UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Stiffness matrix solution for folded plates Payne, Carl Allan 1966

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1966_A7 P3.pdf [ 4.56MB ]
Metadata
JSON: 831-1.0050609.json
JSON-LD: 831-1.0050609-ld.json
RDF/XML (Pretty): 831-1.0050609-rdf.xml
RDF/JSON: 831-1.0050609-rdf.json
Turtle: 831-1.0050609-turtle.txt
N-Triples: 831-1.0050609-rdf-ntriples.txt
Original Record: 831-1.0050609-source.json
Full Text
831-1.0050609-fulltext.txt
Citation
831-1.0050609.ris

Full Text

A STIFFNESS MATRIX  SOLUTION  FOR FOLDED PLATES by C. A l l a n Payne  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE  i n t h e Department o f CIVIL  ENGINEERING  We a c c e p t t h i s t h e s i s as to- t h e r e q u i r e d  conforming  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA F e b r u a r y 1966  In presenting the  t h i s thesis i n p a r t i a l f u l f i l m e n t of  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f  B r i t i s h "Columbia, I a g r e e t h a t available  the L i b r a r y  f o r r e f e r e n c e and s t u d y .  mission f o r extensive  s h a l l make i t f r e e l y  I f u r t h e r agree that  per-  copying of t h i s thesis f o r s c h o l a r l y  p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by his  representatives.  I t i s understood that; copying or p u b l i -  cation of t h i s thesis f o r f i n a n c i a l gain w i t h o u t my w r i t t e n  permission.  Department of  C i v i l Engineering  The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r 8, Canada Date  '  M  a  r  c  h  28  >  1 9 6 6  '  s h a l l n o t be a l l o w e d  A B S T R A C T  A general  a n a l y s i s f o r a simply  plate structure i s presented.  supported prismatic  A s t i f f n e s s matrix  folded  method i s u s e d  w i t h the i n d i v i d u a l p l a t e s taken as s t r u c t u r a l elements. loads are applied  t o the s t r u c t u r e the s o l u t i o n y i e l d s  d e f l e c t i o n s and i n t e r n a l p l a t e f o r c e s .  The e x a c t  When  joint  elasticity  t h e o r y i s u s e d f o r t h e i n p l a n e o r membrane s o l u t i o n and t h e c l a s s i c a l t h i n p l a t e t h e o r y i s u s e d f o r t h e out o f p l a n e o r bending s o l u t i o n . s t i f f n e s s matrix  By m o d i f y i n g a computer program used f o r frame a n a l y s i s , i t i s p o s s i b l e t o s o l v e a f o l d e d  p l a t e s t r u c t u r e w i t h more t h a n two p l a t e s c o n n e c t e d a t t h e same joint.  This  i s d e m o n s t r a t e d by t h e s o l u t i o n o f a n I beam t o r s i o n  problem, which a l s o p r o v i d e s a check of e x i s t i n g t o r s i o n t h e o r i e s . C y l i n d r i c a l b a r r e l v a u l t s o f c i r c u l a r and o t h e r s e c t i o n s a r e a n a l y s e d as f o l d e d p l a t e s t r u c t u r e s . of t h e shape o f t h e c r o s s  A s t u d y i s made o f t h e e f f e c t  s e c t i o n on t h e s h e l l s t r e s s d i s t r i b u t i o n  f o r l o n g , s h o r t , t h i c k and t h i n  shells.  ii.  TABLE OF CONTENTS-  PAGE No. ABSTRACT  ,  CONTENTS  .  NOTATION  viii.  LIST OF FIGURES  II Ill  ix.  INTRODUCTION  "  STIFFNESS METHOD TORSION  i i . i i i .  ACKNOWLEDGMENTS  I  i .  OF I BEAM  IV  ANALYSIS COMPARISON OF BARREL SHELLS OF FIVE DIFFERENT CROSS SECTIONS  26  FIGURES  1. k. 12. 21.  BIBLIOGRAPHY  27.  APPENDIX - COMPUTER PROGRAM  28.  Notation l e n g t h of the s t r u c t u r e plate width . web and f l a n g e p l a t e w i d t h s , r e s p e c t i v e l y JG  torsional rigidity = torsional rigidity  o f one f l a n g e =  torsional rigidity  o f web = J  half the transverse flexural  rigidity  J^G  G  s p a n of b a r r e l  shell  o f one f l a n g e = E I . 3  flexural  rigidity  o f web = E t  2  /l2  (l-v )  c h o r d l e n g t h o f h a l f s h e l l f r o m b a s e t o crown Young's  modulus  distance perpendicular  t o the chord l e n g t h  from i t s midpoint t o the s h e l l  curve  s t r u c t u r e f o r c e v e c t o r of e x t e r n a l loads  F, , P  1'  JP. • i '  0  2  vector of f i x e d  d  joint  F n  edge f o r c e s a c t i n g on t h e  joints. amplitude of the s t r u c t u r e e x t e r n a l j o i n t corresponding  t o the  load  i " * * * d e g r e e of f r e e d o m 1  and t h e m"** h a r m o n i c p l a t e f o r c e v e c t o r of i n t e r n a l j o i n t l o a d s f ^ » f2 ' P^- ' co-ordinates 3 1  i  (J  w  /J)  (M'/b  w  )  n  a  te  1V ,  Notation  (cont'd)  .  2  =  (j /j)(M'/b )  3  =  M'/h  f, 4  =  M/h  &  =  shear modulus  h  =  beam d e p t h  J  =  torsional  =  J  f  ?  f  f  =  E/2(l+v)  constant  of t o t a l c r o s s  section  + 2 J  w  f  J„  =  torsional  constant  o f one f l a n g e = b.,t„ ll  J  =  torsional  constant  o f web = b t  w L  W W  =  triangular  =  L  M  =  t o t a l a p p l i e d external torque  M'  =  the t o r s i o n a l part of M  M"  =  t h e f l a n g e bending p a r t of M  I-ipp M K.J.  = = .=  f i x e d edge moment t o t a l i n t e r n a l torque t h a t p a r t o f M c a r r i e d by t o r s i o n a l  =  that part  «  b e n d i n g moment p e r , u n i t  L  M  1  matrix  such that S  = LL • 1  transposed  of M  carried  by f l a n g e  stress  bending  l e n g t h a s shown  i n P i g . 16 . m  =  h a r m o n i c number o f F o u r i e r  m  =  i n t e r n a l web b e n d i n g moment a s shown i n F i g . 8b  series  fixed  edge a x i a l  force  number o f d e g r e e s o f f r e e d o m o f t h e j o i n t s of the s t r u c t u r e amplitude  of v e r t i c a l roof l o a d i n g per u n i t  area  i n t e r n a l s h e a r b e t w e e n f l a n g e and web a s shown i n P i g . 8b radius of c i r c u l a r rise  of s h e l l  shell  curve  membrane s h e a r f o r c e p e r u n i t l e n g t h a s shown i n P i g . 16 structure fixed  stiffness  matrix  edge l o n g i t u d i n a l s h e a r f o r c e  plate s t i f f n e s s matrix i n plate  co-ordinates  plate s t i f f n e s s matrix i n structure  co-ordinates  transformation matrix f o rtransformation plate co-ordinates I  to structure  from  co-ordinates  transposed  membrane f o r c e p e r u n i t l e n g t h i n t h e l o n g i t u d i n a l d i r e c t i o n as shown i n P i g . 16 membrane f o r c e p e r u n i t l e n g t h n o r m a l t o t h e l o n g i t u d i n a l d i r e c t i o n , as shown i n P i g . 16 plate  thickness  web and f l a n g e p l a t e t h i c k n e s s e s , r e s p e c t i v e l y f i x e d edge s h e a r f o r c e  Notation W  (cont d) 1  =  amplitude of u n i f o r m normal l o a d  th form  harmonic W  ss  amplitude of uniform t a n g e n t i a l m  .X  =  load f o r  th , harmonic  l o n g i t u d i n a l d i s t a n c e f r o m end d i a p h r a g m a s 3,  shown i n P i g s .  7> and 9 .  x  =  abscissa  Y  =  t r a n s v e r s e d i s t a n c e from c e n t e r l i n e as  of b a r r e l s h e l l  =  ordinate of b a r r e l s h e l l  Z  =  L'A  o<  =  difference  between  curves  cp and t h e r o t a t i o n o f  f l a n g e s a s shown i n P i g . 8b  (3-v)/(l+v)  P  =  Y  =  mnb/2a  A  =  structure  d e f l e c t i o n v e c t o r A., , A., ....A.,..A 1'  A_^  =  of p l a t e  shown i n P i g . 7  y  the  curves  amplitude of the structure i n g t o the i'*'*  1  I'  2'  deflection  n  correspond-  degree o f freedom and t h e  m^  harmonic &  =  plate  d e f l e c t i o n v e c t o r 6. , 6 , .... 6  0  i n plate  co-ordinates 6  =  plate  d e f l e c t i o n v e c t o r 6^,  structure £  =  the effect  .... &g i n  co-ordinates o f web b e n d i n g a s d e f i n e d by e q n .  (3s)  vii . Notation 9  (cont'd) =  t h e a n g l e between t h e p l a t e  centerline  and t h e  h o r i z o n t a l , as shown i n F i g . 4 \  =  Et /l2(l-v )a  =  Et/(l+v) a  v  =  Poisson's r a t i o  cp  =3  r o t a t i o n o f beam c r o s s s e c t i o n  X  2  3  2  2  as d e f i n e d i n  Fig. 8 cp  =  t h e a n g l e between t h e c y c l o i d a l s h e l l curve and  cf>£g cp  . *s  t h e h o r i z o n t a l , a s d e f i n e d i n eqn.(47b)  cp o b t a i n e d by t h e G o o d i e r - B a r t o n  . ss cp o b t a i n e d by t h e Timoshenko  method  method.  viii .  ACKNOWLEDGMENTS  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 D r . R.P. H o o l e y  f o r h i s i n v a l u a b l e guidance preparation  of this t h e s i s .  during  the research,  d e v e l o p m e n t , and  The a u t h o r i s a l s o g r a t e f u l t o t h e  N a t i o n a l Research C o u n c i l f o r f i n a n c i a l support i n the form of a n a s s i s t a n t s h i p , and t o t h e s t a f f o f t h e C o m p u t i n g C e n t e r a t t h e U n i v e r s i t y o f B.C. f o r t h e i r  help.  ix.  LIST OF FIGURES FIGURE No. 1  TYPICAL FOLDED PLATE STRUCTURE  '2  SOME FOLDED PLATE CROSS SECTIONS WITH MORE THAN 2 PLATES INTERSECTING AT A JOINT  3  h DEGREES OF FREEDOM OF A JOINT  ' k  8 DEGREES OF FREEDOM OF A JOINT  5  PLATE FORCES f AND DEFLECTIONS £  6  FIXED EDGE FORCES  7  PLATE COORDINATES  8  CROSS SECTION DEFORMATION  • 9  TORSION OF I BEAM  10  APPLIED MOMENT M FOR TIMOSHENKO METHOD I S SUM  11  TIMOSHENKO AND STIFFNESS METHODS COMPARED FOR LOADING AS INDICATED BY EQUATIONS (27) AND (28)  12  APPLIED MOMENT M FOR GOODIER-BARTON METHOD I S SUM OF M' AND M"  13  GOODIER-BARTON AND STIFFNESS METHODS COMPARED FOR LOADING AS INDICATED BY EQNS. (kO) AND (kl)  Ik  DIFFERENCE I N ROTATION </> DUE TO CHANGE FROM TIMOSHENKO LOADING TO GOODIER-BARTON-LOADING  15  CROSS SECTIONAL SHAPES  16  PLATE FORCES PER UNIT LENGTH T , T ^ S AND M^  17  SINUSOIDAL ROOF LOADING  18  VALUES OF 100 M^/pc FOR SHORT THIN HALF ELLIPSE-  19a  T /pa AT MIDSPAN  OF M' AND M"  X  2  19b  T / p a AT MIDSPAN x  r  19c  T / p a AT MIDSPAN  19d  T / p a AT MIDSPAN  20a'  S/pa AT  x  x  •  DIAPHRAGMS  LIST OF FIGURES FIGURE No. 20b  S/pa AT DIAPHRAGMS  20c  S/pa AT DIAPHRAGMS  20d  S/pa AT DIAPHRAGMS  21a  T^/pc AT MIDSPAN  21b  T  21c .  ^/P  c A  MIDSPAN  T  T^/pc AT MIDSPAN  21d  T  ^/P  c  A  MIDSPAN  T  22a  100 M^/pc AT MIDSPAN  22b  100 M^/pc AT MIDSPAN  22c  100 M^/pc AT MIDSPAN  22d  100 M^/pc  2  2  2  2  T  - 3J  2  / T x / ^x c i r c  2k  S/S .  5  T,/T/  ' circ 2  .  p circ  26  M//M/  <p'  AT MIDSPAN  <p  .  circ  1. A STIFFNESS MATRIX SOLUTION FOR FOLDED PLATES  I . Introduction A prismatic  folded  p l a t e s t r u c t u r e , as shown i n F i g . 1, i s a  s h e l l composed o f a s e r i e s o f r e c t a n g u l a r  p l a t e s , completely f i x e d  to. one a n o t h e r a t t h e i r l o n g i t u d i n a l edges o r j o i n t s , and f r a m i n g i n t o and s u p p o r t e d by t r a n s v e r s e A s u r p r i s i n g l y l a r g e number  end d i a p h r a g m s . . . of c o n t r i b u t i o n s  o f f o l d e d p l a t e s may be f o u n d i n t h e l i t e r a t u r e .  t o the a n a l y s i s A fairly  compre-  h e n s i v e summary o f t h e work done i n t h i s f i e l d up t o 1962 i s g i v e n i n t h e P h a s e I R e p o r t on F o l d e d P l a t e C o n s t r u c t i o n which a l s o contains  an e x t e n s i v e  of the A S C E ^ ,  bibliography.  Most o f t h e a u t h o r s o f t h e s e p a p e r s f o u n d t h e a n a l y s i s of f o l d e d p l a t e s t o be a l e n g t h y  and l a b o r i o u s  p r o c e d u r e , and t h e y made  a v a r i e t y o f a p p r o x i m a t i o n s , some o f w h i c h caused s u b s t a n t i a l i n t h e r e s u l t s , i n .an a t t e m p t t o s i m p l i f y t h e s o l u t i o n .  errors  However, a n  (2) e x a c t s o l u t i o n was p r e s e n t e d by G o l d b e r g and L e v e  who u s e d  theory  o f e l a s t i c i t y f o r t h e membrane s o l u t i o n and c l a s s i c a l t h i n p l a t e theory f o r the bending s o l u t i o n . formulated a s t i f f n e s s matrix matrix  elements evaluated  Leve.  