a c t u r e r s s t a t e  following  thickness  i n turn  w h i c h were t o be o f  ('Columbia R e s i n ' ) p l a s t i c ation.  T h e s e m o d e l s were t o  thick  t a k e n as j " .  w i d t h o f a l l members,  cross-section,  inch  and  p.s.i.  p.s.i.  0.0001577  (inches ) - 1  1-31 x 62.4 , 1728 x 300 J  v a l u e s does n o t g r e a t l y  of the r e s u l t s obtained.  Wherever  convenient, the r e s u l t s are c a l c u l a t e d or d i s p l a y e d  i n terms  5  of dimensionless parameters and a t i o n of v a r i a b l e s having t h a t The  s t r a i n was  are thus v a l i d  with s t r a i n was  combin-  ratio.  a r b i t r a r i l y chosen to conform w i t h  b a s i c assumption t h a t the d e f l e c t i o n s are the dimensions of the  f o r any  structure.  The  c o n s i d e r e d f o r one  the  small compared w i t h  v a r i a t i o n of d e f l e c t i o n  s t r u c t u r e and  shown i n Table 11 i n Appendix E.  I t w i l l be  results  seen t h a t  are  the  d e f l e c t i o n v a r i e s d i r e c t l y w i t h the s t r a i n . Once the  s t r a i n i s s e l e c t e d , the uniform s t r e s s i s  determined by the value of E from the  f a m i l i a r equation  stress strain  As  E increases,  constant e.  so w i l l the  T h i s w i l l not  pends o n l y on  by the sidered  stress increase,  e, but w i l l a f f e c t the c r o s s - s e c t i o n a l  weight of the  for  a f f e c t the d e f l e c t i o n , which de-  of each member from equation 3.10 volume and  and  structure.  s p e c i f i c g r a v i t y , p , and  may  areas  thus w i l l a f f e c t This  i s also  conveniently  the  affected be  con-,  i n the equation f o r W  „  .  624  £P  nze E E The  r a t i o — i s a p r o p e r t y of the m a t e r i a l  manufacture and  does not  2  a f f e c t the o p t i m i z a t i o n  used f o r of  struc-  53 t u r a l design  t o perform a s p e c i f i e d f u n c t i o n .  m a t e r i a l the M a x w e l l / M i c h e l l c o n d i t i o n s structure.  F o r any g i v e n  s p e c i f y the optimum  Some t y p i c a l values a r e : -  Material  E x lO-^dynes/cm^  Density  Ratio  gm/cm  3  Percent of , Be  Beryllium  3.1  1.848  1.677  100.0  Silicon  1.31  2.27  0.577  34.4  Pyrex  0.65  2.23  0.291  17 .38  Titanuim  1.15  4.51  0.255  15.20  Aluminum  0.62  2.699  0.230  . 13.69  Steel  1.8  7.9  0 .228  CR39*  0.018  1.31  0.014  13.58 1.37  CR39 i s used f o r model manufacture and i s not a c o n s t r u c t i o n a l material.  The buckling  e f f e c t of the t h i c k n e s s , p a r t i c u l a r l y as regards  of the compression members, i s a l s o d i s c u s s e d Before c o n s i d e r i n g  it  should  the r e s u l t s o f these c a l c u l a t i o n s ,  be emphasized t h a t these s t r u c t u r e s a r e s t i l l  t h e o r e t i c a l i n nature.  They are assumed t o be p i n - j o i n t e d but  no p r o v i s i o n has been made f o r the m a t e r i a l  i n the j o i n t s , •  each member extending only f o r the t h e o r e t i c a l d i s t a n c e joint points. other  later.  The e f f e c t o f the e x t r a m a t e r i a l  approximations r e q u i r e d  i s described  between  i n j o i n t s and  t o manufacture these  structures  i n Chapter 5.  I t should  a l s o be remembered, when c o n s i d e r i n g the  curves by which many of the r e s u l t s are d i s p l a y e d , t h a t .these  54 have no p h y s i c a l m e a n i n g e x c e p t A fractional  Accuracy  a t i n t e g r a l v a l u e s of  number o f f i b r e s h a s  forces,  successive the  is  In e i t h e r  are i n e v i t a b l y  eight  significant  felt  that  significant  s u c h as  e t c . were c a r r i e d  a p p r o x i m a t i o n , o r by  structure.  ties  meaning.  of Results  Most o f t h e c a l c u l a t i o n s , lengths,  no  N.  case  out  a n g l e , member  iteratively  by  a step-by-step process across slight  accumulated  errors  i n basic  i n the procedure.  p l a c e s were u s e d  the f i n a l  fibre  v a l u e s can has  proper-  While  i n most c a l c u l a t i o n s , i t safely  p l a c e s and  this  structures  i n v e s t i g a t e d may  be  quoted  to  four  b e e n done i n t h e t a b u l a t e d  results. The divided  into  1)  the f o l l o w i n g  Symmetrical, A typical  4.1.  S i n g l e Load  s u p p o r t p o i n t s A and  joint,  A L  l o a d o f one (  J  The  N N  )  i  at J  na  supported  B in a vertical hundred  vertically  number o f f i b r e s ,  minimum p o s s i b l e ,  for consideration.  structure of t h i s  I t i s assumed t o be  equal.  groups  c o n v e n i e n t l y be  N  N  Perpendicular to Axis  t y p e i s shown i n F i g u r e in a vertical line.  The  pounds i s a p p l i e d downward N,  t o 20 w i t h one  was  plane  fan r a d i i a t the  direction.  varied  from  case having N =  with  2,  the  100.  are  outer  55  IOO  F i g u r e 4.1  Symmetrical  lb.  C a n t i l e v e r W i t h P o i n t Load  The major s e t o f s t r u c t u r e s had a c o n s t a n t span, L, e q u a l t o 10", w h i l e the s u p p o r t s p a c i n g , D, was 1" t o 10" g i v i n g an  r a t i o range of 1 t o  An a d d i t i o n a l s e t o f 5 f i b r e h a v i n g a c o n s t a n t — r a t i o o f 4.  v a r i e d from  10.  s t r u c t u r e s was  examined  Here the span, L, v a r i e d  from 60 t o 4 i n c h e s , D c o r r e s p o n d i n g l y v a r y i n g from 15 t o 1 inch. The  r e s u l t s o b t a i n e d from these c a l c u l a t i o n s are p l o t t e d  as curves and the f o l l o w i n g c o n c l u s i o n s may  be drawn -  a) F a n  The rapidly  Angle,  &  (see F i g u r e  fan angle  depends l a r g e l y  approaches the value  v a l u e s o f t h e r a t i o |j  for N  for N  larger  than  most c a s e s  there i s l i t t l e  values of N i n p r a c t i c a l  t o be  span o f t h e measured  from  almost  by  using  that i n larger  design.  cantilever  infinite  gained  and  Even a t l a r g e  This indicates  reviewed  values of fan angle,  the r e s u l t i n g  the — r a t i o  f o r 6 becomes  5.  Chan, i n h i s a n a l y s i s  of  on °° .  , the curve  horizontal  convenient  4.2)  fibre  6,  and  [ e q . 1.34  i n Chapter  1,  calculated  the  o r Chan 6 6 ] .  s t r u c t u r e was  a s c a l e drawing of the  not  took volume  The  calculated  corresponding  but  Michell  cantilever. I n t h e p r e s e n t work, t h e this  i s a b a s i c design parameter.  Michell  cantilever  f i b r e s was then  used  value of an  then  having  this  calculated.  i n equation  volume o f  for  1.34  the t h e o r e t i c a l 6 so u s e d  infinite  discussed  i n A p p e n d i x B,  to connect The  ize  s p a n and  dimensionless  data.  1.35  the  f o r N = 20  estimate  of  the  structure.  The  less  than  the a c t u a l  value  and  thus  the  bound on  f o r the  quantity  theoretical  the a c t u a l v a l u e .  i s used  infinite  [-—• L  of  was  fibre  g e n e r a l e q u a t i o n was  fan angle  side of equation the  no  of  as  t h e d e s i r e d number  fan angle  infinite  solution,  been s p e c i f i e d  fan angle  t o o b t a i n an  is slightly  fibre  The  s p a n and  The  optimum volume i s a l s o a l o w e r  left  s p a n has  As  discovered case.  ] d e r i v e d on  the  r  in later  curves  to  normal-  57 b) F i b r e Angle, The  6 (see F i g u r e 4.3)  f i b r e angle, G, d i m i n i s h e s c o n t i n u o u s l y as the  number o f f i b r e s i n c r e a s e s 1  I t e v e n t u a l l y tends t o zero as  N becomes very l a r g e , f o r a l l v a l u e s o f independent  o f s t r u c t u r a l dimensions  L i k e <5, i t i s  and depends o n l y on N  and — . c) Volume Index  (see F i g u r e s 4.4 t o 4.4e)  The dimensionless parameter, ~ — , c a l l e d f o r convenr ' P r ' J_I  ience  the Volume Index,  of — r a t i o s .  i s p l o t t e d a g a i n s t N f o r a range  F i g u r e 4.4 shows a l l the curves on a uniform  small s c a l e w h i l e each curve i s p l o t t e d to b e s t advantage on differing  s c a l e s i n F i g u r e s 4.4a t o 4.4e.  In these  latter  cases, two curves are p l o t t e d showing the e f f e c t of the weight of the s t r u c t u r e i t s e l f . The theory o u t l i n e d i n Chapter  1 p r e d i c t s an optimum  minimum volume which i s shown on a l l these c u r v e s . seen t h a t the "no s e l f weight"  curve i s asymptopic  value and indeed c l o s e l y approximates The  "with s e l f weight"  I t w i l l be to t h i s  i t f o r N as low as 10.  curve i s i n each case somewhat h i g h e r  but a l s o c o u l d be c o n s i d e r e d as a good approximation  to the  u l t i m a t e value f o r N l a r g e r than t e n . Even f o r N = 5, the Volume Index i s w i t h i n 5% of the t h e o r e t i c a l minimum f o r a l l p l o t t e d v a l u e s o f ~ w h i l e the v a l u e s f o r N = 3 are w i t h i n 10% (5% f o r ~ l e s s than 5 ) . f u r t h e r j u s t i f i e s the e a r l i e r statement  This  that p r a c t i c a l  s t r u c t u r e s need c o n t a i n no more than f i v e f i b r e s per f a n and y e t s t i l l be e x c e l l e n t approximations optimum.  t o the t h e o r e t i c a l  Figure  4.2  V a r i a t i o n o f F a n A n g l e w i t h Number o f F i b r e s i n Symmetrical M i c h e l l C a n t i l e v e r s  Figure  4.3  V a r i a t i o n o f F i b r e A n g l e w i t h Number o f i n Symmetrical M i c h e l l C a n t i l e v e r s  Fibres  60  ISO  TKeoK.tical Optimum. 110  - 99'77  i>o -  3«»  40  27-97  zo 9-91  4  +•  6  +'  3 43  8  NUMBER OF FIBRES Figure  4.4  IO N  V a r i a t i o n o f Volume Index w i t h Number o f F i b r e s i n Symmetrical M i c h e l l C a n t i l e v e r s  346  VOLUME  INDEX. 3-45  6$ Self  W^ht.  3-H  Theoretical  343  H  *  4-  Mlnimom  6  3*433  *  i  8  NUMBER, OF FIBRES - N. F i g u r e 4.4a  1  1  'O  Volume Index f o r Symmetrical M i c h e l l Cant Having L , D  62  t O ' O Z  VOLUME INDEX  9S>4  Ji>eofeticol Minimum 9'9o6  4  +  6  +'  JL  8  _4  J  NUMBER OF FIBRES* N Figure  4.4b  Volume Index f o r S y m m e t r i c a l M i c h e l l Having L _ ~ D ~ ^  Cantilevers  63  29-oh  V O L U M E I N D E X  28-5  WirK Self We^ht.  No Self Weigkt.2&Oh Theoretico.1 Minimum. £7 862  275  H  4  i 6  »-  8  NUMBER OF FIBR.ES N. F i g u r e 4.4c  -f  ± io  Volume Index f o r Symmetrical M i c h e l l C a n t i l e v e r s Having L _ . D  64  Figure  4.4d  Volume Index Having L _ _  D  f o r Symmetrical M i c h e l l  Cantilevers  65  180.:  V O L U M E INDEX  5$' I20  • With Self W e i K t . 3  No Self We^brT^^ ^^^ 1  Theoretical Minimum 98-40Z  4  -  H  6  i:  H  NuMBEfe O F FiBRES F i g u r e 4.4e  8  4  ± IO  N.  Volume Index f o r Symmetrical M i c h e l l C a n t i l e v e r s Having L _ ,_ D  Another f a c t o r j u s t i f y i n g t h i s concerns the  total 2  number of members. members.  An N - f i b r e d  framework c o n t a i n s  Manufacturing c o s t s would tend t o be  t o the number of components and  N  proportional  thus c o n s i d e r a t i o n s  manufacturing economy i n d i c a t e t h a t N should  of  be kept to a  minimum. These arguments are r e i n f o r c e d when the between members are c o n s i d e r e d  i n Chapter  EFFECT OF VARIATION OF  The  joints  5.  SPAN  volumes of the s e t of s t r u c t u r e s having e q u a l  but v a r y i n g  span, are p l o t t e d i n F i g u r e  4.5.  T h i s shows  t h a t , f o r the case i n which the e f f e c t s o f the weight o f the s t r u c t u r e i t s e l f are i g n o r e d , p r o p o r t i o n a l to the span.  the volume i s d i r e c t l y  T h i s a r i s e s s i n c e the  structures  are a l l of i d e n t i c a l geometric shape, d i f f e r i n g o n l y i n the scale.  The  f o r c e i n each member i s c o n s t a n t  since  depends o n l y on the geometry, the s t r e s s being uniform.  The  this  everywhere  c r o s s - s e c t i o n a l areas are thus c o n s t a n t  t o t a l volume i s p r o p o r t i o n a l to the span. Volume  =  1.670  (span)  t u r e are i n c l u d e d , the volume i s not d i r e c t l y span but  i n c r e a s e s more r a p i d l y .  by the percentage i n c r e a s e  t  From the r e s u l t s  However, i f the e f f e c t s of the weight of the  to the  and  struc-  proportional  This i s indicated  curve which r e p r e s e n t s  -  67 Percentage Incwase o f Self Weight Vofume a s cpmpafed to Volume i f Self Vie\$hf ne^lecfeb.  e r c e a t a g e irxcrease of S t r u c t u r a l V o l u m e iaci u d m q e f f e c t s ^ self'ooei^T?!- c o m p a r e d t& M O i J m e *|G s e i £ aittoM" is n e o i e c r e d .  20  40  SPAN  inches.  F i g u r e 4.5  68 ( s e l f weight volume - volume w i t h s e l f weight ignored)  x  (volume w i t h s e l f weight ignored) and  the curve c l e a r l y i n d i c a t e s how  this steadily increases.  T h i s r e f l e c t s the w e l l known f a c t t h a t , as b r i d g e i n c r e a s e , the  spans  s t r e s s e s i n the span are more and more due  to  the weight of the s t r u c t u r e r a t h e r than to the a p p l i e d  load.  This  design.  sets  an ultimate  limit  to the  span>  whatever  2) Symmetrical C a n t i l e v e r With T i l t e d End In the p r e v i o u s Michel  s e c t i o n we  c a n t i l e v e r provides  a very  *  Figure  fibre  s a t i s f a c t o r y approximation This p a r t i c u l a r struc-  be used to examine the e f f e c t of a p p l y i n g  ,0".  4.6  Load  have seen t h a t a f i v e  to the t h e o r e t i c a l optimum s t r u c t u r e . t u r e w i l l now  its  the  ^  Symmetrical M i c h e l l C a n t i l e v e r w i t h T i l t e d Load  load  a t angles  other  than  Specifically, t i e s was  p e r p e n d i c u l a r to the a x i s .  a cantilever  the f o l l o w i n g proper-  used:-  Span L  =10"  Support  spacing D  =  Number o f f i b r e s N Fibre  angle  This and  with  2.5" = 5  6  =  18.41°  L D  =  4  s t r u c t u r e was  supported  a s shown i n F i g u r e  s u b j e c t e d t o a one h u n d r e d pound  The d i r e c t i o n  of this  l o a d was  l o a d a t L, j o i n t  varied  between t h e  4.6  J  N N  »  limits  - 4 5 ° < T < +9o° The r e a c t i o n s c o r r e s p o n d i n g i n F i g u r e 4.7, sum  tal  and a r e p e r h a p s o f f a i r l y  of the v e r t i c a l  vertical  component  reactions, A  load,  ( 55  s i  P  to these  nT  H  obvious  c o m p o n e n t s , A^. and B^, must of the l o a d ,  and B , H  resist  the h o r i z o n t a l  ) / p l u s t h e moment r e q u i r e d t o  is  'B'  steadily  fibre  reduced.  w i t h member B ^ further  The  equal the  (P^^ c o s x ) , w h i l e  As x i n c r e a s e s t h e l o a d i s c a r r i e d outer  nature.  the horizon-  thrust  of the  counterbalance  by t h e l o a d , P c o s T x L .  that generated  the  loads are p l o t t e d  and t h e c o r r e s p o n d i n g A t x = 54.2°,  and t h e f o r c e i n A ^  more and more b y outer  'A'  fibre  load  the load i s a l i g n e d drops  to zero.  i n c r e a s e d t h e l o a d on o u t e r A members  (A^g)  If x is becomes  70  —i  -40  1  1  -20  1  1  o  1  1  20  Tilt of L o a d Figure  4.7  1  1  4o  1  « 60  1  J  so  "X. (decrees.)  Reactions a t Supports of a Symmetrical M i c h e l l C a n t i l e v e r w i t h T i l t e d L o a d a t O u t e r End  1  71  Ttlt of Load- c/ejrees Figure  4.8  Volumes o f S y m m e t r i c a l at Various Angles  Michell Cantilevers  Loaded  U l t i m a t e l y , when T becomes +90°, o n l y the  compressive.  members i n the two outer  f i b r e s (A^^ and B ^) a r e i n compresa  s i o n , w h i l e a l l i n n e r members are now i n t e n s i o n . f i b r e s now a c t t o r e s t r a i n the outer  The i n n e r  f i b r e s from becoming  more curved. Figure  4.8 i n d i c a t e s how the volume v a r i e s w i t h  d i r e c t i o n of loading.  As might be expected the s t r u c t u r e has  a maximum volume when x = zero  (pure c a n t i l e v e r ) and d e c l i n e s  s t e a d i l y t o a minimum v a l u e when T = ±90° The  t r a d i t i o n a l method o f d e s i g n i n g  tilted  (pure column).  a mirror  t h a t i s t o be  e n t a i l s a d e t a i l e d a n a l y s i s w i t h the m i r r o r  horizontal.  When t i l t e d  the m i r r o r d e f l e c t i o n s do not exceed  those encountered i n the h o r i z o n t a l p o s i t i o n . corroborates  plane  Figure  4.8  t h i s , i n t h a t the maximum volume i s found a t  T = zero, which corresponds t o the s t r u c t u r e i n a h o r i z o n t a l plane. It  should  be emphasised t h a t two c o n d i t i o n s may be  i d e n t i f i e d when c o n s i d e r i n g given  loads.  In one case a  s t r u c t u r e may be designed t o be optimum f o r a g i v e n  l o a d i n g , say a t x = zero. loads  tilted  a t other  values  I t may however be s u b j e c t e d t o  of x .  C e r t a i n members may become more  h e a v i l y s t r e s s e d than when the s t r u c t u r e i s h o r i z o n t a l . In the other derived  case an optimum s t r u c t u r e may be  t o support a g i v e n t i l t e d  considered  load.  The s t r u c t u r e s  i n t h i s s e c t i o n are optimum f o r a l l v a l u e s  such t h a t the d i r e c t i o n o f the load v e c t o r s e c t o r QLS (Figure 4.6) l i m i t i n g value  of x  l i e s w i t h i n the  Hegemier and Prager  [17] quote the  o f x as -45°_< x <_ 45° which i s t r u e f o r the  72a  theoretical structure.  In the M i c h e l l t r u s s , the p e r m i s s i b l e  8 angle i s somewhat g r e a t e r  8  [-(45+-j) <T<+ (45+-J) ] ,  When the l o a d v e c t o r  f a l l s outside  t h i s s e c t o r both  members meeting a t L c a r r y f o r c e s of the same s i g n and the s t r u c t u r e becomes, i n p a r t , o f t h e Maxwell type. Maxwell theorem p r o v i d e s optimisation  S i n c e the  no geometric c o n d i t i o n o f l a y o u t ,  i s no longer c e r t a i n .  73 3)  Symmetrical C a n t i l e v e r with Point The  this  s e r i e s of loads The  those load  'standard'  fibre  dimensions o f the s t r u c t u r e  i s applied  in  e a c h member  Figure use  i n dealing  are  applied  point  main  combinations of loads.  member may be o b t a i n e d load.  simultaneously  by a d d i n g  Joints  If several the force  loads i n any  t h e e f f e c t s due t o e a c h  The w i d t h s o f members and o t h e r  may t h e n be o b t a i n e d  for .  >•  C a n t i l e v e r w i t h Loaded  to the structure  and,  'E') i s i n t e n d e d  :  4.9  with  are unstressed  However, t h e t a b l e o f f o r c e s  ( d e t a i l e d i n Appendix  I D  uniform.  pound  joints.  -s»  is  with  i n sequence a t each o f t h e twenty f i v e  speaking, redundant.  individual  under  are identical  c a s e , b u t now t h e one h u n d r e d  Members b e y o n d t h e l o a d strictly  c a n t i l e v e r examined  i s shown i n F i g u r e 4 . 9 .  i n the previous  structural  five  Load a t Each J o i n t  by s i m p l e c a l c u l a t i o n s i n c e  properties the stress  74 The placed that  following  a t each  joint  joint  and f o r  Load A t Joint  J  table  lists  i n turn.  the d e f l e c t i o n f o r a  The d e f l e c t i o n i s g i v e n f o r  J -. KC  Deflection At That J o i n t  Deflection At J 55  o, 0 . 0 0 2 5  l , l  J  2,1' 1,2  J  3,1' 1,3  J  4,1' 1,4  J  5,1' 1,5  J  2,2  J  3,2' 2,3  J  4,2' 2,4  J  5,2' 2,5  J  3,3  J  4,3' 3,4  J  5,3' 3,5  J  4,4  J  5,4' 4,5  J  5,5  J  J  J  J  J  J  0.0060  1.1326  0.0012,  0.0043  -0.0003 ,  0.0086  1.4238  -0.0003 , 0.0083  1.6251  0.0059  -0.0004 ,  0.0105  1.9737  0, 0 . 0 0 5 1  0,  0.0104  1.7080  0.0013,  0.0071  0,  0.0142  2.3823  0.0040,  0.0090  0.0002 ,  0.0175  2.9868  0.0009,  0.0100  -0.0010 ,  0.0159  3.3446  0 , 0.0193  3.6795  4) T i l t e d The sections  0.0036,  0.0154  0.0003 ,  0.0238  5.1224  0.0102,  0.0191  0.0057 ,  0.0313  6.3633  0, 0 . 0 2 3 7  0,  0.0296  7.8988  0.0327  0.0074 , 0.0399  10.9196  0, 0 . 0 5 0 1  0 , 0.0501  16.7338  0.0086,  J  Symmetrical  same f i v e  i s shown  fibre  tilted  Cantilevers c a n t i l e v e r as u s e d  i n the v e r t i c a l  It  i s mounted w i t h i t s x a x i s  so  that  parallel  the g r a v i t y t o AB.  0.8335  -0.0002 ,  0, 0 . 0 1 1 0 J  0.00294  0.0034  0.0012,  J  0,  Volume Cu.In.  0.0003,  -0.0009 , 0.0049  J  load  forces  i n two p r e v i o u s  plane i n Figure 4.10.  a t an a n g l e + t o t h e h o r i z o n t a l  due t o s e l f  w e i g h t a r e no  longer  75  Figure  4.10  A one  hundred  to the v e r t i c a l The  Tilted  Symmetrical  pound l o a d  so t h a t  s y s t e m was  Cantilever  i s applied  a t L,  parallel  T = + . analyzed  f o r the range  o ^ t - 1 < 90° and ion,  the r e s u l t s  the,analyses  having  was  t h e same s p a n .  are tabulated  repeated  i n Appendix  f o r a three  fibre  E.  In  addit-  cantilever  As might be expected the are very  f o r c e s and  s i m i l a r t o those obtained  only d i f f e r e n c e i n the two s e l f weight f o r c e s ,  The  s t r u c t u r e volumes  i n Section  2, s i n c e  the  cases l i e s i n the d i r e c t i o n of  same q u a l i f i c a t i o n  the  regarding  optimisation i s also applicable. The r e a c t i o n s d i f f e r c o n s i d e r a b l y changes i n geometry.  In the  d i v i d e d i n t o h o r i z o n t a l and r e f l e c t e d i n the l o a d and  first  however due  to  case the l o a d may  be  v e r t i c a l components which  support r e a c t i o n s .  In the p r e s e n t  case,  h o r i z o n t a l r e a c t i o n s generate a moment t o r e s i s t the  5)  the  the turning  load.  Skew C a n t i l e v e r s We  The  are  s e l f weight f o r c e s are always v e r t i c a l w h i l e  moment caused by the  the  now  t u r n to skew c a n t i l e v e r s which are  fan r a d i i are unequal and  the a x i s .  The  i n Figure  4.11.  The  general  the end  unsyrametrical.  p o i n t L does not  arrangement of such s t r u c t u r e s i s shown  p o i n t C must l i e on the c i r c u m f e r e n c e of a  whose diameter i s AB,  ACB  being  a right  such s t r u c t u r e s are the  circle  angle.  Convenient v a r i a b l e parameters f o r the  fan r a d i i — A  l i e on  f i b r e angle 6 and  s o l u t i o n of  the r a t i o of  (or RADRAT i n the computer programme).  the  The  r  support s p a c i n g equations,  D and  the  values  are assumed t o be  of  specified.  L and  Y i n the  A programme f o r  computer s o l u t i o n of these s t r u c t u r e s i s g i v e n A s e r i e s of these skew c a n t i l e v e r s was a l l w i t h L = 10.0", but having v a r i a b l e v a l u e s  i n Appendix  C.  investigated, of Y from  Figure  4.11  Typical  Rise Figure  4.12  of  Skew M i c h e l l  Cantilever  stVuctuf-e.  V e r t i c a l Reactions a t Supports C a n t i l e v e r s o f V a r i a b l e Skew  of Michell  78 zero  to  2.5  structures  inches i s given  The  that  lineal  manner.  tially  constant,  external  the The  Figure  are  does not  4.13  B ratio —  i n Figure  data  4.12  from  and  components r e m a i n  torque applied change  indicates  The  components change i n a  horizontal the  inch.  these  E.  plotted  vertical  since  load  0.25  i n Appendix  reactions  interesting  the  i n steps of  the  to  the  i t is strictly  substanstructure  by  greatly. way  i n which the  fibre  angle,  r  6,  and  the  r  increases. and at  could least  be  used  a close  The  to determine  this  increasing  structures  f o r s e l e c t i o n of  situations.  skew o f  the  structure  almost l i n e a r  for large  the  variables,  structure  a p p r o x i m a t i o n to the  volume o f  change g r e a t l y ,  the  A  These curves are  The  basis  , change as  actual  structures  by  3%  investigated structures  over the  in this  or  values.  range of under  skews,  does not range.  chapter provide  t o meet a v a r i e t y o f  a  design  7?  Figure  4.13  V a r i a t i o n s o f F i b r e A n g l e and R a d i u s R a t i o i n Some Skew M i c h e l l C a n t i l e v e r s  80 CHAPTER 5 THE DESIGN OF JOINTS AND  OTHER MODIFICATIONS  The r a n g e o f s t r u c t u r e s the  preceding  chapters are approximations  minimum v o l u m e s t r u c t u r e s Their and  design,  considered  however, s t i l l  consideration  particularly  considered  before  and a n a l y z e d  in  t o t h e optimum  by M a x w e l l and M i c h e l l .  requires  further  manufacture could  i f t h e minimum p r a c t i c a l  be  modification undertaken,  volume i s t o be  achieved. The e q u a t i o n s d e r i v e d  i n Chapter  2 and 3 may  be  t o c a l c u l a t e t h e member s i z e s f o r a s t r u c t u r e d e s i g n e d support a s p e c i f i e d system o f loads assumption  that  a l l the j o i n t s  However, sary  t o make t h e a c t u a l  t o have a l e n g t h and  no a l l o w a n c e  having In  equal  a constant fact,  i n a given  used to .  s p a c e , on t h e  are pinned. i s made f o r t h e m a t e r i a l  joints,  t h e members b e i n g  to the distance  neces-  assumed  between j o i n t  points  cross-section.  however, t h e a x i a l  f o r c e a c t i n g a l o n g a,;  member must be t r a n s f e r r e d t o i t s n e i g h b o u r s , p r i m a r i l y t o those  lying  along  t h e same f i b r e  those approximately perpendicular Some e x t r a m a t e r i a l in  the j o i n t s  to permit  members, p a r t i c u l a r l y  this  must  b u t a l s o t o some d e g r e e t o to i t . therefore  be  incorporated  t r a n s f e r o f f o r c e between  i f the uniform  stress  ;  s p e c i f i e d through-  81 out  the t h e o r e t i c a l s t r u c t u r e  i s nowhere t o be e x c e e d e d .  There a r e s e v e r a l p r a c t i c a l of  this  or  local  be  considered.  problem, a l l o f which i n v o l v e increase  significance the  approaches t o the s o l u t i o n  of stress l e v e l .  The l a s t  e i t h e r a d d i t i o n a l volume Some o f t h e s e w i l l  section of this  now  chapter discusses the  o f these approaches i n terms o f t h e f u n c t i o n o f  structure. 1)  Pinned  Joints  E a c h member o f t h e framework c o u l d w i t h ends e x t e n d e d b e y o n d metrical  considerations,  at  the j o i n t  of  the s t r u c t u r e . •  5.1  point, with  Possible  separately  s p e c i f i e d by geo- :  t o accommodate a 'smooth' p i n p l a c e d i t s axis perpendicular  A possible design  Figure  the dimensions  be made  i s indicated  End D e s i g n  to the plane  i n F i g u r e 5.1.  for- T y p i c a l Member  82 For that  the purpose  of this discussion  i t may be assumed  t h e members a r e o f c o n s t a n t t h i c k n e s s t . A t y p i c a l member A ^ i s t h e o r e t i c a l l y  rectangular in  the diagram.  the (  section  and w o u l d t e r m i n a t e a t MN a s shown d o t t e d  A joint  WA  a b  remain  ferred the  or right  compression.  section  uniform.  The a x i a l  imately  o f QK, d e p e n d i n g A fillet  and m i n i m i z e The  and t h e s t r e s s  force w i l l  be t r a n s -  a s t h e member i s i n t e n s i o n  a t F would ease  t h e change o f  any s t r e s s c o n c e n t r a t i o n .  e x t r a volume o f m a t e r i a l p e r end i s g i v e n  approx-  by t h e e q u a t i o n  Excess  »  The  on  fairly  as i n t h e s t r a i g h t p o r t i o n  f r o m t h e member t o t h e smooth p i n o f d i a m e t e r p , t o  left  and,  having a diameter  t h e c r i t i c a l p l a n e QK, t h e c r o s s - s e c t i o n o f t h e -  member i s (WA ^ ) x ( t ) will  shape,  by w i d e n i n g  ). At  or  b e a r i n g may be f o r m e d  end a s shown t o a c i r c u l a r  P +  of uniform  t|^p->WAob(4rt^-G^ at>[2ir-4]| WA  . . . . (5.D  p i n i s p l a c e d i n shear across a d i a m e t r a l plane  from elementary  theory, the shear s t r e s s ,  x, exerted  i t , i s g i v e n by  (5.2)  83 The  2 i n t h e d e n o m i n a t o r w o u l d be  i n c l u d e d i f the  members were d u p l i c a t e d .  As  members, t h e p i n w o u l d be  subjected to a t w i s t i n g  approximately dividing ~  and  a b o u t t h e a x i s QK.  A  or A , into a , D  placing  drawn i n F i g u r e 5.1,  —  two  T h i s may  be  with  single  moment  e l i m i n a t e d by  e q u a l members e a c h o f t h i c k n e s s  3 D  l  them s y m m e t r i c a l l y on  either  s i d e of the  other  bar. The pin  two  diameter  tained  equations  such  throughout The  evaluated Chapter  4,  that uniform the  use  of  the  S e c t i o n 1.  The  actually  tested.  increase  t h e volume s t i l l  to determine  the  i s substantially  ob:  i n c r e a s e i n volume  p i n diameter,  to correspond Selection  with  p, was  t h a t used  on  examined  arbitrarily  i n the  models  i n Table  F i g u r e s 5.2A-5.2D.  the t o t a l  with  the  5.1  and  There are  sub-  a d d i t i o n of  an  volume, i n c l u d i n g  joint  allowance. When j o i n t  allowance  longer decreases  continually  minimum and  rises  then  in  further.  F i g u r e s 4.4A-4E r e p e a t e d indicating  was  o f a l a r g e r v a l u e o f p would  amount o f e x t r a volume i s g i v e n  the v a l u e s are p l o t t e d  upper curve  used  set of structures e a r l i e r  as ~",  stantially  stress  of t h i s  selected  The  be  joint.  significance  by  a b o v e may  i s c o n s i d e r e d , t h e volume as N  rapidly,  i n c r e a s e s but  tending  toward  no  reaches infinity  N becomes v e r y l a r g e . I f s t r u c t u r e s a r e t o be made w i t h p i n j o i n t s ,  a as  there  TABLE  5.1  JOINT ALLOWANCES FOR PINNED SYMMETRICAL (See C h a p t e r  4, S e c t i o n  1 for details  CANTILEVERS  of structures)  L D  N  Net Volume  1  2  8.1807  1.7716  9.9523  2l;66  3  8.1243  2.4297  10.5540  29 .91  4  8.1141  3.1987  11.3128  39.42  5  8.1104  4.0333  12.1437  49.73  7  8.1079  5.8670  13.9749  72.36  10  8.1068  9.0003  17.1071  111.02  2 .  12.5873  4.0605  16.6477  32 .26  3  11.9107  4.4663  16.3771  37 .50  4  11.7966  5.2985  17 .0952  44.92  5  11.7574  6.2789  18.0363  53.40  7  11.7295  8 .4894  20.2189  72.38  10  11.7173  12.2614  23.9787  104 .64  2  20.6597  12.8822  33.5419  62 .35  3  17.3714  11.3477  28.7191  65 .32  4  16.8940  11.7635  28.6575  69 .63  5  16.7338  12.7282  29 .4620  76 .06  7  16.6213  15.3265  31.9478  92.21  10  16.5721  20.0440  36 .6161  120.95  2  24.4237  19 .5259  43.9496  79 .95  3  19.4195  16.3157  35.7352  84.02  4  18.7345  16.2361  34 .9706  86.66  5  18.5071  17.0337  35.5409  92 .04  7  18.3480  19.6439  37.9919  107 .06  10  18.27 87  24.6828  42.9615  135.04  2  42.3208  75.9381  118.2589  179 .43  3  26.6088  57.7703  84 .3792  217.11  4  22.8859  52.2136  77.0995  209 .81  5  24 .3336  50.5002  74 . 8339  207.53  7  23.9534  51.3051  75.2585  214.19  10  23.7877  56.4998  80.2876  237.52  2  4  5  10  Joint Allowance  Gross Volume  % Increase  85  NUMBER, OF FIBRES N F i g u r e 5.2a  E f f e c t of J o i n t Allowance on Volume o f Symmetrical Michell Cantilevers  86  So  With \oint aUou3qr>ce .  5o  >  D  '  VC TJ  8  3o  J  i  i  i  1  1—  NUMBER OF FIBRES - N Figure  5.2b  E f f e c t o f J o i n t A l l o w a n c e on Volume o f Michell Cantilevers - L _ .  20  Symmetrical  D  V  87  NUMBER OP Figure  5.2c  F<3«£S  N  E f f e c t o f J o i n t A l l o w a n c e on Volume o f S y m m e t r i c a l Michell Cantilevers - = 5 D T  88. $oo  T  NUMBER or FIBRES N. Figure.5.2d  E f f e c t of J o i n t Allowance Michell Cantilevers -  on Volume o f  • £ - 1 0  •  Symmetrical .  89  i s an optimum number o f f i b r e s f o r minimum practical  volume.  T h i s number seems t o be between 3 and 5 f o r the range o f ^ considered.  T h i s c o n c l u s i o n i s r e i n f o r c e d by the p r a c t i c a l  c o n s i d e r a t i o n t h a t t h e number o f components should be kept t o a minimum t o reduce manufacturing c o s t s . The  c a l c u l a t i o n s on which the p r e c e d i n g  based r e l a t e t o a s e r i e s of c a n t i l e v e r s having 10 i n c h e s .  T h i s may seem s m a l l t o those  curves a r e a span o f  interested i n  s t r u c t u r a l d e s i g n , but t h i s o r d e r o f s i z e i s o f i n t e r e s t t o o p t i c a l engineers.  I f t h e span i s i n c r e a s e d , the geometric  l a y o u t and member f o r c e s remain unchanged f o r a g i v e n ratio. I f the maximum deflection  i s a l s o t o be kept  constant,  the p e r m i s s i b l e s t r e s s must be reduced i n p r o p o r t i o n t o the change o f span.  T h i s w i l l i n c r e a s e member c r o s s s e c t i o n s  correspondingly  and t h e p r o p o r t i o n o f t o t a l weight a t t r i b u t e d  t o t h e j o i n t s remains  constant.  However, i f the maximum s t r e s s i s unchanged, so t h a t t h e d e f l e c t i o n i s i n c r e a s e d p r o p o r t i o n a l l y , member c r o s s s e c t i o n a l areas are i n c r e a s e d .  5.1.  and  lengths  The e f f e c t o f t h i s r e d u c t i o n i s i n d i c a t e d  5.2, which i s based on data e x t r a c t e d from  Consider  |j equal  their  The p r o p o r t i o n o f m a t e r i a l i n t h e j o i n t s  i s thus reduced. i n Table  are unchanged, although  Table  a c a n t i l e v e r o f one hundred i n c h span, w i t h  to four.  I t c a r r i e s a load of one hundred pounds  a maximum s t r e s s o f three hundred p . s . i .  The Net  89a volume f i g u r e s from T a b l e 5.1 the J o i n t Allowances  are m u l t i p l i e d by ten  are unchanged.  TABLE JOINT ALLOWANCES FOR L=100",  but  £=4,  5.2  PINNED SYMMETRICAL CANTILEVERS Load=100 l b ,  6=300 p . s . i .  Gross Volume  % Increase  12.9  220.5  6.7  173.7  11.3  185.0  6.5  4  168.9  11.8  180.7  7.0  5  167.3  12.7  180.0  7.7  7  166.2  15.3  181.5  9.2  10  165.7  20.0  185.7  12.1  N  Net Volume  2  206.6  3  The  Joint Allowance  5 f i b r e s t r u c t u r e now  has the minimum volume,  whereas w i t h a ten i n c h span 3 f i b r e s was i n d i c a t e s t h a t each combination  optimum.  This result  of parameters should  e v a l u a t e d s e p a r a t e l y t o determine  the optimum case.  be The  c o n c l u s i o n t h a t 5 f i b r e s r e p r e s e n t s a p r a c t i c a l upper is  limit  confirmed. I t i s u n l i k e l y t h a t s t r u c t u r e s would a c t u a l l y  made i n t h i s manner. and  general  The pinned  j o i n t s serve l i t t l e  purpose  i n t r o d u c e p l a y , c a u s i n g e x t r a unwanted d e f l e c t i o n ,  the more complex d e s i g n would be expensive manufacture and  assembly.  be  and  i n labour f o r  89b The concept remains  o f c o n s i d e r a b l e v a l u e as a  b a s i c method o f a n a l y s i s , y i e l d i n g q u i t e a c c u r a t e r e s u l t s .  2)  Rigid  Joints  The complete  M i c h e l l framework c o u l d be c u t from  a s i n g l e sheet o f m a t e r i a l , o r , i n l a r g e s i z e s , made from subassemblies welded  together.  There  i s now no problem  of j o i n t  d e s i g n , s i n c e they are s o l i d , but the a n a l y s i s i s much more complex, s i n c e the framework now i s s t a t i c a l l y  indeterminate.  In such case, the a x i a l f o r c e s i n the members a l r e a d y c o n s i d e r e d g i v e r i s e t o t h e primary s t r e s s e s , w h i l e secondary  s t r e s s e s a r i s e from the bending moments and shear  s t r e s s e s induced a t t h e r i g i d The  joints.  STRUDL programme (STRUctural Design Language)  which has been compiled  as p a r t o f ICES  E n g i n e e r i n g System) f o r the  general  (Integrated C i v i l  s o l u t i o n o f complex  structures  was u s e d  work a l r e a d y The  t o examine t h e s t a n d a r d  considered  as a p i n n e d  fibre  properties  -  Loads a p p l i e d a t L, p a r a l l e l t o AB  S u p p o r t S p a c i n g D = 2.5"  1  Member t h i c k n e s s Number o f F i b r e s F i b r e Angle  The  joints  Member w i d t h s , a s i n t a b l e s 10, A p p e n d i x E  9 = 18.41°  are r i g i d  t = 0.25"  N = 5  and t h e s t r u c t u r e  h a v i n g been c u t from a f l a t The  net-  structure.  c a n t i l e v e r had t h e f o l l o w i n g  Span L = 10"  five  p l a t e one q u a r t e r  member w i d t h s a r e s u c h t h a t with  may be v i s u a l i z e d a s  structure  when l o a d e d  a uniform  s t r e s s o f 300 p . s . i .  inch  i n t h e comparable  thick. > pin-jointed  one h u n d r e d pounds a t L ( J ^ exists at a l l points  ^^)  i na l l  members. To --a  These  were a p p l i e d  loads  varied  s p e c i f i e d member  separately  sizes  a t L, p a r a l l e l  f r o m 10000 pounds t o 0.01 pound  i n each case t h e (constant) weight o f t h e s t r u c t u r e  included  i n the force  represents  loads,  n e g l i g i b l e while  forces  analysis.  t h e extreme s t a t e s  With t h e high are  fixed structure—with  s e r i e s of loads  t o AB. and  this  This  wide range o f  of loading  t o be  loads,  the s e l f  a r e of equal importance with the a p p l i e d I t must be remembered  structure  i n this  i s of f i x e d dimension.  analysis  forces' weight  force. that  Thus t h e s t r e s s to loading.  loads  considered.  the e f f e c t s o f the s e l f weight a t the lowest  was  constant but v a r i e s  from l o a d i n g  however, t h e s t r e s s  s h o u l d be r e a s o n a b l y c o n s t a n t  the  :  i s not,  Within  limits,  through-  out  the structure The  corded data  that are  major d a t a d e r i v e d  i n Table  5.1.  i s reproduced  E a c h member the  f o r any one l o a d i n g .  lower  Since  analysis are r e -  the structure  i s symmetrical,  f o r the upper h a l f o f the s t r u c t u r e  i s paired with  section  from t h i s  only.  i t s corresponding partner i n  [A ^ and  ] --the only  difference  t h e 'B' members a r e i n c o m p r e s s i o n w h i l e  being  t h e 'A' members  i n 'tension'. Similarly  deflections opposite  correspond, having  i n t h e y d i r e c t i o n and e q u a l d e f l e c t i o n s o f .  t a b l e may be d i v i d e d  The f i r s t  three  into ten sections  c o l u m n s name t h e v a r i o u s  show t h e i r w i d t h and c r o s s - s e c t i o n a l a r e a . shows t h e f o r c e s previously The ranging  identical  sign i n the x d i r e c t i o n . The  ation.  J ^ and  and s t r e s s e s  f o r the pinned  f o r examinmembers and  The n e x t  column  structure  examined. next  seven s e c t i o n s  show, f o r a s e r i e s o f l o a d s  f r o m 10000 l b s . t o 0.01 l b s . , t h e f o l l o w i n g  a)  A x i a l Force  b)  Stress  data:-  i n E a c h Member, FORCE  due t o t h a t  force =  .  The n o m i n a l c o n -  AREA stant value c)  i s shown a t t h e t o p o f t h e c o l u m n ,  SHEAR FORCE.  This  i s t h e mean o f t h e s h e a r s a t  e a c h e n d o f member, d)  BENDING MOMENTS.  