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Stability analysis of a spaceframe structure Oberti, Andrea Luca 1969

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STABILITY ANALYSIS OF A SPACEFRAME STRUCTURE  by  ANDREA LUCA OBERTI " L a u r e a " ( E l e c t r i c a l Eng.) P o l y t e c h n i c I n s t i t u t e o f T u r i n , I t a l y , 1965 B.A.Sc. ( C i v i l Eng.) The U n i v e r s i t y o f B r i t i s h Columbia, 1967  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED'SCIENCE i n the Department of CIVIL ENGINEERING  We a c c e p t t h i s t h e s i s as conforming t o the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1969  In presenting  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the  requirements  f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  and  study.  agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r purposes may  be  g r a n t e d by  I t i s understood that gain  s h a l l not  be  the Head of my  Department or by h i s  written  permission.  Andrea L.  Engineering  The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada  September,  1969  I  further  scholarly representatives.  c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l  a l l o w e d w i t h o u t my  Department of C i v i l  that  Columbia  Oberti  i ABSTRACT Two  t h e o r i e s f o r the s t a b i l i t y a n a l y s i s of a spaceframe s t r u c t u r e a r e  presented:  the f i r s t uses o n l y L i v e s l e y s t a b i l i t y  i n c l u d e s i n a d d i t i o n the e f f e c t s due  f u n c t i o n s , the  to chord r o t a t i o n and  second  flexural  end  s h o r t e n i n g of a member. The  critical  c o n d i t i o n i s d e f i n e d by the l o a d which makes the  tangent  s t i f f n e s s m a t r i x of the s t r u c t u r e become s i n g u l a r . Three methods f o r o b t a i n i n g the c r i t i c a l determinant The  p l o t , S o u t h w e l l p l o t , and  load are  load-deflection  curves.  a n a l y s i s i s c a r r i e d out f o r the p l e x i g l a s s model of an  c o n i c a l spaceframe, made of g l u l a m timber, and b u i l t The  presented:  overall c r i t i c a l  actual  f o r the s t o r a g e of  l o a d f o r t h i s s t r u c t u r e i s found  potash.  t o be i n s a t i s -  f a c t o r y agreement w i t h the e x p e r i m e n t a l r e s u l t s o b t a i n e d i n p r e v i o u s model tests. Some a d d i t i o n a l e f f e c t s ,  such as geometric  i m p e r f e c t i o n s i n the  c o o r d i n a t e s and d i f f e r e n t member e n d - f i x i t y c o n d i t i o n s a r e The  concept  investigated.  of e f f e c t i v e l e n g t h f o r a member i s i n t r o d u c e d to p r e s e n t  the r e s u l t s o b t a i n e d by v a r y i n g the h e i g h t / s p a n r a t i o of the  structure.  F i n a l l y some d e s i g n s u g g e s t i o n s a r e g i v e n f o r s t r u c t u r e s of t h i s The  a n a l y s e s were made u s i n g spaceframe programs based  ness method, m o d i f i e d to i n c l u d e s t a b i l i t y e f f e c t s . was  used  f o r the  joint  calculations.  An IBM  type.  on the  stiff-  360/67 computer  ii TABLE OF CONTENTS  Page ABSTRACT  i  TABLE OF CONTENTS  i i  LIST OF FIGURES  iv  LIST OF TABLES  vi  NOTATION  v i i  ACKNOWLEDGEMENTS  ix  CHAPTER I  INTRODUCTION  CHAPTER I I  GOVERNING EQUATIONS AND SOLUTION PROCEDURE .2.1  1  Governing E q u a t i o n s  5  2.2  S o l u t i o n Procedure  8  2.3  Computer Program O u t l i n e  CHAPTER I I I  DESCRIPTION OF THE STRUCTURE AND LOADING SYSTEMS  CHAPTER IV  RESULTS FOR THE ORIGINAL CONE  CHAPTER V  10 15  4.1  Introduction  25  4.2  Determinant P l o t  25  4.3  Southwell P l o t  32  4.4  Load-deflection  4.5  P a r t i a l Loading  44  4.6  C o n c l u s i o n s about S o u t h w e l l P l o t  51  Curves  44  RESULTS FOR SPACEFRAMES WITH DIFFERENT HEIGHT/SPAN RATIO 5.1  C o n i c a l Shapes  54  iii TABLE OF CONTENTS  (Continued) Page  5.2  S p h e r i c a l Shape  CHAPTER V I  CONCLUSIONS AND RECOMMENDATIONS  APPENDIX 1  DERIVATION OF THE NON-LINEAR RELATIONS GOVERNING  64 69  THE DEFORMATION OF A MEMBER  73  APPENDIX 2  EXACT SOLUTION FOR TWO-BAR STRUCTURE  82  APPENDIX 3  NOTES ON THE  85  APPENDIX 4  LIST OF REFERENCES  SOUTHWELL PLOT  88  iv LIST OF FIGURES Page FIG. 1:1  Geometrically Non-linear Effects  2  FIG. 2:1  Member N o t a t i o n  6  FIG. 2:2  Step-by-step  FIG. 2:3  Check o f Computer Program f o r Theory  FIG. 2:4  Two-Bar S t r u c t u r e  13  FIG. 3:1  The O r i g i n a l Cone  16  FIG. 3:2  Member P r o p e r t i e s  17  FIG. 3:3  Connection D e t a i l s  18  FIG. 3:4  Coding o f the S t r u c t u r e  20  H a l f Snow Load  19  " Q u a r t e r Wave" B u c k l i n g  21  FIG. 3:6  Member  21  FIG. 3:7  P e r t u r b a t i o n F o r c e Systems ( " T r i g g e r Systems")  FIG. 4:1  O r i g i n a l Cone.  Solution  10 2  12  FIG. 3:5 (i) (ii)  End-Forces  Determinant  Plots.  F i x i t y Condition 27  (a) FIG. 4:2  23  O r i g i n a l Cone.  Determinant  Plots.  F i x i t y Condition 28  (b) Determinant  Plots  30  FIG. 4:3  Comparison  FIG. 4:4  Southwell P l o t .  E x a c t Geometry  36  FIG. 4:5  Southwell P l o t .  E x a c t Geometry  37  FIG. 4:6  Southwell P l o t .  E x a c t Geometry  38  FIG. 4:7  Southwell P l o t .  Random Geometry  39  V LIST OF FIGURES ( C o n t i n u e d ) Page FIG. 4:8  Southwell P l o t .  Random Geometry  40  FIG. 4:9  Southwell P l o t .  Random Geometry  41  FIG. 4:10  C r i t i c a l Mode Shape.  FIG. 4:11  L o a d - d e f l e c t i o n Curve.  F i x i t y C o n d i t i o n (a)  45  FIG; 4:12  L o a d - d e f l e c t i o n Curve.  F i x i t y C o n d i t i o n (a)  46  FIG. 4:13  L o a d - d e f l e c t i o n Curve.  F i x i t y C o n d i t i o n (b)  47  FIG. 4:14  L o a d - d e f l e c t i o n Curve.  F i x i t y C o n d i t i o n (b)  48  FIG. 4:15  Displacements  FIG. 4:16  O r i g i n a l Cone.  FIG. 4:17  Geometric I n s t a b i l i t y  FIG. 5:1  F i x i t y C o n d i t i o n (a)  along Middle Ring H a l f Snow Load.  Determinant P l o t s  42  49 50  (Snap-through)  52  FC1 Cone.  1/2 O r i g i n a l H e i g h t . D e t e r m i n a n t P l o t  56  FIG. 5:2  FC2 Cone.  1/4 O r i g i n a l H e i g h t . D e t e r m i n a n t P l o t  57  FIG. 5:3  D i f f e r e n t Behaviour o f Adjacent Ribs  FIG. 5:4  C o n i c a l Spaceframes.  62  Effect of Varying Height/  Span R a t i o  63  FIG. 5:5  S p h e r i c a l Spaceframe  65  FIG. 5:6  S p h e r i c a l Dome SF1. D e t e r m i n a n t P l o t  66  FIG. 6:1  E f f e c t o f N o n l i n e a r i t i e s on t h e M a r g i n o f S a f e t y  72  VI  LIST OF TABLES Page TABLE 4:1  Determination of Structure C r i t i c a l  Load  by D e t e r m i n a n t P l o t TABLE 4:2  26  I n f l u e n c e o f Geometric I m p e r f e c t i o n s on A x i a l Loads  TABLE 4:3  31  D e t e r m i n a t i o n o f S t r u c t u r e C r i t i c a l Load by 33  Southwell P l o t TABLE  5:1  TABLE 5:2  Determination of C r i t i c a l  Load by  Determinant  P l o t f o r D i f f e r e n t Height/Span R a t i o s  55  O r i g i n a l Cone FCO.  59  a  = „ }. 3.35 a. = 1/2 a = — 1 o 6.7 a = 1/4 a = \ •, o 13.4 c  o  TABLE 5:3  Cone FC1.  TABLE 5:4  Cone FC2.  60 61  0  2  TABLE 5:5  Comparison o f a C o n i c a l Spaceframe  and a  Spherical 67  vii NOTATION  x, y, z  =  member axes;  X, Y, Z  =  s t r u c t u r e axes;  {P}  =  external load vector;  {U}  =  s t r u c t u r e displacement v e c t o r ;  [K]  =  structure stiffness  [Kj_]  =  s t r u c t u r e tangent s t i f f n e s s  {f}  =  member f o r c e v e c t o r i n member a x e s ;  {F}  =  member f o r c e v e c t o r i n s t r u c t u r e axes;  {u}  =  member d i s p l a c e m e n t v e c t o r i n member axes;  =  member s t i f f n e s s m a t r i x i n member a x e s ;  [k]  =  member s t i f f n e s s m a t r i x i n s t r u c t u r e a x e s ;  [k ]  =  member g e o m e t r i c s t i f f n e s s m a t r i x i n s t r u c t u r e a x e s ;  [k^]  =  member tangent s t i f f n e s s m a t r i x i n s t r u c t u r e a x e s ;  [R]  =  m a t r i x o f d i r e c t i o n c o s i n e s o f a member;  [T]  =  t r a n s f o r m a t i o n m a t r i x f o r a member;  N  =  a x i a l f o r c e i n a member;  X  =  load  =  e x t e r n a l l o a d v e c t o r a t A = 1;  =  l i n e a r a x i a l f o r c e i n a member a t X = 1;  A^j  =  j o i n t d i s p l a c e m e n t component normal t o t h e s u r f a c e o f t h e cone;  o)^  =  j o i n t r o t a t i o n component about an a x i s normal t o t h e s u r f a c e  [k]  {P} N  o  m  o  =  matrix;  factor;  o f t h e cone; a  matrix;  height/span r a t i o ;  NOTATION ( C o n t i n u e d )  L  =  member j o i n t - t o j o i n t  A  =  member c r o s s - s e c t i o n a l a r e a ;  I  =  moment o f i n e r t i a ;  J  =  t o r s i o n a l constant;  r  =  r a d i u s o f g y r a t i o n o f member c r o s s - s e c t i o n ;  E, G  =  e l a s t i c moduli of m a t e r i a l ;  L  =  e f f e c t i v e l e n g t h o f a member;  =  t o t a l applied external load i n v e r t i c a l d i r e c t i o n  =  chord r o t a t i o n .  P P  length;  e  ix ACKNOWLEDGEMENTS  The a u t h o r w i s h e s t o e x p r e s s h i s thanks t o h i s s u p e r v i s o r Dr. R. F. H o o l e y , and t o D r s . D. L. Anderson  and N. D. Nathan o f t h e  C i v i l E n g i n e e r i n g Department, f o r t h e i r encouragement and guidance i n the p e r i o d o f r e s i d e n c e s t u d i e s and d u r i n g t h e p r e p a r a t i o n o f t h i s thesis. The f i n a n c i a l s u p p o r t o f t h e N a t i o n a l Research C o u n c i l o f Canada, i n t h e form o f a P o s t - g r a d u a t e S c h o l a r s h i p , i s g r a t e f u l l y acknowledged.  G r a t i t u d e i s a l s o e x p r e s s e d t o t h e U. B. C. Computing  C e n t r e f o r t h e v e r y generous  allocation  of i t s o u t s t a n d i n g f a c i l i t i e s .  September, 1969 Vancouver,  B. C.  o f computer time and t h e use  STABILITY ANALYSIS OF A SPACEFRAME STRUCTURE  CHAPTER I  INTRODUCTION  Various  s t r u c t u r a l systems may be used t o c o v e r l a r g e a r e a s f o r  r e c r e a t i o n a l o r i n d u s t r i a l p u r p o s e s , e.g. systems o f t r u s s e s and beams, continuous s h e l l s , o r spaceframes. presents  When t h e g e o m e t r i c a l  configuration  an a x i s o f r o t a t i o n a l symmetry, t h e most w i d e l y used systems a r e  e i t h e r continuous s h e l l s , generally i n concrete,  o r framed domes.  I n c o u n t r i e s where t h e c o s t o f l a b o u r i s h i g h w i t h r e s p e c t t o t h a t o f m a t e r i a l s , t h e use o f spaceframe s t r u c t u r e s may show an e c o n o m i c a l advantage o v e r t h e use o f c o n t i n u o u s s h e l l s ; t h e former may a l s o be p r e f e r r e d frosm t h e p o i n t o f v i e w o f speed o f e r e c t i o n and g e n e r a l  construction  q u a l i t y , s i n c e f i e l d work i s reduced t o a minimum w i t h r e s p e c t  t o shop  prefabrication. N o t a b l e examples o f such domes a r e R. B. F u l l e r ' s g e o d e s i c domes, the Astrodome o f Houston, Texas, t h e Schwedler Dome o f B e r l i n . I n t h e d e s i g n o f t h e s e s t r u c t u r e s i t i s r e l a t i v e l y easy t o f i n d the l i n e a r i n t e r n a l f o r c e s and moments due t o a s p e c i f i e d e x t e r n a l  load,  by means o f a l i n e a r s t i f f n e s s a n a l y s i s ( 1 ) .  