Gibson^^  D e P r i e s - S k e n e and S c o r d e l i s ^ ^  method o f s o l u t i o n w i t h s t i f f n e s s  f r o m t h e t h e o r y d e v e l o p e d by G o l d b e r g and  p r e s e n t e d a p a p e r s h o w i n g a method whereby t h e s t r e s s e s  i n a f o l d e d p l a t e s t r u c t u r e a r e d e t e r m i n e d by u s i n g  cylindrical  shell  (5) theory. the  I n a second paper, G i b s o n  v  showed good agreement b e t w e e n  r e s u l t s o f h i s method o f a n a l y s i s , S c o r d e l i s ' method o f a n a l y s i s ,  2 . and  S c o r d e l i s ' e x p e r i m e n t a l work.  F i n a l l y , Mast  a n a l y s i s c o m p a r i s o n of c y l i n d r i c a l s h e l l s and a n a l y s e d by  s h e l l theory,  ordinary  folded  by  a u t h o r ' s o p i n i o n , the  G o l d b e r g and  p l a t e method, and  and  object  t o use  beam and  theory  of  exact t h e o r y  presented  Leve combined w i t h a c o n v e n i e n t s t i f f n e s s method of f o l d e d p l a t e  structures.  o f t h i s t h e s i s i s t o d e v e l o p s u c h a s t i f f n e s s method  i t t o i n v e s t i g a t e two  the  plates  Leve above.  c l a s s i c a l and  presents the best approach f o r a n a l y s i s The  an  V-shaped f o l d e d  e l a s t i c i t y f o l d e d p l a t e method o f G o l d b e r g and I n the  presented  problems, the  torsion  of an  I-shaped  a n a l y s i s c o m p a r i s o n of b a r r e l s h e l l s o f d i f f e r e n t  cross  sections. This two  method d i f f e r s f r o m D e F r i e s - S k e n e and  S c o r d e l i s method i n  respects: 1.  The  method of f o r m u l a t i o n  of the  structure s t i f f n e s s matrix  i s d i f f e r e n t , t h u s .making i t p o s s i b l e t o have more t h a n two connected to the  same l o n g i t u d i n a l j o i n t .  o f f o l d e d p l a t e . s t r u c t u r e s w h i c h may presented i n F i g . 2.  Some examples o f t h e be  solved  w i t h ease  I n a d d i t i o n to  more a c c u r a t e l y and  joint  l o a d s , l o a d s between j o i n t s are  with a smaller  l o a d i n g c a s e s t o be  number of  d e v e l o p m e n t of t h e p r o g r a m i s t i m e c o n s u m i n g , the use  is  are  I n o r d e r t o a n a l y s e any  o n l y n e c e s s a r y t o punch the  required  intro-  handled  plates.  method i s programmed f o r a d i g i t a l c o m p u t e r .  p r o g r a m i s not.-  types  2.  duced, thus p e r m i t t i n g p r a c t i c a l roof  This  now  plates  Although of t h e  the  developed  folded' p l a t e s t r u c t u r e , i t d a t a c a r d s f o r the  program,  t h u s e l i m i n a t i n g t h e t i m e c o n s u m i n g , l e s s e x a c t , and more e r r o r hand  c a l c u l a t i o n s of e a r l i e r  methods.  The t o r s i o n p r o b l e m was s e l e c t e d ability  prone  , because i t demonstrates t h e  o f t h e "method t o h a n d l e more t h a n two p l a t e s f r a m i n g i n a t a  j o i n t , a n d b e c a u s e i t p r o v i d e s a c h e c k f o r p r e v i o u s l e s s e x a c t method of a n a l y s i s  of the problem.  The b a r r e l s h e l l p r o b l e m was s e l e c t e d  because i t p r o v i d e s an  o p p o r t u n i t y ' t o i n v e s t i g a t e t h e e f f e c t o f t h e shape s e c t i o n on t h e s t r e s s e s  i n the s h e l l .  of a s h e l l  cross  II.  u  S t i f f n e s s Method B e f o r e d e v e l o p i n g the  are p r e s e n t e d . and  relationship  (2)  a l t h o u g h the ments a r e  thickness  permitted  may  Now  each j o i n t  longitudinal.  end  boundary c o n d i t i o n s  the  displace-  d i a p h r a g m s shown i n P i g . l ,  f o r the  The  last  ends of t h e  folded  are f o u r  possible  horizontal, vertical, rotational  or l o a d and  a deflection.  Thus f o r t h e  e x t e r n a l f o r c e o r l o a d P.  and  i " ^  is  degree  a deflection A  an  of .  The  i  1  equation  deflections  A = P  r e l a t i o n between the A  of t h e  . external forces  convenient to express the  d i s t r i b u t i o n o f f o r c e s and  forces  and  end  the  deflections i n Pourier  d e f l e c t i o n s i n the  111  for'the  th  S, i t series  harmonic only.  The  longitudinal direction  i s s i n u s o i d a l f o r t h e h o r i z o n t a l , v e r t i c a l , and  s a t i s f i e s the  and  structure s t i f f n e s s matrix  A s o l u t i o n i s then developed f o r the  cosine  P  (la)  structure.  .' B e f o r e a t t e m p t i n g t o f o r m t h e  f r e e d o m and  no  C o r r e s p o n d i n g t o e a c h degree o f f r e e d o m t h e r e  i s an  expresses the  form.  thickness,  s t i f f n e s s method i s d e v e l o p e d .  shown i n P i g . 3 ,  "s  is  superposition  (5)  vary from plate to p l a t e ;  •  stiffness  a linear  and'of u n i f o r m  of a f o l d e d p l a t e s t r u c t u r e t h e r e  d e g r e e s of f r e e d o m as  freedom there  (3)  small;  i s o f f e r e d normal to these planes.  •plate structure.  external force  structure i s isotropic  s t r a i n s , so t h a t  i n the p l a n e s of the  assumption p r o v i d e s the  and  the  geometry changes a r e  each p l a t e i s r e c t a n g u l a r  no r e s i s t a n c e  At  (l)  e x i s t s " between s t r e s s e s and (4)  •is valid;  and  They a r e as f o l l o w s :  homogeneous;  method,' 5 a s s u m p t i o n s  s t i f f n e s s matrix  r o t a t i o n a l d e g r e e s of  l o n g i t u d i n a l "degree o f f r e e d o m .  This  boundary c o n d i t i o n requirements mentioned i n  5.  a s s u m p t i o n (5) and  above.  I t i s also convenient-to describe  forces A  1 s i n m TI X/a The  1  A.  and  P. s i n m  i ' where ' A.  Associated  of f r e e d o m as shown i n P i g . 4»  rrX/a  1  P.,  structure s t i f f n e s s matrix  plate s t i f f n e s s matrices.  there  1  and  by t h e s i n g l e t e r m s o  S  and  P.  i  are amplitudes,  i s formed from the i n d i v i d u a l with  each p l a t e are 8 degrees  C o r r e s p o n d i n g t o each degree of freedom  i s an i n t e r n a l j o i n t f o r c e and a d e f l e c t i o n .  stiffness  and  the d e f l e c t i o n  The i n d i v i d u a l p l a t e  equations  '  s 6 = f  i n p l a t e co-ordinate.s  (lb)  s b = f  i n structure co-ordinates  (lc)  e x p r e s s t h e r e l a t i o n s h i p between t h e i n t e r n a l j o i n t f o r c e s and t h e d e f l e c t i o n s Because there  6  f  or  f  o r 6.  is a rigid  c o n n e c t i o n between  j o i n t s and p l a t e s t h e  4 d e f l e c t i o n s at a' j o i n t a r e t h e "same a s t h e 4 d e f l e c t i o n s a t t h e n e a r edge o f each a d j a c e n t p l a t e . s t i f f n e s s matrix  S  Because  of t h i s  e q u a l i t y , the  structure  c a n be f o r m e d f r o m t h e p l a t e s t i f f n e s s m a t r i c e s  s. B e f o r e t h i s can b e . d o n e , t h e i n d i v i d u a l p l a t e m a t r i c e s must be (2) formed.  G o l d b e r g and L e v e  r e l a t i n g the eight force-vector  f  have d e r i v e d  generalized.forces  to the eight  of t h e - d e f l e c t i o n v e c t o r  6.  the necessary  f^, f g ,  generalized  equations f g of t h e  d e f l e c t i o n s 6^,  &, 9  . . . ,6g  To o b t a i n these r e l a t i o n s h i p s , t h e  i n d i v i d u a l p l a t e p r o b l e m i s . d i v i d e d ' i n t o two p a r t s ,  namely:  1.  The membrane s o l u t i o n r e l a t i n g t h e i n p l a n e f o r c e s and  d e f l e c t i o n s 1, 4 , elasticity 2.  5, and 8 shown i n P i g . 5 u s i n g t h e t h e o r y o f  equations d e f i n i n g the plane s t r e s s  problem.  The b e n d i n g s o l u t i o n r e l a t i n g t h e o u t o f p l a n e f o r c e s and  d e f l e c t i o n s 2, 3, 6, and 7 shown i n P i g . 5 u s i n g t h e c l a s s i c a l  thin  plate theory. These e q u a t i o n s can be c o n v e n i e n t l y e x p r e s s e d i n m a t r i x f o r m as  follows:  1  2  3  2  % 0  e  3  0  f  i  4  b  0  0  5  c  0  6  0  g -h  7  o h  8  d .0  1  4  5  6  7  8 f  \  k  0 -d  a  0  0  e  y 0  0  f  i  0  0  0  JL -b  S  s 6  that i s ,  =  5  =  k  6  s  f  2  f  3  f  4  f  5  f  8  l  (2)  6  f  7  f  8  f  where  a  =  m \  2  , , „ b =. m X_ TT 2 m  TT  [-  cosh  y  Y s e c h Y -3  r cosh Y L~ ; i : Y c s c h Y -3 c o s h Y  —  sinh Y  +  sinh v Y c s c h Y +3 c o s h Y sinh Y  Y s e c h Y +3  sinh Y  + ( l + v)]  7.  c = m  \  TT  2  d = m X  2  k = m X  Y Y seen Y •-3 sinh Y  [  s>j-j.iij  Y  "j  Y c s c h y + 0 cosh y  TT  [  cosh Y Y c s c h Y '- 3 cosh Y  sinh Y Y sech Y + 3 s i n h Y  TT  [  sinh Y Y c s c h Y - 3 cosh Y  cosh Y Y sech Y + 3 s i n h Y  1  s i n h VY coshYY Z = -m \ TT [C + ] . . L_ .—. 2 Y csch Y ~ 3 cosh Y Y sech Y + 3 s i n h y s  i  n  n  c o s n  +  L  3 3 e = m TT X^  —2 a 2 2 \  m f  =  mW  [  —  i  =  -.  m  T  ainh Y - ( i- ) 1 y sech Y ^ s i n h Y  r  cosh Y I Y sech y ~ s i n h Y " sinh Y n Y sech Y ~ s i n h Y  +  r cosh Y • sinh Y i L ; —:—; ~ ; ; — J Y sech Y + s i n h Y Y csch Y - cosh Y  n  [ cosh Y | sinh y •, Y sech Y + s i n h Y Y c s c h y - cosh y  X  1  J  cosh Y Y c s c h Y + cosh Y  cosh Y -\ Y sech Y ~ s i n h y  v  +  . X 1 T  Y _ Y c s c h Y + cosh y c o s h  a 2 2 _ -m TT X^  TT  _  r sinh y "Y c s c h Y cosh Y  n  2  i = m  r sinh Y • Y c s c h Y + cosh Y  and .where _  M  TT  b  .  _  E t  ?  12 (,1-v Ja p  - TiTT)  x  .  2 - 77—1  (1+v) a  Note that the rows 1, 4» 5, and 8 of the m a t r i x r e p r e s e n t the  membrane s o l u t i o n and rows 2, 3, 6, and 7 r e p r e s e n t A l s o , t h e elements i n the j ^ * a unit deflection  6^.  1  the bending  solution,  column a r e the f o r c e s r e q u i r e d t o produce  while a l l other d e f l e c t i o n s of the p l a t e are held  equal to zero. The above e q u a t i o n , s 6 = f , i s i n p l a t e c o - o r d i n a t e s . structure s t i f f n e s s matrix matrices i n P i g . 4.  s  S  Before the  c a n be f o r m e d , t h e i n d i v i d u a l p l a t e  must be t r a n s f o r m e d  i n t o s t r u c t u r e c o - o r d i n a t e s as  shown  T h i s i s a c c o m p l i s h e d by u s i n g t h e t r a n s f o r m a t i o n m a t r i x  where  cosQ - s i n 9  0  0  0  0  0  0  cos9  0  0  0  0  0  0  sin9 0  0  1  0  o•  0'  0  0  0  0  0  1  0  0  0  0  0  0  0  0  cos9 - s i n 9  0  0  0  0  0  0  sin9  cos9  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  1  T =  (3)  Thus  and  6  =  T  6  U)  f  =  T  f  (5)  7  =  I  s  "s  T  =  f  T'  (6) (7)  T,  9.  i s e q u a t i o n ( 2 ) re-expre3sed i n s t r u c t u r e c o - o r d i n a t e s . The  plate s t i f f n e s s matrices  structure s t i f f n e s s matrix  s  a r e now combined t o f o r m  S. A t t h i s s t a g e ,  S  and F  the  i n equation  ( l ) a r e known. The  C h o l e s k i Method ( 7 ) i s c h o s e n t o s o l v e f o r t h e unknowns  b e c a u s e o f i t s speed and a c c u r a c y .  A  I n t h i s method a t r i a n g u l a r m a t r i x  L i s formed such t h a t  "s = L L  (8)  1  where L' i s t h e t r a n s p o s e o f L . S to  i s r e p l a c e d by LL' i n e q u a t i o n ( l a )  give LL  1  A  =  F  Z  =  L» A  (9)  Letting (lO)  equation ( 9 ) i s r e w r i t t e n as LZ = P Because  L  i s triangular,  forward s u b s t i t u t i o n .  ( l l ) Z  c a n be f o u n d f r o m t h i s e q u a t i o n by one  K n o w i n g Z, A i s f o u n d f r o m  ( i o ) u s i n g one  backward  substitution. When t h e s t r u c t u r e d e f l e c t i o n s A a r e known, t h e p l a t e d e f l e c t i o n s 6 f o r e a c h p l a t e a r e known and  the i n t e r n a l j o i n t f o r c e s  f  are  found  by s u b s t i t u t i n g t h e v a l u e o f f f r o m e q u a t i o n (5) i n t o e q u a t i o n (7) t o obtain f  =  I  1  8 5  (12)  . Up t o t h i s p o i n t i n t h e d e v e l o p m e n t o f t h e s t i f f n e s s method, o n l y  10. external joint forces more g e n e r a l  F  have b e e n a p p l i e d t o t h e s t r u c t u r e .  s o l u t i o n which a l s o includes  can be c o n s i d e r e d  The  u n i f o r m l o a d s between j o i n t s  a s t h e sum o f two c a s e s , Case A and Case B, where Cas  A i s t h e f i x e d edge s o l u t i o n i n w h i c h t h e j o i n t s a r e h e l d f i x e d and Cas B i s the s o l u t i o n of . S A = P + Pp wh>ere  P  i s the vector  (13)  of e x t e r n a l  joint forces  and F  i s vector  of  f i x e d edge f o r c e s a c t i n g on t h e j o i n t s . E a c h p l a t e i s t r e a t e d i n d i v i d u a l l y t o o b t a i n t h e f i x e d edge shown i n P i g . 6 ,  as  When W  G o l d b e r g and L e v e (2)  have p r o v i d e d  i s t h e a m p l i t u d e o f t h e u n i f o r m l o a d , and W  forces  the formulas.  i s the amplitude  of t h e t a n g e n t i a l l o a d , t h e f i x e d edge force's a r e as f o l l o w s :  W a „, n M__ = —-—-r PP 2 2 m TT  2  1  r  Y  sinh Y  PP  4 W a t ~ ( l )" PP -T-. r (,1+vJmTT W. a  PP  =  .  Y c s' c h Y^ + 3 c o s h Y n a r > V  v  R  '  1  a  (l v)mTT +  \\^>)  1  V  r L Y  The l a s t s t e p  o n a V l  /-,,-v  a-  sinh Y  SB  P  't  mrrX  v  S  K  4 cosh Y csch Y + 3 cosh  i  n  nmX  Q  / ,\  " a  T  Y  "  U  +  v  )  , N-i °° 3  m-nX S  ~  U  ?  )  i n t h i s method o f f o l d e d p l a t e a n a l y s i s i s t o  mine t h e l o n g i t u d i n a l f o r c e p e r u n i t l e n g t h T . the  ,. , >, (14) ^'  n  Y c s c h Y + cosh Y  nm  N  .  1—  _  v  '  1  2 W a n  ,  S  r 2 cosh v • , -1 . mnX L ;— :— - U sin Y csch Y + cosh Y a  deter-  Again, the s o l u t i o n i s  sum o f t h e Case A s o l u t i o n and t h e Case B s o l u t i o n , and a g a i n , t h e (2)  formulas are provided Por  by G o l d b e r g and Leve  .  t h e f i x e d edge s o l u t i o n , Case A, a t any p o i n t  on t h e p l a t e ,  . \  2W a s i n mnX/a " — ycshY + ecoshY  m  [  ~  C  O  s  h  —  "(vcothY - ^ a i n h  1  1  .  (is)  P o r Case B , 1^ i s a f u n c t i o n o f t h e i n p l a n e edge d e f l e c t i o n s 6 , 4' 5  6  6  a  n  d  6  s  h  o  w  n  i  n  p  8  i g « 5.  U s i n g t h e p l a t e c o - o r d i n a t e system  shown i n P i g . 7 n d a  X  5  (ysechY - 3  =  sinhY)"  S(Y)  1  = s i n h mnY/a  X  6 = ( Y c s c h Y + 3 coshY)"^"  S (Y) - SSI l a  X  7 = (ycschY - 3 c o s h Y ) "  1  C(Y) = c o s h mnY/a  " (ysechY + 3 s i n h y ) "  1  X  8  ^4 ^7  T  2v/(l  =  =  C () H L l l a Y  + v)  k  •(3 + v ) / ( l + v)  =  5  { S  4  {  +6  8  = YcothY  t  &  + 6  inTLY a  c o s h  k' = Y"tanhY  - f T i T v ) a £ i {-SCB.CYKC^.- k + 6  c  =  s i n h ^ a  x 7  [ S  1  ( Y )+  (  f c  )c(Y)] X [C (Y)+(^-k )s(Y)] } +  6  1  c  - k ) c ( Y ) ]+X [ C ( Y ) + ( u - k ) s ( Y ) ] }  V  t  5  1  4  c  t S ( Y ) + ( u - - k ) c ( Y ) > X [ C ( Y ) + (u -k )s(Y) ] } 1  c  ?  8  1  7  t  { X [S (Y) + (n .- k ) 0 ( Y ) ] + X [ C ( Y ) ( i - k ) s ( Y ) ] j 7  1  7  c  8  1  +  t  7  t  (19) The above s t i f f n e s s m a t r i x method p r o v i d e s a f a s t , a c c u r a t e , s t r e s s and d e f l e c t i o n a n a l y s i s f o r f o l d e d this thesis w i l l  plate structures.  show two a p p l i c a t i o n s ;  the analysis  The r e m a i n d e r o f of an I-shaped  beam i n t o r s i o n a n d a n a n a l y s i s c o m p a r i s o n o f b a r r e l s h e l l s o f d i f f e r e n t cross  sections.  12. I I I . Torsion  of I Beam (8)  Timoshenko  d e v e l o p e d a method of a n a l y s i s o f I - s h a p e d beams i n  t o r s i o n which includes but  assumes t h a t t h e  not  d e f o r m i n i t s own  web  and  the f l a n g e  the bending of the f l a n g e s  beam c r o s s  shown by P i g . 8 a .  p l a t e s bend out not  planes  s e c t i o n , a l t h o u g h f r e e t o w a r p , does  p l a n e , as  o f t h e p l a t e edges a r e  i n t h e i r own  A c t u a l l y , both  o f t h e i r p l a n e s and  e q u a l t o one  the  the  rotations  a n o t h e r as shown i n P i g .  8c.  (9) G o o d i e r and o f t h e web The  Barton cross  f o r m u l a t e d a method w h i c h i n c l u d e s  s e c t i o n as  shown by P i g .  the  p l a t e s and  of t h e  p r o b l e m depends upon t h e way  G o o d i e r - B a r t o n methods by  beam c r o s s  one  web  plate.  I t i s a l s o shown t h a t t h e the  simply  analysing  d i s t r i b u t e d torque given  s h o w n - i n P i g . 9.  a  subject  to  presented f o r a  t o an a p p l i e d  Q  external  s i n TiX/a  Since the  ends a r e  (20) prevented from r o t a t i n g  a r e f r e e t o warp) t h e  r e s u l t i n g i n t e r n a l torque i s M, = — M cos ^ t TT o a c r i t e r i o n f o r c o m p a r i s o n i s t a k e n t o be t h e  section,  solution  by  M = M  cross  half  section.  s u p p o r t e d I beam.of l e n g t h  The  section  t o r s i o n a l moment i s a p p l i e d  I n o r d e r t o compare t h e s e m e t h o d s , s o l u t i o n s a r e  • as  previous  beam as a f i v e p l a t e f o l d e d p l a t e s t r u c t u r e - c o m p r i s e d of f o u r  flange  the  deformation  8b.  more p r e c i s e s t i f f n e s s method p r e s e n t e d i n t h e  i s u s e d t o c h e c k t h e Timoshenko and  a  cp, measured as  shown i n P i g .  8.  (but  (21) ' r o t a t i o n of the • v  13.  Timoshenko M e t h o d I n t h i s method i t i s assumed t h a t t h e c r o s s s e c t i o n o f t h e beam does n o t d e f o r m i n i t s own p l a n e . is  The i n t e r n a l t o r s i o n a l moment M  d i v i d e d i n t o two p a r t s ,  M" t  (22)  = C d cp/dX  c a u s e d b y t o r s i o n a l s t r e s s , and  . dX c a u s e d by f l a n g e b e n d i n g , and t h e t o t a l t o r q u e M, = M. where  C  as (24)  + M "  1  t  u  i s expressed  t  = torsional rigidity  D = flexural rigidity  = JC  o f one f l a n g e = E I _, flange E/2 ( l + v) b  G = s h e a r modulus  =  J = J + 2J„ w f J J  and  /3  = b t w w w ' f  =  3  V f / 3 3  b , b^, t w  in P i g . 9.  w  , t ^ , and h a r e c r o s s s e c t i o n a l  P o r M^ a s g i v e n by e q u a t i o n  d i m e n s i o n s a s shown  ( 2 l ) , the s o l u t i o n  of equation  (24) i s cp = cp s i n TrX/a Q  (25)  where  TT  This  solution  distributed  LC +Dh  TT  /2a J  i s v a l i d o n l y when t h e a p p l i e d moment  M i s  o v e r t h e c r o s s s e c t i o n i n a s p e c i f i c manner, t h a t i s , s u c h  14. that  M  i s t h e sum .of two p a r t s , M M'  1  and M",  = (M' /M )M = ^ — . 1 + Dh TT /2Ca t  t  2  2  as shown i n P i g . 10a,  (27) 2  and where  M" = (M"./M )M =  ^—5  *  1 + 2CaVDh  *  where  7—  (28)  V  10c.  as shown i n P i g .  The s e l e c t i o n o f t h e c r o s s s e c t i o n a l d i m e n s i o n s o f t h e t e s t beam, t„ = 2 i n , t' = 1/2 i n . b„ = 8 i n . , and b = 22 i n . , i s d i s c u s s e d 1 w 1 w #  l a t e r i n t h i s paper.  T h i s t e s t beam was l o a d e d w i t h M' and M" and  s o l v e d by b o t h t h e Timoshenko and s t i f f n e s s methods f o r a v a r i e t y o f lengths.  P i g . 11 shows t h e r e s u l t s , w h i c h a g r e e w e l l , e x c e p t f o r s h o r t  lengths.  This discrepancy  i s due t o f l a n g e s h e a r d e f l e c t i o n s , w h i c h  i s not i n c l u d e d i n t h e Timoshenko t h e o r y .  However, when s h e a r  d e f l e c t i o n i s i n c l u d e d t h e agreement•becomes much b e t t e r , b u t i s s t i l l somewhat  i n e r r o r , probably  b e c a u s e t h e r a t i o of M'  t o M" t  used t o t  d e t e r m i n e t h e l o a d d i s t r i b u t i o n o v e r t h e c r o s s s e c t i o n i s a l t e r e d by the e f f e c t of shear d e f l e c t i o n . The t e s t beam was a l s o r u n a s e c o n d t i m e w i t h t h e s t i f f n e s s method but w i t h t h e l a r g e number o f s m a l l c o u p l e s r e p l a c e d by t h e more c o n v e n i e n t shown i n P i g . 10b,  indicated i n equation  s t a t i c a l l y e q u i v a l e n t f o r c e s f ^ and  where  f1 =  (J /j)(M'/b )  (29)  f  (J /j)(M'/b )  (30)  w  w  and 2  =  f  (27)  f  The same:value o f cp i s f o u n d 'for e i t h e r l o a d i n g .  1 5 .  G o o d i e r and B a r t o n Method Tinder some l o a d i n g c o n d i t i o n s , b e n d i n g o f t h e web, w h i c h was n e g l e c t e d i n t h e Timoshenko method, may be i m p o r t a n t .  To a c c o u n t  (9)  f o r t h i s , G o o d i e r and B a r t o n 9  between  a,  the difference  and t h e r o t a t i o n o f t h e f l a n g e s , a s shov/n i n P i g . 8 b .  Equation (2l) M «  i n t r o d u c e the angle  =  becomes C dcp/dX  X  +  2C (dcp/dX - da/dX)  w  (31)  X  where t h e f i r s t term i s t h e moment, c a r r i e d by web t o r s i o n , and t h e s e c o n d , by f l a n g e t o r s i o n , and where • C, = web t o r s i o n a l r i g i d i t y = J G w • w and  f l a n g e t o r s i o n a l r i g i d i t y = J^G  =  As b e f o r e  '  t  and  M. U  =  dX^  d  M ' + M " « t  ( 3 3 )  E q u a t i o n [ 4 ] o f G o o d i e r and B a r t o n ' s p a p e r ^ ^  expresses the bending  moment m o f t h e web as w m  where  D  Equation  w  w  =  6  w  D a / b w  ( 3 4 )  = v/eb f l e x u r a l r i g i d i t y = E t  3 w  /12  •  2 (l-v ). '  [ 2 ] o f t h e same p a p e r e x p r e s s e s t h e t w i s t i n g o f t h e f l a n g e s  by t h e moment m a s w C.  (d cp/dX 2  2  - d a/dX ) 2  2  =  -  ra  ( 3 5 )  16 . When t h e moment Iff the  i n equation  (33-) i s a s g i v e n by e q u a t i o n ( 2 l ) ,  s o l u t i o n o f t h e above e q u a t i o n s (33), (34), a n d (35) i s • <p = cpQ s i n TTX/a  (36)  a = aQ s i n n X / a  (37)  where 2 cp  ° and  22  M  IT " ° 2  2  ( 3 3 )  [C + D h T T / 2 a ] [ l - ] 2  2  2  £  where  =/  2  .  represents equal  [ C . + 6D a / b T T ] [ C 2  f  w w  2  +  t h e e f f e c t o f web b e n d i n g .  to zero, t h i s  s o l u t i o n becomes  Dh TT /2a ] 2  2  2  N o t e t h a t v/hen equal  e  i s set  t o the s o l u t i o n using  t h e T i m o s h e n k o m e t h o d , w h i c h assumes t h a t t h e c r o s s  s e c t i o n does  •not d e f o r m . I n order t o s a t i s f y  t h e assumptions o f t h i s method, M  must be  a p p l i e d s u c h t h a t no t w i s t i n g moment o t h e r t h a n t h e i n t e r n a l moment m ,.as shown i n P i g . 8 b , i s a p p l i e d t o t h e f l a n g e s . equation  (35), w h i c h r e p r e s e n t s  t o be m o d i f i e d .  Therefore,  t h e t w i s t i n g o f a f l a n g e , w o u l d have  when t h e e x t e r n a l l o a d  i n t o t o r s i o n a l and f l a n g e b e n d i n g p a r t s ,  M  1  I n this  case  M  i s divided  and M" r e s p e c t i v e l y ,  as shown i n P i g s . 1 2 ( a ) a n d 1 2 ( b ) r e s p e c t i v e l y , t o t h e web o n l y .  Otherwise  M* must be a p p l i e d  17.  (M '/M,)M = X  oo  X  1 + Dh  TT  M  (4 0 )  o  /2 a  C-  (C^TT /a f 2  M"  ='  (M" /M )M = X  + 6D /h) w  2  M  .X  (41) 2,2  ~„  2C TT. / a f  1 +  (C TT /a 2  Dh TT /2a 2  The  t e s t beam was l o a d e d  the Goodier-Barton  + 6D /h)  2  f  with M  2  1  w  2  a n d M" and s o l v e d by b o t h  and s t i f f n e s s methods f o r a v a r i e t y o f l e n g t h s .  P i g . 