The f i r s t  moment, t h a t n e a r e s t column i s t h a t  c o l u m n shows t h e i n n e r  t o s u p p o r t A, w h i l e  a t t h e outer end,  the second  TABLE 5.3  Member  Width  Cross Area  A  10  B  01  A  l l  B  l l  A  12  B  21  A  13  B  31  A  14  B  41  A  20  B  02  A  21  B  12  A  22  B  22  A  23  B  32  A  24  B  42  A  30  B  03  A  31  B  13  A  32  B  23  A  33  B  33  A  34  B  43  A  40  B  04  A  41  B  14  A  42  B  24  A  43  B  34  A  44  B  44  A  50  B  05  A  51  B  15  A  52  A  25  A  53  B  35  A  54  B  45  92  Pin Joint 1 Rigid Structure 100 l b l o a d - 10000 l b L o a d Force l b s | Force Stress Shear  0.871  0.218  65.29  6989  32120  656 .4  0.759  0.190  56.90  5862  30900  473.6  0.536  0.134  40.22  2817  21000  426 .3  0.358  0.089  26 .83  1530  17110  192.0  0.217  0.054  16.26  7023  12960  1.560  0.390  117.0  11400  29240  1.371  0 .343  102.8  10100  29470  0.988  0.247  74.10  6841  27770  569 .8  0.675  0.169  50.62  4641  27510  238.4  0.422  0.106  31.67  2897  27430  20.41  1.229  0.307  92.15  8405  27360  53.03  1.100  0.275  82.48  7577  27560  12 .04  0.830  0.208  62.26  6071  29260  0.598  0.150  44.88  4454  29770  75.73  0.401  0.100  30 .06  3006  30000  19 .86  0.948  0.237  71.12  6474  27300  0.868  0.217  65.07  6083  28050  43.09  0.690  0.172  51.73  5089  29520  18.43  0.528  0.132  39 .57  3942  29890  18 .05  0.380  0.095  28.52  2858  30050  3.37  1.353  0.338  101.5  9794  28960  1.336  0 .334  100.2  9763  29240  1.267  0.317  95.05  9338  29480  25.96  1.202  0.301  90.17  8871  29510  31.45  1.140  0.285  85.53  8430  29570  85.52  72 .05 1275. 28.76  130 .9  241.1  1048. 170.1  Table Bending Moment  5.3  (continued)  an a  93  Rigid  S t r u c t u r e - 1000 l b L o a d  Stress 3213  Shear 65.65  B.M. 91.71  an (%) 5— 90 .4  70 .14  (%)  Force  916.9  90 .4  Force 699 .2  511.3  69 .0  586.4  3091  47 .38  51.15  69 .0  58.82  250.5  99 .5  281.8  2101  42.66  25.06  99 .6  28 .26  92.15  101 .0  153.0  1712  19 .22  9 .22  101 .0  15.34  29.04  114 .4  1296  7.21  2.91  114 .7  7 .03  2015.  70.24  67 .9  1141.  2925  763.3  33 .1  1010.  2948  763.3  67 .8  684.3  2771  271.2  51 .9  464 .2  23 .3  47.57  113.3  6 .59  127.5  201.6  67 .9  114.5  76.34  33 .1  101.3  57.0  76.34  67 .7  68.62  2752  23 .83  27.12  51 .9  46 .54  289.7  2744  2 .03  4.75  23 .3  29 .03  840.7  2737  5.31  11.35  6 .6  84.29  2.88  286.5  20 .6  758.0  2757  1.21  28 .66  20 .6  75 .99  277.8  33 .1  607.3  2927  13.10  27 .79  33 .1  60 .87  122 .1  27 .5  445.5  2978  7.57  12 .21  27 .5  44 .65  16 .6  300.6  3000  1.98  3.33  16 .6  30 .12  469 .8  45 .9  647 .5  2731  24 .13  47 .00  45 .9  64 .88  147.4  16 .8  608.5  2805  4.31  14 .74  16 .8  60.97  94.11  16 .1  501.1  2953  1.85  9.42  16 .1  51.01  49.48  14 .3  394 .3  2989  1.80  4.95  14 .3  39 .50  285.9  3006  0.33  1.49  8 .2  28.63  106 .3  979 .5  2896  104.9  106 .4  98 .04  524.9  24 .2  976.4  2924  17 .0  52 .49  24 .2  97 .74  294.1  17 .5  933.9  2948  2.59  34 .58  17 .5  93 .46  347 .5  19 .6  887.2  2951  3.15  34.76  19 .6  88.78  274.5  17 .1  843.1  2957  8.55  27 .44  17 .1  84 .34  33.35  14.93 2349.  8 .24  235 .0  Table Rigid Structure Stress  5.3  (continued)  94  - 100 l b L o a d a t J Shear  55 c  B.M.  am/ a  c  Rigid Force  Structure Stress  - 10 l b L o a d  atJ 5 5  Shear  B.M.  am/a  322.3  6.58  9 .20  90.4  7.24  33.3  0.68  0.95  96.0  310.1  4 .76  5.13  69 .0  6 .06  32.0  0.50  0.53  69 .3  210.7  4.29  2.52  99.7  2 .91  21.7  0.45  0 .26  101.  171.6  1.94  0.93  101.3  1.57  17.6  0.21  0.10  104.  129 .7  0.73  0.29  115.4  0.71  13.1  0.08  0 .03  124.  293.4  12.78  20.21  67.9  11.79  30.2  1.30  2 .07  67.6  295.7  0.29  7.65  33.1  10.42  30.4  0.04  0.78  32.8  277.8  5.71  7.65  67.7  7.05  28.5  0.58  0.78  67.3  275.9  2.38  2.71  51.8  4.77  28.3  0.24  0.27  50.9  274 .9  0.19  0.47  23.0  2.96  28.1  0.01  0 .04  19.9  274.4  0.54  1.15  6.6  8.65  28.2  0.07  0.12  7.0  276 .4  0.12  2.87  20.6  7.79  28.3  0.01  0 .29  20 .2  293.4  1.31  2.79  33.1  6.24  30.1  0.14  0.29  33.0  298.5  0.76  1.22  27.5  4 .57  30 .6  0.08  0.12  27 .2  300.6  0.19  0.33  16.4  3.07  30.7  0.02  0.03  14.6  273.6  2.42  4.72  46.0  6.62  27.9  0.25  0 .49  46.6  281.1  0.44  1.48  16.8  6 .22  28.7  0.05  0 .15  16.7  295.9  0.19  0.95  16.2  5.21  30 .2  0.02  0.10  16 .9  299 .15  0.18  0.50  14.3  4.03  30.5  0.02  0 .05  14.9  301.1  0.03  0.15  8.2  2.91  30.6  0.0  0.01  8.0  289 .9  10.52  23.55  106.5  9.89  29 .3  1.08  2.41  292.7  1.70  5.25  24 .1  9.87  29.6  0.17  0 .52  23.9  295.0  0.26  3.46  17.5  9 .43  29.8  0.02  0 .35  17.7  295.3  0.32  3.49  19.6  8.94  29.7  0.38  0 .36  20.0  295.8  0.85  2.74  17.1  8.43  29.7  0.08  0.27  16 .8  108.  T a b l e 5.3 Rigid Force  (continued)  95  S t r u c t u r e - 1 l b Load a t J S t r e s s Shear  5  5  Rigid  B.M.  am/ a  Force  S t r u c t u r e - 0 . 1 l b Load a t J 5 5 S t r e s s Shear  B.M.  am/a  . 0.95  4.34  0.09  0.12  87.6  0.32  1.45  0.03  0.04  82.2  0.79  4.15  0.07  0.07  71.3  0.26  1.37  0.03  0.02  76 .6  0.38  2.81  0.07  0 .04  110.  0.12  0.92  0.03  0 .01  129 .  0.20  2.20  0.04  0.01  126.  0.06  0.66  0.02  0.01  184.  0,08  1.42  0.02  0.01  201.  0.01  0 .26  0 .01  0.0  598.  1.52  3.90  0.15  0.26  65.8  0.49  1.27  0.04  0.08  61.3  1.34  3.90  0.01  0.09  30.5  0.43  1.25  0.01  0.02  25.0  0.89  3.61  0.07  0.09  63.4  0.28  1.12  0.02  0.02  53.8  0.59  3.52  0.02  0.03  43.9  0.18  1.04  0.0  0.0  24.8  0.35  3.36  0.01  0.01  41.3  0.09  0.89  0.01  0.01  1.09  3.54  0.02  0.02  10.2  0.33  1.08  0.01  0 .01  18.7  0.97  3.54  0.01  0.03  17.5  0 .29  1.06  0.01  0.01  10.2  0.77  3.72  0.02  0.04  32.9  0.23  1.09  0.01  0.01  32.6  0.56  3.76  0.01  0.01  24 .9  0 .16  1.08  0.0  0.0  19.2  0.37  3.68  0.0  U. 0  18.7  0.10  0.98  0.0  0.0  67.2  0.80  3.36  0.04  0.07  51.8  0.21  0.90  0 .02  0 .02  67 .8  0.75  3.45  0.01  0.02  16.7  0.20  0.92  0.01  0.0  16 .6  0.63  3.64  0.01  0.02  23.0  0 .17  0.98  0.01  0.01  42 .1  0.48  3.64  0.0  0.01  19 .7  0.12  0.95  0.0  0.0  35 .5  0.34  3.54  0.0  0.0  15 .0  0.08  0.83  0.0  0.0  51.9  1.08  3.20  0.14  0.29  0.20  0 .59  0 .04  0.08  1.08  3.24  0.03  0.05  21.6  0 .20  0.61  0.01  0.0  10 .6  1.02  3.22  0.01  0.04  18.8  0.18  0.57  0 .01  0.01  30.4  0.95  3.18  0.02  0.05  23.7  0.16  0 .52  0.02  0 .01  44.8  0.88  3.09  0 .02  0.02  14.3  0.12  0.43  0.02  0.0  120.  119.  180.  3.0  T a b l e 5. 3 (continued) Rigid Force  96  S t r u c t u r e - 0.01 Load B.M.  am/ a  Force  S t r e s s Shear  0.020  0. 029  80.0  0.246  1.131  77.8  0.201  S t r e s s Shear  - 0.253 1.163  S e l f Weight F o r c e s Only B.M.  am/a  0.020  0.029  79.8  1.057  0.024  0.020  78.1  0.207  1.088  0 .025  0.020  0.098  0.734  0.028  0.012 138.  0.096  0.713  0.028  0 .012  138.  0.046  0.509  0.017  0.006 210.  0.044  0.492  0 .016  0.006  214.  0.008  0.140  0.009  0.003 985.  0.007  0.126  0 .009  0.003 1094 .  0.391  1.002  0.027  0.061  59.5  0.380  0 .973  0.026  0.059  59 .3  0.337  0.983  0.007  0.018  22.9  0.327  0.953  0.007  0.017  22.5  0.214  0.866  0.010  0.018  50.0  0.207  0.839  0 .010  0.017  49.2  0.134  0.792  0.002  0.004  28.6  0 .129  0.764  0.002  0.004  29 .6  0.068  0.644  0.014  0.008 159.  0.065  0.617  0.014  0 .008  0.257  0.835  0.014  0.013  24 .6  0.248  0.808  0.014  0 .013  25.4  0.222  0.807  0.006  0.003  68.8  0.214  0.780  0.006  0 .003  6.4  0.172  0.828  0.005  0.008  32 .4  0.166  0.798  0 .005  0.007  32 .3  0.122  0.813  0.003  0.002  16.5  0 .117  0.785  0.003  0.002  15 .4  0.071  0.708  0.005  0.004  92.8  0.068  0.678  0.005  0 .004  97.0  0.156  0 .656  0.013  0.019  75.7  0.149  0 .629  0.013  0.018  77.2  0.146  0.672  0.007  0.004  16.6  0 .140  0 .644  0 .007  0 .003  16 .8  0.124  0.717  0.006  0.007  51.4  0.119  0.687  0.006  0.007  52.8  0.089  0.677  0.005  0.003  43.3  0.085  0.647  0.005  0.003  45.3  0.053  0.561  0.003  0.003  73.9  0 .051  0.531  0.003  0.003  78.1  0.112  0.331  0.036  0.060 237.  0 .102  0.302  0.034  0 .057  0.116  0.347  0 .011  0 . 001  5.0  0.106  0.318  0.011  0 .001  4.2  0.097  0.305  0.015  0.009  43.6  0.087  0 .276  0.276  0 .015  46.6  0.076  0.254  0.017  0.011  71.2  0.067  0 .224  0 .017  0.011  77.8  0.047  0.165  0.018  0 .003  35.8  0.039  0 .135  0 .018  0 .004  47.8  166 .  249.  97 e) Maximum of  Bending  the a x i a l  a  ,  M  y  this  final  2  =  the  the bending axial  little effects  Since  these  alone,  are p a r a l l e l  ratio  across  and c a n be  t h a t these  t o some v e r y  the s e c t i o n .  f o r a g i v e n member  f o r some members w h i l e a r e such  i t i s noteworthy  a s l a r g e a s 2a o r more a t one o u t e r  uniformly  percentage  give rise  i n others  changes b u t  the s e l f  high  i n the pinned  structure i s constant  i n the r i g i d  when t h e a x i a l  are considered  at  end o f t h e s t r u c t u r e a r e r e a s o n a b l y  but  d e c l i n e continuously along  50%  of the nominal value  loads  percentages.  forces alone the outer  weight  predominate a t the lower  a l l members, i t i s b y no means c o n s t a n t  case).  repeats  implies that the s t r e s s d i s t r i b u t i o n i s  While the s t r e s s for  hand  weight f o r c e s a c t i n g  the various f i g u r e s ,  being  and d e c l i n i n g The  and  this  from u n i f o r m ,  fibre  3  s t r e s s e s i n many members a r e comparable- w i t h  stresses.  added d i r e c t l y , far  "  (WIDTH) 12  external load. In comparing  that  x  .  column a t t h e e x t r e m e r i g h t  information f o r the s e l f  without  percentage  b e i n g larger o f t h e two b e n d i n g moments, WIDTH  t 1  The  as a  stress.  My = — I  g  S t r e s s , expressed  the  (to almost  N  ^  cases.  fibre  The v a l u e s constant  to less  10% i n t h e s e l f  than  weight  98 The  following  table  summarizes t h e d e f l e c t i o n s f o r  J-  t h e end j o i n t  (r  LOAD  DEFLECTION  AT  JOINT  J  (5.5)  STRUDL ' TRUSS' X  *  STRUDL Y  Y  'FRAME 0  1  rads  10000 l b .  -0 .010633  -4.696458  -4 .828454  -0 .592301  Self  -0 .000000  -0.000068  -0 .000095  -0 .000006  10000 + SW  -0 .010634  -4 .696526  -4 .828458  -0 .592306  1000 + SW  -0 .001064  -0.469714  -0 .482940  -0 .059236  100 + SW  -0 .000107  -0.047033  -0 . 048380  -0 .005929  10 + SW  -0 .000011  -0.004765  -0 .004923  -0 .000598  1.0 + SW  -0 .0000013  -0.000538  -0 .000578  -0 .000065  0.1 + SW  -0 . 0000001  -0.000115  -0 .000143  -0 .0000115  0.01 + sw  -0 .0000003  -0.000073  -0 .000100  -o .000006  Weight  * All  'X' d e f l e c t i o n s  f o r Frame a r e e f f e c t i v e l y  Zero.  BIAXIAL STRESS SYSTEMS IN JOINT AREAS  Figure  5.3 i n d i c a t e s  in  a Michell  is  shown d o t t e d and i t w i l l  overlap  framework.  i n the j o i n t  t h e arrangement o f a t y p i c a l  The t h e o r e t i c a l be s e e n t h a t  the j o i n t  shape o f t h e members there  i s considerable  a r e a , QRVW.  Even i f s e c o n d a r y s t r e s s e s of  joint  arising  from t h e r i g i d i t y  are neglected, i t i s obvious that  inside  the area  F i g u r e 5.3  Typical Joint in a Rigid  Michell  Framework  100 QRVW  the material  main  part  o f e a c h member  These Stress  i s subjected  two  Circle,  stress  a s shown  to biaxial  carries  only  systems  may  i n Figure  s t r e s s , whereas the  uniaxial stress, be  compared u s i n g  ±a. Mohr's  5.4.  Tensile  Members a)  UNIAXIAL  STRESS  Klormotl  Sh-cas  b)  BIAXIAL  Figure  5.4  In to The  members uniform  STRESS  Mohrs C i r c l e  Figure  5.4a,  i n tension  f o r U n i a x i a l and B i a x i a l  two  circles  and t h e o t h e r s  uniaxial stress  i s ±o  a r e shown, t o those  one  Stress  relating  i n compression.  and t h e c i r c l e  has a  101 radius of  The maximum s h e a r  The b i a x i a l stresses (±a). shear  state  i sindicated  i n t h e two s e t s b e i n g Mohr's  stress  circle  doubled. ness  T h i s may most e a s i l y  intensity,  system.  i f t h e maximum  a, t h e j o i n t  stress  volume must be  be done b y d o u b l i n g  the t h i c k -  locally. If  is  the  \a\ a n d t h e maximum  for a uniaxial  analysis,  a t any p o i n t i s n o t t o exceed  equal t o  o f equal but opposite  the value simple  i s thus  i n F i g u r e 5.4b,  now h a s a r a d i u s  i s twice  Thus on t h i s  stress  F i g u r e 5.3b i s e x a m i n e d , i t w i l l  be s e e n  that there  c o n s i d e r a b l e o v e r l a p o f t h e t h e o r e t i c a l members i n s i d e t h e  joint  area  QRWV.  The ends o f t h e two A members, A their  s m a l l o v e r l a p a t the bottom being  gap a b o v e .  To a f i r s t  portions o f these Similarly equal  approximation  two members e q u a l s  , .. a n d A , , a b u t , a , £)—J. aJD  equal  t h e volume o f t h e e n d t h e volume o f QRWV.  t h e volume o f t h e ends o f B . , and B , a J., io ao  t h e volume o f t h e j o i n t  joint  o f t h e members i s a b o u t t w i c e  and thus  provide uniform  equals stress  approximate!  area.  Thus t h e t o t a l volume c a l c u l a t e d dimensions  to the triangular  from  the  theoretical  t h e volume o f t h e  t h e amount o f m a t e r i a l n e c e s s a r y t o at a l l points.  T h i s volume h a s a l r e a d y  * I f t h e s t r e s s e s i n t h e two members a r e e q u a l i n s i g n , Mohr's c i r c l e w i l l r e d u c e t o a p o i n t a n d t h e s t r e s s on a l l p l a n e s w i l l be ±cr. T h i s w i l l o c c u r ' f o r c e r t a i n l o a d i n g c o n d i t i o n s a s shown i n T a b l e 5, A p p e n d i x E .  102 been c a l c u l a t e d tables  and  i n Appendix E — a s  work were c u t f r o m from  i s shown i n C h a p t e r the  a single  components o f u n i f o r m  t h i c k n e s s w o u l d have t o be the c o n d i t i o n o f u n i f o r m the  total  4—and  s t r u c t u r a l volume.  section  and  square  RW,  added o v e r  stress  reduce  corners  WV  and  stress  a t Q,  R,  and  each j o i n t  The these it  percentage  fillets  will  VQ  V  probably  sidered real shear  than  The  on  this  same  by  on  uniaxial  throughout  Its effect is. use  variable.  axial  con-  stresses.  The  t h e p r e s e n c e -of b e n d i n g  deflections  .  since  d i s c u s s i n g t h e volume  JOINTS SUBJECTED TO  were c a l c u l a t e d stress  f o r the  specifications,  This i s discussed further  structural  rounded  5 which f o r p r a c t i c a l  of the primary  i s complicated  DEFLECTION OF  provide  volume.  the d e s i g n  paragraphs  o n l y the e f f e c t  stresses.  added t o  In a d d i t i o n the  s h o u l d be  r e p r e s e n t s an u p p e r l i m i t  situation  increase  t o smooth t h e c h a n g e i n c r o s s -  i n c r e a s e the  vary with  preceding  satisfy  i n c r e a s e i n volume n e c e s s i t a t e d by  small f o r N less  The  to  equal  calculated.  i s a f u n c t i o n o f t h e number o f j o i n t s .  certainly  assembled  simultaneously  concentration.  W and  r e a s o n , which would a g a i n  frame-  t h i c k n e s s , a patch p l a t e of  E x t r a volume w o u l d a l s o have t o be a l o n g QR,  appropriate  If a  sheet of m a t e r i a l , or  volume t o t h a t e a r l i e r  fillets  the  i n Appendix  BIAXIAL  considered  and  F.  STRESS  i n Chapter  4  t h e a s s u m p t i o n t h a t t h e members were i n their  length.  When t h e  joints  103 are will  subjected  to biaxial  s t r e s s , the extension  be somewhat i n c r e a s e d  ratio,  assuming  of d i f f e r e n t increase  due t o t h e e f f e c t  that the forces  sign.  Equations  are derived  i n each  o f e a c h member  of  Poisson's  s e t o f members a r e  s p e c i f y i n g t h e amount o f t h i s  i n A p p e n d i x F, and t h e m o d i f i e d  deflec-  t i o n s were c a l c u l a t e d f o r t h e r a n g e o f s t r u c t u r e s d e s c r i b e d i n Chapter  4, S e c t i o n The  1.  results of this  c a l c u l a t i o n are'tabulated i n  A p p e n d i x F and a r e shown g r a p h i c a l l y i n F i g u r e As  m i g h t be e x p e c t e d ,  the e f f e c t  of b i a x i a l  s t r u c t u r e d e f l e c t i o n becomes more s i g n i f i c a n t o f members  increases.  This  follows  5.5.  f o r two  s t r e s s on  a s t h e number  reasons:-  2 a)  The number o f j o i n t s  is N  , and t h i s  is a  joint i  effect. b) As N i n c r e a s e s , and  criteria  stress  thus the p r o p o r t i o n  area  increases.  This  illustrates  enunciated  geometrical  the length  again  i n Chapter  shape a r e b e i n g  systems.  frameworks h a v i n g  o f e a c h member d e c r e a s e s  of the length  i n the j o i n t  the p r e d i c t i o n s of the M i c h e l l 1.  Here s t r u c t u r e s o f l i k e  compared, b u t w i t h  differing  I t c a n be s e e n t h a t t h e s t i f f n e s s biaxial  t h a t o f the systems having  s t r e s s a t the j o i n t s uniform  of the  i s less  than  stresses at a l l points.  Figure  5.5  D e f l e c t i o n o f Some M i c h e l l C a n t i l e v e r s U n i a x i a l and B i a x i a l Stresses  Due  to  105 3) S e m i - r i g i d J o i n t s The  s t r u c t u r a l members i n a M i c h e l l framework form  two s e t s o f members, c o r r e s p o n d i n g linear  to the orthogonal  coordinates of the t h e o r e t i c a l case.  a l o n g a s e t o f these members, a p p r o x i m a t i n g  curvi-  As one moves a curvilinear  c o o r d i n a t e , t h e f o r c e i n c r e a s e s i n each member, toward t h e support p o i n t s .  F o r any g i v e n member, t h e l a r g e r p o r t i o n o f the  f o r c e a c t i n g a l o n g i t i s t r a n s f e r r e d t o i t s neighbours  i n the  same s e t , w h i l e o n l y a s m a l l p r o p o r t i o n i s i n t e r c h a n g e d  with  the a d j a c e n t members o f t h e s e t s c r o s s i n g t h i s f i b r e a t each r  joint. Advantage o f t h i s f a c t c o u l d be taken when a framework.  The s e t o f members forming one f i b r e  from a s u p p o r t p o i n t c o u l d be f a b r i c a t e d as a s o l i d  assembling extending unit>  e i t h e r by c u t t i n g from a s i n g l e s h e e t , o r by w e l d i n g o f subassemblies.  The two 'fans' o f f i b r e s would then be assembled  i n p a r a l l e l p l a n e s , as i n d i c a t e d i n F i g u r e 5.6. The c o n n e c t i o n between t h e f i b r e s a t t h e j o i n t p o i n t s c o u l d be made by g l u e i n g t h e f l a t s u r f a c e s t o g e t h e r , o r by use o f a s m a l l d i a m e t e r joint point.  p i n or r i v e t placed a t the geometrical  The c o n n e c t i o n a c t s m e r e l y t o t r a n s f e r t h e  f o r c e s between s e t s , and i s n o t r e q u i r e d t o c a r r y t h e l o a d as i n t h e case o f t h e pinned  joint.  In p r a c t i c e , t o e l i m i n a t e t w i s t i n g i n t h e p l a n e o f the d r a w i n g ,  one s e t o f members s h o u l d be d u p l i c a t e d w i t h  106  Figure  5.6  Semi-rigid  Joints  i n a M i c h e l l Framework  107 h a l f the t h i c k n e s s  i n each, and p l a c e d on e i t h e r s i d e o f the  other s e t . As  a c l o s e approximation, no e x t r a m a t e r i a l  above t h a t t h e o r e t i c a l l y r e q u i r e d  i s needed  and p r e v i o u s l y c a l c u l a t e d .  I f pinned or r i v e t t e d j o i n t s a r e employed, some i n c r e a s e i n w i d t h i n t h e j o i n t area would be r e q u i r e d t o m a i n t a i n a constant  stress level.  Since  the t r a n s f e r f o r c e s a r e a f r a c -  t i o n o f the t o t a l f o r c e , t h e diameter o f the p i n o r r i v e t c o u l d be kept s m a l l and a l o c a l i n c r e a s e  i n stress  level  c o u l d perhaps be t o l e r a t e d . The  'glued'  j o i n t would be p r e f e r a b l e , r e q u i r i n g l e s s  machining and p r o v i d i n g g r e a t e r r i g i d i t y . between the two s o l u t i o n s d e s c r i b e d an e n t i r e l y s a t i s f a c t o r y d e s i g n  I t i s a compromise  e a r l i e r and should  provide  solution.  ELASTIC BUCKLING OF MEMBERS AND FRAMES  The  members comprising  i n Chapter 4 a r e e x p r e s s l y  the frameworks  designed f o r minimum volume,  Members i n compression a r e t h e r e f o r e elastic  considered  l i a b l e t o f a i l by  buckling. A r o u t i n e check was t h e r e f o r e made on each member i n EI TT  t h i s regard.  As a c r i t e r i o n t h e E u l e r c r i t i c a l  2  load, — ^ —  was c a l c u l a t e d f o r each member. I t was assumed t h a t t h e member was u n c o n s t r a i n e d  at  i t s ends and f r e e t o b u c k l e i n e i t h e r d i r e c t i o n , depending  108 whether  i t s width or thickness  joints  a t t h e end would  member  to buckling  l o w e r bound  was t h e l e s s e r .  increase  the resistance of the  so t h e c a l c u l a t e d v a l u e  to i t s c r i t i c a l  load.  compared, as a p e r c e n t a g e , w i t h  Rigid  This  represented  critical  a  load  the actual a x i a l  was  force  imposed. I n m o s t c a s e s t h e r a t i o was c o n s i d e r a b l y 100%, i m p l y i n g the  design  that  b)  was c o n f i n e d  n  d  B  loads.  the  usually  to their  members w o u l d e l i m i n a t e  the This  from a r e c t a n g u l a r  The l a t t e r  ularly  this  problem.  cross-section to a  case would  ribs,  involve  tubular  w o u l d be  practical  an i n c r e a s e o f  volume.  to their  thickness could  fine  on t h e method o f c o n s t r u c t i o n , r e d i s t r i b u t i o n o f  Many o f t h e s t r u c t u r e s proportion  small  length.  shape, o r t h e a d d i t i o n o f s t r e n g t h e n i n g  structural  N  H e r e t h e c r o s s - s e c t i o n s were v e r y  proportion  material  solutions.  A  e x a m p l e s o f members c a r r y i n g v e r y  Redesign of these c r i t i c a l Depending  long,  N-1,N '  isolated  in  buckling f o r  t o two t y p e s o f c a s e -  where members were v e r y a  resist  than  load.  Buckling  a)  the structure could  smaller  thickness.  was i n v a r i a b l y  give  rise  i n v e s t i g a t e d were v e r y  thin i n  Spans o f 10" were common , giving a ratio  to buckling  o f 40 t o 1.  o f t h e whole frame,  i f t h e l o a d were n o t a p p l i e d  exactly  while  in its  partic-  plane.  109 The  t e s t s i s o l a t e d a c a n t i l e v e r , but i n p r a c t i c e the  framework would n o r m a l l y form p a r t of a l a r g e r s t r u c t u r e w i t h s i m i l a r frameworks l o c a t e d i n p a r a l l e l o r other p l a n e s . These frames would be i n t e r c o n n e c t e d and would p r o v i d e mutual support a g a i n s t b u c k l i n g , as i s common p r a c t i c e i n the d e s i g n of b r i d g e t r u s s e s , c r a n e s , towers and so on. B u c k l i n g was thus f e l t t o be a minor problem i n the use o f M i c h e l l frameworks, p r o v i d i n g no more d i f f i c u l t y i s a t p r e s e n t encountered  i n other designs.  than  I t s incidence  may occur a t somewhat lower s t r e s s l e v e l s s i n c e a M i c h e l l framework i s by i t s n a t u r e , o f minimum volume as compared t o o t h e r s working i n the same space and s u b j e c t e d t o the same loading.  SELECTION OF JOINT TYPE AND DESIGN  The p r e c e d i n g s e c t i o n s o f t h i s chapter d e s c r i b e v a r i o u s ways i n which the j o i n t s o f a M i c h e l l s t r u c t u r e may be designed and a n a l y z e d .  The a c t u a l d e s i g n s e l e c t e d i n a  g i v e n case may w e l l be l a r g e l y determined requirements  by the f u n c t i o n a l  o f the s t r u c t u r e .  In many cases a s t r u c t u r e i s designed t o c o n t a i n a minimum volume o f m a t e r i a l .  In such cases a p e r m i s s i b l e o r  working s t r e s s i s s p e c i f i e d which i s as h i g h as i s c o n s i d e r e d s a f e f o r t h a t a p p l i c a t i o n , being determined  by the l i m i t  1  of p r o p o r t i o n a l i t y o f the m a t e r i a l , sideration of of p r a c t i c e .  1  a maximum s e t by con-  f a t i g u e o r by r e f e r e n c e  to a standard code  Since t h i s s t r e s s i s the maximum  permissible  anywhere i n the s t r u c t u r e , t h i s w i l l determine the v a l u e of the b i a x i a l s t r e s s shown i n F i g u r e the u n a x i a l  5.4b, which i s double  s t r e s s a c t i n g along the g r e a t e r  p a r t o f each  member. Two s o l u t i o n s a r e a v a i l a b l e t o the d e s i g n e r .  He  may e l e c t t o dimension the j o i n t areas i n c o n f o r m i t y w i t h the b i a x i a l s t r e s s , and reduce the c r o s s  s e c t i o n a l area o f the  members so t h a t the u n i a x i a l s t r e s s i s r a i s e d t o the same level. be  A l t e r n a t i v e l y , the member c r o s s  selected  and e x t r a m a t e r i a l  s e c t i o n s may  added i n the area o f the  j o i n t s so t h a t the s t r e s s l e v e l i s maintained constant.  The  and  a  i tis  two  approaches  matter  of  first  yield  convenience  sensibly  the same s o l u t i o n which  is  followed. However, i n o t h e r d e s i g n s i t u a t i o n s , the s t i f f n e s s of the s t r u c t u r e material  i s more important than a b s o l u t e economy o f  (see f o r example, the m i r r o r  i n Chapter 9 ) .  substrates  discussed  I t has been shown i n Chapter 1 t h a t a M i c h e l l  s t r u c t u r e has the l e a s t d e f l e c t i o n as w e l l as the l e a s t volume o f any s t r u c t u r e same space.  s u p p o r t i n g the same loads i n the  In a g i v e n case, once the geometry o f the  system i s e s t a b l i s h e d ,  the d e f l e c t i o n of the s t r u c t u r e i s  d i r e c t l y r e l a t e d t o the (uniform) s t r a i n , as e x p l a i n e d i n  0  Ill Chapter fixed,  2.  The s t r e s s  throughout the s t r u c t u r e  a s s u m i n g i t t o be c o n s t r u c t e d  may w e l l  be a t a l e v e l  minimum volume In  f r o m one m a t e r i a l , and  considerably  below t h a t  used  j o i n t s may w e l l  be o f l i t t l e  the addition  To summarize  stress  level  of extra  material  these a l t e r n a t i v e s ,  i n these  i n T a b l e 5.1.  to 'theoretical' pin-jointed  w h i c h no a l l o w a n c e has b e e n made f o r j o i n t rigid  jointed  structures  i n which e x t r a  'Net V o l u m e  1  structures i n d e s i g n and t o  material  added t o t h e j o i n t s t o m a i n t a i n a u n i f o r m  areas.  i t i s convenient  The v o l u m e s shown i n t h e c o l u m n h e a d e d equally  i n the  s i g n i f i c a n c e and c a n be t o l e r -  to r e f e r t o t h e s t r u c t u r a l volumes l i s t e d  relate  ina  design.  t h i s case the increased  ated without  i s thus  stress  has been level  throughout the s t r u c t u r e . The structures  'gross volume' f i g u r e s  with  j o i n t s designed  apply to p i n - j o i n t e d  to maintain a uniform  stress  level. If  higher  stress  l e v e l s may be t o l e r a t e d  joint  areas  could  be made f r o m a s h e e t o f u n i f o r m  structure  i n a ' d e f l e c t i o n l i m i t e d ' design the structure  volume w i l l  be less  thickness.  this table.  The e f f e c t o f t h i s r e d u c t i o n  in  the following  t a b l e which r e l a t e s  following  The  t h a n t h e 'Net Volume'  in  the  i n the  shown  i s indicated  to cantilevers  having  dimensions:-  L  -  10 i n c h e s  5"  ~  D  -  2.5 i n c h e s  Sheet t h i c k n e s s  4  -  0.25 i n c h e s  Number o f Fibres  Net Volume  Percentage Reduction  20.66  3.99  16.67  19.3  3  17.37  4.48  12.89  25.8  4  16.89  4.78  12.11  28.3  5  16 .73  4.95  11.78  29 .6  7  16.62  5.17  11.45  31.1  16 .57  5.37  11.20  32.4  It w i l l  be n o t e d  t h e number o f j o i n t s  these for  Reduced Volume  2  10  as  Overlap  structures  the i n f i n i t e  the overlap  increases.  volume  The r e d u c e d volume o f  fibre  s t r u c t u r e , but t h i s  excesses  i s only  a d d i t i o n a l saving  of m a t e r i a l  paper, since  has n o t b e e n  elsewhere i n t h i s  to designs  i n which the d e f l e c t i o n i s l i m i t e d  Where i t c a n be a p p l i e d ,  this  i t only  Michell cant or  introduced  designs.  by s e c t i o n c h a n g e s may be  i n a l l types o f design  structures.  in a specific  eliminated  This  to specified-  i s assumed i n t h e above d i s c u s s i o n t h a t  concentrations These o c c u r  applies  increases the  advantage o f M i c h e l l s t r u c t u r e s over o t h e r It  achieved  i n stress intensity.  considered  values.  increases  i s l e s s t h a n t h e t h e o r e t i c a l minimum v o l u m e  because o f the l o c a l This  that  they could  become  a slight  signifi-  p r e s u m a b l y be r e d u c e d  by l o c a l m o d i f i c a t i o n o f s e c t i o n  usually involves only  ignored.  and a r e n o t u n i q u e t o  I f such c o n c e n t r a t i o n s case,  stress  increase  profile.  i n volume.  113  CHAPTER 6 COMPARABLE  In of  the previous  the M i c h e l l  A be  occupying  I t has been  t h e same s p a c e and c a r r y -  load.  series of structures  of various  designs w i l l  e x a m i n e d t o e s t a b l i s h a more q u a n t i t a t i v e b a s i s  assertion. so  the t h e o r e t i c a l s u p e r i o r i t y  a p p r o x i m a t i o n t o t h e t h e o r e t i c a l optimum  minimum volume s t r u c t u r e t h e same  chapters,  s t r u c t u r e has been e s t a b l i s h e d .  shown t o be a c l o s e  ing  STRUCTURES  An i n f i n i t e  range o f d e s i g n  any s e l e c t i o n o f s p e c i f i c  values  for this  parameters e x i s t  can serve  only  as a  d e m o n s t r a t i o n o f t h e s e n e a r optimum p r o p e r t i e s . A  s i m p l e c a n t i l e v e r was s e l e c t e d  structures The  of this  t y p e have a l r e a d y  since  i n Figure  6.1.  support along  The s t r u c t u r e  Where a p p r o p r i a t e , tioned  i s 'built-in'  the y a x i s , the support area  a b o u t t h e a x i s , and h a v i n g  a t the outer  Michell intensively.  system a r e  A l o a d o f 100 pounds i s e x e r t e d  at r i g h t angles t o the x a x i s , a t a d i s t a n c e from t h e o r i g i n .  ,  been s t u d i e d  a r b i t r a r y f i x e d parameters o f the loading  illustrated  now  of ten inches to a  being  a maximum w i d t h o f 2.5  f i x e d support points edges o f t h i s  area.  rigid  symmetrical inches.  A and B a r e p o s i -  114  Figure  All  6.1  Comparative  Structures  s t r u c t u r e s are assumed t o be made from the same  m a t e r i a l , f o r which:Modulus o f E l a s t i c i t y  E  =  300,000 p . s . i .  Maximum S t r e s s  a  =  300 p . s . i .  The space o c c u p i e d by the s t r u c t u r e i s o t h e r w i s e unrestricted.  A l l s t r u c t u r e s are compared, as r e g a r d s volume,  w i t h t h a t o f the t h e o r e t i c a l optimum i n f i n i t e number of f i b r e s . structures  framework, having an  The d e f l e c t i o n of the comparison  i s s p e c i f i e d where p o s s i b l e t o be i d e n t i c a l w i t h  1 X 5  t h a t of the f i v e f i b r e M i c h e l l s t r u c t u r e  (0.050 " ) .  cases the s p e c i f i e d d e f l e c t i o n and s t r e s s are  In some  incompatible.  T a b l e 6.1 compares the r e s u l t s o b t a i n e d from these calculations.  The v a r i o u s  s t r u c t u r e s are d e s c r i b e d  below, w h i l e f u l l e r d e t a i l s and the a s s o c i a t e d  briefly  calculations  may be found i n Appendix G.  STRUCTURE AND TYPE  VOLUME cu.in.  A  I n f i n i t e Fibre Array  16.4826  100.0  B  Five Fibre Michell  16.7338  101.52  C  Three F i b r e M i c h e l l  17.3714  105.39  D  Warren T r u s s  22.8191  138.44  E  2 bar C a n t i l e v e r  43.9233  266.48  Section  50.1000  303.96  Plate  50.1000  303.96  52.7911  320 .28  86 .9854  527 .74  150.3000  911.87  F  • Parabolic  G  Triangular  H  Cylindrical  J  I-beam C a n t i l e v e r  K  Rectangular  Cantilever section  DESCRIPTION OF COMPARATIVE  A.  Infinite  This all  1.19,  STRUCTURES  Array  i s t h e t h e o r e t i c a l optimum s t r u c t u r e  the others  included  Fibre  %  fibre  is listed  a r e compared.  I t h a s an ~ e q u a l  against  t o 4.0,  a n g l e o f 74.26°, and i t s v o l u m e , f r o m i n Table  3 o f A p p e n d i x E a s 16.4826  which  an  equation cu.in.  116 B and C.  5 and 3 F i b r e M i c h e l l  These s t r u c t u r e s Section  Rigid Michell  these volumes c l o s e Chapter  5, S e c t i o n The  Warren It  frameworks c o u l d  t o t h e optimum v a l u e ,  f o r the following  1,  be made, h a v i n g  as d e s c r i b e d i n  fibre  frame 0.0501", i s  structures.  Truss  seems l o g i c a l  that  t h e f o r m o f s t r u c t u r e most  likely  f r a m e , w o u l d have a  a r r a n g e m e n t o f members.  s e l e c t i o n o f one d e s i g n  trary.  i n Table  4,  ,  T h e r e are. many d e s i g n s o f t r u s s the  i n Chapter  2.  t o compete e f f e c t i v e l y w i t h a M i c h e l l similar  fully  are outlined  d e f l e c t i o n of the f i v e  taken as standard  D.  were d e s c r i b e d  1, and t h e i r p r o p e r t i e s  Appendix E.  Cantilevers  The t r u s s  triangular  selected  panels,  i n regular  use,  f o r c o m p a r i s o n was l a r g e l y consists  and arbi-  o f a number o f e q u i l a t e r a l  the d e t a i l s o f which a r e given  i n Appendix  G. Fortituously, the  vertical  height  i f s i x panels  of the truss  a r e used  t o span t e n i n c h e s ,  i s 2.4744" w h i c h c l o s e l y  approximates the s p e c i f i e d spacing  of the support p o i n t s ,  inches. The  volume o f t h e t r u s s  i s 22.8191 c u . i n .  downward d e f l e c t i o n i s 0.06 39 i n .  and i t s  2.5  117  E.  Two Bar C a n t i l e v e r T h i s i s t h e s i m p l e s t multi-member s t r u c t u r e t h a t can  be used t o c a r r y t h e s p e c i f i e d l o a d i n g , and c o u l d be r e g a r d e d as a p r e l i m i n a r y s t e p t o t h e t r u e M i c h e l l framework. an optimum Maxwell  It is  frame, s i n c e t h e s t r e s s i s everywhere  u n i f o r m , b u t does n o t s a t i s f y t h e M i c h e l l c r i t e r i a . . The F i g u r e 6.1.  two b a r s a r e s u p p o r t e d a t A and B and j o i n e d a t L, T h e i r c r o s s - s e c t i o n a l a r e a i s f i x e d , t h e shape  may be o f any c o n v e n i e n t p a t t e r n p r o v i d e d  b u c k l i n g i s con-  sidered Length o f members  -  C r o s s - s e c t i o n a l area  -  (1.4760 i n , s q u a r e  10.0778" 2.179 s q . i n .  o r 1.666  Volume  -  in.diameter)  43.9233 c u . i n .  Note t h a t t h e d e f l e c t i o n i s a f u n c t i o n o f t h e geometry and  thus l i m i t s t h e s t r e s s l e v e l t o 184.98 p . s . i . ^ l e s s t h a n the  s p e c i f i e d 300 p . s . i .  F.  C a n t i l e v e r of Parabolic  Section  This i s a c a n t i l e v e r of constant d e p t h , so p r o p o r t i o n e d  width but of varying  that the s t r e s s i n the outer f i b r e s  i s everywhere 300 p . s . i .  The depth i s zero a t L and a  maximum a t t h e p l a n e AB. I t s major dimensions  are:-  118 Width  -  2.824 i n .  Root Depth  -  2.661 i n .  Volume  -  50.1000 c u . i n .  G.  (constant)  Triangular Plate T h i s i s an a l t e r n a t i v e t o d e s i g n F which aims a t the  same r e s u l t — t o v a r y the c r o s s - s e c t i o n a l p r o p o r t i o n s along the c a n t i l e v e r so t h a t the s t r e s s i n the o u t e r f i b r e i s maintained  constant.  In t h i s case the depth o f the c a n t i l e v e r i s maintained c o n s t a n t w h i l e the width  i n c r e a s e s u n i f o r m l y from L toward  the r o o t . The major dimensions a r e : Depth  (constant)  Width a t r o o t  (max.)  Volume  G are i d e n t i c a l .  1.9960 i n .  -  5.0200 i n .  - 50.1000 c u . i n .  I t w i l l be noted and  -  t h a t the volumes o f s t r u c t u r e s F  T h i s a r i s e s s i n c e they are both  to s a t i s f y the same c r i t e r i a o f uniform s t r e s s i n the outer  fibre.  designed  119 H.J.K. CANTILEVERS OF UNIFORM  The  last  three  structures  uniform cross-sections  and  much o f t h e i r m a t e r i a l  i s stressed  by  a l l the m a t e r i a l  CROSS-SECTION  t o be c o n s i d e r e d  are the l e a s t e f f i c i e n t , below t h e l e v e l  a circular As  in this  c a n be s e e n i n A p p e n d i x G, a c i r c u l a r  cross-section  I f the d e f l e c t i o n i s maintained, the  i s 584.5 p . s . i .  c u . i n . , which is- quoted i s maintained  category:-  meet t h e s p e c i f i e d r e q u i r e m e n t s o f  and s t r e s s .  maximum s t r e s s  attained  c r o s s - s e c t i o n of diameter d  cannot simultaneously deflection  since  i n a M i c h e l l framework.  Three simple cases are considered  a)  have  and t h e v o l u m e i s 52.7911  i n the table.  a t 300 p . s . i . ,  I f t h e maximum  stress  t h e v o l u m e i s r e d u c e d t o 13.907  c u . i n . , b u t t h e : d e f l e c t i o n becomes  0.722".  b) an I beam This  i s more amenable t o c a l c u l a t i o n t h a n t h e c y l i n d e r ,  s i n c e d e p t h and w i d t h may be v a r i e d of and  s e c t i o n s may be c o n s i d e r e d ; flanges  independently.  i f the thickness  A variety  o f t h e web  i s t a k e n as 0.25" t o c o n f o r m w i t h t h e M i c h e l l  frameworks, the f o l l o w i n g d i m e n s i o n s a r e  Width o f f l a n g e s  -  D e p t h o f I-beam  -  Volume  -  16.982 i n . 1.33067 i n . 86.9854 i n  3  .  obtained:-  120 c)  a rectangular A unique  1.33067 i n c h e s  beam  s o l u t i o n may be o b t a i n e d  d e e p ' b y 11.295 i n c h e s  by u s i n g  wide.  a beam  T h i s beam h a s  a volume o f 150.3 c u . i n . These l a s t ity  of a Michell  cases w e l l  framework.  illustrate  the b a s i c  superior-  121  CHAPTER 7 MANUFACTURE AND TESTING OF MODELS  The  p r e v i o u s c h a p t e r s have d e s c r i b e d providing  and d i s c u s s e d  a range o f M i c h e l l  frameworks,  close  to the t h e o r e t i c a l  optimum minimum volume s t r u c t u r e .  d e s i g n s were t h e m s e l v e s  theoretical,  being  from m a t h e m a t i c a l  obtained e n t i r e l y  approximations These  the v a l u e s quoted computation.  Some m o d e l s were t h e r e f o r e made i n c o n f o r m i t y w i t h the d e s i g n requirements e s t a b l i s h e d  i n the previous chapters.  T h e s e were made f r o m a b i r e f r i n g e n t  transparent p l a s t i c ,  Columbia  Resin  polariscope  CR39, and were t h e n examined u n d e r l o a d i n a  t o determine  the i n t e r n a l  stress  distribution.  The m o d e l s were c u t f r o m ~ " t h i c k p o l i s h e d supplied were  by t h e m a n u f a c t u r e r s , t h e H o m a l i t e C o r p o r a t i o n , , a n d  edge m i l l e d  coolant,  a t low r a t e s  to minimize  was  performed  and  details  examining  of feed  edge e f f e c t s  on a n u m e r i c a l l y  o f t h e programming  Time edge e f f e c t s  weight  sheet  and w i t h  ample  due t o h e a t i n g .  controlled milling  This  machine  a r e g i v e n i n Appendix  i n the p l a s t i c  t h e m o d e l a s soon a s p o s s i b l e  l o a d i n g was u s e d a n d t h e f r i n g e  work  H.  were m i n i m i z e d b y after  cutting.  