question-  a b l e r u l e s a r e used t o f i n d t h e a l l o w a b l e  A t p r e s e n t , however,  c o m p r e s s i v e l o a d s i n t h e members.  N o n l i n e a r i t y i n t h e b e h a v i o u r o f t h e s e s t r u c t u r e s under s t a t i c l o a d s , e x c l u d i n g m a t e r i a l n o n l i n e a r i t i e s , a r i s e s from changes i n t h e  ^Numbers i n p a r e n t h e s e s r e f e r t o r e f e r e n c e s  l i s t e d i n A p p e n d i x A.  geometry of the structure  due to i t s deformation:  these e f f e c t s are  p a r t i c u l a r l y important i n those spaceframes where members meeting at a j o i n t are almost coplanar. The object of this thesis i s to determine the e f f e c t of some geometric n o n l i n e a r i t i e s , namely chord rotation and bowing, as shown i n f i g . I l l , on the evaluation  of the e l a s t i c c r i t i c a l load f o r a spaceframe  structure, and to give some design suggestions f o r t h i s type of structure, p a r t i c u l a r l y about the concept of e f f e c t i v e length of a member.  In  addition, an i n v e s t i g a t i o n i s made about the a p p l i c a b i l i t y of the Southw e l l plot f o r the evaluation  of the c r i t i c a l load of the structure.  This w i l l be achieved by analysing an e l a s t i c spaceframe having the shape of a conical s h e l l , with 16 s t r a i g h t r i b s , 4 main rings, and bracing diagonals, as shown i n f i g . 3 t l .  A s t r u c t u r a l model of such a  cone has been used i n the past to predict the c r i t i c a l load  (2).  The  present work w i l l be carried out a n a l y t i c a l l y using two d i f f e r e n t s t r u c t u r a l s t i f f n e s s Tsatrieesj naai&ly (1)  considering  the e f f e c t of the a x i a l force on the bending  s t i f f n e s s of a member, (2)  same as (1), but adding the e f f e c t s of the change i n a x i a l force due to chord rotation and f l e x u r a l end-shortening (bowing), as from f i g . 1:1.  i n i t i a l position FIG.  chord rotation 1:1 - GEOMETRICALLY NONLINEAR EFFECTS  bowing  3 S i n c e the r e s u l t s of the two i t was  not  rotation  a n a l y s e s were always v e r y  deemed n e c e s s a r y to a n a l y s e s e p a r a t e l y  close,  the e f f e c t o f chord  only. I n the s t a b i l i t y a n a l y s i s o f framed s t r u c t u r e s  o f the change i n a member b e n d i n g s t i f f n e s s , due a x i a l f o r c e , has been shown t o be (e.g. L i v e s l e y  importance  to the p r e s e n c e o f  e s s e n t i a l by s e v e r a l  (3), Merchant & H o m e  the  an  investigators  (4)).  I n the f i r s t a n a l y s i s the a x i a l f o r c e i n a member i s d e t e r m i n e d o n l y by the o r t h o g o n a l p r o j e c t i o n s p o s i t i o n o f the member, w h i l e  o f the e n d - d i s p l a c e m e n t s on the  the second a n a l y s i s t a k e s i n t o a c c o u n t  a c t u a l d i f f e r e n c e i n l e n g t h between the f i n a l deformed shape and i n i t i a l configuration.  initial  Thus, w i t h r e f e r e n c e t o f i g . 1:1,  the  the  the  first  a n a l y s i s w o u l d g i v e z e r o a x i a l f o r c e i n the member, whereas the  second  would show the p r e s e n c e of a t e n s i l e a x i a l f o r c e . The  a d d i t i o n a l e f f e c t s considered i n theory  2  c o u l d be  n i f i c a n t f o r s t r u c t u r e s o f s h e l l - l i k e c o n f i g u r a t i o n , w h i c h do not  sigallow  i n e x t e n s i o n a l b e n d i n g , and have been shown to be of e s s e n t i a l i m p o r t a n c e i n the p o s t - b u c k l i n g  domain  (5).  Among o t h e r methods to p r e d i c t the c r i t i c a l l o a d o f a spaceframe dome, t h e r e i s the p o s s i b i l i t y of e s t a b l i s h i n g an e q u i v a l e n t s h e l l , whose c r i t i c a l l o a d may and  experimental r e s u l t s .  homogeneous  be o b t a i n e d from the a v a i l a b l e a n a l y t i c a l  T h i s method may  be u s e f u l when the number o f  members i n the spaceframe i s so l a r g e as t o make a computer a n a l y s i s impossible  o r uneconomical.. The  method, however, f a i l s i f the mesh s i z e  of the spaceframe i s too c o a r s e , o r i f the l a t e r a l s t i f f n e s s of the r i b s i s much s m a l l e r  t h a n t h e i r s t i f f n e s s n o r m a l to the s h e l l s u r f a c e ,  in  w h i c h case b u c k l i n g may o c c u r because o f l a t e r a l b e n d i n g o f t h e members o f the  spaceframe. The n o n l i n e a r  equations governing the deformation of the s t r u c t u r e  may be s o l v e d by v a r i o u s methods (6,7), among w h i c h t h e Newton-Raphson and t h e step-by-step incremental a modified  method have been w i d e l y  used.  I n the present study  v e r s i o n o f t h i s second method w i l l be used, s i n c e t h e Newton-  Raphson method has been r e p o r t e d  to e x h i b i t d i f f i c u l t i e s i n c a l c u l a t i o n  s t a b i l i t y and convergence (8). Other advantages o f t h e i n c r e m e n t a l are the p o s s i b i l i t y o f f o l l o w i n g the p r o g r e s s i v e and  method  deformation o f the s t r u c t u r e  the f a c t that the determinant o f t h e tangent s t i f f n e s s m a t r i x  provides  a t each s t e p a measure o f r e l a t i v e s t a b i l i t y o f an e q u i l i b r i u m c o n f i g u r a t i o n . The f o l l o w i n g a d d i t i o n a l c o n d i t i o n s w i l l a l s o be c o n s i d e r e d i n this investigation: (a)  c h a n g i n g members' end c o n d i t i o n s  so as t o c o n s i d e r  t h e two  cases: i)  P o s s i b i l i t y o f b u c k l i n g about b o t h axes o f member cross-section  ii)  admitted,  P o s s i b i l i t y o f b u c k l i n g about t h e weak a x i s ( i . e . " i n plane" buckling)  excluded.  I n t h e f i r s t case a member p r o v i d e s b e n d i n g r e s i s t a n c e a t a j o i n t with respect  t o b o t h p r i n c i p a l axes o f t h e c r o s s - s e c t i o n  i n t h e second case no b e n d i n g s t i f f n e s s e x i s t s f o r r o t a t i o n s about t h e weak a x i s o f t h e c r o s s ^ s e c t i o n ; (b)  a random v a r i a t i o n o f j o i n t c o o r d i n a t e s , construction  (c)  to simulate  possible  imperfections;  changes o f g e o m e t r i c a l  c o n f i g u r a t i o n , i . e . c o n i c a l and doubly  c u r v e d domes o f d i f f e r e n t h e i g h t / s p a n r a t i o .  CHAPTER I I  GOVERNING EQUATIONS AND SOLUTION PROCEDURE  2.1  Governing Equations As mentioned e a r l i e r , two methods f o r s t a b i l i t y a n a l y s i s a r e  used. in  The f i r s t makes use o f L i v e s l e y s t a b i l i t y f u n c t i o n s , as o u t l i n e d  (3) and ( 4 ) , t o t a k e i n t o account t h e v a r i a t i o n s i n t h e b e n d i n g s t i f f -  ness o f a member due t o t h e p r e s e n c e o f an a x i a l f o r c e . s t i f f n e s s matrix o f the s t r u c t u r e  I n t h i s case t h e  [K] i s a f u n c t i o n o n l y o f t h e a x i a l  l o a d s N_^ i n t h e members. The g o v e r n i n g l o a d - d i s p l a c e m e n t r e l a t i o n s w i l l b e : ' {P} = [K(N.)] {U} . . .  [1]  where {P} i s t h e e x t e r n a l l o a d v e c t o r , and {U} i s t h e s t r u c t u r e placement v e c t o r .  Generally  a few i t e r a t i o n s ,  dis-  a t each l o a d l e v e l , a r e  r e q u i r e d t o a c h i e v e convergence f o r t h e elements o f [ K ] , For s t a b i l i t y a n a l y s i s we can r e w r i t e  [1] i n i n c r e m e n t a l  form  (assuming t h a t t h e member a x i a l l o a d s do n o t change f o r a s m a l l i n c r e m e n t o the e x t e r n a l l o a d s ) :  {AP} = [K(N ) ] ' {AU} . . . ,  0  [2]  Then, by d e f i n i t i o n , a t t h e c r i t i c a l l o a d we have {AP} = {0} f o r some non-zero {AU}, i . e . [K] must become s i n g u l a r , o r  |K|  =o.  I n o t h e r words, a t t h e c r i t i c a l l o a d t h e e q u i l i b r i u m c o n f i g u r a t i o n o f t h e s t r u c t u r e i s not unique.  6 The second analysis, including the effects due to chord rotation and bowing, i s based on the derivation described i n references (9) and (10). The assumptions on which this theory rests are: 1. The material i s linear elastic. 2.  Each member is prismatic and homogeneous.  3. Loads are applied only at the ends of a member. 4. Shear deformations are neglected. 5  e  Linear strains and squares of the rotations are of the same order of magnitude, and small compared to one.  6. Torsion-flexure coupling and warping restraint are neglected. It can be noted that assumptions 4 and 5 correspond exactly to those made by Von Karman i n his "large deflection" theory of plate bending (11). An outline of the derivation of the load-displacement relations for a member, i n the two-dimensional case, i s shown i n Appendix 1. For a general spaceframe member, the member coordinate system, end-forces and end-displacements are shown i n f i g . 211. The member reference frame does not follow the deformation of the member, but i s fixed to i t s undeformed position.  x  The complete s e t o f f o r c e - d i s p l a c e m e n t r e l a t i o n s f o r a member can be w r i t t e n , i n m a t r i x form, a s : { f }  12xl  ^  =  12x12  12xl " V l 2 x l  { u }  ^  {  where { f }  =  { F  =  {  A 1 ' A2' A3' A1' A2' A3' B1' B2' B3' ^ 1 ' ^ 2 ' B 3 F  F  M  M  M  F  F  F  M  }  and T { u }  are  u  A l ' A 2 ' A 3 ' A1» U  U  U  A2'  tt)  > B3 W  }  t h e c o l u m n - v e c t o r s o f member e n d - f o r c e s and e n d - d i s p l a c e m e n t s ; [ k ]  m  i s t h e member s t i f f n e s s m a t r i x , i n c l u d i n g L i v e s l e y s t a b i l i t y  functions:  {fg}  from  i s the v e c t o r o f the n o n l i n e a r geometric terms, a r i s i n g  bowing and c h o r d r o t a t i o n .  I n the present a n a l y s i s i t reduces t o only  two a x i a l t e r m s , s i n c e t h e sway terms U  ( o f t h e t y p e pF^, where  B2 ~ A2 U  p =  - ) , w h i c h appear i n ( 1 0 ) , a r e i n c l u d e d i n L i v e s l e y  stability  functions„ I t s h o u l d a l s o be n o t e d t h a t f i n i t e r o t a t i o n s do n o t obey t h e a d d i t i o n l a w f o r v e c t o r s , b u t t h e e r r o r i s o f t h e o r d e r o f t h e square o f a r o t a t i o n , w h i c h i s n e g l i g i b l e w i t h t h e assumptions used i n t h e p r e s e n t derivation. To o b t a i n t h e system e q u a t i o n s i n s t r u c t u r e , o r g l o b a l , c o o r d i n a t e s , l e t [ R ] 3 3 he t h e m a t r i x o f t h e d i r e c t i o n c o s i n e s f o r t h e X  undeformed member d i r e c t i o n s frame ( X , Y , Z ) . for  (x,y,z) w i t h respect to the g l o b a l r e f e r e n c e  Then i t c a n be shown (1) t h a t t h e t r a n s f o r m a t i o n m a t r i x [T]  a spaceframe member r e s u l t s :  8  [T]  [R] [0] [0] [0]  12x12  [0] [R] [0] [0]  [0] [0] [0] [0] [R] [0] [0] [R]  Then, f o r each member, we have, i n s t r u c t u r e c o o r d i n a t e s : {U} {F}  m  = [ T ] {u} T  = [T]  T  {f}  {F > = [ T ]  T  {f >  G  G  [k] = [ T ] [ k ] T  m  [T]  A d d i n g up t h e c o n t r i b u t i o n s o f a l l t h e members connected t o a j o i n t , t h e system o f j o i n t e q u i l i b r i u m e q u a t i o n s  i s obtained:  [A]  {P} = [K] {U} - {Pg} where  {P} i s t h e e x t e r n a l l o a d v e c t o r ; [K] i s 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 , w h i c h a g a i n i s a f u n c t i o n o n l y o f t h e . a x i a l l o a d s i n t h e members; {U} i s t h e s t r u c t u r e d i s p l a c e m e n t  vector;  {P }, produced, by {F }, i s e q u i v a l e n t t o a l o a d v e c t o r , due t o G G g e o m e t r i c n o n l i n e a r i t i e s , and i s a f u n c t i o n b o t h o f t h e members' a x i a l l o a d s and e n d - d i s p l a c e m e n t s .  