13 shows t h e r e s u l t s , w h i c h , as b e f o r e , a g r e e w e l l e x c e p t f o r short  lengths• The  t e s t beam was a l s o r u n a s e c o n d t i m e w i t h t h e s t i f f n e s s  method b u t w i t h t h e l a r g e number o f s m a l l c o u p l e s equation forces  (40) r e p l a c e d by t h e more c o n v e n i e n t f^  statically  equivalent  shown i n P i g . 12b, where  f, .= The  indicated i n  (42)  M'/h  moments M' and M" a r e added t o f o r m . M , a s  shown i n P i g . 12,  where f. = 4 The  same v a l u e  of  S e l e c t i o n of Test The  M/h  (43)  cp i s f o u n d f o r e i t h e r l o a d i n g . Beam  c r o s s s e c t i o n a l d i m e n s i o n s o f t h e t e s t beam were s e l e c t e d  18.  s u c h t h a t t h e e f f e c t o f web b e n d i n g , and t h e r e f o r e t h e t e r m equation  (39) might be l a r g e .  as t h e l e n g t h zero.  a  Therefore,  Prom t h i s e q u a t i o n  i t i s seen t h a t  a p p r o a c h e s b o t h z e r o and i n f i n i t y , £  e in  e  approaches  w i l l be l a r g e s t f o r some i n t e r m e d i a t e  length  beam, r a t h e r t h a n f o r a l o n g beam o r a s h o r t o n e . To f i n d a beam c r o s s i s best  s e c t i o n f o r which  e  might be l a r g e i t  t o s u b s t i t u t e the values D  =  Et / l 2  C. 'f  = "  b„t f f  (l-v )  3  (44)  2  and  i n t o equation  3  (38) and r e - a r r a n g e  . E  ~  With  £  terms t o o b t a i n  2  '. 7~ U  (45)  (J/3  V j  t *f  (46)  2  3  t  bJ> TT I w  . *f  3  7 77 ,  b b  f  4  a  i n t h i s form, i t i s apparent that f o r £  . b, 2 "f  t o be l a r g e ,  ( t / t ^ ) must be s m a l l , t h a t i s , t h e f l a n g e s must be t h i c k and t h e web t h i n . width  b  Trial should  showed a l s o t h a t t h e beam d e p t h be l a r g e .  h  and t h e web  The t e s t beam c h o s e n s a t i s f i e s  these  requirements.  Comparison of R e s u l t s The d i f f e r e n c e b e t w e e n t h e r e s u l t s o f t h e T i m o s h e n k o and G o o d i e r B a r t o n s o l u t i o n s i s shown i n F i g . 14 where t h e f a c t o r l / ( l - e ) o f equation  (39) i s p l o t t e d v e r s u s  the length  a.  Both these  methods  have b e e n c h e c k e d by t h e s t i f f n e s s method and f o u n d t o be c o r r e c t .  19.  A l t h o u g h t h e a p p l i e d t o r s i o n a l moment i s t h e same i n b o t h c a s e s , i t i s t h e d i f f e r e n c e i n t h e d i s t r i b u t i o n o f t h e a p p l i e d t o r s i o n over the  cross  s e c t i o n t h a t c a u s e s t h e d i f f e r e n c e i n t h e r o t a t i o n cp. The  s t i f f n e s s method r e s u l t s a l s o show t h a t t h e d i f f e r e n c e i n r o t a t i o n b e t w e e n t h e web and t h e f l a n g e s  i s v e r y s m a l l f o r t h e Timoshenko  method l o a d i n g and s u b s t a n t i a l l y l a r g e f o r t h e G o o d i e r - B a r t o n method loading.  G e n e r a l Case I t h a s b e e n shown t h a t f o r b o t h t h e T i m o s h e n k o and G o o d i e r B a r t o n m e t h o d s , t h e a p p l i e d moment cross will  s e c t i o n i n a s p e c i f i c manner. give  inaccurate  The a b i l i t y  r e s u l t s unless  M  must be d i s t r i b u t e d o v e r t h e Change i n t h i s d i s t r i b u t i o n  f o r m u l a r e v i s i o n s a r e made.  of the stiffness'method  t o handle the general  of any d i s t r i b u t i o n o f t o r s i o n a l moment o v e r many c r o s s  case  sections i s  an a d v a n t a g e n o t f o u n d i n t h e o t h e r m e t h o d s .  Conclusions Because o f i t s a b i l i t y t o handle t h i s general  case, the s t i f f -  n e s s method i s t h e most c o n v e n i e n t o f t h e methods c o m p a r e d . Because i t i n c l u d e s and  the e f f e c t of shear d e f l e c t i o n o f the f l a n g e s  t h e e f f e c t o f P o i s s o n ' s r a t i o , and a l l o w s  of t h e beam c r o s s  f o r the free  deformation  s e c t i o n i n c l u d i n g f l a n g e s , t h e s t i f f n e s s method i s  t h e most e x a c t o f t h e methods c o m p a r e d .  I t therefore  provides a  t h e o r e t i c a l check o f t h e o t h e r m e t h o d s . The a b i l i t y  o f t h e s t i f f n e s s method t o a n a l y s e f o l d e d  plate  s t r u c t u r e s w i t h more t h a n one p l a t e i n t e r s e c t i n g a t a j o i n t h a s a l s o  20. been d e m o n s t r a t e d .  Some o t h e r f o l d e d p l a t e s t r u c t u r e s w h i c h may  be s o l v e d w i t h ease a r e i l l u s t r a t e d  i n P i g . 2.  now  I V . A n a l y s i s Comparison of B a r r e l of 5 D i f f e r e n t C r o s s  Shells  Sections  L i t t l e has been w r i t t e n on t h e a n a l y s i s o f n o n - c i r c u l a r cylindrical shells.  However i t i s p o s s i b l e t o o b t a i n a n a n a l y s i s  o f a n y shape o f c y l i n d r i c a l s h e l l by a p p r o x i m a t i n g i t a s a f o l d e d p l a t e s t r u c t u r e and a n a l y s i n g  this  equivalent  p r o c e d u r e i s accurate' p r o v i d e d  t h a t t h e method o f f o l d e d  a n a l y s i s used i s a c c u r a t e a n d p r o v i d e d plate  structure i s properly  s t r u c t u r e . ' This plate  that the equivalent  folded  selected.  U s i n g t h e s t i f f n e s s method p r e s e n t e d i n s e c t i o n I I o f t h i s t h e s i s , a s t u d y i s made o f t h e e f f e c t o f v a r i a t i o n o f t h e c r o s s s e c t i o n a l shape on t h e s t r e s s d i s t r i b u t i o n s i n c y l i n d r i c a l s h e l l s . The  i n v e s t i g a t i o n i s l i m i t e d t o simply  s h e l l s under v e r t i c a l r o o f l o a d i n g .  supported m u l t i p l e  barrel  The r e s u l t s a r e p r e s e n t e d i n  graph form f o r l o n g , s h o r t , t h i c k , and t h i n s h e l l s and c o n c l u s i o n s a r e made r e g a r d i n g To  the r e l a t i v e merits  of the d i f f e r e n t shapes.  o b t a i n good r e s u l t s , t h e e q u i v a l e n t  folded  plate  must be s e l e c t e d w i t h a s u f f i c i e n t number o f p l a t e s .  structure  ^twill  s e q u e n t l y be shown t h a t t h e l o c a t i o n o f t h e j o i n t s i s a l s o Five  important.  c u r v e s a r e s e l e c t e d t o c o v e r a wide r a n g e o f shapes a s  shown i n F i g . 15. as  sub-  The e q u a t i o n s u s e d t o g e n e r a t e t h e s e c u r v e s a r e  follows: half  ellipse  cycloid  x = (<n/2)r  (l-v/l-y /r ) 2  2  x = (r/2)(2cp + s i n 29) + Ttr/2 where  9 = -g a r c cos (^" /(^") T  {hi) (48a) (48b)  22 . circle  segment  parabola  x  sine  x  where  r  TTr/2 - J(l.7337r)  x  i s the r i s e ,  x  t h e a b s c i s s a and  .7337r)  - (y +  2  y  (k3)  2  t h e o r d i n a t e , as  shown i n F i g . 1 5 . L o n g , s h o r t , t h i c k , and t h i n short  s h e l l lengths  shell's a r e t r e a t e d .  The l o n g and  were s e l e c t e d s u c h t h a t t h e r a d i u s t o l e n g t h  r a t i o f o r t h e c i r c u l a r c y l i n d e r , R/a, i s 0.2 a n d 0.6 r e s p e c t i v e l y . The t h i c k and t h i n s h e l l t h i c k n e s s e s  are such that the radius t o  t h i c k n e s s r a t i o f o r t h e c i r c u l a r c y l i n d e r R / t , i s 100 and 200 r e s p e c t ively. its  The edge beam t h i c k n e s s  i s 5.77 t i m e s t h e s h e l l t h i c k n e s s and  d e p t h i s 0.15 t i m e s t h e r i s e , r . The i n t e r n a l f o r c e s c o n s i d e r e d  and  S, a l l f o r c e s p e r u n i t l e n g t h , and t h e b e n d i n g moment p e r u n i t  l e n g t h M^, a s shown i n F i g . 1 6 . the  I n o r d e r t o compare t h e r e s u l t s f o r  d i f f e r e n t shapes, s o l u t i o n s a r e found f o r a s i n u s o i d a l roof  of amplitude  p  sinusoidal line  per unit area load  equal  a s shown i n F i g . 17-  shell.  An a d d i t i o n a l  c  i s h a l f the transverse  the corresponding  tributions.  span o f  V/hen t h e l o a d i n g i s s i n u s o i d a l , t h e d e f l e c t i o n s and s t r e s s e s  a l s o have a s i n u s o i d a l d i s t r i b u t i o n , w i t h t h e e x c e p t i o n and  loading  t o .181 pc i s a p p l i e d t o a c c o u n t f o r t h e  dead l o a d o f t h e edge beam, where the  a r e t h e membrane f o r c e s T , T , x' cp'  l o n g i t u d i n a l d e f l e c t i o n , w h i c h have c o s i n e  When v a l u e s  to the amplitude,  of the shear  S  dis-  a r e g i v e n f o r f o r c e s and s t r e s s e s , t h e y r e f e r  which defines  the entire l o n g i t u d i n a l d i s t r i b u t i o n .  B e c a u s e t h e s h e l l i s assumed t o -be a n i n t e r i o r  section of a multiple  barrel  s h e l l , a d v a n t a g e i s t a k e n o f symmetry so t h a t  s h e l l i s analysed at the  a s shown i n P i g . .15.  only h a l f of  Thus t h e b o u n d a r y  l o n g t u d i n a l edges r e q u i r e t h a t t h e r e be  the  conditions  no h o r i z o n t a l d e f l e c t -  i o n or r o t a t i o n . N o t e t h a t the u n i f o r m r o o f l o a d only.  An a l t e r n a t i v e , but  between the  joints.  values, while low at the  i s a p p l i e d as  joint  more c o m p l e x a t t a c k i s t o a p p l y  loads  the  A d i s a d v a n t a g e of t h i s a p p r o a c h i s t h a t  c o r r e c t f o r the  j o i n t s and  cylindrical shells. does not a r i s e and  p  When j o i n t l o a d s  lines for  only are used, t h i s  c u r v e i s smooth as i t s h o u l d  p  the  f o l d e d p l a t e s t r u c t u r e s , are always  too h i g h a t the p l a t e c e n t e r  the M  load  too  the  problem  be.  Por  this  9 r e a s o n t h e more c o m p l e x method o f l o a d i n g b e t w e e n j o i n t s was S e l e c t i o n of E q u i v a l e n t To  Folded P l a t e  not  Structure  d e t e r m i n e t h e number o f p l a t e s r e q u i r e d f o r an a c c u r a t e  s o l u t i o n s were o b t a i n e d b e t w e e n t h e base and  f o r s h e l l s d i v i d e d i n t o 10,  the crown.  The  constant.  I t was  20 and  j o i n t s between the  l o c a t e d such that the v e r t i c a l d i s t a n c e j o i n t s was  used.  Ay b e t w e e n any  40  analysi plates  p l a t e s were two  found t h a t the r e s u l t s tend to  adjacent converge  t o w a r d s a f i n a l s o l u t i o n as t h e number o f p l a t e s i s i n c r e a s e d .  However  c e r t a i n c a s e s a r e much more s e n s i t i v e t o the  number of p l a t e s u s e d  o t h e r s , the worst case b e i n g  f o r the  ellipse.  This  case i s i l l u s t r a t e d  i s p l o t t e d versus the height p l a t e s a r e not r e g i o n , and  the value  of  short, thin, half  i n P i g . 18, where t h e r a t i o 100  ratio y/r.  The  c u r v e s show t h a t w h i l e  s a t i s f a c t o r y , 20 p l a t e s g i v e good r e s u l t s i n the  j o i n t s are  still  too f a r  apart.  M^/p 10  lower  40 p l a t e s g i v e good r e s u l t s e x c e p t i n t h e u p p e r r e g i o n ,  where i t i s a p p a r e n t t h a t t h e  than  For. t h e f i n a l r u n 45  a d d i t i o n a l j o i n t s were added t o g i v e a  number o f p l a t e s n e a r t h e u p p e r and approximately equal p l a t e lengths The /As  c u r v e c o r r e s p o n d i n g t o t h i s 85  l o w e r b o u n d a r i e s , and  a l s o to  throughout the i n t e r m e d i a t e  h o w e v e r , b e c a u s e of t h e  p l a t e s h e l l i s a l s o shown i n F i g .  b e t w e e n j o i n t s and It of the  i s the  an IBM  to i n t e r p r e t  recommended.  