Dead  p a t t e r n s were r e c o r d e d  photographically. The (flourescent izer,  p o l a r i s c o p e used a d i f f u s e white t u b e s ) and employed  light  source  twelve inch diameter  a n a l y z e r and q u a r t e r wave p l a t e s .  polar-  122 The  various  m o d e l s t e s t e d w i l l n o w be  5 Fibre Pin-jointed  This the  structure  f o l l o w i n g major  Cantilever  i s illustrated  i n Figure  7.1, and h a s  dimensions:-  Number o f f i b r e s  N - 5  Span  L - 20.0 i n .  Support  described.  spacing  D -  5.0 i n .  L/D = 4 Members made f r o m s h e e t j ' t h i c k 1  Pin  joints This  other  - p i n diameter  s t r u c t u r e was made w i t h  models t o p r o v i d e  extra  The  inch  joint  clearance  member  were d r i l l e d  used  of j " dia. steel  to eliminate  thickness  of four,  The members  two p a r a l l e l  t o form each  member.  and t h e n reamed t o a t i g h t  f i t to minimize play.  from l e n g t h s duplicated  i n sets  t h i c k plates being  holes  t w i c e t h e span o f t h e  space f o r assembly.  were g r o u n d t o w i d t h and made quarter-  .  The j o i n t  drill  rod.  p i n s were made The members were  t w i s t i n g a t the j o i n t s - - t h e  thus being  h a l f an i n c h .  :  A typical  total joint  t h u s h a d e i g h t members m e e t i n g a t i t and a s s e m b l y became quite  complex.  aluminum p l a t e s  The f a n members were r e p l a c e d t o ease a s s e m b l y — t h e  w o u l d have b e e n v e r y members o v e r l a p p e d  difficult  by  stresses  t o measure s i n c e  considerably.  solid  i n this  area  the various  123  Figure  7.2A  Stress patterns i n Pinned M i c h e l l C a r r y i n g 110 l b . I n n e r E n d  Cantilever  124 MEMBER S I Z E S IN 5 FIBRE PINNED  Member  A  Width  B  A  31' 13  A  41' 14  A  51' 15  B  B  B  A  22' 22  A  32' 23  Member  A  42' 24  A  52' 25  A  23' 32  1.300  A  33' 33  1.026  A  43' 34  1.583  A  53' 35  A  24' 42  A  34' 43  1.168  A  44' 44  0.9 81  A  54' 45  1.620  21' 12  B  B  CANTILEVER  Width 0.815  B  1.502  B  0.798  B  0.707  B  0.624  B  1.425  B  0.500  B  0.474  B  0.450  B  1.352  B  A l l members a r e i n d u p l i c a t e , e a c h one q u a r t e r i n c h t h i c k . Members n o t l i s t e d i n c l u d e d i n s o l i d m e t a l f a n s .  Fringe in  Figure  m o d e l was  of the  d i a m e t e r o f 12 If are  s e e n i n t h e m o d e l u n d e r l o a d a r e shown  7.2.  The position  patterns  t o o l a r g e t o be s e e n c o m p l e t e l y  p o l a r i s c o p e whose  components h a d a  i n one viewing  inches.  t h e members f o r m i n g  each f i b r e ,  or chain  of  links,  e x a m i n e d i t w i l l be s e e n t h a t t h e c o l o u r r e m a i n s f a i r l y  uniform  along  indicates  them a l t h o u g h  i tdiffers  t h a t the s t r e s s l e v e l  each f i b r e .  Numerically,  between f i b r e s .  remains f a i r l y constant  This along  the d i f f e r e n c e i n s t r e s s l e v e l s i s  125  I Figure  not  7.2B  S t r e s s p a t t e r n s i n Pinned M i c h e l l C a n t i l e v e r C a r r y i n g 110 l b . l o a d . O u t e r end  l a r g e since the blue-purple  members c o r r e s p o n d s  s e e n i n some o f t h e i n n e r  t o the f i r s t  fringe while  p r o m i n e n t i n o t h e r members r e p r e s e n t s level  [ approximately  155 p . s . i .  may be due t o c o m p a r a t i v e l y load not being The ent.  perpendicular  against  structure.  180 p . s . i . ] .  around  the pins  This the  are promin-  i s an i m p r a c t i c a l way  The p u r p o s e o f t h i s m o d e l was  largely  to demonstrate t h a t , w i t h i n l i m i t s ,  uniform  along  each f i b r e  stress  to the c a n t i l e v e r a x i s .  stress concentrations  a real  85% o f t h i s  small e r r o r s i n alignment,  As has a l r e a d y been s t a t e d t h i s  to b u i l d  about  the yellow  t h e s t r e s s was  and g e n e r a l l y c o n f o r m s w i t h t h e  126 predicted figures,  levels.  the s t r e s s should  Solid  sheet  of the three  of i " p l a s t i c  fibre  span was 10.0", s u p p o r t seen when s u b j e c t e d Figure  170 p . s . i .  was c u t t o t h e e x t e r i o r  cantilever described spacing  t o a twenty  2.5", =r = 4. f i v e pound  below. Stress  Its patterns  l o a d a r e seen i n  7.3.  Figure  The lever  have b e e n u n i f o r m l y  i n the  Cantilever  A solid shape  A t 110 l b s . t h e l o a d i n g u s e d  7.3  ' S o l i d ' C a n t i l e v e r - Load  b e n d i n g moment e x e r t e d  increases  linearly  from l e f t  25 l b .  b y t h e l o a d on t h e c a n t i to right.  As t h e s e c t i o n  127 b r o a d e n s t h e s e c o n d moment o f a r e a  increases rapidly  maximum a t t h e w i d e s t  point).  thus d e c l i n e s t e a d i l y  from t h e r e g i o n  widest  increase again  p o i n t and t h e n  r e a c t i o n s which a r e j u s t central  area  distribution  of  10 i n c h e s  Figure  7.4  outside  i s f a r from  will  of the load to the toward  the concentrated  the p i c t u r e area.  i n d i c a t e s a r e g i o n o f low s t r e s s .  Cantilever with  This  The maximum s t r e s s  (to a  The b l a c k  The s t r e s s  uniform.  Lightening  Holes  c a n t i l e v e r i s shown i n F i g u r e and an ^- r a t i o  7.4  having  a span  o f 4.  C a n t i l e v e r With L i g h t e n i n g Holes i n Arrangement A p p r o x i m a t i n g t o a M i c h e l l C a n t i l e v e r - Load 20 l b .  128  The  e x t e r i o r shape i s s i m i l a r t o t h a t of the  f i b r e c a n t i l e v e r d e s c r i b e d below.  The  five  spaces between the  members were approximated by a p a t t e r n of d r i l l e d h o l e s various sizes.  T h i s may  be regarded  as a crude  of  approximation  t o a M i c h e l l framework and as an example of a common method of w e i g h t r e d u c t i o n of s t r u c t u r e s .  ,  I f the spaces between the h o l e s are c o n s i d e r e d  as  e q u i v a l e n t t o the f i b r e s i n a M i c h e l l c a n t i l e v e r , F i g u r e i n d i c a t e s c l e a r l y t h a t the s t r e s s e s are f a i r l y u n i f o r m the  ' f i b r e s ' i n compression r a d i a t i n g from the lower  point.  The  This discrepancy  darker  t o the  fact  i n the body of the c a n t i l e v e r  i n f a c t much below t h a t which i t c o u l d have  safely.  support  stress level.  i s b e l i e v e d t o be due  t h a t the s t r e s s l e v e l a c h i e v e d was  along  c o l o u r of the t e n s i l e f i b r e s i s a l t o g e t h e r  and i n d i c a t e s a somewhat lower  7.4  supported  T h i s i s s i m i l a r t o the case o f the s o l i d c a n t i l e v e r  where s t r e s s l e v e l was  s i m i l a r l y low.  Unfortunately,  the  s u p p o r t h o l e s were p l a c e d r e l a t i v e l y c l o s e t o the edge of the' s t r u c t u r e and caused a s t r e s s c o n c e n t r a t i o n , the  outer  edge of which can be seen a t the upper r i g h t of F i g u r e  7.4.  The model f r a c t u r e d i n t h i s a r e a when a l o a d of 25 l b . was a p p l i e d and no f u r t h e r r e s u l t s c o u l d be The  obtained.  s i g n i f i c a n c e o f t h i s specimen l i e s i n the compar-  i s o n w i t h F i g u r e 7.3.  Although  a s u b s t a n t i a l amount of  the  t o t a l m a t e r i a l has been removed the g e n e r a l s t r e s s l e v e l h a r d l y been changed.  T h i s i n d i c a t e s a more e f f i c i e n t use  has of  129 the  material  as m i g h t be  Five Fibre Rigid  predicted  five  f i b r e s and  It  i s s i m i l a r t o one  4,  Section  of  the  of  the  10"  Michell w i t h an  somewhat a r b i t r a r i l y  generally  ~ ratio  be  decided  0.100".  fragile.  On  the  s e c t i o n s would minimize the  that  the  I t was  during  other  the  4.  changed. width  felt  cutting  hand, the  e f f e c t of  of  i n Chapter  t h e member w i d t h s have b e e n  members t h a n t h i s w o u l d d i s t o r t  w o u l d be  framework,  structures described  n a r r o w e s t member s h o u l d  thinner  fine  a rigid  a span o f  1 except that  I t was  criteria.  Cantilever  T h i s model r e p r e s e n t s having  from the M i c h e l l  that and  use  joint  of  stiff-  ness . The and  dimensions of  i t s general  the  proportions  may  be  of  s t r e s s t h r o u g h a l l t h e members i s e v i d e n t  as  are  Figure is  7.5b  still  shows t h a t  fairly  members h a v i n g remainder.  biaxial the  i n Appendix  seen i n F i g u r e  i s subjected  higher  l b . load.  listed  it  the  t o a 15  model a r e  The  7.5a,  in this  of  a somewhat l o w e r l e v e l  the of  lower  picture,  the  d i s t r i b u t i o n at a load of a few  where  uniform d i s t r i b u t i o n  s t r e s s l e v e l s i n each of  uniform, only  H  joints.  27  lb.  (compression)  s t r e s s than  the  I a)  15 l b . l o a d  I b) Figure  7.5  This the  effect  stated  Five  27 l b . l o a d  Fibre  figure also  Rigid Michell  shows i n many o f t h e i n n e r  o f the secondary bending  i n Chapter  Cantilever  5, t h e e f f e c t  moments.  members  As h a s b e e n  of these bending  moments i s  131 to  increase  it  a t the other.  Figure  t h e s t r e s s a t one s i d e  7.6a  Joint J  3  With  3  o f t h e member and d e c r e a s e  27 l b . L o a d — F i v e F i b r e  Michell  Cantilever  This  i s well  shown i n F i g u r e  view o f j o i n t  ^23'  colour  the j o i n t .  across  A  ^  f°  u r  enlarged of  The A members a r e i n t e n s i o n and  B members a r e i n c o m p r e s s i o n  are  i n s u c h d i r e c t i o n s as t o r o t a t e axis.  i s an  members e x h i b i t a g r a d a t i o n  the  horizontal  7.6 w h i c h  and t h e b e n d i n g  stresses  t h e members t o w a r d t h e  132 This  i s shown d i a g r a m a t i c a l l y  the  axial  the  stress level w i l l  at  and b e n d i n g  stresses  Figure  This colours  a r e added  be h i g h e r  the corresponding points  7.6b  at points  7.6b.  i twill  When  be s e e n  that  A, B. C and D t h a n  E , F, G a n d H.  Stresses  i s substantiated  which reach higher  i n Figure  at Joint J ^  by t h e p h o t o e l a s t i c  fringe  l e v e l s a t the former s e t of  points. The  general  from the j o i n t purple  fringe.  1.5 f r i n g e s and  stress level  i s approximately :  i n a l l f o u r members, 350 p . s . i . ,  i n d i c a t e d by t h e  A t p o i n t s A, B and D a f r i n g e l e v e l  ( c . 500 p . s . i . )  G, t h e s t r e s s l e v e l  remote  of about  i s a t t a i n e d , w h i l e a t B, E , F  i s reduced  to the yellow  c o r r e s p o n d s t o a b o u t 0.8 f r i n g e s o r a l i t t l e  which  l e s s t h a n 300,  133 p.s.i.  The c l e a r l y d e f i n e d  j o i n t area and o f f s h o o t s  first  f r i n g e e n c i r c l e s the  from i t can be seen a t A, B and  D s t r e t c h i n g some d i s t a n c e  away along the member.  c e n t r e o f the j o i n t the pink n e a r i n g the second can  j u s t be seen.  fringes  fringe  The s t r e s s l e v e l here i s about 1.6  (say 600 p . s . i . ) . These f i g u r e s agree w i t h i n  stresses  predicted  about 10% w i t h t h e  by t h e STRUDL c a l c u l a t i o n s shown on  T a b l e 5.1 on page 94 when c o r r e c t e d in  At the  loads.  f o r the d i f f e r e n c e  The h i g h e r f i g u r e s from the p h o t o e l a s t i c  may r e p r e s e n t a d i f f e r e n c e between batches o f o p t i c a l c o e f f i c i e n t f o r CR39.  analysis  the s t r e s s  T h i s was assumed t o be  90 p . s . i . / f r i n g e / i n c h as quoted by manufacturers, b u t some tolerance The full  i s permitted. j o i n t i s shown about t h r e e and one h a l f times  s i z e i n t h i s p i c t u r e , which shows many s m a l l  along t h e edges.  chips  These seem t o be u n a v o i d a b l e when work-  ing with t h i s b r i t t l e material.  I t i s a l s o evident  that  an e r r o r has been made i n c u t t i n g the s i d e o f member B23. The  c u t t e r has been advanced a l i t t l e t o o f a r and has  notched the s i d e o f member A32. small.  The e f f e c t o f t h i s seems  Three F i b r e  Rigid  Cantilevers  Two m o d e l s were made o f t h e t h r e e described ten  i n Section  inches  modified  1 of Chapter  and an ^- r a t i o  as f o r t h e f i v e  joints  with or  a  3  fibre  of i ^ - " radius. . d i a . cutter  s i x times that These  in  the joint  cantilever.  The  t o produce  the f i l l e t  i n the f i r s t  m o d e l s were l o a d e d  area.  Each had a span o f being first  f i l l e t s at  T h e s e c o n d m o d e l was p r o d u c e d  so t h a t  to determine the e f f e c t  cantilever  o f 4, t h e member w i d t h s  m o d e l was c u t w i t h ^ " d i a . c u t t e r the  4.  fibre  radius  3 was -g-"  model. similarly  of f i l l e t  radius  and  examined  on s t r e s s  level  a) Model With g-" Radius  Fillet  F i g u r e 7 . 7 shows t h i s model s u b j e c t e d t o a l o a d o f 25 l b s .  The s t r e s s d i s t r i b u t i o n  i n t h e members i s q u i t e  uniform a t approximately the f i r s t f r i n g e  Figure 7 . 7  (350 p . s . i . ) .  25 l b . Load on Three F i b r e M i c h e l l C a n t i l e v e r With 3 / 8 " Radius F i l l e t s  S t r e s s e s i n t h e v i c i n i t y o f t y p i c a l j o i n t s may be examined i n F i g u r e 7 . 8 which shows j o i n t s twice f u l l  &  an  32  J  a  b°  u t  size.  The p u r p l e f i r s t f r i n g e may be seen i n b o t h  joint  areas and t h e g e n e r a l s t r e s s p a t t e r n i s s i m i l a r t o t h a t seen i n the f i v e f i b r e case. middle o f J  00  The maximum s t r e s s l e v e l i n t h e  i s a p p r o x i m a t e l y e q u a l t o t h a t i n J o i n t J.,-. i n  a)  Joint  b) J o i n t  Figure  7.8  J_„  J  2  2  —  —  27 l b .  27 l b .  J o i n t D e t a i l i n Three F i b r e W i t h 3/8" R a d i u s F i l l e t s  Michell  Cantilever  a) M o d e l  With  Radius  Fillet  Radius  Fillet  I  b) M o d e l  W i t h =- " 16 r  Three F i b r e M i c h e l l C a n t i l e v e r s  - 20 l b .  F i g u r e 7.6a.  At both j o i n t s J ^  an<  -* 22 J  a n  a  r  e  a  °^  s t r e s s i s seen i n the c e n t r e of the l a r g e r a d i u s angled  fillets,  suggesting  zero acute  t h a t the e x t r a m a t e r i a l does not  have much e f f e c t i n t r a n s f e r r i n g the f o r c e s between members. In j o i n t J  3  2  the obtuse comer on the outer s u r f a c e  i c a l l y w i l l be  subjected  to zero s t r e s s .  The  theoret-  colour  f r i n g e s i n d i c a t e the s t r e s s l e v e l i s d e c r e a s i n g r a p i d l y toward the p o i n t but the r e s o l u t i o n i s i n s u f f i c i e n t t o the b l a c k zero  fringe.  F i g u r e 7.9 at  j o i n t ^22'  ^  i n d i c a t e s the e f f e c t of f i l l e t n e  the l a r g e r f i l l e t a first  - -'-'' areas  w  rl  :e  to low  stress.  i f these areas were removed  shape of the model having The  radius  seen on the model w i t h  r a d i u s are s u b j e c t e d  approximation,  be o b t a i n e d .  see  As  the  the s m a l l e r f i l l e t r a d i u s would  large f i l l e t  seems thus t o have i n t r o -  duced more m a t e r i a l which serves  little  or no u s e f u l  purpose. The maximum s t r e s s i n the j o i n t area i s somewhat reduced i n the l a r g e f i l l e t r a d i u s model.  b)  Model With ~16" Two  was in  Radius F i l l e t s  models were made of t h i s s t r u c t u r e .  p o o r l y cut and had many c h i p s and the a r e a of the  joints.  The  an i r r e g u l a r  first finish  b) Figure  7.10  B u c k l e d Shape  Three F i b r e M i c h e l l C a n t i l e v e r  - 25  l b . Load  139 It buckling its  was,  at  25  buckled  however, l b s .  shape,  Complete vided  by  caused  the  by  are  v e r t i c a l  to  a  bucket  due  plane.  i n i t i a t e  D i s t o r t i o n in  Figure  7.10a  are  c l e a r l y  parts  the  stresses  Figure ing  by  7.10b  along  model  only  the load  from  was  the  shape  no  by  the  The  buckling  not  being  applied  model.  individual with  v i s i b l e were  of  i n  and  support was  set  by  This  members  the  very  some  probably  pro-  probably  i n i t i a l l y  adding  could  the  model.  of  lead  have  and  c l e a r l y  shot  swung  bending the  that  be  model. the  one  twisting,  seen  by  I t  l i t t l e  In  other  fringes  into  i s  seen  stresses  blurred  I t  damage.  unloaded  of  high  source  degree  can  high  areas so  the  permanent  when  point,  collapse.  of  plane  this  by  7.10.  averted  model  f a i l e d  remarkable  another. look-  that  returned  to  r e s i d u a l  stress  the  i t s could  observed.  and  7.12. l b .  with The  to  indicates  the  The  15  was  but  at  Figure  non-monochromatic  suffered  o r i g i n a l be  a  i n  tested  elements.  together  which  produced  pattern  The  suspended  to  stress  collapse  twisting  a  and  shown  polariscope  in  sideways  The  loaded  and  the  Figure  fibre 7.7.  model  In  the  f i r s t  25  l b .  may  five  pattern  three  second  fibre  case the  model  with  seen  figure,  be  around  i s  under  shown  joints large  i s  i n  Figures  s t r e s s .patterns  compared. as  load  The i n  Figure  quite  radius  general 7.5  loads  of  s i m i l a r i t y Is  d i f f e r e n t  f i l l e t s ,  for  7.11  obvious.  from  shown  i n  the  I  140  142 Figure pattern  The a n g l e can  to the  be  in stress  seen c l e a r l y  i s not  stresses  n  left  m a n u f a c t u r e and  side  in  )  l  e  colour,  the  outer the  obtuse  zero  black  Values of stated  i n the  fringe  orders.  across  a  r  e  the  s u c h as  e a c h member  evident.  The  bend-  to reduce the  s t r e s s on  the  lower  angle side  of  of not  tested  f o r s e v e r a l days a f t e r  e x h i b i t s c l e a r time-edge e f f e c t s .  CALIBRATION OF  FRINGES IN  TERMS OF  STRESS LEVEL  s t r e s s i n members and  preceding  work by  joints  reference  T h e s e were d e t e r m i n e d by  specimens s u b j e c t e d  dimensions of  mainly  T  level.  at  i s also clearly  and  T h i s model was  tapered  """ d e t a i l .  a l t h o u g h once a g a i n  between them, t h u s r e d u c i n g the  level  variation in stress level  to bending  and  J^2'  resolved.  i n g moments on A^^  A^  joint,  difference in stress  reduction  The due  shows one  i s s i m i l a r i n each p i c t u r e d i f f e r i n g  corresponding  fringe  7.12  to  have b e e n  the  associated  c a l i b r a t i o n of  t o known d e a d w e i g h t l o a d s .  these c a l i b r a t i o n pieces  are  shown i n  The  Figure  7.13. T h e s e s p e c i m e n s were c u t thick and  s h e e t o f CR39 p l a s t i c  were h a n d l e d and  stored  from the  as were t h e i n the  same q u a r t e r  inch  model c a n t i l e v e r s  same manner as  the  actual  143  Figure  7.13  models  tested.  iscope  and  stress  i n  by  simple  The  loaded the  models  to  at  hung  the  any  v e r t i c a l l y  desired  plane  f o r C a l i b r a t i o n of  fringe  could  i n  the  polar-  patterns.  then  be  The  deduced  calculation.  associated  7.14  shows  stress  Error  a  sample  specimen  under  stress  i n t e n s i t i e s  load  and  levels  i n evaluating  l i k e l y  i f the  model  can  viewed  simultaneously,  be  Specimens Material  were  obtain  specimen  Figure the  Tensile Test Birefringent  under  test  and  the  i s  least  c a l i b r a t i o n  so  that  colours  instance  this  was  may  specimen be  compared  d i r e c t l y . In to  the  the  physical  present  limitations  i n  the  impossible  apparatus.  The  due  fringe  144  I  F i g u r e 7.14  F r i n g e Orders i n a Loaded Specimen  p a t t e r n s were r e c o r d e d ing c o n d i t i o n s . compared.  The  The  on Kodachrome I I under c o n s t a n t  light-  r e s u l t i n g t r a n s p a r e n c i e s were then  c o l o u r p r i n t s i n t h i s paper d i f f e r t o some  degree i n the q u a l i t y of t h e i r c o l o u r r e p r o d u c t i o n so t h a t f r i n g e c o l o u r s may  vary s l i g h t l y .  The  s t a t e m e n t s i n the  paper are based on comparison of the o r i g i n a l  transparencies.  The p h o t o e l a s t i c i n v e s t i g a t i o n o f p i n - j o i n t e d and r i g i d s t r u c t u r e s has y i e l d e d r e s u l t s i n s u b s t a n t i a l agreement w i t h those p r e d i c t e d i n C h a p t e r s 4 and  5.  145  CHAPTER 8 TOWER FOR HIGH TENSION TRANSMISSION An I l l u s t r a t i o n the Theory The  structural  the  could  i n the preceding  be a p p l i e d  chapters  as s o l u t i o n s  indicates  f o r many  problems.  Their ing,  o f a P r a c t i c a l Use o f t h e P h i l o s o p h y o f  work d e s c r i b e d  t h a t M i c h e l l frames  LINE  assembly  from s t a n d a r d  r e v e t t i n g or welding  should  steel  sections  r a i s e few p r o b l e m s  by b o l t since  c o n s t r u c t i o n a l t e c h n i q u e s w o u l d be no d i f f e r e n t f r o m  currently  used.  Only t h e s t r u c t u r a l shapes a r e d i f f e r e n t  from those p r e s e n t l y A  those  typical  employed„  example o f a k i n d  of structure,  f o r which  Michell  frames would be e s p e c i a l l y s u i t a b l e , i s t h e t r a n s -  mission  tower employed  loading  on s u c h  purposes, loads  structures  to include  could  any o t h e r This  tance,  since  power l i n e s .  The  i s c l o s e l y s p e c i f i e d f o r design  combined  by e l e c t r i c a l  i c e and w i n d  i n the l e a s t favourable  tower e x i s t s t o s u p p o r t  a t i o n determined structure  tension  the weight of the wire,  and s t r a i n i n g l o a d s , The  than  on h i g h  this  geometrical  requirements.  t h u s be d e t e r m i n e d  way.  configur-  An optimum  containing  less  material  design. last  factor i s of considerable  transmission  lines  practical  are frequently  t h r o u g h m o u n t a i n s and f o r e s t s t o t r a n s m i t  impor-  constructed  electrical  energy  146 from h y d r o e l e c t r i c  power p l a n t s  c e n t r e s of consumption. q u e n t l y h a s t o be moved inaccessible  tower  and  l i g h t weight The  British  to Vancouver practice  was  in this  a c r o s s rugged  f o r the towers  o f ground  the use o f h e l i c o p t e r s an  line  recently  to bring  power f r o m t h e P e a c e  selected  as r e p r e s e n t a t i v e Through  information  shown  50011-To8-B229 which  and i n d i c a t e s  any s t r u c t u r e must  i n this  8.1 i s b a s e d  by an i n s u l a t o r  side of v e r t i c a l  No p a r t o f t h e tower  may  River  project  specifications  installation. on t h e i r  The  drawing  conform. of four  corners o f a square o f e i g h t e e n i n c h  either  completed i n  the geometrical r e s t r i c t i o n s to  Each conductor c o n s i s t s  suspended  work  the courtesy o f I n t e r n a t i o n a l  tower  i n Figure  for this  of current  Power and E n g i n e e r i n g C o n s u l t a n t s , L t d . minimum were o b t a i n e d o f a t y p i c a l  access  advantage.  transmission  field.  fre-  country to comparatively  The d i f f i c u l t y  i s obviously  500 K.V.  Columbia  The m a t e r i a l  sites.  i n c r e a s i n g l y encourages  i n remote a r e a s t o t h e urban  cables placed  side.  This bundle i s  13'2" l o n g , w h i c h may  due t o w i n d , come c l o s e r  a t the  unbalanced  swing  31°  i c e load, e t c .  t o the conductor bundle  t h a n 10 '2" . The spacing rarely  above r e q u i r e m e n t s n e c e s s i t a t e  o f t h e c o n d u c t o r s o f 35'6". less  than  40'  the support g i r d e r s .  a minimum  In p r a c t i c e  horizontal  these are  a p a r t t o p r o v i d e s p a c e a t A and B f o r  Td|a o f Insulator's.  N  \ I /  Extreme- S u i n g o f conductor. Tb|> of insulator Typically  Insulation Clearance 102  T^'above nominal ground  "from n e a r e s t uuih-e i r \ conductor* bundl-e.'  level.  r i 8  Conductor- B u n d l e Con ^ i g u K a f r o n .  Figure  8.1  Clearance  f o r High T e n s i o n  Wires  14 8 Figures which s a t i s f y  8.2  these  a t i o n of Figure  8.1  and  8.3  design  clearance areas  Michell  a t A and  around the  s t r u c t u r e i s t o be  conductor  and  outer panel specified this  criteria.  of  used  having  frame.  critical a t the  f o r the  designs examin-  area  spaces  for  tower, the  i f a  central  inside  s t r u c t u r e s may  large outer panel  the  between  Clearly,  accommodated  A range of  a sufficiently  possible  superficial  conductors.  the  be  to  permit  arrangement. When t h e d e s i g n o f t h e  conductors  i s considered,  the  J„ „ , and N,N-1  joints  J  possible 8.1.  I f these  joints  In F i g u r e cantilever  height will  ductor ation  i s 100  be  J.. . N-1,N  placed c  8.2,  just  and  are  too  the  low  the d e s i g n w i l l  inevitable  B,  Figure  be g r e a t l y spacing  t o w e r i s drawn as a s t a n d a r d  50  i n the outer  panel.  feet,  (jj = 2)  changes  Its It  central  in  three  4.  A more d e t a i l e d  involve slight  must  i n c r e a s e i n volume.  c l e a r a n c e area around the  of the d e s i g n might  The  an  i t s base width  geometry t o i n c r e a s e the  outer  as  as d e s c r i b e d i n S e c t i o n 1 o f C h a p t e r  feet  fits  the  as c l o s e  p a r t o f t h e g a p s a t A and  incurring  seen t h a t the  described  must be  or , e q u a l l y undesirably", the conductor  i n c r e a s e d , thus  fibre  outriggers supporting  i t becomes e v i d e n t t h a t i n a d d i t i o n  to the narrowest  restricted be  B,  i t s c l e a r a n c e must be the  two  A  i n d i c a t e s t h a t the  geometrical design occurs the  illustrates  con-  examin-  this  clearances.  o u t r i g g e r s are  identical  i n S e c t i o n 5 of Chapter  4.  skew c a n t i l e v e r s For  these  as  9 = 28°,  the  149  F i g u r e 8.2  P o s s i b l e Design f o r a Transmission Tower U t i l i z i n g M i c h e l l Framework  150 ratio fall  of their  fan r a d i i  i s 1.526, t h e s p a n i s 42'6" and t h e  i s 17', a s i s t h e b a s e To  complete the s t r u c t u r e ,  J^^  and  of  Figure cables. built  t  n  e  t o w e r and J ^ i -  of p a r a l l e l  in  each  between  outrigger. along the  t h e t o w e r c o u l d be  members o r o f two p l a n e M i c h e l l  frames  t o c o n v e r g e a t t h e t o p t o f o r m an 'A' f r a m e .  s i d e view could  layout  n  s i d e on a t t h e c a b l e s ,  needed t o s u s t a i n f o r c e s a l o n g —the  s t r u t s a r e added  8.2 shows t h e t o w e r a s s e e n l o o k i n g  Looking  inclined  length.  to that  the c a b l e s — a s  If  an a n c h o r  tower  a l s o be a M i c h e l l framework o f i d e n t i c a l  i n Figure  8.2, s u p p o r t e d on f o u r  foundations  t h e f o r m o f a s q u a r e o f 50 f e e t s i d e . . In t h e s e  true  lengths  latter  cases Figure  o f t h e members w h i c h w o u l d be i n c l i n e d  plane o f the drawing. The  8.2 d o e s n o t show t h e ,  .  h o r i z o n t a l frame f o r m e d by t h e two o u t r i g g e r s  seen i n p l a n ,  could  a l s o be a s s e m b l e d  from p a r a l l e l  making a box s e c t i o n g i r d e r , o r c o u l d a g a i n conform t o a M i c h e l l Figure  8.3 i l l u s t r a t e s  an — r a t i o  height  i s 100 f e e t and t h e b a s e " w i d t h  cable  centre  line  geometry.  B eighty  could  members  be l a i d . o u t t o  a tower c o n s t r u c t e d  total  this  when  framework.  f i b r e M i c h e l l c a n t i l e v e r having  although  t o the •  o f 4.  from a f o u r Again the  25 f e e t .  f e e t above n o m i n a l g r o u n d  Here t h e level,  be v a r i e d by m i n o r c h a n g e s i n t h e tower,  151  Figure  8.3  P o s s i b l e Design f o r a Transmission M i c h e l Framework  Tower  152 The o u t r i g g e r s a r e formed from t h r e e t r i a n g u l a r s p r i n g i n g from  and B^^ r e s p e c t i v e l y .  frames  These a r e not l a i d  out i n c o n f o r m i t y w i t h the M i c h e l l c r i t e r i a f o r an optimum s t r u c t u r e and serve t o i l l u s t r a t e one o f many ways i n which the arrangement c o u l d be made. Again, t h i s view i n d i c a t e s the tower p r o f i l e as seen l o o k i n g along the conductors.  The shape seen normal t o the  conductors can be v a r i e d i n the same way as d e s c r i b e d i n t h e first  case. S t r u c t u r a l volumes have n o t been c a l c u l a t e d f o r  these two g e o m e t r i c a l d e s i g n s s i n c e no f i g u r e i s r e a d i l y a v a i l a b l e f o r the standard tower a g a i n s t which i t i s t o be compared.  A c a l c u l a t e d volume may a l s o have l i t t l e  reality  u n l e s s i t i s prepared by a d e s i g n e r f a m i l i a r w i t h the techniques of t h i s s p e c i a l i z e d f i e l d .  Members may u n w i t t i n g l y  be under-designed i n ignorance o f worst l o a d i n g combinations which may be i n c u r r e d , perhaps d u r i n g c o n s t r u c t i o n  rather  than i n s e r v i c e . The q u a l i t a t i v e d e s i g n s i l l u s t r a t e d a r e elementary examples which are r e p r e s e n t a t i v e o f the a p p l i c a t i o n of MaxwellM i c h e l l s t r u c t u r e s t o a t y p i c a l problem.  I t i s n o t suggested  t h a t a complete d e s i g n f o r a tower, which would be r e q u i r e d t o s u s t a i n l a t e r a l and other l o a d s , has been completed and p r e s e n t e d .  153  CHAPTER 9 DESIGN OF A MIRROR SUBSTRATE  A v a r i e t y o f designs  f o r structures u t i l i z i n g Michell  frameworks have been d i s c u s s e d i n p r e v i o u s have a l l had a common p u r p o s e — t o support system and t r a n s f e r i t t o f i x e d The  design  These  an e x t e r n a l f o r c e  foundations.  f i n a l d e s i g n t o be c o n s i d e r e d  s t r a t e f o r a large astronomical  chapters.  concerns the sub-  mirror.  The t r a d i t i o n a l  f o r a mirror u t i l i z e s a c i r c u l a r  (or r e c t a n g u l a r )  blank  of g l a s s , whose t h i c k n e s s i s between one s i x t h and one e i g h t h of i t s diameter.  These p r o p o r t i o n s a r e e m p i r i c a l and have  been found adequate t o r e s i s t t h e f o r c e s t o which t h e m i r r o r i s subject.  These mainly occur d u r i n g manufacture when t h e  s u r f a c e i s b e i n g p o l i s h e d t o shape. Once i n s i t u the m i r r o r i s s u b j e c t e d p r i m a r i l y t o s e l f weight f o r c e s which change d i r e c t i o n as the m i r r o r i s tilted  t o observe and f o l l o w a s p e c i f i e d s t a r .  Thermal  s t r e s s e s are a l s o imposed as t h e ambient temperature changes. These changes a r e normally surrounding  structures.  f a i r l y slow and are minimized by  I t i s o b v i o u s l y d e s i r a b l e t h a t the  m i r r o r s u b s t r a t e should be as l i g h t and y e t as r i g i d as p o s s i b l e so as t o minimize these d e f l e c t i o n s .  Michell struc-  154 tures are has  thus p a r t i c u l a r l y  a p p l i c a b l e t o t h i s p r o b l e m as i t  been demonstrated t h a t they  closely  approach the  minimum volume s t r u c t u r e w h i c h i s s i m u l t a n e o u s l y rigid,  as  shown i n C h a p t e r The  reflecting  some s p e c i f i e d  curve  the  most  1.  surface  with  optimum  an  of  the m i r r o r  accuracy  which  i s formed  i s often  to  required  _g t o be  b e t t e r t h a n one  approximately) across o f as much as accuracy tilted,  t o the  these  A destined  I t may  This  inches diameter  p o s s i b l e to achieve  sag  attitude.  this  When  away f r o m i t s worked of g r a v i t y .  c o m p e n s a t e d by i s the  have a  in a fixed  changing d i r e c t i o n  changes a r e  somewhat d i f f e r e n t  f o r use  and  be  (10  field  In  applying  of  large  local  'Active Optics*! or  Optics'.  satellite.  the  inches.  forces.  'Deformable  force.  s u r f a c e w h i c h may  however, t h e m i r r o r w i l l  correcting  state  the  of a wavelength  f o r a mirror maintained  shape due mirrors  150  twentieth  i n an  problem e x i s t s i n a mirror,  orbiting  astronomical  In s e r v i c e , the m i r r o r w i l l will  not  be  subjected  However, i t s t i l l  i n f l u e n c e of g r a v i t y .  has  be  observatory  in a  weightless  t o changes i n s e l f  t o be  s h a p e d on  A g a i n the  minimum d e f l e c t i o n t o r e d u c e t h e  earth  weight under  s t r u c t u r e should  have  inevitable corrections  a  as  much as p o s s i b l e . The diameter of  mirror 48  considered  inches  have a t h i c k n e s s  of  and,  h e r e as  an  example has  i f made f r o m a s o l i d  8 inches.  The  ing M i c h e l l s t r u c t u r e s f o r support  block,  alternative design  a. would utiliz-  i s indicated i n Figure  9.1.  155  Figure  9.1  Tentative Design  The s u p p o r t permits t  suiting  limited the t  the s o l i d  f o r a Large M i r r o r  of the m i r r o r  mirror  by a d i s t r i b u t i v e  b l a n k t o be r e d u c e d  The d i s t r i b u t i v e  structural  form o f M i c h e l l c a n t i l e v e r s as r a d i a l  i s s e l e c t e d such t h a t  the  highest  the  edge s u p p o r t e d In  around  to a  and l o w e s t  points  of the d i s c i s equal  takes thickness of  to that of  disc.  p a p e r , H, V a u g h a n  d i s c of r a d i u s a supported  a circle  The  diffraction  t h e d i f f e r e n c e between t h e p l a n e s  solid  a recent  support  ribs.  structure  thickness  t h e r e q u i r e m e n t s o f o p t i c a l w o r k i n g and o f  optics.  a uniform  Substrate  o f r a d i u s b.  He  [10] h a s i n v e s t i g a t e d  at discrete points  showed  that  for — =  , an  156 optimum f l a t n e s s o c c u r s 120°  apart.  The deformed  from t h e nominal d e s i g n configuration. will  be u s e d  inches  this  mirror  surface  Using h i s r e s u l t s , three  obtain  a more r e a l i s t i c  section w i l l material.  specific v  For that  specific  cantilevers  of sixteen  6  =  are thus -  in.  0.17  the design  of support r i b s ,  s t r e s s was 1050  was q u i t e  i s i n the mirror  i n passing  directly  e was t a k e n a s 0.0001  p.s.i. that  the s e l e c t i o n of a  a r b i t r a r y and h a s o n l y since  surface.  the great  a minor e f f e c t bulk of  The p e r m i s s i b l e  stress  a f f e c t s the r i bd e f l e c t i o n but the  r e s u l t a n t motion of the mirror  surface  translation,  w h i c h may e a s i l y  adjustment.  The s t r e s s was k e p t  tension.  typical  g r a v i t y = 2.20  strain  deflections  a  p.s.i.  = 0.0795 l b . p e r c u .  the design  the ribs  the c a l c u l a t i o n s  silica,  The p h y s i c a l c o n s t a n t s  t h e volume o f t h e s t r u c t u r e ,  material  to  radial  design,  be made f o r f u s e d  I t may be n o t e d  in  spaced  a r e a t a minimum f o r t h i s  t o support the d i s c a t a radius  pg  on  are used,  shape i s complex b u t t h e d e v i a t i o n s  •E = 10.5 x 1 0  so  supports  [— = rr] . a 3 To  in  i f three  i s a rigid  be c o m p e n s a t e d  by a  body focussing  low t o m i n i m i s e t h e  and i n d e f e r e n c e t o t h e low r e s i s t a n c e  of glass  157 D e t e r m i n a t i o n of T h i c k n e s s t S o l u t i o n s f o r the c e n t r a l d e f l e c t i o n of a u n i f o r m l y loaded p l a t e , edge supported, may t e x t s on e l a s t i c bending  be found i n most standard  of p l a t e s .  For example, i n  "Theory  of P l a t e s and S h e l l s " by Timoshenko and Woinowsky,  Krieger  (McGraw-Hill, 1959), e q u a t i o n 68 on page 57 may  be  used:-  #| W  _ (5+v) p g t a ~ 64 (l+v)D  4  = 0.0000316 i n . f o r the g i v e n d i s c . A paper by R. W i l l i a m s  [12] d e s c r i b e s an exact  s o l u t i o n f o r the d e f o r m a t i o n of a c i r c u l a r d i s c of r a d i u s a, supported at n e q u a l l y spaced p o i n t s around  a c i r c l e of  r a d i u s b. The  formula i s g i v e n as 4 = pga P  [ W i l l i a m s 3.1]  where B i s a f a c t o r depending  on the support  R  u  (9.1)  conditions.  [This e q u a t i o n i s i d e n t i c a l w i t h the s o l u t i o n g i v e n by and Pan  (I.J.M.S., 8_, 1966,  page 336), f o r the same c o n d i t i o n s ,  I t has been n o r m a l i z e d and r e a r r a n g e d so t h a t i t can expressed  Yu  i n the above form].  be  The the  number  flection  are  of  is  given  factor  is  3  supports  required,  in  his  a  n, r,  paper  function and  the  the  radius  expressed  for  of  as  various  radius  at  the  ratio  which  ratio  the  of  , '  de-  — . a  combinations  — a  Tables  these  factors. Using central  and  determined  these  edge by  tables,  the  value  of  b  (—=0, —=1.0)  deflections  interpolation.  This  for  are  which  equal  -=0.68  gives  the  may  which  be agree  a well  with  from  the  oo  Vaughan's  figure  interpolation.  (0.0000316)  determine  the  may  be  new  disc  The  revealed  it  was  supported  outward  for  sixteen  inches.  At  structure The  the  must  spaces  apparatus  between in  This  would  t i l t  the  be  from  of  greater  mirror  so  on  into  to  is  pillar a  the  the  depth  could  be  may  be  for  aligned  drawings,  a Michell  total  is  to  RIBS  type  extending  depth  of  of  inches.  scale  filled  volume  requirements  9.1  and  total  ribs,  where  it  using  arising  value  4.12  construct  within  than  ribs  the  that  the  equation  This  centre  the  applications  depend  a  inches,  the  and  3,  investigation,  planes  difference  D E S I G N OF C A N T I L E V E R  impossible  cantilever,  the  of  thickness.  preliminary  that  value  substituted  GEOMETRIC  A  0.667,  of  of  depth  the  six of  solid  with  the disc.  other  strictly  limited.  space  which  in  in  specific  to  directions. Two c a n t i l e v e r s , each having  t h r e e f i b r e s , were  s e l e c t e d f o r d e s i g n a n a l y s i s and a r e d e s c r i b e d below. changes c o u l d be made t o t h e i r parameters without  Minor  significant  change i n t h e v a l i d i t y o f t h e r e s u l t s . F i g u r e 9.2 - Symmetrical C a n t i l e v e r  a)  T h i s i s a 'standard* i n Chapter 4.  c a n t i l e v e r d e s i g n as d i s c u s s e d  I t s ~ r a t i o i s 2.5.  The support  a t a r a d i u s o f 2.4".  i s on t h e m i r r o r a x i s and t h e o t h e r , D c o u l d be supported The although  on a c o l l a r on the c e n t r a l p i n .  t o t a l depth o f d i s c and r i b i s about 1 2 . 4 " ,  t h i s can be v a r i e d s l i g h t l y .  must be maintained disc.  p o i n t s A^  a t C t o prevent  A minimum c l e a r a n c e  c o n t a c t between r i b and  The r i b c o u l d be r o t a t e d about C.  