2.2  S o l u t i o n Procedure To s o l v e system [4] o f n o n l i n e a r e q u a t i o n s ,  a modified  incremental  method i s u s e d , whereby the e x t e r n a l l o a d s a r e a p p l i e d by f i n i t e  increments,  and an i t e r a t i o n c y c l e by  s u c c e s s i v e s u b s t i t u t i o n s i s c a r r i e d out a f t e r a  number o f s t e p s , t h a t c a n be v a r i e d a t c h o i c e , u s i n g t h e c o m p l e t e , set of n o n l i n e a r equations [ 4 ] ,  "exact",  To o b t a i n t h e i n c r e m e n t a l for  part of the s o l u t i o n , the expressions  the d i f f e r e n t i a l s o f the end-actions  reference  [k ]  where  m  Q  [k ]  Following  (10), we can w r i t e , i n member c o o r d i n a t e s : {Af} = [ k ]  and  o f a member a r e needed.  m  m  '{Au} + [ k ] {Au} m  G  {Au} = -  {Afg},  may be i n t e r p r e t e d as a g e o m e t r i c s t i f f n e s s m a t r i x , w h i c h ,  w i t h o u r a s s u m p t i o n s , happens t o be s y m m e t r i c : [k ]  m  r  i s g i v e n i n A p p e n d i x 1. Then we can w r i t e : • {Af} = [ k ]  m  t  where  [5]  {Au}  [k ] = [k] + [k ] m  m  t  m  G  i s t h e member t a n g e n t s t i f f n e s s m a t r i x .  Assembling the c o n t r i b u t i o n s o f  each member we o b t a i n t h e s y s t e m e q u a t i o n s  i n incremental  form:  [6]  ' {AP} = [ K ] {AU} t  where [ K ] i s now t h e s t r u c t u r e t a n g e n t s t i f f n e s s t  matrix.  A g a i n i n s t a b i l i t y i s reached when e q u a t i o n  [6] has a n o n - t r i v i a l  s o l u t i o n f o r {AP} = {0}, i . e . when |K | = 0. A second s t a b i l i t y c r i t e r i o n can be e s t a b l i s h e d by t h e f a i l u r e i n a c h i e v i n g convergence i n t h e c y c l e o f s u c c e s s i v e s u b s t i t u t i o n s :  for this  method a s m a l l i n c r e m e n t s i z e i s n e c e s s a r y i n t h e n e i g h b o r h o o d o f t h e c r i t i c a l l o a d , i n order to achieve  reasonable  An a d d i t i o n a l method o f d e t e r m i n i n g of S o u t h w e l l  accuracy. t h e c r i t i c a l l o a d i s t h e use  p l o t s i n connection w i t h s e l e c t e d d i s t u r b i n g f o r c e systems.  I t has been shown (12) t h a t i n a p l a n e frame, as l o n g as a d i s t u r b a n c e , e i t h e r a l o a d o r a g e o m e t r i c i m p e r f e c t i o n , e x c i t e s any component o f t h e  10  f i r s t c r i t i c a l mode shape, t h e c o r r e s p o n d i n g S o u t h w e l l p l o t w i l l the r e l a t i v e  c r i t i c a l load.  Thus i t was  d e c i d e d t o check t h e  yield  applicability  o f t h i s method to the p r e s e n t spaceframe s t r u c t u r e .  2.3  Computer Program O u t l i n e A s t a n d a r d spaceframe s t i f f n e s s  program, i n c l u d i n g L i v e s l e y  s t a b i l i t y f u n c t i o n s , has been m o d i f i e d i n o r d e r t o b u i l d  the  tangent  s t i f f n e s s m a t r i x o f the s t r u c t u r e a t each l o a d l e v e l , u s i n g t h e f o r c e s and j o i n t d i s p l a c e m e n t s  axial m  from  t h e assemblage o f w h i c h [K^J i s made up, i s a f u n c t i o n b o t h o f t h e  axial  l o a d and the e n d - d i s p l a c e m e n t s  from the p r e v i o u s s t e p , s i n c e [ k ^ ] ,  o f a member.  Every few s t e p s a c y c l e o f s u c c e s s i v e s u b s t i t u t i o n s is p e r f o r m e d , keeping  t h e e x t e r n a l l o a d s c o n s t a n t , as shown i n f i g .  deflection U FIG. 2:2  - STEP-BY-STEP SOLUTION  2:2.  11 Segment AB c o r r e s p o n d s to  the load-displacement  t o t h e i n c r e m e n t a l s t e p a l o n g t h e tangent  curve, using equation  c o r r e c t i v e i t e r a t i o n c y c l e at constant l o a d . used r e p e a t e d l y :  [ 6 ] , w h i l e BC r e p r e s e n t s t h e I n t h i s c y c l e e q u a t i o n [4] i s  a t each i t e r a t i o n the new v a l u e s o f d i s p l a c e m e n t s  l o a d s a r e used t o b u i l d  and a x i a l  [K] and { P } ; t h e n t h e e q u a t i o n can be w r i t t e n as n  {P} + {Pg} = [K] {U}  [7]  and s o l v e d t o o b t a i n a b e t t e r s o l u t i o n f o r t h e d i s p l a c e m e n t The s t e p s i z e i s reduced  v e c t o r {U}.  i n g e o m e t r i c p r o g r e s s i o n , as t h e t o t a l  load i s i n c r e a s e d , to a l l o w greater accuracy i n the determination o f the c r i t i c a l point„ Only a p r o p o r t i o n a l l o a d s y s t e m i s c o n s i d e r e d , where a t each l e v e l the e x t e r n a l load d i s t r i b u t i o n i s a m u l t i p l e o f the i n i t i a l , l o a d i n g p a t t e r n , w h i c h can be r e p r e s e n t e d by a v e c t o r {P} > Q  s o  basic  that at  each s t a g e we have {P} = X { P } where X i s a n u m e r i c a l p a r a m e t e r .  [8]  Q  Then t h e c r i t i c a l l o a d w i l l be g i v e n  by t h e c o r r e s p o n d i n g v a l u e X•• o f t h e l o a d f a c t o r X. F i n a l l y , t o check t h e program, fig„ 2:3 shows an example ( a p o r t a l frame) p r e s e n t e d i n ( 1 0 ) , w h i c h was used as t h e b a s i c r e f e r e n c e f o r the p r e s e n t  investigation.  I t i s t o be noted t h a t the s o l u t i o n o f r e f . (10) a l l o w e d a 5% convergence t o l e r a n c e , whereas t h e p r o c e d u r e  used by t h e w r i t e r p e r m i t t e d  f u l l convergence, w h i c h may e x p l a i n t h e s l i g h t gap between t h e two c u r v e s . As another s i m p l e example o f a s t r u c t u r e where g e o m e t r i c nonl i n e a r i t i e s a r e i m p o r t a n t , f i g . 214 shows a two-bar s t r u c t u r e w i t h v e r y low r i s e / s p a n r a t i o .  With the given c r o s s - s e c t i o n a l p r o p e r t i e s , geometric  i n s t a b i l i t y w i l l occur before E u l e r b u c k l i n g .  12  LOAD - DISPLACEMENT  CURVES.  50O0  Displacement  FIG.  2*  3  - C H E C K  OF  C O M P U T E R  P R O G R A M  A, inches  F O R  T H E O R Y  2.  •05 10 15 -20 25 30 35  FIG.  2-4-  T W O - B A R  40 45 50  S T R U C T U R E .  14 The two computer a n a l y s e s a r e compared w i t h t h e e x a c t developed  from s t r a i n energy c o n s i d e r a t i o n s :  d e r i v e d i n A p p e n d i x 2.  solution,  the r e l a t i v e expressions are  The s t r u c t u r e was a n a l y s e d as a p l a n e frame w i t h  4 members, as shown below.  I t can be s e e n from f i g . 214 t h a t t h e r e s u l t s g i v e n by t h e o r y a r e v e r y c l o s e t o t h e e x a c t s o l u t i o n ; t h i s i s another o f t h e computer program w r i t t e n f o r t h e o r y  2 .  2  check f o r t h e c o r r e c t n e s s  15 CHAPTER I I I  DESCRIPTION OF THE  STRUCTURE AND  LOADING SYSTEMS  The f i r s t large s t r u c t u r e analysed was  the p l e x i g l a s s model  (1:28 l i n e a l r a t i o ) of an a c t u a l c o n i c a l spaceframe, made of glulam timber and b u i l t i n Esterhazy, Saskatchewan, f o r I n t e r n a t i o n a l Minerals Chemical Corporation  (Canada) L t d . (2).  and  The geometric and e l a s t i c properties  of the s t r u c t u r e are shown i n f i g s . 311 and 3T2.  The angle 8 i s measured  from a v e r t i c a l plane through the axis of a member to~a plane containing the weak p r i n c i p a l axis of the c r o s s - s e c t i o n . The s t r u c t u r e c o n s i s t s of 16 r i b s , 64 r i n g members, and  64  diagonals, which develop the f u n c t i o n of the web  members of a t r u s s :  altogether there are 80 j o i n t s and 192 members.  Thus the s t r u c t u r e i s  s t a t i c a l l y determinate i n i t s primary s t r e s s system, since we have: 80 j o i n t s y i e l d i n g 240 equations of e q u i l i b r i u m , 192 bars p r o v i d i n g unknown a x i a l f o r c e s , and 16 x 3 = 48 unknown foundation  192  r e a c t i o n components.  This y i e l d s 240 unknowns and 240 equations, and by i n s p e c t i o n i t can  be  seen that there are no redundant members„ To simulate the r i g i d foundation  r i n g , each r i b was  p e r f e c t l y f i x e d to the respective foundation  considered  joint.  In the model the members meeting at a j o i n t were held i n place by two aluminum p l a t e s (2), one at the top and one at the bottom, t i g h t l y fastened by b o l t s , as sketched i n f i g . 313.  16  ORIGINAL CONE (Plexiglass  Pion  Model)  View  FIG. 3 ' I - THE ORIGINAL CONE  17  Member  Size (in. x in.)  Area A (in. )  (in!*)  (in?*)  (in. )  1 X 3/16 X 3/4 3/16 X 11/16 3/16 X 9/16  .25 .1406 ,'1289 .1055  .02083 .00659 .00508 .00278  .00130 .000412 .000378 .000309  .00439 .00139 .00125 .000976  0° 0° 0° 0°  2  4  e  Ribs: 1 2 3 4  IM  Rings: 1 2 3 4  1/2 3/8 1/4 3/16  X  1/2 X 3/8 X 1/4 X 1/2  .25 .1406 .0625 .0938  .00521 .00165 .000325 .00195  .00521 .00165 .000325 .000275  .00879 .00278 .00055 .000837  34° 34° 34° 34°  3/16 3/16 3/16 3/16  X  11/16 1/2 X 3/8 X 9/16  .1289 .0938 .0703 .1055  .00508 .00195 .000824 .00278  .000378 .000275 o000206 .000309  .00125 .000837 .000566 .000976  28,2° 25.23° 20.07° 11.41°  Diagonals 1 2 3 4  X  E = 450 k s i  G = 187 k s i  FIG.  3:2 - MEMBER PROPERTIES  18  FIG.  In  3:3 - CONNECTION DETAILS  a d d i t i o n , g l u e and sand were s p r i n k l e d on t h e edges o f t h e  members t o i n c r e a s e f r i c t i o n between t h e members and t h e p l a t e s , so as t o try of  and o b t a i n a r i g i d c o n n e c t i o n w i t h r e s p e c t t o b o t h c r o s s - s e c t i o n a l axes a member. In  t h e computer a n a l y s i s two f i x i t y  c o n d i t i o n s were examined:  i n t h e f i r s t , w h i c h w i l l be c a l l e d c o n d i t i o n ( a ) , a l l t h e members were c o n s i d e r e d p e r f e c t l y f i x e d t o t h e j o i n t s , e x c e p t t h e d i a g o n a l s w h i c h were coded as f i x e d about t h e s t r o n g a x i s , b u t h i n g e d about t h e weak a x i s o f t h e cross-section. the  Because o f t h e way a member s t i f f n e s s m a t r i x i s b u i l t by  computer program, a member coded as h i n g e d a t t h e j o i n t s a t i t s ends,  w i t h no e x t r a degrees o f freedom a l o n g i t s l e n g t h , behaves as a s i m p l e s t r u t , w i t h no b u c k l i n g c o n s i d e r e d .  