s i n c e the  p l a t e s h e l l would a l s o g i v e a s a t i s f a c t o r y c o m p u t e r p r o g r a m g e n e r a t e s i t s own  Thus t h e f o l l o w i n g r e s u l t s a r e  On  joints  o p i n i o n o f t h e a u t h o r s t h a t more c a r e f u l d i s t r i b u t i o n  p l a t e d a t a , t h e o n l y s a v i n g would be  crown of each  18.  p a r a b o l i c d i s t r i b u t i o n of moments  t h i s s y s t e m i s not  j o i n t s f o r t h e 40  s o l u t i o n , but  These r u n s were d i f f i c u l t  roughly  give  region.  w e l l , a n o t h e r s e r i e s o f r u n s w i t h l o a d s a p p l i e d b e t w e e n the  c o n v e r g e d t o t h e same c u r v e .  per  large  joint  and  a s m a l l amount o f computer  f o r 85  time.  p l a t e s b e t w e e n t h e base and  the  shell. 7040 u s i n g d o u b l e p r e c i s i o n i t t o o k a b o u t 1.5  s t r u c t u r e f o r one  load  minutes  case.  Results The 21,  and  r e s u l t s a r e p l o t t e d as d i m e n s i o n l e s s 22.  shape can  I t i s immediately apparent that  cause l a r g e changes i n t h e  reversal.  For  M,  other  f o r the  e x a m p l e , M^  f o r the  24,  25,  changes i n t h e  s t r e s s e s and  19,  20,  shell  even c o m p l e t e s t r e s s  h a l f e l l i p s e has  opposite  sigh to  shapes.  To a i d i n d e t e r m i n i n g 23,  r a t i o s i n Figures  the  e f f e c t of the d i f f e r e n t shapes,  and -26 a r e p r o d u c e d i n w h i c h t h e  Figures  s t r e s s r a t i o s T /T  , circ  S/S circ'  T Vi cp ^cir'c  a n d  M  c/ cp M  a r e  circ  P  1 0  ^ ^ ^ v e r s u s .the p a r a m e t e r 6  e/d.  25. Here  d  i s . t h e c h o r d l e n g t h and  e  i s the d i s t a n c e perpendicular t o  t h e c h o r d f r o m i t s m i d - p o i n t t o t h e s h e l l c u r v e a s shown i n P i g . 1 5 . The  subscript " c i r c " refers to the c i r c u l a r  cylindrical  shell.  Conclusions The of  f o l l o w i n g c o n c l u s i o n s a r e made r e g a r d i n g t h e r e l a t i v e  the different The  shell  merits  shapes.  s i n u s o i d a l and p a r a b o l i c c y l i n d r i c a l  s h e l l s have c o n s i d e r a b l y  l a r g e r i n t e r n a l f o r c e s than t h e other shapes, w i t h t h e e x c e p t i o n of t h e f o r c e T .• However, i n a l l c a s e s T f o r c e which w i l l not govern The  i sa relatively  design.  c i r c l e segment i s b e s t f o r c o m p r e s s i v e  tudinal force T  s m a l l membrane  values of the l o n g i -  and i s a l s o good f o r t h e s h e a r f o r c e  S.  However,  x for long shells,  ' this  a d v a n t a g e i s . l o s t b e c a u s e s i m p l e beam a c t i o n i s  a p p r o a c h e d and t h e v a l u e s o f ' T all  and  S- a r e v e r y n e a r l y t h e same f o r  shapes. The  c y c l o i d h a s t h e l o w e s t maximum v a l u e o f t h e s h e a r f o r c e  a n d , when t h e s h e l l i s long', o f t h e moment M . .' <+' compression,  The s t r e s s e s T  x  S i n  and i n t e n s i o n i n b o t h t h e s h e l l and t h e beam, and T^  i n tension are also r e l a t i v e l y low. The  half ellipse  i s perhaps t h e best s h e l l  t h e l o w e s t v a l u e s of ^ ' i x  compressive  values of T  n  s t r e s s w i s e . I t has  b o t h t h e s h e l l and t h e beam, a l t h o u g h t h e  a t t h e t o p o f t h e s h e l l a r e higher t h a n f o r t  t h e c y c l o i d and t h e c i r c l e segment. l o w e s t v a l u e s of M  The h a l f  e l l i p s e a l s o has t h e  when t h e s h e l l i s s h o r t a n d h a s good v a l u e s f o r  26 . S  a n d T^ i n t e n s i o n . Thus i t appears, t h a t as f a r as s t r e s s e s a r e c o n c e r n e d , t h e b e s t  shapes  a r e i n t h e r e g i o n o f h i g h v a l u e s o f e / d , s u c h as t h e h a l f  ellip  and t h e c y c l o i d , w h i l e t h e w o r s t s h a p e s a r e i n t h e r e g i o n o f low v a l u e of  e / d , s u c h as t h e p a r a b o l a and t h e s i n e . It  cannot be s t a t e d t h a t t h e s e c o n c l u s i o n s w i l l h o l d t r u e f o r  c a s e s o t h e r t h a n t h o s e d i s c u s s e d h e r e , such a s , f o r example, b a r r e l s h e l l s or s h e l l s w i t h o u t edge beams. are  single  However, when s u c h c a s e s  e n c o u n t e r e d , t h e same method may be u s e d t o d e t e r m i n e t h e shape  required f o r the lowest i n t e r n a l  stresses.  FIG. I  TYPICAL FOLDED PLATE STRUCTURE  FIG. 2  SOME THAN  FOLDED P L A T E CROSS SECTIONS WITH 2 P L A T E S INTERSECTING AT A JOINT  MORE  FIG. 3  4  PLATE FIG. 4  DEGREES OF FREEDOM OF A JOINT  COORDINATES STRUCTURE COORDINATES only the left half of the plate is shown  8 DEGREES OF FREEDOM OF A JOINT  FIG. 5  PLATE  FORCES f AND DEFLECTIONS 8  FIG. 6  FIXED  EDGE  FORCES  edge Y = -r- is upper joint  FIG.  7  PLATE  COORDINATES  (a) Timoshenko assumes cross-section does not deform.  FIG.  8  CROSS  SECTION  DEFORMATION  FIG. 9  TORSION  OF  I  BEAM  FIG. 10  APPLIED MOMENT M FOR TIMOSHENKO IS SUM OF M' AND M"  METHOD  by Timoshenko method by Stiffness method by Timoshenko method modified to include effect of flange shear deflection  T+S  4  8 a,  FIG. II  10  12  14  16  18  20  feet  TIMOSHENKO AND STIFFNESS METHODS COMPARED FOR LOADING AS INDICATED BY EQUATIONS (27) AND (28)  FIG. 12 APPLIED MOMENT M FOR GOODIER-BARTON METHOD IS SUM OF M AND M" 1  FIG. 13 GOODIER-BARTON AND STIFFNESS METHODS COMPARED FOR LOADING AS INDICATED BY EQNS. (40) AND (41)  <f>  = <f> by Goodier-Barton method and corresponding loading shown in Fig. 9  a,  feet  FIG. 14 DIFFERENCE IN ROTATION <£> DUE TO CHANGE FROM TIMOSHENKO LOADING TO GOODIER-BARTON LOADING  FIG. 15  CROSS  SECTIONAL  SHAPES  FIG.  17  SINUSOIDAL  ROOF  LOADING  FIG. 18 VALUES OF 100 M^/pc FOR. SHORT THIN HALF ELLIPSE 2  t  FIG.  19c  T  x  /pa  AT  MIDSPAN  pa FIG. I9d.  T /pa x  AT MIDSPAN  FIG.. 2 0 a .  S/pa AT  DIAPHRAGMS  FIG. 20b.  S/pa  AT DIAPHRAGMS  FIG. 20c  S/pa  AT DIAPHRAGMS  FIG. 2lo  T^/pc  AT  MIDSPAN  -1.6 -1.4 -1.2 -1.0 -.8  1±pc  FIG. 21c  T^/pc  -.6  AT MIDSPAN  -4  -.2  0  .2  .4  -  FIG. 2 2 a  3  -  2  1 100  IOOM«£/pc AT 2  MIDSPAN  0  I  2  long Fhick long thin short thick short thin''  thick thin thick thin  FIG.  23  T  x  /T  x c i r c  FIG.  24  S/So.v,  5.0  27. Bibliography  \- -  1.  "Phase I R e p o r t on F o l d e d P l a t e C o n s t r u c t i o n , " R e p o r t of t h e T a s k Committee on F o l d e d P l a t e C o n s t r u c t i o n , Committee on M a s o n r y and R e i n f o r c e d C o n c r e t e , S t r u c t u r a l D i v i s i o n , George P. B o o s s , C h a i r m a n , J o u r n a l of t h e S t r u c t u r a l D i v i s i o n , 'ASCE, • ' V o l . " 8 9 , No. ST6, P r o c . P a p e r 3741, December, 1963, pp. 365. 406..  ?/, /  G o l d b e r g , J .E., and L e v e , H.L., "Theory o f P r i s m a t i c F o l d e d P l a t e S t r u c t u r e s , " P u b l i c a t i o n s of t h e I n t e r n a t i o n a l A s s o c i a t i o n f o r B r i d g e and S t r u c t u r a l E n g i n e e r i n g , V o l . 1 7 , 1557, pp. 58-86 .  3. ' D e F r i e s S k e n e , A. and S c o r d e l i s , A . C , "Direct Stiffness S o l u t i o n f o r F o l d e d P l a t e s , " J o u r n a l of t h e S t r u c t u r a l D i v i s i o n , ASCE, V o l . 90, No. ST/+, P r o c . P a p e r 3994, A u g u s t , 1964, pp. 15-  47•4.  G i b s o n , J . E . , "Computer I n v e s t i g a t i o n of F o l d e d S t r u c t u r a l E n g i n e e r , V o l . 40, No. 5, May, 1962,  Slab Roofs," p.151.  5.  G i b s o n , J .E., "An I n v e s t i g a t i o n of F o l d e d P l a t e S t r u c t u r e s , " S t r u c t u r a l E n g i n e e r , V o l . 42, No. 9, S e p t . 1964, pp. 209-304.  6.  M a s t , P a u l E., ' " A n a l y s i s C o m p a r i s o n f o r S i n g l e C u r v e d S h e l l . R o o f s , " J o u r n a l of t h e S t r u c t u r a l D i v i s i o n , ASCE, V o l . 91,, No. ST2, P r o c . P a p e r 4 2 8 2 , A p r i l , 1965, pp. 1-14.  7.  C h o l e s k i , " C o n t r i b u t i o n s t o t h e S o l u t i o n of Systems of L i n e a r E q u a t i o n s and t h e D e t e r m i n a t i o n of E i g e n V a l u e s , " E d i t e d by O l g a T a u s s k y , N a t i o n a l B u r e a u of S t a n d a r d s , U.S. P r i n t i n g O f f i c e A p p l i e d M a t h e m a t i c s S e r i e s No. 39, S e p t . , 1954, p. 31.  8.  T i m o s h e n k o , S., " S t r e n g t h of M a t e r i a l s , P a r t I I , "D. Company, I n c . , P r i n c e t o n , New J e r s e y , 1962.  9.  10.  Van  Nostran  G o o d i e r , J . N. and B a r t o n , M.V., "The E f f e c t s o f Web B e n d i n g on t h e T o r s i o n , of I-Beams, " J o u r n a l of A p p l i e d M e c h a n i c s , M a r c h , ' 1944. T i m o s h e n k o , S., " S t r e n g t h Company, I n c . , New Y o r k , '  of M a t e r i a l s , P a r t I , " D. 1947  Van  Nostrand  APPENDIX - COMPUTER PROGRAM FPA (FOLDED PLATE ANALYSIS)  D r . R. F. H o o l e y ' s p l a n e f r a m e a n a l y s i s p r o g r a m , ALP I V , was modified  t o solve-the  folded p l a t e problem.  includes  ( a ) r e p l a c i n g t h e c a l c u l a t i o n o f t h e 6 by 6 p l a n e f r a m e  member s t i f f n e s s m a t r i x ness m a t r i x ,  This  modification  by t h e c a l c u l a t i o n o f a n 8 by 8 p l a t e  (b) f o r m i n g t h e s t r u c t u r e s t i f f n e s s matrix  8 matrices instead  o f 6 by 6,  l o a d s between j o i n t s ,  stiff-  f r o m 8 by  ( c ) introducing a routine t o handle  (d) i n t r o d u c i n g  double p r e c i s i o n  arithmetic  t o i m p r o v e a c c u r a c y , and ( e ) c a l c u l a t i n g t h e i n t e r n a l f o r c e N . i x  The p r o g r a m a n d d a t a a r e s t o r e d •of t h e IBM 7040 c o m p u t e r ; operating  time i s v e r y s h o r t .  plate structures input  data,  forces  N  therefore  tapes a r e not required  and t h e  F o r e x a m p l e , 11 d i f f e r e n t f o l d e d  o f 5 p l a t e s e a c h were s o l v e d  deflections, i n t e r n a l joint forces  with printout of and t h e l o n g i t u d i n a l  i n 2g m i n u t e s .  A f l o w sheet and l i s t are  e n t i r e l y w i t h i n t h e c o r e memory  of the f o r t r a n i n s t r u c t i o n s f o r t h i s  a t t h e end o f t h i s a p p e n d i x .  A listing  program  o f d a t a and a p r i n t o u t o f  r e s u l t s f o r a sample f o l d e d p l a t e s t r u c t u r e a r e a l s o i n c l u d e d . ' Whenever d a t a i s r e a d i n , i t i s a l s o p r i n t e d o u t . . The d e s c r i p t i o n o f the i n p u t of operations 1.  The f i r s t  shown i n t h e f l o w data card  data format required  and t h e s e q u e n c e  sheet' a r e a s f o l l o w s :  read contains  s t r u c t u r e d a t a as f o l l o w s :  Format NES  identification  NEJ  number o f j o i n t s  NEM  number .of p l a t e s  NIC  number o f l o a d c a s e s  2.  The s e c o n d d a t a  number 3 110  card  12 0  contains  t h e e l a s t i c p r o p e r t i e s and t h e  l e n g t h o f the s t r u c t u r e as f o l l o w s : E  Young's m o d u l u s p s i  V  Poisson's  AL  structure length f t .  '3.  ratio  N e x t , one d a t a  3 E 1 0.3  card i s read f o r each j o i n t  of the s t r u c t u r e  c o n t a i n i n g j o i n t i n f o r m a t i o n as f o l l o w s : JN(l)  j o i n t number.  ND(l,l) = 1 i fh o r i z o n t a l joint d e f l e c t i o n i s permitted, 0 i f not ND(l,2) = 1 i f v e r t i c a l  joint  deflection i s permitted,  0 i f not  5 1-10  ND(l,3) = 1 i f r o t a t i o n a l j o i n t 0 i f not  d e f l e c t i o n i s permitted, '  ND(l,4) = 1 i f l o n g i t u d i n a l .joint d e f l e c t i o n i s permitted, 0 i f not X ( l ) = x co-ordinate Y  ft  ( l ) = y co-ordinate f t  2 P l 0.5  30. When N D ( l , j ) = 1 a j o i n t d e f l e c t i o n i s p o s s i b l e , t h a t i s , a d e g r e e of freedom e x i s t s .  