Clockwise  rotation  would reduce the o v e r a l l depth but i n c r e a s e t h e l e n g t h o f the s t r u t necessary Conversely  a t L t o support counterclockwise  the d i s c . r o t a t i o n o f the r i b  reduces the l e n g t h o f the s t r u t a t L b u t i n c r e a s e s the overa l l depth.  I f l a r g e r o t a t i o n s a r e made from t h e p o s i t i o n  shown, the r i b parameters would r e q u i r e changing t o m a i n t a i n L at a radius of sixteen inches.  A r i b c o u l d be designed t o f i t w i t h i n t h e s i x i n c h space. In t h i s case one support would be on t h e c e n t r a l a x i s but t h e other would have t o be a t approximately f o u r t e e n inches r a d i u s , almost below the m i r r o r s u r f a c e support p o i n t , Such a s o l u t i o n would merely t r a n s m i t the problem of support t o the main s t r u c t u r e o f the apparatus, and i s not a p r a c t i c a l solution .  Figure  9.2  Three F i b r e Symmetrical Mirror Substrate  Michell Cantilever for  161 b)  F i g u r e 9.3  - Skew M i c h e l l C a n t i l e v e r  The  d e s i g n u t i l i z e s a skew c a n t i l e v e r  second  both support p o i n t s on the c e n t r a l a x i s . a g a i n a l i m i t i n g f a c t o r i n the d e s i g n .  with  Clearance at C i s L may  be p l a c e d  c l o s e r t o the d i s c than i s shown by moving the support p o i n t s A and B downward and  i n c r e a s i n g the depth of the r i b .  As drawn the span i s 16", L  12 ) and the r i s e i s 2.33".  The  support s p a c i n g 6.67",  The  f i b r e angle i s 28°.  o v e r a l l depth of about e l e v e n inches i s somewhat l e s s  than t h a t of the symmetrical  c a n t i l e v e r but the s t r u t at  L i s somewhat l o n g e r . LOADS ON CANTILEVERS AND The were used  MEMBER SIZES  computer programmes d e s c r i b e d i n Appendix C  t o determine  the member l e n g t h s and  maintain a uniform s t r e s s  (of 1050  s i z e s r e q u i r e d to  p . s . i . ) as the m i r r o r  was  moved from the h o r i z o n t a l t o the v e r t i c a l i n f i v e degree s t e p s . The m i r r o r d i s c weighs approximately r i b B s u b j e c t e d t o a l o a d of 197.6 Although the m i r r o r was loads and  593 pounds so t h a t each  pounds.  the maximum t o t a l volume was  r e q u i r e d when  h o r i z o n t a l , i n d i v i d u a l members c a r r i e d  thus had g r e a t e r volume a t o t h e r angular  The maximum c r o s s s e c t i o n a l a r e a of each member was s e l e c t e d t o form a composite  structure.  s t r e s s would not exceed  p . s . i . a t any  1050  larger  positions. thus  In t h i s , the maximum angle but would  a t some a n g l e r e a c h t h i s an optimum s t r u c t u r e  v a l u e i n e a c h member.  f o r any a n g l e b e c a u s e  I t i s not  of this  modifi-  cation. The  composite  structures  analysed using a 'Strudl* secondary The  results  are tabulated  Michell PIN  10 L  ll  '12 A  20  '21 L  22  l  30  k  31  ^32 01 11 21 02 12 22 03 13 23  programme  then  to determine the  stress.  a) S y m m e t r i c a l  Member  t h u s o b t a i n e d were  below:-  Cantilever  JOINTED  (Figure  9.2)  STRUCTURE  Length Ins.  Area In  4. 526  0.109  2. 187  0.093  97.95  2. 187  0.060  63.07  4. 526  0.185  194.30  3. 638  0.162  170.7  4. 409  0.113  118. 8  4. 526  0.199  208.9  5. 281  0.193  202.5  7. 333  0.170  178.9  4. 526  0.097  101.7  2. 187  0.077  80.40  2. 187  0.048  50.47  4. 526  0.169  177.6  3. 638  0.136  143.1  4. 409  0.091  4. 526  0.249  261.0  5. 281  0.241  253.3  7. 333  0.213  223.7  2  Volume - 12.25  cu.in.  Maximum Force 114.5  95.03  Figure  9.3  Three F i b r e Substrate  Skew M i c h e l l C a n t i l e v e r  for  Mirror  164  The  STRUDL a n a l y s i s y i e l d e d t h e f o l l o w i n g a d d i t i o n a l  data:i)  Mirror  Horizontal  R e a c t i o n a t A, H o r i z o n t a l -397.6 l b , V e r t i c a l -52.0 l b . R e a c t i o n a t B, H o r i z o n t a l +397.6 l b , V e r t i c a l +249.6 l b . D e f l e c t i o n a t mirror support p o i n t , L H o r i z o n t a l 0.019", ii)  Mirror  V e r t i c a l 0.053".  Vertical  R e a c t i o n a t A, H o r i z o n t a l +90.7 l b , V e r t i c a l +249.0 l b . R e a c t i o n a t B, H o r i z o n t a l -90.7 l b , V e r t i c a l -51.4 l b . D e f l e c t i o n a t m i r r o r s u p p o r t p o i n t , L. H o r i z o n t a l -0.19",  V e r t i c a l 0.022".  I t i s i n t e r e s t i n g t o note t h a t t h e load i s almost e x a c t l y t r a n s f e r r e d between t h e s u p p o r t p o i n t s as t h e m i r r o r is tilted The  t h r o u g h 90°. volume o f t h i s skew s t r u c t u r e i s o n l y  l a r g e r than t h a t of t h e symmetrical c a n t i l e v e r .  slightly The s l i g h t  e x t r a volume would be a c c e p t a b l e i f i t was d e s i r e d t o s u p p o r t t h e s t r u c t u r e e n t i r e l y from a c e n t r a l The data.  pillar.  STRUDL a n a l y s i s y i e l d e d t h e f o l l o w i n g a d d i t i o n a l  165 b)  Skew M i c h e l l C a n t i l e v e r PIN  Member  A  10  A  l l  A  12  A  20  A  21  . 22 A  A  30  A  31  A  32  B  oi  B  l l  B  21  B  02  B  12  B  22  B  03  B  13  B  23  (Figure  9.3)  JOINTED STRUCTURE  Length Ins.  Area In  3.909  0.109  2.581  0.092  96.53  2.581  0.059  61.95  3.909  0.187  196.7  3.893  0.161  169.2  4.728  0.111  116.8  3.909  0.188  197.8  5.374  0.183  192.1  7.512  0.162  170.2  5.401  0.104  109.3  1.868  0.085  88.9  1.868  0.045  47.29  5.401  0.181  189.6  3.462  0.150  157.4  4.149  0.085  5.401  0.247  259.3  5.261  0.240  251.7  7.162  0.212  222.9  Maximum Force l b s  2  114.6  89.18  Volume - 12.40 c u . i n .  i)  Mirror  Horizontal  Reaction  a t A, H o r i z o n t a l  -443.8 l b , V e r t i c a l  Reaction  a t B, H o r i z o n t a l  443.8 l b , V e r t i c a l  Deflection  at mirror  43.3 l b . 154.3 l b .  support p o i n t L.  Horizontal  0.009",  Vertical  0.059".  166 ii)  Mirror  Vertical  Reaction  a t A, H o r i z o n t a l +37.3 l b , V e r t i c a l  Reaction  a t B, H o r i z o n t a l -37.3  Deflection  at mirror  support  Horizontal The the  above v a l u e s  symmetrical  l a r g e r when t h e m i r r o r than  4 8.1%  ribs  saving  disc  of r i b s  reported f o r  i s h o r i z o n t a l and s m a l l e r  when  vertical  saving  corresponds t o  the t r a d i t i o n a l  thicker solid  volume s a v i n g s  surface,  disc.  d i s c i s used  that these  e x a m p l e s o f t h e way i n w h i c h s t r u c t u r a l  volume r e q u i r e s  other  volume s o l u t i o n s .  are i l l u s t r a t i v e  absolute  minimum  A p a p e r by Duncan  r i b arrangements u s i n g  and d i s c u s s e s  to evaluate  v o l u m e may be  The s o l u t i o n w i t h  further investigation.  illustrates  of a  designs.  be e m p h a s i s e d  r e d u c e d by u s e o f r i b s .  i n place  Any a r r a n g e m e n t o f r i b s  The s o l i d  for specific  should  a r i s e s f r o m t h e u s e o f an a r r a y  the o p t i c a l  show some s a v i n g .  frameworks,  This  o f v o l u m e as compared w i t h  t o support  It  e x a m p l e s i s 6970 c u . i n . ,  design.  considerably  [24]  0.016".  The s a g o f t h e s t r u c t u r e i s  and c e n t r a l p i l l a r .  Much o f t h i s  will  t o those  volume o f e i t h e r o f t h e s e  include disc,  solid  Vertical  f o r c a s e a. The  a  34.4 l b .  point L.  -0.009",  are similar  cantilever.  l b ,V e r t i c a l  163.2 l b .  p o s s i b l e designs  Michell  f o r minimum  V  1 6 7  A l t e r n a t i v e Support Arrangement f o r M i r r o r Two designs  Surface  have been d e s c r i b e d f o r the support  of a c i r c u l a r m i r r o r s u r f a c e , u s i n g three r a d i a l supporting for  the d i s c  ribs  a t two t h i r d s i t s r a d i u s , the p o s i t i o n  minimum t o t a l d e f l e c t i o n f o r a three p o i n t  suspension.  Another p o s s i b l e arrangement c o n s i s t s of supporting  the d i s c a t i t s c e n t r e , and a l s o a t p o i n t s on the  circumference  o f a c i r c l e whose r a d i u s w i l l be somewhat  l a r g e r than t h a t used above  (b = 0.667a).  The deformed  shape o f the d i s c i s i n d i c a t e d i n F i g u r e 9.4.  to,"E-btt  (^of  F i g u r e 9.4  disc.  Deformed Shape o f D i s c  With t h i s arrangement o f supports c o u l d be used w h i l e m a i n t a i n i n g ojedge' c o n s t a n t .  a thinner d i s c  the maximum d e f l e c t i o n ,  F o r purpose o f the p r e s e n t  calculation i t  w i l l be assumed t h a t the d i s c s u r f a c e a t C w i l l be i n the  1 6 8  plane o f the outer supports, S, which i s taken as f i x e d for  the purposes o f d e t e r m i n i n g d e f l e c t i o n .  of  the supports, S, may be regarded  of  the system and may e a s i l y be c o r r e c t e d . The  d e s i g n procedure  The d e r i v a t i o n  of  i n Appendix J .  D e f l e c t i o n o f a D i s c w i t h C e n t r a l and Ring  The at  as a r i g i d body motion  used, where not e x t r a c t e d from standard works,  i s t o be found  1.  deformation  i s somewhat complex and w i l l  be d e s c r i b e d i n a s e r i e s o f s t e p s . equations  Any  Supports  d e f l e c t i o n o f a u n i f o r m l y loaded d i s c  supported  the c e n t r e and c o n t i n u o u s l y around a r i n g o f r a d i u s b  may be determined  by the s u p e r p o s i t i o n o f two l o a d i n g  systems, as shown i n F i g u r e 9 . 5 . The (a)  two loads are r e s p e c t i v e l y : a c e n t r a l p o i n t l o a d on a l i g h t a, supported  (b)  d i s c of r a d i u s  at radius b  a u n i f o r m l y loaded d i s c supported  a t r a d i u s b.  169  A  1  i  4  4  I i  I  I  1  4  1  4  4  1  i  A  1  4  1  1  tS  1  4  4  1  (a)  1 4  4  4  4  I  4  t T  t -  Figure  3.  9.5  Superimposed  D e f l e c t i o n of  standard  example  in  19  of  tests  "Theory  Woinowsky-Krieger Article  (page  Loads  a C i r c u l a r Disc  Solutions most  —  this  on of  Plates  68)  states  the  on  a  *-l  Circular  Central  system  bending  and  (McGraw-Hill,  • Q. —  with  loading  elastic  -  of  may  1959)  Point  be  plates.  S h e l l s " by  Disc  Load  found For  Tomoshenko  equation  deflection at  89 any  in  and  in radius  r  as  /6TTD  a  9'Z  In the' p r e s e n t to  the i n n e r  60,-, -  this  zone o f t h e d i s c ,  equation  applies only  f o r 0 < r < b.  "We  Thus  9-3  J6TTD  The n e g a t i v e is  case,  s i g n i s a p p l i e d s i n c e the c e n t r a l  deflection  upwards.  W  i s the c e n t r a l  c  D  =  E  t  load  3  12(1-v ) 2  From  t h e above  slope  equation  =  df-  In p a r t i c u l a r ,  9.4  4TTD  a t the supports,  r = b and  95  slope =  In t h e o u t e r surface  s e c t i o n b <. r £ a  i s plane.  deflection  i s given  The  slope  ,  there  i s no l o a d and t h e  i s thus constant,  and t h e  by  W b(r-b) 47TTD 0+^) c  9-4  171 4.  D e f l e c t i o n of  This the to  methods obtain  Appendix above side  loading  i s not  described  a  Loaded  solved  i n Chapter  solution.  The  Disc  with  3 of  derivation  that  r e s u l t s being  stated  case,  two  zones  considered,  the  the  CO  =  For  the  support  inner  be  Support  work  [13]  may  i s described  the  must  Ring  i n Timoshenko  J,  For  2 1  Uniformly  below.  As  be  used  in  in  inside  but  the  and  out-  ring.  zone,  0  < r  <  b  97 outer  zone,  b  <  r  <  a  9* 5.  Deflection  of  Disc  with  Combined  Loads »  A the be  convenient  combined zero,  This  is  so  for  that  the  this  requires  a l l support  illustrated In  zero  loading  condition  in Figure  case  value  That i s : -  r  the =  sum  zero.  for  the  the  deflection  under  central deflection  points  l i e in  the  same  9.3  and  to plane.  9.4. of  equations  9.7  is  172  99 This  yields  a  value  f o r the  central  • 3P) - 4(itP) /^|)|  Since  the  the  total  total  weight  reaction  on  of  the  the  ring  support  H. b \ 3  disc,  W,  support  reaction,  9. ,  sA  i s given  is  W  0  by  thus  9-H  The of  the  Inner  0D  Z  Outer  0  equations  plate,  zone  0  under  the  < r  b  <  deflection  combined  loading,  f o r the are  two  zones  thus:-  temp  =  zone  f o r the  b  MPH  < r  <  a  9'*3 -t- Wc  k(T-b'^  173  5.  D e t e r m i n a t i o n o f Support R a d i u s , b For a g i v e n edge d e f l e c t i o n ,  the s u p p o r t s , t h e t o t a l v a r i a t i o n ,  below t h e p l a n e o f  between t h e h i g h e s t and  l o w e s t p o i n t s o f t h e s u r f a c e , w i l l be a minimum i f no p o i n t rises  above t h e s u p p o r t p l a n e .  F o r t h i s t o be s a t i s f i e d  the s l o p e a t t h e support p o i n t s , s,must be zero ( i t w i l l a l s o be zero a t C by symmetry). From e q u a t i o n 9.13, f o r t h e o u t e r zone  4-tt« ^> + 4 oV^v) - 2bVQ +  slope =  dr  9.14 A t t h e s u p p o r t s r = b and t h e s l o p e i s z e r o  Substituting,  v = 0.17,  t h i s may  be s o l v e d  to y i e l d  9*^ which has a r o o t a t  a  6.  Determination The  o f D i s c Thickness  above c a l c u l a t i o n  t  f o r support  r a d i u s was  pendent o f d i s c t h i c k n e s s ,  s i n c e t appeared  'constant'  9.13 may t h u s now be s o l v e d f o r  t  using  term.  equation  deflection,  Equation 9.16  equation  c a l c u l a t e d i n step  r = a i n equation  Substituting  i n the  t o s u b s t i t u t e f o r b and t h e v a l u e o f  31.6 x 10 ^ i n c h e s ,  Putting  only  inde-  1.  9.13  a = 24, ^ = 0.7728,  v = 0.17  this  gives  or The  t  -  3'777  above c a l c u l a t i o n s have been based on t h e u s e  of a c o n t i n u o u s  ring  supports  the c i r c l e  around  on  page 67  is  t h e same  9/7  mc/ie&  (Article  support,  will  load  o f r a d i u s b.  i n fact  be u s e d .  18) n o t e s t h a t  i f a given  around a c i r c l e  although  s i x point  T i m o s h e n k o [13]  t h e central  deflection  i s a p p l i e d a t one o r many The c a l c u l a t i o n  of W  point  and t h e c  superposition described deflections thick  i n Figure  9.5 i s t h u s e x a c t .  calculated i n Section  1 f o r a disc eight  agree c l o s e l y  f o r a continuous  support  The inches  and f o r a r a d i a  175 line  through  lines w i l l  a support.  The edge d e f l e c t i o n  be g r e a t e r t h a n  the uniform  deflection  but i t i s f e l t  calculation  p r o v i d e s an a c c e p t a b l e b a s i s f o r d e s i g n .  Design  o f Support  now  be  comparatively  simple  Ribs  From e q u a t i o n may  radial  here  determined,  7.  that this  on o t h e r  9.17  t h e volume and w e i g h t o f t h e d i s c  calculated 2  Volume  = ua t  =  6835 c u b i c i n c h e s  . . . .  9.18  54 3.4  . . . .  9.19  2 Weight =  (ira t ) p g =  From e q u a t i o n  »  pounds  9.10  21-Q  9'2o  founds  ( l o a d p e r r i b = 13.5 l b . ) Total 543.4 - 81.0 is  load carried =  by s i x r i n g  462.4 p o u n d s .  supports i s  Thus t h e l o a d p e r  77.1 pounds  . . . .  The d e s i g n p a r a m e t e r s c a l c u l a t e d in  F i g u r e 9.6.  The d i s c  i s supported  s i x p o i n t s e q u a l l y spaced inches.  The s u p p o r t  carrying  a l o a d o f 13.5  the o u t e r  end.  support 9.21  above a r e i n d i c a t e d  a t . t h e c e n t r e and a t  around a c i r c l e  of radius  i s p r o v i d e d by s i x r a d i a l  ribs  l b . a t t h e c e n t r e and 77.1  18.55 each l b . at  176  F i g u r e 9.6  The  Support R i b s f o r M i r r o r D i s c  r i b ' c o u l d c o n v e n i e n t l y c o n s i s t of t h r e e M i c h e l l  c a n t i l e v e r s , each h a v i n g t h r e e f i b r e s , s p r i n g i n g from s u p p o r t p o i n t s a t A and B, a t a r a d i u s of 9.275 i n c h e s , h a l f t h a t of the o u t e r d i s c The  supports.  t h r e e c a n t i l e v e r s would have the f o l l o w i n g  functions:(a) Outer l o a d c a n t i l e v e r  -  T h i s i s a skew c a n t i l e v e r  r i s i n g :from A and B t o the o u t e r c a r r i e s a v e r t i c a l l o a d of 77.1 (b) Inner l o a d c a n t i l e v e r  -  support p o i n t S.  It  pounds.  T h i s i s an i d e n t i c a l skew  c a n t i l e v e r , as r e g a r d s geometry, t o ( a ) .  I t also  s p r i n g s from A and B and r i s e s t o the c e n t r a l support C,  c a r r y i n g a l o a d o f 13.5  pounds.  The  proportions  177 of  i t s members must be a d j u s t e d  C equals in  that  equation  these  9.9.  This  sections  coordinates (c)  will  may c o n v e n i e n t l y  will  be  differ  the deflection at  the design  two c a n t i l e v e r s i d e n t i c a l  cross  rib  a t S, t o s a t i s f y  so t h a t  condition specified  be a c h i e v e d  geometrically.  but their  lengths  by making  The member  and j o i n t  identical  Lower moment c a n t i l e v e r - T h e f o r c e s  c a n t i l e v e r s are indicated  i n Figure  a c t i o n on t h e  9.7.  D i f f e r e n t i a l d e f o r m a t i o n o f t h e d i s c and r i b i n the  h o r i z o n t a l plane w i l l  This  may be e l i m i n a t e d  introduce  by a f l e x i b l e  a horizontal force connection  o r by u s e o f d i f f e r e n t maximum s t r e s s e s T h e s e may b e s e l e c t e d  13-5 li  C  to equalise  JS-6  , r  at this  the deformations  ii  a t S.  7 7 / Ik.  Reoct/ on. 9o-6 /b .  9.7  point,  i n d i s c and r i b .  147-5* lb  Figure  a t S.  Force Diagram f o r a t y p i c a l r i b  The be  a t S and C c r e a t e  o p p o s e d by h o r i z o n t a l f o r c e s  supported will at  two l o a d s  by a t h i r d  be i d e n t i c a l  C and M w i l l  a t C a n d M, t h a t  r i b springing  geometrically  a moment w h i c h may  f r o m AB.  with  at M  This  the others.  being  rib  also  The f o r c e s  be g e n e r a t e d b y i n t e r a c t i o n w i t h  the other  ribs. The  s e l e c t i o n of o v e r a l l dimensions o f these r i b s i s  somewhat a r b i t r a r y and c o u l d space the  be d e t e r m i n e d  available f o r the structure.  As we h a v e s e e n  volume o f a s t r u c t u r e , t o c a r r y  decreases  l a r g e l y by t h e  a given  load  earlier,  system,  as t h e space a v a i l a b l e f o r i t s c o n s t r u c t i o n  increases. If  the following values  SPAN  (X SPAN)  9.275"  RISE  (Y SPAN)  2.0"  SUPPORT SPACING then  may be e a s i l y F  m  To  =  complete  be 28.4 pounds  258.3 in  pounds  the next  square, (8.1 rigid  (D)  3.0"  c a l c u l a t e d and i s f o u n d  t o be  147.47 pounds  9.22  the structure  t o b e added t o s u p p o r t will  are selected:-  'A'.  a 'rigid'  The a x i a l  f o r c e on t h i s  ( t e n s i l e ) and t h e t r a n s v e r s e  (as d e t e r m i n e d  section).  from t h e r e a c t i o n s  I f this  axially).  member  force calculated  member i s made one i n c h  i t s d e f l e c t i o n s under t h e s e  x 10 ^ i n c h e s  member AB h a s  loads  I t could  s o f a r as c a n t i l e v e r d e s i g n  will  be  small  be c o n s i d e r e d  i s concerned.  as  179 8.  Calculation of Size  (a)  Simple  of Cantilever  p i n - j o i n t e d design  grammes p r e v i o u s l y  described  Members  -  were u s e d  The c o m p u t e r  pro-  to determine the  g e o m e t r y o f t h e c a n t i l e v e r s and t h e s i z e s o f e a c h member, assuming  the j o i n t s  t o be  The p h y s i c a l rather  than those  pinned.  constants  f o r fused  silica  f o r CR39 a s i n t h e e a r l i e r  were  used  chapters.  The v o l u m e o f e a c h r i b i s t h e n composed  as  follows:-  Volume  of outer  cantilever to A  3.177  Volume  of inner  cantilever to C  6.202  Volume  o f lower c a n t i l e v e r t o M  2.589  Member AB  3.000 T o t a l volume  The t o t a l disc  (6835) p l u s  total  that  o f 6925 c u b i c Since  structure  14.96 8  i s thus  of the r i b s  saving  zontal  ) or a grand  inches.  the d i s c described  i n Section inches  1 of  this  this  represents  o f 52.2%. Analysis  reaction  t h e volume o f t h e  ( 6 x 14.97  c h a p t e r h a s a v o l u m e o f 14,47 5 c u b i c a  cu.in.  of the S t r u d l  at B i s a vertical  f o r c e o f 90.6 p o u n d s  f o r c e o r moment), t h a t  0.000001 i n c h e s  s o l u t i o n shows  the v e r t i c a l  that the (no h o r i -  deflection of A i s  and i t s h o r i z o n t a l d e f l e c t i o n i s 0.00003  180 inches.  In g e n e r a l  the r e s u l t s o f the p i n n e d - j o i n t  r i g i d a n a l y s i s agree w e l l , as was p a r t of t h i s work.  discussed  i n the  and earlier  1  CONCLUSIONS  The array  of loads  requires  stress.  t h a n any o t h e r  The s a v i n g  i n volume c a n r a n g e  given  foundations,  t o t h e same maximum f r o m 30% o r more  t o s e v e r a l hundred per cent  f o r beams o f  section. A M i c h e l l framework w i l l  less  a  type of s t r u c t u r e  t h e same s p a c e , and s u b j e c t e d  open t r u s s e s  solid  t o support  i n s p a c e and t r a n s f e r them t o  less material  occupying  for  use o f M i c h e l l frameworks,  a l s o be s t i f f e r  d e f l e c t i o n u n d e r t h e same c o n d i t i o n s  t y p e o f framework s u b j e c t e d A Michell  five  A  other  framework e x c e e d s t h e optimum minimum s t r u c t u r e by l e s s t h a n 10%.  f i b r e d M i c h e l l frame p r o v i d e s  mation t o the i d e a l , volume.  t h a n any  t o t h e same maximum s t r e s s .  volume o f t h e t h e o r e t i c a l i d e a l A  and e x h i b i t  and i s w i t h i n  greater  five  number o f f i b r e s  a close  per cent need  approxi  o f optimum  not  be  used. The by  reduction  i n v o l u m e o f s t r u c t u r e s made  t h e use o f M i c h e l l frameworks, r e n d e r s economic  use port  wherever m a t e r i a l s i s considerable  to high  rates  cost  o r where t h e s t r u c t u r e s  their  of trans-  are  subjected  of a c c e l e r a t i o n .  Michell designs  are expensive, t h e i r  possible  structures  a l s o have advantages o v e r  i n c a s e s where t h e d e f l e c t i o n i s l i m i t e d .  other In such  182  c a s e s t h e maximum s t r e s s w i l l in  other  be l e s s t h a n t h a t  designs. Manufacture o f M i c h e l l  by and  f r a m e w o r k s has b e e n  t h e i n t r o d u c t i o n o f more s o p h i s t i c a t e d m a c h i n e the use o f numerical Michell-like  and  encountered  other  solid  lightening feature could  by t h e j u d i c u o u s  The u s e o f s u c h h o l e s  o f m a r i n e and a e r o n a u t i c a l  w e l l be e x t e n d e d .  provides outs .  holes.  tools  control.  f r a m e w o r k s c a n be f o r m e d  members  eased  a theoretical basis  placement has l o n g  design,  The c r i t e r i a  i n plates  been a  and t h e i r u s e  described  f o r the layout  of  i n this  o f such  cut-  work  183 REFERENCES  1.  Maxwell,  pp.  J.C.  Scientific  Papers  I I , O.U.P.  1890,  175-177.  2.  M i c h e l l , A.G.M. "The L i m i t s o f Economy o f M a t e r i a l in Frame-Structures." P h i l . Mag. S e r i e s 6, 8(47) 589597, L o n d o n , November 1904.  3.  Cox, H.L. "The T h e o r y o f D e s i g n , " R e s e a r c h C o u n c i l , 19.791, J a n u a r y  4.  Cox, H.L. " S t r u c t u r e s o f Minimum W e i g h t . The b a s i c t h e o r y o f d e s i g n a p p l i e d t o t h e beam u n d e r p u r e bending." A e r o n a u t i c a l R e s e a r c h C o u n c i l , 19.785, January 1958.  5.  Hemp, W.S. "Theory 214, O c t o b e r 1958.  6.  Chan, A . S . L . "The D e s i g n o f M i c h e l l Optimum C o l l e g e o f A e r o n a u t i c s , C r a n f o r d , R e p o r t No. December 1960.  7.  L o v e , A.E.H. "A T r e a t i s e on t h e M a t h e m a t i c a l of E l a s t i c i t y . " C.U.P. 1927.  8.  J o h n s o n , W. "An a n a l o g y b e t w e e n u p p e r bound s o l u t i o n s f o r p l a n e - s t r a i n m e t a l w o r k i n g and minimum w e i g h t two d i m e n s i o n a l frames." I n t e r n a t i o n a l J o u r n a l of M e c h a n i c a l S c i e n c e , 1961, V o l . 3, pp. 239-246.  9.  B a r n e t t , R.L. " S u r v e y o f Optimum S t r u c t u r a l D e s i g n . " E x p e r i m e n t a l M e c h a n i c s , December 1966, p. 19A.  Aeronautical 1958.  of S t r u c t u r a l Design."  AGARD  Report  Structures," 142, Theory  10.  Vaughan, Henry. " D e f l e c t i o n of U n i f o r m l y Loaded C i r c u l a r P l a t e s upon E q u i s p a c e d P o i n t S u p p o r t s . " Journal of Strain Analysis, April 1970.  11.  H i l l , R. "The M a t h e m a t i c a l T h e o r y Oxford, Clarendon Press, 1950.  12.  W i l l i a m s , R. " D e f l e c t i o n Surface of a C i r c u l a r P l a t e with M u l t i p o i n t Support," E n g i n e e r i n g Report 9136, J a n u a r y 1968, P e r k i n - E l m e r .  13.  T i m o s h e n k o , S. and Woinowsky, K r i e g e r . P l a t e s and S h e l l s , " M c G r a w - H i l l , 1959.  of  Plasticity,"  "Theory  of  184 14.  Johnson, W., M e l l o r P.B., E n g i n e e r s , " Van Nostrand,  "Plasticity 1962.  f o r Mechanical  15.  Johnson, W., Sowerby, R. , Haddow, J.B., strain Slip-line Fields." A r n o l d , 1970.  16.  Cox, H.L., "Design of S t r u c t u r e s of L e a s t Weight." Pergamon, 1965.  17.  Hegemier, G.A. and Prager, W., "On M i c h e l l T r u s s e s , " I n t e r n a t i o n a l J o u r n a l of M e c h a n i c a l S c i e n c e , 1969, V o l . 11, pp. 209-215.  18.  Sheu, C.Y. and P r a g e r , W., "Recent Developments i n Optimal S t r u c t u r a l Design," A p p l i e d Mechanics Reviews, No. 21, N o l . 10, October 1968, pp. 985-992.  19.  G h i s t a , D.N., " O p t i m i s a t i o n of S t r u c t u r e s w i t h r e s p e c t to weight," Dept. of C i v i l Eng., Standord, January 1965.  20.  G h i s t a , D.N. " F u l l y - s t r e s s e d design f o r a l t e r n a t i v e loads." ASCE J o u r n a l o f the S t r u c t u r a l D i v i s i o n , V92, No. ST5, October 1966.  21.  Soosaar, K., " O p t i m i s a t i o n of t o p o l o g y and geometry of s t r u c t u r a l frames," Dept. of C i v i l Eng., Massachusetts I n s t i t u t e of Technology, May 19 67.  22.  ICES STRUDL-11, E n g i n e e r i n g Users Manual i n two volumes, Department of C i v i l E n g i n e e r i n g , S c h o o l of E n g i n e e r i n g , Massachusetts I n s t i t u t e of Technology.  23.  Johnson, W. "Upper bounds t o the Load f o r the T r a n s v e r s e Bending o f F l a t R i g i d P e r f e c t l y P l a s t i c P l a t e s , " Int. J . Mech. S c i . 1969, V o l . 11, pp. 913-938.  24.  Duncan, J.P.,"An Optimal concept f o r the support of L i g h t w e i g h t M i r r o r S u b s t r a t e , " Departmental Report, M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia.  "Plane-  O P T I M U M  M I C H E L L  F R A M E S  BY ERIC WILLIAM JOHNSON B.Sc. (Eng), London, 1950 M.Sc. U n i v e r s i t y o f A l b e r t a , 1965 VOLUME 11  TABLE  OF  C O N T E N T S FOR  APPENDIX  Appendix  Page  A  NOTATION  B  EQUATIONS STRUCTURE  C  FOR  STRUCTURES  GOVERNING  1  GEOMETRY  OF 6  COMPUTER PROGRAMMES  FOR  STRUCTURE  DESIGN  17  D  FORCE  SYSTEM  E  DATA  F  BIAXIAL  G  D E T A I L S OF  H  MANUFACTURE  J  D E F L E C T I O N OF  FOR  IN MICHELL  CANTILEVERS  SELECTED MICHELL STRESS  CANTILEVERS  IN JOINTS  COMPARABLE OF  . . . .  . .  STRUCTURES  PHOTOELASTIC UNIFORMLY  . . .  MODELS  25 42 75  . . . . .  83  . . . .  97  LOADED P L A T E S  . . .  118  L I S T OF FIGURES OF APPENDIX Figure  Page  Al  Notation f o r Michell  structures  2  A2  Members i n a t y p i c a l p a n e l  3  Bl  Members i n f a n  6  B2  Typical  7  B3  Joints  B4  Five  B5  Deflection  Dl  Forces acting  D2 D3  F o r c e s a t j o i n t J„„ NN Forces a t a t y p i c a l inner  D4  Forces a t a t y p i c a l  'A  , . . a, ±  34  D5  Forces a t a t y p i c a l  'B' f a n j o i n t , J ^ ^ . .  36  D6  Forces a t j o i n t J ^  D7  Freebody diagrams f o r forces  Fl  Biaxial  F2  Approximation  q u a d r i l a t e r a l panel i n typical Michell cantilever  fibre  . . .  9  symmetrical c a n t i l e v e r  12  of a M i c h e l l  15  frame  on a M i c h e l l c a n t i l e v e r  ...  28 29  J  stresses  1  joint J ^ . . . .  fan joint, J  at a typical  to a b i a x i a l l y  32  37 a t supports. joint  . . . .  .  39 76  stressed  joint  77  F3  E x t e n s i o n o f a t y p i c a l member  80  Gl  Warren t r u s s  84  G2  W i l i o t diagram f o r d e f l e c t i o n o f a Warren truss . . . Two b a r c a n t i l e v e r  86 87  G3  Figure  Page  G4  Cantilever  of p a r a b o l i c  G5  Triangular  plate cantilever  G6  I-beam c a n t i l e v e r  HI  The e f f e c t o f r o t a t i o n  H2  Geometry o f c u t t e r o f f s e t s  H3  Order and d i r e c t i o n of c u t s f o r machining a typical Michell cantilever  Jl  section  Loaded d i s c and supports  89 91 94 101  . . . . . . .  102  .  105  1  APPENDIX A NOTATION FOR STRUCTURES  A j o i n t c o u l d be i d e n t i f i e d by i t s c u r v i l i n e a r c o o r d i n a t e s , a and 3. the angles w i l l ,  T h i s w i l l however prove cumbersome  i n g e n e r a l not be i n t e g r a l  since  values.  A more convenient n o t a t i o n i s used i n t h i s work and i s i l l u s t r a t e d i n F i g u r e s A i and A2.  The support p o i n t s are  i d e n t i f i e d by the l e t t e r s A and B and from these r a d i a t e chains of members.  These are numbered c o n s e c u t i v e l y , commencing with  the base curves ACE and BCD.  In the g e n e r a l case  'a' i s used  to i d e n t i f y p o i n t s on the 'A' chains and 'b' f o r those in, the * :  1  B' c h a i n s .  Joints Joints intersecting letter  are i d e n t i f i e d by the numbers o f the l i n e s  a t t h a t j o i n t , w r i t t e n as s u b s c r i p t s t o the c a p i t a l  ' J ' . The g e n e r a l j o i n t i s thus w r i t t e n ' J ^'. * I t should be noted t h a t the c u r v i l i n e a r c o o r d i n a t e s of the j o i n t s can be o b t a i n e d from these numbers. I f 8 i s the c o o r d i n a t e change between members, (the angle between members), then the c u r v i l i n e a r c o o r d i n a t e of a l l p o i n t s on the c h a i n a i s g i v e n by (a-1) 6 Al } (b-l) 6 Similarly  Figure A l  Notation  for Michell  Structures  3  F i g u r e A2  Members i n a T y p i c a l  Most j o i n t s  are i d e n t i f i e d  symmetrical  structure,  Such j o i n t s  may be i d e n t i f i e d  on F i g u r e A l .  there are pairs  o f 'mirror image  a r e such  Joints with equal subscripts i n a symmetrical  In a 1  joints.  by e x c h a n g i n g t h e s u b s c r i p t s .  T h u s , as an example, J 3 7 a n d  CL  Panel  a pair.  l i e on t h e c e n t r e  line  structure.  Members A member i s i d e n t i f i e d its  inward  lying  end—that  by r e f e r e n c e t o t h e j o i n t a t  nearest the support p o i n t .  a l o n g t h e 'a' c h a i n a r e i d e n t i f i e d  b e r s a l o n g t h e 'b' c h a i n a r e i d e n t i f i e d also refer  to their  lengths.  Members  a s A.^ w h i l e t h e memas  B a  k*  These  symbols  W i d t h s o f Members - WA . — ab The cation  symbol  width  sA  by a d d i n g W t o t h e i d e n t i f i -  f o r t h e member.  Cross-sectional  The  i s indicated  A r e a o f Members - sA , ab  cross-sectional  area i s indicated  b y t h e symbol  , . ab I n most c a s e s t h e members a r e assumed t o be o f  uniform thickness  Then  t.  sA , ab  Force Acting  =  t  (WA , ) ab  on a Member  -  F  A a  E a c h member i s s u b j e c t e d tensile  o r compressive, a c t i n g  indicated its  by t h e symbol FA^.  character,  forces  negative, The  A2.  tensile  3  t o an a x i a l  along  force,  t h e member.  either  This  of the force  force i s  indicates  b e i n g p o s i t i v e and c o m p r e s s i v e  the normal  members o f a t y p i c a l  The q u a n t i t i e s  j  The s i g n  forces  following  . . . . A2  relating  convention.  p a n e l a r e shown i n F i g u r e  t o t h e s e members a r e l i s t e d : ,  below. Normally, the forces and  t h o s e i n 'B' members a r e c o m p r e s s i v e .  however be r e t a i n e d for  i n t h e 'A' members a r e t e n s i l e  clarity  t o minimize confusion,  when c o n s i d e r i n g  oblique  The s i g n w i l l . and i s n e c e s s a r y  loadings.  5 Member  RS  QR  QT  TS  Name  A  ab  B  a,b+1  a ,b  A  a+l,b  Length  A  ab  B  a,b+1  a ,b  A  a+l,b  WA , ab  Width Area (crosssection) Force  s A  ( + ) F A  a ,b+l  ab  s B  a,b+l  (-) FB  ab  (  a ,b+l  -  WB  , a,b  WA ^, , a+1 ,b  sB  , a ,b  sA  ) F B  a,b  (  +  >  F  A  ., ,  a+1 ,b  a l,b +  Members i n F a n s The those  members i n t h e f a n s a r e s i m i l a r l y  d e s c r i b e d above.  b = zero while each f a n a r e A widths,  areas  from a,o  F o r a l l members r a d i a t i n g  B, a = z e r o .  and B  d e s i g n a t e d as  ,. o,b  Thus t y p i c a l  f r o m A,  members i n  The same p r e f i x e s a r e u s e d f o r c  and f o r c e s a s f o r t h e o t h e r members.  6  APPENDIX B  EQUATIONS GOVERNING  GEOMETRY OF STRUCTURE  LENGTHS OF MEMBERS  The derived  a)  lengths  o f t h e members o f t h e s t r u c t u r e may be  as f o l l o w s : -  Fans  Figure  Bl  Members i n F a n  As e a r l i e r  stated  Bo.b »  Bl  From F i g u r e B l  A,.  b  -  8  sinf?  Ba,, = 2r sirx(|) A  B2  7 b)  Remainder of  Figure  B2  shows a t y p i c a l  the  structure, typical  The  general  titles  of  Figure  B2  Construct to  Structure  of the  TW  q u a d r i l a t e r a l panel  a l l panels  outside  joints  s i d e s are  and  the  base  of  fans,  indicated.  T y p i c a l Q u a d r i l a t e r a l Panel  perpendicular  t o RS  and  QX  perpendicular  QR. Then f r o m t h e AaM.b  geometry of ~  figure  Aq,t> + Ba.b siaB cos  Ba.b*i =  the  Aa.b  Q  SitxQ +  cos &  B3  Bo.t.  8  Once t h e l e n g t h s o f t h e c i r c u m f e r e n t i a l A  ( l  b)  a  n  d  each p a n e l  B  ( a 1)'  a  r  e  known e q u a t i o n s  i n turn starting  f a n members,  B3 may be a p p l i e d t o  f r o m C and w o r k i n g o u t w a r d  ina  s y s t e m a t i c manner.  Joint  Coordinates  A typical Michell The  c o o r d i n a t e s o f e a c h j o i n t may be f o u n d  each f i b r e used  framework i s shown i n F i g u r e B3.  i n sequence from  for this  angles  purpose,  o f both  follows;  but the sign  sets i s indicated  for determining The  the support.  by w o r k i n g Either  convention  since these  along  s e t may be  used  f o r the  are also  needed  deflections.  formulae  different  for calculating  formulas  ^ and ^  B a  b  a  r  e  a  s  being required f o r the f a n  members.  'A' Members  TAa.o VAa.b 'B'  =[(a-l)8-pj  =[(Q-b)&-(P+|)]  f a n members  only  . . B4 other  members  Members f a n members  only  . . B5  •VB . = [(b-a)0*(p-f)] a  b  other  members  Figure B3  Joints in Typical Michell  Cantilever  10  J Once these angles are determined any  the c o o r d i n a t e s o f  joint follow.  XJ . Q  B  =  Q  + baO  B6  1 A.b sia(^A .b) a  0  baO  '  SPAN AND RISE OF STRUCTURE  Equations B6 may be used t o express  the span, L, and  r i s e , Y, i n terms o f member l e n g t h s and angles i|>A and hence u l t i m a t e l y by equations B l t o B4 i n terms o f 8, r,. and r_.. A  F i g u r e B4 shows a f i v e f i b r e symmetrical of each member are t a b u l a t e d  below.  structure.  B  The lengths  To save space s i n 8 and  cos 8 are w r i t t e n r e s p e c t i v e l y as s and c w h i l e  (2r s i n 8/2)  i s w r i t t e n as f . From these t a b u l a t e d v a l u e s , e x p r e s s i o n s may be• d e r i v e d f o r the span, L, i n terms o f the member l e n g t h s , which i n t u r n reduce The  t o the f a n r a d i u s r and the f i b r e angle 8.  angle ACO i s 45° by d e f i n i t i o n  w h i l e the angle n between  the a x i s and each s u c c e s s i v e A member i s g i v e n by n =  (45-8/2).  Such e x p r e s s i o n s may be w r i t t e n f o r v a r i o u s v a l u e s o f N by c o n s i d e r i n g K, M, N and L i n s u c c e s s i o n as the o u t e r p o i n t o f the span.  11  Member A i.o  4  Length  o As.c  BI.I "fe /(L+s) - f c  A^.i  A3.1  / ( s + 2s + s + sc + c) 4 - c  A .z  z  3  3  J ( s + s + s ' c + c*) + c  B3.2  3  2  A4.I  f () + 5 + S C - r S C )  C  J  As..  3  i ( l + s+ s c + s c + s c ) - j - c z  3  J (S + S* + S*C * S C*+ C ) -r C Z  3  3  4  4  j ( l + S + S + 2 s + 3 + SC->2s*c) - r C i  3  4  J- ^2s +3s*+ s + sc •+• s c + c ) 3  B4.3  2  4  3  2  } (2s + 4s*> 2s •+ sc + 2s c + s c * 5 c*+ c ) -r c 3  3  / ( l + 2s + 2s*+ 3 s + 3s + 3  4  3  3  5  5  s c •+ 3sc+ 2s c) -r c 3  5  J^2s+5s*> 5s +4s + 35 +6 + s c + 3 s c + s V + 3s*c + 3 s ^ + c ) -i- c 3  B4.4  4  £  4  3  A54  4  f(s+ 3s + 3s -!- s + s c + 2s c + s?c\ s*c+ c 3 ) -r c  A5.2  A 5.3  c  3  4  6  j£l+2a + 3s + 6s + 6 s 2 s * + s c + 4s*c + a  ^(  2 s  3  4  + sc^sV+s^e)  * 7 s * IOs + 9s +7s + 4 s * + s + sc + 4 s c + s c V ^ ' c ] z  3  4  s  7  igure B 4  F i v e F i b r e Symmetrical  Cantilever  N»2. fsU5 L ~  t-cos45 +cos(45-|)|I (s 2s +s +sc tcos> 45 + cos  s  +  a  2  sc H-c c )]  a  l  +  +  ?  | 2 L (2s + S + Sc + c + i)J 2  N- + L = > c o s 4 5 + c o s ( 4 5 - | j — 11 + 2s + 2s + 3s-*- 3s*+ s + sc+ sc' z  3  5  U 3sc+2&\?£c* sc + s\ 3  = t COS 45 + c o s ( 4 5 - | )  -  "f C O S 45 + cas(+S-QJ |  3  •+ 2 , s c  7  + s  l  I  L = t cos 45 + cos(4S-|)  sc< c + 3  +  +  ,  [2+2s+2s*+ s -»- 2 s - s - ( - 2 s c  £  N=5  z  4  3  s  c + c  ks f 7&S /o^+9^+75 +4s^s +sc + 2sc :>  7  2  *2s\\ 3s*c -*3sV+ sc* 2sV»+ sfc*+ s V 5  + c^-* c \  +  c^-*-  I + 4&i + 4sc -c 4sh -»•  + 2s»<^ s  Over t h i s l i m i t e d range of N no coherent p a t t e r n i s v i s i b l e i n these e q u a t i o n s , e i t h e r as d e r i v e d ( f i r s t l i n e ) o r  2  2  a f t e r r e d u c t i o n by use o f the equation s i n 8 + cos 6 = 1 . I t may w e l l be t h a t a more e x h a u s t i v e a n a l y s i s o f the geometry w i l l y i e l d a g e n e r a l equation but the v e r y nature o f the cons t r u c t i o n seems t o make t h i s r a t h e r u n l i k e l y .  sc */ 4  14 D e f l e c t i o n o f the Structure The  change i n l e n g t h  o f e a c h member i s known o n c e t h e  g e o m e t r y o f a framework h a s b e e n d e t e r m i n e d . strain  i n a l l members i s a d e s i g n  length  o f a member i s s i m p l y  Ae the  s i g n being  = ± £A  Since  uniform  r e q u i r e m e n t , t h e change i n  • ..... sa  Qb  t h e same a s t h a t o f t h e f o r c e FA ^ a c t i n g o n t h e  same member. The  d e f l e c t i o n may t h u s be d e t e r m i n e d b y t r i g n o m e t r i c a l  s o l u t i o n o f a W i l i o t diagram. framework a s shown i n F i g u r e t h a t member A , , (a,b-l) compression.  joint will  joints J  will  i s i n tension  represented  J , , , and J , , (a,b-l) (a-l,b) 4  diagram  fixed origin  of  this  point  t h e members w i l l  suffer  b y PQ and PS r e s p e c t i v e l y and 1  r e l a t i v e to the  The d e f l e c t i o n s o f t h e s e J ^.  