Thus t h e p o s s i b i l i t y o f l a t e r a l  19 b u c k l i n g o f the l o n g d i a g o n a l s , which i n the model were h e l d i n p l a c e by s t r i n g s s t r e t c h i n g from r i b t o r i b , and i n the a c t u a l s t r u c t u r e by p u r l i n s , was  excluded. In the second c o n d i t i o n , c a l l e d  considered  f i x e d to the j o i n t s  condition  ( b ) , a l l members were  f o r b e n d i n g about the s t r o n g a x i s , i . e . i n  a d i r e c t i o n normal to the s u r f a c e o f the cone, and h i n g e d about the weak a x i s , i . e . f o r bending i n the s u r f a c e o f t h e cone. perhaps, r e p r e s e n t s  more c l o s e l y the a c t u a l b e h a v i o u r o f the j o i n t s o f  the model, and c e r t a i n l y a l l o w s  a better s t a b i l i t y  s t r u c t u r e , where p u r l i n s and d e c k i n g prevent " i n plane" The  T h i s second c o n d i t i o n ,  provide  a n a l y s i s o f the a c t u a l  adequate l a t e r a l b r a c i n g to  buckling.  f o l l o w i n g f i g . 3:4  shows the a c t u a l c o d i n g  o f the s t r u c t u r e  f o r the computer a n a l y s i s . Because o f symmetry, o n l y h a l f o f the s t r u c t u r e was a n a l y s e d : j o i n t s l y i n g i n the plane o f symmetry were p e r m i t t e d that plane, half their  and  only displacements i n  and the members c u t by the p l a n e o f symmetry were e n t e r e d  with  stiffnesses. T h i s procedure s t i l l  allowed  o f b u c k l i n g modes not p o s s e s s i n g  the a n a l y s i s o f p a r t i a l  symmetric about <fc.  I FIG.  3:5  loadings  r o t a t i o n a l symmetry, as s k e t c h e d  i n f i g . 3:5.  ( i ) - HALF SNOW LOAD  the  FIG.3-4  ~ C O D I N G  OF  THE STRUCTURE  21  symmetric about  I  • up X down  FIG. 3 : 5 ( i i ) - "QUARTER WAVE" BUCKLING  For each j o i n t t h e computer a n a l y s i s p r o v i d e s d i s p l a c e m e n t s and r o t a t i o n s i n t h e X, Y, Z d i r e c t i o n s ( s t r u c t u r e a x e s ) , the i n t e r n a l f o r c e s shown i n f i g . 3 : 6 .  FIG. 3 : 6 - MEMBER END-FORCES  and f o r each member  22 The b a s i c l o a d i n g c o n d i t i o n examined was a u n i f o r m l y d i s t r i b u t e d snow l o a d , w h i c h was s i m u l a t e d w i t h c o n c e n t r a t e d according  to the pressure  l o a d s a t each  a t and t h e s u r f a c e a r e a t r i b u t a r y t o t h e j o i n t .  I n a l l t h e f o l l o w i n g graphs t h e v a l u e o f P the t o t a l v e r t i c a l  joint  q  shown w i l l  represent  l o a d due t o such U.D.L. when t h e l o a d f a c t o r X i s e q u a l  t o one. The d i f f e r e n t d i s t u r b i n g f o r c e systems used ( " t r i g g e r systems") are shown i n f i g . 3:7. contained  The magnitude o f t h e s e p e r t u r b a t i o n l o a d s was  i n a range from 4% t o 8% o f t h e g i v e n e x t e r n a l l o a d , a c t i n g  a t t h e same j o i n t .  23  FIG.  3  7-  PERTURBATION  F O R C E  S Y S T E M S  ( TRIGGER  S Y S T E M S )  24  FIG. J  7  (Cont.d)  25 CHAPTER IV  RESULTS FOR  4.1  THE  ORIGINAL CONE  Introduction The model t e s t s c a r r i e d out i n the p a s t (2) y i e l d e d a v a l u e o f  904  ± 80 l b . f o r t h e t o t a l s t r u c t u r e c r i t i c a l l o a d , determined  by  S o u t h w e l l and L u n d q u i s t p l o t s , u s i n g j o i n t d e f l e c t i o n s normal t o the s u r f a c e o f the cone. In  the p r e s e n t i n v e s t i g a t i o n , d e t e r m i n a n t  plots,  Southwell  p l o t s , and l o a d - d e f l e c t i o n curves were used f o r the e v a l u a t i o n o f the c r i t i c a l load.  4.2  Determinant P l o t T a b l e 4.1  c o n t a i n s the r e s u l t s from the d e t e r m i n a n t  plots,  w h i c h a r e shown i n f i g s . 411 and k°.2 f o r f i x i t y c o n d i t i o n s (a) and  (b)  respectively. The  random v a r i a t i o n o f j o i n t c o o r d i n a t e s , s i m u l a t i n g g e o m e t r i c  i m p e r f e c t i o n s , was geometry" was The  c o n t a i n e d i n a range o f +.5%.  used i n a l l runs thus  The same "random  labelled.  t o l e r a n c e shown i n the v a l u e s o f c r i t i c a l l o a d depends  e s s e n t i a l l y on the a c c u r a c y o f the a n a l y s i s i n the n e i g h b o r h o o d o f the c r i t i c a l load, i . e .  on the magnitude of the i n c r e m e n t a l s t e p and on  number o f i t e r a t i o n s  performed a t each l o a d l e v e l .  The v a l u e s o b t a i n e d by the d e t e r m i n a n t  the  graphs are seen t o be  i n good agreement w i t h t h e p r e v i o u s e x p e r i m e n t a l d e t e r m i n a t i o n o f the  26  TABLE 4.1.  DETERMINATION OF STRUCTURE CRITICAL LOAD BY DETERMINANT PLOT  (a)  A l l members, e x c e p t b r a c i n g d i a g o n a l s , p e r f e c t l y  fixed  to t h e j o i n t s a t t h e i r ends.  Exact  geometry  Random geometry  Theory 1  Theory 2  920 ± 10 l b .  915 ± 10 l b .  927 ± 10 l b .  928 ± 10 l b .  Theory 1 :  Stability  functions only.  Theory 2 :  Stability  f u n c t i o n s p l u s c h o r d r o t a t i o n and bowing.  (b)  A l l members f i x e d t o t h e j o i n t s about t h e s t r o n g b u t h i n g e d about t h e weak a x i s .  Theory 1 Exact  geometry  Random geometry  Theory 2  1048 ± 15 l b .  1035 + 15 l b .  996 ± 15 l b .  972 ± 15 l b .  axis,  27  Load  FIG.  4-1-  ORIGINAL FIXITY  Factor  C O N E . CONDITION  \(P =l20lb.) g  DETERMINANT (A)  P L O T S .  28  7 60  7 80 800 820 8-40 8 60 Load Factor \ (P = 1201b) 0  FIG.  4'  2-ORIGINAL FIXITY  CONE.DETERMINANT CONDITION  P L O T S . (B)  880  29  c r i t i c a l load, e s p e c i a l l y considering the l i k e l y differences between the model and the structure here analysed,  owing to material n o n - l i n e a r i t i e s  (creep of p l e x i g l a s s ) , actual j o i n t behaviour, a p p l i c a t i o n of the ext e r n a l loads, and actual construction  imperfections.  The higher c r i t i c a l load f o r f i x i t y condition (b) i s due to the fact that i n this case the l a t e r a l buckling of i n d i v i d u a l members about t h e i r weak axis i s excluded from the a n a l y s i s .  I t i s as i f the  members were a c t u a l l y restrained against motion i n the plane of the s h e l l , since no degrees of  f r e e d o m  a member between panel points.  a r e  c o n s i d e r e d  a l o n g  the length of  I t can also be argued that i f member  buckling about the weak axis of the cross-section i s allowed f o r , as i n condition (a), above a c e r t a i n load l e v e l some members bring  negative  contributions to the s t i f f n e s s matrix of the structure. The influence of geometric imperfections  and of d i f f e r e n t pertur-  bation force systems on the zeros of the s t i f f n e s s matrix determinant appears to be minor as shown i n f i g . 4J3.  This could be explained by  the fact that the structure i s s t a t i c a l l y determinate i n i t s primary stress system:  hence the d i s t r i b u t i o n of the a x i a l loads i n the members  should not change much for small imperfections, neighborhood of the c r i t i c a l  except probably i n the  load.  In e f f e c t table 4.2 points out some of the biggest differences i n a x i a l loads between the structure with exact geometry and that with random geometry, at about 1/3 of the c r i t i c a l load and near buckling. It can be seen that the differences are magnified as X ->- X , but since cr* b  our geometric imperfections  are d i s t r i b u t e d at random, t h e i r o v e r a l l  e f f e c t on the structure s t i f f n e s s determinant probably tends to zero.  30  O R I G I N A L  8 0  FIG.  CONE.DETERMINANT  8-20 Load Factor 4: 3-  COMPARISON  P L O T S .  8 40 \ (P = 1201b.)  8 60  a  DETERMINANT  P L O T S  31 TABLE 4.2. INFLUENCE OF GEOMETRIC IMPERFECTIONS ON AXIAL LOADS  (a)  Members perfectly fixed to the joints. (Theory  Member No. 3 5 6 13 34 39 63 65 76 85  1 . Trigger system 3).  load factor X = 2.67 Random geom. Exact geom. 16.4 l b . 16.4 32.3 5.65 5.46 4.1 1.4 1.4 2.1 -1.3  16.2 lb„ 13.6 33.7 7.7 5.9 5.4 .77 1.87 2.75 -2.  X - 7.70. P = 120 l b . cr o  Exact geom. X = 7.52  Random geom. X = 7.60  48.9 l b . 48.8 90.4 14.9 15.9 10.2 4.18 4.17 5.6 -3.4  52.4 l b . 36.6 99.5 22.6 18.8 18.3 2.1 6.3 10. -8.  Axial loads: +ve compression, -ve tension.  (b) Members fixed about strong axis, hinged about weak axis. (Trigger system 9)  Member No. 2 3 6 8 15 23 34 42 87  X - 8.50. P = 120 l b . cr o  X = 2.67  Exact geom.  Random geom  lb.  32.8 l b . 15.9 34.3 32.6 5.7 20.0 6.1 3.7 -.75  31.9 16.0 32.5 33.3 6.0 20.9 5.6 4.5 -1.5  0  X = 8.15  Exact geom. 96.2 49.6 95.6 102.4 14.8 68.2 16.8 14.5 -5.  lb.  Random geom. 107.6 67.9 124.0 90.0 30.7 49.6 27.8 -11.4 4.2  lb.  32  4.3  Southwell P l o t The second method used f o r t h e e v a l u a t i o n  was  the S o u t h w e l l p l o t :  table  o f the c r i t i c a l  load  the r e s u l t s from the graphs a r e summarized i n  4.3. The d e f l e c t i o n components p l o t t e d were d i s p l a c e m e n t s  r o t a t i o n s i n a d i r e c t i o n normal to the s u r f a c e o f the cone:  and  A., and  u„  N  N  r e s p e c t i v e l y , as s k e t c h e d below.  A few t y p i c a l S o u t h w e l l p l o t s a r e shown i n f i g s . 4;4  and  following. F o r these d e f l e c t i o n components ( i . e . A^ and to ) i t was observed 4.3,  t h a t i n a l l graphs, the r e s u l t s o f w h i c h a r e shown i n t a b l e  i f the S o u t h w e l l p l o t was  a s t r a i g h t l i n e o v e r a range o f  of a m p l i t u d e e q u a l t o a t l e a s t 60% o f the v a l u e o f t h e l a r g e s t  deflections deflection  used, on the s i d e o f the l a r g e s t d e f l e c t i o n , then the c o r r e s p o n d i n g c r i t i c a l l o a d was w i t h i n 15% o f the v a l u e g i v e n by the d e t e r m i n a n t for  plot  the same c a s e . This condition  could  the use o f the S o u t h w e l l p l o t .  t h e n be t a k e n as a p r a c t i c a l c r i t e r i o n f o r A c t u a l l y , i n the m a j o r i t y  u s i n g A„ and u>„, i f the above c o n d i t i o n was  met,  d e t e r m i n a n t p l o t s gave v a l u e s i n agreement w i t h i n  o f the p l o t s  the S o u t h w e l l 10%.  and  33 TABLE 4.3.  DETERMINATION OF STRUCTURE CRITICAL LOAD BY SOUTHWELL PLOT  (i)  P l o t t i n g j o i n t deflections normal to the surface of the cone (A^) A)  Members' end f i x i t y condition (a) ( f u l l y  fixed)  Random geometry  Exact geometry C r i t i c a l load (lb.) Theory  Theory  1  2  Joint  Trigger System  C r i t i c a l load (lb.)  *  Joint  Trigger System  2500  28  9  1360  19  1  2560  18  6  1742  19  1  2960  18  3  1712  18  6  3200  8  2  1792  18  10  2080  18  8  1680  18  7  1824  18  3  5240  23  1  2340  28  9  1552  18  3  3680  19  4  1850  18  4  1900  19  4  2256  23  4  Theory  1 :  S t a b i l i t y functions  only.  Theory  2 :  S t a b i l i t y functions plus chord r o t a t i o n and bowing.  ^Torsional r i g i d i t y of members put equal to zero  (GJ = 0),  (cont'd  ...)  34  B)  Members' f i x i t y c o n d i t i o n  (b) : f i x e d about s t r o n g  axis,  h i n g e d about weak a x i s .  Exact C r i t i c a l load (lb.) Theory  Theory  Random geometry  geometry Joint  Trigger System  C r i t i c a l load (lb.)  Joint  Trigger System  1  2750  18  3  950  14  6  •  3700  18  7  960  8  6  2160  8  11  980  13  3  1665  23  11  985  28  9  1010  13  6  1010  13  9  1038  18  3  1040  18  7  1086  18  6  1200  19  6  2  1152  13  9  960  28  9  1166  28  9  1032  13  9  1230  18  5  1090  18  9  1584  39  5  1104  38  9  1648  38  5  Theory  1 :  Stability  functions  only.  