These d e g r e e s o f f r e e d o m a r e numbered c o n s e c u t i v e l y i n  t h e o r d e r i n w h i c h t h e ND v a l u e s a r e r e a d i n and  are c a l l e d  F o r each N D ( l , j ) = 0 t h e c o r r e s p o n d i n g  placement numbers.  the  dis-  displacement  number i s s e t e q u a l t o z e r o .  A f t e r a l l t h e d a t a c a r d s have b e e n r e a d  • i n and  have been numbered, t h e t o t a l number of  a l l the displacements  d i s p l a c e m e n t s , Nil., i s p r i n t e d  4.  One  out.  data card i s read f o r each p l a t e of the s t r u c t u r e c o n t a i n i n g  p l a t e i n f o r m a t i o n as  follows: Format  MN  ' p l a t e number  JNX  lesser  JNII  greater joint  T  plate thickness  The  j o i n t number  4 displacement  3  110  number F 3 0 . 3  inches  numbers a s s o c i a t e d w i t h t h e l e s s e r j o i n t f o l l o w e d  by. t h e 4 f o r t h e g r e a t e r j o i n t f o r m t h e 8 p o s i t i o n numbers NP plate.  The  d i f f e r e n c e b e t w e e n t h e l a r g e s t and  number o f a p l a t e i s NB1  of e a c h  smallest position NB1  f o r t h a t p l a t e , and t h e l a r g e s t v a l u e of  d e t e r m i n e s t h e band, w i d t h o f t h e s t r u c t u r e s t i f f n e s s m a t r i x .  After  each  p l a t e data card i s read, the p l a t e s t i f f n e s s m a t r i x elements are  cal-  c u l a t e d d i r e c t l y i n s t r u c t u r e c o - o r d i n a t e s and  Also,  s t o r e d i n memory.  c o n s t a n t s r e q u i r e d l a t e r i n t h e p r o g r a m f o r c a l c u l a t i o n of f i x e d moments and. i n t e r n a l p l a t e s t r e s s e s a r e c a l c u l a t e d 'and s t o r e d . all out.  t h e p l a t e d a t a c a r d s a r e r e a d , t h e maximum band w i d t h '  end After  is.printed  31. 5.  The p o s i t i o n ' n u m b e r s NP a r e u s e d t o f o r m t h e s t r u c t u r e s t i f f n e s s  m a t r i x , S,. f-i'bm t h e e l e m e n t s o f t h e i n d i v i d u a l p l a t e s t i f f n e s s ¥ i s s>^-'red i n band f o r m as a s i n g l y  matrices.  subscripted variable.  (6) ,4.  The C h o l e s k i  equation if  o r s q u a r e r o o t method i s u s e d t o s o l v e t h e m a t r i x  SA = F f o r t h e unknown d e f l e c t i o n s A.  i t s s t i f f n e s s matrix  'S = L L , 1  S i s positive definite.  S i s positive definite  a r e a l l non z e r o and p o s i t i v e . number o f t h e d i s p l a c e m e n t and  7 -  NJL 8. ing  I n t h i s m e t h o d , where  o n l y when t h e d i a g o n a l e l e m e n t s of I f this  o c c u r s i s p r i n t e d out  unstable.  card i s read c o n t a i n i n g the s i n g l e entry  number o f ' l o a d e d  NJL d a t a c a r d s  Format 110  joints  a r e r e a d , one f o r e a c h l o a d e d  joint,  each c o n t a i n -  the f o l l o w i n g data: I 6,4  JNU  j o i n t number  PP(l)  amplitude-of  PP(2)  amplitude  of v e r t i c a l load l b . / f t .  PP(3)  amplitude  of r o t a t i o n a l load f t . - l b . / f t .  Pp(4)  amplitude  of l o n g i t u d i n a l load l b . / f t .  The l o a d v e c t o r  h o r i z o n t a l load  P- i s f o r m e d f r o m t h e s e  One d a t a c a r d i s r e a d NPL  containing'the  number o f l o a d e d  plates  X  lb,/ft. /,. F 1 0 . 3  joint loads, positive direct-  i o n s o f w h i c h a r e shown i n P i g . . 3.  9.  L  c o n d i t i o n i s not f u l f i l l e d , t h e  where t h e i n s t a b i l i t y  the s t r u c t u r e i s r e j e c t e d as being One d a t a  A structure i s stable  s i n g l e entry 110  32. 10.  NPL .data c a r d s a r e r e a d , one f o r e a c h l o a d e d p l a t e , e a c h  containing MM  the f o l l o w i n g data:  ' I 6, h X  p l a t e number 2  • WH  a m p l i t u d e o f h o r i z o n t a l p l a t e l o a d l b , / f t .. o f h o r i z o n t a l " p r o j e c t i o n on t h e l o n g i t u d i n a l v e r t i c a l plane  2 F  1 0 . 3  2 WV  amplitude of v e r t i c a l p l a t e load  lb . / f t .  of  v e r t i c a l p r o j e c t i o n on t h e h o r i z o n t a l p l a n e . WH  and  WV  are converted to  plate loads r e s p e c t i v e l y .  WW  and  WT,  t h e n o r m a l and t a n g e n t i a l  The f i x e d edge f o r c e s  e a c h p l a t e and c o n v e r t e d t o s t r u c t u r e  co-ordinates.  - a p p l i e d t o t h e j o i n t s as a d d i t i o n a l j o i n t l o a d s . load vector 11.  P  and t h e s t r u c t u r e  the  That i s , t h e j o i n t  s t i f f n e s s matrix  S, a r e  The e q u a t i o n "  is  These l o a d s a r e  i s augmented by t h e f i x e d edge f o r c e s .  At t h i s point  known.  are calculated f o r  solved  for A  S A = P  by t h e C h o l e s k i  method a s p r e v i o u s l y  explained,  and  d e f l e c t i o n s o f each j o i n t a r e p r i n t e d out w i t h p o s i t i v e d i r e c t i o n  a s shown i n P i g . 3. 12.  Prom t h e s t r u c t u r e d e f l e c t i o n v e c t o r 6  deflection vectors  are formed. f  the  =  8 i n t e r n a l joint forces  f o r each p l a t e .  ,f  and  6  7  A, t h e i n d i v i d u a l p l a t e  By t h e e q u a t i o n  "& .  f i n structure co-ordinates  are found  are converted to forces  and  f  3 3  delections  6  i n plate co-ordinates  and t h e f i x e d edge f o r c e s  pre-  v i o u s l y c a l c u l a t e d and s t o r e d i n memory a r e added t o t h e f o r c e s obtain the t o t a l i n t e r n a l joint forces,  -f^ ,  •  f  to  which a r e p r i n t e d o u t .  P o s i t i v e d i r e c t i o n i s shown i n P i g . 5.  13.  The l o n g i t u d i n a l p l a t e f o r c e  fixed  T^  i s t h e sum o f t h a t  edge c o n d i t i o n and t h a t due t o t h e d e f l e c t i o n s  6.  due t o t h e This  force i s  c a l c u l a t e d and p r i n t e d out f o r t h e u p p e r and l o w e r numbered edges and t h e c e n t e r l i n e o f each p l a t e .  14.  P o s i t i v e forces are t e n s i l e .  I f a n o t h e r l o a d c a s e i s t o be a n a l y s e d ,  t h e program r e t u r n s t o  r e a d the"new l o a d d a t a a n d p r o c e e d s w i t h t h e a n a l y s i s a s b e f o r e . that i t i s necessary t o convert  S  once, because t h e s t i f f n e s s matrix  15.  present;  or repeats  only  cases.  t h e program checks i f  f o l d e d p l a t e s t r u c t u r e , stops  i f none i s  t h e e n t i r e a n a l y s i s - f o r t h e new s t r u c t u r e .  Thus  number o f s t r u c t u r e s c a n be r u n a t t h e same t i m e i f t h e r e a r e no  error  messages. The  For  i s t h e same f o r a l l l o a d  A f t e r a l l l o a d c a s e s have b e e n a n a l y s e d ,  data i s present for-another  any  t o L by t h e C h o l e s k i method  Note  u n i t s r e q u i r e d f o r t h e d a t a i n p u t have b e e n i n d i c a t e d a b o v e .  p r i n t e d o u t p u t , t h e u n i t s a r e a l l i n pounds and f e e t . When t h e p r o g r a m i s u s e d - t o a n a l y s e - b a r r e l s h e l l s by a p p r o x i m a t i n g  them w i t h f o l d e d p l a t e s t r u c t u r e s , i t c a n be m o d i f i e d  to calculate the  j o i n t ' a n d member d a t a i f t h e shape o f t h e s h e l l c a n be e x p r e s s e d i n closed  form.  34. EPA  PLOW SHEET  Start  >  $$ p r i n t s t r u c t u r e data ( Head and m mga t e r i a l p r o p e r t i e s  J_  ( Read and p r i n t j o i n t data  I Assign, displacement n o s . N~D{1 ,J) 1 , 2 , , . . .NTI= no. of unknown displacements . .I < I f t o o many degrees of f r e e d o r Q  / p r i n t e r r o i N .— yes•»\^message J-*- Stop  No  ~ "  u s 6 Xi  -—  ] -- ,^  [Read £ p r i n t member data")  •  C a l c u l a t e p o s i t i o n numbers NP one t o eight £ f i n d max. band w i d t h t o date  o  C a l c . member s t i f f n e s s elements i n structure co-ordinates C a l c . t s t o r e member c o n s t a n t s r e q u i r e d l a t e r - f o r c a l c u l a t i o n of f i x e d end moments and i n t e r n a l plate stresses  ( P r i n t max. band w i d t h )  I  P_orm s t r u c t u r e s t i f f n e s s m a t r i x S i n band form Convert 3 t o 1 by C H o l e s k i ' s Method S = LL* <^ i f any S(L) < 0^112  <  output s t r u c t u r e unstableN at displacement L J Stop  •p a"  •H ^  (Head $ p r i n t  d  Form l o a d v e c t o r  CD O  CV'  O  due t o j o i n t  loads  1  H  CD -P  .fl  05 rH  03 0)  tS  O  .joint loads')  Y  O  P,  /Seaad $ p r i n t p l a t e l o a d s f o r p l a t e s V llooaa d e d b e t w e e n j o i n t s  '  o cS  £ o  O  I  C a l c u l a t e f i x e d edge r e a c t i o n s and a p p l y t o j o i n t s i n s t r u c t u r e c o - o r d i n a t e s by augmenting load vector  CD CD  H  Knowing s t r u c t u r e s t i f f n e s s m a t r i x S and l o a d v e c t o r F, s o l v e f o r j o i n t deflections A by C h o l e s k i Method  t (Print  joint  deflections  Aj  1'  S o l v e f o r plajte f o r c e s f .f = s 6 and c o n v e r t t o p l a t e f o r c e s i n plate co-ordinates f and add t h e p l a t e f o r c e s due t o t h e f i x e d edge case t o o b t a i n the t o t a l p l a t e forces f  CD -P  05 P,  o  a  CD  t  £  (Print t o t a l plate  o  (33 CD CD O  a o  CD -P  03 H  CL,  —  ^  forces)  Knowing p l a t e d e f l e c t i o n s and p l a t e l o a d s between j o i n t s , f i n d i n t e r n a l p l a t e s t r e s s e s f o r each p l a t e  T  (Output i n t e r n a l p l a t e  stresses')  go t o s t a r t f o r a new  structure  THIS JOINT FIXED HORIZ., VERTICALLY 8L0N6IT., BUT FREE TO ROTATE  E = 3,500,000 psi  320 lb/ft  LOAD CASE 2,200 lb/ft  20 lb/ft  LOAD  CASE 2  JOINT AND PLATE NUMBERS PLATE THICKNESSES EXAMPLE  STRUCTURE  STRUCTURE NO. N RS  6  _  .  YOUNG'S MODULUS E PS1 ""• ~l> 7'0~'0  0. "  JOINT NO. J N U)  I  .7  IF  MOVE , O VERTICAL  IF 4>  RESTRAINED ROTATION  O  ND(1,J) LONGIT.  1  3 4  1 1  1 1  1 .1  1 1  1  1  1  1  NO.  LESSER JNT. NO. J N L  1 3 4  5  6 5  TO  —c  1  I 2  2  '/'  ' 4 0.  1  MN  3  -  FREE  DATA L I S T I N G F O R 5AMPLE. STRUCTURE _SHEE-T—I—  LENGTH AL FT.  1  PLATE  10 3  o  H0RIZ  0  ij_  C A S E ' S NL.C  5  POI S S O N ' S RATIO V -  —  L  LOAO  . . — ..  _  6  u  NO. O F  NO. OF JOINTS NO. OF PLATES NftJ NftM  —  3  5  —-  G R E A T E R JNT. WO. JWU  i  \  COOROINATE  XCO 0. ~  FT-  2. "" 3. 15. 15T ' 23 .  P L A T E THICK. T IN-  2'  -4-  3 4  4 . 4.  6  4.  COORO.IN A T E i(r> FT.  0.  '  o» 13. 14. 9. 9.  8 6 0! Ti z\  1  i  fI S 9  $ JOB S-TIME SIBFTC  17094  PAYNE THESIS "10 FPAIII DECK WRITE ( 6 , 6 1 ) -9 9 CONTINUE ' FPAIII C .A.PAYN.E " . C C FOLDED P L A T E A N A L Y S I S D I M E N S I O N J N ( 105 ) , N D ( 1 0 5 , 4 ) , NP( 90.'8) D I M E N S I O N NN22(.20 ) »ELAST (20 ) » J N L L ( 9 0 ) , J N U U ( 90 ) D I M E N S I O N D2( 9 0 ) ' D I M E N S I O N . S C K 9 0 , 3 ) , SC2 ( 90»3)» S C 3 ( . 9 0 , 3 ) , S C 4 ( 9 0 , 3 ) , S Y ( 3 ) D I M E N S I O N SIY(3)»'CY(3) »CIY(3)» WTT (' 90 ) »CAS ( . 90 , 3 ) DOUBLE P R E C I S I O N X ( 10 5 ) > Y'( 105 ) , FF ( 4 ) , XMM ( 90),YMM(' 9 0 ) , T T ( _9_0.)