i s shown a s F i g u r e  ( t h e s u p p o r t p o i n t s A and B ) . diagram represents  J  joints  A p o r t i o n of. t h e B5b.  showing d e f l e c t i o n o f each j o i n t  on t h i s  assume  a n d member B, , is in (a-l,b)  c a u s e a f u r t h e r movement o f j o i n t  a vector  of a Michell  For convenience,  move t o a new p o s i t i o n J  corresponding W i l i o t diagram  a  B5a.  As a r e s u l t o f t h e s e f o r c e s  changes i n l e n g t h the  Consider a portion  The  This i s  relative to coordinates  the d e f l e c t i o n of  * the  corresponding  point  i n t h e framework.  In t h e s e d i a g r a m s iJ;A and ibB a r e b o t h shown p o s i t i v e as d e f i n e d i n F i g u r e B3. I t may be shown t h a t t h e e q u a t i o n s s t a t e d b e l o w a p p l y e q u a l l y f o r c a s e s where t h e s e a r e n e g a t i v e .  Figure  B5  Deflection  of  a Michell  Frame  From t h e f i g u r e : -  Xab = *a.b-. Aa,b-. cos^Aa.b-, + QR sinS'Aa.b-! + £  Xo...b-^B ./.fc>s/Viiy6 . .6 5 R cosVBo-i.6 +  s  0 /  a  B9  Yab= Ya.b-/ - HAo.b., S/Vl^Ap.bw + QR^StPAo.b., = Yo.,J, eBa.,.6 «« *B^.+ SR sin 4>B«W.A +  6  In t h e s e  equations  a l l t e r m s a r e known e x c e p t t h e  l e n g t h s QR a n d SR w h i c h may b e e l i m i n a t e d b y c r o s s - s u b s t i t u t i o n . The  following  X |>= Xo.b-, f A . b - , +  Q  r e s u l t s a r e thus  C o s q J A  Q  obtained:-  o i > - , - K eih. q > A . - , Q  b  . . . . Y  q b  = Y .b,, q  -EAa.b., siatfVWb-,  where K r e p r e s e n t s  K -  K «»» 4>A .b-i Q  the following:-  (Xq-i.t> -X .b-i)sinq>e>. Q  -  q (i>  -  EBq-i.i, + (Tra.b-i~YQ..)oosrBo..t l  (  « » • (H»Aa.b-i + SV&a-i.b)  These e q u a t i o n s joint  i nturn,  are unstressed, equal  t o zero.  BIO  are applied  J  t h e same e q u a t i o n s  Bll  sequentially  working outward from C ( ] _ j ) -  b  t o each  *f Y a n  may b e u s e d w i t h  members e placed  /  17  APPENDIX  COMPUTER PROGRAMMES FOR  The  'DESIGN OF  equations described  Appendices A  and  B were d e r i v e d  structures.  The  data  compiled  w i t h the  C  MICHELL STRUCTURES  i n C h a p t e r s 2 and  from study o f  for six representative  a i d of a desk c a l c u l a t o r .  were t h e n c h e c k e d by  large  scale graphical  3  and  generalized structures  was  These r e s u l t s solutions  wherever  possible. I t was impractical  evident  f o r the  s o l u t i o n of  analyzed  i n Chapter  designed  to produce the  tions being  used  The  4.  routines  of  The  the  wide range o f  Computer programmes were required  to check these  R a t h e r e a c h s e c t i o n was calculation  t h a t t h e s e m a n u a l methods w o u l d  were n o t  data,  written  comments.  calcula-  as  a single unit. to  the  quantities.  f o l l o w i n g programmes a r e  e a c h segment.  thus  earlier  proceeding  s i n c e many s u b t i t l e s have b e e n i n c l u d e d of  structures  programmes.  proved before  succeeding  the  be  Some n o t e s a r e  fairly  straightforward,  to e x p l a i n  the  appended t o a m p l i f y  purpose  these  18 PROGRAMME 1 - SYMMETRICAL  As  explained  i n Chapter  framework i s c o m p l e t e l y spacing, fibre  9 , may be u n i q u e l y d e t e r m i n e d  f o r s t u d y may The  considered Steps  support  1-32  these  the  f o r c a l c u l a t i o n and t h e manner i n w h i c h i s printed.  commence  at step  reset  33.  of  f o r e a c h new  structure  H e r e t h e memory  areas  t o z e r o and N, L and D a r e s p e c i f i e d ,  Member t h i c k n e s s , are  nature  t h e s i z e o f memory a r e a s s e t  The c a l c u l a t i o n s  are  be  sections:-  specifying  data  33-87  variables  t h i s r e s u l t may  T h e r e a r e o f an a d m i n i s t r a t i v e  aside  The  calculated.  programme w h i c h a c h i e v e s  i n the following  from  w i d t h s and a l l o t h e r  t h e n e a s i l y be  nature  b)  Michell  s p e c i f i e d once t h e span, L,  p a r a m e t e r s and member f o r c e s ,  a)  2, a s y m m e t r i c a l  D, and number o f f i b r e s , N, have b e e n s t a t e d .  angle,  required  CANTILEVERS  a l l stated  o, and s t r a i n , e,  here as i s t h e s p e c i f i c g r a v i t y  the material.  weightless,  T, s t r e s s ,  I f the structure  SPGTY  i s s e t equal  i s assumed  to zero.  The  l e n g t h s o f t h e f a n members, A , » and B, , > (a,o) (o,b) 3  are  c)  88-120  calculated,  This is  here  since  these are independent of  i s t h e 'heart' that  o f t h e programme  as i t  8 i s calculated.  A value of 6 i s  assumed a n d t h e c o r r e s p o n d i n g  span, TL, c a l c u -  19 lated If  ( s t e p 101 ).  smaller than  TL i s t h e n  compared w i t h L .  L , 8 i s i n c r e a s e d by 6 8  (DELTHE) and t h e programme r e t u r n e d If  TL i s l a r g e r  reduced  than  t o s t e p 72 .  L, 8 i s t o o l a r g e so i t i s  and DELTHE i s d i v i d e d b y TEN b e f o r e  returning  t o s t e p 72 .  I n t h i s way t h e e x a c t  value o f 8 i s approached iterative  process  from below.  continues  until  This  t h e ERROR o r  L TL —  —^—  i s less  been  repeated. As  than  part of this  members  The a n g l e s defined  the 144-209  T|»A  b  The l a s t  (PSIA) a n d iJ;B  (PSIB) a s This i s  forces.  i n A p p e n d i x D. this N  ab  preliminary to the c a l c u l a t i o n of  f r o m t h e somewhat u n w i e l d y  N  steps of t h i s  lengths.  The f o r c e s i n a l l members  J  c y c l e s have  i n F i g u r e B3 a r e c a l c u l a t e d .  an e s s e n t i a l  e)  these  forty  loop, the lengths of a l l  are calculated.  section print d) S t e p s 121-143  0.0001%,or  arecalculated  equations  The l o a d s a r e f i r s t  described read i n ;  programme p r o v i d i n g f o r a s i n g l e i n any d i r e c t i o n .  s t e p s 167-169 p e r m i t s tiple  load a t  A simple m o d i f i c a t i o n o f t h e i n t r o d u c t i o n o f mul-  loadings, or loads a t other  j o i n t s , as  may be d e s i r e d . f)  210-257  J o i n t c o o r d i n a t e s , X J , , . a n d Y J , , and j o i n t d e f l e c t i o n s , X, , a (ab) n d Y, , c a n now be ' (ab) ( a b ) ' (ab) w  J  s  w  calculated.  Although  the d e f l e c t i o n i s inde-  pendent o f the magnitude o f the f o r c e on  a member, t h e n a t u r e  or  compressive—must  of the f o r c e — t e n s i l e  be known t o d e t e r m i n e  w h e t h e r t h e member e l o n g a t e s o r Steps sign  acting  contracts.  215-222 c h e c k t h i s a n d a s s i g n t o t h e change i n l e n g t h  the correct  and t h u s t o t h e  deflection. g) S t e p s  258-End  The r e m a i n d e r o f t h e programme member w i d t h s , properties  cross-sectional  a simple  The  final  t h e volume o f t h e t e n s i l e  o f t h e c o m p r e s s i v e members  sum, a f t e r w h i c h t h e t o t a l * found.  for  a r e a s and o t h e r  and t h e s u p p o r t r e a c t i o n s .  segment c a l c u l a t e s and  calculates  The s p e c i m e n programme i n c l u d e s c a n t i l e v e r f o r w h i c h N = 2.  and o f t h e i r  s t r u c t u r a l weight i s  a printout  of data  PROGRAMME 2 - SKEW MICHELL CANTILEVER  In parameters, are  a skew s t r u c t u r e  t h e x and y c o o r d i n a t e s o f t h e o u t e r end J  c a l l e d XSPAN  variables,  t h e span L i s r e p l a c e d  and YSPAN  N  N  -  These  i n t h i s programme w h i c h c o n s i d e r s  0, and t h e r a t i o o f t h e f a n r a d i i , Several  by two  methods were t r i e d  THETA and RADRAT w i t h v a r y i n g  success.  called  two  RADRAT.  f o r the c a l c u l a t i o n of B o t h XSPAN  and YSPAN a r e  unknown f u n c t i o n s variable first  could  used  oscillated  o f THETA and RADRAT, and c h a n g e s i n e i t h e r  affect  failed  both  a b o u t them, r e p e a t i n g  Trial  The i t e r a t i v e  procedures  t o c o n v e r g e on t h e c o r r e c t v a l u e s and u s u a l l y  t e r m i n a t e d by a s a f e t y  examples  lengths.  t h e same c a l c u l a t i o n s  counter.  and e r r o r g r a p h i c a l  indicated  until  that  small  o f some  typical  c h a n g e s i n THETA p r i m a r i l y  XSPAN w h i l e YSPAN was c o r r e s p o n d i n g l y I t was t h e r e f o r e  solutions  decided  affected  s e n s i t i v e t o RADRAT. that  the c o r r e c t i o n s  t o be  made t o t h e e s t i m a t e d v a l u e s o f THETA and RADRAT s h o u l d be related  t o the corresponding An  error  errors  i n XSPAN and YSPAN.  term f o r each o f these  lengths  was  thus  calculated  ERRX»  XSPAN - TLX XSPAM  ERRY*  YSPAN-TLY  . . . .  YSPAK  where TLX and TLY a r e t h e d e r i v e d and  rise  values  of the s t r u c t u r e .  N o r m a l l y ERRX  v a l u e s o f t h e span  and ERRY w i l l  have  i n t h e range  "l-O ^  E1RRX <  a minus s i g n large.  ci  This  error  +  l ' 0  . . . . C2  i n d i c a t i n g that, t h e e s t i m a t e  i s too  t e r m was t h e n u s e d t o c a l c u l a t e t h e c o r r e c t i o n  t o THETA o r RADRAT by u s e o f t h e t y p i c a l  statement  22  THETA * THETA -> (ERRX * DELTHE)  ••••  DELTHE i s s p e c i f i e d i n t h e programme p r e f a c e c o n v e n i e n t s m a l l a n g l e ( t y p i c a l l y 1.0 d e g r e e ) .  0 3  as a  S i m i l a r pro-  cedure a p p l i e s t o RADRAT. Provided  t h a t THETA and RADRAT a r e c l o s e t o t h e  t r u e v a l u e , t h i s r o u t i n e converges q u i t e r a p i d l y and a u t o m a t i c a l l y c o r r e c t s f o r over and under e s t i m a t e s ,  since the correc-  t i o n changes s i g n w i t h t h e e r r o r term. However, i f t h e assumed v a l u e s o f THETA and RADRAT . are remote from t h e t r u e v a l u e , t h e e r r o r term c a n be l a r g e ( g r e a t e r than ±1.0) and t h e c o r r e c t i o n i s c o r r e s p o n d i n g l y coarse.  T h i s may be so l a r g e t h a t t h e new v a l u e o f THETA o r  RADRAT i s even f u r t h e r from t h e ' r o o t ' b u t on t h e o t h e r of i t .  side  I f t h i s o c c u r s t h e e r r o r f u n c t i o n w i l l be a l t e r n a t e l y  p o s i t i v e and n e g a t i v e procedure i s divergent The  and o f ever i n c r e a s i n g s i z e .  The  and no s o l u t i o n i s o b t a i n e d .  programme was thus m o d i f i e d  t o check t h i s , by •'  comparing t h e magnitude o f t h e e r r o r f u n c t i o n s i n s u c c e s s i v e cycles. divergent  I f e i t h e r increase, i n d i c a t i n g the s t a r t o f a s o l u t i o n , the next c o r r e c t i o n i s halved.  This  has been found t o be most e f f e c t i v e a l t h o u g h i t i s p o s s i b l e t h a t t h e use o f a v e r y poor p r e l i m i n a r y e s t i m a t e  o f THETA  and RADRAT c o u l d would this  still  diverge.  regard.  1 a n d may  generate e r r o r functions Use o f F i g u r e  The programme  be s i m i l a r l y  4.13  so l a r g e t h a t should  i s otherwise  they  assist in  similar  to  Programme  d i v i d e d , as f o l l o w s :  a  1-33  Administrative  b  34-66  S p e c i f i c a t i o n o f N, XSPAN, YSPAN, SIGMA, STRAIN and SPGTY„  c  67-149  Iterative described  d  150-191  PSIA(ab)  e  192-234  Forces  f  235-283  Joint coordinates  g  284-END  Member p r o p e r t i e s , r e a c t i o n s and v o l u m e s .  to these  W,  f o r THETA and RADRAT  and d e f l e c t i o n s .  Variants  was p r o d u c e d b y m i n o r now  be  modifications  described:-  Structures  a t the beginning  o f s e c t i o n e , was  placed  equal  zero.  Variable The entering ing  T,  i n members.  programmes, w h i c h may  Weightless  to  data  routine above.  D,  andPSIB(ab).  Minor Specialized  and Format.  Strain  results  i n Table  the d e s i r e d value  11, A p p e n d i x E were o b t a i n e d  of strain  from i t the corresponding  value  as an i n p u t of s t r e s s .  and  by  calculat-  24 Direction The  o f S e l f Weight  equations  i n A p p e n d i x D, a l l o w  Forces  f o r the forces for a variable  attraction,  by t h e i n t r o d u c t i o n  (THGRAV) .  The s t e p s  incorporating  this  i n section  described  was o b t a i n e d  of  of a variable  i n Chapter  i s a part  System), a very  Unlike  angle,  e were r e p l a c e d  o f ICES  matrix  (Integrated  by  Civil  complex computer r o u t i n e  capable  types o f s t r u c t u r e . the routine  described  of the structure  a known s t r u c t u r e  above which  as i n p u t  together with  b u t i t does t h i s  inversion, determining  moments and r o t a t i o n s Full  structures  (STRUctural  shear  calculate  the j o i n t the c r o s s - s e c t i o n a l  a r e a a n d s e c o n d moment o f a r e a o f e a c h member.  by  by a s e t  5, d a t a f o r r i g i d  member p a r a m e t e r s , STRUDL r e q u i r e s  analyses  + ,  STRUCTURE  b y u s e o f t h e STRUDL programme  s o l v i n g many  coordinates  d i r e c t i o n of g r a v i t a t i o n a l  4 - RIGID JOINT  D e s i g n Language) which Engineering  derived  value.  PROGRAMME  As  i n members, a s  I t thus  comprehensively  forces,  bending  o f t h e members a t e a c h r i g i d  d e t a i l s o f STRUDL may be f o u n d  MIT, where t h e s y s t e m was  joint.  i n t h e handbooks  developed.  published  FORTRAN  IV  G COMPILER C C  0001  0002 0CC3 0004 0005  1 3  0006 00C7 0008  9  0009 0010  13 15  0011 0012  17 19  0C13  21  0014  23  0015  25  0016 0017 0018  27 29 31  0019  33  0020 0021 0022  35 37 39  11  MAIN  04-29-70  19:28:53  PAGE  0001  PROGRAMME F O R T H E S O L U T I O N O F S Y M M E T R I C A L M I C H E L L C A N T I L E V E R S . I N C L U D I N G THE E F F E C T OF S E L F W E I G H T F O R C E S . O I M E N S I C N A < 2 5 , 2 5 ) , E ( 2 5 1 2 5 ) , X( 2 5 , 2 5 ) , Y ! 2 5 , 2 5 ) , P S I A ( 2 5 , 2 5 > , 1 PSIB( 25,25) ,FA125,25),FB!25 ,25),T ILT125,25),SULCLAI 25,25), 2 LOAD! 2 5 , 25 ) , X J ( 2 5 , 2 5 ) , Y J < 2 5 , 2 5 ) , W A ( 2 5 , 2 5 ) , W B < 2 5 , 2 5 1 , A R E A A ( 2 5 , 2 5 3 ),AREABI25,25),MIZA!25,25),MIZBI25,25I,EULCLB(25,25), 4 H T I N A 1 2 5 , 2 5 ) , W T I N B ! 2 5 ,2 5 ) , E A ( 2 5 , 2 5 ) , E B ( 2 5 , 2 5 ) REAL L . L O A O , I T E H L , I T E M R , M I Z A . M I Z B FORMAT, 13) FORMATIF15.6) FORMAT(1H0,4HN = , I 3 , 5 X , 4 H L = , F 1 0 . 6 , 5 X , 4 H 0 = . F 1 0 . 6 . 5 X , 1 6HL/D = ,F10.6,5X,12HTHICKNESS = ,F10 .6,5X,10HSP.GTY. = , F 6 . 3 ) _ F O R M A T ( !>IO,_8Hj;HETA_ ,F 2 0 . 6 , 1 5 X , 8H0ELTA = , F 2 0 . 6 , 1 5 X , 8 H E R R 0 R = , 1 F20.6,/) F O R M A T ! 1 H C 5 0 X , " L E N G T H S OF M E M B E R S ' / I F O R M A T I 2 5 X , 2 H A t ,I 2 , I H , , I 2 • 4 H ) = , F 1 5 . 6 , 2 0 X , 2 H B ( , 1 2 , 1 H , , 1 2 , 4 H ) = , I F15.6) F O R M A T , 1 H C 2 5 X , ' J O I N T C O O R D I N A T E S ' , 4 0 X , " J O I N T C E F L ECT I O N S « / ) F0RMAT(6X,3HXJ( , 1 2 , I H , , 1 2 , 4 H ) = , F 1 2 . 6 , 8 X , 3 H Y J ( , I 2 , 1H,,12,4H) = , 1 F12.6 ,9X,2HX<, 1 2 , I H , , 1 2 , 4 H ) = , F l 2 . 6 , 8 X , 2 H Y ( 12,1H, , 12,4H) = 2 F12.6) F0RMATI2F15.6) F0RMAT(1H0,20X,5HL0AD(,I2,1H,,I2,4H) = ,F15.6,20X,5HT ILT(,12,IH,, 1 12,4H) = . F 1 5 . 6 , / ) FORMAT ( I H 0 , 6 X , ' M E M B E R ' , 7X , ' F O R C E ' , 7_X , • H I D I H ' . 7X , ' A R E A ' , 3 X , 1 * SECOND MOMENT* , 8 X , ' M E M B E R ' , 7 X , " F O R C E ' . 7 X , ' W 1 D T H ' , 7 X , ' A R E A * , 3 X , 2 'SECOND MOMENT'/) F O R M A T ( 1 H 0 , 3 X , 3 H A 1 , 12 , 1 H , , 1 2 , 1 H ) , I X , 4 F 1 2 . 6 , 9 X , 3 H B ( , 1 2 , I H , , 1 2 , 1 IH),1X,4F12.6) F O R M A T ( 1 H 0 , 5 X , ' V O L U M E T E N S I L E MEMBERS= • , F 9 . 6 , 5 X , 1 * VOLUME C O M P R E S S I V E M E M B E R S ' , F 9 . 6 , 5 X , ' T O T A L VOLUME= • , F 9 . 6 ) F0RMATIF15.6) F0RMAT(F15.6) F O R M A T { 1 H 0 . 2 4 H M G D U L U S OF E L A S T I C I T Y = . F 1 0 . 1 , 1 X . 6 H P . S . 1 5X, 1 17HUNIF0RM STRESS = , F 1 0 . 6 , 1 X , 6 H P . S . I . , 5 X , 2 1 7 H U N I F G R M S T R A I N = , F 1 0 . 6 , IX , 1 2 H INC HE S / 1 N C H . ) . . FORMAT(2X,'MEMBER' ,6X. ' F O R C E ' , 5 X , ' W I O T H ' , 4 X , ' A R E A ' , 5 X , ' I' , 5 X , 1 ' W T / I N ' , 3 X , ' R A T I O • , 5 X , ' M E M B E R • , 6 X , ' F O R C E • , 5 X , •WlDTH' ,4X, • A R E A ' , 2 5X,'I», 5X,'WT/IN«,3X,'RATIO',//) FORMAT! 1 H 0 , 5 0 X , 'WEIGHT= ' . F 1 5 . 6 ) FORMAT(F15.6) FORMAT!IH0.40X,'THE FOLLOWING RESULTS ARE C A L C U L A T E D • , / , 4 I X , 1 I F O R . A S T R U C T U R E MADE FROM CR39 PLASTIC',/,43X, 3  2 3 4  0023 0024  "41 43  0025  45  0026  49  0027  51  'RAVING A SPECIFIC GRAVITY  OF  i.3i',7,42x,  'THE EFFFCTS OF SELF WEIGHT FORCES•,/,43X, ' A R E INCLUOED IN THESE RESULTS.',//) F O R M A T ! 1 H 0 . 4 5 X , ' R E A C T I O N S AT S U P P O R T P O I N T S ' ) F O R M A T l 1 H 0 . 2 5 X , ' A T A , H O R I Z O N T A L COMPONENT = « , F 1 2 . 6 , 1 0 X , i_ ' V E R T I C A L COMPONENT = ' , _ F 1 2 . 6 I _ _ _. F O R M A T l IH0.25X,'AT B , H O R I Z O N T AL COMPONENT = • , F 1 2 . 6 , 1 0 X , " 1 ' V E R T I C A L COMPONENT = « , F 1 2 . 6 ) F O R M A T ( 2 H A ( , 1 2 , I H , , 12,2H» , F 1 1 . 6 , F 9 . 6 , 3 F 8 . 5 , F 8 . 4 , 3 X , . 2HBI , I 2 , 1 H , , I 2 , 2 H ) ,F11.6 , F 9 . 6 , 3 F 8 . 5 , F 8 . 4 ) FORMAT! 1 H C 5 9 X , ' N O T E ' , / , 4 5 X , ' T H E TERM RATIO I N THE F O L L O W I N G ' , 1 / , 4 3 X , * TABLE INDICATES THE PERCENTAGE RAT 1 0 • , / , 4 5 X ,  FORTRAN  IV  G COMPILER  MAIN  04-29-70  19:28:53  MEMBER TO I T S ' , / , 4 5 X , ' C F T H E FORCE IN EACH FOR BUCKLING',/,52X, 'EULER CRITICAL LOAD • < E * I * l P l » * ( P I ) ) / ( L * L > ') 53 FORMAT!1H0,//) PI=3.14159265 PI4=0.25*PI R00.T=S.QK.T..(.2...aL 990 CONTINUE 999 CONTINUE K 1 = 2 Ll = 2 K=l A L L T H E A R R A Y S A R E Z E R O E D OUT F I R S T . DO 10 1 = 1 , 2 5 DO 10 J = l , 2 5 A(I,J)=0.0 B ( I , J ) = C.O XI I , J ) = 0 . 0 2 3 4  0C28 0029 0030 _0.O3.l-. 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 -J0_04.2_ 0043 0044 0045 0046 JIOAJL 0048 0049 0050 -0.0.5JL 0052 0053 0054 _a.05.5_ 0056 0057 0058 0059 _Q.0A0_ 0061 0C62 0063 0064 _0..0.63_ 0066 0067 0068  _y.u.t..J.i=o_..o. PSIAII,J)=0.0 PSIBII,JJ=0.0 10 CONTINUE N I S N U M B E R OF F I B R E S READ ( 5 , 1 ) N N.1.= N± .1 N2=N+2 NN=N-1 L IS LENGTH OF SPAN. READ(5,3)L D IS SUPPORT S P A C I N G READJ.5,3).D._... R A f 10 = L / D T I S T H I C K N E S S OF A L L M E M B E R S I N I N C H E S . T=0.25 C ELMOD IS MODULUS OF ELASTICITY. ELM0D=300000.C SXR_AJN_=__0_..OCJ . SIGMA = 3 0 0 . 0 W R I T E ( 6 , 3 1 1 EL M O D , S I G M A , S T R A I N C T H E S P E C I F I C G R A V I T Y ( S P G T Y 1 OF C R 3 9 P L A S T I C I S 1 . 3 1 SPGTY = 1 .31 WRITE(6,5)N,L,0,RATI0,T,SPGTY C=ISPGTY«62.4)/1728.0 R A O I A L I S L E N G T H O F A L L M E M B E R S R A O I A T ING FROM A AND B RADIAL=D/SORT(2.0) DO 2 0 1 = 2 , N l 11=1-1 J=l _J.J_LJ_!..  20 C  0069 _C  A(I , J)=RADIAL 8 ( J , I l=RADIAL CflNT I N U E E P S I S P E R C E N T ERROR EPS = 0.000001 C A L C U L A T I O N CF FAN ANGLE  PAGE  0002  FORTRAN  IV  G COMPILER C C  0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081 0082 0083 0084 0085 0086 0087 0088 0Q89 0090 0091 0092_ 0093 0094 0095 0096 _0097_ 0098 0099 0100 0101 0102 __0_103_ 0104 010? 0106 0107 0108 0109 0110 0 111 0112 0113 0114  90  3C  _50 40 C  04-29-70  19:28=53  F A N A N G L E , T H E T A , I S G E N E R A T E D BY AN I T E R A T I V E P R O C E S S , THE ' A P P R O X I M A T I O N ' T H E T A = Z E R O THETA = o . O DELTA = 1 0 . 0 CONTINUE K = K+1 I F I K . G T . 4 0 ) GO TO 1 0 0 TR=THETA*pi>180 .0 TR2=0.5*TR FACT=PI4-TR2 FACTl=PI4+TR2 A(2,2)=D*R00T*SIN(TR2) DO 3 0 I = 2 , N B(I,2)=A< 2,2) A<2, I I = A ( 2 , 2 ) CONTINUE DO 4 0 J = 2 , N J1=J+1 J2P_50 I=J1_,NL 11=1-1 A I I , J ) = ( A ( I 1 , J ) + B ( I I , J )*SIN(TR) )/COS(TR • B ( J , I) - A( I , J I B ( I I , J 1 l = ( A < I I , J »*SIN(TR)+B(I I,J))/C0S(TR> A ( J l , I I )=B< I I , J 1 ) CONTINUE _ A ( J l , j l )=B( J 1 , J I I CONTINUE TL I S C A L C U L A T E D L E N G T H OF M A I N P A R T O F C A N T I L E V E R ITEML=RADIAL*C0S(PI4-(TR*FL0AT(NN))I ITEMR=0.0  70  D0_70 J =2,N IT EMR=ITEMR + A ( N l , J ) * C O S ( F A C T 1 + ( T R * F L O A T ( J - N 1 ) » ) CONTINUE TL=0.0  80  100  _0.U.5_ 0116 0117 0118 0119 0120 0121  MAIN  130  TL=TL+ITEML+ITEMR GAP=L-TL ERROR=GAP/L I F ( A 8 S ( E R R O R ) . L T . E P S ) GG T G 1 0 0 I F ( G A P . G T . 0 . 0 ) GO TO 8 0 THETA=THETA-DELTA DELTA=0.1*0ELTA I F ( D E L T A . L T . C . 0 C 0 C 1 ) GO TO 1 0 0 THETA=THETA +DELTA GO TO 9 C THETA=THETA + DELTA GG TO 9 0 WRITEI6,7)THETA.DELTA.ERROR WRITE ( 6 , 3 9 1 WRI TE ( 6 , 9 ) _ . . 0 0 130 1 = 2 , N l 11=1-1 DO 1 3 0 J=1,N JJ=J-1 WRITE(f>,ll)li,JJ,A(I,J),JJ,II,B(J,I) CONTINUE  PAGE STARTING  0003 WITH  FORTRAN  IV G C O M P I L E R  0167 0168 0169  .  04-29-70  19:28:53  PAGE  0004  C A L C U L A T E T H E I N C L I N A T I O N S OF T H E VARIOUS MEMBERS P S F A ( I , J ) AND P S I B < I , J ) BETAA=P14 B E TR 2 = F A C T1 BETR2M= FACT DO 1 8 0 J=1,N Jl=J+l JJ=J-1 00 180 1 = 1,N I 1=l-I 11=1+1 I F I J . E Q . l ) GO TO 1 5 0 PSIA(I1,J)=(<TR*FLOAT( 11-J ) J-BETR2) I F ! I . G T . l ) G 0 TO 1 4 0 P S I B ( I , J l ) = < < T R * F L O A T ( J l - 2 ) ) + BETAA) GO TO 1 7 0 P S I B ( I , J l ) = ( I TR * F L O A T ( J l - I ) ) + B FT R2 M ) 140 GO TO 1 7 0 PSIA(Il,JI=HTR*FL0AT(Il-2))-BETAAI _. .150 I F ( I . G T . l ) G O TO 1 6 0 PS I B ( I , J 1 ) = ( I T R * F L O A T ( J l - 2 ) J + B E T A A ) GO TO 1 7 0 P S I B I I , J l l =((TR*F LOATIJl-I))+BETR2M) 1 6C CONTINUE 170 CO.NT.I N.UE _.. . . . . . ..1 8.0 WRITE ( 6 , 5 1 ) DO 1 9 0 I = 1 , 2 5 00 190 J = 1 , 2 5 LOAD(I,J)=0.0 T I LT ( I , J > =0 . 0 FA(I,J)=0.0 FBI I,J1 = 0.0 XJ(I,JJ=0.0 YJ(I,J)=0.0 WA( I , J ) = 0 . 0 WB( I , J ) = 0 . 0 . .. A R.E A A ( I , J ) = . . 0 . . C . . .. AREAB(I.J) = 0.0 MIZA(I,J) = 0.0 MIZBII,J) = 0.0 WTINA(I , J )= 0 . 0 WTINBII,J)=0.0 EUL.CL A( I , J . ) = C . O EULCLBI I , J ) = 0 . 0 EA ( I , J ) = 0. 0 EB( I , J ) = 0 . 0 CONTINUE 190 THE F O L L O W I N G S E C T I O N C A L C U L A T E S T H E F O R C E S I N T H E MEMBERS, C . ...... C . A L L O W I N G FOR T H E E F F E C T S O F S E L F W E I G H T . F I R S T S P E C I F Y A P P L I E D L O A D S , L O A D ( I , J ) , AND T H E A N G L E , T I L T (I,J), C THAT E A C H L O A D M A K E S W I T H THE V E R T I C A L . REA0(5,17)LGAC(N1,N1),TILT(N1,N1) WRITE(6,19)N ,N , L O A O ( N l , M ) , N ,N , T I L T ( N 1 , N 1 ) TILT<N1,N1)=(TILT<N1,N1)*PI)/180.0 W I S 7 F R 0 I N T H I S C A S E WHERE T H E M E M B E R S A R E A S S U M E D TO B E C C  0122 0123 0124 0125 0_12_6 0127 0128 0129 0130 0131 .0132 0133 0134 0135 0136 0137 __0.13.8 0139 0140 0141 0142 0143 _D.14_4 0145 0146 0147 0148 0149 .___0_1.5-C 0151 0152 0153 0154 0155 0.15.6 0157 C158 0159 0160 0161 0162 0163 0164 0165 0166  MAIN  c  c  FORTRAN  017C 0171 0172  IV G COMPILER C  MAIN  04-29-70  19:28:53  PAGE 0 0 0 5  WEIGHTLESS. W = 0.0 D0WN1 = C 0 S I T R ) - ( W * I C O S ( F A C T ) ) * ( A ( N 1, N)+B I N, N 1 ) ) ) FA(Nl»N 1 = <LCAO(N1,NI)*(COS(FACT+TILT(N1,N1))+W*(<8(N,N1)* SINITILTINl,N1)))I)J/DCWN1 FB(N,N1I = - ( L O A 0 ( N l , N l ) * C O S ( F A C T - T I L T ( N l , N i n - W * ( A ( N l , N I * S I N 1 ( T I L T I N l , N l ) l ) ) /0GWN1 DO 210 1=1,NN Nl I=N + l - I N2I=N+2"I DO 210 J=1,NN NJ=N-J N1J=N+1-J N2J=N+2"J I F ( J . G T . l ) GO TO 200 I F ( I . E Q . 1 ) GO TG 210 P S I = ( I 0.5*PI J-t-PSI A( N2I ,N1J) ) PSIT=PSI-TR D 0 W N 2 = ( C 0 S I T R ) - W * l A I N 2 I , N 1 J ) * C O S ( P S I T ) + BIN1 I , N 2 J ) * S I N ( P S I 1)) FAIN2I,N1J) = ((FA(N2I,N2J)*(1.0+1W*(A(N2I,N2J)*COS(PS IT)-B(NlI,N2J 1 )*SIN(PSIT ) ) ) ))-(FB(N2I,N2J)*(SIN(TR)+(W*(BIN2I,N2J)*COS<PSIT)+ 2 B ( N 1 I , N 2 J ) * C 0 S ( P S I ) ) 1 1) + ( L G A D ( N 2 I , N 2 J ) * ( C C S I P S IT+T I L T ( N 2 1 , N 2 J ) J 3 +(W*B(N1I,N2J)*SIN(TILT(N2I,N2JI)))))/D0WN2 F B I N i l , N 2 J ) = - ( ( F A ( N 2 I , N 2 J ) * IS INITR > + ( W * ( A ( N 2 I , N 2 J ) * S I N ( P S I ) + 1 A I N 2 I , N 1 J ) * S I N ( P S I T ) ) ) ) )-(FBIN2I,N2J)*<1.0"(W*(A(N2I , N1J)*CCS( 2 P S I 1 - B I N 2 I , N 2 J ) * S IN(PS I ) ) ) 1 j + ILOAC1N2 I» N 2 J ) * ( S I N ( P S I + T I L T I N 2 I , 3 N 2 J ) ) - ( W * A ( N 2 I , N 1 J 1 * S I N ( T I L T ( N 2 I , N 2 J I > ) ) > )/00WN2 CONTINUE DO 220 1=1,NN 1  0173 0174 0175 0176 0177 0178 0179 018C 0181 0182 0183 0184 0185 0186  200  0187  0188 0189 0190 C191 0192 0193 0194 0195  210  NlI=N*1-I  N2I=N+2-I  P S I = ( 0 . 5 * P I « - P S I A ( N 2 l ,1) ) PS IT=(PS I-TR2 )  0196  0197 0198 0199 0200 0201 0202 02C3 0204  0205  220  0 0 W N 3 = ( C G S ( T R 2 ) - ( W * ( A ( N 2 I , 1 ) * C O S I PS I T ) + B ( N 1 1,2 )*S I N ( P S I ) ) ) ) F A I N 2 I , ! ) = ( I F A ( N 2 I , 2 ) * ( 1 . 0+( W*( A ( N2 I , 2 ) *C 0 S ( P S I T ) - B ( N l I ,2 >* S I N ( 1 P S l T ) M ) ) - ( F B ( N 2 I , 2 ) * ( S I N ( T R ) + (W*(B(N2I,2)*C0S(PSIT) + 2 B ( N l l , 2 ) * C O S ( P S I + T R 2 ) ) ) ) ) * < L O A D ( N 2 I , 2 ) * ( C O S ( P S IT+ T I L T I N 2 I , 2 ) ) 3 + < W * B ( N 1 I , 2 ) * S I N I T I L T I N 2 I , 2 ) ) ) > ) ) /DOWN 3 FeiNlI,2>=-l(FA(N2I,2)*(SIN(TR2)*(W*(A(N2I,2)*SIN(PSI)+A(N2I,l)* 1 SIN(PSIT)))))-(FB(N2I,2)*(C0S(TR2)-(W*(A(N21,1)*C0S(PSI+TR2 )2 B(N2I,2)*SIN(PSI)))))+(L0AD(N2I,2)*(SIN(PSI+TILT(N2I,2))-IW* 3 A(N2I,1)*SIN(TILT(N2I,2)))))J/DOWN3 CONTINUE DO 230 J=1,NN N1J=N+1-J N2J=N+2-J PSI = (0.5*PI + P S I A ( 2 , M J H PS I T = ( P S I - T R 2 ) D0WN4 = ( C C S ( T R 2 ) - ( W * ( A ( 2 , N 1 J ) * C 0 S ( P S I T ) + B 1 1 , N 2 J ) * S I N ( P S I > 1 ) ) FA(2,N1J )=((FA(2.N2J)*{COS(TR2)•(W*IA(2,N2J)*C0S<PSIT>-B(1,N2J)* 1 SINI P S I - T R ) ) ) ) ) - ( F B ( 2 , N 2 J ) * ( S I N I T P 2 t + ( W * ( B ( 1 , N 2 J ) * C C S ( P S I ) + 2 B ( 2 , N 2 J > * C O S I P S I T ) ) )> ) + ( LOAD! 2 ,N2 J) * ( COS ( PS IT+T IL T I 2, N2J ) ) + 3 (W*B(1.N2J)*SIN(TILT(2,N2J)I)))>/D0WN4 FBI 1 , N 2 J ) = - ( ( F A ( 2 , N 2 J I * ( S I N I T R ) + 1 W * I A I 2 , N 2 J ) * S I N ( P S I ) + A ( 2 . N I J I * 1 SIN(PSI-TR)))))-(FB(2,N2J)*(1.0+(V.*(B(2,N2J)*SIN(PSI)-A(2,N1J)*  FORTRAN  IV  G COMPILER 2 3  0206 0207 0208  0209  021C 0211 0212 0213 _J1214 0215 0216 0217 0218 0219 _Q22JD 0221 0222 0223 0224 0225 0776 0227 0228  0229 _0.23.0_ 0231 0232 0233 0234 0235 0236 0237 0238 0239 0240 0241 0242 0243 0244 0245 0246 024.7  MAIN  04-29-70  19:28:53  PAGE  0006  COSIPSI)))))+(LOAD(2,N2J)*iSIN(PSI+TILT(2,N2J)J-(W*AI2,N1J»*  S IN(TILTI7.N2J)))>>I/D0WN4 230 CONTINUE DOWN 5= ! 1 . 0 - t W * ( A ( 2 , l ) * S l N ( B E T A A ) » B ! l , 2 ) * C 0 S ! B E T A A ) ) ) ) FA(2,1)=((FA(2,2)*(CCS<TR2)+(W*(A<2,2)*SIN(BETAAJ-B(1,2)*COS<BETA IA+TR2)I> > > - ( F B I 2 , 2 ) * ( S I N ! T R 2 ) + ( W * ! B < 1 , 2 ) * S I N l B E T A A - T R 2 I + B12«2)* ...2_....SJ N ( B E T A A ).) ) ).) r.( L 0 A OJ 2_,. 2 ) * ( S I N ( T I LT ( 2 , 2 ) - B E T A A 1 - ( W*B ( 1 , 2 ) * S I N I 3 T I L T 1 2 , 2 ) ) ) ) ) ) /D0WN5 F B I l , 2 ) = - ( < F A < 2 , 2 ) * ( S I N ! T R 2 > + t W * ! A l 2 , 2 ) * C 0 SI B E TAA I + A { 2 , 1) * C O S ( 1 B E T A A + T R 2 ) ) ) ) ) - 1 F B ( 2 , 2 ) * I COS 1 T R 2 ) - ! W » ( A ! 2 . 1 ) * S I N ( B E T A A - T R 2 ) 2 fit 2 , 2 ) * C O S ( B E T A A ) ) ) ) ) + 1 L O A D I 2 , 2 ) * ( C O S ( T I L T ( 2 , 2 ) - B E T A A ) - ( W * A ( 2 , 1 3 )*SIN(TILT(2,2)))))I/DGWN5 .C CAL.CUJ. AT 1 ON._Q.F_ P J . N _ J 0 1 NT C O O R D I N A T E S , X J I I , J ) AND Y J ( I , J ) , AND J O I N T C DEFLECTIONS, X ( I , J ) AND Y ( I , J ) . DO 3 0 0 I =1,N1 DO 3 0 0 J = 1 , N 1 I F ( J . G T . l ) GO TO 2 6 0 I F ( I . E Q . D GO TO 3 0 0 2 6(3 .C-QNJJ NU E_ I F | F A ( I , J ) . L T . 0 . 0 ) GO T C 2 7 0 E A I I . J ) = STRAIN*A(I , J) GO TO 2 8 0 270 E A I I . J ) = -ISTRA IN*A( I , J ) ) 280 I F I F B I I , J ) . G T . O . O ) GO TO 2 9 0 E^J_,JLL_Ji^RAIJi*Ej(J.,A) GO TO 3 0 0 290 E B I I . J ) = - I S T R A IN*BI I,J) ) 300 CONTINUE DO 3 1 0 J = 1 , N J l = J+1 D.0_3.1.0_I_^_l.i.K '. II = I+l F A C T O R = l ( X C I , J 1 ) - X I 1 1 , J ) ) * S I N I P S I B ( I , J 11)+1Y( I 1 , J ) - Y I I , J 1 ) ) * C C S < 1 P S I B I I , J l ) ) - E B I I , J l l - E A l I I , J ) * S I N t P S I B I I , J l l + P S IA(I 1 , J ) ) ) / C O S ( 2 PSIAI 11,J )+PSIB(I,J1)1 X I I 1 , J 1 ) = ( X I I 1 , J ) + E A ( I 1 , J ) * C 0 S I P S 1 A < 1 1 , J ) ) - F A C T O R * S I N ( P S I A ( 11 , J ) ) ) Y l I 1 , J l ) = ( Y ( I 1 , J ) - E A 1 I.1.,.JJ * S l N ( P S J A ( J J ,. . J ) ) - F A C T C R * C O S ( P S I A I 1 1 . i _ J J L U _ CONTINUE 310 C O O R D I N A T E S OF THE P I N J O I N T S . C WRITE!6,131 DO 3 2 0 1 = 1 , N 11 =1+1 XJI11,1)=0.0 X J 1 1 . I 1 )=0.0 YJ!I l,l)=D/2.0 YJ( 1,11)= - D / 2 . C 320 CONTINUE DO 3 3 0 1 = 1 , N 11=1+1 X J ! 11,2) = A!11,1)*COS(PS I A! 11,1) ) YJ!U,2)=YJII1,1)+!A(I1,1)*SIN!PSIA(I1 ,1 ) ) ) CONTINUE 330 DO 3 4 0 1 = 1 , N l DO 3 4 0 J = 2 , N J1=J+1  FORTRAN 0248 0249 0250 0251 0252 0253 0254 0255 0256 0257 0258  0259 0260 0261 026? 0263 0264 0265 0266 C267 0268 0269 027G 0271 0272 0273 0274 0275 0276 0277 0278 0279 028C 0281  0282 0283 0284 0285 0286 0287 0288 0289 0290  IV  G COMPILER  340  MAIN  04-29-7C  19:28:53  PAGE 0 0 0 7  X J I I , J l ) = X J < I , J ) + A( I , J J f C O S I P S I A l I , J ) ) Y J I I , J l )=YJ1 I , J ) + A ( I , J ) * S I N ( P S I A 1 I , J ) ) CONTINUE DO 3 6 0 1 = 1 , N l 11=1-1 DO 3 6 0 J = 1 , N 1 JJ=J-1 I F I J . G T . l ) GO TO 3 5 0 I F 1 I . E Q . 1 ) GO T O 3 6 0  3 50 WRITE(6 ,15) I I , J J , X J I I , J ) , 1 1 , J J , Y J I I , J ) , 1 1 , J J , X ( I , J ) , I I , J J , Y ( I , J ) 360 CONTINUE C C O M P I L E P R O P E R T I E S OF E A C H M E M B E R . C F O R C E S AND W E I G H T S I N P O U N D S , W I D T H S I N I N C H E S , C R O S S S E C T I O N A L A R E A S C IN S Q U A R E I N C H E S A N D M O M E N T S OF I N E R T I A I N I I N C H E S ) * * 4 00 4 0 0 1=1,N ' 11 = I + l 1 1 = 1 - 1 00 400 J = l , N J 1= J + 1 JJ=J-1 WAIIl.J) = IABSIFAIU,J)1)/(T*SIGMA) WBII.Jl) = IABSIFBI I , J l ) ) ) /1T*SIGMA) AREAAI11,J)=(T*WAI 11,J)) AREABII,J1)=(T*WB<I , J 1 ) ) M I Z A I U . J ) =IIWAI I 1 , J ) * * 3 ) * T 1 / 1 2 . 0 MIZBli.Jl) = ((We(I,JlJ**3)*Ti/12.0 WTINAI U , J ) = I A R E A A I I 1 , J ) * S P G T Y * 6 2 . 4 ) / 1 7 2 8 . 0 WTINeiI,J1)=(AREABII,Jl)*SPGTY*62.4)/1728.0 I F ! W A I I 1 , J ) . G T . C . 2 5 ) GO TO 3 7 0 E U L C L A I 1 1 , J ) = ( A B S ( F A ( I 1 , J ) ) * A I 1 1 , J ),*A I 1 1 , J I * 1 0 0 . 0 ) / 1 IFLM0D*MIZAI11,J)*PI*PI) GO TO 3 8 0 370 EULCLA(U,J)=IA8S(FA(I1,J))*AII1.J)*A(II.J I * 7 6 8 0 0 . 0) / 1 ELMOD* 1 WAII 1 , J ) * P I * P I ) 380 IFIW8II,J1).GT.0.25) GU TO 3 9 0 EULCLHII,Jl) = (AES(FBI I , J l ) ) * B ( I , J l ) * B < I , J l ) * 1 0 0 . 0 ) / 1 (ELMOO*MIZB(I , J 1 ) * P I * P I ) GO TO 4 0 0 390 EULCLB( I , J l ) = I ABSIFB( I , J l ) ) * B I I , J 1 ) * B < I , J 1 1 * 7 6 8 0 0 . 0 ) / ( E L M O D 1 *WB( I , J 1 ) * P l * P I ) 400 CONTINUE C R E A C T I O N S AT S U P P O R T S . C T H E H O R I Z O N T A L C O M P O N E N T S OF T H E R E A C T I O N S A T A AND B A R E C M E A S U R E D P O S I T I V E A C T I N G T O T H E R I G H T , AND T H E V E R T I C A L C COMPONENTS ARE MEASURED P O S I T I V E A C T I N G UPWARD. FAH=0.0 FAV=0.0 DO 4 1 0 1 = 2 , N l F A H = F A H - I F A I I , 1 ) * C O S I PS I A ( 1 , 1 ) ) ) FAV=FAV+(FA(I,l)*SIN(BETAA-(TR*FL0AT(I-2))))+IABS(FA(I,l))* 1 A(I,l)*W) 410 CONTINUE FBH=0.0 FBV=0.0 DO 4 2 0 J = 2 , N 1  FORTRAN  IV  G COMPILER  0291 0 2S2 0293 0294 0295 _0.2.9.6 0297 0298 0299 0300 0301 _0.3.0.2 0303 0304  420  0305  450 C  0 30 6 0307 0308 0309 0310 JQ.3.L! 0312 0313 0314 0315 0316 0317 0318 0319 0320 03 21 0322 .03.23 0324 0325 0326 0327 0328 0329 0330 0331 0332 0333 0334 0.3.35. 0336 0337 0338 0339 C34C 034.1  460 470 48G  490 500  900  MAIN  04-29-7C  19:28:53  PAGE  0008  FBH=FBH( F B ( 1 , J ) * S I N { P S I B < 1 , J ) )) FBV=FBV-(FB(1,J) *C0S(BETAA+(TP*FLOAT(J-2)))>+(ABS(FB(l,J))* 1 B<l,J)*W) CONTINUE WRITEI6,41) WRITE<6,43)FAH,FAV WR I . T E ( 6 , 4 5 ) f B H . F 3 V WRITE ( 6 , 3 3 ) 00 450 1=1,N II = I - l II = I+l DO 4 5 0 J=1,N .J-l__=_J±J. . JJ = J-l WRITE(6 ,49) I , J J , F A ( I I , J ) , W A ( I I , J ) , A R E A A ( I I , J ) , MIZA( I I , J ) , 1 WTINA(I 1 , J ) . E U L C L A l I l , J ) , I I , J , F B ( I , J l ) ,W B ( I , J l ) , A R E A B l I , J l 1 , 2 MIZB(I,Jl),WTINBII,Jl),EULCLBI I,J1) CONTINUE F I N A L S T E P I S TO D E T E R M I N E V O L U M E AND W E I G H T OF S T R U C T U R E . VOLUME = 0 . 0 VOL TEN = C O VOLCOM = 0 . 0 WEIGHT=0.0 DO 4 8 0 I =1,N DO 4 3 0 J = 1 , N 11=1+1 J1=J+1 I F ( F A ( 11 , J ) . L T . 0 . 0 I GO TO 4 7 0 V O L T E N = V O L T E N + ( A ( 11 , J ) * A R E A A ( 1 1 , J ) ) I F ( F B ( I . J D . L T . O . O ) GO TO 4 7 0 V O L T EN = V O L T E N + I B I I , J 1 ) * A R E A B < I , j l )) GO TO 4 8 0 I F ( F 8 ( I , J 1 ) . L T . C C ) GO TO 4 8 0 VOLTEN = VOLTEN + ( B ( I , J 1 ) *AREAB( I,J 1 ) ) CONTINUE DO 5 0 0 I = 1 , N DO 5 0 0 J = 1 , N 11 = I+l J l = J+1 I F ( F A I I l . J I . G T . O . O ) GO TO 4 9 0 VOLCOM = V O L C O M + I A ( I l , J ) * A R E A A l 1 1 , J I 1 IF(FB(I,J1).GT.0.0) GO TO 4 9 0 VOLCOM = VOLCCM + (B(I,Jl)*AREAB(I,Jl)) GO TO 5 0 0 I F ( F B ( I, J l ) . G T . 0 . 0 ) GO T O 5 0 0 VOLCOM = VOLCCM + ( B ( I , J 1 ) * A R E A B ( I » J l 1 ) CONT I N U E VOLUME = VOLUME + V C L T E N + VGLCOM WFIGHT = WEIGHT + ( V O L U M E * C ) WRITE(6,25) VCLTEN,VOLCOM,VOLUME WRITE ( 6 , 3 5 ) WEIGHT WRITE(6,53) GO T O 9 9 0 CONTINUE END  EXECUTION  »RUN  TERM[NATED  -LOAD»  FX 6 C U T I O N  5=*S0URCE«  _MODULUS-.0F.. E L A S T I C I T Y N  =  2  THETA  =  6=»SINK«  BEGINS  L  =  =  300000.0  10.000000  0  P.S.I..  =  UN1F.0RM. STRAIN...:.  . U N I F O R M _ST.ReSS.j_.300.000.000 _P..S...I ....  2.500000  L/0  =  4.000000  DELTA  64.942383  0.250000  THICKNESS  0,.00100.0_I.NCHES/.I.NCH.  SP.GTY.  =  1.310  0.000010  THE F O L L O W ING RESULTS ARE CALCULATED FOR A S T R U C T U R E MADE F R O M CR39 PLASTIC HAVING A S P E C I F I C GRAVITY OF 1.31 ~TJiE EFFECTS OF SELF WEIGHT ' F O R C E S ARE INCLUDFD IN THESE RESULTS.  LENGTHS A(  1,  A( A(  2, 2,  01  =  OF  MEMBERS  1.767767 1 .898141 1.767767 8.541646  1. I I 0, 21 1,21  1.767767 l..8?81_41_ 1.767767 8.541646  =•_ = =  NOTE THE TERM RATIO IN T H E FOLLOWING TABLE INDICATES THE PERCENTAGE RATIO_ OF T H E FORCE IN E A C H M E M B E R TO I T S EULER CRITICAL LOAD FOR BUCKLING ( E » H I P I 1*1 P I I ) / I L « L I LOAD I  2,  21  JOINT XJ ( XJ( XJ( XJI XJ{ XJI XJI XJI  c. 0. 1, 1. 1. 2. 2, 2.  =  -  YJI YJ( YJI YJI YJ I YJI YJI YJI  -FORCE  .  A.  AT B. .WIDTH  21  JOINT  0, 0. 1, 1. 1, 2, 2. 2,  11 = 21 = 0) = 1) = 2) = 0) = 11 = 2)  HORIZONTAL  XI XI XI XI XI XI XI XI  ! .250000 -1.250000 1 .250000 0.000001 -1 .852940 1.250000 1.852942 0.000008  REACTIONS AT  2.  COORDINATES  o.c 0.0 0. 0 1.25000C 1.661765 0. C 1.661764 10.000010  1 ) 21 = 01 1) = 2) = 0) 11 21  TILT!  100.000000  COMPONENT  -  HORIZONTAL COMPONENT « AREA I WT/IN  AT  SUPPORT  0, 0. 1. 1. 1, 2. 2. 2.  1) 2) 01 1) 21 01 1) 21  = = = =  .  -  DEFLECTIONS Y( Y( Y( Y( Y( Yl Yt Y(  0.0 0.0 0.000000 -0.003799 0.0 0.003799 0.000000  0. 0. 1, 1. 1. 2, 2, 2,  11 21 01 1) 21 0)  1)  2)  = = =  0.0 0.0 0.0 0.002500 0.005289 0.0 0.005289 0.061761  POINTS  -400.000488  VERTICAL  COMPONENT  400.000244 R A T I O . . ...MEMBER  VERTICAL FORCE  COMPONENT HJO.T.H  49.999954  .AREA  50.000214 I  H.T./_IN_  A(  1.  01  202.490082  2.699867  0.67497  0.41000  0.03193  6.0793  Bl  0.  II - 2 0 2 . 4 9 0 0 8 2  2.699867  0.67497-0.41000  0.03193  A( A( A(  1, 2, 2,  11 01 11  146.674e66 273.200684 230.488968  1.955665 3.642675 3.073186  0.48892 0.91067 0.76830  0.15583 1.00698 0.60468  0.02313 7.0090 0.04308 6.0793 0.03634141.9335  B( B( Bl  0, 1, 1,  21 1) 21  3.642675 1.955665 3.073186  0.91067 0.48892 0.76830  0.04306 6.0793 0.02313 7.0090 0.03634141.9335  VOLUMF  TENS I L F MEMBERS^  10.2.93581  VOLUME  COMPRESSIVE WEIGH T=  «»  FATAL  FCRTRAN  * END-OF-FILE ERROR O C C U R E O ERROR R E T U R N  ERROR:  ENCOUNTERED ON U N I T 5  ON  READ  OPERATION  MEMBERS=  -273.200684 -146.674866 -230.488968  10.293581  0.973886  TOTAL  VOLUME=  1.00698 0.15583 0.60468  20.587158  6.0793  FORTRAN  IV  G COMPILER  MAIN  04-29-70  19:3_!39  PAGE  C C 0001  P R O G R A M M E FOR S O L U T I O N OF M I C H E L L C A N T I L E V E R S W I T H D I F F E R I N G F A N R A D I I . LOWER R A D I U S , R A D B , I S A S S U M E D TO B E L A R G E R T H A N R A D A • DIMENSlONA!25,25),B(25,25),X!25,25),Y(25,25),PSIA(25,25), 1 P S I B I 7 5 , 2 5 ) ,FA! 2 5 , 2 5 ) , F B ( 2 5 , 2 5 ) , T I L T ( 2 5 , 2 5 ) , E U L C L A 1 2 5 , 2 5 ) , 2 LOAO(25,251, XJ(25,25),YJ(25,25),WA{25,25),WB(25,25),AREAA(25,25 3 ) ,AREAB(25,25),MIZA(2 5,25),MIZB(25,25 ),EULCLB(25,25), 4 W T I N A ( 2 5 , 2 5 ) , W T I N B l 2 5 , 2 5)_, E A ( 2 5 , 2 51 , E B ( 2 5 , 2 5 ) , , 5 ERX(80|,ERY(80> REAL LOAD, ITFML.ITEMR,ITEMY1,ITEMY2,MIZA,MIZ8 1 FORMAT(13) 3 F0RMAT(F15.6) FORMAT! 1 H 0 , 1 5 X , 4 H N = , I 3 , 1 0 X , 4 H D = ,F10.6,10X,8HXSPAN = , 5 1 F10.6,10X,8HYSPAN = ,F10.6,//) 7 FORMATJIHO,•THFTA= ' , F 8 . 5 , 4 X , ' T L X =. • , F 8 . 5 , 3 X , • T L Y = ' , F 8 . 5 , 1 2X,'RADRAT = ' , F 8 . 5 , 3X,'RADA = ' ,F8.5,3X,•RADB = ' , F 8 . 5 , 2 4X , ' B E T AC = • , F 8 . 5 , / / ) 9 F O R M A T ! 1 H C . 5 0 X , ' L E N G T H S OF M E M B E R S ' / ) FORMAT!25X,2HA< , 12, 1 H , , 1 2 , 4 H ) = ,F15.6 , 20X,2HB(,I 2,IH,,12,4H) = , 11 1 F15.6) F O R M A T ! 1 H 0 . 2 5 X , • J C I NT C O O R D I N A T E S ' , 4 0 X , " J 0 I N ' T C E F L ECT I O N S ' / ) I 3 F0RMAT(6X,3HXJ( , 12, 1 H , , I 2 , 4 H ) = , F 1 2 . 6 , 8 X , 3 H Y J ( , I 2 , I H , , 1 2 , 4 H ) = , 15 1 F 1 2 . 6 . 9 X . 2 H X ! , 12,IH , ,12,4H) = , F 12.6,8X,2HY( 1 2 , I H , , 12,4H) = , 2 F12.6) 17 F0RMATI2F15.6) 19 F O R M A T ! 1 H 0 . 2 0 X , 5 H L C A D I , 12 , 1 H , , . I 2 , 4 H ) = , F 1 5 . 6 , 20 X , 5HT I L T I , I 2 , I H , , I 12,4H) = ,F15.6) FORMAT(1 H O , 6 X , • M E M B E R ' , 7 X , ' F O R C E ' , 7 X , ' W I D T H • , 7 X , • A R E A , 3 X , 21 1 ' S E C O N D MOMENT' ,8 X , ' M E M B E R ' , 7 X , ' F O R C E ' , 7 X , • W I D T H • , 7 X , ' A R E A ' , 3 X , 2 'SECOND MOMENT'/) 23 F0RMATUH0,3X,3HA( , I 2 , I H , , I 2 , 1 H ) , 1 X , 4 F 12 . 6 , 9 X , 3HB ( , I 2 , 1 H , , I 2 , I 1H),1X,4F12.6) 25 " FORMAT 1 I H O , 5 X » ' V O L U M E T E N S I L E M E M B E R S = ' , F 9 . 6 , 5 X , 1 ' V O L U M E C O M P R E S S I V E MEMBERS= • , F 9 . 6 , 5 X , • T O T A L VOLUME= ',F9.6) F0RMATIF15.6) 27 FORMAT(F15.6) 29 F O R M A T ! I H O , 2 4 H M O D U L U S OF E L A S T I C I T Y = , F 1 0 . 1 , 1 X , 6 H P . S • I . , 5 X , 31 1 17HUN I FORM S T R E S S = ,F10.6,1X,6HP.S.I.,5X, 2 1 7 H U N I F 0 R M S T R A I N = , F 1 0 . 6 , I X , 1 2 H INC H E S / I N C H . ) FORMAT!IHO , 5 X ,'MEMBER' ,4X , ' F O R C E ' , 3 X , ' W I D T H ' , 4 X , ' A R E A ' , 5 X , 33 1 • I ' , 5 X , ' W T / I N . ' , 3 X , ' R A T I O ' , 5 X , ' M E M B E R ' , 4 X , « F O R C E ' , 4X 2 • W I D T H ' , 4 X , • A R E A ' , 5 X , ' I ' , 5 X , ' W T / I N . • , 3 X , ' R A T 10 ' ) 35 FORMAT! I H O , 3 X . 2 H A ! , I2,1H,,I2,1H),1X,F9.4,5F8.4,5X,2HB1,I2,1H,, 1 12,IH), IX,F9.4,5F8.4) 37 FORMAT(Fl5.6) 39 FORMAT! I H O , 4 0 X , 'THE FOLLOWING RESULTS ARE CALCULATED' , / , 4 1 X , 1 ' F O R A S T R U C T U R E MADE FROM CR39 PLASTIC•,/,43X, ? 'HAVING A S P E C I F I C GRAVITY OF 1.31',/,42X, 3 "THE EFFFCTS OF SELF WEIGHT FORCES',/,43X, 4 'ARE INCLUDED IN THESE RESULTS.') F O R M A T ! 