Theory  2 :  Stability  functions plus  chord r o t a t i o n and bowing.  (cont'd  ...)  (ii)  Plotting joint  r o t a t i o n s about an a x i s normal t o t h e s u r f a c e  o f t h e cone (co^) A)  Members' f i x i t y c o n d i t i o n (a)  Exact C r i t i c a l load (lb.) Theory  1  geometry Joint  Random geometry Trigger System  C r i t i c a l load (lb.) 1120 960 930 1000 1048* 1000* 1016* 1040* 890 912 918 934 960 968 980 1040 1070  9 24 34 39 9 24 34 39 14 18 18 18 13 13 18 38 8  1 1 1 1 1 1 1 1 1 1 3 10 3 1 7 1 1  880 930 980 990 1000 1030 1096 1168  28 18 34 24 9 8 39 9  1 3 1 1 4 1 1 1  890 910  23 28  9 9  928 924 922 936 922 915 920  13 17 18 19 20 23 28  8 8 8 8 8 8 8  28 9 8  9 1 1  Joint  Theory  2  916 1100 1120  Theory  1 :  Stability  functions  Theory  2 :  Stability  f u n c t i o n s p l u s c h o r d r o t a t i o n and bowing.  *Torsional r i g i d i t y  only.  o f members e q u a l z e r o  (GJ = 0 ) .  Trigger System  JOINT  No. 18  AN 4*(x/o ) 3  D/'sp fa cement,  inches  FIG. 4'4-SOUTHWELL PLOT. EXACT GEOMETRY.  JOINT  No. 18  c a  ^(xio ) 3  20  5/5  Fixity Condition (a) Trigger System 8 Theory (7) Po  320 lb.  o X cr  \ 10  I x5 Per  320 x 2-88 =922lb.  OV  U> (xto ) 3  10  20  Potation,  FIG. 4; 5-SOUTHWELL  30  Radians  40  50  60  PLOT'. EXACT GEOMETRY.  N  38  Joint No. 18 AN  FIG.  4  = 6-  S O U T H W E L L  PLOT.  EXACT  GEOM.  39  FIG. 4 '7 - SOUTHWELL PLOT. RANDOM GEOM.  JOINT Note  0  • & + ve down N  40  ward  80  FIG. 4-8 - SOUTHWELL  No. 13  A  N  120 160 Displacement, inches  PLOT.  200  240  RANDOM GEOMETRY  o  ^  JOINT No. 28  FIG. 4 •• 9 - SOUTHWELL PLOT .RANDOM GEOMETRY.  42 Other d e f l e c t i o n components were i n v e s t i g a t e d , namely t h e m e r i d i o n a l and c i r c u m f e r e n t i a l components, b u t f o r t h e s e t h e above c r i t e r i o n does n o t a p p l y , i . e . some S o u t h w e l l p l o t s , s t r a i g h t over a range o f d e f l e c t i o n s as s p e c i f i e d above, y i e l d e d a v a l u e o f c r i t i c a l l o a d more than 30% h i g h e r than t h e c o r r e c t v a l u e . T h i s may be e x p l a i n e d by t h e f a c t t h a t these l a t t e r were n o t components o f t h e l o w e s t c r i t i c a l mode shape.  deflections  Actually, f o r  the s t r u c t u r e w i t h e x a c t geometry and members p e r f e c t l y f i x e d t o t h e joints  (condition  ( a ) ) , i t was p o s s i b l e  to find a disturbance  force  system, namely system 8, c o n s i s t i n g o f two c o u p l e s about t h e Y - a x i s a c t i n g a t j o i n t s 18 and 28, e x c i t i n g a mode shape i n w h i c h a l l t h e j o i n t r o t a t i o n s p l o t t e d ( s e e t a b l e 4.3 ( i i ) ) y i e l d e d p r a c t i c a l l y t h e same value o f c r i t i c a l load.  The c o r r e s p o n d i n g mode shape i s g i v e n below  i n f i g . 4:10.  V  z  FIG.  4:10 - CRITICAL MODE SHAPE.  FIXITY CONDITION (A)  43 The increase  f a c t t h a t the j o i n t r o t a t i o n s showed a l a r g e r a t e  of  at a p p r o x i m a t e l y the same c r i t i c a l l o a d i n d i c a t e s t h a t  i s a g l o b a l phenomenon, and  buckling  the s t r u c t u r e as a whole d i s t o r t s .  I n t h i s case ( i . e . f i x i t y c o n d i t i o n ( a ) ) b u c k l i n g seems t o i n i t i a t e d by b e n d i n g of the main members about the weak a x i s o f c r o s s - s e c t i o n , w h i c h may why  be c a l l e d " i n p l a n e " b u c k l i n g .  f o r t h i s c o n d i t i o n the normal d e f l e c t i o n  be  the  T h i s may  explain  i s not a p p l i c a b l e , i n the  sense t h a t , i n the range of l o a d s a p p l i e d t o the s t r u c t u r e f i r s t c r i t i c a l l o a d ) , i t does not s a t i s f y the p r e v i o u s l y  (below  the  mentioned  c r i t e r i o n , i . e . S o u t h w e l l p l o t s are not s t r a i g h t l i n e s o v e r a wide r a n g e . The  v a l u e s o f c r i t i c a l l o a d shown i n t a b l e 4:3  obtained  f o r t h e s e cases were  by d r a w i n g a s t r a i g h t l i n e i n t e r p o l a t i n g t h r o u g h the l a s t  few  p o i n t s o f the g r a p h , as shown i n f i g . 4:4. There i s the p o s s i b i l i t y t h a t , i f the a n a l y s i s c o u l d be  carried  beyond the f i r s t c r i t i c a l l o a d , t h e s e graphs would a l s o become s t r a i g h t l i n e s o v e r a wide range, i n d i c a t i n g a h i g h e r  c r i t i c a l mode.  p o s s i b l e t o p e r f o r m t h i s check by d e t e r m i n i n g the f i r s t few  I t should  eigenvectors  o f the s t r u c t u r e s t i f f n e s s m a t r i x ;  the p r o c e s s , however, would be  g i v e n the s i z e of the m a t r i x  analysed.  I n a few  t o be  cases the t o r s i o n a l r i g i d i t y of the members was  e q u a l to z e r o , to see whether t h i s a f f e c t e d the g r a p h s , s i n c e S o u t h w e l l method was  be  d e v e l o p e d from the e i g e n v a l u e f o r m u l a t i o n  lengthy  put  the of  the  beam-column problem (4, 1 2 ) , where the t o r s i o n a l r i g i d i t y o f a member, GJ, does not i n t e r v e n e .  T h i s had  p r a c t i c a l e f f e c t f o r to .  a c e r t a i n e f f e c t on one  p l o t w i t h A.,, N  no  44  4.4  Load-deflection  curves  F i g s . 4J11 and f o l l o w i n g show some t y p i c a l l o a d - d e f l e c t i o n c u r v e s f o r t h e two f i x i t y c o n d i t i o n s  (a) and ( b ) .  These c u r v e s a r e c o n s i s t e n t w i t h t h e p r e v i o u s  r e s u l t s , because  i n the f i r s t case ( p e r f e c t l y r i g i d - j o i n t e d spaceframe) t h e d e f l e c t i o n component  appears t o be l a r g e l y u n a f f e c t e d by t h e f i r s t c r i t i c a l  whereas the r o t a t i o n  load,  c l e a r l y i n d i c a t e s t h e onset o f b u c k l i n g .  Under c o n d i t i o n (b) i n s t e a d ,  becomes a good i n d i c a t i o n o f  b u c k l i n g , and a change o f s i g n f o r t h e d e f l e c t i o n , p r o b a b l y depending on t h e t r i g g e r system u s e d , can be n o t e d as t h e l o a d approaches t h e critical  level. Fig.  4:15  shows t h e p a t t e r n o f t h e d i s p l a c e m e n t s  along the  m i d d l e r i n g , n e a r b u c k l i n g , f o r f i x i t y c o n d i t i o n (b) and v a r i o u s t r i g g e r systems.  I t i s r e m a r k a b l e t h a t t h e same wavy p a t t e r n , e s s e n t i a l l y due  to t h e p a r t i c u l a r arrangement o f t h e d i a g o n a l s , was a l s o o b s e r v e d i n t h e experimental  tests.  I t can a l s o be n o t e d t h a t a s m a l l d i s t u r b a n c e  system can cause  a l a r g e d i s t o r t i o n i n the displacement p a t t e r n .  4.5  Partial  loading  To c o n c l u d e t h e r e s u l t s f o r t h e o r i g i n a l cone, an i n v e s t i g a t i o n was made o f a U. D. L. c o v e r i n g o n l y h a l f o f t h e s t r u c t u r e . Fig.  4:16  shows t h a t t h e v a l u e o f  i s s l i g h t l y l e s s than f o r  the c o r r e s p o n d i n g case o f a U. D. L. o v e r t h e whole s t r u c t u r e , namely  JOINT No. Id AN  FIG. 4 •• If - LOAD - DEFLECTION CURVE. FIXITY (A).  JOINT No. 18 N  •critical load (by Def.)  80  I) and (2 Random Trigger  (T) Stability 40  •  Geometry System  3  Functions only -t- Chord Rototion and Bowing  II-  250  5  0  0  , *5° Rotation  0  7  FIG. 4 •• 12 - LOAD-DEFLECTION  1  0  0  0  io Radians Nt  250  f  1500  CURVE. FIXITY (A).  JOINT  4»  No. 13  -critibol lood levels  £  T'~  7  8 0.  60/Random Geometry  •i  ^  *  Trigger System 9 (7) Stability (2)  o  40  //—  Functions only //  + Chord Rototion and Bowing  Note • A + ve down ward N  -A  N  20  40  60 80 100 Displacement, inches  FIG. 4* 13-LOAD DEFLECTION CURVE. FIXITY(B).  120  (xlO ) 3  JOINT No. 28 AN  critical  loads  80 ond  @  Rondom Geometry Trigger System 9 &&0  •—  (7) Stability Functions only  v.  (2)  //—  //  +  Chord Rototion and Bowing  40 -  •A (xtO ) 3  20  40 60 80 Displacement, inches  FIG. 4- 14 - LOAD-DEFLECTION  100  CURVE. FIXITY(B).  H  00  49  Fixity (b) Theory I Stability Functions only P = 120 lb 0  (  X ~ - 8-30) c  ondom Geometrf Trigger System 9  Exact Geometry No Triggers —•Linear Ano lysis No Triggers Exact Geometry  w  Angular Position Note  Positive Down word  FIG. 4- 15 - DISPLACEMENT A  N  ALONG MIDDLE RING.  50  Fixity Condition (b) Exact Geometry  NTN C\j  \  2 0 i  1  1  1  1  1  (7) Stability Functions only (2) //— // + Chord Rotation and Bowing  Load  Factor  \ (R= 1201b.)  FIG 4 • 16- ORIGINAL CONE .HALF SNOW LOAD. DETERMINANT PLOTS  51 X =8.20 cr  v s . 8.62 f o r t h e o r y  2  X =8.31 cr  v s . 8.72 f o r t h e o r y  1  J  J  , and '  I n c o n c l u s i o n then,the s t r u c t u r e b u c k l e s a t about t h e same l o a d p e r square f o o t whether on t h e whole o r h a l f dome.  4.6  C o n c l u s i o n s about S o u t h w e l l p l o t The  l o a d - d e f l e c t i o n curves a l l o w another e x p l a n a t i o n o f the o r to^:  d i f f e r e n t r e s u l t s shown i n t a b l e 4.3, d e p e n d i n g on t h e use o f  i n f a c t i t i s shown i n A p p e n d i x 3 t h a t t h e S o u t h w e l l p l o t b e i n g a s t r a i g h t l i n e i m p l i e s the corresponding l o a d - d e f l e c t i o n curve being a r e c t a n g u l a r h y p e r b o l a h a v i n g t h e c r i t i c a l l o a d f o r h o r i z o n t a l asymptote. seen t o o c c u r t o <A^ f o r f i x i t y c o n d i t i o n ( a ) , and t o  This i s  for fixity  c o n d i t i o n (b) o n l y . T h i s shape o f l o a d - d e f l e c t i o n c u r v e ( i . e . r e c t a n g u l a r h y p e r b o l a ) i s a l s o c o n s i s t e n t w i t h t h e r e s u l t s o b t a i n e d (Home and Merchant ( 4 ) , Timoshenko (17)) by u s i n g t h e c r i t i c a l modes f o r t h e s e r i e s e x p a n s i o n o f any d e f l e c t i o n component.  Then each term o f t h e  s e r i e s i s i n d e p e n d e n t l y m a g n i f i e d as t h e e x t e r n a l l o a d approaches t h e corresponding c r i t i c a l load. By analogy w i t h t h e l o a d - d e f l e c t i o n c u r v e s r e l a t i v e t o t h e E u l e r column problem,  i t c o u l d a l s o be s a i d t h a t t h e S o u t h w e l l p l o t i s w e l l  a p p l i c a b l e t o i n s t a b i l i t y cases o f t h e b i f u r c a t i o n t y p e .  I n s t e a d , from  the two-bar s t r u c t u r e example (Appendix 2 ) , i t appears t h a t when g e o m e t r i c instability  (snap-through)  dominates,  t h e S o u t h w e l l p l o t would n o t be a  s t r a i g h t l i n e and s h o u l d n o t be a p p l i e d ( s e e f i g .  4117).  52  FIG. 4:17 - GEOMETRIC INSTABILITY (SNAP-THROUGH) In f a c t the equations governing geometric i n s t a b i l i t y  differ  from t h o s e g o v e r n i n g E u l e r b u c k l i n g p r e c i s e l y f o r t h e p r e s e n c e o f t h e a d d i t i o n a l terms due t o chord r o t a t i o n and bowing.  As l o n g as t h e s e  a d d i t i o n a l e f f e c t s a r e n e g l i g i b l e , i t must be p o s s i b l e t o use t h e S o u t h w e l l p l o t f o r t h e e v a l u a t i o n o f the c r i t i c a l l o a d , p r o v i d e d t h e p r e v i o u s l y mentioned  c r i t e r i o n i s s a t i s f i e d , and a c o r r e c t s e t o f  d e f l e c t i o n s and d i s t u r b a n c e systems i s used. However, f o r t h i s type o f s t r u c t u r e , t h e r e may be d i f f i c u l t i e s  53 i n d e t e c t i n g the l o w e s t c r i t i c a l mode and c o r r e s p o n d i n g c r i t i c a l l o a d , as " i n p l a n e " and "out o f p l a n e " b u c k l i n g a r e c o u p l e d , because o f t h e c u r v a t u r e of  t h e cone s u r f a c e .  