_' DOUBLE "PRECISION E , V , A L » E M U » E M U 4 , E M U 7 , T , E A L DOUBLE P R E C I S I ON DM » EL » EM » ALPHA » EALR »SINH »COSH,CSCH » S ECH » ACSCH DOUBLE P R E C I S I O N A S E C H , V S I N H , V C O S H , C l , C 2 ,C3>C4,C5,C6> C7»C8 »C9 t C 1 0 DOUBLE P R E C I SI ON C 11» C 12 »C 13 ».C 14'» C 15-» C l 6 » ELAM1 » ELAM2 * S 1 1 » S 4 1 » S 5 1 DOUBLE P R E C I S I O N S 8 1 * S 4 4 » S 5 4 » S 8 4 » S 5 5 » S 8 5 » S 8 8 , S 2 2 » S 3 2 » S 6 2 » S 7 2 » S 3 3 DOUBLE PR.ECLSi.O.N_.S..63„rS73-,.SA6_,_S7_6_>_S7l7_>_S^ DOUBLE P R E C I S I O N • S M M ( 9 0 » 3 6 ) , D ( 3 1 5 ) , S(2345)»PI,PI 2 , P I 3,AR DOUBLE P R E C I S I O N D E X P » D S Q R T , X M » Y M , F P N ( 9 0 ) > F P S ( 9 0 ) DOUBLE P R E C I S I O N A L P H A A ( 9 0 ) > S I N H H ( 9p'),COSHH( 9 0 ) > C S C H H ( 9 0 ) DOUBLE P R E C I S I O N DEL ( 8 ) , DEM ( 8 ) » XN ( 3 ) » F ( 8.V» FPM ( 90)»FPV( 9 0 ) DOUBLE P R E C I S I O N ACSCHH( 90)»VCOSHH( 9 0 ) , F F L ( 4 ) >FFU ( 4 ). FORMAT • ( I X » I 9 » 5 I 10) 2 FORMAT ( F 1 0 . 1 »5F10.3 )• 4 'FORMAT ( 1X» I9»4I10»2F10..6) 11 ' FORMAT ( 1 X , 2 5 H J 0 I N T DLSP IN WRONG ORDER) 13 F 0 R M A T ( 1 X » I 9 >110»I 10»I 1 0 > 1 1 0 » F l 0 . 3 » F 1 0 . 1 ) 15 ' FORMAT ( I X » 3 9 H J O I NT NUMBERS IN WRONG ..ORDER ONMEMBER 1 4 ) FORMAT( 1X» I 5 »4X'» 4 F 1 0 . 3 ) • -' 12 36 S T R E S S FROM FORCE I2-..11H • AT J O I N T 1 4 , 43 F O R M A T ( 1 X . 3 3 H . Ji 115H N O T ' I N C L U D E D . ) '• 10 '• F 0 R M A T ( 1 X » I 9 »I10»E10.''3) 9 51 •60FORMAT ( I X » 25HORDERREVERSED BY MACHINE) 8 61 FORMAT(1H1) 7 7 1 F O R M A T ( I X ? 1 0 H T H E R E ARE I5»21H' '-DEGREES OF FREEDOM.) 5 72 FORMAT ( 1 X , 2 4 H T H E TOTAL BAND WIDTH I S 15) 5 JOINTS,15,9H PLATES ) FORMAT ( I X » 13HSTRUCTURE NO. I 3 » 6H - - HAS I'5',9H 4 73 I5,18H LOAD CONDI FORMAT ( I X » 3'HAND I 5 ? 27H - E L A S T I C C O N S T R A I N TWITH S 3 74  8 6  joy n  jzi 1"A{  . __.jxmN.Sjj_.. 75 76 77  _ _...._.  F O R M A T ( I X ? 70H JO IMT NUMBER IX COORD Y COORD) - FORMAT ( 1X.67H . PLATE 1CKNESS ) F O R M A T ( 1 X » 58H NUMBER  X  1  DISP  LOWER JOINT  Y DISP  ROTATION  Z DISP  UPPER  LOWER  UPPER  JOINT  TYPE  TYPE  THI I  78 F O R M A T ( I X ? 38H JOINT TYPE E L A S T I C CONSTRAINT) 7 9' F O R M A T ( 1 X ? 4 9 H J 0 I N T NO.' X FORCE Y FORCE MOMENT Z FORCE) FORMAT ( 1 X > 1 5 H L 0 A D C O N D I T I O N 14,6H HAS I 4 , 1 6 H LOADED J O I N T S . ) 80 90 FORMATdX?/) 91 F O R M A T ( I X » 2 8 H T H E MODULUS OF E L A S T I C I T Y I S F 1 1 . 1 , 4 H P S I ) 98 FORMAT ( I X ? 30HTQ.O. MANY DEGR EES, OF R EJJLDjOM,._j3 YJ A ) — - _ . . . . 104 FORMAT ( IX » 1 1 H O V E R L A P OF I 6 » 1 9 H I N S T I F F N E S S MATRIX) .213 F O R M A T ( 1 X » 3 0 H S T R U C T U R E IS U N S T A B L E AT D I S P 14) _4D0 FORMAT ( 1 X, 1 5HI PAD C O N D I T I O N 14.6H HAS I 4 . 1 6 H L.O.A.D.E.D_P_LAXE.S.._»./_L 401 F O R M A T ( I X ? 3 0 H P L A T E NO. HOR LOAD VERT LOAD*/) 403 ,FORMAT(IX»/,15H PLATE ' STRESSES?/) _tQA_._FO.Ri1 AT.LlXt_.it8J1MJEMJ3JE.R.__J0LNT NO. ...UONGI.T.. ST.RESS. ,NX_, .. . ...).. 405 FORMATdX, 48HNUMBER JL JU AT J N L AT C L PL AT J N L ) 411 FORMAT(IX?I4?I8?I6?3F10.2) 415 FORMAT ( 1 X ? 8 F 1 5 . 5 ) __ 416 FORMAT(IX?1P8E15.7) 505 FORMAT < 1 X ? 1 7 H J 0 I N T D E F L E C T I O N S ) 506 FORMAT ( I X ?98H JO, INT_ J_ UM B E , X „ D E _ F L ECT.1Q N„... .1... Q L L F L E C J J ON. ROTAT 10 IN Z DEFLECTION FINAL X F I N A L Y)  516 518 519  .>ORMAt(lX»I7»8XtlPE10.3»3E15'.3iOP2F15.3)  FORMAT(IX>12HPLATE FORCES) FORMAT(IX?131HMEMBER J O I N T NO.. , A X I A L FORCE NORMAL SHEAR 1 MOMENT ' L 0 N 6 I T . SHEAR A X I A L FORCE NORMAL SHEAR MOMENT 12___ 2 L O N G I T . SHEAR ) „ „ . _ . _ . ______ i i 520 FORMAT(IX?131HNUMBER JL JU AT J L AT J L 10 1 AT J L .AT J L AT J U AT J U ' AT J U 2 AT J U ) 8 535 F0RMAT(1X?I4?4X? I 4?2X?I 4?IX?1P8E14.3) 7 536 FORMAT ( I X ? 3 7 H S T R E S S A N A L Y S I S FOR S T R U C T U R E NUMBER 14) 5 6 00 FORMAT ( 1 X »_1 7 HP 0,1 S_SON__F_ A T.I Q .IS .__». F5...3J „_ 5 601 . F O R M A T ( I X ? 2 3 H L E N G T H OF S T R U C T U R E IS ? F l 0 . 3 , 4 H F T . ) 4 95 READ 1 ? N R S ? N R J ? N R M ? N R E ? N L C 3 PRINT 7 3 ? N R S ? N R J ? N R M • .  G  6 'OLi Ti  JP.RUM.L -9JL... PRINT READ PRINT PRINT PRINT  74,NRE,NLC 2»E,V,AL 90 91»E 600, V  . ERi_N .T_ ^ j P _ l _ t A L _ „ EMU=(3.D0-V)/(1.+V) EMU4=(2.*V)/(1,+V) EMU7=(3.D0+V)/(l.+V) E = E * 144.DO PRIN T 90 P R I N T~7'¥"" " DO 5 I = 1 ,N R J READ 4,JN( I ) » N D ( I » 1 ) » N D ( I 2 ) > ND ( 1 , 3 ) > ND ( 1 , 4 ) i X ( I ) , Y( I ) PRIN T 4,JN ( I ) , N D ( I , 1 ) , N D ( I , 2 ) , N D ( I , 3 ) , N D ( I , 4 ) , X ( I ) , Y ( I ) A S S I GN D I S P L A C E M E N T NUMBERS _PJ2«5, I F ( N D ( I , J ) •1) ND ( I. J ) = N NU = N U + l  9 5  .1 12  10 16_  11 ~12  10 C 3 7  97  5~96~ 4 3  5,7  GO T 0 5 NUM = N D ( I , J ) I F J N UM-I_)„.9,_lp,10 ND( I V J ) = N D ( N U M , J ) CONT INUE GO T 0 1 2 PR I N T 1 1 P R I N T 1 , JN ( I ) G O T 0 9 5_  "NU='N U-T N U I S NUMB E R N 3 3 = NU-3 1 5  OF  .1 F ( N3 3 ) - 9 , 69 6 , 9 7 PRIN T 98,N J 3 0 _T 0 9 5 _ 3 3 P R I N T Tl",N U PRIN T 90 WRIT E ( 6 , 6 1 )  DISPLACEMENTS  s  14  C 17 18 19  MEMBER DATA .... _ . „_ NB = 0 PRINT 76 PRINT 77 DO 3 K=1»NRM READ 13, MN,JNL,JNU,KL,KU,T PR I NT„, 1.3^, M N,, JN L , J NU.,_K L. ,,K U ,. T. T=T/12.0DO IF(JNU-JNL) 14,14,17. PRINT 15,MN PRINT 60 NUM = JNL _J.NL = . J.N.U.. . JNU = NUM NUM = KL _XI ^_K.U KU '= NUM .CALCULATE POSITION NUMBER NP ONE TO EIGHT HO 18 1 = 1 »iL,_._.:. - .„ NP(K,I)=ND(JNL,I) DO 19 1=5,8 NP ( K , I ) = ND ( JNU, 1-4 ) . . FIND BAND WIDTH NB=(TOTAL WIDTH + ONE)/TWO MAX =0 DO -23 1 = 1,8 IF(NP(K,I ) ) 23,23 ,31 I F(NP(K,I )-MAX) 20,20, 21 MAX= .NP ( K , I ) IF(NP(K,I)-MIN) 22,23,23  _MJLH_-=_51.PJX.^J„L ho 9 8 25 i7 24 6 , 5  CONTINUE NB1=MAX-MIN . IF(NB1-NB) 24,24,25 NB=NB1 XM=X(JNU)-X(JNL) J_M=XLJ NUJ.lY-i J-N,L J , _ _ _ _ _ _ DM = DSQRT ( ( XM*XM ) + ( YM*YM') ) EL=XM/DM EM = YM/DM .  '.till  9 Li  .AR=AL^.D^ P1=3.1415926536 PI2=PI*PI  12  _  •1 .0 9  C  P I3= P I2*P I ALPHA=PI /(2.*AR) EAL=DEXP(ALPHA) EA L R = l_. /_EA.L ,... ... 'S I N H = . 5 * ( EAL-EALR ) COSH=.5*(EAL+EALR) CSCH=1./SINH __ 5ECH=l./COSH ACSCH=ALPHA*CSCH A$E£h=kLP„dA*S_£r_JH ,_..., ., VSINH=(3.DO-V)*SINH/(l.+V) . VCOSH=(3.D0-V)*COSH/{l.+V) C l = PT *(COSH/(ASECH+SINH)-SINH/(ACSCH-COSH)) C2=-PI *(COSH/(ASECH + SINH)+SINH/(ACSCH-COSH) ) *(COSH/(ACSCH+COSH)-SINH/(ASECH-SINH)-(l.-V) C5 = P I 2 1 \- __. C_5_L-P.I_2.__,. , _» (. C 0S H / ( A CSC H+C0_SH_),+ SXNH/_.(_/_ALQl-^ C3=C5/AL C4=C6/AL C7 = P I 3 *(SINH/(ACSCH+COSH)-COSH/(ASECH-SINH))/AL C8 = - P I 3 #(SINH/(ACSCH+COSH)+COSH/(ASECH-SINH) )/AL C9 = PI *(-COSH/(ASECH-VSINH)+51NH/(ACSCH+VCOSH)) C_P.=„-P I_____ * ( . COSH /J A S_ ECH - V S I NH H_S I. N H_/ J_ A C SCH +V_C 0 S H ) L -___-_.. C l l = PI * ( COSH/(ACSCH-VCOSH)-SINH/(ASECH+VSINH)+(l.+V)) C12=-PI * ( COSH/(ACSCH-VCOSH)+SINH/(ASECH+VSINH)) C13=C11 C14=C12 *(-SINH/(ACSCH-VCOSH J+COSH/(ASECH+VSINH) ) C15= P I *( SINH/(ACSCH-VCOSH)+COSH/(ASECH+VSINH)) C16= PI ELAM1=E*T**3/(12.D0*(l.-V**2)*AL) ELAM2=E*T/((1.+V)**2*AL) FORM MEMBER S T I F F N E S S ELEMENTS IN MEMBER COORDINATES S l l = E LAM2 #C 9 S41= ELAM2*C13 _S5_1 = -ELAM2*C1_0 S81=-TLAM2^C14  S44= S54=  ELAM2*C15 ELAM2*C12  8  6  OH  IT  z\  - - i i  .S. UI U V O O  2 t'i tCl < Cf  UJ  Q  QL  o o u UJ  a: u Z>  i—  CM LO r\j r\i  = !  r-  ! i  LU  i  _l  LO LO LU I—I  *  vO| O CO in rH u_ r- co oo <r rH v£) CM r-U CO U. U U_ UIU u u u u u.; u u u U j|t >j<,; jjc i—i * * *j * * 1  LO  CM  * LU  * *  _J LU +  5: _J  LU LU  I  rH  *| s  LO  *  CM CM LO * CM  * 51*  LO  LO  +1 ^ i-H rH' LO CM CO*  CM  vO vO  * z LO' * s: LU CV LU *l * -1 - S* LU  i,  < \  CM  CM  + I  rH rH  CO vh  00  LO  m m c\ rH >Jc CM! rH LO LO co CM cn <(-  * *  LO  CM  CM  LO  LO  * *| s: CM  LU 0,1 * _J _J LU LU' ~L, T  I  T- |  +  m co! LO  *I  O CO CvO LO CO  rH r- oo  CM 5: CM LO * LO! LO LU ' LO CO * * • ! * *  * l * * CO o * co o <H * * LU _ l r- • <t;-i s: Cm *! * * * f—I rH r—ij rH rH UJ LU LO * l * LUUJiUJUJUJLULULOO LU LU CO O CO, LU UJ O * l _J LU LU LU LU l| CMJ r \ j c\] c \ j I < < < < < < < S S 2 2 21 _J - J _l or — w III II- II II 111 II II llj II II III II II II II II —' —' —' < ) < < < _J UJ LU LU LU LU UJ UJ LU < < UJ UJ UJ UJ UJ LU I CO O O l H IM CO <f m vn r— co O o >-H CM I co <r m I': . rH; in vO III CIIM CO •—i II II ' II II II II II II) II II II II <f(m m co' in oo LOi LO  LO  CM, CM co co co vO' r- co LO io) co LO LO  UJ  CM  I  LO  LO  LOl  LO  Si  CM  LU  CM  H  LO  i—I  H  CO  CO  LO. LO  LO  51 s:5:si25s:s: LOsJ CO5:LO s: CO  CO  CO  co col co  LO  CM  CM  CMJ CM  CM  CM  5 s s ; 3 5 : i 5 :  LO  LO LO! CO  LO  CO] CO  CO  LO  N  n  ^  n  o LL.  U  cn o  _S.MJ„2.6.)_s-_S.8_k. ,._ . SM(27)= EL#*2*S55+EM**2*S66 SM(28)= EL*EM*S55-EL*EM*S66 SM(29)=-EM*S76 SM.( 3 0 ) = E L * S 8 5 SM(31)= EM**2*S55+EL**2*S66 £N&AZlsJE.lJ*£2&-. S M ( 3 3 ) = EM*S85 SM(34)= S77 S M ( 3 5 ) = 0.0 SM(36)= S88 TANH=SINH/COSH ,_CQIH^.03Jd/ALN.H_. AKT=ALPHA*TANH. AKC=ALPHA*COTH AMDA5=1./(ALPHA*SECH-EMU*SINH) AMDA6= 1./(ALPHA*CSCH+EMU*COSH) AMDA7=1./(ALPHA*CSCH-EMU*COSH) -&&QA&=LaJLL A L £ J i A * S E,CH,+E.M U_*.S.I.IilJ_L L__ SY(1)=SINH SY(2)=0. SY(3)=-SINH SIY(1)=ALPHA*SINH SIY(2)=0. SJX< 3 L=S_LYJ. 1.) ^ CY ( i ) =CO~SH CY(2)=1 . CY(3)=COSH CIY(1)=ALPHA*COSH CIY(2)=0.  SJU JL ~ 1 J DO 4 i 2 " ' I = l % 3 " 3  412  =  C  Y  1]  „  „  ' "" ~ "~ " ~~ ~ SC1(K . I )=AMDA5*(SIY( I) + (EMU4-AKT)*CY( I ) ) S C 2 ( K » I )= A M D A 6 * ( C I Y ( I ) + (EMU4-AKC >*SY(I ) ) SC3(K» I ) = A M D A 7 * ( S I Y ( I ) + ( E M U 7 - A K C ) * C Y ( I ) ) S C 4 ( K >I ) = A M D A 8 * ( C I Y ( I ) + ( E M U 7 - A K T ) * S Y ( I ) ) D2JJC) = E * T * 0 . 5/( l . + V ) _ _ ACA=A"LPHA*C6f H-2 . *V/( i ' . " + V ) " ' ACC = A L P H A * C S C H + (.3 .-V ) *COSH/ ( l . + V ) DO 4 1 3 1=1.3  s ' &  •  [  ;  •  ;  S S  L 413 28  C A S ( K t I ) = ( C I Y ( J ) - A C A * S Y ( I ) )/ACC J N L L ( K ) = JNL ' ' J N U U ( K ) = JNU XMM(K) = XM : YMM(K) = YM DMM(K) = DM _ ALPHAAJ. K.L=ALPHA..„ . " ' SINHH(K ) =SINH C O S H H ( K ) = COSH C S C H H ( K ) =CSCH TT ( K) =T ACSCHH(K)=ACSCH  _ ._...  •  .  '. —  c  5 0! :» Zi ! f  :  FPM(K)=0. FPV(K)=0. FPN(K)=Q. FPS(K)=0. DO 3 5 1 = 1 , 3 6  _  SMM(• K, ij.=j>mxx.,„. -, . CONTINUE IF(NRE) 53,53,52 _5_2 P.BJLNJ__a.O P R I N T 78 DO 5 4 1 = 1 * N R E R FAD 51 , N 2 2 , N ? 3 , F I A S TJ XJ.,, PRINT.51»N22,N23»ELAST(I) 54 NN22(I)= ND(N22,N23) m. 5 3 PRINT 90 NB=NB+1 MIKE = ( 2 * N B ) - 1 PRINT 7 2 , MIKE 11 PRINT 90 10 C • F I L L IN STRUCTURE S T I F F N E S S 9 . NV = ( NU*NB ) +NU + NB-2 0 N0=NV-2345 7 IF(NO) 102,102,103 5 _1J!3: PRINT 104,NO 5 GO TO 9 9 4 102 MS = NU*NB 3 DO 10 5 1=1 ,MS ' 35  3  V  _  1 2  '  .  .  MATRIX  .  .  ;  8.  £  1  £ /.I  _1.03_  117  112  150 151  .J.I.3.. 114 115 107 "l2 2 123 121 C  ,1 2 16  3  .__„o...a  J 4 = ( ( NB+ 1 )*NU) + 1 J 5 = J4+NB -3 DO 1 17 -I=J 4 , J 5 S( I ) = 0.0 NB1 = NB-1 0.7 ..Lai..'N.R.M..... . DO 1 07 J = l »8 I F ( PN ( L » J) 1)0 7 »107 »112 J l = ( J - l ) *1(6 - J ) /2 DO 1 07 I = J ) I F ( N P ( L . I )107»107»106 PJ. L__J.A -NPJ_L.* IJ.) 113.».l 50 ,.1 14.. I F( I- J ) 15 1.113,151 I )-1 )*NB1 + NP.( L» J ) K= ( P(L» N IJ1 = I+Jl S ( K)= S ( K+)( 2 . 0 * S M M ( L * I J 1 ) ) GO T 0 107 K.= (NP( L».J.) - 1 . ) * N B..1 +APJ.L ,,I ) . „ GO T 0 1 1 5 ) )*NB1+NP<L,J) K= ( PN ( L * I-1 I + J l IJ1 = S ( K ) =S ( K ) +SMM ( L , U l ) CONTINUE I F ( N R E J 121 • _1_2 2_2 b o " l 2 3 " " i = l »NRE K=((NN22(I ).-l)*NB)+l S(K) = S(K )+(ELAST(I)*12.0) CONT I NUE SOLV E KASE = 1 L1=N B + l K3 = N B - l L2= ( NB#NU) -K3 NBS = K 3 * K 3 NB2 = NB-2 L = 1 TF'C'S T l ) )"2 1 2,2 1 2 , 2 1 6 S ( 1=DSQRT ) (S(1)) DO 2 02 1=2 ,NB  6  ; o.(  IT  =3JAJJ-Si J . J „ - „ DO 2 0 3 L = L 1 » L 2 » N B K1=L-NBS K 1 1 = K1 K2=L-K3 IF(K1)204,204,205  _2JJ_2,  _SJJL)  204 205 2 06  K l = ( ( L - l )./NB)+l DO 2 0 6 K = K 1 » K 2 . K 3 S(L)=S(L)-S(K)*S(K) IF(S(L))212.?12.207 S(L)=DSQRT (S(L ) ) DO 2 0 8 N = 1 , N B 2  207  Ml=Kll+(K3*N) I F (Ml") 2 0 9 , 2 0 9 . 2 1 0 M1=K1 I N = l +N DO 2 1 1 M = M 1 , K 2 , K 3 MN=M+N M l N ) = M l Ml^(.SJ-MJ.*S,(.M.M.) S(LN)= S(LN)/S(L) S(LN+1)=S(LN+1)/S(L) G.0_L0_2-L4 PRINT 2 1 3 . L • . GO TO 9 9 CONTINUE BACK S U B S T I T U T E ONE J2 = (NB*NU)+1 -J3=( NR + 1 ) * N U  .209 _2_L0  ? 