1 H C 4 5 X , ' R E A C T I C N S AT S U P P O R T P O I N T S ' ) 41 FORMAT I I H O , 2 5 X , ' A T A , H O R I Z O N T A L C O M P O N E N T = ',F12.6,10X, 43 1 ' V E R T I C A L COMPONENT = ',F12.6) FORMAT! I H O , 2 5 X , ' A T B , H O R I Z O N T A L COMPONENT = ',F12.6,10X, 45 1 ' V E R T I C A L COMPONENT = «,F12.6) 47 FORMAT!IHO,59X,'NOTE'./,45X,«THE TERM RATIO IN T H E FOLLOWING', :  0002 0003 00C4 0005 ~0006  0007 0008  "~ocoT~ 0010  0011 0012 0C13  0014 * 0015 0016 0017 0C18  0019  0020 0C2 1 0022  0 02 3 " " 0024 0025 _ Q C_26  0001  1  FORTRAN  IV  G COMPILER 1 2 3 4  0027  '~0028 0029 0030 0031 0032 0033 0034 0035 0036 003 7 0038 0C39 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0C50 0051 0052 0053 0054 0055 0056 0057 0058 0059 0060 0061 0062 0063 0064 0065 0.06.6  MAIN  C4-29-7C  19:36:39  PAGE  /,43X,'TABLE INDICATES THE PERCENTAGE RAT 1 0 • , / , 4 5 X , ' O F THE F O R C E IN E A C H MEMBER TO ITS',/,45X, 'EULER CRITICAL LOAD FOR BUCKLING',/t52X, ' ( E * I * ( P I ) * ! P I ) )/(L*L) ' ,/) 49 F O R M A T ! 1 H 0 , 4 2 X , ' E I G H T Y C Y C L E S OF C A L C U L A T I O N OF T H E T A • , / , 4 3 X , 1 • A N D R A D R A T C O M P L E T E D . ERROR REMA I N S • , / , 4 5 X , ' G R E A T E R THAN S P E C I F I E D MAXIMUM',/) ... 2 F t ) R M A T ( l H 0 , 5 0 X , «WEIGHT= '.F15.6) 51 FORMAT!IHO,//) 53 FORMAT(2HA(,I 2 , I H , , I 2 . 2 H ) , F 11. 6 , F 9 . 6 , 3 F 8 . 5 , F 8 . 4 , 3 X , 55 I 2HB(,12,1H,,12,2H) ,F11 .6,F9 . 6 , 3 F 8 . 5 , F 8 . 4 ) FORMAT! I H O , 1 5 X , • T H E T A ' ,9X,'DELTHE',9X,'RADRAT•,9X,•DELRAT•, 61 1 10X, ' G A P X * , 1 1 X , ' G A P Y ' , / ) 63 FORMAT!10X,6F15 .8) P 1 = 3 . 14 1 5 9 2 6 5 PI4=0.25*PI R00T=SQRT<2.CJ CONT I N U E 990 999 CONTINUE K= 1 A L L T H E A R R A Y S A R E Z E R O E D OUT F I R S T C DO 1 0 1=1,25 0 0 10 J = 1 , 2 5 A! I,J 1 = 0 . 0 B( I , J ) = 0 • 0 X!I,J)=0.0 Y( I , J ) = 0 . 0 PSIAU ,J)=0.0 PSIB11,J)=0.0 10 CONTINUE DO 16 I = 1,80 ER X ( I ) = 0.0 ER.YU) =0.0 CONTINUE 16 N I S N U M B E R OF F I B R E S C READ ( 5 , 1 ) N N1=N+1 N2=N+2 NN=N-1 X S P A N . Y S P A N , A R E C O O R D I N A T E S OF THE O U T E R END OF T H E S P A N . C READ(5,3)XSPAN READ!5,3)YSPAN C D IS S U P P O R T S P A C I N G READ!5,3JD WRITE!6,5 IN,D,XSPAN,YSPAN INCHES. T I S T H I C K N E S S OF A L L M E M B E R S IN C T=0.25 EL MOD IS MODULUS OF ELASTICITY. C ELMOD=300000.0 SIGMA=300.0 STRAIN = 0.001 WRITE(6,31IELMOC,SIGMA,STRAIN 1.31 c THE S P E C I F I C G R A V I T Y ( S P G T Y ) OF C R 3 9 P L A S T I C I S SPGTY=1.31 C=(SPGTY*62.4>/172 8.0  0002  FORTRAN  0067  0068 0069 0070 0071 0C72 0073 0074 0075 0076 0077 0 078 0079 0C80 0081 0082 0083 0084 0085 0086 0087 0088 0C89 0090 0091 0092 0093 0094 0095 0096 .00.97 0098 0099 0100 0101 0102  0103 0 104 0105 0106 0107  oios  0109 0110 0111 0112 0113  IV  G COMPILER  MAIN  04-2~9-7CT  '  19:36:39  PAGE  0003  C C A L C U L A T I O N OF FAN P A R A M E T E R S . RADR IS RADIUS OF FAN CENTRED ON A,AND C RADB IS RADIUS OF FAN CENTRED ON 8 . RADB I S GREATER THAN RADA. C THETA IS FAN ANGLE AND IS GENERATED BY AN I T E R A T I V E P R O C E S S . S T A R T I N G C WITH AN APPROXIMATE VALUE CF T H E T A . C DELRAT AND DELTHE ARE S P E C I F I E D INCREMENTS FOR RADRAT AND T H E T A . C RADRAT EQUALS RADB DIVIDED BY RADA RADRAT = 1 . 0 DELRAT = 0 . 1 THETA = 1 6 . 0 DEL THE = 1 . 0 ERR = 0 .01 EPS=0.00001 CONTINUE 19. KK = K - 1 I F 1 K . G T . 8 0 ) GC TO 120 0 E N O M = S « R T ( 1 . 0 + ( R ADRA T*RADRA T1) RAC A = D/DENOM RADB=(D*RADRAT)/DENOM TR=THETA*PI/180.0 TR2=0.5*TR RADCON=(180.0/PI) BETAA=ATAN(1.0/RADRAT1 BETAD=(180.0*BETAA)/PI BETR2=(BETAA+TR2) BETP2M=I BET 4 A - T R2) 00 20 I = 2 , N 1 A l l , 1) = R A D A B( I , I l = RADB CONTINUE 20 A<2,2> = 2.0*RA03*SIN<TR2( B ( 2 , 2 ) = 2.0*RADA*SIN<TR2> DO 30 I = 3 , N A(2, I)=A(2,2) 3(1,21=6(2,2) 30 CONTINUE DO 40 J = ? , N Jl=J+l 00 40 1=3 , N 1 11=1-1 A( I , J l = ( A ( I I , J ) + B ( I I , J ) * S I N ( T R ) )/COS(TR ) B( [ I , J 1 ) = (A( I I , J ) * S I N ( T R ) + B ( I I , J ) ) / C O S ! T R ) 40 CONT INUF TLX IS C A L C U L A T E D LENGTH OF. C A N T I L E V E R . C C TL Y IS C A L C U L A T E D R I S E OF C A N T I L E V E R . ITEML=RADA*COS((TR*FLOAT(NN)l-BETAA) ITFMY1=(0.5*D)*(RADA*SIN((TR*FLOAT(NN)l-BETAA)) TLX=ITEML TLY=ITEMY1 DO 50 J = 2 , N TLX =TLX • A(N 1 , J ) * ( C O S ( ( T R * F L 0 A T ( N 1 - J ) ) - B ETR 2 ) ) TL Y =TLY +A(N1tJ)*SIN((TR*FLOAT(Nl-J)1-BETR2) CONTINUE 50 GAPX=XSPAN-TLX ERR X=GAPX/XSPAN GAPY _= YSPAN - . TLY _ .  FORTRAN 0114 0115 0116 0117 0118 0119 _0J.2J3__ 0121 0122 0123 0124 0125 .0.126 0127 0128 0129 0130 013] 0.1.3.2 0133 0134 0135 0136 0137 0138 0139 0140 0141 0142 0143 - 0144 0145 0146 0147 0148 0149  0150 0151 0152 0153 0154 0155 0156 0157 0158 0159 0.160 0161 0162 0163 0164 0165 . .0.1.66  IV  G COMPILER  60  70  80  100  120  130 C C  140 150  MAIN  04-29-70  19:36:39  ERRY = G A P Y / Y S P A N E R X ( K ) = ERRX E R Y ( K ) = ERRY I F ( A B S ( E R R X ) . L T . E P . . A N D . A B S ( E R R Y ) . L T . E P S ) GO TO 1 2 0 I F ( A B S t E R X ( K ) ) . GT . A B S (EP.X ( KK ) ) ) GO TO 6 0 IF(ABS(ERY(K)).GT.A6SIFRY(KK)))G0 TG 80 IHET A = .THETA + ( DELTHE*ERRX.) . . RADRAT = RADRAT+<DELRAT*ERRY) K = K+1 GO TO 9 9 THETA = THETA+IOELTHE*0.5*ERRX) I F ( A B S ( E R Y ( K I ) . G T . A B S ( E R Y I K M ) ) GO TO 7 0 RADRAT = R A D R A T + ( O E L R A T * E R R Y ) K = K+1 GO TO 9 9 RADRAT = RADRAT+ I 0 . 5 * D E L R A T * E R R Y ) K = K+ 1 GO TO 9 9 RADRAT = RADRAT+(0.5*DELRAT*ERRY) I F ( A B S ( E R X ( K ) I . G T . A B S I E R X ( K K ) ) ) GO TO 1 0 0 THETA = THETA+<DELTHE*ERRX) K = K+ 1 GO T O 9 9 THETA = THETA+(G.5*DELTHE*ERRX) K = K+1 GO TO 9 9 WRITE ( 6 , 7 ) THETA,TLX,TLY,RADRAT,RAOA ,RACB,BETAC WRITE 16,53) WRITE (6,39) WRITE(6,9I 00 130 1=2,Nl 11=1-1 DO 1 3 0 J=1,N JJ=J-l WRITE<6,11)I I , J J , A ( I , J ) , J J , I I , B ( J , I ) CONTINUE C A L C U L A T E THE I N C L I N A T I O N S OF T H E VARIOUS MEMBERS PSIA (I,J) AND P S I B (I , J ) DO 1 8 0 J=1,N J1=J+1 JJ=J-1 DO 1 8 0 1=1,N 11=1-1 11=1+1 IF ( J . E O . l ) GO TO 1 5 0 P S I A d l , J ) = ( ( T R * F L O A T ( 11 - J ) ) - B E T R2 ) I F ( I . G T . l ) G 0 TO 1 4 0 PS I B ( I , J l ) = ( ( T R * F L O A T ( J 1 - 2 ) ) + B E T A A ) GO TO 1 7 0 PSIB(I,J1)=(ITR*FL0AT(J1-ID+BETR2M) GD TO 1 7 0 P S I A ( I l , J ) = ((TR*FLOAT(11-2)l-BETAA) I F ( I . G T . l ) G 0 TO 1 6 0 PSlB(I,Jl)=((TR*FLOAT(Jl-21l+EETAAl GO TO 1 7 0  PAGE  0004  FORTRAN  IV  G COMPILER  0167 0168 0169 0170 0171 0172 0173 0174 0175 0176 0177 0178 0179 0180 0181 0182 0183 0184 0185 0186 0187 0188 0 189 0190. 0191  160 170 180  0195 0196 .019 7 0198 0199 0200 0201 0202 0203 0204 0205 0206 0207 0208 0209 0210 0211  c c  ....  04-29-70  19:36:39  PAGE  0005  PSIB(I,J1)=((TR*FL0AT(J1-I))+BETR2M) CONTINUE CONTINUE WRITE (6,47) DO 1 9 0 I = 1 , 2 5 DO 1 9 0 J = 1 , 2 5 LOADII,J)=0.0 TILTH,J)=0.0 FA(I ,JI=0.0 FBI I , J 1 = 0 . 0 XJI I , J ) = 0 . 0 YJI I , J ) = 0 . 0 WA( I , J ) = 0 . 0 WBII , J ) =0.0 AREAA I I , J ) = O . C AR F AB1 I , J ) = 0 . 0 MIZAII.J) = 0.0 M I Z B I I, J 1 = C O WTINAII,J1=0.0 WTINBII , J 1 = 0 . 0 E U L C L A I I , J 1=0 . 0 EULCLBI I , J ) = 0 . 0 EAI I , J ) = 0 . 0 E B I I, J ) = 0 . 0 CONTINUE 1 90 C THE F O L L O W I N G S E C T I O N C A L C U L A T E S T H E F O R C E S I N T H E MEMBERS, A L L O W I N G FOR T H E E F F F C T S O F S E L F W E I G H T . F I R S T S P E C I F Y A P P L I E D L O A D S , L O A D I I . J ) , AND T H E A N G L E , T I L T II,Jl, THAT E A C H L O A D M A K E S W I T H THE V E R T I C A L . RCAD(5,17)LCAniNl,Nll,TILT|Nl,Nl) WPITEI6,19)N,N,LOAD!N1,N1),N,N,TILT (N1,N1) TILTIN1,N1) = ITILTIN1,N 1 )*PI)/180.0 MEMBERS ARE ASSUMED W E I G H T L E S S FCR T H I S C A L C U L A T I O N S IS S E T EQUAL TO Z E R O . W = 0.0 D 0 W N 1 = C 0 S ( T R 1 - W * I A l N l , N ) * SIN1 BETAA+TR2) + D ( N , N l ) * C O S ( B E T A A - T R 2 1 ) F A ( N l , N ) = I L O A D I N 1 , N 1 1 * ( S I N ( B ETA A+TR 2 ~ TI L TI N 1 , N 1 ) ) «•{ W*B { N , N l ) * _. I S I M T I L T I M , N 1 ) ) ) 1 )/D0WN1 F B I N , N 1 ) = - ( L 0 A D ( N 1 , N 1 ) * ( C 0 S I B E T A A - T R 2 - T I L T ( N 1 , M ) ) - 1 W*A ( N l , N ) * 1 S I N I T I L T I N 1 . N 1 ) 1) I J/DOWNl DO 2 1 0 1=1,NN N l I= M + 1- I N2I=M+2"I DO 2 1 C J = 1 , N N NJ=N-J N l J = N+1 - J N2J=N+2-J I F I J . G T . 1 ) GO T C 2 0 0 I F I I . F 0 . 1 ) GC T C 2 1 0 200 P S 1 = 1 I 0 . 5 * P I )+P SI A l N 2 I , N U ) 1 PS 1T = PS l - T R DOWN2=ICOSITR)-W*I A I N ? I , N l J I * COS(PS IT)+B(N1I,N2J)*SIN(PSI111 FA(N2I,N).J) = ( ( F A ( N 2 I , N 2 J ) * < 1 . C - M W * ( A ( N 2 I , N 2 J ) * C 0 S I P S I T ) + BIN1I ,N2J 1 I * S I N I P S I T | | ) ) ) - I F H ( N 2 I , N2 J >* ( S I N ( T R ) + ( W* ( B < N2 I, N2 J ) * C O S ( P S IT ) • 2 B ( N 1 1 , N 2 J ) * C O S ( P S I I I ) ) ) +1LOAD 1 N 2 1 , N 2 J I * I COS I P S I T + T I L T < N 2 I , N 2 J J )  c c c  ' 0192 0193 0194  MAIN  FORTRAN  IV  G COMPILER  0221  0222 0223 0224 0225 0?2.6_ 0227 0228 0229  0230  0231 ,0232 0233  0234  0235 _J) 23.6 0237 023 8 0239 0240 0241 .0 242  04-29-70  19:36:39  PAGE  0006  3  0212  0213 _0.2.1A_ 0215 0216 0217 0218 0219 _022Q_  MAIN  210 _  +{W*B(N1I,N2J)*SIN{ T I L T I N 2 I , N 2 J ) ) ) ) ) ) / D 0 W N 2 FBI N i l , N 2 J ) = - ( ( F A I N 2 I , N 2 J ) * l S I N ( T R ) + ( W * < A ( N 2 I , N 2 J )*SIN(PS1 ) + 1 A(N2I,N1J)*SIN(PSIT)))))-(FB(N2I,N2J)*(1.0-(W*<A(N2I,N1J)*CCS( 2 P S I ) - B ( N 2 I , N 2 J i » S I N ( P S I ) ) ) ) ) » I L O A C 1 N 2 I , N 2 J ) * t S I N ( P S I+TILT1 N 2 1 , 3 N2J))-(W*A(N2I.N1J)*SIN<TILTIN2I,N2J>)))))/DCWN2 CONTINUE DO. 220 1=1,.N.N . _. . Nl I = N +1-I N2I=N+2-I PSI=(0.5*PI+PSIA(N2I,1) ) PS I T = ( P S I - T R 2 ) D0V.N3 = ( C C S ( T R 2 ) - ( W * ( A ( N 2 I » 1 )*COS< PS IT )+B ( N 1 1 , 2 ) * S IN(PS I ) ) ) >  F A ( N 2 I . 1) = ( (FA(N2I,2)*(1.0+(W*(A(N2I,2)*C0S(PSIT)-B(N1L,2)*SI_NL_ PSIT ) ) ) ) ) - ( F B ( N 2 I , 2 ) # ( S I N ( T R ) + ( W * ( B ( N 2 I » 2 ) * C 0 S I P S I T ) + BIN II » 2 ) * C 0 S ( P S I + T R 2 ) ) ) ) ) + ( L O A D ( N 2 1 , 2 ) * ( C C S ( P S IT+T I L T I N 2 1 , 2 ) ) + ( W « B ( N 1 I . 2 . * S I N ( T I L T ( N 2 I , 2 . ) ) . ) . /DOWN3 FB(N1I,2)=-((FA(N2I,2I*(SINITR2)+(W*(A(N2I,2)*SIN(PSI)+A(N2I,1)* 1 SIN(PSIT)))))-(F8(N2I»2)*(COS(TR2)-(W*(A(N2I,l)*COS(PSI+TR2>2 BIN2I , 2 ) * S I N ( P S I ) ) ) )1 + I L 0 A C I N 2 I , 2 )*(S IN(PS I+T I L T I N 2 I , 2 ) J r < W * 3 AIN2I,1)*SINITILTIN2I,2))))))/00WN3 220 CONTINUE DO 230 J = 1 , N N N l J=N + 1 - J N2J=N+2-J PSI=(0.5*PI+PSIA(2»N1J) I PS I T = ( P S 1 - T P 2 ) D 0 W N 4 = ( C C S ( T R 2 ) - IW*I A 1 2 , N l J ) * COS(PS IT ) + B | 1 , N ? J )*S IN I PS I > ) > . FA(2,NlJ)=t(FA(2,N2J)*(C0S<TR2)»(W*lA(2,NlJ)*C0S(PSIT)-Btl,N2J)* 1 SIN|PSI-TR)))))-IFB(2,N2J)*ISINITR2)+(W*(B(1,N2J.*C0S(PSI)+ 2 B ( 2 , N 2 J ) * C O S ( P S I T ) ) ) )) + ( L O A D ( 2 , N 2 J ) * ( C 0 S ( P S I T + T I L T ( 2 , N 2 J ) ) + 3. (W*B(I,N2J)*SIN(T I L T ( 2 . N 2 J ) ))))I/C0WN4 _ F B I 1 , N 2 J ) = - ( ( F A ( 2 , N 2 J ) * ( S I M T P ) + ( W * ( A ( 2 , N 2 J ) * S INI (PS I I + A ( 2 , N 1 J )* 1 SIN(PSI-TR))) ))-(FB(2,N2J)*( 1.0+<W*(B(2,N2J)*SIN(PSI)-A(2,NU)* 2 COS(PSI)))))+(L0A0(2,N2J)*(SIN(PSI+TILT(2,N2J) ) - ( W*A ( 2 . N 1 J ) * 3 S I N I T I L T I 2 . N 2 J > ) ) ))I/D0WN4 230 CONTINUE DOWN 5= ( 1 . 0 - ( W A ( A ( 2 , l ) * S I N ( B E T A A ) + B ( l t 2 ) * C OS (BETAA )_)))_ _ _ F A ( 2 , 1 ) = ( ( F A ( 2 » 2 ) * ( C 0 S ( T R 2 ) + ( W * ( A ( 2 , 2 ) * S I N ( B E T A A ) - B ( 1 , 2 )*COSIBETA 1A+TR2)))))-(FB(2,2)*(SIN(TR2)+ <W*(EI 1 , 2 ) * S I N ( B E T A A - T R 2 ) + E ( 2 , 2 ) * 2 SIN(BETAA) ) ) ) ) - ( L O A D ( 2 , 2 ) * ( S T N ( T 1 L T ( 2 , 2 ) - B E T A A ) - ( W * B ( 1 , 2 ) * S I N ( 3 TILT(2,2)))))1/D0WN5 FBI 1 , 2 ) = - ( ( F A ( 2 , 2 ) * ( S I N ( T R 2 ) + ( W * ( A ( 2 , 2 ) < = C 0 S ( B E T A A ) + A ( 2 , U * C 0 S ( _ l _ EETAA+TR2 ) ) ) ) ) - ( F B ( 2 , 2 ) * ( C O S ( T R 2 ) - ( W * ( A ( 2 , 1 ) * S I N I B E J A A - T R 2 ) 2 B ( 2 , 2 ) * C 0 S ( B E T A A ) ))) I + ( L O A D ( 2 , 2 ) * ( C O S ( T I L T < 2 , 2 ) - B E T AA)-(W*A112,1 3 ) * S I N I T I L T ( 2 , 2 ) )) ) ) J / 0 0 V . N 5 C A L C U L A T I O N CF PIN JOINT COORD I N A T E S , X J ( I , J ) AND YJ < I , J ) , AND J O I N T C D E F L E C T I O N S , XI I , J ) AND Y ( I , J ) . DO 300 I = 1 , N 1 _ . . . DO 300 J = 1 ,N1 IF ( J . G T . 1 ) GO TO 260 I F ( I . E Q . 1 ) GC TO 300 260 CONTINUE IF(FA(I,Jl.LT.O.O) GO TO 270 EA( I , J ) = STPAIN*A< I , J ) G.O TO 2 80 1 2 3  FORTRAN 0243 0244 0245 0246 0247 0248 0249 0250 0251 0252 0253  0254 0255 0256 0257 0258 0259 0260 0261 0262 0263 0264 0265 0266 0 267 0268 0269 JO 2 7 0 0 271 0272 0273 0274 0275 _0276 0277 0278 0279 0280 0281 0282 0 28 3  0284 0285_ 0286 0287 0288 0289 0290 0 291_  IV G COMPILER 270 280  290 300  310  320  330  340  350 360 C C C  MAIN  04-29-70  19:36:39  PAGE  0007  EA(I,J) = -(STRAIN*A(I,J)) I F I F B I I , J l . G T . O . O ) GO T O 2 9 0 EB(I ,J) = STRAIN*8(I , J) GO T O 3 0 0 EB( I , J I (STRAIN*B(I,J)) CONTINUE 00 310 J = 1 , N J l = J+1 DO 3 1 0 I = 1 , N II = I+l FACTOR=((X(I,J1)-X(I1,J))*SIN(PSIB(I,JI)I+(Y(I1,J)-Y(I,Jl))*COSt 1 PSIBI I , J l ) ) - E B ( I , J l > - E A ( 1 1 , J ) * S I N ( P S I B I I , J l ) + P S I A ( I I , J) ) )/COS( 2 P S I A I I I , J 1 + PS I B ( I , J 11 I __ XI I I , J 1 l = ( X ( I I , J ) + E A ( I I , J ) * C O S ( P S I A ( 1 1 , J ) 1 - F A C T O R * S I N ( P S I A ( 1 1 , J 1 1 ) Y ( U , J I ) = ( Y ( U , J I - E A ( I 1 , J ) * S I N ( P S I A ( I I , J ) ) - F A C T 0 R * C O S ( P S I A( I 1 , J ) ) 1 CONTINUE COORDINATES OF TEE P I N J O I N T S . WRITE(6,131 DO 3 2 0 1 = 1 , N ii =i+i x j < i i , i i=o.o XJ(1,111=0.0 Y J ( II, 1 1 = 0 / 2 . 0 Y J l 1 , II1= -D/2.0 CONTINUE DO 3 3 0 1 = 1 , N t 1=1+1 X J ( II,2 ) = A( I 1 , 1 ) * C O S ( P S I A ( I I , 1 ) 1 YJ(I1,2)=YJ(I1,11+(A(I1,11*SIN(PSIA(I1,1>») CONTINUE' DO 3 4 0 I = 1 , N 1 00 340 J=2,N J1=J+1 XJ(I,J1)=XJ(I,J)+AII,J)*C0S(PSIA(I,J)1 YJ( I , J l ) = Y J ( I , J 1 + A ( 1 , J ) * S I N ( P S I A ( I , J I 1 CONTINUE DO 3 6 C 1 = 1 , N l I I = 1-1 DO 3 6 0 J = 1 , N 1 JJ=J-1 I F ( J . G T . l ) GO TO 3 5 0 I F ( I . EQ . 1 1 GO TO 3 6 0 W R I T E ( 6 , 1 5 ) LL >JAJXJl I , J l , II,J J , Y J ( I , J l , I I , J J , X ( I , J l , II,JJ,Y( I,J1 CONTINUF C O M P I L E P R O P E R T I E S O F EACH M E M B E R . F O R C E S AND W E I G H T S I N P O U N D S , W I D T H S I N I N C H E S , C R O S S S E C T I O N A L A R E A S IN S Q U A R E I N C H E S A N D MOMENTS O F I N E R T I A I N ( I N C H E S 1 * * 4 DO 4 0 0 1 = 1 , N II = I+l II = I - l DO 4 0 0 J = 1 , N J1=J+1 JJ=J-1 WA( I 1 , J ! = ( A B S ( F A ( I 1 , J 1 1 ) / ( T * S I G M A ) WB( I , J 1 1 = ( A B S ( F B I I , J l 111 / ( T * S I G M A )  FORTRAN  IV  G COMPILER  0292 0293 0294 0295 0296 0297 _Q.29.8. 0299 0300 0301 ...0302 0303 0304 0305 _03.0.-__  . 370 380  390 400 C C C C  0307 0.3.0 fi 0309 0310 0311 0312 _0.3.13 0314 0315 0316 . 0317 0.3.1.8 0319 0320 0321 0322 0323 0.32.4 0325 0326 0327 0328 0329  0330 0331 0332 0.3.3.3  410  420  450 C  MAIN  04-29-70  19:36:39  PAGE  A R E A A I [ 1 , J ) = ( T * W A ( 11 , J ) ) AR E A B ( I , J 1 > = ( T * W B ( I i J l l l M I Z A I 1 1 , J ) = ( ( W A ( 11 , J > * * 3 ) * T ) / 1 2 . 0 MIZ8(I,J1) = (< WB ( I » J 1 ) * # 3 ) * T ) / 1 2 . 0 WT I N A ( I 1 , J ) = ( A R E A A I I I , J > * S P G T Y * 6 2 . 4 ) / 1 7 2 8 . 0 W T 1 N B I I , J 1 ) = ( A R E A B ( I , J l ) * S P G T Y * 6 2 .4 )/1728 .0 I F ( W A ( 1 1 , J ) . G T . 0 . 2 5 ) GO TO 3 7 0 EULCLA(Il,J)=(ABS(FA(ll,J))*AtIl,J)*A(ll,J)*110.0)/ I IELMOD*MIZA<I 1,J)*PI*PI) GO TO 3 8 0 EULCLA(I1,J) = ( A B S ( F A ( U , J ) ) * A ( I 1 , J ) * A < 11,J1*76800.0 )/(ELMOD* 1 WAII 1 , J ) * P I*P I ) I F ( W B ( I , J 1 1 . G T . 0 . 2 5 ) GO TO 3 9 0 EULCLBII,J1)=(ABSIFB(I,J1))*B(I,J1)*B(I,J1)#100.0)/ 1 <ELMOD*MIZB<I,J 1 ) * P I * P I ) GO TO 4 0 0 E U L C L B I I , J 1 ) = ( A B S ( F 8 ( I , J 1 ) ) * B < I , J 1 ) * B ( I , J 11*76800.0)/<ELMOO 1 *WB(I , J 1 ) * P I*PI ) CONTINUE R E A C T I O N S AT S U P P O R T S . T H E H O R I Z O N T A L C O M P O N E N T S O F T H E R E A C T I O N S AT A AND B A R E M E A S U R E D P O S I T I V E A C T I N G T C T H E R I G H T , AND T H E V E R T I C A L COMPONENTS ARE MEASURED P O S I T I V E A C T I N G UPWARD. FAH=0.0 FAV=0.0 DO 4 1 0 I=2,N1 FAH=FAH-<FA(I,l)*COS(PSIAU,l))) FAV=FAV+(FA{I,1)*SIN(BETAA-(TR*FLOAT(1-21)))+(ABS(FA(1,1))* 1 A(I,1)*W) CONTINUE FBH=0.0 FBV=0.0 DO 4 2 0 J = 2 , N 1 FBV=FBV-(FB(1,J )*COS(BETAA*(TR*FLOAT(J-2))))+(ABS(FB(l,J>>* 1 B(1,J)*W) FBH=FBH(FB(1,J) *SIN(PSIB(1,J))1 CONTINUE WRITE(6,41) WRITE(6,43)FAH,FAV WRITE(6 , 4 5 ) F B H , F B V WRITE t 6 , 3 3 ) DO 4 5 0 1= 1 , N II = I-l 11 = I+l DC 4 5 0 J=1,N J l = J+1 JJ = J - l W R I T E ( 6 , 5 5 ) I , J J , F A ( I l . J ),WA( I I , J ) , A R E A A I I 1 , J ) , M I Z A l I 1 , J ) , 1 WTINA(II.J) ,EULCLA( I1,J),II,J,FB(I,J1),WBII,J1),AREAB(I,J1), 2 MIZB( I , J 1 ) , W T I N B ( I , J 1 1 , E U L C L B ( I , J 1 ) CONTINUE F I N A L S T E P I S TO D E T E R M I N E V O L U M E AND W E I G H T OF S T R U C T U R E . VOLUME = 0 . 0 VOL T E N = 0 . 0 VOLCOM =0.0  0008  FORTRAN 0334 0335 0336 0337 0338 0339 0340 0341 0 34 2 0 343 0344 0345 0346 0 34 7 0348 0349 0350 0351 0352 0353 0354 0355 0356 03.5 7 0358 0 359 0360 0361 0362 0363 0364 0365 0366 TCTAL  IV  G COMPILER  MAIN  04-29-70  WEIGHT=0.0 DO 4 8 0 I =1,N DO 4 8 0 J =1,N 11=1+1  460 4 7C 480  490 5.C0 ..  ""900  MEMORY  J1=J+1 I F I F A ( 11 , J l . L T . O . O ) GO TO 4 7 0 VOL T E N = V O L T E N + I A I 1 1 , J ) * A R E A A 1 1 1 , J ) > I F ( F B ( I , J 1 ) . L T . 0 . 0 ) GO TO 4 7 0 VOL T E N = V O L T E N + t fit I, J l ) * A R E A B ( I , J 1 ) 1 GO TO 4 8 0 I F < F B ( I , J 1 ) . L T . O . O ) GC TO 4 8 0 VOL TEN = VOL T E N + ( B ( I , J 1 ) * A R E A B ( I , J l ) ) CONTINUE 00 500 I =1,N DO 5 0 0 J = 1 , N 11 = I+l J l = J+1 I F I F A ( 1 1 , J l . G T . 0 . 0 1 GO TO 4 9 C V O L C O M = V O L C C M + l A( I i , J ) * A R E A A ( H , J I | I F ( F B ( I , J 1 1 . G T . 0 . 0 1 GO TO 4 9 0 V O L C O M = V O L C O M + ( B < I , J 1 ) * A R E A B ( I , J 1) ) GO TO .500 I F ( F B I I , J 1 ) . G T . 0 . 0 ) GO TO 5 0 0 VOLCOM = VOLCOM + ( B l I , J 1 ) * A R E A B ( I . J l ) ) CONTINUE V O L U M E = V O L U M E + VOL T E N + V O L C O M WEIGHT=WEIGHT+IVOLUME*C) W R I T E ( 6 , 2 5) VOLTEN,VOLCCM,VOLUME WRITE ( 6 , 5 1 ) WEIGHT WRITE(6,53) GO TO 9 9 0 CONT I N U E END  REQUIREMENTS  013200  BYTES  19:36:39  PAGE  0C09  EXECUTION  TERMINATED  SRUN - L O A C H 5 = « S O U R C E « EXECUTION BEGINS N  MODULUS THETA-  OF  =  3  ELASTICITY  35.839*9  6=»SINK,»  0  =  TLX  =  2.5OOOO0  300000.0  P.S.I.  9.83861  TLY  =  UNIFORM =  =  XSPAN  1.00368  10.000000  STRESS  -  300.000000  R AOR AT  =  1. 2 5 6 7 1  . .. .  YSPAN„_  P.S.I. RAO A  1.000000.  UNIFORM  -  STRAIN  =  0.001000  INCHES/INCH.  BET.»0.__3.8...S.02O3.  .1.5.5636..... RADB..?....1.. 9 5 6 4 7  THE FOLLOWING RESULTS ARE CALCULATED FOR A S T R U C T U R E MADE F R O M CR39 PLASTIC HAVING A S P E C I F I C GRAVITY OF 1.31 THE EFFECTS OF SELF WEIGHT FORCES ARE' INCLUDED IN THESE RESULTS. .... LENGTHS A(  1 . 01  AI Al Al Al Al  1 . 11 1. 21 2 . 01 2 . 1) 2 , 21 3 , 01 3, 1) 3 , 21  A( A( Al  OF  MEMBERS  =  1.556356  81  _  1.203424  = = = =  1.203424 1.556356 2. 1 7 5 2 2 6  Bl B< Bl Bl Bl  2.963503 1.556356 3.373761 5.640590  =  0.  11  =  1.956466  1,  11 1 I 21  = =  0.957316 0.957316 1.956466  2. Or  2) 21 0 . 31 1 . 31 2 , 31 1. 2,  61 B( Bl  = = ? . . . .  2.049350 2.750836 1.956466 3.396167 5.531818  NOTE THE TERM RATIO I N THE FOLLOWING TABLE INDICATES THE PERCENTAGE RATIO OF T H F FORCE IN E A C H M E M B E R TO I T S EULER CRITICAL LOAD FOR BUCKLING IE» I « | P I > * ( P I I I / I L * L )  LOACI  3,  31  JOINT X Jl XJ( XJ 1 XJ( XJ( XJI X Jl XJI  XJI XJI XJI XJI XJI XJI XJ I  Al Al Al A( A( Al Al A( A(  0. Or C, 1, 1, 1, 1, 2i  0) 11 21 31  YJI  -z  = = =  01  -  TENSILE  FORTRAN  • END-OE-FILE ERROR CCCURED ERROR RETURN  tSIGNOFF  1 ) 21 • 31  -1.250000 -1 . 2 5 0 0 0 0 -1.250000 1 .250000 0 .281102 -0.721413 -1 .923919  =  1 .250000  = = = = = =  1.177264 0.412272 -2.056483 1.250000 2 .100945 2.987380  11 2) 3) 0)  0.0 0.0 0.0 I.217982 1.883706 1.836732 0.0  YJI YJI YJ 1 YJI YJI YJI N<  1.554655 3.59092 5  YJI YJ< YJI YJI YJI. YJI  2, 1 ) 2 , 21 2 . 31 3 , 01 3 , 11 3. 2)  YJI  3,  3>  3)  =  0.0  01  = =  XI XI XI XI  11 21 Ot 0 . 31 1 . 01 X 1 1, 1) XI 1. 2 ) XI 1, 3 ) XI 2, 0 1 0.  XI X 1 XI XI X I XI XI  31 = i .003675 R E A C T I O N S A l SUPPORT  2. 2. 2 . 3, 3 . 3 . 3.  = = = = = = = =  1) 2) 3) 0)  = = = =  1) 2) 3)  = = =  DEFLECTIONS  0.000000 -0.000780 -0-004115 0.0  0.001369 0.001086 -0.008104 0.0 0.004573 0.011826 0.005429  HORIZONTAL  COMPONENT  =  -393.544434  VERTICAL  COMPONENT  AT  R i, H O R I Z O N T A L  COMPONENT  =  393.544434  VERTICAL  COMPONENT  2.084218 1.111179 1.775552 1.689494 1.369884 MEMBFP.S =  I 0.08229  WT/IN. 0.01869  RATIO 4.7122  B(  0.31676 0.15341 C. 6 3 1 5 7 0.52105 0.27779 0.44389 0.42237 0.34247  0.04238 0.00481 C.33589 0.18862 0.02858 0.11662 0.10047 0.05356  0.01498 O.O0726 0.02988 0.02465 0.01314 0.02100 0.01998 0.01620  2.8173 2.8173 ,4.7122 9.2047 17.0849 4.7122 22.1427 61.8943  Bl Bl 81 Bl B( 81 Bl Bl  8.168051  ERROR: ON  REAP  OPERATION  VOLUME  COMPRESSIVE  Y( Y( YC Yl Yl YI Y( Yl  0, 0. 0.  11 21 31  = =  0.0  1. 1,  0)  =  0.0 0.002500 0.004462 0.005536 0.0  Yl Yl Y( Yl Yl Yl VI  2, 2. 2. 3, 3, 3, 3,  1. 1,  1) 21 31  2  01  t  1) 2) 31 01 1) 2) 31  -  = =  = = = =  0.0 0.0  0.004037 0.010976 0.020636 0(iO 0.004157 0.017950 0.051016  PC I NTS  A,,  AREA 0.39519  .  0.0 0.0 0 .0 0.0  AT  WIDTH 1.580773 1.267C59 0.613632 2.526271  ENCOUNTERED ON U N I T 5  TILT!  JOINT  0, 0 . 0. 1. I. 1. 1 > 2 ,  5.230312 0.0 1.303126 4.558352 9.838614  MEMBER FORCE 118.558060 l i r 01 1,, 1 1 95.0294B0 46. 022369 1 i 21 189.470322 2 i. 0 ) 2 . 1 1 156. 316360 2 . 21 83. 338516 133.166428 3,, 01 126.712097 3 . 1) 102. 741394 3 i 21  FATAL  100.000000  -  2 • 1 1 = 2 < 21 = 2 . 31 = 3 • 01 3 . 11 3 . 21 = 3 . 31  VOLUME  «»  11 21 31  =  COORDINATES  MEMBER 0, 1 1 0. 0. 1.  21 3) 11 1 . 2) 1. 31 2 . 11 2 . 2) 2. 3)  MEMBERS-  -  FORCE 116.271820 187 . 6 5 5 0 4 5 149.641769 -91.479309  WIDTH 1.550290 2.502067 1.995223 1.219724  151.623413 142.388916 -40.955383 -74.163086 115.452560  2.021645 1.898519 0.546072 0.988841 1.539367  8.837169  TOTAL  =  9.852905 90.147156  AREA 0.38757 0.62552 0.49881 0.30493 0.50541 0.47463 0.13652 0.24721 0.38484  VOLUME=  . I WT/IN. .RATIO 7.4464 0.07762 0.01833 0.32633 0.02959 7.4464 7.4464 0.16548 0.02360 0 . 0 3 7 8 O 0. 0 1 4 4 2 1.7826 0.17214 0.02391 8.1702 0 . 1 4 2 5 6 0. 0 2 2 4 5 2 2 . 4 3 7 8 1.7828 0.00339 0.00646 0.02014 0.01169 14.7208 0 . 0 7 5 9 9 0. 0 1 8 2 1 59.5302  17.005219  25  APPENDIX D FORCE SYSTEM IN  The members o f following  and  s o l u t i o n of the  techniques  determinate  structure.  f o r each j o i n t  L,  separately. and  the  i s then s o l v e d , It  (J  N N  )  lated  the  A  This the  Each j o i n t  appropriate  Once t h e  two  f o r c e polygon  members A^ _-^ N  forces  next inner  and  constructed;  s i n c e o n l y .two  procedure  constructed,  N  ,  are  J ,„ , > may (N-1,N)  the  last  t o be  support  ' f a m i l i e s ' of  considered.  points. toward  dealt with.  . >  reactions.  i n which the  f r e e b o d y d i a g r a m s and  be  2  s y s t e m a t i c a l l y , working  r e g u l a r way  s i m i l a r i n shape f o r need t o be  N  unknown f o r c e s e x i s t a t t h e s e  i s followed  the  B _-^  joint  i n t h e s e members have b e e n c a l c u -  j o i n t s J .... ... and . (N,N-1)  Because o f the  only  is isolated  trignometrically .  support p o i n t s which are  are  any  f r e e body d i a g r a m i s drawn  S o l u t i o n o f these p o i n t s y i e l d s the  is  straightforward,  f o r a n a l y s i s of  J  solved  various  i s most c o n v e n i e n t t o commence a n a l y s i s a t  where o n l y  involved.  i s quite  standard  considered  this  f o r c e s a c t i n g i n the  a Michell cantiliver the  statically  TYPICAL MICHELL CANTILEVER  force  joints  framework polygons  and  s i x cases  26  There a r e : a) J o i n t  NN. The  o n l y two member j o i n t  the f i r s t  t o be a n a l y z e d .  b) A l l o t h e r j o i n t s NOT  on t h e b a s e f a n s .  For inner j o i n t s , The  forces  before  i n t h e s t r u c t u r e and  f o u r members meet a t e a c h  i n two o f t h e s e must t h u s be  the j o i n t  c a n be s o l v e d .  known  For joints  on  o r J _ r t h e same e q u a t i o n s  the o u t e r f i b r e s , may be u s e d ,  point.  N  placing  the values f o r the missing  o u t e r member a t z e r o . c) J o i n t s on t h e 'A' b a s e f a n , EXCEPT F o u r members meet a t t h e s e between them d i f f e r thus  separate solutions  joint, d) J o i n t s  from  those  are similar  c a r e f u l use o f the 'C  is  less  J-j^-  T h e y c a n be s o l v e d equations but error  own e q u a t i o n s a r e u s e d .  In s y m m e t r i c a l  frameworks, w i t h  loadings,  i s unnecessary.  this  The o u t e r  t o t h o s e on t h e 'A' :  by  i f their  (b) and  members.  on t h e 'B' b a s e f a n , EXCEPT  likely  i n case  are required.  f a n , b u t o f o p p o s i t e hand.  e)  j o i n t s but the angles  J ^ ^, h a s o n l y t h r e e  These j o i n t s  J-^l*  symmetrical <  Joint J - ^ This its  joint own  i s unique  special  i n i t s g e o m e t r y and r e q u i r e s  solution.  27 f)  Support P o i n t s  A & B.  These a r e the l a s t (N) In  points  t o be s o l v e d .  members c o n v e r g e a t t h e s e  the following  Many  joints.  s o l u t i o n s , g e n e r a l i t y has been  m a i n t a i n e d by t h e i n c l u s i o n o f t h e f o l l o w i n g v a r i a b l e s ond  form i s t h a t used  dicated  on F i g u r e  i n computer  solutions).  (sec-  These a r e i n -  Dl:-  3 - o r BETAA a n g l e ABC  For  a symmetrical  6 = 45°  structure  A tan g = — B r  More g e n e r a l l y  ;  -  :  (2.2).  r  T , , . o r TILT , . (a,b) (a,b) u  A n g l e between d i r e c t i o n o f a p p l i e d load,  P, , o r LOAD, , , and (a,b) (a,b) N  s  perpendicular W or W  A constant gravity the  relating  (y a x i s )  the s p e c i f i c  o f the s t r u c t u r a l m a t e r i a l ,  uniform  u n i t s used. forces  t o span a x i s  s t r e s s and t h e s y s t e m o f For lengths  i n inches,  i n pounds Specific  Gravity  Uniform Stress  + o r THGRAV  The  62.4 2 x 1728  a n g l e between t h e d i r e c t i o n o f  gravity  and t h e p e r p e n d i c u l a r  span a x i s  (y a x i s ) .  to the  28  Figure Dl  Forces Acting  on  Typical Michell  Cantilever  The  f r e e body d i a g r a m s and  type of j o i n t  a r e now  F i g u r e D2  The D2,  Forces  t o be  assumed t o be  compressive.  is  by  determined  numerical  at  at j o i n t  a sketch of the  i n A_, ., , i s assumed N,N-1  the  f o r each  c o n s i d e r e d i n sequence:-  forces acting  together with  f o r c e polygons  tensile The  J  J  N  N  N  for  are  shown i n F i g u r e  f o r c e polygon. while  The  t h a t i n B„  t r u t h of these  sign of equations,  v a l u e s are used  N  , is N-1,N  assumptions  when t h e  computation.  force  actual  30 SELF WEIGHT FORCES The  s e l f weight  force,  (SWT), i s t a k e n a s h a l f t h e  w e i g h t o f e a c h o f t h e members m e e t i n g for  a g i v e n member A ^  i t s weight  a t the j o i n t .  Thus  i s g i v e n by  Weight of A b = Aob*(WA . *l)x 0  0  b  - Aab * fFAob) x =  2 W  *>»62-4  Dl  [AabFAob]  Thus f o r t h e j o i n t  NN t h e s e l f  weight f o r c e  i s given  by t h e e q u a t i o n  (SWT)  =: WJA .^FA .H.,-»- BH...MFBH.,. ] N  H  M  D2  SOLUTION OF FORCE POLYGON  From F i g u r e D2 t h e f o l l o w i n g e q u a t i o n s may be d e r i v e d : TV  -  T U+UV  FB H. 5ia(:3+§)^( ^  . . D3  S  M  N  LU  *  LR+fcU =  M P * Q V  31 C o m b i n i n g D3 and D4  FA  sinfr+f-t)+ ( S W T ) C O S 0  D5  F E W * - F> COS ( H - T ) * ( cos 0 N  E q u a t i o n D2 may from  s w t  ^ ° c  3  (ft-|-T)  t h e n be u s e d  to eliminate  these equations to o b t a i n the f i n a l  (SWT)  result:-  RON N.M-l  D6  FBM_,.M—  P„„ [cos^-f-t) -W[A . -, sintt - Iff] n  (cos0 - W[AN.N-» SI n (p+f  h  EV..N  cos (p-f-fj])  32 b)  Typical  Inner  Joint  b)  Force  Polygon  (y+t-0) a)  Freebody  Diagram  l^tf-^  Swrz  Figure  D3  Figure in  the main  FB.^ Nb J  aN  Forces  D3  body  i s zero  illustrates  since  t h i s member i s zero.  1  FA  i s omitted  FA . ab  determined  angle from  shown  the  joint  joints  while  and FB , ab  for joints J  a r e known  forces  determined.  a s lj. i n t h e f r e e  equation  Joint  at a typical  F o r edge  , , and FB , , a r e t o be a,b-I a-i,b The  Inner  the forces  of the structure.  s i m i l a r l y FA aN  while  at a Typical  body  diagram  may  be  H>=  90  + [(a-b)e- 1/3+f)]  D7  which i s d e r i v e d d i r e c t l y from e q u a t i o n B4. The s o l u t i o n o f the f o r c e polygon f o l l o w s a s i m i l a r pattern to that described  i n s e c t i o n A.  (SWT) = W AobFbb+Aa.b-.FA .b-<-* B . F B c b + B - , . b F B . , . J ] Q  and  Q  b  a  A  D8  thus  FAa.t>  |[ w|Aoi>co ^f-9)-Bo-,.i> +  S  sin(?+f-9)]]  ± B . b C c o a + t - 9 ) + W[B .,, s i a ^ f j l j Q  b  cos D9  FAo.b |>ne +W[A sin(H>+t) + A .b-.sin(4M-0)]J +FB. C »~ tN»>-' o»foH} - B o b siH t)J] qb Qb  Q  w  cos 0-w(A  0  .bH  cos(y+t-Q)+ B .,. sin(^+^jj a  b  c) T y p i c a l The  'A'  Fan  Joint  forces at a t y p i c a l  F i g u r e D4.  Joint  J  c o n s i d e r e d by  N  ^ may  case,  90°:-  be  the angle  y  =  11  i|> may  'A'  is specifically  be  fan j o i n t excluded  o m i t t i n g f o r c e FB^ obtained  from  Forces  at a Typical  shown i n  while ^.  equation  90 + ( ( a - l ) 0 - p )  F i g u r e D4  are  joint  In.this B4  by  adding  . . . . . . D10  'A'  Fan  Joint  35 The  s e l f weight f o r c e i s now g i v e n by:-  (SWT)=w[Ao.*FAa,^^  • • •  Equations f o r the unknown f o r c e s , FA^  q  and  may be d e r i v e d from the f o r c e polygon and equation to e l i m i n a t e  (SWT).  FA  PA , a  0  =  a J  The y i e l d s  J + W [ A  Q  F  B  (  a  _ T _  0  1  1  2.) '  D l l used  eventually -  , C O S ^ T - ! ) -  BO-...  sin^+t-IJ  +FB... si'n6+W[BQ.,cos(4'+t-f)-+ Bo-,., cosfy+t+i)~ "cos(y »t-t)+w [BQ-,., s>n  (y-*JD  D12  R\a..{staf • w[Aa.oS»'nty+f-1) + Ao.. sin (H>+•*)]] +FB ..[cosf-W[Aa.o oos(<M+f) - Bo, a  Sin  (4>+i)JJ  . • R ; . , [ s i n ( y ^ - W ^ q . o sin ( £ - jjflj cos I - W[A O COS (H» 0  +B -,., sin q  *• f)|  36 d)  a)  Typical  Freebody  'B'  are  shown  shown  Joint  Diagram  Figure  The  Fan  D5  forces  Forces  a t a  i n Figure  i n Figure  b)  D5,  D4.  < B'  typical  fan joint,  and  For  a t a  Force  may  this  be  Polygon  'B'  Joint  excluding  compared  with  joint J - ^ / those  case:-  (SWT) * W [ A , r A , M , . , F A . - + B . F B . i - B o . b F B . J ] t  b  b  )  b  l  i  <>  l  b  0  D13  37  cos | + W[A ,. cos (jM- f) - Bab s.'n (4>+*-0)]  FAi.«>  FA l.fcrl —  +  F B , .  b  D13  b  cos^t-|) + w[Bob Sin (Y-*)]]  cosf -  w[A,.t>-, cos (M-f)+Bobs*n(y+t)]  FAi.b C^9 W[Ai. sin(4»+^)+Ai.b-» si a (y+ 4-0)]] •FB,. [ I -W[A,.M«>sC^)-*B»bsin(V+^)J] V P,.b CsmC^t) - W[A..b-, Sin (T-4)]] +  b  FB  AB  b  cos f - W JA,.b-j  e) J o i n t J  a) Freebody  (q>+*-|) t Bob sin ^>+*)]  11  Diagram Figure D6  b) Force Polygon Forces a t J o i n t J  D14  The F i g u r e D6.  forces  From  acting  a t C, J o i n t  are indicated i n  the diagrams  (SWT) = vs/^ FA -»-A FA, -»-B FB + B F B j ll  n  Solution results  l0  0  0l  01  M  ....  D I S  of the force polygon y i e l d s the following  f o r t h e unknown f o r c e s  FA.  and F B  cosf -  n  n  .  cos  W|A„sin^©+ S o , (f-MX FA„ FB., s » n | - w|BIIsin('HD+ B .,sin(f-13+|J d3i sin Ct-p) - W JBo.,-smCf-.4j]] +  0  i + WJA,. S U \ ( * -e) - Bo.» 0  D16  FA,,  FB , 0  _sin|+W A,.o cos(P |->0 + A„«>s^-^) +  +FB cos|j!-W A,.o s i a ( ) 3 - f - 8 i , co*(pHj L+R, M  [A „ O Sin  Bo.,  P)]  f)  Support  Figure  D7  Points  Freebody Diagrams  No e x t e r n a l body  diagrams  effect  i n Figure  on t h e s t r u c t u r e .  included order  applied  D7 s i n c e  such f o r c e would  The s e l f w e i g h t f o r c e  have  no  i s however  an o v e r a l l c h e c k on t h e a c c u r a c y o f compu-  The e x t e r n a l  forces  should  be i n e q u i l i b r i u m  errors  i n t h e complex  across  the s t r u c t u r e . Considering  force  f o r c e , P, i s shown i n t h e f r e e -  i n t h e c a l c u l a t i o n o f t h e r e a c t i o n components i n  to provide  tation.  f o r Forces a t Supports  i s g i v e n by:-  a c t i n g on t h e e n t i r e  and t h i s  chain  provides  structure  a t e s t f o r any  of c a l c u l a t i o n c a r r i e d out  t h e s u p p o r t p o i n t A, t h e s e l f  weight,  40  SWT  =  w  D17  £(A*.FAO.^ Qal  The  various  forces  and  vertically  a c t i n g a t A may  to y i e l d  the  be  reaction  resolved horizontally components.  Thus:-  D18  A.  - I[*  Similar point  e q u a t i o n s may  be  derived  p e r h a p s f i n a l l y be  s y s t e m depends p r i m a r i l y on rather  size  components a t  support  B:-  I t may  the  f o r the  than i t s span.  forces and  the  shape o f  If self-weight  i n a l l members a r e  depend o n l y  pointed  on  the  the  forces  completely  angular  out  that  the  structure, are  neglected,  independent of  shape o f  force  the  span  structure.  The i n t r o d u c t i o n o f s e l f weight f o r c e s m o d i f i e s s i n c e the weight of a s t r u c t u r e i n c r e a s e s w i t h span. e f f e c t may  o r may  this  This  not be important depending on the r e l a t i v e  magnitude of the a p p l i e d e x t e r n a l l o a d i n g s of the s t r u c t u r e supporting  them.  and the weight  42 APPENDIX E DATA ON SELECTED MICHELL CANTILEVERS  A range of M i c h e l l c a n t i l e v e r s were analyzed w h i l e subjected The  t o a v a r i e t y o f loads a p p l i e d a t the pinned  c h o i c e o f parameters and the c o n c l u s i o n s  a study of t h i s data are d i s c u s s e d explained  joints.  reached from  i n Chapter 4.  As  t h e r e i n , a l l s t r u c t u r e s were assumed t o be c u t  from sheet CR39 one q u a r t e r i n c h t h i c k .  Thus t h r e e p a r a -  meters are common t o a l l s t r u c t u r e s : Youngs Modulus  E = 300,000 p . s . i .  Specific Gravity  p - 1.31  Thickness  t = 0.25  inches.  For a l l but one s e r i e s o f c a l c u l a t i o n s , the s t r a i n was taken as 0.001, which t h e r e f o r e  f i x e d the uniform  stress  as 300 p . s . i . The pounds.  u n i t load applied  i n a l l cases was one hundred  T h i s was an a r b i t r a r y s e l e c t i o n but permits o f r a p i d  c o n v e r s i o n of r e s u l t s t o correspond w i t h any other  loading  desired. The  data i s presented as a s e r i e s o f t a b l e s i n the  same sequence as the s e c t i o n s o f Chapter 4. present a summary o f the i n f o r m a t i o n  The t a b l e s  actually calculated  • since  the  sheer bulk of  t h i s work. member was lated  43  For  output prevents  i t s inclusion in  example f o r e a c h s t r u c t u r e  considered  separately  and  the  investigated,  following  items  each calcu-  for i t : member l e n g t h  ,  coordinates  of  i t s ends  deflections  of  i t s ends  cross-sectional  (before  loading)  area  width s e c o n d moment o f ratio load  of  force  weight per Little is  of  this  information,  structures  since A  i t can  a l l be  printout  a u t h o r on  table  having  buckling  of  r e l a t e s to Figure  the  following  be  Support Spacing  D =  1.0",  2.0"  Number o f  N =  2,  4,  the  Load  and  lists  data  for  properties:10"  without  quite  d a t a may  - End  4.1  L =  shown w i t h and  calculated  a l l the  SPAN  Fibres  from d e f l e c t i o n s ,  request.  In e a c h c a s e r e a c t i o n s , are  values)  critical  apart  TABLE 1 - S y m m e t r i c a l C a n t i l e v e r s  This  two  i n member t o E u l e r  s i m p l y when r e q u i r e d . from the  ( l e s s e r of  inch  represented here  obtained  area  3,  2.5", 5,  7,  5.0", 10,  10.0"  20  d e f l e c t i o n s , v o l u m e s and effect  of  self  weight.  weights  the  'with s e l f weight' f i g u r e being g i v e n  case, in  i n upright p r i n t .  The  first  'weightless'  values  follow  italics.  TABLE 2 - 5 F i b r e Symmetrical C a n t i l e v e r - End  T h i s t a b l e a l s o r e l a t e s to F i g u r e  4.1  a second s e r i e s of c a n t i l e v e r s of v a r y i n g ~ ratio. as  i n each  A l l had  f i v e f i b r e s but  P o i n t Load  and  concerns  span but  constant  the other p r o p e r t i e s  varied  follows:L  -  60",  D  -  15.0", 10.0", 5.0",  L D  N  40",  20",  10",  5",  4".  2.5",  1.25",  1.0".  4 =  }  i n a l l cases.  5'  TABLE 3 - T h e o r e t i c a l Optimum  Structure  These s t r u c t u r e s are the t h e o r e t i c a l optimum or minimum volume s t r u c t u r e s c o r r e s p o n d i n g to each s e t of M i c h e l l frameworks considered  i n T a b l e s 1 and  2.  The  quan-  t i t i e s quoted i n t h i s t a b l e are c a l c u l a t e d from equation 1.20  using values derived  from F i g u r e  4.2.  These v a l u e s  are p l o t t e d as the u l t i m a t e minimum volumes i n f i g u r e s 4.4  to  4.4e.  TABLE 1 DATA FOR  SYMMETRICAL MICHELL CANTILEVERS WITH 1 0 0 POUND LOAD > t = 0.25, a = 300 p . s . i . , E = 300,000  J ,  Note - F i g u r e s L D 1  L 1 0 .0  D 10.0  in italics N 2  r e f e r to 'weightless'  9  22.6199  Radius 7.0711  Deflecion J ^ N 0.0245  2 3 3 4  11.3532  0.0243  7.5741  0.0243  5 5 7 7 10  5.6819  0.0243  3.7886  0.0243  20  -100.0000 49. 9999  99.9999 50.0000  -100.1851 50.1918  2.5259  -100.1845 50.1917  -100.1850 50.1916  -100. 0006 49.9999  —  0.0243  -100.1849 50.1914  -100.0007 49.9998  10 20  100.1859 50.1935  -100.1850 50.1921  1.1965.  - 0.0243  Weight  B  -100.1859 50.1934  -100.0001 49.9999  —  Volume  Reactions A  -100.0006 49.9999  4  p.s.i.  structures.  -100.0003 49.9999  —  AT  Specific Volume  8.1807  0.3870  3.4708  8.1667  0.3863  3. 4648  100.1850 51.1921  8.1243  0.3843  3.4468  100.002 50.000  8.1106  0.3837  3.4410  100.1851 50.1919  8.1141  0.3838  3.4425  100. 0005 50. 0000  8.1004  0.3832  3.4367  100 .1845 50.1918  8.1104  0.3 8 3 7  3.4409  100.0000 50.0000  8. 0968  0.3830  3.4352  100.1850 50.1917  8.1079  0.3836  3.4399  100. 0006 50.0000  8. 0943  0.38 29  3.4341  100.1849 50 . 1 9 1 5  8.1068  0.3835  3.4394  8.0931  0.3829  3.4336  8.1059 . .  0.38 35  3.4390  ' 0.3828  3.4333  100.0006 49.9999  -100..1847 50.1910  , 100.. 1 8 4 6 50 . 1 9 1 0  ' -100.000 6 49.9996  100.0006 49.9996  ' 8.0923  (Table 1 - continued) L D  L  2  10.0  D  Radius  Deflection J  2  5.0  2 3 3 4 4 5 5 7 7 10 10 20 -  4  9  N  10.0  2.5  20 2 2 3 3  46.3972  3.5355  NN  0.0377  23.9915  0.0357  —  16.0919  0.0353  12.0946  0.0352  8.0753  0.0351  5.3871  0.0351  —  2.5528  0.0351  —  64.9424  1.7678  0.0618  —  35.8997 —  0.0520  Reactions A  Volume  Weight  Specific Volume  12 .5873  0 .5954  10.6808  12.5594  0.5941  10. 6571  11.9107  0.5634  10.1067  11.8880  0.5624  10.0874  11.7966  0 .5580  10.0098  11.7 746  0.5570  9.9912  11.7574  0.5562  9.9766  11. 7356  0.5552  9.9581  11.7295  0.5549  9 .9529  11 .707 8  0.5538  9.9345  11.7173  0.5543  9 .9426  11. 6958  0.5533  9.9243  11.7095  0 .5539  9.9359  11.6880  0.5529  9.9177  20 .6597  0 .9773  35.0600  20.5872  0. 9739  34.9370  17.3714  0.8218  29.4797  17.3332  0.8200  29.4149  B  -200.5097 50.2976 -199.9992 49.9999 -200.4681 50.2815 -200.0001 49. 9998 -200.4631 50.2787 -200.0010 49.9998 -200.4613 50.2776 -200. 