54 CHAPTER V  RESULTS FOR  SPACEFRAMES WITH DIFFERENT  HEIGHT/SPAN RATIO  5.1  C o n i c a l shapes To check the i m p o r t a n c e of g e o m e t r i c n o n l i n e a r i t i e s , t h e  l o a d l e v e l f o r a U. D. L. o v e r the whole s t r u c t u r e was  critical  d e t e r m i n e d f o r two  o t h e r cones o f l o w e r h e i g h t / s p a n r a t i o but w i t h the same g e o m e t r i c configuration.  Only the d e t e r m i n a n t p l o t was  Let a  Q  used i n t h e s e c a s e s .  be the h e i g h t / s p a n r a t i o f o r t h e o r i g i n a l cone,  w i l l be c a l l e d FCO;  which  the o t h e r two s t r u c t u r e s a n a l y s e d , c a l l e d FC1 and 1  r e s p e c t i v e l y , w i l l have h e i g h t / s p a n r a t i o s  = -^a  FC2  1 and  = Tp ^* 3  T a b l e 5.1 shows the v a l u e s o f c r i t i c a l l o a d f o r the t h r e e s t r u c t u r e s , o b t a i n e d by d e t e r m i n a n t p l o t s , w h i c h are p r e s e n t e d i n f i g s . 511 and 512.  Only f i x i t y  c o n d i t i o n (b) was  a n a l y s e d , s i n c e i t was  assumed  t h a t l a t e r a l b u c k l i n g o f t h e members would be p r e v e n t e d by p u r l i n s and decking. A g a i n i t can be noted t h a t the d i f f e r e n c e s i n t h e v a l u e s o b t a i n e d by t h e o r i e s  1  and  2  are n e g l i g i b l e .  However the i n f l u e n c e o f non-  l i n e a r i t i e s i s a p p r e c i a b l e , s i n c e the o v e r a l l c r i t i c a l l o a d drops f a s t e r t h a n the s i n e o f the s l o p e a n g l e , as i t would happen f o r a l i n e a r t h e o r y , since  t i e s i z e s o f the members, hence the maximum l o a d t h e y c o u l d  carry,  have been k e p t c o n s t a n t t h r o u g h o u t . From a d i m e n s i o n a l a n a l y s i s p o i n t o f v i e w i t can be s a i d t h a t for  cones h a v i n g the same g e o m e t r i c c o n f i g u r a t i o n and arrangement o f  web  members, the c r i t i c a l a x i a l l o a d i n a member, N  . w i l l be a f u n c t i o n  55 TABLE 5.1.  DETERMINATION OF CRITICAL LOAD BY DETERMINANT PLOT FOR DIFFERENT HEIGHT/SPAN RATIOS  Fixity  c o n d i t i o n ( b ) : members f i x e d t o t h e j o i n t s i n t h e s t r o n g d i r e c t i o n , h i n g e d i n t h e weak direction.  E x a c t geometry:  U. D. L  0  o v e r t h e whole s t r u c t u r e .  Theory  1  Theory  FCO  (a ) o  1048 l b .  FC1  (h  494 l b . •  486 l b .  FC2  C  218 l b .  214 l b .  a  o  2.21 7.43  2  1035 l b .  1 3.35  Theory  1  Livesley s t a b i l i t y  functions.  Theory  2  Livesley stability  functions plus chord r o t a t i o n  and bowing.  DETERMINANT PLOT.  3-85  3-90  395 Load  Fixity condition ( b) Exact geometry No Triggers  400 4-05 Factor \ I P = 1201b.)  4-/0  0  Theory (J) • Stability Functions only Theory \2) • "— '/ + Chord Rotation and Bowing  FIG. 5-/-FCI  CONE  1/2 ORIGINAL HEIGHT  4-15  DETERMINANT PLOT.  57  No Triggers Fixity con dition (b) Exact geometry  Load Factor Theory (7) •' Theory (2) •  \(P =l20lb) o  Stability Functions only //  FIG. 5-2-FC2  '/  + Chord Rotation and Bowing  CONE -1/4 ORIGINAL HEIGHT  58 o n l y o f i t s s t i f f n e s s , i t s l e n g t h and t h e h e i g h t / s p a n a.  r a t i o o f t h e cone,  Thus we can w r i t e : N  c r  = f [ A E , E I , L, a ]  [*]  where L = j o i n t - t o - j o i n t l e n g t h o f a member, a = height/span  r a t i o o f t h e cone.  H a v i n g a r e l a t i o n s h i p among 5 v a r i a b l e s , by Buckingham's n-theorem, t h i s can be reduced t o a r e l a t i o n s h i p i n v o l v i n g o n l y 3 dimensionless  p a r a m e t e r s ; f o r i n s t a n c e we may choose:  cr  AEL  —J3I~  N  L  L — r  a ,  ,_ . .  obtaining  2  -^ff-  where  , -gj- ,  - g[~ ,  a]  [**]  i s the slenderness  2 r a t i o o f t h e member, b e i n g A r = 1 .  Now we can d e f i n e an e f f e c t i v e l e n g t h o f a member, L  = kL, e  such as  2  N  =  "  n EI  =• .  a )  S u b s t i t u t i n g i n t o [**] we o b t a i n t h e s i m p l e  2  e  relationship  L  e  — JLi  L = g' [— , a]  , w h i c h can be used t o p r e s e n t  the e f f e c t s of  TC  a change i n h e i g h t / s p a n  r a t i o f o r s i m i l a r cones.  T h i s i s shown n u m e r i c a l l y i n t a b l e s 5.2 t o 5.4 and g r a p h i c a l l y i n f i g . 5:4. I t i s t o be n o t e d t h a t r i b members behave i n two d i f f e r e n t ways, because o f t h e arrangement o f t h e d i a g o n a l s .  These would be  5  TABLE 5.2.  Fixity  ORIGINAL CONE FCO. a  c o n d i t i o n (b).  Member Number  Rib  =  E x a c t geometry.  L r  N  9  -ATF  Theory  (lb.) cr  1  K  " L  Members: Type  1  2  41.6  105.8  2.46  27  55.5  47.0  2.08  52  60.9  11.8  3.63  77  74.4  Type  4.66  4.28  2  3  41.6  51.5  3.52  28  55.5  34.8  2.41  53  60.9  24.25  2.53  78  74.4  13.4  2.52  18  93.5  67.9  1.37  43  88.3  58.1  1.17  68  78.5  35.45  1.125  13  98.4  18.92  1.77  38  114.5  14.61  1.47  63  130.0  4.05  2.13  R i n g Members:  Diagonals:  N  cr  = X N , where N = a x i a l f o r c e i n a member a t l o a d cr o o  X = 1, d e t e r m i n e d by l i n e a r L = member l e n g t h  spaceframe a n a l y s i s .  (joint-to-joint)  ;  L  /~EI— = II ^ —— cr  level  60  TABLE 5.3.  CONE FC1. a  1  1 2 o a  1 6.7  F i x i t y condition (b). Exact geometry. Theory 1 .  Member Number  L r  L N  (lb.) cr  Rib Members: Type  1  2  36.5  85.0  3.12  27  48.6  39.8  2.57  52  53.3  8.6  4.85  77  65.2  4.33  5.05  Type  2  3  36.5  37.0  4.75  28  48.6  27.2  3.1  53  53.3  19.5  3.22  78  65.2  12.3  3.0  18  93.5  49.95  1.6  43  88.3  55.5  1.2  68  78.5  32.4  1.18  13  94.0  15.27  2.06  38  107.5  14.7  1.56  63  118.7  4.6  2.19  Ring Members:  Diagonals:  61  TABLE 5.4.  Fixity  CONE FC2. a  13.4  E x a c t geometry. Theory 1 .  condition (b).  Member Number  4 o  L r  N  (lb.)  L  cr  Rib Members: Type  1  2  35.15  64.6  3.73  27  46.9  33.72  2.9  52  51.3  8.36  5.11  77  62.6  4.69  5.05  Type  2 35.15  26.1  5.85  28  46.9  21.92  3.6  53  51.3  16.11  3.68  78  62.6  11.0  3.3  18  93.5  27.76  2.14  43  88.3  43.9  1.35  68  78.5  27.7  1.28  13  92.8  9.85  2.6  38  105.5  11.73  63  115.6  4.41  3  R i n g Members:  Diagonals:  1.79 2.3  62 s t r e s s l e s s and e v e r y r i b (e.g. AB and CD from f i g . 5:3) would  behave  i d e n t i c a l l y , i f t h e s t r u c t u r e were a n a l y s e d as a space t r u s s , because o f the r o t a t i o n a l symmetry o f t h e s t r u c t u r e and o f t h e l o a d i n g system (UDL).  I f t h i s i s done, i t can be seen t h a t t h e j o i n t s o f a r i b l i k e AB move outward from t h e s u r f a c e o f the cone, w h i l e t h e j o i n t s o f a d j a c e n t r i b s l i k e CD move i n w a r d .  I t i s t o be n o t e d t h a t , even f o r s m a l l  s t r a i n s , as must o c c u r i n t h e e l a s t i c r a n g e , t h e j o i n t d i s p l a c e m e n t s normal t o t h e cone s u r f a c e a r e q u i t e l a r g e , because a d j a c e n t r i n g members a r e almost c o l l i n e a r . To r e s t o r e s l o p e c o n t i n u i t y o f t h e r i n g members ''(fSame a c t - i o n ) , s h e a r f o r c e s , a c t i n g normal t o t h e s u r f a c e o f the cone, a r e r e q u i r e d a t the ends o f t h e s e members.  The magnitude o f t h e s e f o r c e s i s about 25% o f  EFFECT OF VARYING HEIGHT/SPAN RATIO. ( For L/r shown in Tobies 52 to 5-4)  L  2  Nu? Xcr No  L - member length  FIG. 5=4 - CONICAL SPACEFRAMES  64 the c o r r e s p o n d i n g e x t e r n a l j o i n t l o a d .  Because o f t h e s m a l l a n g l e between  the p l a n e determined by two d i a g o n a l s m e e t i n g a t a j o i n t  (say No. 22) and  the p l a n e tangent t o t h e cone a t t h e same j o i n t , t h e s e s h e a r f o r c e s  arising  from frame a c t i o n produce r a t h e r l a r g e c o m p r e s s i v e l o a d s i n t h e d i a g o n a l s of  t h e two l o w e r a r r a y s .  These l o a d s c a r r i e d by t h e d i a g o n a l s , i n t u r n ,  modify t h e d i s t r i b u t i o n o f a x i a l l o a d s i n two a d j a c e n t r i b s l i k e AB and CD. In  summary t h e n , t h e secondary s t r e s s e s s e t up i n t h i s  continuous  space t r u s s a r e much l a r g e r t h a n t h e secondary s t r e s s e s i n an o r d i n a r y p l a n e t r u s s because members a t a j o i n t a r e almost c o p l a n a r . T h i s a c c o u n t s f o r t h e two t y p e s o f r i b members shown i n t h e f o l l o w i n g t a b l e s and f i g u r e s :  type  1  f o r r i b s l i k e CD and type  2 for  r i b s l i k e AB. The h i g h v a l u e s o f k o b t a i n e d i n d i c a t e t h a t , under t h e g i v e n c o n d i t i o n s o f l o a d i n g and member f i x i t y , b u c k l i n g o f t h e s t r u c t u r e i s a g l o b a l phenomenon, i n v o l v i n g t h e whole s t r u c t u r e r a t h e r than i n d i v i d u a l members. In  some c a s e s , e.g. f o r t h e upper r i b members o f type  1  , the  h i g h v a l u e o f k i s due t o o v e r d e s i g n , i . e . t h e member i s n o t s t r e s s e d t o i t s f u l l capacity.  T h i s happens because t h e a c t u a l cone had t o s u p p o r t  a s u p e r s t r u c t u r e h o u s i n g m e c h a n i c a l systems  f o r the h a n d l i n g of potash ( 2 ) .  S i n c e i t has been shown t h a t t h e e f f e c t o f g e o m e t r i c n o n l i n e a r i t i e s i s always minor f o r t h e s e c o n i c a l s p a c e f r a m e s , an e i g e n v a l u e a n a l y s i s c o u l d be c a r r i e d o u t t o d e t e c t t h e a c t u a l c r i t i c a l mode shapes:  this could help  e x p l a i n the r e s u l t s obtained f o r the e f f e c t i v e lengths.  5.2  S p h e r i c a l shape To i n v e s t i g a t e t h e s t r u c t u r a l advantages  o f f e r e d by a d o u b l y  65 c u r v e d shape w i t h out  r e s p e c t t o a s i n g l y c u r v e d one, an a n a l y s i s was c a r r i e d  f o r a s p h e r i c a l spaceframe dome, h a v i n g t h e same h e i g h t and span as  the c o n i c a l spaceframe FC1 ( i . e . a = T^J)  FIG.  as shown i n f i g . 515.  5:5 - SPHERICAL SPACEFRAME  T h i s s p h e r i c a l spaceframe w i l l be c a l l e d FS1. The i n f i g . 516.  c r i t i c a l l o a d was d e t e r m i n e d by d e t e r m i n a n t p l o t s , shown Again theory  1 and t h e o r y  2  y i e l d p r a c t i c a l l y t h e same  value. T a b l e 5.5 compares the r e s u l t s o b t a i n e d by t h e o r y  1  f o r the  c o n i c a l and s p h e r i c a l dome r e s p e c t i v e l y . A g a i n i t can be n o t e d t h a t k i s g e n e r a l l y i n d i c a t i n g that buckling  involves  g r e a t e r t h a n one,  t h e whole s t r u c t u r e  and n o t i n d i v i d u a l  members. For  t h e r i b members t h e r e d u c t i o n  s p h e r i c a l dome w i t h  i n the values of k f o r the  r e s p e c t t o t h e c o n i c a l would p e r m i t a r e d u c t i o n i n  member s i z e s o f t h e o r d e r o f 50%.  