11 208 203 212 214 C 310  1 2  li 10 9 8 7 S 5 4  38 '  WRITE(6,61) PRINT 536.NRS DO 5 0 7 K A S E = 1 .Nl C DO 3 8 L = 1 , N U FK(L)=0.0 R FAD l . N J I PRINT 90 PRINT 8 0 , K A S E , N J L P R I N T _90 IF(NJL.EQ.O) GO TO 4 1 4 PRINT 79  _„  ._,„  '  ....., ,  ;  ) , _ _ _ „ ,  _ ....  :  • £  ! n  Js  1 s i £i  40  JkZ. 41 39 414  -RE.AD.._3.6j J.N U . . F F (.l..).,.FE JLZ.L.. FF.( 3 )..,.F F ( , 4 ) . P R I N T 3 6 ' . J N U , F F ( 1 ) , F F ( 2 ) , FF ( 3 ) » F F ( 4 ) DO 3 9 L = l » 4 J>N.D_LJ_NU_»_L. ) I F ( I ) 4 0 , 40 , 4 1 IF(FF(L ) ) 42,39,42  i iir ~Zl  J?-R I.NT_43.,_I_ .?.JNU,_ . 6 0 TO 3 9 FK( I ) =FK( I ) + F F ( L ) CONTINUE READ 1 , NP L  PRINT 90 _ „ , ™ „ P _ R LNX..4.0-Q,, K.A S.E„,..NJB-I DO 4 1 8 1=1 »NRM FPM( I = 0 .  _EJEV_LL = H . _ FPS(I FPN( I  „4ia_™W,II,iX  I 12.  Jl 10  9  8 7  5. 5  4 3  =0. =0.  I F ( N P L . E Q . O ) GO TO 37 PRINT 401 DO 4 0 2 1 = 1 , N P L READ 3 6 , M N ,WH ,WV P R I N T 3 6 , M N.WH.WV J N L = J N L L (.MH>___.„ JNU=JNUU(M N ) XM = XMM ( MN.) YM=YMM(MN) DM=DMM ( MN) E L = X M / DM EM = YJ^/ DM ~ALPHA = A T P H A A ( M N ) S I N H = S INHH (MN ) C 0 S H = C OSHH ( MN ) C S C H = C S C H H (MN) A C S C H = A C S C HH(MN ) V C O S H = V C O S HH(MN) T = TT ( M N )""'"" WVT=WV *XM WHT=WH *YM  11  i  S  ^2yij&z.m/J.*E.EM.., ...,______._.___.„_,___.. TWT=-WVT*EM-WHT*EL WN=TWN /DM WT = TWT/DM , ____ • P R I N T 4 1 5 > WN ,WT WTT(MN)=WT ' •• __JLPJ^-U_ Ni,_f_LWN*,A LJ.*_2 _/_£I_2 JJM.JjufJ^_0SH / _ ( A C S C H + C O S b j J ™ L • ___________ FPV(MN)=(2.*WN*AL/PI)*SINH/(ACSCH+COSH) F P N ( MN ) = ( . 4 . # A L * W T * S I N H ) / ( ( l . + V ) * P I * ( A C S C H + VCO'SH ) ) FPS(MN)=(AL*WT/( (1.+V)*PI) )*(4.*C0SH/(ACSCH+VCDSH)-( l . + V ) ) FFL(1)=-FPN(MN)*EL+FPV(MN)*EM F F L ( 2 ) = - F P N ( MN ) * E M - F P V ( MN ) * E L  _,.._,_  .  . 420 . 402 '37 m  T * 311 52 Ji 10 9 8 7 5. 3 0 4 5 4 303 3  JL =ND(JNL»'L) JU=ND(JNU,L) I F ( J L . E Q . O ) GO TO 4 2 0 F K(JL)=FK(JL)+FFL(L) I F ( J U . E Q . O ) GO TO 4 0 2 FK(JU)=FK(JU)+FFU(L) CONTINUE PRINT 90 DO 3 11 1 = 1 . N U : KK = J 2 + I - 1 S(KK)=FK(I) MJ.2J^S±J2JLL5±XJ__^m. ' DO 3 0 2 L = L 1 . L 2 » N B J 1 = J 2 + ( ( L - l ) /NB ) J =J 1 - K 3 K1=L-NBS • IF(K1) 304.304.303 K l = l + ( ( L - l ) /NB ) _._,„_ J =J2 K2=L-K3 DO 3 0 1 K = K 1 » K 2 » K 3  ...  ;  _i  ___ _  „  «  _  , ' 'p j£  ___„_-___„.____™ '  ;  .  .  '  _.____._  ,  :  -  1  ,  '  \  ,  .  ;  J  .  .  ,  \  u  Z l  :  ___________  C  !__ ,  ' :  _ J  6  :  FFL(4')=FPsfMN) FFU(1) =F F L ( 1 ) EFJJJ.2J__.EELJ.2J FFU(3)= FPM(MN) FFU(4)=-FPS(MN)  __,:.,.._  \  12  1  :  1  £  £  /.I  .SJ.J1.) =S.( J l ) - S ( K ) * S J J L - . 301 J = J+1 302 S(Jl)= S(Jl)VS(L) C B A C K S U B S T I T U T E TWO L3=L2+1 S(J3) =S(J3)/S(L2) .DCL.3.0.8 ...LR__Li_xL2_» N.B L= L 3 - L R Jl= J2+((L-l)/NB) J =J1+1 K1=L+1 K2=L+K3 . J 3 . 0 _ 3 - P J ? „ . K =J<; . 1 , X 2 S ( J l ' ) =S ( J l ) - S ( K ) * S ( J ) 309 J =J +1 S ( J 1 ) = S ( J 1 ) / ,S ( L ) _3_0_8_ DO 3 1 2 1 = 1 . N U KK=J2+I-1 PRINT 90 PRINT 505 PRINT 90 PRINT 506 DO 6 0 4 K = 1 , N R J .N2 = NDX K » 1 J N3 = ND~ ( K > 2 ) N 4 = ND ( K » 3 ) N 5 = ND ( K » 4 ) IF (N2) 508.509.50J 508 DX=D(N2) 12 XF = X(K)+DX ii ' GO TO 5 1 0 iO 5 0 9 DX=0.0 9 . XF = X ( K ) IF(N3) 511.512.511 8 510 •/ 5 1 1 DY=D(N3) _YF_=Y( K H - D Y GO TO" 5 1 3 ' " " DY=0.0 4 512 YF=Y(K) 3  —I  8  6 OH  IT ZL  13  'j g L  -5JL.35 14 5 15 507 602  r  I F .(.N4J_5J.J4_,3-L5,I.5.1_4, ROT=D(N4) GO TO 5 0 7  6  Ol|  'IT ~:z:  ROT=0.0  2 ,603,602  ,K,DX,DY,ROT,DZ,XF,YF PRIN T 518 PRIN T 90  _EBJ,N.  605  _5_2.2_ 523 52 _5_2it_ J i 525 ho 5 2 6 9 8 527 > 7 528 5 ,5 2 9 5  530  PRIN T 520 BACK S U B S T I T U T E THREE _D_0_5_ 2_1_K=. 1_,J\|.R.M_ DO 6 0 5 I = 1 , 8 F ( I.) = 0 . 0 .N.L5.N. .P_LK.».l,J L — , N2 = NP ( K , 2 ) N3 = N P ( K , 3 ) PJ-K-tA ) N5 = N P ( K , 5 ) N6 = N P ( K » 6 ) N8 = N P ( K , 8 DO 5 2 2 1 = 1 , 3 6 J S L M J J . .LHLS.MM L K . , _ L L I F ( N 1 ) 523 , 5 2 5 , 5 2 3 DO 5 2 4 1 = 1 , 8 F ( I ) = £J&LI J J t Q t fcLU." I F ( N 2 ) 526 , 5 2 8 , 5 2 6 F ( 1 )= F ( 1 )+ S M ( 2 ) * D ( N 2 DO 5 2 7 1 =2 , 8 F ( I ) =F( I ) +SM( 1 + 7 ) * D ( N 2 ) IF(N3) 529,531,529 F ( 1 ) = F ( 1 ) + S M ( 3 )*D (_N 3 i . _ „ . F ( 2 ) = F ( 2 ) +SM ( 1 0 ) *D~( N 3 ) DO 5 3 0 1 = 3 , 8 F(I)=F(I)+SM(I+13)*D(N3)  14  _5-3J„..—LE .(_N 4-L_5.3.2.».5...3A., 5.3.2 532  533 534 606  F(1)=F<1)+SM(4)*D(N4) F(2)=F(2)+SM(11)*D(N4) F(3)=F(3)+SM(17)*D(N4) D O 533 1 = 4 , 8 F ( I )=F( I ) + S M ( 1 + 1 8 ) * D ( N 4 )  J_FJ_N5,L_ 6_0J3.» J =- 3 DO  607  6 0 9 . , 6.06  1 = 1 , 4  J = J + 9 - I 607  I)+SM(J)*D(N5) 1=5,8 £ n _ L = F J . - L L ± S M J .L±221 * B I N 5J„ IF(N6) 610,613,610 F(  I )=F(  DO  _6_0 8_ 609 610  608  J = - 2  _ D _ 0 _ 6 _ U _ 1=1,5 J = J + 9 - I 611  F ( I )= F ( I.)+SM(  J ) * D ( N 6 )  612 613 614  F ( I } =F ( I ) + S M ( 1 + 2 5 ) * D ( N 6 )  IF(N7 ) J = - l  6 1 4 , 6 1 6 , 6 1 4  DO  1 = 1 , 6  6 1 5  J = J + 9 - I 615  _ F J J J_= F X J J J L S M J , J ) * D . ( . N 7 _ ) _ . F  (T) = F7 7 )  + SM (  34  j"*6 ( N 7 )  F ( 8 ) = F ( 8 ) + S M ( 3 5 ) * D ( N 7 ) 616  I F ( N 8 )  6 1 7 , 6 1 9 , 6 1 7  617  J=0 D O 618  1=1,7  12,  a  i i 618  10 9 619 8 7 S 5  J=J+9-I F'.( I ) = F ( F ( 8 ) = F (  AX I A L L =  T7+TMTJT*D TH8 ) 8)+SM(36)*D(N8) (F(1)*XMM(K)  AXIALU= ( F ( 5 ) * X M M ( K ) PLSHL = - ( F ( 1 ) * Y M M ( K ) PL5HU = . ( F ( 6 | * X M M ( K ) BML = F ( 3 ) " ' " BMU= F ( 7 ) ZSH'L = F ( 4 )  + F ( 2 ) * Y M M ( K ) ) / D M M ( K ) + F ( 6 ) * Y M M ( K ) ) / D M M ( K ) - F ( 2 ) * X M M ( K ) ) / D M M ( K ) - F ( 5 ) * Y M M ( K ) ) / D M M ( K )  15  f S  " s  ' c  521  __ _ _ _ „ 407  Iv  12  ii liO 9 8 '/  8 , 5 4 3  410 406  YoT  A X I A L L = A X I A L L + F P N '( K ) AX I A L U = A X I A L U + F P N ( K ) PLSHL=PLSHL+FPV(K) P L S H U = P L S H U + F.PV(K) BML=BML+FP M ( K ) JMUr-B.MU-FP. ZSHL=ZSHL- FPS(K) ZSHU=ZSHU+ F P S ( K ) • PRINT 5 3 5 . K > J N L L ( K ) >JNUU(K) »AX I A L L » P L S H L . B M L . Z S H L .AX I ALU,PLSHU•B .MU. ZSHU PRINT 403 _P_RI NT ,40.4, PRINT 405 DO 4 0 6 K = l » N R M DO 4 0 7 1 = 1 .,_8 DEL( I ) =0. N = NP< K » I ) JJLLN...N E.. OJ. D E L ( I ).= D ( N ). CONTINUE E L = X M M ( K ) / DM M( K ) EM = Y M M ( K ) / DM M ( K ) DEM(1)= D E L ( 1 ) * E L + D.EL(2)*EM DEM(2)=-DEL(1)*EM+DEL(2)*EL _D,E.MX3 )..=... D E L . (.3.) DEM ( 4 ) = D E L ( 4 ) DEM(5)= DEL(5.)*EL + DEL(6)*EM DEM(6)=-DEL(5)*EM+DEL(6)*EL DEM ( 7 )= DE L ( 7 ) DEM ( 8 ) = DE L ( 0^1=1..v3 X N ( I ) = - ( D2 ( K ) * P I / AL.) * < DEM( 1 > * ( - S C I ( KI , ) + S C 2 ( K » I ) ) + D E M ( 5 ) * ( S C 1 ( K , I ) . + S C 2 ( K . I ) ) + D E M ( 4 ) * ( S C 3 ( K , I ) - S C 4 { K , I ) ) + DEM ( 8 ) * ( S C 3 ( K , I ) + S C 4 ( K , I ) ) ) X N F = - 2 . * A L *W T T ( K ) * C A S ( K » I ) / P I XN ( I ) = X N ( I ) + X N F P R I N T 411» K >J N L L ( K ) , J N U U ( K ) » X N ( 1 ) . X N ( 2 ) » X N ( 3 ) W.R.I TE. 1.6x61 ) C O N T I NUE GO TO 9 9 STOP  " 6 Ol|  i TT " zi  .  T  . _  I  8  DATA L I S T I N G  NO. OF LOADED JOINTS NJL  FOR  .. 5 A M P L E . . 5 T R U G T U R E SHEET 2-  MOMENT LONGIT. LOAD H0RI2. LOAD VERTICAL LOAD FF(0 — LB./FT. FF(2.) f LB./FT. FF(3) 3 FT.LB/fT FF(4)\ LB. /FT.  J O I N T NO. JNU i  -430 .  2 2 0 0. "'-32 0  LOAD CASE T  NO. OF LOADED PLATES NPL  NO. OF LOAOED JOINTS N J L  0  NO. OF LOADED PLATE'S NPL  LOAD CASE 2  I"  PLATE NO. MN 2  LL  7  5 _ S  3 ...  HORIZ. LOA.D VERTICAL LOAD ^LB/FT - WV"TW L6./FT. 2  2  20-.  -30 .  ^  , ,  6 0  T  "  r  ~ 1  STRUCTURE  AND  - 0  NO.162  HAS  6  ELASTIC  CONSTRAINTS  5  JOINTS,  2  WITH  PLATfcS  '  LOAD C O N D I T I O N S .  ^ I  THE MODULUS OF E L A S T I C I T Y POISSON RATIU I S 0.200 L E N G T H OF S T R U C T U R E I S  JOINT  NUMBER  X  DISP  1 2  1 1  3  I  5 "'6  0  THERE ARE  IS  I  21  I  D E G P E E S OF  3500000.0  PSI  .  *O.CO0 F T .  Y DISP I 1 1 1 0  ,  ROTATION L  1  1  I 1 1 I  Z DISP  l  X COORD  o.oooooo  1 2.000000 1 8.000000 1 15.000000 0 15.000000 I 23.000000  .  .  .  . •  I j \  Y COORD  o.nooooo .  8.000000 13.000000 14.000000 9.000000 9.000000  FRFEDOI'.  J  PLATE NUMBER 1 2_ 3 4 5.  V,  THE  TOTAL  BAND  LOWER JOINT. 1 2 3 A 5 WIDTH  I S  UPPER JOINT 2 2 A 5 6 H  LOWER UPPER THICKNESS TYPE TYPE INCHES 0 0 4.000 _0 ___ A.uOO -0 -0 4.COO -0 -0 6.000 r.O -0. 4.0.00_  STRESS  LOAD  ANALYSIS  CONDITION  J O I N T NO. I  LOAO  CONDITION  JOINT  NUMBER 1 2 — - - -• :  3  4 5 6  PLATE  .  PLATE MEM8ER NUMBER 1 2  5  — ^_  Y  2  FORCE -0.000 -320.OOC  HAS  X  D E F L E C T I CM 3.9I8E-01 3 . i 24E-02. -7.603E-03 -3.797E-C3 0.G00E-39 -0.00OE-39  J O I N T NO. JL JU 1 2 2 3 .  4 5  3  HAS  KUMBER  162  LOADED  j  •II  JOINTS.  MOMENT -480.000 -0.000  0 . L0ACF3  "  2  FORCE -0.000 -0.000  I'LAfE:.  y..REFi.F.cnc; -1.1236-01 r2.30r>c-02 2.7l«c-02 1.335E-05 0.0006-39 -2.2586-02  '  I  ROTATION 5.382E-02 ... 2 . 6 7 6 E - 0 2 -3.241E-03 -6.049E-05 6.645E-04 -4.5536-03  CEFLECTION 4 •268E-03 - 4 .08 1 6 - 0 3 2 .2156-03 -3 .3106-05 0 .0006-39 -0 .0006-39  FINAL X 0.392 2.034 7.992 14.996 15.000 23.000  F  i NAL  Y  -0.112 7. ) 7 7 13.027 14.000 9.000 8.977  FORCES  KEMRER NUMBER 1 2  4  1  STRUCTURE  DEFLECTIONS  JOINT V-"- -.-T.r,  3  1  X FORCE 2 2 0 C . COO - 0. 0 0 0  6  R  PGR  3  4 5  5 6  A X I A L FORCE AT J L 5.3366 02 4 . 8 7 7 6 03 -2.053E 02 -7.?.10t 02 Q.QOOE - 3 9  NORMAL S H E A R aT J L - 2 . 13':-E 0 3 9.8R66 0 2 9.5046 0 2 - L . 2 6 0 E 03 3.2666 02  MOMF.NT AT J L - 4 . 8 0 0 6 02 1.082c 04 2 . 3 8 3 E 03 - 3 . 3 5 7 E 03 2.203E 03  L O N G I T . SHEAR AT J L 3.6386-12 - 4 . 9 3 8 6 02 6 . 8 6 1 6 03 - 3 . 0 4 1 E 02 O.O00E-39  A X I A L FORCE AT J U -4.5186 0 ) - 3 . 2 4 3 E 02 1 . 1 3 7 E 03 4.385E 0? 0 . 0 0 0 6 --39  NORMAL S H E A R AT JU 2.03 4 6 03 - 9 . 1 6 6 6 02 - 9 . 5 1 3 6 02 1.2536 03 - 3 . 2 0 0 6 02  STRESSES J O I N T NO. . LCNGIT.. STRESS NX JL AT J N L AT CL P L AT J N L JU 56206.75 778.95 - 5 4 7 4 3 . 5 1 1 2 - 11574.34 29167.00 2 3 -54320.44 13438.82 -209.39 29272.93 3 -204.94 97.71 4 5 -499.00 O.QC _ 0.00 5 6. -.0.00 .  •  '  .'.  .  ,_•  . .  --  .  X  - -~-  . ..  SHCAK MDMhNT LO^ . 0 1 T . AT J U A r ju 4. >j)3c 0 2 . 0 8 2 E 04 - 2 .3386 03 - 6 . 8 6 1 6 03 3 . 3 5 7 6 03 3 . 0 4 1 6 02 7 . ^ 4 1 6 >;2 - 2 .2036 03 -0 . 0 0 0 6 - 3 9 - 0 . 0 0 0 6 - S~>  LOAD  CONDITION  LOAOEO JOINTS.  HAS  y LOAD C O N D I T I O N PLATE 2  NO.  HOR  2  HAS  I.QAC  VCKT LOAD  20.000 25.90164  -JO.000 4.91303  JOINT  DEFLECTIONS  JOINT  NUMBER  A  DEFLECTION  -l-.-4_9_-._3' "  1.634E-04 3.033E-04 2.101E-04... O.000E-39 -0.000E-39  PLATE MEHBER NUMBER  PLATE MEMBER NUMBER  1  2  1  LOADED  PLATES.  Y  DEFLECTION ROTATION 7.3S5E-05 -1.3216-04 -3-325E-04 -2.432E-04 -5.0216-04 1.726E-04 ; 6.735E-07 . ..-3 . 2 0 4 E - 0 5 .. 0.0006-39 -5.153E-05 -3.424E-04 -3.950E-05  1 -DEFLECTION 1 .1446-04 -3.628E-05 -7.S5SE-05 . 1.611E-05 0.000E-39 -0.0O0E-39  FINAL X  -0.001 " 2.000 8.000 ...15.000. 15.000 23.000  FINAL Y 0.000 a .ooo 12.999 14.000 9.0C J 9.000  FORCES JOINT  JL  NO. JU  A X I A L FORCE AT J L 2.3696-13 1.359E 02 1.920E 02 1.0346-01 0.0006-39  NORMAL. SHEAS AT J L -7.9626-13 9.874E 01 - 1 . 7 6 5 E 00 1.414E 0 1 -1.1246-01  STRESSES JOINT JL 1 2 3 4  5  NO. JU 2 3 4  •5 6  L C H G I T . S T R E S S NX AT J N L AT J N L AT CL PL 453.67 1509-69 -512.32 -1069.12 -711.03 -505.90 -397.09 209.75 -1075.03 164.75 26.13 318.30 0.00 -o.co . . 0.00  •IGMcMT AT J L -7. I 0 5 t - l 4 2.344E 01 4.593E 01 5.1376 01 -5.993E 00  LONG I T . SHEAR A X I A L FORC^ AT J L AT J U 8 . 5 2 7 E - 14 -1.679E 02 -3.035E 02 -1.625E 02 1.482E 02.__ -1.393E 01 3.7536 02 1.3066 02 0.000E-39 0.000E-39  NORMAL SHEA* A r JU 4.3945-01 1.023c 02 . 2.101c 00 -1.333= 01 3.553E-15  MOMENT AT JU  -2.3446  01  -4.5986  01  - 5 . 13 76_ 01 5 . 9 9 3 . 00 -0.0006-33  LOUSIT. SHEAR AT JU  3.0 35E 9 2 - 1 . 4 8 2 c 02 -3.753E 02 -3.0966 02 -0.000E-39  I  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0050609/manifest

Comment

Related Items