0011 49.9997 -200.4587 50.2770 -199.9998 49.9996 -200.4603 50.2766 -200.0022 49.9997 -200.4588 50.2750 -200.0014  200.5097 50.2977 199.9992 50.0000 200 .4680 50.2817 200.0000 50.0000 200.4631 50.2788 200.0020 49.9999 200.4612 50.2778 49.9998 49.9998 200.4587 50.2771 199.9998 49.9997 200 .4603 50.2767 200.0021 49.9998 200 .4588 50.2760 200.0014  -401.5833 50.4886 -400.0005 50.0000 -401.1079 50.4107 -399.9812 49.9999  401.5835 50 .4888 400.0002 50.0002 401.1079 50 .4110 399.9812 50.0002  (Table 1 - continued) L D  L  D  o  5  10.0  2.0  N  6  4  24 .3895  4  -  5  18 .4137  5  -  7  12 .3342  7  -  10  8 .2402  10  -  20  3 .9084  20  0  2  69 .3904  2  -  3  39 .4075  3  -  4  26 .9230  4  -  Reactions  Deflection Radius  J  NN  0.0506  0.0501  0.0498  0.0496  0.0495  1.4142  0.0730  0.0581  0.0561  A  B  Volume  Weight  Specific Volume  -401.0779 50.3993  401.0779 50.3996  16 .8940  0.7992  28.6695  -399.9990 49.9998  399.9990 50.0001  16.8592  0 . 7975  28.6105  -401.0662 50.3952  401.0662 50.3955  16 .7338  0.7916  28.3977  -400.0022 49.9997  400.0024 50.0000  16.7000  0.  28.  -401.0549 50.3922  401.0552 50.3925  16 .6213  0.7863  28.2068  -400.0017 49.9995  400.0015 49.9997  16.5882  0. 7 847  28.1506  -401.0552 50.3906  401.0552 50.3909  16 .5721  0.7840,  28.1233  -400.0063 49.9993  400.0063 49.9996  16.5394  0. 7 824  28.0678  -401.0444 50.3891  401.0444 50.3894  16.5410  0.7825  28.0705  -399.9990 49.9989  399.9990 49.9992  16.5084  0.  28.0152  -502.3442 50.5774  502 .3442 50.5777  24 .4237  1.1554  51.8110  -499.9978 49.9999  499.9978 50. 0002  24.3203  1.1505  51.5916  -501.4792 50.4591  501.4792 50 .4595  19 .4195  0.9186  41.1954  -500.0000 49.9998  500.0002 50.0002  19.3748  0.9165  41.1006  -501.3994 50.4427  501.3994 50 .4432  18.7345  0 .8862  39.7423  -500.0032 49.9997  500.0034 50.0002  18.  0.8844  39.  6948  7900  7809  3403  6580  .  (Table 1 - c o n t i n u e d ) L D  10  1  10.0  D  1.0  N  6  5  20.3674  5  -  7  13.6627  7  -  10  9.1336  10  -  20  4.3338  20  -  2  79.1194  2  -  3  49.1555  3  -  4  34.2914  4  -  5  26.1422  5  . -  Radius  Deflectior J  Reactions  NN  -501.3752 50.4372  501.3752 50 .4375  18 .5071  0.8755  39 .2599  500.0042 50.0000  18.4689  0. 8737  39.1788  -501.3533 50 .4329  501.3535 50.4333  18 .3480  0 . 8680  38 .9224  -500.0000 49. 9994  499.9998 49.9998  18.3108  0. 8662  38.  -501.3533 50.4308  501.3535 50 .4311  18 .2787  0.8647  38.7754  500.0071 49.9995  18.2419  0. 8629  38.6973  -501.3398 50.4282  501.3398 50 .4286  18 .2353  0.8626  38.6833  -1008.5251 1008 .5256 51.0006 51.0011  42 .3208  2.0020  179.5537  41 . 9809  1.9859  17  26.6088  1.2587  112.8927  26.5373  1.2554  112.5893  -1003.0139 L003.0139 50.5876 50.5883  24.8859  1.1772  105.5829  24.  1.1745  105.3386  -1002.9141 L002 .9143 50 .5739 50 .6748  24.3336  1.1511  103.2397  -1000.0001 49. 9990  24. 27 99  1.1486  103.0119  0.0549  0.0547  -500.007 3 49.9941  0.0546  -500.0000 49.9983  -  999.9873 49.9993  -1000.0012 49.9994  0.7071  0.0728  500.0000 49.9987  999.9873 50.0000  -1003.3850 1003.3850 50 .6248 50.6291 -1000.0005 49.9993  0.0745  Specific Volume  B  -500.0042 49.9997  0.0796  Weight  A  0.0554  0.1259  Volume  1000.0005 49.9998  1000.0015 50.0001  1000.0061 49.9998  18.1987  8283  . 0.8609  8434  38.6056  -  8.1116  (Table L D  L  1 - continued) D  N  0  Radius D e f l e c t i o n NN J  7  17.6353  0.0717  7 10  11.8193  0.0712  10 20  5.6171  0.0709  20 -  Reactions A  B  -1002.8503 50.5648 -1000.0090 49.9991 -1002.8147 50.5599 -1000.0032 49. 9984 -1002.8008 50.5537 -1000. 0105 49.9966  1002 .8501 50.5657 1000.0093 49.9997 1002.8147 50.5606 1000.0034 49.9991 1002 .8008 50.5544 1000.0107 49.9973  Volume  Weight  Specific Volume  23.9534  1.1331  101.626 6  23.9021  1.1307  101.4090  23.7877  1.1253  100.9236  23.7374  1.1229  100.7102  23 .6857  1.1205  100.4909  23.6361  1.1181  100. 2804  TABLE 2 Data F o r Symmetrical F i v e  Fibre Michell Cantilevers  L o a d a t J,-j. - 100 l b .  L D  t = 0.25, a = 300 p . s . i . ,  Ins.  D  N  8  Ins.  Radius Ins.  60.0 15.0  10.6066  40.0 10 .0  7.0711  J  NN Ins.  1.3006  0.2004  A  20.0  10.0  5.0  2.5  5  18.414  3.5355  1.7678  0.1002  0.0501  lbs.  4.0  1.25  1.0  0.8839  0.7071  0.0251  0.0200  Support  p.s.i.  Volume  lbs.  cu.in.  weight  Specific Volume  lb  -406 .4343 52.3984  406 .4341 101.4232 52.3987  4 .7979  28.6868  -400. 0.022 49.999?  400.0024 50.0000  4.7400  28.3409  -402.2778 51.5923  402 .2776 51.5927  67 .3423  3.1857  28.5708  -400.  40.0  66.8002  3.1600  28.3408  0022  .0024  100.  2003  50.0000  -402.1335 50.7927  402.1335 50 .7930  33.5352  1.5864  28.4558  -400. 0022 49.9997  400.0024 50.0000  33.4000  1.5800  28.3411  -401.0662 50.3952  401.0662 50 .3955  16 .7338  0.7916  28.3977  -400.0022  400.0024  16.  0.7900  28.3403  49.9997  5.0  E = 300 ,000  B  49.9997  4  Span and F i x e d  Reactions  Deflection L  of Variable  7000  50.0000  -400.5337 50.1971  400 .5337 50 .1974  8.3585  0.3954  28 .3692  -400.0022 49.9997  400.0024 50.0000  8.3500  0.3950  28.3403  -400.4272 50.1576  400 .4275 50.1579  6 .6854  0.3163  28 .3640  -400.0022  400.0024  6.6800  0.3160  28.3411  49.9997  50.0000  Ln  o  51 TABLE • Data  for Cantilevers  With  Fan Angles  with  but  Fibre Michell  i n Previous  Table  6  o  D  F i b r e Networks  t h e Twenty  Detailed  9  L  Infinite  Equal to  Structures  3  o  P  For  L  r  1  1.1965  22.7335  8.0927  3.43342  2  2.5528  48.5032  11.6835  9.91390  4  3.9084  74.2596  16 .4826  27.9714  5  4.3338  82.3422  18.1571  38.517 4  10  5.6171  23.5147  99.7653 .  a l l  structures,  106.725  = 20, L = 10 .0", t = 0.25, a = 300  N  Load  Data  for Series Networks  of  (P ) T  Cantilevers  Equivalent  Cantilevers  to  L D  e  Five  lb.  With  6  Infinite  Fibre  Fibre Michell  With Variable  Detailed  L  = 100  p . s . i .  Span  Previously  m  r  V  ?Lr  m  60  4  18.414° 73.656  27.8624  10.6066  98.5084  40  4  18 .414° 73.656  27.8624  7.0711  65.6726  20  4  18.414° 73.656  27.8624  3.5355  32.8358  10  4  18.414° 73.656  27 .8624  1.7678  16.4183  5  4  18.414° 73.656  27.8624  0.8839  8.2092  4  4  18.414° 73.656  27.8624  0.7071  6 .5672  Note:  V  = 1.6418  L  in this  case.  52 TABLES 4 and 5 - S y m m e t r i c a l C a n t i l e v e r W i t h T i l t e d  The The  s t r u c t u r e and l o a d i n g  framework h a s t h e f o l l o w i n g  End Load  a r e shown i n F i g u r e  4.6.  measurements:-  Span L  =10.0".  Support Spacing D  =2.5".  fr N  = 5.  Load y axis.  100 l b a t L  T varies  from  (Jg ) unclined 5  a t an a n g l e  -45° t o + 9 0 ° i n s t e p s  T to the  of 5°.  T a b l e 4 summarizes t h e r e a c t i o n s , v o l u m e , w e i g h t a n d d e f l e c t i o n o f L f o r t h i s range o f l o a d i n g s , illustrates  t h e mass o f d a t a  investigations. force  This  thus the f o r c e  i n quite  simple  f o r e a c h member, t h e a x i a l  i s symmetrical  Note t h a t a l - -  the loading  i s n o t and  i n member A ^ i s n o t e q u a l n u m e r i c a l l y  to  i n B, . ba These f o r c e s  table, rical  lists  i s derived  f o r each angular p o s i t i o n o f the l o a d .  though the s t r u c t u r e  that  table  that  while Table 5  when t h e t i l t positions.  are equal i s zero  f o r only  two l o a d i n g s  and 9 0 ° , s i n c e  i n this  t h e s e a r e symmet-  TABLE  4  S Y M M E T R I C A L M I C H E L L C A N T I L E V E R S WITH SKEW END L O A D I N G N = 5, L = 1 0 . 0 " , 2 . 5 " , E = 3 0 0 , 0 0 0 p . s . i . , a = 300 p . s . i . , £ = 0 . 0 0 1 , S p g t y 1.31. D e f l e c t i o n 0.00549", L o a d 100 p o u n d s at various points Direction of load  A.Hor.  A.Vert.j  Reactions B.Hor.  Volume  Weight  -45 40 35 30 25  -318.916 -339.341 -357.184 -372.308 -384.599  3.695 9.579 15.389 21.082 26.614  248.190 275.049 299.813 322.297 342.328  67.651 67.717 67.267 66.306 64.839  11.8308 12.8172 13.7060 14 .4906 15.1649  0.5597 0 .6063 0 .6484 0 .6855 0.7174  20 15 10 - 5 0 + 5 10 15 20 25  -393.962 -400.328 -403.647 -403.893 -401.066 -395.187 -386.300 -374.473 -359.795 -342.380  31.945 37.032 41.837 46.323 50.395 54.208 57.545 60.445 62.885 64.846  359.753 374.441 386.278 395.176 401.066 403.904 403.668 400.360 394.005 384.651  62.880 60.441 57 .543 54 .207 50.396 46 .325 41.840 37.036 31.950 26 .621  15.7237 16.1629 16 . 4 7 9 1 16 . 6 6 9 9 16.7338 16 . 6 7 0 3 16.4800 16.1643 15.7255 15.1671  0.7438 0.7646 0.7796 0 .7886 0.7916 0.7886 0.7796 0.7647 0.7439 0.7175  30 35 40 45 50  -322.359 -299.885 -275.128 -248.278 -219.533  66.313 67.276 67.727 67.662 67.083  372.370 357.255 339.421 319.003 296.158  21.090 15.398 9 .589 3.707 - 2.204  14.4932 13 .7090 12 . 8 2 0 5 11.8344 10.7583  0.6856 0 .6485 0.6065 0.5598 0.5089  55 60 65 70 75  -189.127 -157.277 -124.229 - 90.237 - 55.557  65.993 64.407 62.331 59.780 56.774  271.059 243.898 214.880 184.226 152.171  - 8.097 -13.929 -19.655 -25.231 -30.615  9.7324 9.3453 8.8872 8.4026 7.9381  0 .4604 0.4421 0.4204 0 .3975 0.3755  80 85 90  -  53.336 49.493 45.274  118.957 84.838 50.073  -35.767 -40V645 • -45.205.  7 .4691 "7 .0708 6.9496  0.3533 0.3345 . -0-.3286  20.455 14.803 49.948  Deflections *  0 .0  ,  0.0501  .0170 .0170 .0170 .0186 .0186  , , , , ,  0.0265 0.0265 0.0265 0.0225 0.0225  -0 . 0 1 9 3 -0 ; o 2 o i -0 .0208  , , ,  0.0193 0.0132 0.0  -0 -0 -0 -0 --0  U)  TABLE 5 Forces  i n Members o f a F i v e  Subjected  Fibre  Symmetrical  t o a One Hundred Pound L o a d Forces'  In  'A'  Michell  Cantilevers  a t Various Angles  Members  Tilt  Volume  -45  11.831  104  110  116  122  123  5.63  17.5  30.6  45.1  -40  12.817  105  111  117  123  125  8.63  20.8  34 .2  49.0  -35  13.706  105  111  117  123  125  11.6  23.9  37.5  52 .6  -30  14.491  105  111  117'  123  124  14.4  26.9  40.6  55.7  -25  15.165  104  110  115  121  123  17.2  29.6  43.3  58.4  -20  15.724  101  107  113  119  120  19.8  32.1  45.7  60.7  -15  16.163  98.6  104  110  115  117  22.2  34 .4  47 .8  62.5  -10  16.479  94.9  100  106  111  113  24.5  36.4  49 .5  63.9  - 5  16 .670  90.6  95.5  101  106  107  26.6  38.1  50.8  64 .7  Zero  16.734  85.5  90.2  95.0  100  101  28.5  39.6  51.7  65.1  + 5  16.670  79.8  84.2  88.7  93.5  94.7  30.2  40.7  52 .3  64 .9  10  16.480  73.5  77.5  81.7  86.1  87.2  31.7  41.5  52 ;4  64.3  15  16.164  66.7  70.3  74.1  78.1  79.1  32.9  42.1  52.1  63.2  20  15.726  59.3  62.5  65.9  69.4  70.3  33.8  42.3  51.5  61.6  25  15.167  51.5  54.3  57.2  60.3  61.1  34.5  42.1  50.4  59.5  30  14.493  43.2  45.6  48.1  50.7  51.3  35.0  41.7  49.0  57.0  35  13.709  34.7  36 .6  38.6  40.6  41.2  35.2  40.9  47.2  54 .0  40  12.821  25.9  27.3  28.8  30.3  30.7  35.1  39 .8  45^0  50.7  45  11.834  16.9  17.8  18.8  19.8  20.0  34.7  38.5  42.5  46.9  50  10 .758  7.76  8.18  8.62  9 .08  9.20  34.1  36 .8  39 .7  42 .8  55  9 .732  -1.44 -1.52 -1.60 -1.68 -1.71  33.2  34 .8  60  9.345  -10.6 -11.2  -11.8 -12.6 -12.6  32.1  32 .6  36 .6 33.2  38.4 33.6  65  8.887  -19 .7 -20.8 -21.9 -23.1 -23.4  30.7  30.2  29.5  28 .7  70  8.403  -28.7 -30 .2 -31.9 -33.6 -34 .0  29 .1  27 .5  25.6  23 .4  75  7.938  -37.4  -41.6 -43.8 -44.4  27 .2  24 .6  21.5  18.1  80  7.469  -45.9 -48.3 -51.0 -53 .7 -54 .4  25.2  21.5  17 .3  12.6  85  7 .071  -54 .0 -56 .9 -60 .0 -63.2 -64 .0  23.0  18 .2  12.9  6 .94  90  6 .946  -61.6 -65.0 -68.5 -72.2  20.6  14.8  8.42  1.27  A  54  A  53  -39.4  a  52  A  51  A  50  -73.1  A  44  A  43  A  42  A  41  55 Table 5  A  40  A  (continued)  34  A  33  A  32  A  31  A  30  A  24  A  23  A  22  A  21  A  20  52.0  5.93  19 .1  34.9  53.7  63.1  6 .25  20.8  39 .6  63.4  75 .7.  56.1  9.10  22.9  39 .4  59.0  68.7  9.58  25.1  45.1  70.3  83.3  59.7  12 .2  26.5  43.6  63.8  73.8  12.8  29.3  50.3  76.7  90.2  62.8  15.2  29.9  47.5  6 8.1  78.3  16.0  33.2  55.1  82.5  96 .4  65.5  18.1  33.1  51.0  72.0  82.3  19.1  36.9  59 .5  87.6  102  67.7  20.9  36.0  54.1  75.3  85.6  22.0  40.3  63.4  92.1  107  69.3  23.4  38.7  56.8  78.0  88.3  24.7  43.4  66.9  95.9  110  70.5  25.8  41.1  59.1  80.1  90.2  27 .2  46 .2  69 .8  99 .0  114  71.1  28.1  43.1  60.9  81.6  91.5  29 .6  48.6  72.2  101  116  71.1  30.1  44.9  62 .3  82.5  92.2  31.7  50.6  74.1  103  117  70.6  31.8  46.3  63.1  82.7  92.1  33.5  52.3  75.4  104  117  69.6  33.4  47.3  63.6  82.4  91.3  34.2  53.5  76.1  104  117  68.1  34.6  48.0  63.5  81.4  89.8  36.5  54 .4  76 .3  103  116  66.0  35.7  48.3  62.9  79.7  87 .6  37.6  54 . 8 75.8  101  113  63.4  36.4  48.2  61.9  77 .5  84.8  38.4  54 .9  74 .8  98 .7  110  60.4  36.9  47.8  60.4  74.7  81.3  38.8  54.5  73.2  95.6  106  56.8  37.1  47 .0  58.4  71.3  77.2  39 .0  53.7  71.1  91.7  102  52.9  37.0  45.9  56.0  67.4  72.5  38.9  52 .5  68.4  87.2  96.0  48.5  36.6  44 .4  53.2  62.9  67.2  38.5  50.8  65.2  81.9  89 .8  43.8  35.9  42.6  49 .9  58 .0  61.5  37 .9  48.8  61.5  76.1  82 .8  38.8  35.0  40.4  46.3  52.7  55.3  36.9  46.5  57 .4  69 .7  75 .2  33.4  33.8  38 .0  42 .3  46.9  48.6  35.6  43.7  52 .8  62.7  67.1  27.8  32.3  35.2  38.0  40.8  41.6  34 .1  40.7  47.7  55 .3  58 .4  22.0  30.6  32.2  33.4  34 .4  34 .3  32 . 3  37 .3  42.4  47 .4  49 .3  16.0  28.7  28.9  28.6  27.7  26.7  30.3  33.7  36.7  39 .2  39.9  9.92  26.6  25.4  23.5  20.8  18.9  28.0  29.7  30 .7  30.7  30 .1  3.74  24.2  21.8  18.3  13.7  11.0  25.5  25 .6  24.5  22 .0  20 .1  -2.46  21.7  17.9  12 .9  6.54  2.94  22.8  21.3  18.1  13.1  9.91  56  Table 5  A  14  A  13  (continued)  A  12  A  l l  A  10  •. 4 5 B  B  35  B  25  B  1 5  B  05  B  44  3.21  10.8  21.0  34.3  41.3  -16 .9 -17.8  -18.7  -19 .8 -20.0  -34.7  4.92  13.1  24.1  38.2  45.6  -25.9  -27.3  -28.8  -30.3  -30.7  -35.1  6.60  15.4  26.9  41.8  49.5  -34.7  -36 .6 -38.6  -40.6  -41.2  -35.2  8.22  17.5  29.6  45.1  53.1  -43.2  -45.6  -48.0  -50.6  -51.3  -35.0  9.78  19.4  32.0  48.0  56.3  -51.5  -54.2  -57.2  -60.3  -61.0  -34.5  11.3  21.3  34.2  50.6  59.0  -59.3  -62.9  -65.9  -69.4  -70.3  -33.8  12.7  22.9  36.1  52.8  61.3  -66 .7 -70.3  -74.1  -78.1  -79 .1 -32 .9  14.0  24.4  37.8  54.6  63.1  -73.5  -77.5  -81.7  -86 .1 -87 .2 -31.7  15.2  25.7  39.2  56 .0  64.4  -79.8  -84.2  -88.7  -93.5  -94 .7 -30.2  16.3  26.8  40.2  56.9  65.3  -85.5  -90.2  -95.0  -100  -101  -28.5  17.2  27.7  41.0  57.4  65.6  -90.6  -95.5  -101  -106  -107  -26 .6  18.1  28.4  41.4  57.5  65.5  -94.9  -100  -106  -111  -113  -24 .5  18.7  28.9  41.6  57.1  64.9  -98.6  -104  -110  -115  -117  -22.2  19 .3  29 .2  41.4  56.3  63.7  -101  -107  -113  -119  -120  -19 .8  19.7  29.2  40.8  55.1  62.1  -104  -109  -115  -121  -123  -17 .2  19 .9  29 .0  40.0  53.5  60 . 0 -105  -111  -117  -123  -124  -14.4  20.0  28.6  38.9  51.4  57.4  -105  -111  -117  -123  -125  -11.6  20.0  28.0  37 .5  49 .0  54.5  -105  -111  -117  -123  -125  -8.64  19.8  27.1  35 . 8 46.1  51.1  -104  -110  -116  -122  -123  -5.63  19 .4  26 .1  33.9  43.0  47.3  -102  -108  -114  -120  -121  -2.59  18.9  24.8  31.6  39.5  43.1  -100  -105  -111  -117  -118  + 0.48  18.3  23.4  29.2  35.7  38.6  -96.2  -101  -107  -113  -114  + 3.54  17.5  21.8  26.5  31.6  33.9  -92.0  -97 .0 -102  -108  -109  + 6 .58  16.6  20 . 0  23.6  27.3  28.8  -87.2  -91.9  -96 .9 -102  -103  +9 .56  15.5  18.1  20.5  22.8  23.6  -81.7  -86.1  -90.8  -95.7  -96.9  +12 .5  14.4  16.0  17.3  18.1  18.1  -75.6  -79.7  -84 .0 -88.5  -89.7  +15 .3  13.1  13.8  13.9  13.3  12.6  -68.9  -72.6  -76.6  -80.7  -81.7  +18.0  11.7  11.5  10.5  8.35  6 .91 -61.7  -65.0  -68.5  -72.2  -73.2  +20 .6  57 Table 5  B  34  -38.5  B  24  (continued)  B  14  B  04  B  43_  B  33  B  23  B  13  B  03  B  42  B  32  -42.5  -46.9  -48.5  -36.6  -44.4  -53.1  -62.9  -67.2  -38.5  -50.8  -39 . 8 -45.0  -50.7  -52.9  -37.0  -45.9  -56.0  -67.4  -72.5  -38.9  -52 .4  -40.9  -47.2  -54.0  -56.8  -37.0  -47.0  -58.4  -71.3  -77 .2 -39 .0 -53.7  -41.7  -49.0  -57.0  -60.4  -36.9  -47.8  -60.4  -74.7  -81.3  -38.8  -54 .5  -42.1  -50.4  -59.5  -63.4  -36.4  -48.2  -61.9  -77.5  -84.7  -38.4  -54.9  -42.3  -51.5  -61.6  -66.0  -35.7  -48.3  -62.9  -79 .7 -87 .6 -37 .6 -54 .8  -42.1  -52.1  -63.2  -68.1  -36.6  -48.0  -63.5  -81.4  -89.8  -36 .5 -54.4  -41.5  -52.4  -64.3  -69.6  -33.4  -47.3  -63.6  -82.4  -91.3  -35.2  -53.5  -40.7  -52.3  -64.9  -70.6  -31.8  -46.3  -63.1  -82.7  -92.1  -33.5  -52 .3  -39.6  -51.7  -65.1  -71.1  -30.1  -44 .9 -62.3  -82.5  -92.2  -31.7  -50.6  -38.1  -50.8  -64 .7 -71.1  -28.1  -43.1  -60.9  -81.6  -91.5  -29 .6 -48.6  -36.4  -49.5  -63.9  -70.5  -25.8  -41.1  -59.1  -80.1  -90.2  -27.2  -46 .2  -34.4  -47.8  -62.5  -69.3  -23.4  -38.7  -56 .8 -78.0  -88.3  -24.7  -43.4  -32.1  -45.7  -60.7  -67.7-20.8 -36.0  -54.1  -75.3  -85.6  -22.0  -40.3  -51.0  -29 . 6 -43.3  -58 .4 -65.5  -18.1  -33.1  -72.0  -82.3  -19 .0 -36 .9  -26.9  -40.6  -55.7  -62.8  -15.2  -29 .9 -47 .5 -68.2  -78.4  -16.0  -33.2  -23.9  -37.6  -52.6  -59.7  -12.2  -26.5  -43.6  -63.8  -73.8  -12.9  -29.3  -20.8  -34.2  -49.0  -56.1  -9 .10 -22.9  -39.4  -59.0  -68.7  -9.60  -25.1  -17.5  -30.6  -45.1  -52.0  -5.94  -19.1  -34.9  -53.7  -63.1  -6.26  -20.8  -14.1  -26 .8 -40.9  -47.6  -2.73  -15.1  -30.1  -48.0  -57.0  -2.87  -16 .3  -10.6  -22 . 8 -36 . 3 -42 . 8 + 0.51 -11.1  -25.1  -42.0  -50 .5 + 0.53 -11.6  -6.95  -18.6  -31.5  -37.7  + 3.73 -6.94  -20.0  -35.6  -43.6  -3.30  -14.2  -26.4  -32.3  +6.93 -2 .75 -14.6  -29.0  -36 .3 + 7 .30 -2.13  + 0.39 -9.80  -21.1  -26.7  + 10.1 +1.47 -9.17  -22.1  -28.8  +10.6 +2.66  + 4.07 -5.27  -15.6  -20.8  +13.1 +5.67 -3.65  -13.1  -21.0  +13.9 + 7.43  + 7.72 -0.71  -10.1  -14.8  +16 .1 + 9.83 + 1.89 -7.91  -13.1  +17.0 + 12 .1  + 11.3 + 3.86 -4.43  -8.63  +19 .0 +13.9 + 7.43 -0.70  -5.13  +20.0 + 16 .8  + 14.8 + 8.40 + 1.25 -2.49  + 3.93 -6 .90  +21.7 + 17.9 +12.9 + 6 .51 +2.91 +22.8 +21.3  Table 5  B  22  B  (continued)  12  B  02  B  41  B  31  B  21  B  l l  B  01  -65.2  -81.9  -89.7  -19.8  -27.1  -35.8  -46.1  -51.0  -68.4  -87.1  -96.0  -20.0  -28.0  -37 .5 -48.9  -54.4  -71.1  -91.7  -102  -20.0  -28.6  -38.9  -51.4  -57 .4  -73.2  -95.5  -106  -19 .9 -29 .0 -40.1  -53.4  -60.0  -74.8  -98.7  -110  -19.7  -29 .2 -40.9  -55.1  -62.1  -75.8  -101  -113  -19.3  -29.1  -41.4  -56 .3 -63.7  -76.3  -103  -116  -18.7  -28.9  -41.6  -57 .1 -64.9  -76.1  -103  -117  -18.0  -28.4  -41.4  -57.5  -75.4  -104  -117  -17.2  -27 .7 -41.0  -57 .4 -65.6  -74.1  -103  -117  -16.3  -26.8  -40.2  -56 .9 -65.3  -72.2  -101  -116  -15.2  -25.7  -39.2  -56 .0 -64.4  -69.8  -99.0  -114  -14.0  -24.4  -37 .8 -54.6  -63.1  -66.9  -95.9  -110  -12.7  -22.9  -36.1  -61.3  -63.4  -92.1  -107  -11.3  -21.3  -34 .2 -50.6  -59.5  -87.6  -102  -9.79  -19 .4 -32.0  -48.1  -56.3  -55.1  -82.5  -96.4  -8.22  -17 .5 -29.6  -45.1  -53.1  -50.3  -76.7  -90.2  -6.60  -15.4  -26 .9 -41.8  -49 .5  -45.1  -70.4  -83.3  -4.92  -13.1  -24.1  -38.2  -45.6  -39.6  -63.5  -75.8  -3.21  -10.8  -21.0  -34.3  -41.3  -33.8  -56.1  -67.7  -1.48  -8.42  -17.8  -30.2  -36.7  -27.7  -48.3  -59 .0 +0.27 -5.95  -14.5  -25.8  -31.8  -21.4  -40.1  -50.0  + 2 .02 -3.44  -11.1  -21.2  -26.7  -14.9  -31.6  -40.5  + 3.75 -0.90  -7.52  -16 .5 -21.3  -8.33  -22 .9 -30.7  +5.45 + 1.65 -3.93  -11.6  -1.69  -14.0  -20.8  +7 .11 + 4.18 -0.31  -6 .68 -10.2  -10.6  + 8.72 +6.68 + 3.31 -1.67  + 4.96 -5.0  + 11.6 + 4 .04 -0.37  -52.8  -65.5  -59 .0  -15.9 -4.56  + 10 .3 + 9 .13 + 6 .91 + 3.34 + 1.17  +18.1 + 13.0 +9.87 +11.7 +11.5 + 10.5 + 8.33 + 6.89  59  TABLE 6 - L o a d a t E a c h J o i n t o f a S y m m e t r i c a l  The jected  previous  pound the  (J  5 5  each o f these translations  force  be n o t e d  a t each j o i n t .  and o f t h e o u t e r  and o t h e r  table.  i t will  For a given  :  in.  rigid-body,  to the d e f l e c t i o n  loadings  may be  member, t h e t o t a l  be t h e sum o f t h e i n d i v i d u a l f o r c e s  considered  separately.  The c r o s s  sectional  d a t a may t h e n be c a l c u l a t e d f r o m t h e known. >  stress. d e f l e c t i o n o f the s t r u c t u r e under m u l t i p l e  n o t d e t e r m i n e d by a d d i t i o n . recorded  provided  end o f t h e  structure.  and d e f l e c t i o n s f o r m u l t i p l e  from t h i s  The  tion  In a d d i t i o n  t h a t many members a r e u n l o a d e d  and r o t a t i o n s , t h e y c o n t r i b u t e  a c t i n g along  uniform  i n each  t o a one h u n d r e d  s t r u c t u r e s , b u t because o f t h e i r  c a u s e d by e a c h l o a d area  the force  sub-  ) are recorded.  Forces  no o t h e r  For  It will  load  outside  t o t h a t most r e m o t e  example,  to the structure  lies  suppose t h e r e  shown i n F i g u r e  supports being  at J^c-  loading  be e q u a l t o t h e d e f l e c -  f o r t h e l o a d most remote from t h e s u p p o r t  members l e a d i n g  the  joint  the remainder o f t h i s  calculated  6 itemizes  i n succession  deflections of that  It w i l l  is  Table  fik>re c a n t i l e v e r , s u b j e c t e d  load placed  structure  of  t a b l e s have d e a l t w i t h c o n t i l e v e r s  t o an e n d l o a d .  member o f a f i v e  Cantilever  the area  enclosed  points,  by t h e  load. are several  loads  4.9, t h e most r e m o t e  I f a l l the other  loads  applied from  a r e at; j o i n t s  TABLE 6  Load a t Joint  Volume  5.1 5.2 5.3 5.4 5.5  1.9737 3.3446 6.3633 10.920 16.734  4.1 4.2 4.3 4.4 4.5  1.6251 2.9868 5.1224 7.8988 10.920  3.1 3.2 3.3 3.4 3.5  1.4238 2 .3823 3.6795 5.1224 6.3633  2.1 2.2 2.3 2.4 2.5  1.1326 1.7080 2.3823 2.9868 3.3446  1.1 1.2 1.3 1.4 1.5  0.8335 1.1326 1.4238 1.6251 1.9737  ,  l  A  54  85.53  61.66 90.17  52  31.49 65.00 95.05  A  51  -1.93 33.19 68'.50 100 .2  A  50  -33.73 -1.93 33.62 69 .39 101.5  A  44  43  28.52  33.53 39.57  35 .11 42.19 51.73  100.6  85.51 106.1  61.65 90 .14 111.8  42  o  Table 6  A  40  28.45 33.43 42.75 55.92 71.12 -1.829 31.90 65.83 96.26 119.4  (continued)  A  A  A  34  30.06  20.56  105.4  A  33  35.35 44.88  28.50 32.84  100 .6 111.1  ————— A  32  A  31  A  30  37.02 48.21 62.26  34 .89 46.60 63.10 82.48  28.45 36.97 51.04 70.17 92.15  33.51 39.53 46.40  35.10 42.17 51.67 61.32  31.91 37.19 46.10 57 .29 68.25  85.50 106.1 117.1  61.65 90.13 111.8 123.5  30.26 62.46 91.31 113.3 125.1  A  24  A  23  A  -  31.67  21.66  10.49  99.39  37 .26 50.62  30.04 36 .89  20 .54 22 .76  105 .4 104 .8  22  A  21  A  20  39 .02 54.73 74.10  36.78 53.23 75.99 102.8  28.46 40.70 60.18 86.52 117.0  35.33 44 .83 54 .95  37.00 48.17 62.19 76.15  31.92 40.95 5 4.37 70.63 86.46  28.48 32.82 36 .32  33.50 39.51 46.36 51.28  32.10 37.15 44.71 52 .78 58.37  100.6 111.1 110.4  85.50 106.0 117 .1 116 .4  59.26 86.63 107 .4 118.6 118.0  Table 6  (continued)  A  * 1 4  16.26  11.12  5.84  -0. 31  88.90  A  13  19.13 26.83  15.42 19 .51  10.55 11.96  5.11 5.06  99.70 88.91  A  12  A  l l  A  10  20.04 29.09 40.22  18.89 28.37 41.45 56.91  14.24 21.33 32.52 47.66 65.29  18.15 23.81 29.76  19.00 25.67 33.91 42.09 .  15.97 21.45 29.38 38.91 48.19  14.63 17.40 19.96  17.21 21.04 25.22 28.17  16.06 19.46 24.15 29.00 32.35  10.01 10.81 10.73  13.88 15.45 16.82 16.70  14.51 15.92 18.00 19.64 19.51  100.3 99.71 88.92  90.60 100.3 99.72 88.93  70.73 89.45 99.02 98.45 87.80  B  45  -85.53  B  35  B  25  B  15  B  05  -90.17  -95.05  -100.2  -101.5  -61.66  -65.00  -68.50  -69.39  -31.49  -33.19  -33.62  +1.90  +1.93  B  44  -100.6 -28.52  +33.73 CTl  Table 6  1  Bs4  (continued)  B  24  B  14  B  04  B  43  B  33  B  23  B  13  B  03  -117.9 -119.4 -65.07 -71.12  -105.4 ,|-111.1 -20.56 -32.84 -30.06 . -44.88  -117.1 -46 .40 -62.26  -123.5 -61.32 -82.48  -125.1 -68 .25 -92.15  -85.51 -90.14 -33.53 -42.19  -95.02 -96.26 -51.69 -55.92  -100.6 -28.50 -35.35  -106.1 -39.53 -48.21  -111.8 -5167 -63.10  -113.3 -57.29 -70.17  -61.65 -35.11  -64.99 -65.83 -40.50 -42.75  -85.50 -33.51 -37.02  -90.13 -42.17 -46.60  -91.31 -46 .10 -51.04  -61.65 -35.10 -34.89  -62.46 -37.19 -36 .97  -106.1 -111.8 -39.57 -51.73  -31.49 -31.90 -33.10 -33.43  +1.83 -28.45  B  42  -99.39 -10.49 -21.66 -31.67  B  32  -104.8 -22.76 -36.89 -50.62 -105.4 -20.54 -30.04 -37.26  -30 .26 -31.91 -28.45  (Ti CJ  Table 6  (continued) , ...  B  22  B  12  B B  02  -110.4 -36.32 -54.95 -74.10  -116.4 -51.28 ' -76.15 -102.8  -118.0 -58.37 -86.46 -117.0  -111.1 -32.82 -44.83 -54.73  -117.1 -46.36 -62.19 -75.99  -118.6 -52.78 -70.63 -86.52  -100.6 -28.48 -35.33 -39.02  -106.0 -39.51 -48.17 -53.23  -107.4 -44.71 -54.37 -60.18  -85.50 -33.50 -37.00 -36.78  -86.63 -37.15 -40.95 -40.70 -59.26 -32.10 -31.92 -28.46  41  -88.90 +0.31 -5.39 -11.12 -16.26  B  31  B  21,  B  ll  B  01  Load at Joint' ,  -88.91 -5.06 -11.96 ! -19.51 -26.83  -88.92 -10.73 -19 .53 -29 .76 -40 .22  -88.93 -16.70 128.17 -42.09 -56.91  -87.80 -19.51 -32.35 -48.19 -65.29  5.1 5.2 5.3 5.4 5.5  -99.70 -5.11 -10.55 -15.42 -19.13  -99.71 -10.81 -17 .40 -23.81 -29.09  -99.72 -16.82 -25.22 -33.91 -41.45  -98.45 -19.64 -29 .00 -33.91 -47.66  4.1 4.2 4.3 4.4 4.5  -100.3 -10.01 -14.63 -18.15 -20 .04  -100.3 -15.45 -21.04 -25.67 -28.37  -99.02 -18.00 -24.14 -29.38 -32.52  3.1 3.2 3.3 3.4 3.5  -90.60 -13.88 -17.21 -19.00 -18.89  -89.45 -15.92 -19.46 -21.45 -21.33  2.1 2.2 2.3 2.4 2.5  -70.73 -14.51 -16.06 -15.97 -14.24  1.1 1.2 1.3 1.4 1.5  •C-  65 Table 6  (continued)  D e f l e c t i o n a t Loaded J o i n t  Deflection at L i  Jr-  r  0.001 0.001 0.010 0.009 0.0  0.006 0.010 0.019 0.033 0.050  0.0 -0.001 0.006 0.007 0.0  0.011 0.016 0.031 0.040 0.050  -0.001 0.004 0.004 0.0 -0.009  0.005 0.009 0.015 0.024 0.033  0.0 0.0 0.0 0.0 -0.007  0.008 0.018 0.024 0.030 0.040  0.001 0.001 0.0 -0.004 -0.010  0.004 0. 007 0.011 0.015 0.019  0.0 0.0 0.0 0.0 -0.006  0.009 0.014 0.019 0.024 0.031  0.0 0.0 -0.001 -0.004 -0.001  0.003 0.005 0.007 0.009 0.010  0.0 0.0 0.0 0.0 0.001  0.006 0.010 0.014 0.018 0.016  0.0 0.0 -0.001 0. 001 -0.001  0.003 0.003 0.004 0. 005 0.006  0.0 0.0 0.0 0. 0 0.0  0.003 0.006 0.009 0.008 0.011  o  J  , s u c h t h a t a < 3 and b < 5, t h e s t r u c t u r e ab — — '  be e x a c t l y one  t h e same a s i f o n l y  load d i d not s a t i s f y  at J _ , 0  J-r-  deflection will  were l o a d e d .  these conditions  If,however,  b u t was p l a c e d , s a y  t h i s would modify t h e s t r u c t u r a l d e f o r m a t i o n .  members A--, A - , A _ , A ^ ,  A^, A^,  and  the d e f l e c t i o n of j o i n t  3  B ^ - are a l l unstressed,  to the load the  4  joints  there i n this  would modify area.  slightly  behaviour  t h e imposed  Member c r o s s - s e c t i o n s  the  s t r e s s , and t h u s t h e s t r a i n ,  constant.  TABLE 7 - T i l t e d  Symmetrical  various  table contains  five  i n the v e r t i c a l  Span  L = 10"  Support Spacing  D = 2.5"  L D Number o f F i b r e s  = 4 N = 3.  to maintain  Cantilever  regarding  a  three  plane but t i l t e d a t  4.10.  framework p r e v i o u s l y  follows:-  stresses  ;  c a n t i l e v e r dimensions are g e n e r a l l y  fibre  J r . 2 due.  loadings.  are adjusted  information  a n g l e s a s shown i n F i g u r e The  the  Three F i b r e  c a n t i l e v e r placed  , Bj-  , B ^ ,  i s due t o t h e r e q u i r e m e n t t h a t  uniform.  fibre  4  the d e f l e c t i o n s of a l l  are  This  3  In such c a s e s t h e d e f l e c t i o n s would  have t o be c a l c u l a t e d t o s a t i s f y This  B33, B  Although  considered  similar to  and a r e a s  TABLE 7 3 FIBRE SYMMETRICAL MICHELL Tilt Angle  Reaction at A A  H  A  v  CANTILEVER  WITH T I L T E D LOAD  Reaction at B B  H  B  v  Deflections Volume  Weight  -45 -40 -35 -30 -25  -221.121 -252.364 -282.663 -311.095 -336.797  230.065 227.373 219.298 206.084 188.131  221.121 252.365 282.663 311.095 336.797  -129.484 -126.744 -118.625 -105.372 - 87.387  12.2778 13.3023 14 .2256 15 .0407 15.7414  0 .5808 —1 0.6293 0.6730 0.7115 0 .7447  -20 -15 -10 - 5  -358.986 -376.986 -390.249 -398.372 -401.108  165.986 140.322 111.919 81.642 50.411  358.986 376.986 390.249 398.372 401.108  - 65.214 - 39.528 - 11.110 19.177 50.411  16.3221 16.7785 17.1070 17.3051 17.3714  0.7721 0 .7937 0 .8093 0.8186 0.8218  + 5 10 15 20 25  -398.372 -390.249 -376.986 -358.985 -336.797  -  19.177 11.110 39.529 65.215 87.387  398.372 390.249 376.986 358.985 336 .797  81.642 111.920 140 .323 165 .987 188.132  17.3051 17.1070 16 .7785 16 .3221 15.7414  0 .8186 0 .8093 0.7937 0 .7721 0 .7447  30 35 40 45 50  -311.095 -282.663 -252.364 -221.121 -189.883  -105.372 -118.625 -126.744 -129.484 -126.764  311.095 282.662 252.364 221.121 189.883  206.084 219.298 227 .373 230.065 227.292  15.0407 14.2256 13.3023 12 .2778 11.1600  0.7115 0.6730 0.6293 0.5808 0 .5279  55 60 65 70 '75  -159.599 -131.189 -105.515 - 83.354 - 65.378  -118.670 -105.451 - 87.510 - 65.394 •'- 39.7 80  159.599 131.189 105.515 83.354 ' 65.378  219 .141 205.862 187.858 165.680 140.002  9.9575 8.6795 7 .7690 7 .4223 7.0193-  80 85 90  - 52.130 - 44.010 - 41.259  - 11.449 18.734 49.854  52.130 44.010 41.259  111.605 81.354 50.172  6.5630 6.2115 6.1264  Zero  y  X  0  0.0520  0.4710 0.4106 0.3675 0.3511 0.3320  -0'. 0165  0.019 7  0 .3105 0 .2938 0.2898  -0. 0177 -0. 0184  0.0110 0  TABLE 8 5 FIBRE SYMMETRICAL MICHELL CANTILEVER WITH T I L T E D LOAD tilt Angle  Reaction at A A  H  *V  Reaction at B B  H  B  v  Deflections Volume  Weight  -45 -40 -35 -30 -25  -223.088 -253.984 -283.945 -312.061 -337.475  228.083 225.417 217.426 204.355 186.598  223.088 253.984 283.945 312.061 337.475  -127.524 -124.811 -116.779 -103.670 - 85.881  11.8283 12.8151 13.7043 14.4894 15.1641  0 .5595 0.6062 0 .6483 0.6854 0 .7173  -20 -15 -10 - 5 Zero  -359 .416 -377.214 -390.329 -398.361 -401.066  164.696 139.314 111.224 81.281 50.395  359.416 377.214 390 .329 398.361 401.066  +  63.952 38.550 10.445 19.507 50.396  15.7234 16.1629 16.4792 16.6700 16 .7338  0 .7438 0 .7646 0.7796 0 .7886 0.7916  + 5 10 15 20 25  -398.361 -390.329 -377.214 -359.416 -337.475  -  19.507 10.466 38.550 63.953 85.882  398.361 390.329 377.214 359.416 337 .475  81.281 111.224 139.314 164.696 186.598  16.6700 16.4792 16.1629 15.7234 15 .1641  0 .7886 0.7796 0.7646 0.7438 0 .7173  30 35 40 45 50  -312.061 -283.945 -253.984 -223.088 -192.197  -103.670 -116.779 -124.811 -127.524 -124.838  312.060 283.945 253.984 223.088 192.197  204.355 217.427 225.417 228.083 225.346  14.4893 13.7043 12.8151 11.8283 10.7517  0 .6854 0 .6483 0.6062 0.5595 0.5086  55 60 65 70 75  -162.248 -134.152 -108.759 - 86.842 - 69.062  -116.837 -103.764 - 86.021 - 64.152 - 38.822  162.248 134.152 108.759 86.842 69.062  217.291 204.162 186.362 164.432 139 .039  9 .7276 9.3400 8.8814 8.3972 7.9326  0 .4602 0.4418 0 .4201 0 .3972 0 .3753  80 85 90  -55.957 -47.923 -45.200  - 10.805 + 19.047 + 49.826  55.957 47.923 45.200  110.957 81" .'040 50.201  7.4638 7.0656 6.9404  0 .3531 0 .3342 0 .3283  x  0.0501  0.0170  0.0265  0.0186  0.0225  •0.0193 •0.0201 •0.0208  0.0193 0 .0131 0  CO  69 TABLE 8 - T i l t e d This the  structures  thus in  data  loading tion  Fibre  i n this table  differing  i n t h e i r angular  this table  Five  only  Symmetrical  i s s i m i l a r to t h a t . i n Table  i n t h e number  layout.  It will  of the s e l f weight  Cantilevers  increased  were  span  (L = 10") b u t o f  increasing  from  z e r o t o 2.5" i n t e n e q u a l  s t e p s and t h e k e y d a t a f o r t h i s s e r i e s o f s t r u c t u r e s in Table  It a complete  of  i s listed  9.  TABLE 10 - F i v e  The  The  from the d i r e c -  fibre Michell cantilevers  a l l of equal  The r i s e  4.  forces.  A series of f i v e  rise.  the f i g u r e s  i n Table  i d e n t i c a l apart  TABLE 9 - Skew M i c h e l l  investigated  7,  o f f i b r e s and  be s e e n t h a t  are c l o s e l y s i m i l a r to those  situations are v i r t u a l l y  Cantilevers  Fibre  Symmetrical M i c h e l l  seems a p p r o p r i a t e  to conclude  Cantilevers  t h i s appendix  s e t o f d a t a f o r one o f t h e s t r u c t u r e s  framework c h o s e n i s t h a t  having  five  investigated.  f i b r e s and a s p a n  10", l o a d e d w i t h one h u n d r e d pounds p e r p e n d i c u l a r  cantilever  axis.  Being  symmetrical,  with  to the  figures are given  here-  f o r h a l f t h e members. Table 10B c o n t a i n s  10A l i s t s  d e t a i l s of forces  w i d t h s and r a t i o load  member l e n g t h s  a r e shown  of a x i a l force  i n Table  10C.  and j o i n t  coordinates,  and d e f l e c t i o n s w h i l e to Euler  critical  t h e member  buckling  TABLE 9 SKEW MICHELL X Span In.  Y Span  Theta  Radrat.  CANTILEVERS Deflection  Volume  Vertical A lbs.  Reactions B lbs.  10 .0  0.250  18.420  1.058  0.0501  16 .742  40 .350  60.459  10.0  0.500  18.439  1.120  0.0502  16.757  30.319  70 .489  10.0  0.750  18.470  1.185  0.0502  16.781  20.288  80.520  10 .0  1.000  18.513  1.255  0.0503  16.816  10 .257  90 .551  10 .0  1.250  18.569  1.329  0 .0505  16 .860  0 .226  100.582  10.0  1.500  18.636  1.409  0 .0506  16 .913  -  9.805  110.613  10.0  1.750  18.715  1.494  0.0508  16 .976  -19 .836  120.646  10.0  2.000  18.806  1.587  0.0510  17 .047  -29 .878  130.689  10.0  2.251  18.908  1.687  0.0513  17.128  -39.921  140.734  10.0  2.501  19.020  1.796  0.0515  17.217  -49 .958  150.774  The  horizontal  reactions  For a l l structures  a t A and B a r e +401 p o u n d s i n a l l c a s e s .  N=5,  D=2.5",  a = 300 p . s . i . ,  t = 0.25", e = 0.001,  E= Load  300,000  p.s.i.  100 l b . a t J,.,..  TABLE 10A  TABLE OF MEMBER LENGTHS AND Five Fibre Michell  b =  0  1  JOINT COORDINATES Cantilever  2  3  4  5  a  A 1  —  1.7678  B  0.5657 0.0 1.25  A 2  4  5  '  -  0.7845 0.5657  0.8574  0.0  1.5808  2.2172  1.25  0.4588  0.0  1.7678  1.0152  1.1891  0.5657  0.9342  1.3805  0.0  1.7498  2.7187  3.6832  1.25  0.9987  0.6955  0.0  A  1.7678  1.2583  1.5643  1.9146  B  .-  0.5657  1.0151  1.5907  2.3140  0.0  1.7396  2.9977  4.4906  6.0436  1.25  1.5643  1.5870  1.1198  0.0  A  1.7678  1.5146  1.9867  2.5475  3.2092  A  0.0  1.5513  2.9794  4.9657  7.3969  B  1.25'  2.0977  2.6020  2.6379  1.8769  B  A 3  1.7678  1.25  . B  -  -  10.000 0.0  72 TABLE  10B  FORCES IN MEMBERS AND Five  b ->-=  Fibre  1  65.293  DEFLECTIONS  Symmetrical M i c h e l l  0  FA  JOINT  Cantilever  2  3  4  5  56.905  FB  -56.905  1 to  FA  0.0  0.0  0.0  0.0025  117.020  FB 2 CO  FA  FA  0.0  0.0016  0.0013  FA  82.483  62.257  44.876  -26.827  -50.624  -44.876  0.0  0.0016  0.0006  0.0  0.0  0.0011  0.0015  0.0020  FB CO  -74.104 0.0  71.122  4  -40.219 0.0012  FB CO  74.104  0.0  92.151  3  102.793  65.070  51.728  39 .572  28.519  -16.261  -31.673  -30.059  -28.519  0.0  0.0019  0.0013  0.0009  0.00  0.0  0.0005  0.0014  0 .0024  0.0033  101.475  100.177  95.048  90 .171  85.528 .  FB 5 CO  For  0.0  0.0020  0.0022  0.0020  0.0015  0.0  0.0  -0.0001  0.0010  0.0023  0.0039  0 .0055  a structure  x = 10.0", J,,;  t =  i n which  D = 2.5",  0.25".  |  =4.  Load  100 pound a t  TABLE IOC MEMBER WIDTHS AND CRITICAL BUCKLING LOAD RATIOS FOR 5 FIBRE SYMMETRICAL MICHELL CANTILEVER  a  b ->-=  0  1  WA  0.8706  0.7587  4,  WB 1 Q.  0.501  2  3  4  0.7587 0.068 0.068  WA  1.5603  WB 2  g. *o  WA  0.156  1.2287  WB 3  O,  0.252  WA  0.9483  WB 4  0, "0  0.423  WA WB 5  1.3530  0.208  -  1.3706  0.9880  0.5362  0.9880  0 040  0.092  0.135  0.092  1. 0998  0.8301  0.5984  0.3577  0.6750  0.5984  0.104  0.250  0.647  0.304  0 .233  0.647  0.8676  0.6897  0.5276  0.3803  0.2168  0.4223  0.4008  0.3803  0.256  0.625  1.601  4.502  0.828  0.703  1.915  4.502  1.3357  1.2673  1.2023  1.1404  0.156  -  -  0.299  0.546  0.963  -  -  -  5  74 TABLE 11 - V a r i a t i o n o f D e f l e c t i o n  A the  final  as  detailed  Strain  t e s t was made t o d e t e r m i n e t h e e f f e c t o f  s t r a i n i n the s t r u c t u r e .  with a ten inch  With  The s t a n d a r d  five  varying  fibre cantilever  s p a n was e x a m i n e d w i t h a r a n g e o f s t r a i n s ,  i n T a b l e 11.  Uniform Strain  Uniform Stress  Deflection (J )  0.00001  3.0  0.000501  2053.399  0.00005  15.0  0.002505  347.830  0.0001  30.0  0.005010  170.415  0.0002  60.0  0.010020  84 .348  0.0004  120.0  0.020040  41.961.  0.0006  180.0  0.030060  27.927  0.0008  240.0  0.040080  20 .928  0.0010  300.0  0.050100  16.734  0.0020  600.0  0.100200  8.358  0.0040  1200.0  0 .200400  4.177  0.0060  1800.0  0.300600  2 .784  0.0080  2400.0  0.400800  2 .088  0.0100  3000.0  0.501000  1.670  This proportional imately is  table  demonstrates that  proportional  the d e f l e c t i o n i s d i r e c t l y t h e volume i s o n l y  to the s t r a i n .  due t o t h e e f f e c t s o f t h e s e l f w e i g h t  been n e g l e c t e d  !  5 5  to the s t r a i n , but that  inversely  Volume  approx-  The d i s c r e p a n c y  forces.  t h e c o r r e s p o n d e n c e would have been  I f t h e s e had complete.  75  APPENDIX F  BIAXIAL STRESS IN JOINTS  The rigid  state of stress  M a x w e l l framework h a s b e e n d i s c u s s e d  manner i n C h a p t e r  5.  vicinity  of a r i g i d The  need n o t n e c e s s a r i l y  the  actual  the stresses  bending the  the (c)  at a specific  Fl.  joint.  Further, f o r  and d i s p e r s e d  along  to act.  t y p e s o f s t r e s s may be  considered:-  stresses of  w h i c h a r e assumed t o a c t i n c o n -  i n e a c h member, a r i s i n g f r o m  of the j o i n t .  the section, reaching  These v a r y  linearly  maximum v a l u e s a t  fibres.  shear s t r e s s e s parabolic  i n Figure  i t i s a t t h e edges o f t h e q u a d r i l a t e r a l  stresses  outer  i n the  directions;  rigidity  across  stresses  i n magnitude o r d i r e c t i o n  the u n i a x i a l tensile/compressive  ventional (b)  represent  they a r e considered  magnitude a  acting  a r e shown s e p a r a t e d  Three d i s t i n c t (a)  are indicated  of stress  members a l t h o u g h  QRWV t h a t  i n a simplified  shown i n t h i s d i a g r a m a r e i l l u s t r a t i v e  and  clarity,  The s t r e s s e s  joint  stresses  state  of j o i n t s i n a  The e f f e c t s o f t h e s e c o n d a r y  r e m a i n s t o be c o n s i d e r e d .  the  i n the v i c i n i t y  i n e a c h member.  manner a c r o s s  maximum a t t h e n e u t r a l  These v a r y  the section axis.  ina  reaching  a  ULLLlUJli. Distribution of S h e a r Sttess  across section,  Distribution of Bending StfesS across section.  iLUilUu  Uniaxial Stresses a'.=>ncj m e m b e r s .  Figure F l  Biaxial  Stresses at a T y p i c a l  Joint  The these in  sum  various  since the  the  effect  section.  of  Further,  as  i t s geometric  c o u l d be  an  as  element i s  representative joint  stresses varies  area  across joint  c e n t r e , J j^, the  will  they  approximation to the  considered  i f the  by  e l e m e n t i s moved f r o m t h e  i n some complex manner as As  e l e m e n t QRWV  element of the  bending  the  the  t o draw a  d i a g r a m f o r an  s h e a r and  on  to zero,  I t i s impossible  freebody  boundary toward vary  forces exerted  s t r e s s e s must add  equilibrium.  quantitative  of the  a rectangle  Q  stresses  f l o w between the real as  s t a t e , the  members. joint,  shown i n F i g u r e  F2.,  R  + V  Figure  F2  Approximation  w  to B i a x i a l l y  Stressed  Joint  ,  :  , The  of  78  sides of this  rectangle  a r e t a k e n as t h e average  t h e w i d t h s o f t h e a d j a c e n t members;  QR  =; V W =  (WBqb  that i s  * WBo-.-b^  2 . . . . = R W =  QV  (WA  Q  b  F l  + WAo.b-i)  2 The normal  stress,  actually at  axial  force  a l o n g e a c h member  a, on e a c h f a c e o f t h i s  be c c o s 8/2, a s s u m i n g  rectangle.  the rectangle  a uniform This should  t o be o r i e n t e d  0/2 t o t h e a x i s o f e a c h o f t h e f o u r members c o n v e r g i n g a t  joint  J  a b  «  However, c o s 8/2 i s c l o s e  9 encountered i n these c a l c u l a t i o n s ,  will  to 1 f o r the values of :  and may be d r o p p e d  N-5.  