66  DETERMINANT PLOT.  Fixity Condition ( b) Exact Geometry No Triggers U. D. L .  (7)  Livesley Stobility Functions only  (2) Chord Rotation and Bowing  5 • -  10 O  10 2  10-4 Load  10-6 10 8 Factor \(P = 1201b.) a  FIG. 5 6 - SPHERICAL DOME SF1 :  I/O  67 TABLE 5.5.  COMPARISON OF A CONICAL AND A SPHERICAL SPACEFRAME  F i x i t y condition (b).  E x a c t geometry.  S p h e r i c a l dome FS1  C o n i c a l dome FC1 C r i t i c a l load factor, X ' cr (P = 120 l b . ) o  U. D. L.  10.81  4.11  Overall c r i t i c a l load (vertical)  1300  494 l b .  L r  Member Number  L N  (lb.) cr  L r  lb  •  L N cr  ( l b .)  "  I  T  R i b Members: . 2 27 53 78  36.5 48.6 53.3 65.2  85.0 39.8 19.5 12.3  3.12 2.57 3.22 3.0  37.2 48.75 53.5 66.5  93.5 88.3 78.5  49.95 55.5 32.4  1.6 1.2 1.18  95.4 91.5 82.1  94.0 107.5 118.7  15.27 14.7 4.6  2.06 1.56 2.19  95.2 109.1 120.3  150.2 96.0 51.8 35.5  2.3 1.65 1.98 1.73  42.0 72.6 61.8  1.71 1.02 .82  25.6 17.15 7.37  1.57 1.43 1.71  R i n g Members: 18 43 68 Diagonals: 13 38 63  a =  height/span r a t i o N  cr  = X N cr o EI N cr  :  N  o  6.7  = a x i a l f o r c e i n a member a t X=l d e t e r m i n e d ., spacerrame a n a l y s i s  by l i n e a r  68 However, t h i s advantage from a s t r u c t u r a l p o i n t o f v i e w would be p a r t i a l l y o f f s e t by f a b r i c a t i o n c o m p l i c a t i o n s due t o t h e f a c t t h a t the r i b s would no l o n g e r be s t r a i g h t t h r o u g h o u t . A l s o , p a r t i a l l o a d i n g s may cause t e n s i o n i n t h e main members, i n c r e a s i n g the complexity  of the connections.  F i n a l l y a s p h e r i c a l shape would r e q u i r e a g r e a t e r q u a n t i t y o f d e c k i n g m a t e r i a l and l o n g e r members t h a n t h e c o r r e s p o n d i n g (but i t would p r o v i d e a g r e a t e r volume o f c o v e r e d ,  c o n i c a l shape  a v a i l a b l e space).  I n c o n c l u s i o n , t h e p o s s i b i l i t y o f u s i n g a s p h e r i c a l dome i n s t e a d o f a c o n i c a l one seems worthy o f d e t a i l e d i n v e s t i g a t i o n .  69  CHAPTER VI  CONCLUSIONS AND RECOMMENDATIONS  The o v e r a l l c r i t i c a l load for the o r i g i n a l cone determined i n this analysis was found to be i n s a t i s f a c t o r y agreement with the value obtained  i n the previous model t e s t s . The r e s u l t s of this i n v e s t i g a t i o n show also that, for a space-  frame of this type and for p r a c t i c a l values of height/span r a t i o s , a s t a b i l i t y analysis i n the e l a s t i c range up to the f i r s t c r i t i c a l load can be successfully c a r r i e d out using only Livesley s t a b i l i t y functions: the additional e f f e c t s due to chord rotation and bowing are generally negligible.  A c t u a l l y i t should be noted that Livesley's version of  s t a b i l i t y functions  ( 3 , 4)  contains sway terms of the type Np, where  N i s the a x i a l load i n a member and p i s the chord rotation, as shown i n Appendix 1.  S t r i c t l y speaking, these terms should be included i n the  geometric part of the s t i f f n e s s matrix, as i t i s done i n r e f . ( 1 0 ) . This may explain p a r t l y the closeness  of r e s u l t s obtained by theories  2 , since Livesley s t a b i l i t y functions were used for theory  1 and  1  An i n v e s t i g a t i o n of the post-buckling behaviour of this type of spaceframes ( s t a t i c a l l y determinate i n t h e i r primary stress system) seems to be of l i t t l e p r a c t i c a l i n t e r e s t , since the structure does not possess any adequate reserve strength beyond the f i r s t c r i t i c a l load. For other spaceframes, namely those with redundant members, i t may be i n t e r e s t i n g to follow the deformation of the structure past the lowest c r i t i c a l load:  then displacements and rotations may become very  70 l a r g e and the p r e s e n t a n a l y s i s would no l o n g e r be a p p l i c a b l e . In such a c a s e , as w e l l as i n the g e n e r a l case o f s t r u c t u r e s e x h i b i t i n g t r u l y l a r g e d e f l e c t i o n s w h i l e r e m a i n i n g e l a s t i c , the b e s t to  way  t a c k l e the p r o b l e m seems to be t o r e s o r t t o a f i n i t e element formu-  l a t i o n w i t h an i n c r e m e n t a l s t e p - b y - s t e p s o l u t i o n by means o f T a y l o r ' s expansion  theorem, as shown i n r e f e r e n c e s (13) and  (14).  I f the p r o b l e m r e q u i r e s i n v e s t i g a t i o n , the  lateral-torsional  s t a b i l i t y o f an i n d i v i d u a l member can be checked s e p a r a t e l y i n a second s t a g e o f the a n a l y s i s , u s i n g f o r i n s t a n c e the p r o c e d u r e r e f e r e n c e s (15) and  outlined i n  (16).  From the p r e s e n t i n v e s t i g a t i o n i t appears a l s o t h a t , when p e r f o r m i n g a n u m e r i c a l s t a b i l i t y a n a l y s i s , the S o u t h w e l l p l o t s h o u l d o n l y be used t o g e t h e r w i t h t h e d e t e r m i n a n t  p l o t , s i n c e i t can be m i s l e a d i n g ,  e s p e c i a l l y f o r the d i f f i c u l t y , i n a complex s t r u c t u r e , o f f i n d i n g a c o r r e c t d i s t u r b i n g f o r c e system and a c o r r e s p o n d i n g s e t o f d e f l e c t i o n s , t r u l y r e p r e s e n t a t i v e o f the l o w e s t b u c k l i n g mode. set  of geometric  For t h i s purpose, a  imperfections together w i t h s e v e r a l d i f f e r e n t p e r t u r -  b a t i o n f o r c e systems s h o u l d be used f o r a r e l i a b l e i n v e s t i g a t i o n .  If  the S o u t h w e l l p l o t s thus o b t a i n e d a r e s t r a i g h t l i n e s upward from about 40% o f the v a l u e o f the l a r g e s t d e f l e c t i o n used, and y i e l d c o n s i s t e n t v a l u e s f o r the c r i t i c a l l o a d , then such a v a l u e can be r e l i e d upon. t h e s e cases the S o u t h w e l l  In  p l o t can g i v e u s e f u l i n f o r m a t i o n about t h e  s e n s i t i v i t y o f the s t r u c t u r e t o d i f f e r e n t p e r t u r b a t i o n s , and can be r e garded as an a l t e r n a t e method o f f i n d i n g the e i g e n v e c t o r  corresponding  to the l o w e s t eigenvalue, o f the s t r u c t u r e s t i f f n e s s m a t r i x . I t can be n o t e d t h a t the d i f f i c u l t y mentioned above o f  finding  71 a correct disturbance  s y s t e m , does n o t o c c u r i n an a c t u a l model t e s t ,  since natural imperfections  i n t h e c o n s t r u c t i o n o f t h e model o r i n t h e  l o a d i n g system u s u a l l y p r o v i d e the Southwell  a s u f f i c i e n t p e r t u r b a t i o n f o r t h e use o f  plot.  The  concept o f e f f e c t i v e l e n g t h o f a member w i t h  associated  a l l o w a b l e s t r e s s can be u s e f u l i n the p r e l i m i n a r y d e s i g n o f a s t r u c t u r e of t h i s t y p e .  However, s i n c e b u c k l i n g i s g e n e r a l l y n o t due t o f a i l u r e o f  an i n d i v i d u a l member, b u t r a t h e r i s a g l o b a l phenomenon i n v o l v i n g t h e whole s t r u c t u r e , i t cannot be s a f e l y assumed t h a t t h e u n s u p p o r t e d  length  o f a member i s t h e j o i n t - t o - j o i n t l e n g t h , as i t i s o f t e n t a k e n f o r p l a n e trusses.  F o r i n s t a n c e , f o r t h e c o n i c a l spaceframes a n a l y s e d  i n the present  Le work, t h e r a t i o j — i s seen t o v a r y a p p r o x i m a t e l y from 2 t o 4 f o r t h e r i b members and from 1 t o 2 f o r t h e r i n g members, a c c o r d i n g ratio.  to the height/span  Thus f o r each g i v e n s t r u c t u r e o f t h i s type (spaceframe domes) i t  i s advisable to carry out a s t a b i l i t y a n a l y s i s w i t h L i v e s l e y s t a b i l i t y functions. An i n v e s t i g a t i o n s h o u l d  a l s o be made o f p a r t i a l l o a d i n g s and,  p o s s i b l y , o f the e f f e c t o f g e o m e t r i c i m p e r f e c t i o n s , s i n c e they can cause a considerable  change i n t h e a x i a l l o a d d i s t r i b u t i o n i n t h e members,  e s p e c i a l l y c l o s e t o the c r i t i c a l l o a d To c o n c l u d e , w i t h r e f e r e n c e reference  level. t o f i g . 6 t l , and q u o t i n g  from  ( 4 ) , p. 45, " S i n c e , i n s t r u c t u r e s s u b j e c t t o i n s t a b i l i t y , t h e  margin o f s a f e t y as measured by a l o a d f a c t o r may be markedly and dangerously  s n a l l e r than t h e ' f a c t o r o f s a f e t y ' measured by a s t r e s s  72  •4  Maximum s t r e s s  FIG.  6:1 - EFFECT OF NONLINEARITIES ON THE MARGIN OF SAFETY  r a t i o , i t i s important that load used i n d e s i g n . " .  f a c t o r s , n o t s t r e s s f a c t o r s , s h o u l d be  73 APPENDIX I  DERIVATION OF THE NON-LINEAR RELATIONS GOVERNING THE DEFORMATION OF A MEMBER (10)  The assumptions  made i n t h i s d e r i v a t i o n , as l i s t e d i n  C h a p t e r 2, a r e : 1.  The m a t e r i a l i s l i n e a r  elastic.  2.  Each member i s p r i s m a t i c and homogeneous.  3.  Loads a r e a p p l i e d o n l y a t t h e ends o f a member.  74 4.  Shear d e f o r m a t i o n s a r e n e g l e c t e d .  5.  L i n e a r s t r a i n s and squares o f t h e r o t a t i o n s a r e o f the same o r d e r o f magnitude and s m a l l compared t o one.  6.  T o r s i o n - f l e x u r e c o u p l i n g and w a r p i n g r e s t r a i n t a r e n e g l e c t e d .  W i t h t h e s e assumptions  i t can be shown ( r e f . ( 1 1 ) , page 466)  t h a t t h e E u l e r i a n and L a g r a n g i a n  d e s c r i p t i o n of the deformation d i f f e r  o n l y by h i g h e r o r d e r i n f i n i t e s i m a l s . Then we can w r i t e , f o r t h e t w o - d i m e n s i o n a l c a s e : (e)^ = e - 5 K 9u . 1 £  = -r— +  -7T  9x  2  2 iii  fi n  9(0  9v 9x  CO = - r -  where e = t h e l o n g i t u d i n a l s t r a i n o f t h e c e n t e r - l i n e o f t h e member, u,v = t r a n s l a t i o n components w i t h r e s p e c t t o x,y d i r e c t i o n s , co = r o t a t i o n o f a c r o s s - s e c t i o n about t h e z - a x i s , K = c u r v a t u r e o f the c e n t e r - l i n e o f t h e member. 9u W i t h o u r assumptions  2 co  << 1.  Then t h e a x i a l f o r c e absence o f i n i t i a l  N and b e n d i n g moment M r e s u l t :  ( i n the  strains)  N = AE £ . M = EI K and t h e t r a n s v e r s e s h e a r f o r c e V i s :  [2]  75 We the  can now  a p p l y the p r i n c i p l e o f v i r t u a l d i s p l a c e m e n t s , i n  form: 6W.  = 6W  1  I n our  , where 6W e '  case:  L /  denotes v i r t u a l work.  (N6e  + M6K )dx = F  o  6u„  Bx D  +  F  J3  Ax  6 U  6v„ + M L 6a> B a ii  + F_, By  A  +  V  V  D  A  +  M  A%  ' ' '  M  Note t h a t , u s i n g the L a g r a n g i a n d e s c r i p t i o n , where a l l v a r i a b l e s are r e f e r r e d to the i n i t i a l s t a t e , we L.H.S. o f  can p e r f o r m the i n t e g r a t i o n at  [4] a l o n g the undeformed p o s i t i o n o f the member.  