5 ^ 4 , 0 « 18*41*, e o a | * o - 9 « 7  M«3,  5-4,  If  are  imposes  small  cross-sectional be n e g l e c t e d The although  i n general  the shear f o r c e s  and t h e a v e r a g e  shear s t r e s s  a t each  joint  (shear f o r c e /  a r e a ) i s s m a l l compared w i t h a.  as a f i r s t  It will  thus  approximation.  b e n d i n g moments however, c a n n o t be so n e g l e c t e d  their  structure.  = o-9S\  t h e v a l u e s i n T a b l e 1, C h a p t e r 5, a r e e x a m i n e d i t  be s e e n t h a t quite  6 ' 35-90°.  >  relative  The b e n d i n g  w i t h t h e normal  stress  effect  varies  stresses  greatly  across the  are of-comparable  scale  i n the area of the fans but r a r e l y  79 exceed  30% o f t h e b e n d i n g  stress  i n the outer parts o f the  structure. Thus t h e n o r m a l s t r e s s e s o n t h e f a c e s o f t h e r e c t a n gular block read  c o u l d be m o d i f i e d  (a ± -j) .  are outer  limits  Even these joint  area, If  limiting then  This should  the c e n t r a l these  values  be r e a d  on t h e v a l u e  extreme v a l u e s  extreme combinations the v a r i o u s cases  only apply  i n the value  simple  case  locally practice  a t the  f o r most  of values.  ±c. a s r e p r e s e n t i n g the-  f o r the  By d r a w i n g M o h r s c i r c l e f o r 1  i t w i l l be f o u n d  t h a t t h e maximum  shear  w h i c h i s an i n c r e a s e o f one  already determined stress.  i n Chapter  5 f o r the  T h i s maximum w i l l  i n two o f t h e o u t e r c o r n e r s could probably  joints.  corners of the  s t r e s s may be d e t e r m i n e d  of b i a x i a l  these  a t the majority o f outer joints>  s t r e s s v a r i e s from y - t o - j — , third  of the stress  are accepted  of stress  i n F i g u r e F2 t o  i n t h e sense t h a t  s t r e s s e s being  values  t h e maximum s h e a r  as i n d i c a t e d  be t o l e r a t e d  of the j o i n t  occur area  as a l o c a l i z e d  and i n  stress  concentration. In p r a c t i c e , the  stress  duce t h e s e  flow around  i ffillets  were p r o v i d e d  a t the j o i n t s ,  these c o r n e r s might w e l l s u f f i c e  maximum v a l u e s  toward  the normal.value  a.  to r e -  80 DEFLECTION OF STRUCTURES CONTAINING BIAXIALLY STRESSED JOINTS  The considered uniaxial jected  deflections  i n Chapter  stresses  o f the M i c h e l l  i n a l l members.  to b i a x i a l stress  and w i l l  the  The j o i n t  a r e a as b e i n g u n i f o r m l y  areas are sub- • •  F o r purposes o f c a l c u l a t i o n  ±o  members, as shown i n F i g u r e  the stresses  indicated  i n Figure  i n the  [ t h e v a l u e s on t h e a x e s o f F2.  A t y p i c a l member may t h u s be d i v i d e d as  assuming  t h u s h a v e a somewhat l a r g e r  t h e magnitude o f t h i s change, c o n s i d e r  joint  already  4 and A p p e n d i x E a r e c a l c u l a t e d  d e f l e c t i o n due t o t h i s e f f e c t . of  cantilevers  into three  zones  F3.  V  B i o * ially Sl>es6ed  Zones.  Jab • i Untax i d l y  Stressed  "Zone • .Ny V v v N. v v v V  A db  Figure  F3  Extension of a Typical  Member  The change i n l e n g t h o f ( J  a  b  J  m &  ^ y thus be  c a l c u l a t e d as the sum of the changes o f the three zones. the  AL,  From  figure  ~  E  (l •*• V ) (wB b+WSo-//>) q  2. F2  Aob - (WBab-^WBo-i-b') -.(WBQ,^, + WB<W.6+/) 2  "2.  Total change , AA fe = Al,"* AL +ALa, Q  2  F3  E  The change o f e x t e n s i o n due t o b i a x i a l s t r e s s i s thus g i v e n by: -  Chpe (mcreose) ,n expansion o" ncib  ^/wB, +WB ., *WBa.b ^WB -, J • F4 2. E. V / b  q  b  +  0  A s i m i l a r e x p r e s s i o n may be w r i t t e n f o r the g e n e r a l bar i n the other s e t .  82 Change i n e x t e n s i o n o f B  2 E  J  The d e f l e c t i o n s , c a l c u l a t e d i n t h i s manner, o f the range o f c a n t i l e v e r s e a r l i e r d e s c r i b e d are t a b u l a t e d pin jointed  i n Chapter 4, S e c t i o n  below and compared w i t h the d e f l e c t i o n s o f the structures. Deflections  L D  N  1  2 3 4 5 7 10  0.0245 0.0243 0.0243 0.0243 0 . 0243 0.0243  0.0254 0.0253 0.0253 0.0253 0.0253 0.0253  2  2 3 4 5 7 10  0.0377 0.0357 0.0353 0.0352 0.0351 0.0351  0.0415 0.0393 0.0389 0.0388 0.0387 0.0387  4  2 3 4 5 7 10  0.0618 0.0520 0.0506 0.0501 0.0498 0.0496  0.0799 0.0673 0.0652 0.0644 0.0639 0.0637  5  2 3 4 5 7 10  0.0730 0.0581 0.0561 0.0554 0.0549 0.0547  0.1029 0.0827 0.0793 0.0781 0.0772 0.0768  10  2 3 4 5 7 10  0.1259 0.0796 0.0745 0.0728 0.0717 0.0712  0.2634 0.1879 0.1739 0.1687 0.1648 0.1630  Uniaxial  Stress  Biaxial  Stress  1,  83  APPENDIX G DETAILS OF COMPARABLE STRUCTURES  A range o f s t r u c t u r e s are d e s c r i b e d subjected  t o the  same l o a d i n g and  i n Chapter 6, a l l  support c o n d i t i o n s .  These  i l l u s t r a t e the s u p e r i o r i t y of the M i c h e l l type o f framework in a typical situation. The specified  M i c h e l l s t r u c t u r e s conform t o the  i n Chapter 4,  namely:-  E  = 300,000 p . s . i .  Maximum s t r e s s a  =  Load 100  l b a t 10"  Where a p p r o p r i a t e , the t h i c k n e s s i s 0.25".  parameters  300  p.s.i.  from plane of  supports.  the maximum support spacing  i s 2.5",  of members, normal to the plane of the  The  and  structure  d e f l e c t i o n of each of the f o l l o w i n g s t r u c t u r e s i s  s p e c i f i e d as 0.0501", t h a t o f the f i v e f i b r e M i c h e l l c a n t i l e v e r of s i m i l a r — r a t i o wherever t h i s i s compatible w i t h  the  specified stress. The detail  design  of these s t r u c t u r e s i s now  ( s t r u c t u r e s A,  i n Chapter  B, and  discussed  C having a l r e a d y been  in  discussed  4):-  D. Warren Truss The  general  i n F i g u r e G l and  arrangement of the t r u s s s t u d i e d  i s based on the s i m i l a r s t r u c t u r e  by Chan i n r e f e r e n c e  1.6.  i s shown  discussed  84  irJOO Its  Figure Gl  Preliminary  Warren Truss  a n a l y s i s i n d i c a t e d t h a t the s t r u c t u r e  shown above c o u l d be designed w i t h a span o f 10", an i n d i v i d u a l member l e n g t h o f 2.8571" and an o v e r a l l depth o f 2.4744" which i s c l o s e to the 2.5 i n c h support spacing lever.  o f the M i c h e l l c a n t i -  T h i s seemed a reasonable compromise  not f o l l o w that t h i s a r b i t a r y d e s i g n  although i t does  i s optimum o f i t s type.  The f o r c e s i n the members were c a l c u l a t e d by means o f a Maxwell diagram, s o l v e d t r i g n o m t r i c a l l y on the f i g u r e above.  and are i n d i c a t e d  85  The c r o s s - s e c t i o n a l area o f each member may be a d j u s t e d so t h a t the s t r e s s i n each i s u n i f o r m l y 3 0 0 p . s . i . , whether i n t e n s i o n or compression. The volume o f the s t r u c t u r e may then be c a l c u l a t e d and i s found t o be 2 1 . 1 7 c u . i n . S i n c e the s t r e s s i n each member i s u n i f o r m the s t r a i n will of  s i m i l a r l y be u n i f o r m and i s  £ = 0 . 0 0 1 .  The e x t e n s i o n  each member i s thus known and may be c a l c u l a t e d by use of  a W i l i o t diagram as shown i n F i g u r e G 2 . From t h i s diagram the downward d e f l e c t i o n o f L i s 0 . 0 6 3 9  inches.  *  In p a s s i n g i t may be noted t h i s w e l l i l l u s t r a t e s the f i n d i n g s o f Chapter 1 . I t was t h e r e p r e d i c t e d t h a t the M i c h e l l s t r u c t u r e (or r a t h e r the optimum frame t o which the M i c h e l l s t r u c t u r e i s a c l o s e approximation) i s l i g h t e r y e t s t i f f e r than any o t h e r . I f the framework i s i n c r e a s e d i n c r o s s - s e c t i o n t o reduce the d e f l e c t i o n t o 0 . 0 5 0 1 " , the volume becomes N  0 . 0 6 3 9  I  21.17  x  Q  0 5 Q 1  O  '  O  R  N  N  N  2 7 . 0 0  cu.  in.  o| 1 »  o-oio"  Changes ia member lengths ^Sxx, are all ecjaal -fc> 0-ooa24" Excepr £8Qa>hx'k osjuals Q 00/43".  DOWNWARD  OF L  O ' 4- ' 8  IZ  -=>  '6  SCALE - THOUSANDTHS OP I N C H E S .  * gure G2  ftJ>  A.B (ORIGIN)  W i l i o t Diagram f o r D e f l e c t i o n of Warren Truss  (4)  87 Two  Bar  Cantilever  tlOOlb. a) G e n e r a l  Arrangement  b) D e f l e c t i o n F i g u r e G3  Two  Bar  of  Structure  Cantilever  The F i g u r e G3.  general  arrangement of the s t r u c t u r e i s shown i n  From the geometry ALO = BLO = 7.1° and, by  elementary s t a t i c s , the f o r c e a c t i n g along each member i s 50 ±  s  in  i  n  =  7  ±(403.11) l b , AL being i n t e n s i o n w h i l e BL i s  compression. From the geometry o f the d e f l e c t e d s t r u c t u r e , and  assuming t h a t the s t r a i n i s t o be the same i n both members, by  Pythagoras  Gl  From which, s i n c e y i s known  o-osoi  The  6  =  G2  e ^  000621"  x =  o-000/3"  s t r a i n i n AL and BL i s thus  O- Q Q 6 Z 1 IO-07782  -  +  =  £.£ =  C r o s s - s e c t i o n a l area =  4 0 3 'M  Uniform s t r e s s  <B498  0-0006/7  I84*'98 -  G3  G4  2*1792  i . e . 1.476 inches square o r 1.666 inches d i a . Volume o f frame  =' 43.9233 c u . i n .  G5  89 The compression member LB c o u l d f a i l by b u c k l i n g .  If  EITT^  the E u l e r c r i t i c a l l o a d , —=  , i s taken as a c r i t e r i o n , the  a x i a l l o a d f o r f a i l u r e i s 11,500 pound which f a r exceeds the 403 pound f o r c e i n the member. C a n t i l e v e r With P a r a b o l i c  F i g u r e G4  Thus b u c k l i n g i s very  unlikely.  Section  C a n t i l e v e r of P a r a b o l i c  Section  The g e n e r a l arrangement of t h i s s t r u c t u r e i s shown i n F i g u r e G4.  I t has a constant  i s a f u n c t i o n of l e n g t h , x. bending s t r e s s , i n an outer  width t and a depth h, which  I t i s s p e c i f i e d t h a t the maximum f i b r e , a, i s t o be constant a t  all  planes. Consider  d i s t a n c e x from  an e l e m e n t o f t h e c a n t i l e v e r , MNQP, a t a  the l o a d .  Applying  the elementary  equation  Jl I  3  i±?)  ,12 /  01-  If  G6  a i s t o be c o n s t a n t , e q u a t i o n G6  shape o f t h e c a n t i l e v e r T h i s may to  determine  i s parabolic.  be u s e d  s l o p e and  to solve the d i f f e r e n t i a l  equation  deflection:i  .2 EX  implies that the  y^E  !2  I  \ pj fzr  G7  Thus G8  dx LO  G9  The c o n s t a n t s G and s l o p e and d e f l e c t i o n  Thus  GO  =s  may  be d e t e r m i n e d  since the  a r e z e r o a t x = L.  • G10  91 In the present  case co = 0.0501 a t x = zero, a = 300 p . s . i .  Thus from  G10,  t = 2.8228"  and  G6,  H = 2.661"  from  Volume  =  | HLt  =  Gil  50.100 c u . i n ,  G12  Triangular Plate Cantilever T h i s s t r u c t u r e i s shown i n F i g u r e G5.  The c a n t i l e v e r  depth, h, remains constant w h i l e the width i n c r e a s e s from zero a t the t i p , L,  A IOOlb. 'igure G5  Triangular Plate  Cantilever  uniformly  to a maximum, T, a t the r o o t AB.  By t h i s means the s t r e s s i n  the outer f i b r e s may be maintained  constant a t a l l p l a n e s .  From the formula  a  '  R "  G13  1  s i n c e c and y a r e constant, the r a d i u s o f c u r v a t u r e , R, i s /  constant and thus the beam bends t o a c i r c u l a r shape, I Thus ...  assuming u  T  -u  \  G14  i s s m a l l so t h a t u)  \  may be neglected.  T  Li  Thus from G13,  -FT  I '9960 »N. • • - . c i 5  or A l s o , from the same equation,  PL  or T F i n a l l y the volume =  &PI?  -j  6-0200/^.  . . G16  C0u £ ^ThL  -  5 O I 0 0  CU.IW.  . . . G17  93 G.H.J. CANTILEVERS OF UNIFORM CROSS-SECTION  For a simple c a n t i l e v e r , the d e f l e c t i o n due t o a p o i n t end l o a d i s g i v e n by the f a m i l i a r e q u a t i o n PL 3EI 3  W  L  =  •  . . . .  G18  In the p r e s e n t case a l l v a r i a b l e s a r e known except I which then may be c a l c u l a t e d  Further  I  =  2.21778 i n .  Y  =  Sr- = M  4  . . . .  G19  . . . .  G20  from 0.66533  or the t o t a l depth o f s e c t i o n i s 1.33067 i n s . These equations apply, whatever the form o f the c r o s s section.  Three cases may now be c o n s i d e r e d : -  a) C y l i n d r i c a l C r o s s - s e c t i o n , Diameter d From G20, TTD  the diameter i s a maximum o f 1.33067 i n .  4  and thus I = -g-^— = 0.154 which i s c o n s i d e r a b l y l e s s than the v a l u e s p e c i f i e d by G19.  Thus a c y l i n d r i c a l  cantilever  cannot completely s a t i s f y the c o n d i t i o n s . I f G19 i s s a t i s f i e d D = 2.5926 i n . Volume  = 52.7911 c u . i n .  Maximum s t r e s s  = 584.5 p . s . i .  . .  G21  Alternatively  i f the diameter  maintain the stress 13.9069  cu.in.,  0.7220 i n .  as 300 p . s . i . ,  but the d e f l e c t i o n  The f i r s t  b) I Beam  i s fixed  a t 1.33067 i n . t o  t h e volume i s r e d u c e d t o i s greatly  case i s quoted  a s more  increased to typical.  Cantilever  ''103 lb. Figure G 6  The s e c t i o n convenience are  I Beam  studied  width,  W.  i s shown i n F i g u r e G16.  the thickness o f both  t a k e n as e q u a l , and b o t h  Cantilever  flanges  flanges  For  and t h e web, t ,  a r e assumed  t o have e q u a l  The  second moment, cf area o f the c r o s s - s e c t i o n may  now be c a l c u l a t e d about the n e u t r a l a x i s .  ±H%  2  \Z  wt"  G22  iz  Equations G19, G20 and G22 may be combined t o connect the three v a r i a b l e s H, t and W.  Any one o f these may be  s e l e c t e d a r b i t r a r i l y , and the other two c a l c u l a t e d from  these  equations. F u r t h e r the volume o f the s t r u c t u r e i s g i v e n by ,  V = ZWLt + HLt  • - • • G23  T h e o r e t i c a l l y a minimum volume i s obtained as t tends to  zero, but t h i s i s i m p r a c t i c a l s i n c e the f l a n g e s would not  be s e l f s u p p o r t i n g and would buckle  [ f o r t = 0.1 i n . , the  f l a n g e s are over 30" wide]. A t h i c k n e s s o f 0.25" was t h e r e f o r e a r b i t r a r i l y  selected  as being equal t o t h a t of the members o f the M i c h e l l framework. In  such case:-  Thickness o f a l l Members  =0.25"  Width o f Flanges W  = 16.982"  Height o f Web, H  = 1.329"  Volume o f S t r u c t u r e V  = 86.9854 c u . i n .  . . . .  G24  c)  Beam o f The  Rectangular^Section  I beam o f t h e p r e v i o u s  to a rectangular it G19  s e c t i o n may be c o n v e r t e d  s e c t i o n by p l a c i n g H e q u a l  becomes a l l f l a n g e ,  o r may b e s o l v e d  and G 2 0 , k n o w i n g t h a t  _ bh  3  to zero,  directly  so t h a t  from  equations  .  " 12  1  I n e i t h e r c a s e t h e same u n i q u e s o l u t i o n i s o b t a i n e d : Height  1.33067  Width Volume  in.  1 1 . 2 9 5 1  in.  1 5 0 . 3 0 0 0  in.  .  s e c t i o n use t h e i r m a t e r i a l  planes,  close  specified  that  volume t h a n t h e r e c t a n g u l a r  \  fibre  near the r o o t .  'dead wood' i s e l i m i n a t e d .  Q 2 5  inefficiently.  a x i s does l i t t l e  i n the outer  stress level  .  these c a n t i l e v e r s o f uniform  content very  to the n e u t r a l  while that  .  3  I t may be o b s e r v e d  Material  .  only  work a t a l l  approaches the  The I beam h a s l e s s  s e c t i o n s i n c e much o f t h i s \  97  APPENDIX H MANUFACTURE OF•PHOTOELASTIC MODELS  The  techniques  f o r the manufacture o f good q u a l i t y  models from b i r e f r i n g e n t p l a s t i c s f o r p h o t o e l a s t i c i n v e s t i g a t i o n are w e l l e s t a b l i s h e d . Cuts should be made w i t h low feeds u s i n g sharp and g r e a t care must be taken t o prevent  tools  l o c a l heating with  the i n t r o d u c t i o n of permanent s t r e s s p a t t e r n s a t zero load. The models d i s c u s s e d i n Chapter complicated  shape compared t o f a m i l a r  7 were of r e l a t i v e l y t e n s i l e and bending  t e s t specimens.  I t was decided t o m i l l these from sheet  CR39 p l a s t i c ,  t h i c k , using a numerically c o n t r o l l e d  ing machine. The  mill.; ,  t a b l e movements were operated by synchronous  (SLOSYN) motors which respond  t o e l e c t r i c a l p u l s e s , each p u l s e  causing the motor t o r o t a t e by a f i x e d amount.  Smooth s u r f a c e s  could thus be c u t i n two p e r p e n d i c u l a r d i r e c t i o n s — a l o n g the machine t a b l e , o r a t r i g h t angles t o i t — b y a p p r o p r i a t e motor.  driving the,  Cuts c o u l d a l s o be made a t 45° t o these  d i r e c t i o n s by d r i v i n g both motors  simultaneously.  Surfaces not a l i g n e d with these d i r e c t i o n s c o u l d only be produced i n a s e r i e s o f steps r e s u l t i n g i n a rough  finish  98 which was  undesirable.  I t was  t h e r e f o r e decided  to mount  the model blank on a r o t a r y t a b l e so t h a t each member c o u l d be turned p a r a l l e d with the x or y axes procedure was  p a r t i c u l a r l y convenient  for cutting.  for Michell cantilevers  s i n c e a l t e r n a t e members are p e r p e n d i c u l a r . F i g u r e 2.2,  CF, FK,  KH and  c o u l d thus be c u t i n one The  For  HL are a t 90° to one  example,;in another  s e t t i n g of the t a b l e .  and ?  procedure used to make the models d e s c r i b e d i n  Chapter 7 was  as f o l l o w s : -  a) J o i n t C o o r d i n a t e s , The  This  Member S i z e s , e t c .  c o o r d i n a t e s o f the j o i n t p o i n t s were found from  the computer p r i n t o u t d e s c r i b e d i n Chapter 4 and Appendix  C.  The member widths were a l s o obtained  but  these were m o d i f i e d models.  as d e s c r i b e d i n Chapter 7 to o b t a i n s u i t a b l e  The v a l u e s used f o r design purposes, and  a c t u a l l y achieved and  from the same source  j  those  on the models, are t a b u l a t e d i n Tables:1  2 f o r the three and  ;  f i v e f i b r e M i c h e l l C a n t i l e v e r s . Mem;  bers whose widths are not shown were not modelled s e p a r a t e l y but were merged i n t o the s o l i d areas of the  fans.  b) Allowance f o r A n g u l a r i t y The model blank,  as has  mounted on a r o t a r y t a b l e and the machine axes.  a l r e a d y been s t a t e d ,  was;  turned to a l i g n each cut w i t h  In t h i s process  a point i n i t i a l l y  at  99  TABLE 1 THREE FIBRE MICHELL CANTILEVER  Member  A  10  A  l l  A  12  A  20  A  21  A  22  A  30  A  31  A  32  B  01  B  l l  B  21  B  02  B  12  B  22  B  03  B  13  B  23  Width (300 p . s . i . )  Nominal Width  A c t u a l Width ~"  fillet -  |  fillet  1.5925  0.300  1.2640  0.238  0.240  -  0.5878  0.110  0.113  •-  2.5540  0.480  0.482  -  2.0846  0.392  0.393  0 . 393  1.0638  0.200  0.204  0.203  1.9048  0.358  0.359  1.8122  0.340  0.341  0 .343  1.4674  0.276  0.277  0.278  1.5925  0.300  1.2640  0.238  0.240  0.5878  0.110  0.114  2 .5540  0.480  0.482  2.0846  0.392  0.394  0.394  1.0638  0.200  0.203  0.204  1.9048  0 .358  0.360  1.8122  0.340  0.341  0.342  1.4674  0.276  0.279  0.279  -  -  -  -  -  .-  100  TABLE 2 FIVE FIBRE RIGID MICHELL CANTILEVER Member  A  Design Width 300 p.s.i.  10' 01 B  A  20  R  02  30' 03 A.„, B„.. 40' 04 5 f i ' rm A  B  l l '  A  l l  B  A  12' 21  A  13' 31  A  14' 41  A  51' 15  A  52' 25  A  53' 35  A  54' 45  A  4 1 ' 14  A  4 2 ' 24  A  43' 34  B  B  B  B  B  B  B  B  B  B  44  44  A  31' 13  A  32' 23  B  B  A  R  33'  33  A  34' 43  A  2 1 ' 12  B  B  A  A A  R  22'  R  22  23' 32 24' 42 B  Nominal Width  Actual  Width  •A'  •B'  0.8706 1.5603 1.2287 0.9483  Merged  together in s o l i d 'fans'  1.3530 0.7587 0.5362 0.3577 0.2168 1.3357  0.351  0 .357  0.356  1.2673  0.333  0.336  0.338  1.2023  0.316  0.319  0.319  1.1404  0.300  0.300  0.303  0.8676  0.228  0 .230  0.229  0.6897  0.181  0.182  0.182  0.5276  0.139  0.140  0.140  0.3803  0.100  0.103  0 .102  1.0998  0.289  0.290  0.290  0.8301  0.218  0.218  0 .218  0.5984  0.157  0.160  0.161  0.4008  0.105  0.105  0.105  1.3706  0 .360  0.360  0.360  0.9880  0.260  0.260  0.260  0.6750  0.177  0.177  0.177  0.4223  0.110  0.110  0 .108  101 c o o r d i n a t e s x, y w i l l move t o a new p o s i t i o n X, Y. j o i n t coordinates referred calculated  The  to i n S e c t i o n A must then be r e -  t o allow f o r t h i s r o t a t i o n as shown i n F i g u r e H i .  F i g u r e HI  The E f f e c t o f R o t a t i o n  The p o i n t P has c o o r d i n a t e s x, y with r e f e r e n c e t o the axes x, y.  I f now the axes are r o t a t e d through an angle  3 t o new d i r e c t i o n s , X, Y, the c o o r d i n a t e s o f P w i t h r e f e r ence t o the new axes are X, Y. X  •=  sr  O M = ON + NM  :>c £ 3 3 ;3 + LfS/nft  . . . .  HI  102  Y -a, MP s -  NPcos/3 •» (y -xtnnp) 3 cos^  —  cosy*  smjS  . . . .  These equations enable the new c o o r d i n a t e s  H2  o f each  j o i n t p o i n t to be found as the model i s r o t a t e d t o d i f f e r e n t positions. c) O f f s e t s f o r C u t t e r Radius and Member Width The  coordinates  o f the c u t t e r a x i s may now be d e t e r -  mined f o r the b e g i n n i n g and end o f each c u t .  Allowance must  be made f o r the widths o f the members and f o r the c u t t e r r a d i u s as i n d i c a t e d i n F i g u r e H2.  Cutter"  F i g u r e H2  Geometry o f C u t t e r  Offsets  A t y p i c a l j o i n t i s a l i g n e d so t h a t member F J i s p a r a l l e l t o the Y a x i s . equations HI and H2.  The c o o r d i n a t e s o f J are known from The c o o r d i n a t e s o f 0 are r e q u i r e d f o r  the programme f o r numerical c o n t r o l o f the m i l l i n g  machine.  C l e a r l y the s e t o f f i n the X d i r e c t i o n , Sx i s  £x =  W  k+  r  =. W. + D  X Q - X J + W,  or  +  * . . . .  2  •~"Z  H3  The s e t o f f i n the y d i r e c t i o n , S , i s g i v e n by Y  5y = Pl^l - MN  2co4©  a  o  3  I f the angle FJG i s acute  6  . . .  3  [(90-6)  H4  i n s t e a d o f (90+6)  as drawn], the c o o r d i n a t e s o f 0 become  Z  Y * Q  ~ 2 *fa+ s»>x Q) + W, sine + 2 co*a  .  .  . .  H5  104 d)  Backlash  The tion  - Order of  e l e c t r o n i c c o n t r o l s have p r o v i s i o n f o r  (more s t r i c t l y  correction)  machine l e a d s c r e w . points  are  Cuts  This  approached  i s done by  i n the  machine r e v e r s e s  location  i n the  the  ensuring  stopping  d i r e c t i o n and  milling  that a l l  point  If is  approaches the  the  overshot,  desired  positive direction.  Care t h e r e f o r e the  i n the  positive direction.  machine i s moving n e g a t i v e l y , the  of backlash  elimina-  has  t o be  p o s i t i v e d i r e c t i o n s only  w o u l d o t h e r w i s e be  cut  t a k e n t o make a l l c u t s  to eliminate  i n members due  the  to t h i s  notches  in  that•  uncontrolled  motion. Figure made and  the  routines  of  Programme  were  cutting  the  m a c h i n e may  now  be  slightly  system.  programmes  a t the  are:-  cuts  f o r the  r o u t i n e , which v a r i e s  end  of  f o r the  this  three  s t r u c t u r e s , which  are  appendix, c o n s i s t of a s e r i e s o f  commands, e a c h o f w h i c h c o n t a i n These  i n which the  followed  programme f o r c o n t r o l o f  from system to  listed  of  f o l l o w i n g a simple  The  order  models.  Compilation The  written  i n d i c a t e s the  is typical  o f a l l the e)  H3  up  to  f o u r words o r  'bits'.  105  Figure  H3  Order and D i r e c t i o n o f Cuts f o r Machining a Typical Michell Cantilever  106 i)  t h e s t e p number. the  control  iii)  the x  on t h e  cabinet,  coordinate, These a r e s t a t e d i n u n i t s o f  t h o u s a n d t h o f an i n c h .  One o r b o t h  c o o r d i n a t e s may be l i s t e d . movement  iv)  for identifying  I t i s d i s p l a y e d on a c o u n t e r  the y coordinate. one  i s only  s t e p and p l a y s no p a r t i n t h e m a c h i n i n g  process.  ii)  This  of  these  These govern the  of the c u t t i n g t o o l .  In f a c t ,  i t is  t h e work t h a t i s moved p a s t  a stationary cutter,  but  i t i s usual  the t o o l  The  c o n t r o l gear allows  to consider  control instructions tions' .  for this  known as  as moving.  fact,  'miscellaneous  func-  T h e s e have t h e f o l l o w i n g e f f e c t s : -  02  Rewind t a p e .  T h i s ends a programme.  f  06  T o o l change.  A t t h e end o f t h e s t e p  i n which  this  a p p e a r s , t h e machine i s stopped.  provides  an o p p o r t u n i t y  for a tool  changed o r a machine s e t t i n g 52  T o o l Down.  53  T o o l Up. raised  55  t o be  This  t o be altered. .  T h e s e commands c a u s e t h e t o o l  or lowered  Fast Traverse.  to a preset This  i s used  between c u t t i n g p o s i t i o n s .  t o be  level. t o move t h e t o o l  NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL  P R E P A R E D D  A  T  E  B Y ;  EWJ  30 J a n u a r y  C H E C K E D  B  P A R T  W O R K  O R D E R  N O .  70 R E M A R K S :  S H E E T  O  F  1  S E Q . N O .  N A M E  COLUMBIA  Y  D A T E  T A P E  E N G I N E E R I N G , U N I V E R S I T Y O F BRITISH  '  6  N O .  T A B O R E O B  + O R  "X" I N C R E M E N T  T A B O R E O B  5 F I B R E MICHELL  CANTILEVER.  (Origin  left  + O R  8.000"  "Y " I N C R E M E N T  • T A B O R , E O B  1/8" c u t t e r .  of turntable  " M" F U N C T .  centre).  I N S T R U C T I O N S  E . O . B .  E  %  E  1  T  +  8000  T  2  T  -  3760  T  3  T  +  2801  T  4  T  5  T  6  T  7  T  +  1724  E  8  T  -  1654  T  9  T  +  1590  T  10  T  11  T  12  T  13  T  +  946  E  14  T  -  855  T  L5  T  +  765  T  L6  T  17  T  L8  T  L9  T  +  534  E  10  T  -  390  T  11  T  +  287  T  12  T  13  T  -  >4  T  +  >5  T  1 E C H . E N G . 3 - 6 9 • O R M 7 2 8  +  +  +  226  282  384  -  2713  T  55  E  T  0655  E  T  52  E  535  E  52  E  T  535  E  T  52  E  T  +  2039  E  T  -  1981  T  T  +  1912  T • "  +  264  +  1147  E  T  -  1071  T  535  E  T  +  1002  T  52  E  +  342  T  535  E  52  E  •  T  +  548  E  T  -  437  T  535  E  T  +  334  T  52  E  +  484  T  535  E  T  52  E  T  +  207  E  4414  T  -  4365  T  2240  T T  T +  1270  E  324.20°  :  T  T  Rotate  06535 52  E E  Rotate +  18.41°  NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF BRITISH COLUMBIA P R E P A R E D •  B  Y  P A R T  N A M E  W O R K  O R D E R  N O .  A T E  C H E C K E D  B  Y  D A T E  R E M A R K S :  S H E E T  T A P E  „ Z  O  F  . O  N O .  +  T A B O R E O B  O R  26  T  +  27  T  28  T  +  1376  E  29  T  -  1291  T  30  T  +  1213  T  31  T  32  T  33  T  34  T  +  827  E  35  T  -  706  T  36  T  +  610  T  37  T  38  T  -  39  T  +  40  T  41  T  42  T  43  T  +  1103  E  44  T  -  1003  T  45  T  +  931  T  46  T  47  T  -  48  T  +  49  T  50  T  S E Q. N O .  M E C H . F O R M  E N G . 3 - 6 9 7 2 8  +  +  -  230  302  +  T A B O R E O B  O  T  -  1210  T  535  E  T  +  1158  T  52  E  +  306  T  535  E  T  52  E  "X" I N C R E M E N T  R  "Y " I N C R E M E N T  E . O .  +  652  E  T  -  569  T  535  E  T  +  492  T  52  E  +  414  T  535  E  T  52  E  +  148  3083  T  -  2782  1672  T  T-  06535  E  52  E  T  +  665  E  T  -  604  T  535  E  T  +  550  T  52  E  +  352  T  535  E  T  52  E  T  06535  E  T  52  E  06535  E  +  207  1879  T  -  1649  1220  T  B .  I N S T R U C T I O N S  -  E  T  T  4928  »M" F U N C T .  T  T  236  T A B O R E O B  Rotate +  18.41°  Rotate +  18.41°  Rotate +  141.14°  E  T  +  173  E  T  -  173  T  109  NUMERICAL TAPE CONTROL PROGRAM D E P A R T M E N T O F M E C H A N I C A L ENGINEERING, UNIVERSITY O F BRITISH  P R E P  A R E DB  Y  P A R T  N A M E  W O R K  O R D E R  COLUMBIA  N O .  D A T E C H E C K E D B  Y  D A T E  R E M A R K S :  S H E E T  T A P E  3  O  F  6  N O .  +  "X"  +  "Y"  "M "  T A B O R E O B  O R  51  T  +  1220  T  T  52  E  52  T  -  1220  T  T  535  E  53  T  52  E  54  T  +  55  T  56  T  57  T  58  T  59  T  60  T  -  1103  61  T  +  62  T  63  S E Q . N O .  I N C R E M E N T  T A B O R E O B  O R  I N  C R E M E N T  T A B O R E O B  F U N  C  T.  E . O .  T  +  173  T  608  T  +  306  T  +  1672  T  T  -  52  E  1672  T  T  535  E  665  T  52  E  T  E  T  -  604  T  535  E  T  +  550  T  52  E  T  T  535  E  1103  T  -  T  52  E  1031  T  T  535  E  T  +  931  T  T  52  E  64  T  -  931  T  T  535  E  65  T  52  E  66  T  +  67  T  68  T  69  T  70  T  71  T  72  T  _  1376  73  T  +  74  T  75  T  M E C H . E N G . 3 - 6 9 F O R M 7 2 8  -  06535  236  +  .352  T  +  207  T  2417  T  -  1049  T  +  2240  T  T  52  E  —  2240  T  T  535  E  06535  E  T  +  1270  T  52  E  T  _  1210  T  535  E  T  +  1158  T  52  E  T  T  535  E  1376  T  T  52  E  _  1298  T  T  535  E  +  1213  T  T  52  E  230  +  306  B .  I N S T R U C T I O N S  Rotate +  18.41°  Rotate +  18.41°  NUMERICAL TAPE CONTROL PROGRAM D E P A R T M E N T O F M E C H A N I C A L ENGINEERING, U N I V E R S I T Y O F BRITISH C O L U M B I A  PREPARED BY • ATE  PART NAME  CHECKED BY • ATE SHEET ,  OF -  4  6  WORK ORDER NO.  REMARKS: 1  TAPE NO.  + "X" OR INCREMENT  SEQ. NO.  TAB OR EO B  76  T  77  T  78  T  79  T  80  T  -  827  81  T  +  82  T  83  1213  TAB OR EOB  + "Y" OR 1N C R EM EN T  T  TAB O R EOB  »M" FUN C T.  E.O. B.  T  535  E  T  +  652  T  52  E  T  -  569  T  535  E  T  +  492  T  52  E  T  T  535  E  827  T  T  52  E  -  731  T  T  535  E  T  +  610  T  T  52  E  84  T  -  610  T  T  535  E  85  T  86  T  +  87  T  88  T  89  T  90  T  91  T  92  T  93  T  94  T  95  T  96 97  T T  98  T  99  T  100  T  M E C H . F O R M  E N G . 7 2 8  3 - 6 9  -  302  +  414  T  +  148  T  52  E  4226  T  —  3086  T  06535  E  +  2801  T  T  52  E  -  2801  T  T  535  E  _  + +  -  226  T  + 2026  T  52  E  T  _  1968  T  535  E  T  +  1912  T  52  E  1724  T  T  535  E  1724  T  T  52  E  1660  T  T  535  E  1590  T  T  52  E  1590  T T  +  1147  T T  535 52  E E  T  -  1071  T  535  E  T  +  1002  T  52  E  T  535  E  282 946  T  +  264  INSTRUCTIONS  Rotate + 18.41°  r  NUMERICAL TAPE CONTROL PROGRAM D E P A R T M E N T O F M E C H A N I C A L ENGINEERING, U N I V E R S I T Y  PREPARED BY DATE CHECKED BY DATE SHEET 5  PART NAME  O F BRITISH C O L U M B I A  WORK ORDER NO.  REMARKS: OF 6  TAPE NO.  TAB OR EO B  + "X" OR INCREMENT  TAB OR EO B  101  T  +  946  T  102  T  -  856  T  103  T  +  765  104  T  -  765  105  T  106  T  107  T  108  T  -  534  109  T  +  110  T  111  SEQ. NO.  + "Y" OR INCR EM EN T  TAB OR EO B  "M" FUNCT.  E.O. B.  T  52  E  T  535  E  T  T  52  E  T  T  535  E  +  342  T  +  548  T  52  E  T  -  437  T  535  E  T  +  334  T  52  E  T  T  535  E  534  T  T  52  E  -  431  T  T  535  E  T  +  287  T  T  52  E  112  T  -  287  T  T  535  E  113  T  52  E  114  T  -  115  T  116  -  384  +  484  T  +  207  T  5120  T  -  2077  T  +  8000  T  T  +  4584  T  117  T  +  978  T  118  T  119  T  -  978  E  120  T  -  7973  T  121  T  +  2689  T  122  T  -  1150  T  123  T  +  2850  T  124  T  -  1125  T  125  T  +  2250  T  MECH. ENG. 3-69 FORM 728  T  -  562  55  E  T  55  E  T  52  E  1124  E  —  3699  T T  409  T T  -  139  E  T  +  +  06535  T T  06535 52 06535 52 06535 52  E  INSTRUCTIONS  Rotate +  324.20°  R o t a t e + 144.2°  E E  R o t a t e + 18.41°  E E E  Rotate +  18.41°  NUMERICAL TAPE CONTROL PROGRAM DEPARTMENT  O F M E C H A N I C A L ENGINEERING, UNIVERSITY O F BRITISH  PREPARED BY DATE CHECKED BY DATE SHEET , 6  PART NAME  COLUMBIA  WORK ORDER NO.  REMARKS: OF , 6  TAPE NO.  SE Q. NO.  TAB OR EO B  + "X" OR INCREMENT  TAB OR EO B  126  T  -  1225  T  127  T  +  1850  T  128  T  -  850  T  129  T  +  1157  T  130  T  +  1413  T  131  T  +  1092  T  132  T  133  T  -  1067  E  134  T  -  7552  T  135  T  +  1157  T  136  T  -  850  T  137  T  +  1850  T  138  T  -  1200  T  139  T  +  2225  X  140  T  -  1050  T  141  T  +  2775  T  142  T  -  1050  T  143  T  +  2589  T  144  T  -  3389  T  145  T  -  8000  T  M E C H . F O R M  E N G . 7 2 8  3 - 6 9  T  + "Y" OR 1N C R EM ENT  • -  -  +  497  370  1922  TAB OR EOB  "M " FUNCT.  T  06535  E  T  52  E  T  06535  E  T  52  E  T  06535  E  T  52  E  E.O. B.  +  3624  E  -  5546  T  06535  E  T  52  E  T  06535  E  T  52  E  T  06535  E  T  52  E  T  06535  E  T  52  E  T  06535  E  T  52  E  T  06535  E  T  0255  E  + + +  +  370  497  139  409  3137  INSTRUCTIONS R o t a t e + 18.41°  R o t a t e + 9.20°  Rotate  331.4°  Rotate  151.4°  R o t a t e + 9.20°  R o t a t e + 18.41°  R o t a t e + 18.41°  R o t a t e + 18.41°  R o t a t e + 324.20°  NUMERICAL TAPE CONTROL PROGRAM D E P A R T M E N T O F M E C H A N I C A L ENGINEERING, U N I V E R S I T Y  PREPARED BY DATE 1  4  F  e  b  CHECKED BY DATE SHEET  1  EWJ _ 7  TAB OR EO B  WORK ORDER NO.  PART NAME  3 Fibre M i c h e l l Cantilevers  0  REMARKS: OF  1/16" Radius F i l l e t s  3  O r i g i n 8000 L e f t of Turntable Centre.  TAPE NO.  SEQ. NO.  O F BRITISH C O L U M B I A  + "X" OR INCREMENT  TAB OR EO B  + "Y " OR INCR EM EN T  TAB OR EO B  "M"  FUNCT.  E.O. B.  INSTRUCTIONS  E %  E  1  T  +  8000  T  2  T  -  4060  T  3  T  +  5103  T  4  T  5  T  6  T  7  T  8  T  9  T  10  T  +  2013  E  11  T  +  140  12  T  13  T  _  14  T  +  15  T  16 17  T T  18  T  —  19  T  20  + -  i  —  2073  T  55  E  T  0655  E  T  52  E  T  +  2606  E  669  T  +  342  T  535  E  1446  T  T  52  E  T  +  T T  -  737  E  3480  T  535  E  +  2225  T  52  E  T  +  636  T  535  E  T  +  430  T  52  E  3509  T  _  3739  T  06535  E  2935  T  T  52  E  1789  T  +  623  E  T T  _  +  564 442  T T  535 52  E E  851  T  _  1313  T  06535  E  +  716  T  T  52  E  T  +  540  T  T  06535  E  21  T  +  716  T  T  52  E  22  T  -  649  T  T  535  E  23  T  +  699  T  T  52  E  24  T  —  1443  T  T  06535  E  25  T  +  698  T  T  52  E  MECH. ENG. 3-69 FORM 728  +  236  + + +  2382 606 2579  Rotate + 332.95°  Rotate + 35.90°  Rotate + 17.95° Rotate + 324.10°  Rotate + 324.10°  , "  I I  NUMERICAL TAPE CONTROL PROGRAM D E P A R T M E N T O F M E C H A N I C A L ENGINEERING, UNIVERSITY O F BRITISH  PREPARED BY • ATE CHECKED BY DATE SHEET  2  PART NAME  COLUMBIA  WORK ORDER NO.  REMARKS: OF  3  TAPE. NO.  r* TAB OR EOB  SEQ. NO.  26  T  27  T  28  +  "X"  OR INCREMENT  TAB OR EOB  +  "Y"  OR INCREMENT  TAB OR EOB  FUNCT.  06535  "M"  E.O. B.  7220  T  T  +  698  T  T  T  -  1443  T  29  T  +  699  T  30  T  -:  649  T  . 31  T  +  716  T  32  T  +  540  T  33  T  +  716  T  34  T  -  615  T  35  T  2935  T  T  52  E  36  T  2935  T  T  535  E  37  T  38  T  39  T  40  T  +  41  T  42  T  43  T  44  T  45  T  46  ' T  47  -  -  2579  606  2382  T  812  06535  E  52  E  T  535  E  T  52  E  T  T  06535 52 06535  E  E  +  623  T  52  E  T  -  564  T  535  E  T  +  442  T  52  E  2141  T  -  258  T  +  5103  T  T  52  E  -  5103  T  T  535  E  2606  T  52  E  2401  T  535  E  2225  T  52  E  236  T _  326  +  T  —  T  +  06535  E  —  2013  T  T  535  E  T  +  2013  T  T  52  E  48  T  -  1789  T  T  535  E  49  T  +  1446  T  T  52  E  50  T  _  1446  T  T  535  E  M E C H . F O R M  E N G . 3-69 728  +  518  Rotate + 324.10°  R o t a t e + 324.10°  E  T  -  Rotate + 270.00°  E  T  T  +  52  E  INSTRUCTIONS  R o t a t e + 17.95°  Rotate + 35.90°  NUMERICAL TAPE CONTROL PROGRAM D E P A R T M E N T O F M E C H A N I C A L ENGINEERING, UNIVERSITY O F BRITISH  PREP A R ED BY DATE  PART NAME  CHECKED BY DATE SHEET  us COLUMBIA  WORK ORDER NO.  REMARKS: ^  ° 3 F  TAPE NO.  SE 0. NO.  TAB • + "X" OR OR INCREMENT EO B  51  T  52  T  53  T  54  T  -  -  364  4478 INNER CU':s  TAB OR EO B  "Y"  IN CREMENT  TAB OR EO B  "M"  FUNCT.  E.O. B.  T  +  737  T  52  E  T  -  619  T  535  E  T  +  430  T  52  E  T  -  1423  T  ON * [ODE L  55  T  +  8000  T  56  T  +  4584  T  57  T  +  854  T  58  T  59  T  -  800  60  T  -  8522  T  61  T  +  5290  T  62  T  -  1812  T  63  T  +  3572  T  64  T  -  1166  T  65  T  +  1400  T  66  T  +  1263  T  67  T  +  837  T  68  T  69  T.  -  797  E  70  T  -  8103  T  71  T  +  1400  T  72  T  -  1216  T  73  T  +  3772  T  74  T  -  1912  T  75  T  +  5240  T  76  T  -  3884  T  77  T  -  8000  T  MECH. ENG. 3-69 FORM 728  + OR  T  -  06535  E  INSTRUCTIONS  R o t a t e + 332.95°  C0MPLET ED  438  T  55  E  T  55  E  T  52  E  +  876  E  -  2913  T  E.  T  -  307  T T  -  830  T T  +  1921  T T  T  -  3382  E  5303  T  T  +  830  T T  +  307  T T  +  2475  06535 52 06535 52 06535 52 06535 52  06535 52 06535 52 06535 52  E  R o t a t e + 152.95°  E E  R o t a t e + 35.90°  E E  R o t a t e + 17.95°  E E  R o t a t e + 333.20°  E  E  Rotate +  153.20°  E E  R o t a t e + 17.95°  E E  R o t a t e + 35.90°  E  T  06535  E  T  0255  E  R o t a t e + 332.95°  NUMERICAL TAPE CONTROL PROGRAM D E P A R T M E N T O F M E C H A N I C A L ENGINEERING, U N I V E R S I T Y  PREPARED DATE  BY  7  gT^J  Feb.  PART  NAME  WORK O R D E R NO.  Three  70  Michell  C H E C K E D BY DATE  O F BRITISH C O L U M B I A  Fibre  Cantilever  REMARKS:  SHEET  1  °  F  With  2  3/8" R a d i u s  Origin  8.000"  Cutter.  left  of Turntable  Centre.  T A P E NO.  SEQ. NO.  +  TAB OR EOB  OR  E  -  "X"  INCREMENT  TAB OR EO B  %  E  1  T  +  8000  T  2  T  -  3448  T  3  T  +  4179  T  4  T  5  T  +  6  T  +  7  T  8  T  9  T  10  T  +  1468  E  11  T  -  2524  T  12  T  +  2011  T  13  T  14  T  -  15  T  16  T  17  T  18  T  +  19  T  20  T  21  T  22  T  23  T  24  T  -  25  T  26  T  M E C H . E N G . 3-69 F O R M 728  -  + OR  -  "Y"  1N C R E M E N T  1761  TAB OR EOB  "M"  FUNCT.  E.O. B.  T  55  E  T  0655  E  T  52  E  T  +  2086  E  1593  T  +  862  T  535  E  522  T  T  52  E  1165  T  +  216  E  T  -  2660  T  535  E  T  +  1302  T  52  E  -  2049  T T  06535 52  E E  T  +  123  E  5272  T  -  123  T  +  2011  T  T  52  E  -  2011  T  T  535  E  52  E  T  +  123  T.  1905  T  +  120  T  +  4179  T  -  5129  T  +  T  06535  06535  E  E  T  52  E  504  T  535  E  +  1302  T  52  E  T  -  1806  T  535  E  T  +  2086  T  52  E  2418  T  -  280  T  535  E  +  1468  T  T  52  E  -  1165  T  T  535  E  +  950  +  1142  INSTRUCTIONS  NUMERICAL TAPE CONTROL PROGRAM D E P A R T M E N T O F M E C H A N I C A L ENGINEERING, U N I V E R S I T Y  PREPARED BY DATE  PART NAME  CHECKED BY DATE SHEET  2  117  O F BRITISH C O L U M B I A  WORK ORDER NO.  REMARKS: 0  F  2  TAPE NO.  SEQ. " NO.  TAB OR EOB  OR INCREMENT  TAB OR EO B  -27  T  +  522'  28  T  -  522  ; 29  T  30  T  -  31  T  32  +  "X"  TAB OR EOB  FUNCT.  T  T  52  E  T  T  535  E  + OR  "Y"  INCREMENT  "M"  E.O. B.  +  216  T  52  E  5154  T • -  1403  T  D6535  E  +  8000  T  T  55  E  T  +  4904  T  T  55  E  33  T  +  846  T  T  52  E  34  T  35  T  -  846  E  36  T  -  8987  T  37  T  +  5489  T  38  T  -  1806  T  39  T  +  3566  T  - 40  T  -  1166  -  41  T  +  1000  T T  42  T  +  1447  T  +  43  T  + ;  1303  T  44 45  T T  +  4300  E  -  1303  T E  - 46  T  -  7447  T  -  6074  T  47  T  +  1000  T  : 48  T  -  966  T  49  T  +  3522  T  50  T  -  1762  T  • 51  T  +  5289  T  • 52  T  -  4083  T  53  T  -  8000  T  MECH. ENG. 3-69 FORM 728  T  T  -  625  +  1250  E  -  3412  T  D6535  E  T  52  E  T  36535  E  T  52  E  830  T T  D6535 52  E E  1774  T  06535  E  T  52  E  -  307  T +  830  T T  +  307  T T  f  2787  06535 52 06535 52 06535 52  INSTRUCTIONS  E E E E E E  T  06535  E  T  0255  E  -  

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