Moreover  can s u b s t i t u t e everywhere t o t a l d e r i v a t i v e s f o r p a r t i a l d e r i v a t i v e s , u,v,w, b e i n g d i s p l a c e m e n t components o f p o i n t s now  functions  of x  Now  l e t n(x)  and  ct(x) be  ( s m a l l ) v a r i a t i o n s o f u(x)  toa.  a l s o have: 6K  3(x) be  = 6(4~) dx  = 4^ dx  J and  the v a r i a t i o n of v ( x ) , we a(x)  we since  are  only.  t h e n <§e = <5(T~ + TT U ) = — h dx 2 dx We  of the c e n t e r - l i n e ,  the  = 6a) =  letting have:  6(4dx ^) =dx4^ •  S u b s t i t u t i n g i n the L„H.S of e q u a t i o n [4] we  obtain;  and  w(x);  76 L  L  /• (N<Se + M6<)dx = / o  [ N ( ^ + toa) + M^-] dx dx dx  o  i n t e g r a t i n g by p a r t s + Ma  = Nn  — J Cn -5 dx  o  Ntoa + a — ) dx  o  ... We now n o t i c e t h a t n(o) = 611^ , n(L) = 6 u a ( L ) = 6co_ , 6(0) = 6 v /  A  a(Nco - ^ )  dx  o  , g(L) = 6v„ , dx = /  [5]  , a ( o ) = 6to^ ,  and t h a t  I - (Nco - ^ )  dx = g(Nco -  2  dx  o  B  dx  dx  dx  o  Thus we g e t : ( s u b s t i t u t i n g back i n t o [ 5 ] )  /  (NSe  dx = N_6VL - N . 6 u .  + M6K )  B  o  , ,. dM. , + (Nco - ^ ) 6 v T  B  B  B  A  , dM,. . - (Nco - ^ ) 6 v A  A  A  + M ( L ) S<o  B  -  M(Q)6co  A  A  . dN , - / n ^ dx o  - / [3 4" (Nco - f ^ ) ] d x dx dx  [6]  o  Comparing [6] w i t h [ 4 ] , f o r a r b i t r a r y v i r t u a l d i s p l a c e m e n t s , and by Lagrange's lemma ( e . g . r e f . ( 1 1 ) , p. 2 7 3 ) , we must have:  and  the boundary F  = -N A  Ax  dM  = -(Nco) + ( - )  F  A  "Ay  M  conditions:  - -(N.)  A  - V  A  \ A  \ at x = 0  = -M(0)  A  and F F  not  .  =N B  Bx  By  =  ( N W )  B  ^  = M(L)  It  should  present  "  ,dM, fe>B - < >B Nu  +  f o r F, and F„ . Ax Bx  r  t h i s i s a consequence of the i n i t i a l du _,_ 1 . dx 2  If  at x=L  B  be n o t e d t h a t the term i n v o l v i n g t h e shear f o r c e V i s  i n the e x p r e s s i o n s  o r t h a t a>  V  22  assumptions: and  K =  dx  I t can be shown t h a t  2  V = -  dM dx  << 1 .  the f u l l  expressions  f o r e and K are used, and t a k i n g V = -  dM ds  then a shear f o r c e component w i l l appear i n the h o r i z o n t a l components o f the  end-forces. The  above d e r i v a t i o n shows one o f the advantages o f the v a r i a t i o n a l  methods i n o b t a i n i n g a u t o m a t i c a l l y boundary c o n d i t i o n s  f o r a given  the d i f f e r e n t i a l e q u a t i o n s and the  problem.  We can now f o l l o w r e f . ( 1 0 ) N = const.  N e  AE  =  =  = F  g  x  i n obtaining:  [from [ 7 ] ]  du , 1 dx 2"  2  W  Hence: "Bx  =  AE L~  ( U  , B " V  s +  •AE  2 L  f  2  r  W  o  d  x  . . . .  [9]  78 The  second term a t t h e r i g h t - h a n d s i d e o f [9] i n c l u d e s t h e e f f e c t s o f  c h o r d r o t a t i o n and bowing. Now [8] becomes: •~•2 dx or  -N-^ dx  2 2 d M „ d v — j ~ — 2 dx dx N  =0  n  =  2 , d v where M = E I — j dx T  >  »  w h i c h has f o r s o l u t i o n :  I  h  - "  < 21 D  s  i  *  n  I  +  D  22  c  o  s  * !>  +  D  23 I  +  °24 ' *  [ 1 0 ]  where: ,2  22  D  " BX F  L  2  " 21 A " 22 B  =  K  ~21 "  W  K  * KB -  1-C  v  W  +  K  23  P  »*) -l(A) D, A - G ' 22 y  V  D „ = to. + 23 A cj)  cb(S-Ccb) 2(1-C)-Scj)  21  K  and  23  =  K  21  +  K  . '  _ cb(cb-S) 2(1-C) -Scb  22  22  C = cos <j> c  •  V  .  P  =  B ~ A V  1  •  S = s i n <j> Note t h a t p r e p r e s e n t s  t h e " c h o r d r o t a t i o n " o f a member.  I f F„ i s p o s i t i v e , i . e . t h e member i s under t e n s i l e Bx l o a d , <J) becomes i m a g i n a r y , trigonometric functions.  axial  and h y p e r b o l i c f u n c t i o n s s u b s t i t u t e t h e  79 Using equation  [10] i t i s now p o s s i b l e t o p e r f o r m t h e i n t e g r a t i o n  a p p e a r i n g i n [ 9 ] , namely  AE 2L  J  757Q  .  L  2• AE dx = ^ t - / 2L  .dv.2 . (-j—) dx dx  L  to  r  Q  ~f  {  °23  +  X  2  2*  +  d-C)D ] 2 2  +  SC,  *  2  Then we can e x p r e s s  21  . - s i  \ p 2  +  I-SD  the end-forces  on a member as f u n c t i o n s o n l y  of the end-displacements, i . e . 2 1^ = M(L) = E I  j  EI j  =  _  +  k  ^  p  ]  ^  dx M -M(0)  [K  A  EI By ^2 23  F  =  K  [  2 1  ~ A W  CO  K CO  A +  22  +  2  p  ]  +  - K  B  p F  2 3  p]  Bx >  F  = -F By  Ay  _ Bx  F  Ax  [11]  =  AE . L" B " V ( U  . +  AE 2L  "  f  U  d  2, x  Q  = -F Bx  The r e l a t i o n s f o r t h e t h r e e - d i m e n s i o n a l case a r e o b t a i n e d a s s o c i a t i n g bending  i n the (x,y) plane, bending  unrestrained torsion.  These r e l a t i o n s h i p s  i n t h e ( x , z ) p l a n e and  can be p u t i n m a t r i x  form,  80 i n member c o o r d i n a t e s , as f o l l o w s : {f} = [ k ] {u} - { f >  [12]  m  G  where  { f } = v e c t o r o f member  end-forces,  {u} = v e c t o r o f member e n d - d i s p l a c e m e n t s , [k]  m  = member s t i f f n e s s m a t r i x , i n c l u d i n g t h e s t a b i l i t y  {f  } = v e c t o r o f g e o m e t r i c a l l y n o n l i n e a r terms  - ' i f - ' j ; . L  where  A =  /  (co  0  2  0,0,0,0,0, -  2 + c o ) d x  s h o u l d be n o t e d t h a t the s t a b i l i t y  .A  and  [k ]  f u n c t i o n s s and s c d e s c r i b e d i n r e f . ( 4 ) , page 52. [12] becomes  m  and  m  {Au} = - { A f }  G  - [k] + [k ] m  m  L  J  [ k  lG 3x3 ]  L  0  1  0 0  where  {Au}  ;  m  G  Tk l G 12x12  m  ,  G  t  In fact i t  p r e v i o u s l y defined correspond e x a c t l y  m  m  T  ,m f u n c t i o n s i s used i n [ k ] .  {Af} = [ k ] {Au} + [ k j {Au} = [k ] [k ]  ,o,o,o,o}  ,  I n i n c r e m e n t a l form, e q u a t i o n  being  0  f  y  i f L i v e s l e y ' s form o f s t a b i l i t y  to  f-  t k  functions,  1  A  u  lG 3x3 ]  1  _  —  AEp  3  L AEp,  _  1 T 1 " T  0  1  °  1  ° AEp  1 1 1  3  L  h  | 1  AEp  3  L ^ 2 ^  0  I  0  I  1 [ k  ]  0  0  0  lG 3x3  AEp  2  1 ' L HAEp P 1 2  1 |  0  L . AEp'  3  V w  i  t  h  p  3  B  —  " L  V  A  " >  w — wA B A W  " P  2  =-^~L  81  82 APPENDIX 2  EXACT SOLUTION FOR TWO-BAR STRUCTURE  Because o f t h e symmetry, t h e p r o b l e m has one degree o f freedom, say u. By t h e p r i n c i p l e WSu = 6U  o f v i r t u a l work:  , where U i s t h e s t r a i n energy o f t h e s t r u c t u r e .  But now 6U = ~ <5u du  , hence  W = f du For  [1]  t h e two b a r s :  AE 2 U = - — (AL) Li o AL = L  o  , where  - L  Therefore U = — L  [2L + u o 2  o  2  - 2uh - 2L o  Ao 2  + u  2  - 2uh ]  83 Hence dU 2AE , j . u - h . W = 7- = — [u - h - L ZZZZZZZZZZZT" J du L o 7^5 7. IT  r  °  Therefore  A + 2  u  2  r  , L J 2  - 2uh  W = 0 f o r u = 0, h , 2h  W = W  dW = 0 du  cr  when 4~ = 0 du  for u = u  u  cr  (tangent s t i f f n e s s equal to zero)  , where cr'  = 1 ±  5  h  / /  (  Lax 2/3  a  -1  [3]  We see t h a t i n d e e d t h e r e a r e two c r i t i c a l p o i n t s , symmetric w i t h r e s p e c t t o u = h.  84 To f i n d W W  , we s u b s t i t u t e [3] back i n t o [ 2 ] , obtaining:  = 2AE[l-(f- ) / ] / o 2  cr  3  3  2 [ 4 ]  For — = .10, we have a ' u - r "  1000 W «  4 2 3 6  '  - 2 A F T ^  =  -  1  9  °  5  4  APPENDIX 3  NOTES ON THE SOUTHWELL PLOT  The S o u t h w e l l p l o t w i l l be a s t r a i g h t l i n e i f f d(|) , do f  = a constant = c  Expanding the d i f f e r e n t i a t i o n :  i  or  <7>  d  — - c = - o P  S e p a r a t i n g the v a r i a b l e s : d6  4>  d6  86 Integrating:  In 6 = In (j - c) + k  1  where k^ i s a constant of integration. Therefore In 6 (j - c) = - k  6 (-|- - c) = e  or  x  ^ = c^ ( a constant)  [1]  which i n the plane (P,<5) represents a rectangular hyperbola with a horizontal asymptote for P = P cr J  6 =  r  = — , since c '  1 p " c  To see t h i s better, consider the t r a n s l a t i o n of axes: i 6 = 6 ' - — c c  P = -P' + c and substitute back into [1]: l 1 (6' - - i ) ( -P« + C  —  c  or  H  -  i  - c) = c  1  (6' - — ) ( ! + cP' - 1) = -C.P' +  whence 6'cP'  or  - c,P' = -c.P' + — 1 1 c  6'P' = —TT = a constant,  then  87 The above e q u a t i o n i s the usual form f o r a r e c t a n g u l a r h y p e r b o l a , i n the p l a n e ( P ' , 6 ' ) , r e f e r r e d t o i t s asymptotes. Note t h a t c was the s l o p e o f the S o u t h w e l l l i n e , and we have thus shown t h a t  1 C  =  F~ cr  *  88 APPENDIX 4  LIST OF REFERENCES  J . M. Gere & W. Weaver:  A n a l y s i s o f Framed S t r u c t u r e s ,  D. Van N o s t r a n d Co., New Y o r k ,  1965.  B. Madsen: "Unique D e s i g n i n G l u l a m " , E. I . C. E n g i n e e r i n g J o u r n a l , April  1962.  R. K. L i v e s l e y :  M a t r i x Methods o f S t r u c t u r a l A n a l y s i s , Pergamon  P r e s s L t d . , London, 1964. M. R. H o m e , W. Merchant: Ltd.,  The S t a b i l i t y o f Frames , Pergamon P r e s s  London, 1965.  J . H. A r g y r i s :  " C o n t i n u a and D i s c o n t i n u a " , From t h e P r o c e e d i n g s  o f the Conference on M a t r i x Methods i n S t r u c t u r a l M e c h a n i c s , h e l d a t W r i g h t - P a t t e r s o n A i r F o r c e Base, O h i o , 26-28 O c t o b e r , 1965. F. B. H i l d e b r a n d :  I n t r o d u c t i o n to Numerical A n a l y s i s , McGraw-Hill  Book Co., I n c . , New Y o r k ,  1956.  R. H. M a l l e t t and L. A. S c h m i t , J r . : " N o n l i n e a r S t r u c t u r a l A n a l y s i s by E n e r g y S e a r c h " , ASCE, J o u r n a l o f the S t r u c t u r a l D i v i s i o n , June  1967.  W. Merchant, D. M. B r o t t o n , M. A. M i l l a r : the  Analysis of Nonlinear E l a s t i c  "A Computer Method f o r  P l a n e Frameworks" , Paper  10, I n t e r n . Symposium on the Use o f E l e c t r o n i c D i g i t a l  Computers  i n S t r u c t u r a l E n g i n e e r i n g , U n i v e r s i t y o f N e w c a s t l e on Tyne, E n g l a n d , 1966.  No.  89 (9)  M. S. Zarghamee and J . M. Shah:  " S t a b i l i t y o f Spaceframes"  ASCE, J o u r n a l o f t h e E n g i n e e r i n g Mechanics D i v i s i o n , A p r i l (10)  J . J . Connor, R. D. L o g c h e r , S h i n g - C h i n g Chan:  1968.  "Nonlinear  A n a l y s i s o f E l a s t i c Framed S t r u c t u r e s " , ASCE, J o u r n a l o f the (11)  Structural Division,  Y. C. Fung:  June 1968, pp.  F o u n d a t i o n s o f S o l i d Mechanics ,  1525-1545. C h a p t e r 16.  P r e n t i c e - H a l l , I n c . , Englewood C l i f f s , N. J . , 1965. (12)  S. T. A r i a r a t n a m : "The S o u t h w e l l Method f o r P r e d i c t i n g  Critical  Loads o f E l a s t i c S t r u c t u r e s " , Q u a r t . J o u r n . Mechanics and A p p l i e d Mathematics , V o l . XIV, P t . 2, 1961. (13)  J . T. Oden:  "Numerical Formulation of Nonlinear E l a s t i c i t y Problems",  ASCE, J o u r n a l o f t h e S t r u c t u r a l D i v i s i o n , June 1967. (14)  N. D. Nathan:  F i n i t e Element F o r m u l a t i o n o f G e o m e t r i c a l l y N o n l i n e a r  Problems o f E l a s t i c i t y , • (15)  Seattle,  Ph.D.  T h e s i s , U n i v e r s i t y o f Washington,  1969.  R. De V a i l : The L a t e r a l - T o r s i o n a l B u c k l i n g o f d o u b l y - s y m m e t r i c w i d e - f l a n g e SecfciQ.ns M.A.Sc. T h e s i s , U n i v e r s i t y o f B. y  Vancouver, A p r i l (16)  B. A. Z a v i t z :  C,  1968.  A s t i f f n e s s m a t r i x f o r t w i s t bend b u c k l i n g o f  narrow r e c t a n g u l a r s e c t i o n s , M.A.Sc. T h e s i s , U n i v e r s i t y o f B. C , (17)  Vancouver, May  S. P. Timoshenko:  1968.  Theory o f E l a s t i c S t a b i l i t y  H i l l Book Co., I n c . , New Y o r k , 1961.  , 2nd ed., McGraw-  

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